untitled-the-hong-kong-polytechnic-university

Proceedings of the 5th Cross-strait Conference on Structural and Geotechnical

Engineering (SEG-5)

VOLUME 2

13-15 July 2011, Hong Kong, China

Edited by

Y. Q. Ni, J. H. Yin and X. W. Ye

The Hong Kong Polytechnic University

Organised by

The Hong Kong Polytechnic University

Co-organised by

Zhejiang University

National Taiwan University

Authors retain all proprietary right in any process, procedure, or article of manufacture described in the

Work. Authors may reproduce or authorize others to reproduce the Work, material extracted verbatim from

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persons as a result of operation or use of this publication and/or the information contained herein.

ISBN: 978-988-15439-2-9

Published by: Faculty of Construction and Land Use, The Hong Kong Polytechnic University, Hong Kong,

China.

SCIENTIFIC COMMITTEE

Chairman

Jan-Ming KO

Vice-chairmen

Shi-Lin DONG

Jin-Guang TENG

Yeong-Bin YANG

Members

Andrew, Ka-Ching CHAN

Siu-Tack CHAN

Kuo-Chun CHANG

Kam-Tim CHAU

Yun-Min CHEN

Zu-Yu CHEN

Moe MS CHEUNG

Kin-Kuen CHOY

Reuben Pui-Kwan CHU

Xue-Yi FU

Xiu-Run GE

Ji-Ping HAO

Albert Ngai-Leung HO

Vai-Pan IU

Wei-Liang JIN

Chang-Hua KE

Sritawat, KITIPORNCHAI

Albert K H KWAN

Kin-Kei KWAN

Ching-Kwong LAU

Chack-Fan LEE

Liang-Jenq LEU

Andrew YT LEUNG

Christopher, K Y LEUNG

Chien-Chung LI

Ching-Lung LIAO

Hung-Jiun LIAO

Kim-Meow LIEW

Chi-Chang LIN

Xi-Liang LIU

Chin-Hsiung LOH

Ke-Jian MA

Za-Chieh MOH

Jian-Guo NIE

Jui-Lin PENG

Ji-Ping RU

Zu-Yan SHEN

Shi-Zhao SHEN

The Hong Kong Polytechnic University

Zhejiang University

The Hong Kong Polytechnic University

National Yunlin University of Science & Technology

Ove Arup & Partners

Housing Department, the Government of the HKSAR

National Center for Research on Earthquake Engineering

The Hong Kong Polytechnic University

Zhejiang University

China Institute of Water Resources and Hydropower Research

The Hong Kong University of Science & Technology

Buildings Department, the Government of the HKSAR

The Hong Kong Institution of Engineers

China Construction Design International (Shenzhen)

Institute of Rock and Soil Mechanics, Chinese Academy of Sciences

Xian University of Architecture & Technology

Hong Kong Geotechnical Society

University of Macau

Zhejiang University

Beijing Institute of Architectural Design

City University of Hong Kong

The University of Hong Kong

Ove Arup & Partners

Maunsell Consultants Ltd.

The University of Hong Kong

National Taiwan University

City University of Hong Kong

The Hong Kong University of Science & Technology

CECI Engineering Consultants, Inc., Taiwan

China Engineering Consultants, Inc.

National Taiwan University of Science and Technology

City University of Hong Kong

National Chung Hsing University

Tianjin University

National Taiwan University

Guizhou University

Moh and Associates, Inc.

Tsinghua University

National Yunlin University of Science & Technology

National Natural Science Foundation of China

Tongji University

Harbin Institute of Technology

SCIENTIFIC COMMITTEE

James C TAI

Leslie George THAM

Keh-Chyuan TSAI

Chi-Sing WAI, JP

Chung-Yue WANG

Hok-Ning WONG

Shi-Lang XU

You-Lin XU

De-Qing YI

De-Yu YIN

Ji-Da ZHAO

Jian ZHENG

Ying-Ren ZHENG

Dai ZHOU

Xu-Hong ZHOU

T.Y. Lin Taiwan Consulting Engineers, Inc.

The University of Hong Kong

National Taiwan University

Development Bureau, the Government of the HKSAR

National Central University

Civil Engineering and Development Department, the HKSAR

Zhejiang University

The Hong Kong Polytechnic University

Zhejiang Provincial Institute of Architectural Design & Research

Taiyuan University of Technology

China Academy of Building Research

Ministry of Railways, The People’s Republic of China

Logistical Engineering University

Shanghai Jiao Tong University

Lanzhou University

ORGANIZING COMMITTEE

Co-chairmen

Yi-Qing NI (Structural Engineering)

Jian-Hua YIN (Geotechnical Engineering)

Members

Chih-Chen CHANG

Chien-Chou CHEN

Zhi-Hua CHEN

Yung-Ming CHENG*

Kwok-Fai CHUNG

Jian-Guo DAI*

Hua DENG

Shang-Hsien HSIEH

Eddie LAM

Siu-Seong LAW

Qiu-Sheng LI

Zong-Jin LI*

Han-Long LIU

Man-Hoi LOK

Yao-Zhi LUO

Wai-Meng QUACH*

Li-Zhong WANG

Yuk-Lung WONG

Wen-Hwa WU

Yu-Fei WU*

Yong XIA

Xue-Yu XIONG

Jun YANG*

Ben YOUNG

Quentin Z Q YUE

Ka-Veng YUEN

Li-Min ZHANG*

Yang ZHAO*

Wan-Huan ZHOU*

Song-Ye ZHU*

The Hong Kong Polytechnic University

The Hong Kong University of Science & Technology

National Yunlin University of Science & Technology

Tianjin University

The Hong Kong Polytechnic University

Zhejiang University

National Taiwan University

The Hong Kong Polytechnic University

City University of Hong Kong

The Hong Kong University of Science & Technology

Hohai University

University of Macau

Zhejiang University

University of Macau

Zhejiang University

The Hong Kong Polytechnic University

National Yunlin University of Science & Technology

City University of Hong Kong

The Hong Kong Polytechnic University

Tongji University

The University of Hong Kong

University of Macau

The Hong Kong University of Science & Technology

Zhejiang University

University of Macau

The Hong Kong Polytechnic University

(* Executive Committee Members)

Secretaries

Xiao-Wei YE

Tao YU

Hua-Fei ZHOU

The Hong Kong Polytechnic University

iii

PREFACE

Riding on the success of the previous 4 conferences respectively held in 1994 (Hangzhou), 1997

(Hong Kong), 2003 (Taipei) and 2007 (Hangzhou), we are delighted to convene the 5th Cross-strait

Conference on Structural and Geotechnical Engineering from 13 to 15 July, 2011 at The Hong Kong

Polytechnic University.

This conference is organized by The Hong Kong Polytechnic University and co-organized by

Zhejiang University and National Taiwan University. This event is not only to provide a forum for

structural and geotechnical engineering professionals and academia from the Chinese mainland,

Taiwan, Hong Kong and Macau as well Chinese scholars from other countries to meet together and

share new ideas, achievements and experiences through presentations and discussions, but also to

review trends in research development and engineering applications.

The proceedings of the conference comprise 8 keynote papers (or lectures) and 20 invited papers (or

lectures) as well as 92 regular papers. These papers have covered a wide range of issues concerning

structural and geotechnical engineering. Showcasing diversity and quality, these papers report the

current state-of-the-art and point to future directions of research and applications in this exciting area.

The success of the conference is due to the dedication and support of many individuals and

organizations. On behalf of the Organizing Committee, we would like to thank all authors for careful

preparation of their papers, and all speakers of keynote papers, invited papers, and regular papers for

sharing their work, experience and insight at the conferencing. All papers submitted to the conference

were reviewed by members of the Scientific Committee and the Organizing Committee. We are

grateful to all of them for their important contributions to the conference. In addition to sharing the

paper review work, members of the Organizing Committee have also been most generous with their

time in the organization work. As chairs of the Organizing Committee, we are indebted to all of them.

The financial support from Kwang-Hua Fund for College of Civil Engineering, Tongji University is

acknowledged with heartfelt gratitude.

On behalf of the Organizing Committee, we would like to express many thanks to colleagues from

Faculty of Construction and Land Use for their secretarial support, in particular, Miss Liz Lau, Miss

Cindy Li, Ms Connie Man, and Mr Jason Au, who have been involved in the production of the

proceedings and preparation of the conference.

One of the founders of this series of conferences, Professor Wen-Lu Jin ( 金问鲁教授 ) has passed

away. We have written one Chinese article, following this preface, to memorialize Professor Jin for

his contributions to this series of conferences and to his achievements in research and practices in

structural and geotechnical engineering.

Prof. Y.Q. Ni and Prof. J.H. Yin

Chairs of the Organizing Committee of SGE-5

The Hong Kong Polytechnic University, Hong Kong, China

TABLE OF CONTENTS

Scientific Committee

Organizing Committee

Preface

Table of Contents

iii

Volume 2

Parallel Session – Structural II

Revitalization of Historic Buildings – “Conversion of Yau Ma Tei Theatre and the Red Brick 525

Building into a Xiqu Centre”

K.Y. Ma, Y.K. Chan & C.Y. Wong

Temperature Effect on Variation of Frequency of Beams: A Comparative Study 535

X. Q. Zhou, B. Chen & Y. Xia

Topology Optimization Using Stochastic Search Methods 545

J.Y. Guo & L.J. Leu

Analysis of the Kinematic Path of Load-Bearing Cable-Bar Mechanisms 557

H. Deng, Y.Z. Zu, X.S. Wu & B.W. Jiang

Application of Recursive Stochastic Subspace Identification in On-Line Bridge Monitoring 569

System

J.H. Weng, C.H. Loh, S.S. Chao, K.C. Lu & C.H. Chen

Development of Multifunctional Laminar Shear Container for Shaking Table Test 581

H.F. Sun, L.P. Jing, N.W. Wang & X.C. Meng

Experimental Study on Crack Mode in Reinforced Concrete Structures with Rebar Corrosion 587

Y.X. Zhao, J. Yu & W.L. Jin

Experimental Study on Seismic Behavior of Autoclaved Aerated Concrete Block Composite 596

Walls with Structural Columns

P.C. Ling, J. Zhao, B.Z. Zhou, H.G. Wu, Q.S. Miao, Y.Y. Zhu & J. Tu

Seismic Resistant Design and Analysis of Vertical Boundary Elements in Steel Plate Shear Walls 602

K.C. Tsai, C.H. Li, J.T. Chang & C.H. Lin

Application of Singular Spectrum Analysis to Health Monitoring of Bridge Structure 611

S.H. Chao, C.H. Loh, J.H. Weng, P.Y. Lin & C.H. Chen

FEM Analysis of Shear Connector Behavior of Continuous Steel-Concrete Composite Girder 618

Bridge

H.Y. Cao, Z.J. Chen, H.P. Zhu & H.Y. Yang

A Study on Damage Assessment of the Scoured Bridges 627

C.Y. Wang & Y.C. Sung

Wind Tunnel Test and Wind-Induced Structural Dynamic Response Analysis for a Steel 636

Velarium Roof

H.F. Zhu, W.J. Chen, Y.L. He & S.L. Dong

Load-Deformation Analysis of Reinforced Concrete Columns Including Shear Effects 643

Q. Zhang & J.X. Gong

Systematic Categorization of Structural Components in Stonecutters Bridge 649

K.C. Lin, X.W. Ye, Y.Q. Ni & K.Y. Wong

Stability Behavior of Fixed-Roof Tapered-Wall Steel Tanks under Differential Settlement 655

Z. Wang, X. Lei, Y. Zhao & X.Y. Xie

Parallel Session – Geotechnical II

Preliminary Study of the Influence of Underground Structures on Groundwater in the Centre 661

of Guangzhou City

H. Cao & G.Y. Luo

Reflection of Seismic Waves at Boundary of an Unsaturated Poroelastic Half-Space 670

W.Y. Chen, T.D. Xia & X.B. Kong

Earthquake Loss Estimation of Building with Shallow Foundation due to Soil Liquefaction 677

C.C. Lu, J.H. Hwang, C.R. Huang & Y.D. Lu

A Quantitative Evaluation on Bridge Scouring Safety 686

H. Wang, C.Y. Wang, T.R. Wu, S.C. Hsieh, Y.Y. Ko, J.S. Chiou,

M.Z. Chen, T.Y. Chen, C.H. Chen & C. Lin

Influence of Pore Pressure on Tunnel Face Stability 697

W. Liu, X.W. Tang, H.Y. Wang & B. Huang

Installation Load and Working Capacity of Jacked Piles: Some Experiences in China 701

F. Yu & J. Yang

Effects of Presence of a Soft Soil Layer on Liquefaction Evaluation 706

C. Li, W.W. Yang, Y.C. Lo, S.H. Yung & C.H. Wong

Resistance of Soil-Root System of Selected Native Plants in Hong Kong 716

F.T.Y. Leung, B.C.H. Hau, L.G. Tham & W.M. Yan

Guided Wave Interpretation of Impulse Responses for Deep Foundations with Emphasis on 720

Cross-Sectional Profiles

H. Wang, B.T. Chen & T.P. Chang

Establishment of the New Failure Criterion of Soils 728

R.Q. Xu & X.C. Wang

Numerical Evaluation of the Seismic Performance for a Ductile Fiber Reinforced Concrete 731

Coupled Wall System

C.C. Hung

Geotechnical Properties of Colluvial and Alluvial Deposits in Hong Kong 735

K.W. Lai

Parallel Session – Structural III

Multi-Located Tuned Mass Dampers for Vibration Mitigation of Tower-Like Structures with 745

Whipping Effect

Y.F. Duan, Y.Q. Ni & J.M. Ko

Nonlinear Analysis and Calculation of Second-Order Effect of Sway-Restricted Columns 756

J. Xu & J.X. Gong

Nonlinear Stability Analysis for a Concrete-Filled-Steel-Tubular Arch/Continuous Beam Bridge 763

Including Defects in Arch Rib

J.L. Hu, Q.S. Yan, Z. Chen & Y.H. Gao

Numerical Simulation of Wind Pressure and Wind-Resistant Optimization on Concave Roofed 769

Low-Rise Building with Eaves

D. Zhou, J.H. Tu & J.L. Li

Trace Analysis of Mechanical Response of Defect Member During Losing Overall Structural 777

Stability

J.M. Guo, W.L. Xue, X.Q. Zhao & X.S. Wu

Introduction for Program Saptrans to Convert SAP2000 Model to ABAQUS and NASTRAN 784

X.Q. Yang, X.Y. Fu & Y.J. Huang

Walking Comfort Analysis on Station Building Floor System of Shen Zhen North Station 792

B. Wu, X.Y. Fu, M.L. Meng, Z.H. Chen & J.X. Qu

Integrate On-Line Recursive SSA and SSI-COV Algorithms for Operational Modal Analysis 797

of Structures

Y.C. Liu & C.H. Loh

Progressive Collapse Analysis of Multi-Storey Frame Structures 806

Z. Wang, J.R. Pan & H.P. He

The Approximate Formula of Vertical Vibration Fundamental Frequency of Three-Tower 810

Suspension Bridge

B. Liu, Y. Zhang, K.L. Chen & J.X. Shen

Evaluation of the Potentiality of Bridge Scouring 818

S.H. Hung, C.Y. Wang, W.F. Lee, H.P. Lien & C.K. Huang

Recommending a Patent Technology of Reducing the Noise Induced by Rail Transit 828

F. Lin, J.F. Gu & G.Z. Qian

Development of a Digital Camera System for Monitoring of Structural Displacement 832

F. Xu, Y.Q. Ni & X.W. Ye

RC Moment Frame Buildings Column Loss Analysis: The Effect of Masonry-Infill Wall 838

S. Li, S.P. Liu, C.H. Zhai & L.L. Xie

Seismic Response Analysis of Shen Zhen North Station under Multi-Dimension and 846

Multi-Support Seismic Excitation

B. Wu, X.Y. Fu, M.L. Meng & J.X. Qu

vii

Axial Load Carrying Capacity of Circular Reinforced Concrete Columns 851

Ivy F.Y. Ho & Eddie S.S. Lam

Parallel Session – Structural IV

Ultimate Strengths of Concrete Bridge Deck Slabs Reinforced with GFRP Bars 861

Y. Zheng, Y.F. Pan & G.Y. Yu

Seismic Upgrade of Reinforced Concrete Beam-Column Joint Under High Level Axial Load 868

B. Li & Eddie S.S. Lam

Structural Health Monitoring of RC Structures Subjected to Seismic Loading Using 880

Piezoceramic-Based Sensors

W.I. Liao, Y.C. Sung, K.C. Chang & J.S. Hwang

Smart Elasto-Magneto-Electric (EME) Sensors for Stress Monitoring of Steel Bars Using 886

Magneto-Electric (ME) Sensing Units

R. Zhang, Y.F. Duan, Y. Zhao, S.W. Or & K.Q. Fan

Some Considerations on Five Technical Specifications of Steel-Concrete Composite Structure 893

Z.T. Tu, G.M. Teng & G.Z. Qian

Space Analysis of Long-Span Curved Continuous Rigid Frame Bridges with High Pier Under 897

Gravity Load

Y.X. Yang, X.W. Hao & L. Sun

Stability Analysis Method for Lattice Shells Accounting for Member Buckling 903

W. Tian , Y. Zhao & S.L. Dong

Structure Design of Shanxi Olympic Gymnasium 913

X.B. Yang, Y. Gao, X.Y. Fu, X.M. Cui, T. Wang & J.R. Zhou

Study on the Mineral Admixtures Influence on the Electric Flux and Chloride Ion Migration 925

Coefficient of Concrete

Y. Li, L. Qiao, C. Yan, Y. Zhang & X.L. Du

Applications of Non-Contact Measurement Techniques to Bridge Health Inspection 930

C.S. Wang, C.Y. Wang, Y.C. Sung, C.C. Cheng & P.Y. Hung

Vortex Shedding Suppression of Cylindrical Structures Near Plane Wall 938

J.S. Cui, F.P. Gao, Z.P. Zang & X.T. Han

Revitalization of Existing Building Group by Using Dampers at Top Floor Level and Base 946

Isolators

Z.D. Yang & Eddie S.S. Lam

Summary on the Structural Design of Shenzhen North Train Station 955

X.Y. Fu, B. Wu, Z.H. Chen, M.L. Meng, M. Guo, C. Sun, H.B. Jiang,

Y.W. Feng, J.W. Shao, J.R. Zhou, J.X. Qu & Y.L. Liu

Numerical Modelling of the Cyclic Behaviour of RC Columns Retrofitted with FRP Jackets 970

J.G. Teng, J.Y. Lu, L. Lam, G. Lin & Q.G. Xiao

Direct Tensile Test of Normal Strength Concrete at Elevated Temperature 978

Eddie S. S. Lam & S. Fang

viii

An Innovative and Proven Solution for Repelling Water Ingress out of Concrete Structures 987

E.C. Chang

Static Performance Analysis of Schwedler Elliptic Suspendome 990

F. Li, C.G. Wang, X.Z. Guo & L.Y. Wang

Monitoring Analysis of Deep Foundation Pit for Xinguoguang Commodity Houses in 996

Wenzhou

C.J. Zhai & T.D. Xia

The Influence of Inhomogeneous Initial Stress Field on the Propagation of Longitudinal 1002

Perturbation in Elastic Continuum

W.T. Hu, T.D. Xia & W.Y. Chen

Parallel Session –

Structural II

The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

REVITALIZATION OF HISTORIC BUILDINGS – “CONVERSION OF YAU MA TEI

THEATRE AND THE RED BRICK BUILDING INTO A XIQU CENTRE”

K.Y. Ma 1 , Y.K. Chan 2 , and C.Y. Wong 2

1 Hong Kong University Master of Arts in Chinese Historical Studies Alumni Association, HKSAR, China

2 Architectural Services Department, HKSAR, China. Email: chanyk1@archsd.gov.hk

ABSTRACT

In response to the raising aspirations from the local public on heritage conservation, the Chief Executive

announced in his 2007/08 Policy Address a set of initiatives for enhanced conservation of historic or heritage

sites and buildings in Hong Kong. The initiatives require an impact assessment on sites or buildings from the

implementation of government capital works projects so that their conservation will be given due consideration.

As Grade II and I historic buildings graded by the Antiquities and Monuments Offices (AMO) respectively, Yau

Ma Tei Theatre (YMTT) and the Red Brick Building (RBB) were decided to have revitalization and they would

be converted to a Xiqu Centre and supporting venue after being vacant for more than a decade. This paper aims

at providing a case study on the engineering solution in the conversion of historic buildings into a new

functional use. The first part will describe the history of the YMTT and RBB that were built in 1930 and 1895

respectively. The second part will depict how to strike a balance between the upkeep of the cultural significance

of the historic building and its modification works to satisfy current statutory requirements, operation and

construction needs. The project exemplifies the difficulties encountered in the construction works at YMTT

within a congested site. This includes: the construction of basement and foundation works within the peripheral

vulnerable masonry walls of the theatre with extensive temporary shoring system and stringent site controls on

dewatering, vibration and settlement monitoring; the extensive temporary stabilization works on the masonry

structures during demolition and re-construction works; and the installation of new steel roof trusses with

internal and external constraints. By revitalizing the old theatre to a robust and well-performing building, the 80

year-old theatre with a valuable memory of the Yau Ma Tei district was transformed to a well-equipped Xiqu

Centre and as a result the buildings will continue to serve the Public.

KEYWORDS

Historic Building, Essential Elements, Structural Assessment, Asbestos Removal, Basement Construction,

Masonry, Steel Truss, Brickwork Restoration.

SOME HISTORICAL BACKGROUNDS ON HONG KONG THEATRE

Early Western Theatre

The first attempt to erect a theatre in Hong Kong was Messrs Dutronquoy & Co. who announced in October

1842 that the Theatre Royal “is advancing most rapidly towards completion” (Friend of China, October 1842).

It would be located behind the London Hotel in Queen’s Road. Unfortunately, Mr. Gaston Dutronquoy departed

from Hong Kong suddenly on the 17 December and nothing was heard about the theatre thereafter (Smith 1982).

During the winter of 1845-46, western drama performances were carried out at Aqui’s Theatre (the area

presently bounded by Hiller Street, Morrison Street, Queen’s Road and Bonham Strand) in the Lower Bazaar

(the theatre is located in the area presently bounded by Hillier Street, Cleverly Street, Jervois Street and Bonham

Strand), which was built in late 1845.

George Duddell (1821-1887) built the Victoria Theatre in 1848 at the lot up Wyndham Street on a hill behind the

Hong Kong Club. The building was described as large, well ventilated, and brilliantly lighted by the China Mail.

The first performance was conducted on 1 November 1848 with the Chief Justice Major Caine in the seat. In

1859, an auction advertisement on the Theatre next to the Oriental Hotel was last seen before there was nothing

heard (Smith 1982).

Theatre Royal in City Hall 1869 was opened by the Duke of Edinburgh during his Royal Visit to Hong Kong on

2 November 1869 11:15 am (China Mail, 3 November 1869). Duke also attended the virgin performance of the

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theatre on 3 November 1869. After that, the theatre had served the western society till the Second World War.

Tung Hing ( 同慶 ) Theatre at the junction of Po Yan Street and Po Hing Fong (now Po Hing Mansion opposite

Tung Wah Hospital) was completed in 1867. On 5 November 1869, the theatre was honoured by welcoming the

Duke of Edinburgh, the first Royal Visit of Hong Kong, and the performance was the renowned Cantonese

Opera called “Loh Kwoh Tung Seong” ( 六國大封相 ) (Daily Press, 6 November 1869). In 1892, the theatre was

re-constructed with red brick throughout, an iron roof, and all fire proof floors, with four entrances turning it to

become a fire-proof theatre (Daily Press, 12 May 1892). The architect was Albert Denison. The name of the

theatre was also changed to Tsung Hing ( 重慶 ) Theatre (Daily Press, 6 May 1892).

Importation of “Moving Pictures”

On 28 April 1897, Professor Maurice Charvet demonstrated the first film in Hong Kong in the old City Hall

(China Mail, 28 April 1897). On 16 October 1897, he brought a better machine to demonstrate the film show

again in the Royal Theatre of City Hall, yet some of the pictures still had a haziness of outlines (China Mail, 18

October 1897). It started the importation of western films to Hong Kong.

Merging of Opera and Film in Chinese Theatre

From 1903 onwards, a double show scheme was proposed by Ko Sing ( 高陞 ) Theatre, that is, after the

Cantonese Opera Show, a short film followed. On 22 July 1909, Ko Sing’s screen was thrown a film about the

Chinese Imperial funeral by the American Cinematograph Company. Since then, both western film and

Cantonese Show were performed.

From Moving Picture to Sound Film

The first public exhibition of the Edison Kinetophone took place on the evening of 28 April 1913 in the Theatre

Royal. Mr Thomas A. Edison explained his inventions in the first part before demonstrating the show. His

Excellency the Governor, Sir Henry May, was one in the seats though the quality of sound had much to be

improved (China Mail, 29 April 1913). The technology took sixteen years before the talkie film could be landed

properly on the market. On 4 November 1929, China Mail used “the birth of the talkie” as the title to report the

first “talkie” to be produced in the Colony.

From Black and White to Colour Film

In 1921, China Mail used “voice and colour in moving pictures” to introduce the interview with the American’s

largest film exporting Company Far East Manager Dr. Howells. He expected that Hong Kong could see the film

of the States just one month after its show. In 1926, the Technicolour process was used throughout the film

called “the Black Pirate” which was shown in the Queen’s Theatre on 1 October (China Mail, 1 October 1926).

THEATRE IN YAU MA TEI (YMT) AND THE BIRTH OF YMT THEATRE

The earliest theatre in YMT is called Po Hing ( 普慶 ) Theatre (now Nathan Road Eaton Hotel); however, in 1917

it was once called “Yau Ma Tei Theatre” by the Government. In 1916, the Kwong Chi ( 廣智 ) Theatre was

completed (now the YMT Car Park Building in Temple Street). 1921, the Number One ( 第一 ) Theatre came into

YMT (now YMT Jockey Club Polyclinic in Public Square Street). In 1928, Tai Wah ( 大華 ) Theatre was

completed (now Novotel Hotel on Nathan Road). In 1929, Po Hing Theatre was re-constructed, and in 1932 the

Kwong Ming ( 光明 ) Theatre was completed (now Henry Leung Community Centre in Public Square Street). On

1 February 1934, China Mail used “Colony’s Latest Theatre” as the title to describe the opening of the Alhambra

( 平安 )Theatre (now Alhambra Building on Nathan Road), and it then said that the theatre was “the largest and

most up-to-date cinema house in the Colony”. The dress circle was the largest in the Colony having a clear span

of 110 feet and a depth from back to front of 52 feet”, “the whole of the reinforced concrete framework passed

all tests in May 1932 to the satisfaction of the Building Authority”, and it could accommodate 1795 seats. The

architect was Wong Tai Cho. The date of birth of YMTT is controversial. According to the Director of Public

Works Report 1930 (Report of the Director of Public Works for the Year 1930, Paragraph 36), YMTT was

completed during the year. The earliest YMTT advertisement noted in the newspaper was in April 1931 (Kun

Sheung Yat Po, 27 April 1931). During such time, the film industry in Hong Kong was prosperous, and the

colour video and audio technology was mature and watching film became one of the entertainments for Hong

Kong people.

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YMT Theatre History

From 1931 to 1941, the theatre showed Mandarin films. In 1935, it had been used as a Boxer and Magic Show.

After 1941, it showed English films as well as Cantonese films. After the Second World War, it resumed

business in 1948 and joined with the New World, Pei Ho and Sun Wah Theatre to form an alliance of film

showing, mainly for Cantonese films from Ling Kwong Film Production Co ( 嶺光出品 ). 1970, the theatre

changed to show Mandarin films of the Shaws Film Production Co ( 邵氏公司 ). In the same year, the

Government started changing the rent on an annual basis, as the site was marked to develop a market under

Public Works Category C (Hong Kong Public Record Office, HKRS716-1-17). Somehow, the market site was

identified in Mong Kok Road, and YMTT could therefore continue its business. In the 1980s, the theatre was

bounded to change its marketing strategy by showing Japanese pornographic films due to the common use of

video tape and disc players at home, until the 1990s where Western pornographic films were shown. It had

gained the name of “Theatre for the Adults” after twenty years of showing such films. On July 31 1998, it closed

down and left unused. In 2003 when the revitalization of the theatre was approved by the Government, its

notorious name still remained irrespective its function was crowned with the cultural heritage of Cantonese

Opera where the practice of “appointing of a Prince Minister for six states”( 六國大封相 ) would still be trained.

YMTT was born in the era of prosperous film industry in a lower class region in Kowloon, and it had not been

served any important historical persons nor famous in its layout. After revitalization, its function remains close

to that of those in the 19 th century Chinese theatres. The glory of the Cantonese Opera Play during the Royal

Visits perhaps is the only historical memory apart from its notorious name of “Theatre for the Adult”.

THE HISTORY OF YMT PUMPING STATION

The original use of the RBB was a pumping station at the end of the 19 th century. Due to the significant increase

in population in Kowloon, an investigation of water supply was conducted by the Resident Engineer, Mr.

Francis Alfred Cooper (later the Director of Public Works) in 1890 and a report was submitted in 1891. After a

series of testing, the construction of a pumping station in YMT commenced in 1894 and it started operation on

the Christmas Eve of 1895. The pumping station consisted of three components: the office, the pumping

equipment and the staff quarter. The pumping station could not keep pace with the growth of population, and in

1903, government started to build the Kowloon Reservoir. Although the project was completed in 1911, the new

service of the water supply only started operation on the Christmas Eve of 1906, exactly 11 years after its

operation, and the pumping station was thus abandoned. In 1915, Government spent HK $1,493.54 to renovate

the pumping station office and turned it to a post office. The YMT Post Office was opened on 1 July 1915.

The Kowloon Central Post Office was opened on 10 August 1967, and the YMT Post Office was then moved to

the new building. On 12 August 1967, the old post office was closed. The unused post office was marked for the

construction of a refuse collection point (RCP), latrine and bathrooms. The project was awarded to Tai Chung

Decoration and Building Construction Company at a cost of HK $108,049.33 in 1967 (Hong Kong Public Record

Office, HKRS156-1-7075). Unfortunately, at the end of 1969, the progress of the project could only achieve 15%,

and the Government had to terminate the contract and re-tender the works. Works re-started in 1970 and

completed within one year. In 2000, the RCP, latrine and bathrooms were moved to their opposite site of Shanghai

Street just next to the Home for the Street Sleepers.

After the Second World War, the Hawker’s License Office in Leighton Road was found insufficient. In 1952, the

Pumping Station Staff Quarter was used as the Hawker’s License Office Kowloon Branch to facilitate hawkers

in Kowloon (Hong Kong Public Record Office, HKRS621-1-59). On 27 May 1965, the Urban Services

Department Headquarters in Sai Yee Street opened and the Office was moved to the new premises with the

Pumping Station Staff Quarter left vacant. The Mong Kong Kai Fong Association Limited rented part of the

Quarter at a rate of $10 per year in 1969. Two years later, the Society rented the whole building and it became

the Youth Centre and Study Rooms for the students. The Year of 1987 was the international Street Sleepers Year.

The Street Sleepers’ Shelter Society Trustees Incorporated rented the building and turned it into the Home for

the Street Sleepers in November 1988 as the first stage of the project. In December 1989, the project was

completed with 34 beds. It could accommodate 78 persons. Unfortunately, in September 1990, the occupants

turned the building into a drug centre, and the House was closed.

The history of the RBB could be found nearby: the Post Office in Nathan Road was just a few minutes’ walk;

hawkers are found everywhere near the RBB; the latrine, RCP, bathrooms, and the Home for the Street Sleepers

are on the other side of Shanghai Street. The memories of the pumping station could only be revitalized after the

opening of the new training centre when trainers walk back to the YMT Cantonese Opera Training Centre in

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summer with full dresses and make up where they naturally want a drop of water!

SIGNIFICANCE OF BUILDING ELEMENTS AT YMTT AND RBB

YMTT was graded by Antiquities Advisory Board (AAB) as a Grade II historic building in December 1998

while RBB was graded by AAB as a Grade I historic building in June 2000. Figure 1 shows the location of the

buildings.

Although YMTT is not invaluable in terms of construction technology, it is the only surviving pre-war cinema

building in the urban areas of Hong Kong. The building is divided into three basic sections: the front section, the

middle section and the back section. The scale and overall architecture of the building had been assessed to be

exceptional significant in the Heritage Impact Assessment (HIA) Report. The front facade facing Waterloo Road,

gable walls, stage setting for the proscenium arch, interior walls, roof profile and timber purlins are all of high

significance (Architectural Conservation Office 2008). The architecture of YMTT (Photos 1 and 2) presents a

unique example of cinema in Hong Kong. The type of construction with peripheral masonry wall supporting

structural steel roof trusses, roof tiles supported on timber purlins and the reinforced concrete floor supported on

load bearing masonry wall was common in the 1930s. The amalgamation of construction materials of the 19 th

and early 20 th century in a small building has certain historical values. Any modification works to this project

within the graded building needs the approval from AMO.

The architecture of RBB is of functionalism design and exceptional significance according to HIA report. The

external brick wall, pitched roof of double layer Chinese tiles, brick chimney stack and flue openings are also

exceptional significant. Photo 3 shows the appearance of RBB.

Figure 1 Location plan

of YMTT and RBB

Photo 1 External view

of Yau Ma Yei Theatre

(before conversion)

Photo 2 Internal view

of Yau Ma Yei Theatre (before

conversion)

Figure 2 Conversion works at YMTT

Photo 3 External view of Red Brick Building

PROJECT DEVELOPMENT

The project includes the conversion of the YMTT and RBB to provide a performing and practicing venue for

small to medium size Cantonese Opera troupes for the promotion and preservation of this intangible world

cultural heritage. The Xiqu Centre includes a timber flooring stage suitable for Cantonese Opera performance,

an auditorium of 302 seats with a new orchestra pit, a foyer, air conditioning & lighting control rooms and

ancillary facilities. The structural alteration works are unavoidable to alter the original building fabrics in order

to meet the current safety standards. In order to fulfill the functional and statutory requirements of the new

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development and preserve the two graded historic buildings, skilful techniques have been applied to strike a

balance and ultimately the conversion proposals were agreed and approved by the AMO.

According to the Code of Practice (COP) for Fire Resisting Construction, a safety curtain is required to separate

the stage from the auditorium. This curtain requirement makes it unfit for the Xiqu Centre. A fire engineering

approach has been carried out to provide with a drencher installation as an alternative solution instead of using

prescriptive method. Separate fire exits are also provided for the auditorium and stage. The fire engineering

approach also justifies the provision of one staircase for the control room on the first floor. Figure 2 summarizes

the conversion works on the existing building.

The RBB is modified to function as a practice venue for small scale Cantonese Opera performances and training

of artists. Owing to spacious constraint, a fire engineering approach has also been carried out to justify the

omission of hose reel system. New building services requirements require openings to be formed on the brick

wall.

STRUCTURAL SURVEY AND ASSESSMENT

There are no records of structural drawings of the YMTT nor on the modifications or extension works in the past.

A structural survey has been carried out, including site measurements, identification of significant structural

defects, condition of material deterioration and foundation investigation. Testing of materials has also been

carried out in order to assess the structural adequacy and stability of the existing structures. The findings and

assessment are summarized below:-

Description of the existing building (YMTT)

The theatre building is approximately 45m long by 18m wide on plan, with the apex and eaves of the roof at

about 11m and 7m above external ground level respectively. It has a pitched roof with asbestos cement sheets

sitting on circular timber purlins, which are supported by a series of steel roof trusses. The front part of the

building comprises two storeys of reinforced concrete 1/F and roof. The rear part has a concrete roof floor. Two

corner rooms enclosed with a brick wall at the building rear may be an extension to the building.

Roof truss and timber purlins

The roof consists of seven identical steel trusses, each of them built in a triangular shape spanning between

masonry piers. The top chords consist of 2 nos. 152x76 channels and the bottom chords consist of 2 no. 90x10

steel plates in the middle and 2 no. 100x75x10 angles at both ends. Sections of steel members with marks from

different manufacturers were seen, e.g. “Froding Han”, “Port Talbot”, “Dorman Long”, “Cargo Fleet”,

“Skinningrove” (these being British steel manufacturers) and “BHP” (which is the largest Australian steel

manufacturer). As different sections were butt welded randomly to form a single member, this revealed that

steel materials were expensive and sometimes economical solution outweighed the design consideration. The

members are riveted together with gusset plates. The steel trusses have been painted but there are signs of paint

peeling and surface rusting due to inadequate maintenance after the theatre was closed down in 1998.

There are 344 nos. timber purlins with average diameter of 150mm (varying from 100mm to 170mm). The

fixing details and defects including sagging, wet/dry rot and split shakes are shown in Photo 4. Termite

infestation was not observed.

It was originally intended to take down one of the purlins for bending test; but the proposal was dropped after a

close examination of the connection between the purlins and the roof asbestos sheeting. Structural calculation

revealed that the purlin was structurally inadequate to withstand the wind uplift pressure of the current wind

code, nor to support a vertical imposed load of 0.75kPa. Although the timber purlins were decided not to support

the new roof system, they are to be preserved for their historical values and remained on the roof after

preservation treatment according to HIA.

The existing steel members of the roof trusses have been tested with yield strength similar to that of Grade S275,

BS EN 10025. The roof steel trusses have been checked to be just sufficient to carry the self weight of the

existing roof system but are inadequate to satisfy the current wind code, which has been enhanced since 2004.

The angles at both ends of the bottom chord have a FOS of 0.7 (but is adequate in accordance to the COP on

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Design Wind Effects HK-1983) whilst the steel plates at the middle of the bottom chord are too slender and with

lower FOS. The truss is insufficient to carry the additional loads of the new roofing system and building

services/stage facilities. In order not to intervene the existing truss, a new roof system supported on new steel

frame was designed.

Photo 4 Fixing details of

timber purlins

Reinforced concrete members

Photo 5 Concrete spalling and

rusting of rebars at 1/F soffit of

YMTT

Photo 6 Granite pier and pad

footing

Photo 5 shows the condition of concrete spalling and rusting of rebars on the first floor soffit at the front part of

the YMTT. Tests including coring test, carbonation depth test, chloride content test, rebound hammer test,

open-up inspection of rebars and cover meter survey have been carried out for the concrete floors.

The areas of spalling surfaces varied from 100mm by 100mm at the front entrance to 2000mm by 1000mm at

the back stage. Coring tests indicated the estimated in-situ cube strength ranging from 19.5MPa to 30.0MPa.

Depth of carbonation was found exceeding 60mm. The chloride content showed the results from

latter value is calculated according to the soil parameters obtained from the ground investigation and laboratory

test results.

Red Brick Building

The existing record drawing of the pumping station building is shown in Figure 3. Most parts of the building

have been demolished leaving the remaining RBB. RBB is mainly constructed in brickwork. The external walls

are all load bearing solid brick walls supporting the floor, roof loads and resisting all lateral forces, which are

supported on a concrete strip footing. The first floor is constructed in timber with a ‘tongue and groove” timber

planks supported on timber joists. At 1/F verandah, the floor comprises rolled steel joists with mass concrete

encasement. The pitched roof comprises a typical Chinese clay tile construction supported by timber rafters,

which are supported by timber purlins at the brick wall. Two samples of the mortar had been taken for chemical

analysis. The cement: lime: sand content is 1: 3.7: 2.8 and 1: 0.3: 4.9 at the outer and inside layers respectively.

An ultrasonic test for timber was also carried out and there was no evidence of any cavities within the timbers.

The hardwood of strength class of SC6 in BS 5268:Part 2:1991 has been used for structural checking for the

timber joists, which are supported on brick walls at both ends. The timber floor was found capable of supporting

an allowable imposed load of 5.0kPa. Strengthening works for RBB are therefore unnecessary. The roof and

stability of the existing structure have been checked and found adequate to resist the current design wind load.

SITE CONSTRAINTS AND CONSTRUCTION DIFFICULTIES

The YMTT was built up to the site boundaries and is located within the urban area abutting busy main streets,

see Figure 1. There is a lack of loading/unloading spaces, it is difficult in getting access into the building for

excavation and soil backfilling works and there is no space for setting up site facilities. The lifting operations for

coping at Grid 2, installation of structural steel frames and delivery of heavy plants need road closure which

requires close coordination with the relevant authorities and private parties.

The demolition works at Shek Lung Street, re-construction works of the concrete floors at the front and back

parts of the building and construction of basement require substantial temporary works design and construction

such that the stability of the existing masonry wall, proscenium arch, adjacent structures and services will not be

affected. The existing stage was protected by erecting a temporary access ramp steel deck platform, which also

forms the internal haul road leading to the inside and low level of the building. Good planning of works

sequence is critical to avoid conflicts between different trades.

CONSTRUCTION WORKS AT YMTT AND RBB

Removal of Asbestos Roof Corrugated sheeting (ACCS)

Before tendering of the works contract, an asbestos investigation report has been carried out and a preliminary

asbestos abatement plan (AAP) has been accepted by the Environmental Protection Department (EPD). The

contractor had to adopt the approved preliminary plan so that the removal of asbestos could start on site at the

early stage of the contract. According to the plans and requirements by EPD, the contractor is required to

provide with the following:-

• a scaffolding to be erected surrounding the building with at least 2m higher than the ACCS roof and the roof

to be covered with tarpaulin sheet which fully enclose the ACCS and prevent any disturbance to the

surrounding during the asbestos removal work

• a platform to be provided on the roof for workers’ movement and workers are not allowed to walk on the

ACCS and to avoid any accidental damage of the ACCS

• a scaffolding platform to be erected inside the theatre just underneath the whole roof area to allow access

Photo 7 shows the sheltered platform being erected in the middle portion of the roof for removal of ACCS while

the preceding platform is being dismantled. The temporary scaffolding platform, spanning at about 9m, was

installed with supports at both ends of the building and at vertical posts through the existing vent opening. The

platform is a 2m deep truss formed by 50mm diameter steel tubes.

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RBB

Engine House Boiler House

(Pumping

Equipment)

Figure 3 Record drawing of

Pump Station Building

Photo 7 Removal of Asbestos Roof

Corrugated Sheeting

Photo 8 Lifting of coping with

steelwork support

Re-construction of Coping, External Wall/RC floor at Shek Lung Street and RC floor at the Front Entrance

The 18m long coping, which was originally believed to be made of granite stone, is to be retained after the

supporting wall underneath has been re-constructed. It was later found out that the coping is composed of

various types of brick work with sand rich mortar. The coping, after approval by AMO, was divided into

sections (each at approximately 2 tons) and lifted away, as shown in Photo 8, at a maximum reach of 20m by a

30t crane sitting in Reclamation Street. After the reinforced wall at Grid 2 had been completed, the coping was

lifted and restored in position with made up brick and cement sand.

The design of the temporary works has considered (a) the stability of masonry wall with consideration of the

excavation and lateral support (ELS) for basement construction and (b) the stability of the proscenium arch,

which is then exposed to wind load after the external wall in Shek Lung Street is removed. The whole structure

is also required to be protected from weather.

Extensive temporary works can be seen in Photo 9. The concrete floor was temporarily supported by scaffolding

during its demolition. The junction of the concrete floor and the remaining masonry wall structure was saw cut

by hand held tools so that the impact of the demolition works on the existing structures was reduced to minimal.

Design Consideration and Construction of Basement

A new basement, approximately 25m x 10 m on plan (the bottom slab being extended outwards by 1m to resist

water uplift) and at a depth of about 4 to 5m below ground, is about 2.8m away from the peripheral masonry

wall. The information of the existing ground investigation boreholes adjacent to the site was obtained from the

Geotechnical Information Unit of the Civil Engineering Development Department to determine the geology of

the site. Five additional boreholes within the building were also carried out to confirm the ground conditions and

laboratory testing was also carried out to confirm the assumed design soil parameters.

Sheet pile wall (FSPIII) with staged excavations has been designed to penetrate 6m to 9m below ground level

and with toe grout to -11.0mPD as water cut-off system to facilitate dewatering. Structural checking, toe

stability and hydraulic failure checking have been carried out. The predicted maximum ground settlement due to

lateral deflection of the sheet pile wall and dewatering are about 6.2mm and 2.2mm respectively. In order to

verify the adequacy of the depth of grout curtain and the design assumptions, a full scale pumping test,

instrumentation and monitoring have been carried out before commencement of excavation work. The layout of

pump wells, observation wells, re-charged well and monitoring stations is shown in Figure 4. The sheet piles

were installed by Giken Method with minimum ground vibration. Photo 10 shows the Silent Piler, which works

on top of the reaction piles and self moves to the next position by pressing in the former sheet piles. Photo 11

shows the temporary works for basement construction.

Figure 4 Excavation and Lateral Support and Instrumentation Layout

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Unfortunately, some boulders were encountered at depths of some 3m below ground from grids 6 to 11. Pipe

pile wall (219mm OD at 270mm spacing) was employed in this sectional length to overcome boulder

obstruction with the use of pre-boring method. In order to minimize the vibration effect, the following

precautionary measures have been carried out prior to installation of pipe pile:-

• Strengthening the soil mass underneath the pad footing of the adjacent masonry wall by pressure grout

• Reducing the power of pre-boring machine so as to cause minimal vibrations

The construction of basement was carried out smoothly and the maximum ground water drawdown was within

the tidal ranges as recorded before the pumping test. The measured maximum vibration peak acceleration was

1mm/sec at a distance of 5m from the vibration source. There were no observable changes in settlement markers

and tilting markers after the completion of the basement and footings.

Photo 9 Temporary works at

Masonry external walls and arch

Photo 10 Silent piler Photo 11 Temporary works for

basement construction

Photo 12 Roof truss installation Photo 13 Removal of salt at brick wall Photo 14 Damp proof

course injection

NEW ROOF AND ITS INSTALLATION

The new aluminum roof system comprises a proprietary roofing system, sound barrier panels with thermal

insulation system and a profiled glass-fibre reinforced plastic sheeting. As identified in HIA report, all the

existing roof steel trusses should be preserved (though they have no functional use) except at Grids 4 and 9. The

removal of these two trusses is to give a sufficient headroom at the stage area and to provide a space for the new

control room. The configuration of the new truss should be identical to the existing truss.

The new stanchion, with the web and outer flange embedded in concrete, was also designed to support the

granite pier laterally. The concrete casing also acts as fire protection for the steel stanchion. The fabrication of

steelwork was carried out in the mainland of China and was delivered to site for assembly by bolt and nut

connections. Through the cooperation of the contractor with the neighbourhood and relevant authorities, the

contractor successfully obtained an extra works area in Reclamation Street. The installation of the stanchions

and roof steel trusses, originally planned at 6 week duration, was completed in 22 consecutive hours. Photo 12

shows the installation of roof trusses and stanchions.

RESTORATION OF BRICKWORK AT RBB

Efflorescence is one of the most common forms of surface stains on brick wall. It occurs when soluble salts in

the mortars become soluble and leach out of the wall, forming surface deposits. Water can enter the brick wall

through the joint separation cracks, other construction defects or may be absorbed into brick wall in contact with

the ground through capillary action and migrating upward into the brickwork.

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The soluble salts inside the brick wall can be removed by the application of paste comprising chemical detergent

agent and a grade filter paper with high porosity. Salts are absorbed into the paper and removed, as seen in Photo

13. The rising damp problem can be resolved by the application of an injected liquid damp proof course into the

base of the wall, forming an impermeable barrier, as seen in Photo 14. Hydraulic lime, which accepted by the

AMO, is used for the repair of mortar joint and internal rendering.

CONCLUSION

This project exemplifies the contribution of structural engineers in building conservation project. The successful

completion of the project is a concerted effort among different professional disciplines and the contractor, who

have resolved all technical difficulties for the completion of the project. Most importantly, the contractor need to

understand the construction sequence and to ensure that the existing structure is safe at all stages of the work

with indicative solutions provided by the project team. By revitalizing the old theatre to a robust and

well-performing building, the 80 year-old theatre with a valuable memory of the Yau Ma Tei district was

transformed to a well-equipped Xiqu Centre, and as a result the buildings will continue to serve the Public.

ACKNOWLEDGEMENT

The authors would like to record their thanks to the Director of Architectural Services for her kind permission of

publishing this paper. The authors would also like to thank the Chief Structural Engineer/1 for his

encouragement to prepare the manuscript.

REFERENCES

Architectural Conservation Office (2008). Heritage Impact Assessment Report of Yau Ma Tei Theatre and Red

Brick Building (Hong Kong: Architectural Conservation Office) (avaialble:

http://www.lcsd.gov.hk/CE/Museum/Monument/form/YMTT_RBB_HIA%20Report.pdf, accessed: 14

April 2011).

Beall, C. (2004). Masonry Design and Detailing: for Architects and Contractors (New York: McGraw Hill, 5 th

ed).

BMMK Ratcliffe Hoare & Co Ltd (2006). Red Brick Building 344 Shanghai Street Yau Ma Tei Kowloon

Structural Investigation Report (Hong Kong: BMMK Ratcliffe Hoare & Co Ltd) (Unpublished Report).

Bristish Standards Institution (1991). BS 5268: Part 2: Structural use of timber part 2. Code of practice for

permissible stress design, materials and workmanship (London: BSI).

Ho Tin & Associates Consulting Engineers Limited (2004). Structural Survey of Yaumatei Theatre Assessment

Report (Hong Kong: Ho Tin & Associates Consulting Engineers Limited) (Unpublished Report).

Ove Arup & Partners Hong Kong Ltd (2000). Yaumatei Theatre Preparation of Heritage Tourism Development

Structural Survey Report (Hong Kong: Ove Arup & Partners Hong Kong Ltd) (Unpublished Report).

Smith, C. T (1982). “The Hong Kong Amateur Dramatic Club and its Predecessors”, Journal of Hong Kong

Branch of Royal Asiatic Society, 22, 217-252.

CVs OF THE AUTHORS

Ir MA Koon-yiu

Ir Ma was former a Senior Structural Engineer in Architectural Servcies Department before his retirement in

2005. He is now the honorary secretary of the Building Discipline Advisory Panel of the Hong Kong Institution

of Engineer and the vice- chairman of the HKU Master of Arts in Chinese Historical Studies Alumni.

Ir CHAN Yu-kai

Ir Chan was graduated from the University of Hong Kong and has obtained a Master degree from the Hong

Kong University of Science and Technology. He is a Member of the Hong Kong Institution of Engineers. Ir

Chan is a Structural Engineer in Architectural Services Department and is the project structural engineer for the

project of “Conversion of Yau Ma Tei Theatre and Red Brick Building into Xiqu Centre”.

Mr. WONG Chun-yam

Mr. WONG is now an assistant engineer in the Architectural Services Department. He is responsible for the

structural analysis and design for the project.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

TEMPERATURE EFFECT ON VARIATION OF FREQUENCY OF BEAMS: A

COMPARATIVE STUDY

Xiao-Qing Zhou 1 , Bo Chen 2 , and Yong Xia 2

College of Civil Engineering, Shenzhen University, Shenzhen, China.

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China.

ABSTRACT

Changing environmental conditions, especially temperature, have been observed to be a practical yet

complicated factor affecting vibration properties of structures. This study aims to compare the effect of

temperature on vibration frequencies of different structures made of different materials. Three models, a steel

beam, an aluminium beam, and a reinforced concrete (RC) slab, were constructed and placed outside the

laboratory. Temperature of the three models was measured continuously over one whole day under the identical

temperature condition, together with a series of forced modal testing to extract the natural frequencies. The

relation of frequencies to temperature of the three models is established. It shows that all of the identified natural

frequencies have a negative correlation with temperature. Variation of the natural frequencies per unit

temperature change is about 0.02%, 0.03% and 0.30% for the steel beam, aluminium beam and RC slab,

respectively, by linear regression technique. The thermal coefficients of modulus for the three materials are then

identified and are close to the available results. It is found that thermal coefficient of modulus of RC is much

larger than steel and aluminium, which indicates that RC structures are more sensitive to ambient temperature

than steel structures and aluminium ones, although the temperature variation of the RC structures is usually

smaller than the metallic structures under the same environmental temperature condition.

KEYWORDS

Temperature effect, modal testing, natural frequency, linear regression.

INTRODUCTION

Vibration-based structural condition assessment has been widely investigated across the world during the last

decades (Doebling et al. 1998). A successful condition assessment heavily relies on the accuracy of the

measured vibration properties of the structures such as natural frequency and mode shape. Some studies have

found that the changes in structural vibration properties due to temperature variation could be more significant

than those due to a medium degree of structural damage (Salawu 1997) or under normal operational loads (Xu et

al. 2010). If temperature effects are not fully understood, then false structural condition identification may occur

(Xia et al. 2006, Giraldo et al. 2006).

Thermal effects on structural dynamic properties have been investigated since 1970s. In their pioneer study of

the effect of temperature on structural vibration properties, Adams et al. (1978) investigated the relation between

temperature and the axial resonant frequency of a bar. Cornwell et al. (1999) investigated the thermal variation

of dynamic properties of the Alamosa Canyon Bridge and found about 5% daily changes of the first three

natural frequencies. Ni et al. (2005) extracted the modal properties of three cable-supported bridges in Hong

Kong using one-year monitoring data. Liu and DeWolf (2007) reported that the maximum frequency changes of

a curved concrete box bridge is about 6% subjected to a yearly peak temperature range about 70°F (39°C).

Similar studies have been also made by Peeters and De Roeck (2001) on the Z24-Bridge, Macdonald and

Daniell (2005) on the Second Severn Crossing Bridge, and Desjardines et al. (2006) on the Confederation

Bridge.

The previous studies demonstrate that the natural frequencies of structures decrease with the increasing

structural or ambient temperature while the thermal effect on mode shapes is negligible. For example, the first

two modal frequencies identified from the Bill Emerson Memorial Bridge (Song and Dyke 2006) monotonically

decreased as temperature went up in a linear way, while the mode shapes did not show a significant change. Xia

et al. (2006) investigated the effects of temperature and humidity on a continuous concrete slab for nearly two

years and observed that the frequencies had a strong negative correlation with temperature and humidity.

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However, no clear correlation of mode shapes with temperature can be observed as the ambient temperature

affects the entire structure in a uniform manner. As the effect of temperature on damping ratio is concerned, no

consistent conclusion has been drawn. One possible reason is that the damping ratio of structures is more

difficult to be accurately measured. In their laboratory study on the RC slab, Xia et al. (2006) observed that

damping ratios increased when temperature went up.

Although extensive studies on temperature effect of structures have been made, most of these investigations

mainly focus on the temperature-induced frequency variation of concrete structures. Thermal effect on structures

made by other construction materials such as steel and aluminium has not yet to be systematically investigated

and compared with each other. In addition, study on quantitative relation between the dynamic properties and

temperature is rare. This paper investigates the effect of temperature on vibration frequencies of different

structures made of different materials, in particular, steel, aluminium, and RC. Experimental results demonstrate

that RC structures are more sensitive to ambient temperature than steel structures and aluminium ones, although

the temperature variation of the RC structures is usually smaller than the metallic structures.

EXPERIMENT DESIGN

Description of the models

To investigate the variation of natural frequencies due to temperature change, a RC slab, a steel beam, and an

aluminium beam were constructed outside the laboratory and exposed to solar irradiation. Configuration of the

models is illustrated in Fig. 1. The simply-supported RC slab is 3,200 mm long, 800 mm wide, and 120 mm

thick with 100 mm overhang at each end (Fig. 1a). It is assumed that the temperature variation along the

horizontal direction of the slab is not significant. Consequently temperature gradient may occur in the vertical

direction (thickness direction) only. Seven thermocouples (T1 to T7) with an evenly interval of 20 mm in

vertical direction were embedded into the RC slab to collect the time dependent temperature data. Three

accelerometers (A1~A3) were firmly mounted on the bottom surface of the slab to collect the acceleration time

histories and extract the natural frequencies. The Young’s modulus and density of the concrete were measured as

2.1×10 10 N/m 2 and 2.316×10 3 kg/m 3 , respectively.

The steel cantilever beam measures 750 mm long, 50.8 mm wide, and 3.0 mm thick (Fig. 1b). As the thermal

conductivity coefficient of steel is rather large and the thickness of the beam is rather small, it is expected that

the temperature difference between the top surface and bottom surface is small. Therefore, only the top and

bottom surfaces of the beam were mounted with thermocouples (T8 and T9) to measure the temperature. Two

accelerometers (A4 and A5) were placed on the steel beam for modal testing. The Young’s modulus of the steel

is 2.06×10 11 N/m 2 and the density is 8.027×10 3 kg/m 3 .

As shown in Fig. 1c, the aluminium cantilever beam measures 750 mm long, 50.0 mm wide, and 5.5 mm thick.

Similar to the steel beam, two thermocouples (T10 and T11) were mounted on the top and bottom surface of the

beam and two accelerometers (A6 and A7) were installed on top surfaces for modal testing. The Young’s

modulus of the aluminium is 4.6×10 10 N/m 2 and the density is 2.881×10 3 kg/m 3 . An additional temperature

sensor (T12) was employed to measure the air temperature.

Experimental procedure

A previous study on a RC slab indicates that the mode shapes are not dominantly affected by the temperature

changes (Xia et al. 2006). Consequently, this study only investigates the changes of natural frequency with

respect to temperature. The temperature variation was recorded from the embedded thermocouples every 1

minute from 7:00 to 21:00. The modal testing was carried out once per half an hour from 7:45 to 20:45 in the

day. Seven accelerometers were used to measure the vibration responses of the concrete slab, the steel beam,

and the aluminium beam under the impact from an instrumented hammer. 30,000 data points were collected at a

sampling rate of 1,000 Hz. Three to four tests were performed for each configuration and an average taken. The

frequency response function (FRF) of the measured data was calculated in a frequency range of 0 to 500 Hz

with a frequency resolution of 0.033 Hz. Natural frequencies were then extracted from the FRFs with the global

rational fraction polynomial method (Ewins 2000).

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Accelerometers

Support

Thermocouples (T1~T7)

100 mm

750×4=3000 mm

Bottom

surface

100 mm

50 mm

750 mm

20×6=120 mm

(a) RC slab

Accelerometers

Top surface

250 mm 500 mm

50.8 mm

Thermocouples (T8~T9)

3.0 mm

(b) Steel beam

Accelerometers

Thermocouples (T10~T11)

Top surface

50 mm

T10

T11

5.5 mm

500×2=1000 mm

Figure 1 Configuration of the RC slab and beams

TEMPERATURE DATA

It is observed that the RC slab and the two metallic beams were exposed to sun irradiation in the morning only

and shaded by buildings from about 12:00. The temperature variation of the models during the day is shown in

Fig. 2, together with the air temperature.

Temperature variation of the RC slab

Fig. 2(a) plots the temperature variation of the RC slab. The data of the thermocouple T6 are abnormal and thus

not shown here. It can be seen that the temperature at the slab (T1 to T7) increased from 7:45 to about 12:00 and

then decreased after that. Moreover, the temperature near the top surface (T1 and T2) went up quicker and more

significantly in the morning than the temperature inside and the temperature near the bottom surface. In the

afternoon, the former dropped down quicker than the latter. This thermal inertia is due to the fact that the

thermal conductivity of RC is low (about 1.5 W/(m.K) ).

Temperature variation of the steel beam and aluminium beam

The temperature measured at the surface of the steel beam (T8 and T9) and the aluminium beam (T10 and T11)

are drawn in Fig. 2(b). It is observed that the temperature of two beams arose substantially until about 11:30.

During this period, the temperature of the two beams was much higher than the air temperature. After that time,

the direct solar irradiation was blocked and the structural temperature dropped down very quickly and

approached to ambient temperature. It is also shown that the temperature difference between the top and bottom

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temperature of the beams is small, as expected, because the thermal conductivities of steel (54 W/(m.K) ) and

aluminium (177 W/(m.K) ) are higher than that of RC.

Comparing the temperature variation of RC slab and two metallic beams, one can find that the temperature of

the RC slab arose slower than the metallic beams and was lower than the latter before 11:30. In the afternoon,

the situation reversed. This is because the thermal diffusivity of RC is smaller than steel and aluminium.

Consequently, the temperature of the RC slab is less sensitive to the ambient temperature change.

Temperature (degree)

T12 Ambient temp.

Temperature (degree)

T12 Ambient temp.

T8 Steel beam top

T9 Steel beam bottom

T10 Aluminum beam top

T11 Aluminum beam bottom

07:45 09:45 12:15 14:45 17:15 19:45

Time

(a) RC slab

Figure 2 Temperature variation of the models

VARIATION OF FREQUENCY WITH RESPECT TO TEMPERATURE

(b) Metallic beams

The dynamic properties of the models were obtained per half an hour. The first four frequencies of the two

metallic beams and first two frequencies of the RC slab were extracted from the collected data at different time.

Natural frequency of the steel beam

3.67

23.72

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(a) 1st frequency

1st freq.

3.66

3.65

3.64

3.63

3.62

Frequency (Hz)

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(b) 2nd frequency

2nd freq.

23.70

23.68

23.66

23.64

23.62

Frequency (Hz)

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66.4

139.6

139.5

Temperature (degree)

3rd freq.

07:45 09:45 12:15 14:45 17:15 19:45

66.3

66.2

66.1

66.0

Frequency (Hz)

Temperature (degree)

4th freq.

07:45 09:45 12:15 14:45 17:15 19:45

139.4

139.3

139.2

139.1

139.0

138.9

138.8

138.7

Frequency (Hz)

Time (Hour)

(d) 4th frequency

Figure 3 Variation of natural frequencies of the steel beam with respect to temperature

Temperature (degree)

1st freq.

2nd freq.

3rd freq.

4th freq.

0.6

0.4

0.2

0.0

-0.2

-0.4

Freq. change (%)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

Figure 4 Variation percentage of natural frequencies of the steel beam with respect to temperature

Fig. 3 shows the first four natural frequencies of the steel beam with respect to the surface temperature. For

clarity, the temperature axis is placed on the left in reverse order and the frequency axis on the right. It is

observed that the variation trend of the first four natural frequencies matches the surface temperature very well

although the frequency variation is very small. Also, the natural frequencies drop down with the temperature

increase and vice versa, as many researchers have found. The variation percentage of the natural frequencies

with respect to the surface temperature is shown in Fig. 4. The relative variation ranges of the first four natural

frequencies are −0.44% to 0.31%, −0.18% to 0.06%, −0.29% to 0.05%, and −0.31% to 0.03%, respectively. This

indicates that different modes have similar relative frequency change, which will be quantitatively analysed

later.

Natural frequency of the aluminium beam

Variation in the first four natural frequencies of the aluminium beam with respect to the temperature is plotted in

Fig. 5. Again variation trend of all of the four natural frequencies agrees with the surface temperature well. It is

also observed that the higher modes match with the temperature better than the lower modes do. One reason is

that the variation magnitude of the lower frequency is smaller than the variation magnitude of the higher

frequency under the identical percentage change. For example, the absolute variation of first frequencies is 0.03

Hz only, while the counterparts for the second, third and fourth frequency are 0.25 Hz, 0.30 Hz and 0.51 Hz,

respectively. Consequently the measurement noise will have more significant effect on the lower modes than on

the higher modes. Nevertheless, the experimental results still show the strong correlation between the

frequencies and temperature. Similar to the observation from the steel beam, the natural frequencies of the

aluminium beam decrease with increase of the surface temperature before 11:30 and increase slowly after that

-0.6

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time with decrease of the surface temperature. The relative variation ranges of the natural frequencies are

−0.38% to 0.59%, −1.33% to −0.04%, −0.47% to 0.02%, −0.44% and 0.22%, respectively. The figure is not

drawn here to avoid repetition.

3.24

20.4

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(a) 1st frequency

T10

T11

1st freq.

3.23

3.22

3.21

3.20

3.19

3.18

3.17

3.16

Frequency (Hz)

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(b) 2nd frequency

T10

T11

2nd freq.

20.3

20.2

20.1

20.0

19.9

19.8

19.7

Frequency (Hz)

59.8

116.2

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

T10

T11

3rd freq.

59.6

59.4

59.2

59.0

Frequency (Hz)

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(d) 4th frequency

T10

T11

4th freq.

116.0

115.8

115.6

115.4

115.2

115.0

114.8

Frequency (Hz)

Figure 5 Variation of natural frequencies of the aluminium beam with respect to temperature

Natural frequency of the RC slab

The first two natural frequencies and the structural temperature at top and bottom surfaces are illustrated in Fig.

6. The relative change is shown in Fig. 7. It is seen that the variation trend of the first two frequencies are quite

close to that of the surface temperature. In addition, one can find that frequency variation of the RC slab is much

larger than those of the metallic beams under the same temperature condition.

18.0

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(a) 1st frequency

1st freq.

17.5

17.0

16.5

16.0

Frequency (Hz)

Temperature (degree)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

(b) 2nd frequency

2nd freq.

Frequency (Hz)

Figure 6 Variation of natural frequencies of the RC slab with respect to temperature

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Temperature (degree)

1st freq.

2nd freq.

-2

-4

-6

-8

Freq. change (%)

07:45

09:45 12:15 14:45 17:15 19:45

Time (Hour)

Figure 7 Variation percentage of natural frequencies of the RC slab with respect to temperature

QUANTITATIVE RELATION BETWEEN FREQUENCY AND TEMPERATURE

The linear regression technique is utilized to examine the relation between the frequencies and the structural

temperature. A linear regression model has the form of (Kottegoda and Rosso 1997)

-10

= β + βt+ ε

(1)

0 t f

where f is the frequency, β 0 (intercept) and β t (slope) are the regression coefficients, ε f is the regression error, and

t is the temperature explanatory variable. In the regression analysis, the averaged temperature measured across

the structural cross section is adopted. For the RC slab, the temperature measured at six thermocouples is

averaged. For the steel beam and the aluminium beam, the temperature at top and bottom surfaces is averaged.

The correlation coefficients of the three models are calculated and listed in Table 1. It is seen that the correlation

coefficients of the first four natural frequencies of the steel beam are −0.932, −0.990, −0.957, and −0.967,

respectively, implying a very good linear correlation between the average temperature and natural frequencies.

The fitted regression lines of the steel beam with the 95% confidence bounds are plotted in Fig. 8. The

regression results of the aluminium beam and the RC slab are shown in Fig. 9 and Fig. 10, respectively.

Although the correlation coefficients of the latter two models are smaller than the counterparts of the steel beam,

good linear correlation between the natural frequencies and average temperature can still be observed from

Table 1 and Figs. 9 and 10.

Table 1 Correlation coefficients of natural frequencies and averaged temperature

Mode Steel beam Aluminium beam RC slab

1 -0.932 -0.876 -0.832

2 -0.990 -0.650 -0.822

3 -0.957 -0.855

4 -0.967 -0.958

The regression coefficients of the three structures are also obtained and listed in Table 2 for each mode. The

slope β t represents the frequency change with respect to temperature and the intercept β 0 represents the

frequency at 0 ◦ C. For comparison of different modes, β t is normalized to β 0 and listed in Table 2 as well. β t /β 0

indicates the percentage of the frequency change when the structural temperature increases by one degree

Celsius. It can be seen that the changes of the first frequency of the two metallic beams are larger than those of

higher modes. This is because the measurement noise has larger effect on the lower modes, as discussed above.

Except the first mode of the metallic beams, the modal frequencies of the steel beam, the aluminium beam and

the RC slab decrease by about 0.015% to 0.020%, 0.027% to 0.042%, and 0.24% to 0.34%, respectively, when

the structural temperature increases by one degree Celsius. Obviously the natural frequencies of the RC slab are

much more sensitive to temperature than the metallic beams.

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1st Frequency .(Hz)

3.70

3.68

3.66

3.64

3.62

Observation

Fitted data

95% Bound

2nd Frequency .(Hz)

23.74

23.72

23.70

23.68

23.66

23.64

Observation

Fitted data

95% Bound

3.60

25 30 35 40 45

Temperature (degree)

(a) 1st frequency

23.62

25 30 35 40 45

Temperature (degree)

(b) 2nd frequency

3rd Frequency .(Hz)

66.4

66.3

66.2

66.1

Observation

Fitted data

95% Bound

4th Frequency .(Hz)

139.5

139.4

139.3

139.2

139.1

139.0

Observation

Fitted data

95% Bound

66.0

25 30 35 40 45

Temperature (degree)

138.9

25 30 35 40 45

Temperature (degree)

(d) 4th frequency

Figure 8 Relation of natural frequencies to temperature (the steel beam)

Table 2 Regression coefficients of the beams

Mode

Steel beam Aluminium beam Concrete slab

β 0 β t β t /β 0 β 0 β t β t /β 0 β 0 β t β t /β 0

1 3.69 -0.0014 -3.78×10 -4 3.28 -0.0024 -7.31×10 -4 19.34 -0.066 -3.4×10 -3

2 23.79 -0.0035 -1.47×10 -4 20.37 -0.0086 -4.22×10 -4 82.15 -0.200 -2.4×10 -3

3 66.64 -0.0132 -1.98×10 -4 60.01 -0.0163 -2.72×10 -4

4 140.06 -0.0253 -1.81×10 -4 117.63 -0.0470 -4.00×10 -4

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1st Frequency .(Hz)

3.30

3.25

3.20

3.15

Observation

Fitted data

95% Bound

2nd Frequency .(Hz)

20.6

20.4

20.2

20.0

19.8

Observation

Fitted data

95% Bound

3.10

25 30 35 40

Temperature (degree)

(a) 1st frequency

19.6

25 30 35 40

Temperature (degree)

(b) 2nd frequency

3rd Frequency .(Hz)

59.9

59.7

59.5

59.3

Observation

Fitted data

95% Bound

4th Frequency .(Hz)

116.4

116.2

116.0

115.8

115.6

115.4

Observation

Fitted data

95% Bound

115.2

59.1

25 30 35 40

Temperature (degree)

115.0

25 30 35 40

Temperature (degree)

(d) 4th frequency

Figure 9 Relation of natural frequencies to temperature (the aluminium beam)

1st Frequency .(Hz)

Observation

Fitted data

95% Bound

2nd Frequency .(Hz)

Observation

Fitted data

95% Bound

25 30 35 40

Temperature (degree)

(a) 1st frequency

25 30 35 40

Temperature (degree)

(b) 2nd frequency

Figure 10 Relation of natural frequencies to temperature (the RC slab)

CONCLUSIONS

This paper investigates the temperature effect on variation of natural frequencies of three structures constructed

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espectively with RC, steel, and aluminium. The time dependent temperature of the three structures was

measured continuously in one day, together with a series of modal testing. The relation of frequency to

temperature change of the three models is quantitatively studied and compared. The following conclusions can

be drawn from the results:

1. Temperature of the metallic beams increases quicker than the RC slab in the morning and drops down

quicker than the RC slab in the afternoon, as the thermal diffusivity of RC is smaller than steel and

aluminium. In addition, the temperature gradient of the RC slab is larger than the two metallic beams as the

heat conductivity of concrete is smaller than steel and aluminium, as expected.

2. The first a few natural frequencies of the three testing structures are well-correlated with temperature

change. In particular, the frequency decreases with the increasing temperature in the morning and increases

with decreasing temperature in the afternoon, although the variation of the first mode of the metallic beams

is very small.

3. The modal frequencies of the steel beam, the aluminium beam, and the RC slab decrease by about 0.02%,

0.03% and 0.30%, respectively, when temperature increases by one degree Celsius, regardless of modes.

This implies that frequency of concrete structures is more sensitive to temperature change than steel

structures and aluminium structures. The quantities can be used to eliminate the temperature effect on

structural frequencies.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support provided by the Hong Kong Polytechnic University

(Project No. A-PJ14) and the Shenzhen University Basic Research Grant (Project No. 201045).

REFERENCES

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visualization for structural monitoring of the confederation bridge”, Advances in Structural Engineering,

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Hertfordshire, UK.

Giraldo, D.F., Dyke, S.J. and Caicedo, J.M. (2006). “Damage detection accommodating varying environmental

conditions”, Structural Health Monitoring, 5(2), 155-172.

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ambient loads”, ASCE Journal Structural Engineering, 133(12), 1742-1751.

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Peeters, B. and De Roeck, G. (2001). “One-year monitoring of the Z24-Bridge: Environmental effects versus

damage events”, Earthquake Engineering and Structural Dynamics, 30(2), 149-171.

Salawu, O.S. (1997). “Detection of structural damage through changes in frequency: A review”, Engineering

Structures, 19(9), 718-723.

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considering temperature effects”, Proceeding of fourth World Conference on Structural Control and

Monitoring, San Diego, USA.

Xia, Y., Hao, H., Zanardo, G. and Deeks, A.J. (2006). “Long term vibration monitoring of a RC slab: Temperature

and humidity effect”, Engineering Structures, 28(3), 441-452.

Xu, Y.L., Chen, B., Ng, C.L., Wong, K.Y. and Chan, W.Y. (2010). “Monitoring temperature effect on a long

suspension bridge”, Structural Control and Health Monitoring, 17x(6), 632-653.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

加入隨機搜尋之拓樸最佳化

郭哲宇呂良正

國立臺灣大學土木工程學系碩士生

國立臺灣大學土木工程學系教授

摘要 : 本文首先探討結合基因演算法之結構最佳化演進法 (GESO) 及元素交換法 (EEM),

比較此兩種包含隨機化之拓樸最佳化方法與結構最佳化演進法 (ESO) 及雙向結構最佳化演進法

(BESO) 之結果差異 , 主要是觀察隨機化的加入對於改善原先無隨機化之拓樸最佳化結果是否有

改善 , 一般而言隨機程序的加入是有好處的 , 這兩種方法尤其以 EEM 觀念簡單 , 很容易實作。本

文進而針對 EEM 之缺點提出改良型元素交換法 (MEEM) , 主要是將元素交換個數隨演進步數由線

性改為雙線性 , 另外初始設計領域可以是任意一個體積比 , 不用是目標體積比。

關鍵詞 : 拓樸最佳化隨機搜尋基因演算法元素交換法改良型元素交換法

結構最佳化演進法雙向結構最佳化演進法

TOPOLOGY OPTIMIZATION USING STOCHASTIC SEARCH METHODS

J. Y. Guo 1 and L. J. Leu 2

1,2

Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan

Abstract: Two stochastic search methods for topology optimization including the genetic evolutionary

structural optimization(GESO) and element exchange method(EEM) methods are first presented. Results

obtained by the two methods are compared with those from the evolutionary structural optimization(ESO) and

bi-directional evolutionary structural optimization(BESO) methods. It is found in general stochastic search

methods can obtain better results. Among the two stochastic search methods, the EEM is better because it is

simple and easy to impement. To make EEM more efficient, a modified element exchange method (MEEM)

is proposed in this paper. The MEEM adopts a bi-linear relation to determine the number of exchange elements

during the evolution process instead of a linear relation. In addition, the volume ratio of the initial design

domain in the MEEM can be any value.

Keywords: topology optimization, stochastic search, genetic algorithm, element exchange method, modified

element exchange method, evolutionary structural optimization, bi-directional evoluitionary etructural

optimization.

一、前言

佳化之結構設計時 , 拓樸最佳化是其中的第一步 [1] , 拓樸最佳化之結果決定了結構的主要型式、

樣貌與桿件分布 , 因此一個良好的拓樸最佳化結果影響結構整體設計極大。在拓樸最佳化的方法中 ,

主要使用結構最佳化演進法 (Evolutionary Structural Optimization, ESO) [2] , 本方法流程簡單、觀念

容易 , 計算時間與負荷亦較其他方法為小 , 容易於實際工程設計應用。而雙向結構最佳化演進法

(Bi-directional Evolutionary Structural Optimization, BESO) [3] 針對此部分問題加以改良 , 以平滑化

及投影之概念 , 使結構最佳化演進法除了移除外亦能夠做填回之動作 , 使方法具有更大之自由程度 ,

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並也允許不同的初始設計。

在進行最佳化時 , 並非皆能夠找到全域最佳解 , 常會找到區域最佳解 [4] , 拓墣最佳化中也會遇

到此種問題。因此若是加入隨機化步驟 , 希望進入區域最佳解時 , 能夠因為隨機使得當時的解能夠

重新有機會再度找到全域最佳解。在許多的最佳化方法中 , 如基因演算法 (Genetic Algorithm, GA)

[5,6] 、模擬退火法 (Simulated Annealing Method, SA) [7] 、蟻群最佳化法 (Ant Colony Optimization, ACO)

[8]

等 , 皆擁有隨機之特性能夠使求得的解達到全域最佳解。因此本研究運用結合基因演算法之結構

最佳化演進法 (Genetic Evolutionary Structural Optimization, GESO) [9] 以及元素交換法 (Element

Exchange Method, EEM) [10] 此兩種包含隨機步驟之方法 , 嘗試改善原先 ESO 以及 BESO 方法之結果 ,

並因能更趨近全域最佳解而讓拓樸結果能夠更加具有適用性。

本文於第二節介紹 GESO [9] 方法 , 將此方法和原先 ESO [2] 方法進行比較 , 並在 ESO [2] 方法裡加入

GESO [9] 方法中可以用來做改良之部分並觀察其效果。在第三節中 , 介紹元素交換法 (EEM) [10] ,

並探討此方法與 BESO [3] 方法之比較以及其優缺點。第四節中 , 將原本元素交換法中不足的部分加

以改良。改良後之元素交換法將可更加具有泛用性及適用性 , 並可針對不同設計領域之題目 , 尤其

是由全滿領域以及全空領域進行最佳化設計之結果一致性。第五節將此改良型元素交換法運用於 3D

結構上 , 觀察於 3D 之情況時 , 可能需改進或是與原先 2D 不同之地方。最後一節總結本研究之成果

以及可能繼續改進之處。

二、結合基因演算法之結構最佳化演進法 (Genetic Evolutionary Structural Optimization, GESO)

結合基因演算法之結構最佳化演進法 (GESO) [9] [5,6]

將基因演算法中諸如突變以及交配之概念

結合入結構最佳化演進法 (ESO) [2] 中 , 進行結構設計時 , 將設計領域離散為有限元素網格 , 並賦

予每個元素一串全部為 1 之基因碼。進行有限元素分析後依照應變能排序 , 將最後 m 個元素執行突

變步驟 , 隨機將選中之元素基因碼某一碼由 1 改為 0, 之後執行交配步驟。若元素之基因碼全部為

零時 , 則判定此元素移除。若基因碼只有一碼為零之元素 , 將此減少厚度 , 此舉可以使此元素因為

應力集中使得此元素在下一步分析後之排序能夠提高順位進而使此元素不被移除。

例題一 ( 如圖 1), 為 200 mm × 100 mm 之設計領域離散為有限元素網格 , 每個元素為 2 mm × 2

mm 之 Q4 平面應力元素 , 厚度為 1 mm, 楊氏模數 E 為 207 GPa, 伯松比 ν 為 0.3。力量加載在下方

中央及四分點位置 , 各為 10 kN, 左側及右側皆為固定端 , 因為對稱採用半領域進行設計。移除比

率各為 50%、60%、70%, 比較 GESO 及 ESO 之結果 , 由表 1 可看出使用 GESO 方法之結果稍微

比 ESO 方法好些。表 1 及本文之其他結果 ,C 為應變能 ( 單位為 kN-mm), 結構受力固定之下 , 應

變能越小相當於勁度越大 , 也即 C 越小越好。

例題二之結構同例題一 , 但是將 ESO [2] 方法加入厚度改變。即原先於 ESO [2] 方法中被判定為需

移除之元素 , 不先移除而先改變厚度 , 待下次若再被判定為移除時再行移除。觀察運用此方法所改

善的幅度和例題一的結果對照。表 1(ESO, 加入厚度變化 ) 結果 , 可看出加入厚度改變後 , 可以

稍微改善原先的結果但效果並沒有 GESO 這麼好 , 因此 GESO 方法可以達到比較好的結果 , 因此可

以看出加入隨機化可以有改進的效果。

圖 1 例題一之結構示意圖

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表 1 例題一 GESO、ESO 及 ESO 加入厚度變化之比較

RV50% RV60% RV70%

GESO

C = 2.69 kN-mm C = 3.11 kN-mm C = 3.88 kN-mm

ESO

C = 2.69 kN-mm C = 3.12 kN-mm C = 3.96 kN-mm

ESO( 加入厚度變化 )

C = 2.70 kN-mm C = 3.11 kN-mm C = 3.88 kN-mm

三、元素交換法 (Element Exchange Method, EEM)

元素交換法 (EEM) [10] 運用簡單的概念將有限元素分析後之各元素依照空心元素以及實心元素

分別依照應變能進行排序 , 實心元素所佔之比例即為體積比 , 整個演進過程中體積比是固定的。本

法將實心元素中應變能表現較差之元素以及空心元素中應變能表現較佳之元素進行交換 , 藉由持續

交換最後得到最佳化結果之拓樸圖形。元素交換法之基本原理如下 :

(a)

(b)

圖 2 元素交換法之基本結構 (a) 交換前 (b) 交換後

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圖 2(a) 為一有串聯及並聯彈簧之系統 , 其中彈簧 K s 之勁度為彈簧 K v 勁度之 1000 倍 , 若將 K s

中應變能表現較差者 ( 彈簧 1) 及 K v 中應變能表現較好者 ( 彈簧 4) 兩者交換 , 交換後之系統見圖

2(b)。系統之總應變能變化如下 :

交換前 :

1 1 F

= × + × + × + × ≈ =

2 2 2

s −6 −3 0 −3 2 2

⎡

⎣1 10 1 10 1 10 1 10 ⎤

⎦ δ K δ s

(1)

交換後 :

−4 −4 0 0 2 2 1 F

= ⎡2.5 10 2.5 10 1 10 1 10 Ks

2 ⎣ × + × + × + × ⎤

⎦ δ ≈ δ =

4 K

(2)

要注意的是交換前 δ ≈ F / K

, 交換後 δ ≈ F /2K

, 可以看到交換兩個彈簧因而使系統之總應變

能下降 , 因此在做完有限元素分析後隨著計算 ( 演進 ) 步數增加漸漸減少交換元素的個數 , 得到穩定。

最後得到的拓樸即為所要的答案。而交換個數隨演進步數之變化如下 (int= 取整數 ):

⎡ ⎛M

M = int M −k

⎢⎣

⎝

− M

k EE−max

EE−min

EE ⎢ EE−max

⎜

Nmax

⎞⎤

⎟⎥

⎠⎥⎦

(3)

為在第 k 演進步時之元素交換個數 , N max

為最大步數 ( 總步數 ), 一般可採 N

max

= 500 。依

照此式 , 每步之交換個數呈線性遞減。 M EE

之最大值 ( M

EE − max

) 及最小值 ( M

EE − min

) 一般分別在目標體

積比總元素的 5% 至 10% 以及 0.2% 至 0.4% 間。依照應變能進行交換後 , 進行隨機交換步驟。隨機

交換個數隨演進步數之變化如下 :

⎡ ⎛M

M = int M −k

⎢⎣

⎝

− M

k RS −max

RS −min

RS ⎢ RS −max

⎜

Nmax

⎞⎤

⎟

⎠⎥⎦

(4)

隨機交換之個數變化 (4) 式與交換個數之個數變化 (3) 式有相同形式 , 最大值以及最小值也相同於

交換個數時之設定。

元素交換法皆是在進行交換 , 因此必須於一開始即設定好目標體積比之元素 , 和 ESO [2] 方法從

全佈滿元素之初使設計領域或是 BESO [3] 方法可從任意體積比之初始設計領域進行最佳化不同。

由於必須於一開始即設定好目標體積比之元素來當作初始設計領域 , 因此可以在不同之設計領

域之下得出相近的答案。

例題三如見圖 3 為 160 mm × 100 mm 之設計領域離散為有限元素網格 , 每個元素為 5 mm × 5 mm

之 Q4 平面應力元素 , 厚度為 1 mm, 楊氏模數 E 為 200 GPa, 伯松比 ν 為 0.3, 力量加載在右側下

方 10 kN, 左側為固定端。本例採 M

EE − max

= 6.25% ( 相當於 20 個 ) , M

EE − min

= 0.625% ( 相當於 2 個 ),

演進步 1000 步。比較固定體積比 50% 下三種不同初始領域下之結果 , 結果見表 2。由表 2 可看出

在不同初始領域下可得到幾乎相同的拓樸 , 但是因為隨機步驟的關係 , 造成最後的拓樸會有少數游

離元素產生。

圖 3 例題三之結構示意圖

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表 2 例題三不同初始設計之結果

初始領域

( 體積比 50%)

拓樸結果

應變能 C = 11.09 kN-mm C = 10.95 kN-mm C = 11.09 kN-mm

例題四如圖 4 為 12000 mm × 6000 mm 之設計領域離散為有限元素網格 , 每個元素為 100 mm ×

100 mm 之 Q4 平面應力元素 , 厚度為 100 mm, 楊氏模數 E 為 210 GPa, 伯松比 ν 為 0.3, 力量加載

在下方中央 30 kN, 下方兩四分點位置各 15 kN, 左側端為鉸支承 , 右側端為滾支承。針對目標體積

比 30% 時不同初始設計領域和 BESO [4] 方法之比較 , 且由於結構對稱性因此此例題將隨機步驟取消。

本例採 M

EE − max

= 6.94% ( 相當於 150 個 ) , M

EE − min

= 0.28% ( 相當於 6 個 ) 。

由表 3 可看出和 BESO [3] 比較也得出一致的結果 , 只有在最後一個初始領域下產生有些許的不

同。

圖 4 例題四之結構示意圖

四、改良型元素交換法 (Modified Element Exchange Method, MEEM)

前一節所使用之元素交換法 [10] , 在不同的設計領域下可以達到不錯的一致結果。但是 , 原先之

方法仍有幾個問題 : 一、此種方法必須在一開始即將目標體積比設定好 , 演進中體積比不變。無法

如 BESO [3] 方法可以設定任意體積比。二、由於隨機步驟造成最後拓樸之結果產生游離元素 , 見表 3。

三、由於結構對稱性之問題 , 因此在某些情況下必須將隨機步驟取消 , 因而造成結果不夠一致 , 如

表最後的初始領域產生稍微不一致之結果。因此本文提出改良型元素交換法即是針對前述之問題加

以解決。

元素交換法之元素交換個數遞減方式為線性 , 在此將之改為雙線性 , 在一開始時採用較大之遞

減率 , 以上如下列兩式所描述 (int= 取整數 ):

If k ≤ N′ :

k ⎡ ⎛M′

−MEE−max

⎞⎤

MEE

= int ⎢M′

−k⎜

N′

⎟⎥

(5)

⎣ ⎝

⎠⎦

If k > N′ :

⎡

M = int M − k − N

⎢⎣

⎛M

− M

−max −min

( ′) k EE EE

EE ⎢ EE−max

⎜

Nmax

− N′

⎝

⎞⎤

⎟

⎠⎥⎦

(6)

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以上式子 M ′ 採用 0.2 倍至 0.3 倍之目標體積比元素數量 , N′ 設定為總步數 N 之十分之一。

例題五為針對此雙線性之收斂函數之結果 , 所使用之結構算例同例題四 , 採用參數為

EE − max

= 6.94% ( 相當於 150 個 ) , M

EE − min

= 0.28% ( 相當於 6 個 ) , M ′ = 25% 。本結構見圖 4, 結

果見表 4。由表 4 可看出所提之 MEEM 針對不同設計領域 , 可得到一致之拓樸。

max

表 3 EEM 方法以及 BESO 方法在初始設計之比較

初始領域 ( 體積比 30%) 拓樸結果 (EEM) 拓樸結果 (BESO)

初始領域 ( 體積比 30%)

表 4 例題五之結果

拓樸結果

例題六如圖 5 為 12000 mm × 6000 mm 之設計領域離散為有限元素網格 , 每個元素為 100 mm ×

100 mm 之 Q4 平面應力元素 , 厚度為 100 mm, 楊氏模數 E 為 210 GPa, 伯松比 ν 為 0.3, 力量加載

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在下方均布載重 10 kN/m, 左側端為鉸支承 , 右側端為滾支承。演進時相關參數設定同前題 , 結果

如表 5, 可知在此均佈力作用下 , 仍然可以得到相當一致的拓樸。

圖 5 例題六之結構示意圖

初始領域 ( 體積比 30%)

表 5 例題六之結果

拓樸結果

針對解決原本元素交換法必須在固定體積比下進行交換 , 使元素交換法能夠更加擁有彈性。依

照當前步數下之元素與目標體積比時元素之比值當作倍率來調整當前之交換值 , 實心元素以及空心

元素之交換數量如下 :

−

= × (7)

k k Solid

EE Solid

MEE Obj

NSolid

−

= × (8)

k k Void

EE Void

MEE Obj

NVoid

藉由以上式子 , 在每步中 , 實心元素以及空心元素的交換數量不會一樣 , 因此兩者之交換數量

差便會使得體積比改變。而將比值開四次方是希望整體領域在初期不會有太大的體積比變化 , 希望

能使變化較緩慢。

例題七為 12000 mm × 6000 mm 之設計領域離散為有限元素網格 , 每個元素為 100 mm × 100 mm

之 Q4 平面應力元素 , 厚度為 100 mm, 楊氏模數 E 為 210 GPa, 伯松比 ν 為 0.3, 此一設計領域共

分為四種不同之加載形式 , 如圖 6 所示。每一種加載初始設計領域皆有全滿領域以及全空領域兩種 ,

藉由交換比值使結果最後可達到 30% 體積比 ( 目標體積比 ); 採用參數為 M

EE − max

= 6.94% ( 相當於

150 個 ) , M

EE − min

= 0.28% ( 相當於 6 個 ) , M ′

= 25% 。

結果見表 6, 此例中也加入收斂準則 , 若本演進步與前面一步之應變能 C 相對差異小於 1/10000

即停止 , 該步數於表 6 中以 S 表示 , 最大演進步為 500。比較此兩種不同設計領域之結果可以看出

有很好的一致性 , 也可看到運用此方法可以解決在進行全空領域進行最佳化設計時 , 難以做出如從

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全滿領域之結果 , 或是需要運用其他輔助方法來達成 [3] 。本方法不需進行其他設定即可同時滿足從

全滿領域以及全空領域進行設計之結果。

圖 6 例題七之四種加載示意圖

表 6 例題七全滿與全空兩種初始設計領域結果比較

( 全滿 ) ( 全空 )

C = 2.35 kN-mm (S=300)

C = 2.43 kN-mm (S=464)

C = 2.14 kN-mm (S=188)

C = 2.24 kN-mm (S=463)

C = 1.29 kN-mm (S=189)

C = 1.32 kN-mm (S=499)

C = 3.25 kN-mm (S=409)

C = 3.34 kN-mm (S=373)

在原來 EEM 方法中 , 每次執行實心與空心元素交換後 , 緊接著必須進行隨機步驟 ( 見方程式 (4)),

此使得最後之拓樸產生游離之元素。為了解決此問題本研究將隨機元素個數隨演化步驟變化由 (4)

式改為如下 :

If k ≤ N′ :

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If k N′ > :

k ⎡ M ′

EE ⎤

MRS

= int ⎢M′

− k×

N′

⎥

⎣

⎦

(9)

M = 0

(10)

以上之隨機元素個數 (9) 及 (10) 式係配合元素交換個數 (5) 及 (6) 式 , 主要是考量當交換元素個數多

時 , 也就是演化前期 , 隨機步驟將有助於找到較佳的拓樸雛形 , 而於演化後期 , 由於交換元素個數

較少 , 解 ( 拓樸 ) 已經趨於收斂 , 若加入隨機步驟將反而不利解的收斂。

但針對擁有對稱性之結構設計時 , 隨機之加入會引起不對稱。如果要維持對稱性 , 必須事先

加入對稱性的考量。如圖 7 為一個 6 × 6 有限元素網格 , 首先給予各個元素編號 , 假設此一結構沿

著 Y 軸對稱 , 因此在隨機步驟時 , 若是 11 號元素被選中 , 則元素 8 號也必須同時被選中 , 若 20 號

元素被選中 , 則 23 號元素也必須同時被選中 , 依此類推。依照以上的對稱考量進行隨機程序 , 可

以使得最終之拓樸結果仍然可以保持對稱性。

圖 7 對稱性之範例示意

例題八之結構形式同例題四 ( 見圖 4) , 由於此結構對 Y 軸對稱 , 因此加入對稱性考量後此題即

可加入隨機步驟。此題採用已進入區域最佳解之拓樸當作初始領域 , 並比較有無隨機化之差別。結

果見表 7, 由表 7 可以明顯看出加入隨機化後可以對進入區域最佳解之情況達到改善。

表 7 例題八隨機化對區域最佳解之改善

有加入隨機化

無加入隨機化

初始領域

拓樸結果

例題九見圖 8 為 12000 mm × 6000 mm 之設計領域離散為有限元素網格 , 每個元素為 100 mm ×

100 mm 之 Q4 平面應力元素 , 厚度為 100 mm, 楊氏模數 E 為 210 GPa, 伯松比 ν 為 0.3。力量加載

在右側下方 30 kN, 左側端固定端 , 目標體積比為 30%。此題比較多種不同初始領域包括全滿領域、

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全空領域以及四種在目標體積比之領域 , 此外也比較不加入隨機化及加入隨機化之結果。演進相關

參數設定同例題七 , 收斂準則也同。

結果如表 8, 由表 8 可看出隨機化可以使拓樸改變 , 雖然並非所有結果皆是往好的方向改善 ,

一部份結果反而變較差 , 但整體來看加入隨機化之改善仍然存在。而在全滿領域以及全空領域之初

始設計兩者比較上 , 隨機化可得到更好且更一致的結果。

圖 8 例題九之結構示意圖

表 8 例題九之結果

初始領域 ( 全空、全滿及四種體積比 =30%)

拓樸結果 ( 無隨機化 )(C 單位為 kN-mm)

C = 2.35 (S=300) C = 2.43

C = 2.52

C = 2.54

C = 2.46

C = 2.38

(S=464)

(S=500)

(S=445)

(S=395)

(S=473)

拓樸結果 ( 加入隨機化 )(C 單位為 kN-mm)

C = 2.33

C = 2.45

C = 2.43

C = 2.68

C = 2.85

C = 2.44

(S=402)

(S=456)

(S=360)

(S=500)

(S=473)

C = 2.33

C = 2.38

C = 2.36

C = 2.42

C = 2.59

C = 2.44

(S=374)

(S=338)

(S=476)

(S=410)

(S=229)

(S=500)

五、改良型元素交換法 (MEEM) 於 3D 之應用

使用改良型元素交換法 (MEEM) 進行 3D 結構之設計時 , 需要針對結構之對稱性在初期設定好 ,

三維空間之對稱情況可同時有對 XY 平面、XZ 平面、YZ 平面 , 也可同時對兩平面對稱等。因此在

進行一開始的對稱性設計時 , 必須先大致對最後的拓樸結果有初步的概念才能夠設定對稱性。

例題十如圖 9 為 32 mm × 32 mm × 16 mm 之設計領域離散為有限元素網格 , 每個元素為每邊長

1 mm 之 B8 實體元素 , 楊氏模數 E 為 200 GPa, 伯松比 ν 為 0.3。力量加載在 X 軸遠端處 YZ 平面

之中心 10 kN,X 軸近端處之 YZ 平面為固定端 , 目標體積比為 20%。分別有三種不同初始領域 ,

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包括全滿領域、全空領域以及固定體積比 20% 之領域 , 此外也比較有無隨機化的影響。採用參數為

EE − max

= 7.81% ( 相當於 128 個 ) , M

EE − min

= 0.24% ( 相當於 4 個 ) , M ′

= 25% , 收斂準則同例題

七。

結果見表 9, 由表 9 可以看出在 3D 情況下 , 有無加隨機化以及在不同初始領域下皆可得到很好

的一致性 , 因此為了減少計算時間 , 在 3D 的例子中可以省略隨機化之步驟。

圖 9 例題十之結構示意圖

表 9 例題十之結果

初始領域 ( 全滿、全空、體積比 =20%)

拓樸結果 ( 無隨機 )

C = 2.39 kN-mm

(S=465)

C = 2.40 kN-mm

(S=500)

拓樸結果 ( 加入隨機化 )

C = 2.41 kN-mm

(S=500)

C = 2.39 kN-mm

(S=478)

C = 2.42 kN-mm

(S=462)

C = 2.43 kN-mm

(S=242)

六、結語

[10]

將隨機化加入可以對拓樸結果產生改善 , 尤其是在元素交換法 (EEM) 中更可以顯著的看見。

而在改良型的元素交換法 (MEEM) 中 , 由於加入了對稱性 , 因此可以針對擁有對稱性的結構進行設

計。本研究提出之改良型元素交換法 (MEEM) 可以解決原本元素交換法

[10]

(EEM) 無法達到的部分 ,

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特別是可以允許設計領域有不同體積比。

在 3D 的情況之下 , 由於不加入隨機化的結果一致性已經很好 , 因此為了減少運算時間 , 可以

考慮不加入隨機化 , 但此仍需更多例題來確認。

參考文獻

[1] 張容慈 . 結合拓樸、尺寸及形狀最佳化之橋梁設計 . 臺灣大學土木工程學研究所碩士學位論文 , 2008 年 .

[2] Xie, Y. M. and Steven, G. P. Evolutionary Structural Optimization. Springer, 1997, Berlin.

[3] 呂其翰 . 雙向結構最佳化演進法及多重材料拓樸最佳化之探討 . 臺灣大學土木工程學研究所碩士學位論文 , 2010

年 .

[4] Arora, J. S. Introduction to Optimum Design. McGraw-Hill, 1989, Singapore.

[5] 林豐澤 . 演化式計算上篇 : 演化式演算法的三種理論模式 . 智慧科技與應用統計學報 . 2005, 3(1): 1-28.

[6] 林豐澤 . 演化式計算下篇 : 演化式演算法的三種理論模式 . 智慧科技與應用統計學報 . 2005, 3(1): 29-56.

[7] 蘇穎香 . 應用最佳化設計系統於版、殼結構 . 臺灣大學土木工程學研究所碩士學位論文 , 2006 年 .

[8] Kaveh, A., Hassanil B., Shojaee S. and Tavakkoli, S.M. Structural topology optimization using ant colony methodology.

Engineering Structures, 2008, 30: 2559–2565.

[9] Liu, X., Yi, W.-J., Li, Q.S., and Shen, P.-S. Genetic evolutionary structural optimization. Journal of Constructional Steel

Research. 2008, 64: 305–311.

[10] Rouhi, M., Rais-Rohani, M., and Williams, T.N. Element exchange method for topology optimization. Struct Multidisc

Optim, 2010, 42: 215–231.

-556-

The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

受荷索杆机构系统的运动形态分析

邓华

, 祖义祯 , 伍晓顺 , 蒋本卫

( 浙江大学空间结构研究中心 , 浙江杭州 310027)

摘要 : 模拟索穹顶、Pantadome 系统在建造中的张拉或提升成形过程均面临一个受荷索杆机构

系统的运动分析问题。本文将此类索杆机构系统的运动形态离散为由一系列随驱动索长度变化的平

衡构型组成。对应运动路径上的每个离散点 , 将其平衡形态按一个已知构件原长为条件的找形问题

来处理 , 并利用动力松弛法来求解。该方法不建立刚度矩阵 , 避免了体系几何不稳定性所引起的刚

度矩阵奇异问题。索单元采用了可考虑大垂度效应的悬链线模型 , 并推导了反映其原长和内力关系

的协调方程。根据系统切线刚度矩阵的正定性条件 , 通过跟踪切线刚度矩阵最小特征值的变化来分

析系统运动形态的稳定性。最后模拟了一个索穹顶的张拉成形过程和一个折叠柱面网壳的提升成形

过程 , 计算结果表明了本文提出的运动分析策略能够有效地进行受荷索杆机构的施工成形模拟分

析。

关键词 : 索杆机构运动分析施工模拟悬链线索单元动力松弛法

ANALYSIS OF THE KINEMATIC PATH OF LOAD-BEARING CABLE-BAR

MECHANISMS

H.Deng *, Y.Z.Zu, X.S.Wu and B.W.Jiang

Space Structures Research Center, Zhejiang University, Hangzhou 310027, China

Abstract: The pretensioning analysis of cable domes or the hoisting simulation of Panta domes during erection

is involved in analyzing the kinematic path of a load-bearing cable-bar mechanism. In this paper, the kinematic

path of such cable-bar mechanism system is regarded to consist of a sequence of discrete equilibrium

configurations with the change of active cable lengths. For each discrete point in kinematic path, the solution of

its corresponding equilibrating configuration can be treated as a kind of form-finding problem with prescribed

unelongated member lengths, and the dynamic relaxation method is employed for solution. Since no stiffness

matrix is established in this method, difficulties caused by matrix singularity due to the geometrical instability of

such mechanism assembly can be avoided in numeric computation. Catenary element is adopted to exactly

consider the large-sag effect of slack cable. A new compatible equation of catenary cable element, which reflects

the relationship between unelongated cable length and its tensile force, is put forward. According to the positive

definition condition of tangent stiffness matrix, the minimal eigenvalue of tangent stiffness matrix is employed

to determinate the stability of equilibrium configuration in kinematic path. The pretensioning process of a cable

dome is simulated by using this method. A foldable reticulated cylindrical shell (a Panta-dome system) is also

illustrated to analyze its deploying state during lifting. The results show that the computational strategy put

forward in this paper is valid for the erection simulation analysis of load-bearing cable-bar mechanism.

Keywords: Cable-bar mechanism, kinematic analysis, construction simulation, catenary cable element; dynamic

relaxation method.

基金项目 : 国家自然科学基金资助项目 (50978226)

通讯作者 : 邓华 , 博士 , 教授。从事空间结构和钢结构研究。

E-mail: denghua@zju.edu.cn, Tel: 0571-88208696

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一、前言

现代结构工程中面临着越来越多的 “ 机构 ” 分析问题 , 如可展天线 [1,2] 、开启式屋盖 [3] 、张拉成

形过程的索杆结构 [4] [5]

、Pantadome 系统等。运动路径跟踪是这些 “ 机构 ” 系统分析的主要工作 , 但

由于系统自身刚度的奇异性和形状变化的大位移特征 , 对于常规的结构分析方法而言是困难的问

题。实际上到目前为止 , 机构系统运动路径分析在方法上依然存在不同的求解思路 , 并没有统一。

对于简单的单机构位移模态体系 ( 如可展结构 ), 一般直接利用几何关系来确定其刚体位移运动路

径 [6] 。但对于大多数土木工程中的机构分析问题来说 , 运动路径往往与荷载相关 , 机构位移模态众

多且弹性变形和刚体位移相耦合 , 因此仅利用几何关系还不足以满足求解的条件。目前的研究中 ,

一种做法是将机构运动形态分析作为一类包含刚体位移的特殊结构分析问题来看待 , 通过在常规的

有限单元法中引入一些能够处理刚度矩阵奇异性的数学策略来实现刚体位移的求解 , 如 “ 广义逆 ”

方法 [7] 。也有文献基于多体动力学理论 [8] , 借助动力学方程对机构运动进行数值仿真计算。近期 ,

文献 [9] 针对索杆张力结构的施工成形分析提出了一种新的运动形态分析思路。该方法指出了驱动机

构运动的原因是主动索长度的减小 , 因此将索杆张力结构的成形过程描述为体系在张拉索长度变化

下对应于一系列施工步的平衡形态求解问题 , 那么对于每一个特定状态 ( 施工步 ), 可以归结为一

个已知构件原长的松弛索杆体系在特定荷载 ( 如自重 ) 作用下的找形问题。文献 [9] 中提出了一种基

于 “ 力密度 ” 法的求解策略 , 索段按抛物线索单元进行计算。但是当初始迭代形状与最终平衡形态

偏离较大时 , 基于力密度法的求解策略需要限制节点位移增量的步长来保证计算的收敛性 , 而步长

的减小进一步导致迭代次数的增加和计算效率的降低。同时由于采用抛物线索单元 , 对于大垂度索

单元将产生较大误差。本文从求解方法的角度 , 提出了一种基于动力松弛法的受荷松弛索杆体系的

找形策略 , 同时索单元采用了精确的悬链线单元以便考虑在松弛状态下索段易出现大垂度的情况。

文中还通过一个索穹顶结构张拉成形分析算例和一个折叠展开式网壳的提升成形分析算例 , 从数值

上考察该计算方法的正确性和有效性。

二、运动分析的描述

无论是索穹顶的张拉成形分析还是 Pantadome 的提升成形分析 , 实际上就是要跟踪体系在施工过

程中的形状和构件内力。而施工过程体系形态变化来自于运动的驱动 , 即张拉 ( 提升 ) 某些拉索使

得其长度缩短 , 因此运动驱动变量在数学上可以描述为张拉 ( 提升 ) 索长度的变化。但是从工程需

求的角度来看 , 索杆系统的成形过程并不需要严格描述一个完整连续的形态变化 ( 展开 ) 过程 , 而

只是要求解一系列特定施工步骤或时刻下 , 体系的形状以及相应的构件内力。也就是说 , 可将施工

成形过程离散为一系列随驱动变量变化 ( 特定张拉索长度 ) 上的平衡形态求解问题问题。当然 , 足

够密的驱动变量离散点取值 , 也是施工成形过程的近似连续描述。

应该注意的是 , 对于某个特定施工步骤或时刻 , 构件的原长 ( 即无伸长长度 ) 是已知的 , 即

s =s -Δ s+s. (1)

0 1

式中 : s 1

为初始平衡态 ( 成形后的状态 ) 的几何长度 ; Δ s 为初始平衡态构件的弹性伸长量 ( 或缩短

量 ); s t

为张拉索的牵引长度 , 对于非张拉构件为零。张拉索的 s

在张拉过程中是变化的 , 其值需不

断扣除千斤顶已拔出长度。

索杆张力结构的施工过程可以认为是将一些已知放样长度的构件进行组装的过程。对于某个特

定的时刻 , 已组装好的构件体系在其自重作用下所达到的平衡状态即是需要求解的形态。从这方面

理解 , 索杆张力结构施工过程中某个特定时刻的体系形态求解 , 实际为一些已知原长的构件根据特

定的连接方式在其自重作用下所达到的平衡形态求解问题 ( 即找形问题 )。因此对于某个特定的施

工步 , 这个基本找形问题的数学描述如下 :

(1) 已知条件

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该时刻各索段和杆的原长 s

; 构件的截面面积 A 和弹性模量 E ; 构件之间的连接关系 ( 即拓扑

关系 ) 以及边界约束情况 ; 索上横向荷载 q ( 通常仅为索的自重 ) 和节点荷载 p ( 包括实际节点重

量以及可能的外挂荷载 )。

(2) 求解内容

在当前结构自重作用下的平衡状态 , 具体包括结构形状 ( 以节点坐标描述 ) 和构件内力。

应该注意的是 , 这类找形问题与常规张力结构的找形问题有所不同 , 主要反应在在施工过程中

体系的几何不稳定 ( 可变 ) 性。应该澄清一个概念 , 这种几何不稳定性并不能理解为体系不存在平

衡状态。在特定的荷载作用下 , 任何受力体系都会通过形状和内力的调整来达到与当前荷载相适应

的某个平衡状态。从能量角度来看 , 这实际上是体系达到系统势能最小的客观要求。

三、单元分析

3.1. 基本假定

(1) 索为理想柔索 , 不受压且无弯曲刚度 ;

(2) 杆为两节点单元 , 杆自重等效为两端节点荷载 ;

(3) 索上只有沿索长均布的竖向荷载 ( 包括自重 );

(4) 构件材料符合虎克定律 , 且满足小应变假定。

3.2. 悬链线索单元分析

如图 1 所示空间索段 , 定义其局部坐标系 o% − xz % % , 其中 x % 轴平行于索段在整体坐标系 o- xy 平面上

的投影线 ,z % 轴和 z 轴与索上均布荷载同向。根据悬索结构的基本理论 [10] , 索的平衡曲线为悬链线 ,

曲线方程为 :

qx%

−1

qc qL

zx %( ) = [coshα

−cosh( −α)]

, α = sinh ( ) + . (2)

sinh

2 H

式中 : L 是悬索两端节点的水平距离 ;c 是悬索两端节点的竖向高差 ;q 为沿索长的均布荷载 ;

H 是悬索张力的水平分量 , 为常数。

将式 (2) 对 x % 进行求导 , 可得悬链线斜率方程为

z% '( x) = sinh( α − % ) , (3)

当 x% =0 时 , 有 z% '(0) = sinhα

。索端张力 T,

T 与 z% '( x% ) 、 H 的关系式为

T H z

= 1 + (% '(0)) ,

T H z L

在局部坐标系 o% − xz % % 下 , 索端张力在各坐标轴上的分量为

i i j j i

x% z% x%

z% 0 z%

= 1 + (% '( )) . (4)

F =− H, F =−H ⋅ z% '(0); F = H, F =−q⋅s −F

(5)

在整体坐标系下 , 索端张力对索两端 i 、 j 节点产生的节点力分别为

i i i

F = H( x − x )/ L, F = H( y − y )/ L, F = Hz% '(0). (6)

x j i y j i z

F =−H( x − x )/ L, F =−H( y − y )/ L, F = q⋅ s + F %

(7)

j j j i

x j i y j i z 0 z

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上两式中 : ( x , y , z )、( x , y , z ) 分别是单元两端节点在整体坐标系下的坐标。

i i i j j j

根据几何条件 , 变形后索长 s 可按下式计算

L H qc qL

s = + z x dx = − = −

sinh

2 H

' 2 −1

∫ 1 ( % ( )) (sinhα sinh β), β sinh ( ) .

(8)

引入小应变的假定 , 索由于张力引起的变形伸长量 Δ s 可由下式求得

H L ds 2 % H L

( ) (1 ' (%)) % H

Δ s = dx = + z x dx = (2Lq+ Hsinh(2 α) −Hsinh(2 β )).

EA 0

dx%

EA 0

4qEA

∫ ∫ % (9)

根据 s0

+Δ s = s的变形协调条件 , 可以得到悬链线索单元的变形协调方程式为

gHqLcs ( , , , ,

) = (sinhα −sinh β) − (2Lq+ Hsinh(2 α) −Hsinh(2 β)) − s0

= 0. (10)

4qEA

式 (10) 反映了索单元原长 ( s 0

)、索张力水平分量 ( H )、形状参数 ( L , c )、索上均布荷载 q 以及索

的材料特性 ( E , A ) 之间的关系。在实际结构分析中 , 如果悬索的两端节点坐标、材料特性以及索上

均布荷载确定 , 那么利用式 (10) 可以解决以下两方面的问题。

(1) 已知索张力水平分量 H , 求解出索原长 s

。在索杆张力结构预应力态 ( 施工成形后的终态 )

分析时 , 由于索中预张力 T 通常很高 , 索的平衡曲线接近于直线 , 因此可以近似认为索的预张力沿

索长均匀分布 , 即为常数。进而可将 T 的水平分量近似认为就是 H 值。于是利用方程 (10), 便可计

算出索单元在预应力态的原长 s

, 以作为结构施工成形分析的基本前提条件。

(2) 已知索原长 s

, 求解索张力水平分量 H 。如前所述 , 索杆张力结构的施工成形分析可认为

是一个已知构件原长的受荷索杆体系的找形问题。在数值分析时 , 当假定体系处于某种形状时 , 便

可以通过方程 (10) 求解某已知原长的索段在当前形状下的张力水平分量 H , 进而利用式 (6) 和 (7) 求解

出该索单元对两端节点提供的节点力 , 于是可以用于考察该节点的平衡条件。

应该注意的是 , 公式 (10) 是一个形式较为复杂的超越方程 , 在已知 s

求解 H 时 , 一般需要进行

数值迭代求解。根据作者的经验 , 采用常规的二分法或黄金分割法便可以有效求解。另外可以分析 ,

给定 s

时 , 通过方程式 (10) 求解出的 H 只有一个正根。

T i

z %

F i

o %

z %

F x %

T j

图 1 悬链线索单元

图 2 空间杆单元

3.3. 杆单元分析

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对于空间杆单元 ( 如图 2), 轴力 T 和杆原长 s

及当前变形长度 s 的关系为

在整体坐标系下 , 杆轴力对两端 i 、 j 节点产生的节点力分别为

T = ( s− s0

). (11)

i i i

F = T( x − x )/ s, F = T( y − y )/ s, F = T( z − z )/ s.

(12)

x j i y j i z j i

j j j

F = T( x − x )/ s, F = T( y − y )/ s, F = T( z − z )/ s.

(13)

x i j y i j z i j

四、形态求解和稳定性跟踪

4.1. 动力松弛法

[11]

动力松弛法是根据达朗贝尔原理 , 将静力问题转化为一个拟动力问题来寻求系统平衡形态的

一种数值方法。其基本思想是 , 将受力系统离散为在空间节点位置上具有一定虚拟质量的质点 , 在

不平衡力的作用下 , 这些离散的质点必将沿不平衡力方向运动 , 从宏观上使系统的不平衡力趋向减

小。进一步通过引入虚拟阻尼 , 使得系统运动动能逐步减小并趋近于零 , 并最终达到静力平衡状态。

在运动过程中任意时刻 t , 离散结构上任意单元节点 k 在坐标 l = x, y,

z 方向上的动力平衡方程

为

Rkl = Mklv& kl

+ Cklvkl

, (14)

式中 : R kl

、 M kl

、 C kl

、 v& kl

和 v

分别为 t 时刻节点 k 在坐标 l 方向上的不平衡力、虚设质量、虚设阻

尼系数、加速度和速度。

k 节点的不平衡力 R 可按下式取值

上式中 :

Rkl = pkl + ∑ Fl

, l = x, y,

z. (15)

p kl

是节点 k 上外荷载在整体坐标系三个坐标轴方向的分量 ; n 为与节点 k 相连的单元数 ;

而 F 可分别根据索、杆单元类型由式 (6)、(7) 或 (12)、(13) 计算。

给定迭代时间步 Δ t , 可得到中心有限差分形式

R =

( v

t+Δt/2 − v

t−Δ t/2 ) +

( v

t+Δt/2 + v

t−Δt/2

), (16)

kl Δt

kl kl 2 kl kl

整理上式得

那么 t+Δ t 时刻节点 k 的坐标为

⎛M / Δt−C

/2⎞

t t/2 t t/2

kl kl

+Δ

= v

−Δ ⎜ ⎟+

kl kl ⎜M / Δ t+ C /2 ⎟ kl M / Δ t+

C /2

⎝ kl kl ⎠ kl kl

t +Δ t t t +Δ t /2

kl kl kl

, (17)

x = x + v Δ t. (18)

上世纪七十年代中期 ,P.A.Cundall [12] 提出了 “ 运动阻尼 ” 的概念 , 即将体系看成为无阻尼的自

由振动 , 令 C

= 0 。而系统振动的收敛通过判别系统的总动能来实现。如果系统总动能达到动力峰

值 , 令节点的速度分量为零 , 体系在新的位置重新开始振动 , 直到下一个动能峰值的出现。“ 运动

阻尼 ” 的思想已广泛应用于张力结构的动力松弛法找形分析中 , 如果采用运动阻尼 , 那么式 (17)

可以写为

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t t/2 t t/2

Rkl

+Δ −Δ kl

= + . (19)

假定结构体系在 t = 0

时刻从静止状态开始振动 , 即 ν = 0 , 按直线差分可以得到

=− ν , 因此

−Δt/2 Δt/2

t/2 t 0 t

ν −Δ kl

( Rkl

)

= Δ

=− . (20)

理论上讲 , 动力松弛法可以采用任意虚设节点质量。根据 Barnes [11] 的建议 , 节点各个方向的虚

设质量可以相等 , 即统一取 M = M (= l x,y,z)

。但为保证算法的收敛性和稳定性 , 迭代时间增量 Δt

和 M

需满足关系 Δt ≤ 2 M / K , 其中 K

为刚度矩阵在第 k 节点自由度方向对应的主元元素 , 为

便于计算 , 一般可取各节点的最大可能刚度值 K

k max

。因此

M = K Δ t . (21)

kmax

式中 :λ 为刚度增大系数。通常计算时 , 可取 λ = 1, 当计算不收敛时 , 可以适当增大 λ ; 当计算收

敛但非常慢时 , 可以适当减小 λ , 以加快收敛速度。

对于节点 k , 节点最大可能刚度 K

k max

度组成。其中所有杆单元提供的最大可能刚度

而所有索单元提供的最大可能刚度为

n c

分别由与其相连的所有杆单元和索单元提供的最大可能刚

n b

∑ [( )/ ] ,

(22)

K = EA + T s

k max 0

K = ∑ ⎡ ⎣

−α cos γ −α sin 2γ −α sin γ⎤

⎦

. (23)

2 2

k max 1 2 4

i=1

上两式中 : n b

、 n c

分别为与节点 k 相连的杆单元数和索单元数 ; 参考杆单元最大可能刚度的物理意

义 , 本文中 K 取所有相连索单元的弦向刚度之和。索单元弦向刚度的推导可参见文献 [14]。

k max

由于引入运动阻尼的思想 , 在迭代过程中 , t 时刻结构所有虚设质量点的总动能

3 t 2

l kv

= kl

= ∑∑ 。当

1 ( )/2

< e 时 , 结构在 t− 3 Δ t/2和 t+ Δ t/2之间的某个 t 时刻达到局部

t+Δt/2 t−Δt/2

动能峰值 , 假设 t 时刻距 t 时刻为 Δ

进而可得 t 时刻节点 k 的坐标为

, 则可用下式按抛物线插值近似确定

* t−Δ t/2 t+Δt/2 t−Δ t/2 t+Δt/2 t−3 Δt/2

Δ t =Δ⋅ t ( e −e )/(2 e −e − e ) =Δ⋅ t η,

(24)

x = x −Δ t + + . (25)

2 t

t* t +Δ t t t/2

t kl

kl kl

(1 ην )

+Δ kl

2 M

确定系统动能峰值后 , 需将各节点坐标重置为 t 时刻坐标 , 速度置零 , 重新开始迭代。

采用运动阻尼的动力松弛法进行松弛索杆体系找形计算的步骤可总结如下 :

(1) 给定索杆体系的拓扑关系、构件原长 s

、截面面积 A 、弹性模量 E 、索上均布荷载 q 、

节点荷载

等初始数据 , 假定体系的初始形状 , 并将初始迭代速度置零 ;

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(2) 根据构件原长 s

按式 (10) 和 (11) 计算出索张力水平分量 H 和杆单元内力 T , 然后根据式

(6)、(7) 或 (12)、(13) 求出 F , 进而根据式 (15) 求出节点不平衡力 R ;

(3) 根据式 (22) 和 (23) 求出各节点最大可能刚度 K

k max

, 再根据式 (21) 计算节点虚设质量 M

进而根据式 (20) 确定 v −Δ t ;

(4) 由式 (19) 和 (18), 可得 t+Δ t/2时刻节点速度

(5) 由节点速度

t t/2

v +Δ

定 t 时刻 e 最大时的节点坐标 x

始迭代计算。否则 ,

计算此时体系总动能

t t/2

e +Δ

t t/2

v +Δ

, 并将速度置零 , 然后以 x

t t

和 t+ Δ t时刻节点坐标 x +Δ ;

; 如果出现 e

t+Δt/2 t−Δt/2

< e , 按式 (24) 和 (25) 确

为初始节点坐标 , 返回第 2 步重新开

(6) 经计算求得 t+ Δ t 时刻节点坐标后 , 结合初始数据 , 再计算各构件内力对两端节点产生的

节点力及节点不平衡力 ; 重复 2~6 步 , 直到节点不平衡力满足给定精度 ε

。

4.2. 稳定性判别

动力松弛法仅实现了连杆机构的平衡形态求解。值得指出的是 , 索杆机构本质上是几何不稳定

( 可变 ) 系统 , 必须依靠内力 ( 由荷载或预应力产生 ) 提供的几何刚度来维持平衡形态的稳定性 [13] 。

在后面的算例分析中可发现 , 当提升过程中某个平衡构型的内力并不能保证当前形态的稳定性时 ,

连杆机构易于发生形状突变而偏离预定的运动路径 , 造成不能提升到设计构型。因此 , 跟踪提升过

程中各施工步平衡形态的稳定性是连杆机构运动分析的重要内容。

平衡系统的稳定性可采用 Lagrange-Dirichlet 能量准则来进行定量描述 , 即对于稳定的平衡状态 ,

其相应的系统势能 Π 处于最小值。一般情况下 , 可利用势能二阶变分 δ Π > 0 来作为稳定性的判别

条件。对于一个平衡系统 , δ 2 Π 可以表示为节点位移变分 δ u 的二次型 [12] , 即

2 T

δ Π δ

Tδ

= ( u K u )/2, (26)

式中 : K T

为系统当前平衡构型的切线刚度矩阵。根据有限元理论 , K T

=K E

+K G

, 其中 : K E

为当

前平衡构型对应的线弹性刚度矩阵 , 与机构自身的几何和材料刚度相关 ; K G

为当前平衡构型对应

的几何刚度矩阵 , 来自于平衡内力的贡献。文献 [8] 给出了 K

的具体表达式 , 并对杆系机构的稳定

性条件进行了详细讨论。

既然 δ Π 可以表示为节点位移变分 δ u 的二次型 , 因此可通过考察各施工步平衡构型 K 的正定

性 , 来实现连杆机构提升过程的形态稳定性判别。由矩阵理论易知 , 判别 K

正定性的方法很多 ,

如 K

的所有特征值是否均大于零。考虑到被简化后的连杆机构自由度较少 , 因此本文利用 K

的最

小特征值 λ

min

来作为稳定性判别指标 , 具体如下 :

⎧λmin

> 0,

⎪

⎨λmin

= 0,

⎪

⎩λmin

< 0,

稳定状态 ;

临界状态 ;

不稳定状态 .

(27)

五、算例

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5.1. Geiger 型索穹顶的施工成形模拟分析

图 3 所示为一轴对称的 Geiger 型索穹顶结构 , 跨度为 100m , 施工成形后预应力态的几何参数

3 3

如图 3(b) 所示。施工过程仅考虑构件自重 , 索和杆的材料密度为 7.85× 10 kg / m , 索的弹性模量为

11 2

1.7 × 10 N/m , 杆的弹性模量为 2.1× 10 N/m 。构件的截面面积和成形后形态的设计内力值 ( 包括

自应力效应和自重作用效应 ) 列于表 2。

外压环梁

上径向索

下环索

压杆

斜索

(1)

(4)

16.5

+7.500

(3) (9) +6.200

(2)

(7) (5) (8) (6) 6

(11) 4

(10)

+0.200

+4.500

16.5 17 17

16.5

+3.600

±0.000

-5.400

16.5

(a) 立体图

图 3 Geiger 型索穹顶

100

(b) 剖面图 ( 单位 :m)

根据式 (10) 和 (11) 分别计算预应力态下各索杆单元的原长 ( 非伸长长度 , 见表 2), 以作为体系

施工成形的前提条件。考察该索穹顶施工过程的六个特定步骤 , 具体说明如下 :

第一步 : 安装脊索 1、2、3, 桅杆 7、8、9, 环索 10、11。然后将下斜索 4 安装 , 此时其原长

为 23.3152m( 原长 17.3152m, 预留牵引长度 6m);

第二步 : 张拉下斜索 4 使其到达初始态原长 17.3152m;

第三步 : 安装下斜索 5, 此时其原长为 20.8185m( 初始态原长 16.8185m, 预留牵引长度 4m);

第四步 : 张拉下斜索 5, 使其到达初始态原长 16.8185m;

第五步 : 安装下斜索 6, 此时其原长为 19.0705m( 初始态原长为 17.0705m, 牵引长度 2m);

第六步 : 张拉下斜索 6, 使其到达初始态原长 17.0705m, 此时索穹顶施工完成 , 到达设计位置。

对于每一个施工步 , 均是一个松弛索杆体系的找形分析问题。应用本文方法 , 计算出的各施工

步平衡形态的节点坐标值见表 2, 单元内力见表 3( 索单元用 T

表示 )。同时各施工步的平衡形态见

图 4。

(a) 第一步

(b) 第二步

(d) 第四步 (e) 第五步 (f) 第六步

图 4 索穹顶施工过程示意图

从表 3、表 4 可以看出 , 采用本文方法计算的结构成形后 ( 第六步 ) 的节点坐标和单元内力与

设计预应力态相比 , 其误差很小。

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表 1. Geiger 索穹顶各组单元计算数据

单元组编号 1 2 3 4 5 6 7 8 9 10 11

初始预应力 kN 1924.2 1001.0 535.24 1527.1 909.89 457.39 -453.14 -172.73 -517.91 2803.8 1721.6

截面面积 mm 2 3205 3205 3205 3391 3205 3205 25819 25819 25819 7996 4250

原长 m 16.8285 16.6729 17.0329 17.3152 16.8185 17.0705 9.0008 6.0002 3.0003 17.3051 8.7789

表 2 各施工步节点 x、z 向坐标 (m)

节点

步骤

2-x 2-z 3-x 3-z 4-x 4-z 5-x 5-z 6-x 6-z 7-x 7-z

一 32.1977 -12.8544 33.6393 -3.9698 16.1588 -11.5214 17.0383 -5.5860 0.0000 -8.7874 0.0000 -5.7871

二 33.4463 -5.1141 33.6199 3.8849 16.1582 -3.8393 17.0373 2.0961 0.0000 -1.1271 0.0000 1.8732

三 33.4473 -5.1193 33.6183 3.8797 16.7033 -3.9197 17.0377 2.0712 0.0000 -1.1437 0.0000 1.8566

四 33.4896 -5.3461 33.5232 3.6540 16.9946 0.4161 17.0567 6.4158 0.0000 3.3575 0.0000 6.3578

五 33.4900 -5.3482 33.5229 3.6519 16.9949 0.4093 17.0560 6.4091 0.0000 3.3135 0.0000 6.3138

六 33.5000 -5.4001 33.5000 3.6000 17.000 0.1999 17.0001 6.1999 0.0000 4.4998 0.0000 7.4998

表 3 各施工步单元内力 T(kN)

单元

步骤

1 2 3 4 5 6 7 8 9 10 11

一 231.46 222.48 220.00 4.8503 / / 20.115 8.9251 2.9789 11.998 2.9053

二 203.12 201.05 198.41 351.08 / / -81.607 8.9260 2.9804 643.76 2.9043

三 212.004 208.01 206.06 371.49 3.6680 / -87.739 9.8352 2.9790 681.18 4.8838

四 1569.0 771.63 762.81 1300.4 787.41 / -378.56 -140.41 2.9788 2386.1 1489.2

五 1563.7 758.25 747.22 1309.2 795.57 3.5787 -381.42 -142.19 21.926 2402.2 1504.6

六 1924.5 1001.1 535.39 1528.2 910.57 457.74 -453.28 -172.79 -518.13 2804.5 1722.1

5.2. 七连杆机构的非对称提升

图 5 为一正放四角锥三心圆柱面网壳的设计构型剖面图及其几何参数 , 其中上弦杆将网壳剖面

的三段圆弧等分为 39 份。根据 Panta-dome 施工要求 , 沿柱面母线方向拆除多道上弦或下弦杆 , 可

将网壳划分为如图 6(a) 和 (b) 所示的可动机构系统 , 并分别简化为五连杆或七连杆模型。简化模型中 ,

设各杆轴向刚度 ( EA ) 均为 4.12× 10 N , 机构法施工时提升吊索 ( 如图 7) 的轴向刚度均取 1.7 × 10 N 。

网壳总重 W = 2500kN , 假定沿着圆弧方向均匀分布 , 故各杆重量可按比例确定 , 并等分到两端节

点上。不考虑吊索重量。

3.50

40.00

α α

110.00

α 1

= 24.487

= 100.00

= 85

= 115

(a) 五连杆机构

(b) 七连杆机构

(b) Seven-bar linkage

图 5 三心柱面网壳 ( 长度单位 :m)

图 6 简化的连杆机构模型

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考察图 6b 所示的七连杆机构 , 设置 4 根非对称布置的吊索并进行提升分析 ( 假定吊点 12 相对

于原对称吊点 9 发生了 2m 的水平偏差 )。体系的单元和节点编号见图 7, 初态和设计态的节点坐标

见表 4。各杆重量取 G1 = 339⋅ G、 G 2

= 5/39⋅ G 、 G 3

= 6.5 / 39 ⋅ G、 G 4

= 10 / 39 ⋅ G。同样对提升过

程进行求解 , 表 5、表 6 和图 7 分别给出了第 0、8、16、24、32、40 步平衡构型的节点坐标、单元

内力以及形状。参照图 7, 可以发现该七连杆机构的提升过程存在以下主要特点。首先 , 由于吊索

非对称布置 , 预先设定的初态构型 ( 图 8a) 实际上并不是一个平衡构型 , 而真实的平衡构型如图 9a

所示。其次 , 在提升过程的前期体系基本呈一个稳定的运动形态 , 这从图 7 所示的 λ

min 变化曲线也

可以看出。但从第 37 步开始 , λ min 值急速下降 , 在第 39 和 40 步间发生突变。因此进一步对第 39

和 40 步间的吊索长度进行细化 ( 取 100 步 ), 发现两个施工步间部分区段的平衡构型 λ

min 值已经出

现了负值 , 并随后又如图 7 所示急剧增加。相应地从图 9f 也可以看出 , 此时节点 2 和 7 突然发生 “ 塌

落 ”, 从而偏离了预期的运动路径。这也反映了对机构法施工的网壳进行提升过程的形态稳定性分

析的重要性。

λ min ( ×10 3 )

1500

1200

900

600

300

-300

10 20 30 40

提升步

图 7 最小特征值变化曲线 ( 七连杆机构 )

9 10 11 12

8 9 10 ○11

3 4 5

z/m

x/m (a) 初态

G 3

4 5 6

2 G2

z/m

x/m (b) 设计态

G 4

G 1

图 8 七连杆机构的杆件和节点编号

(a) Step 0 (b) Step 8 (c) Step 16 (d) Step 24 (e) Step 32 (f) Step 40

图 9 七连杆机构提升过程

表 4 七连杆机构初态和设计态各节点坐标 ( 单位 :m)

节点号 1 2 3 4 5 6 7 8 9 10 11 12

初始态

x 0 -7.950 11.243 36.272 73.728 98.757 117.950 110 10 30 80 102

z 0 10.254 6.412 4.333 4.333 6.412 10.254 0 40 40 40 40

设计态

x 0 2.468 11.583 36.272 73.728 98.417 107.532 110 10 30 80 102

z 0 12.738 30.060 34.665 34.665 30.060 12.738 0 40 40 40 40

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表 5 七连杆机构各提升步的节点坐标 ( 单位 :m)

提升步 x2

0 1.510 12.887 20.480 8.063 45.584 7.308 82.884 3.902 107.838 6.744 119.039 -9.308

8 -4.224 12.268 15.338 11.595 40.450 11.220 77.865 9.457 102.938 10.904 121.752 5.500

16 -6.985 10.935 11.953 15.885 37.068 15.997 74.522 15.585 99.637 15.640 118.358 9.935

24 -5.703 11.654 11.712 20.591 36.798 21.802 74.252 21.444 99.343 20.340 116.628 11.155

32 -3.186 12.578 11.673 25.319 36.657 27.878 74.111 27.487 99.110 25.074 113.983 12.349

40 6.536 11.208 11.729 30.081 36.394 34.811 73.849 34.525 98.555 30.008 103.051 10.957

表 6 七连杆机构各提升步的单元轴力 ( 单位 :kN)

提升步 /step 1 2 3 4 5 6 7 8 9 10 ○11

0 -250.697 -30.096 96.857 367.335 404.742 1110.846 -912.446 404.061 625.164 456.095 1350.713

8 -274.438 89.400 159.444 358.937 324.248 353.232 -374.847 379.422 583.388 499.492 490.043

16 -260.722 145.063 173.516 332.031 213.148 179.596 -266.664 410.845 561.118 542.904 428.495

24 -228.171 112.728 137.357 340.315 173.738 131.123 -226.683 420.051 580.420 563.464 431.370

32 -217.304 70.299 100.496 401.072 145.852 85.296 -211.150 411.761 625.496 600.876 422.742

40 264.973 503.324 279.023 1006.170 350.012 419.608 179.965 818.488 942.827 886.049 760.565

六、讨论

无论是索穹顶的张拉成形还是折叠网壳的提升成形 , 如果将其运动形态分析简单地按常规的大

位移小应变问题来理解 , 那么对于传统的结构理论来说并不是一个易于解决的问题。然而以吊索长

度作为驱动体系运动的控制参数 , 将提升过程看作为由一系列随控制参数变化的平衡构型组成 , 那

么该运动分析问题便可转化为对应于各控制参数离散点的平衡形态找形问题。

(1) 由于索杆张力结构施工过程体系几何上的非稳定性 , 体系自身刚度是不完善的 ( 刚度矩阵

奇异 ), 因此采用常规的刚度分析方法在求解此类找形问题时将会在数值计算上出现困难。相比之

下 , 动力松弛法将这个静力分析问题转化为一个拟动力问题进行求解 , 并不需要直接考察体系的刚

度矩阵 , 因此避免了常规刚度方法的局限性。

(2) 理论上讲 , 悬链线是承受沿索长均布荷载索单元平衡形状的精确解 , 并不受索单元跨中垂

度大小的影响 , 因此能够充分模拟索杆张力结构在施工过程中索段通常易于出现大垂度的情况。本

文推导的悬链线索单元变形协调方程的表达式虽然还是较为复杂 , 但是与 Jayaraman [15] 提出的变形

协调方程式相比 , 已经非常简洁 , 也使 H 和 s

的互相换算变得简单直接。同时该方程采用一些常规

的数值迭代方法便可求解。另外 , 由于通常的索杆张力结构的构件数量并不多 , 单元和节点规模不

大 , 因此在计算过程中并不会出现计算时间上的困难。

(3) 从算例可以看出 , 本文方法能够很好地模拟索穹顶结构的张拉成形和折叠柱面网壳的提升

成形过程。

(4) 值得强调的是 , 索杆机构本质上依然是一种易于出现不稳定平衡构型的系统 , 同时提升过

程中体系的形状和内力紧密关联 , 在保证形态稳定性方面很难给出明确的规律 , 因此对提升过程

进行稳定性跟踪和判别是非常重要的。对于自由度较少的索杆机构 , 文中直接将当前平衡构型切

线刚度的最小特征值作为运动形态稳定性的判别指标。形式虽简单 , 但物理意义清楚而且非常有

效。

参考文献

[1] Onoda, J. Two-dimensionally deployable truss structures for space application. Journal of Spacecraft and Rockets, 1988,

25(2): 109-116.

[2] Guest, S. D. and Pellegrino, S. A new concept for solid surface deployable antennas. Acta Astronautica, 1996, 38(2):

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103-113.

[3] Gantes, G. C. Deployable structures: analysis and design, WIT Press, Southampton, 2001.

[4] Geiger, D. H., Stenfaniuk, A. and Chen, D. The design and construction of two cable domes for the Korean Olympics. In:

Proceedings of the International Association of Space Structures-American Society of Civil Engineering. Int. Symp, Osaka:

1986, 2: 265-272.

[5] Kawaguchi, M. Engineering aspects of space frames, Current and Emerging Technologies of Shell and Spatial Structures,

In: Proceedings of the International Association of Space Structures Colloquium, Madrid, 1997:51-62.

[6] Kumar, P. and Pellegrino, S. Computation of kinematic paths and bifurcation points. International Journal of Solids and

Structures, 1999, 37(2000): 7003-7027.

[7] 张其林 , 罗晓群 , 杨晖柱 . 索杆体系的机构运动及其与弹性变形的混合问题 . 计算力学学报 , 2004, 21(4): 470-474.

[8] 谢铁华 , 关富玲 , 苏斌 , 石卫华 . 空间索杆式展开结构的动力学研究与分析 . 空间结构 , 2004, 10(3): 48-54.

[9] Deng, H., Jiang, Q. F. and Kwan, A. Shape finding of incomplete cable-strut assemblies containing slack and prestressed

elements. Computer and Structures, 2005, 83(21-22): 1767-1779.

[10] 沈世钊 , 徐崇宝 , 赵臣 . 悬索结构设计 , 北京 : 中国建筑工业出版社 , 1997.

[11] Barnes, M. R. Form Finding and Analysis of Tension Structures by Dynamic Relaxation. International Journal of Space

Structures, 1999, 14(2): 89-104.

[12] 钱若军 . 张力结构的分析、设计、施工 , 南京 : 东南大学出版社 , 2003.

[13] Calladine, C. R. and Pellegrino, S. First-order infnitesimal mechanisms. International Journal of Solids and Structures,

1991, 27: 505-515.

[14] 伍晓顺 , 罗挺 , 邓华 . 悬链线索单元的协调方程和弦向刚度矩阵 . 空间结构 , 2006, 12(4): 32-35.

[15] Jayaraman, H. B. A curved element for the analysis of cable structures. Computers and Structures, 1981, 14(3-4):

325–333.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

應用遞迴式隨機子空間識別於橋梁即時監測系統之研究

翁健煌

1 、羅俊雄

2 、趙書賢

1 、盧恭君

1 3

、諶佳慧

台灣大學土木工程學系博士後研究員、 2 台灣大學土木工程學系教授、 3 台灣大學土木工程學系專任研究助理

摘要 : 本研究之目的在於發展隨機子空間識別的遞迴演算法 (Recursive Stochastic Subspace

Identification) 以降低電腦計算量 , 使其運算速度滿足即時監測條件下的要求 , 然後將此識別理論透

過軟硬體整合技術實踐於硬體監測平台 , 最後經由縮尺橋梁的沖刷實驗驗證該系統的準確性與穩定

性。

隨機子空間識別為近年發展成熟的系統識別技術之一 , 適用於微震環境中的多輸出線性動力系

統 , 可經由時間域的資料處理識別出結構的自然頻率、阻尼比與模態振形。該識別方法的優點在於

利用 LQ 分解與 SVD 分解降低電腦計算量同時提高精確度 , 然而 , 其計算速度目前尚無法完全達成

多自由度系統的即時監測要求 , 有鑑於此 , 本研究推導的理論可使隨機子空間識別在即時監測使用

時達到遞迴演算 LQ 分解的目的 , 省去其重新計算的時間 , 因此有助於改善運算即時性的問題。本

研究的長期目標為建立即時監測系統以偵測橋梁受沖刷損壞的行為變化 , 並以遞迴式隨機子空間識

別作為主要的分析工具。為了解真實橋梁的行為反應 , 本研究進行了一系列的橋梁現地實驗與實驗

場實驗 , 其中 , 現地實驗選擇了一座七孔簡支梁型式的鋼筋混凝土橋做為長期觀測的目標 , 藉由平

時與颱風洪水期間的振動資料比對以釐清橋梁受洪水沖刷時的特性變化 ; 而實驗場實驗則設計了一

座四孔簡支梁形式的縮尺鋼橋 , 分別進行靜態與動態實驗 , 靜態實驗之目的在於了解橋墩埋入深度

與模態振形之關係 , 而動態實驗則模擬全橋結構受激烈洪水沖刷之情況 , 目的在於了解橋梁可能產

生的損壞形式與其損壞過程中的結構特性變化。

由上述實驗之分析結果驗證 , 橋墩的埋入深度確實與整座橋梁的自然頻率和模態振形有絕對的

關係 , 沖刷愈嚴重的橋墩可提供的水平側向勁度愈小 , 因而使模態振形在該橋墩位置的振幅放大 ,

此現象有助於判斷橋梁沖刷損壞的發生位置。而在實驗場的沖刷實驗中 , 本研究所整合的即時監測

系統確實可達成每秒更新識別結果的分析要求 , 而橋墩沖刷深度與橋梁自然頻率的相關性也可從監

測系統中做即時觀察。

關鍵詞 : 系統識別結構健康診斷橋梁監測系統

APPLICATION OF RECURSIVE STOCHASTIC SUBSPACE IDENTIFICATION IN

ON-LINE BRIDGE MONITORING SYSTEM

Jian-Huang Weng 1 , Chin-Hsiung Loh 2 , Shu-Hsien Chao 1 , Kung-Chin Lu 1 and Chia-Hui Chen 3

1 Post-doctor, Dept. of Civil Eng. National Taiwan University, Taipei, Taiwan.

2 Professor, Dept. of Civil Eng. National Taiwan University, Taipei, Taiwan.

3 Research Assistant, Dept. of Civil Eng. National Taiwan University, Taipei, Taiwan.

Abstract: This paper presents a recursive stochastic subspace identification (RSSI) technique for on-line bridge

monitoring system. The RSSI technique reduces the computation time so that the monitoring system can update

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the analysis results more quickly. In this study, the RSSI program is successfully integrated into the hardware

platform of the on-line bridge monitoring system. Finally, the accuracy and stability of the monitoring system

are verified through the scoring test of a reduced-scale steel bridge.

The stochastic subspace identification (SSI) is a well-developed technique for system identification. It is used to

identify the modal properties of the time-invarying linear system. Natural frequencies, damping ratios and mode

shapes can be identified by using only output response of the structure under ambient-vibration environment.

SSI has the advantage of low memory requirement and high accuracy due to the use of singular value

decomposition (SVD) and LQ decomposition (LQ). However, traditional SSI is not suitable for on-line analysis

due to the computational complexity of SVD and LQ decomposition. For on-line structural health monitoring

the recursive identification technique should be used. This paper provides a different recursive algorithm for LQ

decomposition to reduce the computation time of RSSI. The Givens rotation as well as the matrix operation

appending a new data set are derived so as to extract the time-varying features of the system without loss of

generality and to establish on-line and almost real-time system identification. The result from the RSSI

technique is applied to structural health monitoring of bridge system.

In order to understand the dynamic behavior of real bridges, this study conducted both field and laboratory

experiments. The field experiment collected data of a real bridge during normal operation and typhoon. The

change of natural frequency of the bridge during typhoon was observed from the result of RSSI. In the

laboratory experiment, a reduced-scale steel bridge was constructed in artificial riverbed to simulate the bridge

scour. An on-line monitoring system with the RSSI analysis was established to detect the change of modal

properties of the bridge. It is found that the loss of embedded depth of the piers will decrease the lateral stiffness

of the bridge. Therefore, the natural frequencies will also drop down. Furthermore, the amplitude of the mode

shape close to the damage location will increase due to the loss of the constraint force.

Keywords: system identification, structural health monitoring, bridge monitoring system.

一、前言

子空間識別 (Subspace identification, SI) 是目前發展成熟的線性系統識別技術中的一種 , 在其理論

發展的歷史中應用到許多重要的線性代數計算 , 如最重要的奇異值分解法 (Singular value

decomposition, SVD) 與 LQ 分解法 (LQ decomposition) 等 , 這兩種矩陣分解方法在線性代數的發展歷

史中算是較新的理論 , 在此之前 , 多數傳統的系統識別理論皆與最小二乘法有關 (Least-square

method)。早期的子空間識別法習慣使用結構系統的脈衝響應 (Impulse response) 來識別結構特性

[1,2,3] , 在狀態空間模型的推導中結構的脈衝響應又可稱之為馬可夫參數 (Markove parameter), 雖然使

用結構的脈衝響應可以得到不錯的分析結果 , 然而結構的脈衝響應卻不容易製造 , 尤其對於大尺寸

的結構物而言是件相當困難的挑戰 , 因此 , 後續有更多的理論被發展出來解決類似這樣的問題。部

份的研究學者發展以系統的輸入外力及輸出反應作為系統識別的依據 , 並考慮量測噪訊與系統誤差

之影響而提出結合確定與不確定因素的動力系統模型 (Combined deterministic-stochastic problem) 以

建立輸入與輸出訊號之間的關係 [4,5] ; 另一部分學者則嘗試單純由結構的輸出反應來重建系統特性 ,

但前提是此時結構的輸入外力來源必須是隨機訊號的模式 (Stochastic realization framework) [6] 。近十

年來尚有數種與子空間識別法相關的理論被陸續發展出來 , 其主要差異在於投影處理過程中採用了

不同型式的加權矩陣 (Weighting matrix), 其中較為知名的理論有 CVA、N4SID、MOESP 與 IV-4SID

等

[7,8,9] ,1996 年 Van Overschee [10] 對上述各種子空間識別的理論做了一系列完整的比較 , 並將前述的

SVD 分解與 LQ 分解應用於子空間識別法的演算過程中 , 同時也由理論推導證明了子空間識別法與

卡式過濾理論 (Kalman filter) 的相關性 , 其貢獻確立了子空間識別法於結構系統識別領域的重要性。

-570-

隨機子空間識別法為子空間識別理論中的一種 , 適用於微震環境中的多輸出線性動力系統 , 可

經由時間域的資料處理識別出結構的自然頻率、阻尼比與模態振形。該識別方法的優點在於利用 LQ

分解與 SVD 分解降低電腦計算量同時提高精確度 , 然而 , 其計算速度目前尚無法完全達成多自由

度系統的即時監測要求 , 有鑑於此 , 近幾年有許多提升電腦計算速度的遞迴式演算法被提出以解決

即時監測的需求 , 這些方法被稱之為遞迴式隨機子空間識別 (Recursive stochastic subspace

identification, RSSI)。隨機子空間識別法的運算量多集中與處理 SVD 分解與 LQ 分解 , 因此有部分

研究是採取避開 SVD 分解的方式來節省運算時間 [11,12,13] , 有部分研究則採用遞迴式最小二乘法取代

LQ 分解 [14] , 基本上所有的類似方法都在加速隨機子空間法的兩個運算步驟 :(1) 更新 LQ 分解的結

果 ;(2) 更新觀測矩陣 (Observability matrix) 的識別結果。1996 年 Golub 提出利用吉文斯旋轉 (Givens

rotation) 的方式達成 LQ 分解的更新 , 但其推導僅限於更新單一次取樣後的分析結果 , 因此分析結果

的更新頻率必須等於資料量測的取樣頻率 , 在應用上仍會有計算速度趕不上取樣頻率的問題 , 本研

究因此推導將多次取樣資料進行一次更新的方法 , 大幅降低更新 LQ 分解所需要的運算量 , 同時達

到更新速率可任意調整的目的。

本研究推導的理論可使隨機子空間識別在即時監測使用時達到遞迴演算 LQ 分解的目的 , 省去

其重新計算的時間 , 因此有助於改善運算即時性的問題。在實際應用方面 , 研究的長期目標為建立

即時監測系統以偵測橋梁受沖刷損壞的行為變化 , 並以遞迴式隨機子空間識別作為主要的分析工

具。為了解真實橋梁的行為反應 , 本研究進行了一系列的橋梁現地實驗與實驗場實驗 , 其中 , 現地

實驗選擇了一座七孔簡支梁型式的鋼筋混凝土橋做為長期觀測的目標 , 藉由平時與颱風洪水期間的

振動資料比對以釐清橋梁受洪水沖刷時的特性變化 ; 而實驗場實驗則設計了一座四孔簡支梁形式的

縮尺鋼橋 , 分別進行靜態與動態實驗 , 靜態實驗之目的在於了解橋墩埋入深度與模態振形之關係 ,

而動態實驗則模擬全橋結構受激烈洪水沖刷之情況 , 目的在於了解橋梁可能產生的損壞形式與其損

壞過程中的結構特性變化。

二、遞迴式隨機子空間識別的理論推導

隨機子空間識別法的優點在於利用 LQ 分解與 SVD 分解降低矩陣的運算量 , 其中 LQ 分解使用於

矩陣的投影計算而 SVD 分解則用於分解投影結果以萃取系統的觀測矩陣 :

li li j

⎡ Y li L 0 Q Y Y L Q ΓiXˆ

p ⎤ = ⎡ 11 ⎤ ⎡ ⎤

11 ⇒

f p

21 11

(1)

⎢

T ⎥

⎥

⎢

L21

⎥

⎣ ⎦ ⎣

22⎦

⎣Q21⎦

T ⎡ ⎤⎡V

⎤

= USV = [ U1

U2] ⎢ ⎢ ⎥ ≈ U1S1V1

⇒ Γ = U1

0 S

⎥ T

(2)

⎣ 2⎦⎣V2

⎦

(1) 式說明了將新的資料矩陣 Y 正投影至舊的資料矩陣

Y 上所得的投影結果等於 Γ Xˆ

i i

Γ i

為系統的

觀測矩陣而 Xˆ 為系統的近似狀態 , 此近似狀態與利用卡式過濾理論所求得的系統狀態相等 , 一般而

言 , 矩陣的正投影可依其定義進行較複雜的運算 , 或者也可直接採用 (1) 式的 LQ 分解得到相同的結

果同時節省電腦的記憶空間。根據隨機子空間識別的基礎理論可知 , 由 L 的行向量所組成的子空

間與 Γ 的行向量所組成的子空間相等 , 因此可再利用 (2) 式的 SVD 分解求得

L 的主要行向量組成的

子空間 U , 最後 ,

U 1

即等於系統的觀測矩陣 Γ , 而系統的頻率、阻尼與模態振形等資訊則可由觀

測矩陣再行求得。

(1) 式中資料排列的方式被稱之為漢克爾矩陣 (Hankel matrix), 實際上是將結構反應的歷時紀錄

分段截取並重疊排列的架構 :

-571-

⎡

⎢

⎥

⎢

K y

⎥

⎢ M M O M ⎥

⎢

⎥

⎡Yp

⎤ ⎢ y

K y

j-1

⎥

2li×

⎢ ⎥ ≡

= [ h1

K h

] = H1:

∈R

⎢

⎥

(3)

⎣Y

⎦ y

K y

⎢

⎥

⎢y

K y

1 ⎥

⎢ M M O M ⎥

⎢

⎥

⎢⎣

2i+

K y

2i+

j-1

⎥⎦

為了方便解說因此先定義 h 為漢克爾矩陣內的第 k 個行向量 , 而

H 則是含有 1~j 個行向量的矩陣 ,

因此當新的取樣資料加入時則 (3) 式需要再新增一個行向量 h , 而識別結果則反映出符合 1~2i+j 時

j+ 1

間段結構特性的等值線性系統 , 由此定義本研究中遞迴式隨機子空間識別理論所要解決的問題如

下 :

= L1Q1

(4)

H = L ( p )

(5)

1+ p : j+

p 2Q

< j

已知 H 的 LQ 分解結果如 (4) 式 , 當新資料

1: j

H 補進而且舊資料

j + 1:

H 移除時 , 如何以遞迴演算的方

1: p

式由舊的分析結果 { L 1

,Q 1

} 結合新的資料片段 H 來計算新的 LQ 分解結果

j + 1:

{ L 2

,Q 2

}。首先 , 為了移

除 L Q 中舊資料

1 1 H 的部分必須將 (4) 式的分解結果做退偶 (decouple) 的處理 , 因此使用吉文斯旋轉

1: p

法則將 Q 的前 p 個行向量上三角化 :

⎡ε

σ ⎤

1Q1

= ⎢ ⎥

(6)

⎣0

Q1⎦

根據 Weng [15] 的證明 , 由於 G 與

Q 皆為正交矩陣 (Orthogonal matrix) 因此 (6) 式的結果又等於 :

⎡I

0 ⎤

G1Q1

= ⎢ ⎥

(7)

⎣ 0 Q1⎦

I 為矩陣大小為 p 的單位矩陣 , 藉由已知的 G 可將 (4) 式的結果退偶 :

⎡I

p ⎤

H1:

= L1Q1

= ( L1G1

)( G1Q1

) = [ H1:

] ⎢ ⎥ (8)

⎣ 0 Q1

⎦

(8) 式的退偶結果將舊資料 H 由 { }

1: p

L 1

,Q 1

中移出並保留新的 LQ 分解所需要的資料片段 :

⎤

1 1 p + 1:

接下來引入新的資料 H 完成新 LQ 分解的雛形 :

j + 1:

1: j

L Q = H

(9)

⎡ 0 I

p ⎤

[ H

1: j p

L1] L2

p : j+

+ ⎢ = Q

⎥

(10)

⎣ ⎦

(10) 式中的 { L 2

,Q 2

} 並不滿足 LQ 分解的定義 , 雖然 Q 為正交矩陣但

L 並不是完全的下三角矩陣 , 因

此必須再藉由吉文斯旋轉對 L 進行下三角化的處理 :

p j p

= L G G Q = L

(11)

(

2 2

)(

2 2

)

2 2

: +

整體而言 ,LQ 分解的遞迴式演算法則如圖 1 所示 , 藉由兩次的吉文斯旋轉來更新 LQ 分解的結果 , 根

據實際計算顯示 , 當 (3) 式中的漢克爾矩陣為方陣時 (2li=j),(5) 式的 LQ 分解運算需要執行 j(j-1)/2 次的

吉文斯旋轉才可完成 , 然而 , 使用上述的遞迴式演算法則需要 p(2j-p-1) 次即可完成 , 一般而言 , 資

料的更新長度 p 會遠小於整段分析長度 j, 在此項限制之下 , 使用本研究之演算法可節省相當多的時

間。此外 , 如果在分析上有需要加快監測系統的反應速度時則可於 (10) 式中加入遺忘因子 (forgetting

factor):

L = H μ

(12)

[ ]

2 j + 1: j+

遺忘因子 μ 的功用在於降低舊資料占分析結果的比重 , 每一次的 LQ 分解更新會使舊的資料乘上一次

遺忘因子 , 愈舊的資料乘的次數愈多 , 當分析長度 j 較長的時候識別結果會與實際結構情況有時間上

的延遲 , 使用遺忘因子可以削減時間延遲的長度。

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圖 1 LQ 分解的遞迴演算示意

在理論驗證及實際應用方面 , 本研究以橋梁沖刷實驗的範例作為解說 , 並分為現地實驗與實驗

室實驗兩部分 , 現地實驗為本研究針對具有沖刷損壞威脅的真實橋梁所作的長期監測實驗 , 此實驗

資料包含三個部分 :(1) 為期半年的橋梁常態監測資料 ;(2) 颱風期間水位高峰時段的連續監測資料 ;

(3) 單個橋墩基礎 4m 開挖深度實驗的前後資料 , 此現地實驗之目的在於建立實際橋梁的長期監測資

料庫 , 一方面可由大量的背景資料識別橋梁常態的特性同時也可監測環境因素改變時對於橋梁特性

的影響程度。另外在實驗室實驗部分 , 為了補足現地實驗無法任意模擬沖刷損壞的缺點 , 本研究根

據國道高架橋梁的一般設計方式規劃了一座 4 跨每跨簡支總長 4.5m 的縮尺鋼橋模型 , 此實驗資料

包含兩個部分 :(1) 靜態的橋墩基礎開挖資料 ;(2) 動態的全橋沖刷資料 , 實驗目的在於模擬橋墩基礎

埋設深度的改變 , 並期望藉由系統識別的技術找出橋梁於此種損壞模式下的行為特徵 , 另一方面 ,

本研究所發展的遞迴式演算法已順利藉由 LebVIEW 軟體開發環境與硬體監測平台結合 , 達到即時

結構安全監測的功能 , 恰好可經由縮尺橋梁的沖刷實驗來驗證此套即時監測系統的性能。

圖 2 牛鬥橋照片與微振量測佈置圖

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三、現地橋梁長期監測資料分析

根據歷史新聞資料顯示 ,2009 年 10 月 5 日芭瑪颱風侵襲造成蘭陽溪上游日累積雨量達到

579mm, 造成蘭陽溪水位暴漲 , 洪水嚴重淘刷牛鬥橋橋墩基礎並將位於橋梁下游測的攔砂壩保護工

沖毀 , 經橋梁養護單位評估後暫時封閉牛鬥橋往來交通至洪水退去。本研究自 2010 年 4 月開始至

10 月為止近半年的時間針對牛鬥橋雙向橋梁共進行過 10 次的微振量測 , 其中一次為 2010 年 9 月

19 日凡那比颱風侵襲期間 , 圖 2 為跨越宜蘭縣蘭陽溪的牛鬥橋照片與結構圖示 , 該橋為 7 跨每跨簡

支的鋼筋混凝土橋 , 每跨長度 36.6m 總長 256.2m, 橋面板於每個跨之間皆有伸縮縫 , 墩柱高度約為

10.5m 而基礎埋設深度則超過 10m。在橋梁振動量測規畫部分 , 由於本研究之目的在於監測水流沖

刷所造成的結構特性改變 , 因此規劃於每個橋面板上量測 3 個點位 , 每個點位量測其水流方向的速

度反應 , 全橋共有 21 個量測點 , 因感測器數目有限故採分階段量測方式完成全橋量測。

圖 3 Fanapi Typhoon 期間量測方案 ( 無線感測器網路 )

本現地實驗初期規劃是採取定期收集橋梁反應數據的方式實施 , 並非建立一套固定式的監測系

統長期連續收集資料 , 圖 3 為凡那比颱風期間監測系統的硬體配置說明 , 颱風期間的橋梁量測解決

方案需要考慮到三個重點 :(1) 結構尺度的限制 ;(2) 實驗操作人員的安全性 ;(3) 電源供應問題 , 因

此 , 在硬體配置上可將整個監測系統分為感測端、主機端與使用端三個區塊 , 感測端使用型號

VSE-15D 的速度計 , 該速度計為微振量測等級 , 敏感度 1000V/m/s, 可量測範圍 ±0.1m/s, 收集到的

類比訊號經由本團隊所開發的 NTU-WSU 模組以無線傳輸方式將數位化資料傳至主機端 , 主機端的

cRIO-9022 控制器具有資料集錄與分析功能 , 搭配 LabVIEW Real-Time 軟體可線上進行結構健康診

斷分析 , 透過 DIR-455 3G 無線路由器加上 3G SIN 卡可將分析成果上網公布 , 而使用端可由網頁瀏

覽分析成果或下載實驗數據。對於大尺寸結構物而言無線感測技術大大減少了人員工作量同時也不

必擔心有線傳輸造成訊號衰減的問題 , 主機端就近設置於河岸旁安全處以確保實驗人員安全同時可

方便進行檢修與觀測橋梁現況 , 感測端與主機端皆採用電池供電 , 由於颱風路徑與登陸時間預測不

易 , 因此以連續供電 3 天的要求來設計電池容量。

圖 4 為牛鬥橋平時與颱風期間頻率的監測比較 , 此分析結果為使用遞迴式隨機子空間法進行離

線 (Off-line) 分析所得 , 圖左為 D05 點位所測得資料的分析結果而圖右為 D14 點位的結果 ,D05 點位

結果中的上部三個小圖為橋梁平時觀測所對應的頻率變化情形 , 而下部觀測時間較長的圖則是颱風

洪峰期的分析結果 , 由 D05 的觀測結果看來橋梁在水流運動方向上有三個主要頻率被識別出來 , 同

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圖 4 平時與颱風期間頻率監測結果 (D05 與 D14 點位資料 )

時颱風期間的頻率與平時量測幾乎相等 , 無任何變化發生 ; 然而 , 由 D14 的分析結果卻可清楚觀察

到在颱風期間 1000sec 附近橋梁頻率確實發生輕微的降低 , 且四個觀測到的頻率皆有此下降的現

象 , 而隨著洪水退去橋梁頻率又逐漸歸回原位。本現地實驗並未量測水位或基礎沖刷深度 , 但根據

現場實驗紀錄可知 , 凡那比颱風當天洪水集中靠近於河流的左岸 , 約為 S3 至 S5 橋面板的範圍沖刷

最為嚴重 , 其餘孔位的水量明顯較小 , 因此可推估 , 橋梁水平向頻率降低的現象與水流沖刷的力道

有關 ,D05 點位底下的橋墩受沖刷力道較小造成頻率變化不明顯 , 而 D14 的沖刷力道較大所以頻率

下降則較明顯 , 然而 , 因為沒有沖刷深度資料作為佐證 , 因此無法將頻率變化與沖刷深度做連結 ,

唯一可以確定的是 , 洪水退去後橋墩基礎的埋設狀況並無明顯的改變 , 因此橋梁頻率又回復至一般

標準。

除了颱風期間的資料之外 , 本研究尚有橋墩基礎開挖後的微震量測資料 , 由於舊有的牛鬥橋載

重設計愈來愈無法滿足目前的交通現況 , 在加上歷年颱風侵襲造成基礎淘空與橋台邊坡流失 , 因此

公路總局已在橋梁舊址下游側興建新橋取代既有的牛鬥橋 , 同時在舊橋拆除之前同意交由國家地震

工程研究中心進行一系列的橋梁耐震性能評估實驗 , 而橋墩基礎開挖即為其中一項。本研究選擇靠

近河川左岸的 P6 墩柱進行開挖 , 目標開挖深度為 4m, 圖 5 中的照片即為怪手開挖過程 , 同時於開

挖前與開挖後分別對橋梁進行完整的微振動量測 , 並以隨機子空間法識別橋梁的模態振形 , 圖 5 下

半部即為開挖前後橋梁前兩個可識別模態的振形比較。結果顯示單一橋墩基礎 4m 的開挖量對於整

體橋梁的特性影響微乎其微 , 橋梁開挖前後的頻率、阻尼與模態振形幾乎完全一致。此次開挖實驗

說明了利用真實橋梁進行損壞模擬的困難度 , 但橋梁結構安全監測系統的開發確實需要配合損壞模

擬實驗得以證明系統的精確性及可靠度 , 因此橋梁縮尺實驗對於本研究而言勢在必行。

圖 5 橋墩基礎開挖照片及模態振形分析結果比較

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四、實驗室縮尺橋梁實驗

有鑑於真實橋梁的損壞資料取得不易 , 因此本研究決定自行規畫有關橋梁沖刷損壞的模擬實

驗 , 圖 6 為根據國道一般跨河橋梁為原型所設計的縮尺模型 , 全橋為 4 跨每跨簡支 , 橋面板由 2cm

厚的鋼板構成 , 中間兩跨的尺寸為 100cm*47cm, 另兩邊跨的尺寸為 125cm*47cm, 橋面板與帽梁接

合處置有 2mm 厚的橡膠墊作為緩衝以模擬支承墊效果 , 帽梁與橋柱部分為空心鋼構 , 兩者合併高

度 29.5cm, 橋柱直徑 7cm, 橋墩基礎為管狀壓克力構造 , 高度 50cm, 直徑 15cm, 內部放置可調式

質量塊與攝影鏡頭 , 質量塊之作用在於調整質量比例已符合原型比例 , 而攝影鏡頭則用在記錄沖刷

深度的改變 , 作為後續分析比對用途。實驗場地為水利署新店辦公區的水工構造實驗場 , 該實驗場

可進行大型流場模擬 , 本試體全長 4.5m, 河道寬度設計為 4m, 橋墩基礎以細石埋設 , 初始埋設深

度設定為 33cm。

圖 6 縮尺橋梁實驗及儀器佈設圖

在儀器佈設方面 , 本實驗仍然沿用牛鬥橋現地實驗的配置方式 , 每座橋面板設置 3 個點位量測水流

方向的運動 , 由於感測器數目足夠因此採 12 個點位一次量測的方式進行監測。圖 7 為橋梁縮尺實

驗的硬體配置情形 , 主要以有線傳輸的方式進行量測並將監測系統分為感測端與主機端 , 感測端使

用 AS-2000 加速度計量測橋面板的加速度反應 , 該感測計屬於一般振動量測等級 , 敏感度 5mV/Gal,

可量測範圍 ±2000Gal, 為使價格較低的 AS-2000 加速計取代 VSE-15D 速度計的性能 , 本團隊另行

設計一組類比帶通濾波器 (Band-pass filter) 與放大器 (Amplifier) 強化擷取 0.02~50Hz 波段的訊號 , 經

初步微振實驗成果顯示 ,AS-2000 經濾波及放大處理後的加速度積分結果可與 VSE-15D 的量測吻

合 , 唯獨 VSE-15D 在低頻範圍 (

踐以遞迴式隨機子空間法即時監測橋梁頻率變化 , 在使用 4 個點位資料進行分析的狀況之下 , 當單

次分析資料長度為 5sec 時可達到每 1 秒更新頻率識別結果的速度。

圖 7 縮尺橋梁實驗的量測方案

本研究分別對縮尺橋梁進行兩種不同型式的實驗 , 其中一項為靜態的基礎開挖實驗 , 而另一項

為動態的橋樑沖刷實驗 , 兩者的目的皆在於探討基礎埋設深度改變對於橋梁頻率及模態振形的影

響 , 希望藉由模擬實驗的方式找出橋梁受沖刷損壞時所可能產生的行為特徵。靜態的基礎開挖實驗

安排如表 1 所述 , 共分為 9 個開挖階段 ,Case1 為橋梁的初始狀態 , 三座橋墩基礎的埋設深度皆為

33cm,Case2~Case6 模擬橋梁左側基礎持續淘空的情形 ,Pier1 的埋設深度由 33cm 逐漸降至 8cm,

Pier2 由 33cm 降至 23cm,Pier3 則維持不變 ;Case7~Case9 則模擬橋梁右側基礎也開始遭受沖刷直

至所有 Pier 的埋設深度皆沖刷至 8cm 為止。

表 1 縮尺橋梁靜態開挖實驗 - 埋設深度及模態識別結果

Case 1 2 3 4 5 6 7 8 9

Embedded

depth (cm)

Pier 1 33 28 23 18 13 8 8 8 8

Pier 2 33 33 28 23 23 23 18 18 8

Pier 3 33 33 33 33 33 33 33 18 8

Identification results from SSI

Freq. (Hz)

Damp. (%)

Mode1 12.57 12.21 11.62 10.91 10.63 9.91 10.27 9.49 7.4

Mode2 20.53 20.52 19.42 20.00 19.50 -- 19.57 17.76 17.61

Mode1 2.1 1.7 2.4 1.6 2.7 3 2.4 2.8 2.3

Mode2 3.4 3.0 2.3 2.8 3.0 -- 2.3 1.1 1.9

表 1 下半部為利用隨機子空間法對橋梁頻率及阻尼所做的識別結果 , 由頻率的變化方式可以推

測基礎埋設深度的減少確實會導致橋梁整體水平勁度的喪失 , 進而造成頻率下降。雖然橋梁的沖刷

損壞確實可經由頻率的改變觀察得知 , 但是 , 如同牛鬥橋現地開挖實驗的結論一樣 , 整體橋梁的自

然頻率對於局部埋設深度的改變並不敏感 , 如 Case1 與 Case2 之間的比較 , 單一基礎開挖約 1/6 的

深度只造成第一模態頻率約 2.9% 的下降 , 同時 , 頻率的降低只說明了橋梁整體勁度的損失 , 並無法

幫助判斷沖刷發生的位置。對於判斷損壞的發生位置而言 , 模態振形的改變或許可提供有用的資

訊 , 圖 8 為各個開挖階段模態振形的識別結果 , 左圖為 Case1~Case5 模態振形的變化情形 , 隨著橋

梁左側基礎的埋設深度逐漸下降 , 橋梁第一模態振形也開始往左傾斜 , 證明當左側的勁度損失較為

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嚴重時左側的振幅也會跟著變大 , 整體看起來就像是振行往左傾斜 ,Case6~Case9 右側埋設深度也

逐漸開挖至相同深度 , 因此振形又逐漸恢復至平衡狀態。整體而言 , 頻率與模態振形或許對於局部

輕微的橋梁損壞不甚敏感 , 但是對於整體的結構特性改變確實有達到偵測的效果 , 從頻率與模態振

形的變化情形確實可以協助判斷橋梁可能的損壞狀況。

圖 8 靜態開挖實驗之模態振形變化

在動態沖刷實驗部分 , 為了模擬與靜態開挖實驗具有相同損壞模式的情況 , 河道被修改成集中

沖刷橋梁右側基礎的型式 , 目的在使右側基礎產生嚴重淘空 , 並測試即時監測系統的分析是否能正

確反應出與靜態實驗相同的結論。圖 9 為沖刷實驗的記錄資料與即時分析結果展示 , 其中下圖為

Node6 的速度反應紀錄 ; 中圖為橋墩基礎的埋設深度紀錄 ; 上圖為遞迴式隨機子空間法的即時分析

結果。實驗記錄時間全長為 5000sec, 水流實際到達橋墩基礎的時間點為 600sec 左右 ( 如紅色虛線標

示位置 ),600~700sec 時段基礎埋設深度急遽下降 , 橋梁右側基礎 Pier2 與 Pier3 的沖刷程度較為嚴

重 , 約沖刷了 10cm 的深度 , 而 Pier1 僅沖刷了 5cm, 隨後沖刷情況趨於穩定 , 除了 Pier2 繼續沖刷

至埋設深度僅剩 10cm,Pier1 與 Pier3 皆有發生輕微沖刷回填的現象 ,5000sec 後沖刷情況完全穩定 ,

埋設深度幾乎不再有任何改變。由頻率的變化觀察 , 當水流尚未到達橋墩基礎時 , 橋梁的第一模態

頻率約為 12Hz 左右 , 與靜態實驗未損壞前的結果相同 ,600~700sec 時段頻率急遽下降至 10Hz 左右 ,

到 5000sec 沖刷結束時頻率約降至 8Hz。此外 , 由圖 10 沖刷前後的模態振形比較可到與靜態實驗相

同的結論 , 由於沖刷集中於橋梁的右側 , 故無論從第一模態或是第二模態的振形來看 , 沖刷嚴重的

位置振幅都會變大 , 使得振形皆往右傾斜。

圖 9 縮尺橋梁動態沖刷實驗 - 沖刷深度及頻率變化

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圖 10 動態沖刷實驗之模態振形變化

五、結論

本研究之目的在於發展隨機子空間識別的遞迴演算法以降低電腦計算量 , 使其運算速度滿足即

時監測條件下的要求 , 然後將此識別理論透過軟硬體整合技術實踐於硬體監測平台 , 最後經由縮尺

橋梁的沖刷實驗驗證該系統的準確性與穩定性。在縮尺橋梁的沖刷實驗中 , 藉由 PXI 硬體平台搭配

LabVIEW 圖形化程式編寫軟體的方式 , 本研究確實建立了一套以遞迴式隨機子空間識別法為基礎的

橋梁即時監測系統 , 並達成即時監測橋梁頻率、阻尼與模態振形變化的目標。此外 , 透過現地橋梁

及縮尺橋梁的實驗成果顯示 , 雖然頻率與模態振形對於局部輕微的橋梁損壞不是非常敏感 , 但是對

於整體的結構特性改變確實具有偵測的效果 , 從頻率與模態振形的變化情形確實可以協助判斷橋梁

可能的損壞狀況。最後 , 在未來研究方向上 , 完整的橋梁安全監測系統並非單獨一種分析方法即可

完成 , 需整合不同型式的感測器及其他訊號處理的技術得以趨於完善 , 本研究將朝此方向持續邁進。

誌謝

本研究承國科會補助 ( 計劃編號 :NSC 98-2625-M-002-018-MY3), 僅此致謝 ; 另外也感謝國家

地震工程研究中心於橋梁實驗上的極力協助 , 使整體研究得以順利完成。

參考文獻

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[3] Moore, B. C. Principal component analysis in linear systems: Controllability, Observability and Model Reduction. IEEE

Transaction on Automatic Control, 1981, 26(1): 17-32.

[4] Larimore, W. E. Canonical variate analysis in identification, filtering and adaptive control. Proc. of 29 th Conference on

Decision and Control, Hawaii, USA,1990, 596-604.

[5] Verhaegen, M. and Dewilde, P. Subspace model identification-Part I. International Journal of Control, 1992, 56(5):

1187-1210.

[6] Akaike, H. Markovian representation of stochastic processes by canonical variables. Siam Journal on Control and

Optimization, 1975, 13(1): 162-173.

[7] Larimore, W. E. The Optimality of Canonical Variate Identification by Example. Proc. of SYSID ’94, 1994, 4-6 July,

Copenhagen, Denmark, 2, 151-156.

[8] Overschee, P. V. and Moor, D. B. N4SID: Subspace Algorithms for the Identification of Combined

Deterministic-Stochastic Systems. Automatica, 1994, 30(1): 75-93.

[9] Verhaegen, M. Identification of the deterministic part of MIMO state space models given in innovations from

input-output data. Automatica, 1994, 30(1): 61-74.

[10] Overschee, P. V. and Moor, D. B. Subspace identification for linear systems: Theory-Implementation-Applications.

Dordrecht, Netherlands: Kluwer Academic Publishers, 1996.

[11] Goethals, I., Mevel, L., Benveniste, A. and Moor, D. B. Recursive output only subspace identification for in-flight

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flutter monitoring. Proc. of the 22 nd International Modal Analysis Conference, Dearborn, Michigan, 2004.

[12] Lin, P., Zhang, N. X. and Bin, N. On-line Modal Parameter Monitoring of Bridges Exploiting Multi-core Capacity by

Recursive Stochastic Subspace Identification Method. Proc. of 2008 American Control Conference (WeA18.5), Seattle,

Washington, USA, 2008.

[13] Chang, C. C. and Li, Z. Recursive Stochastic Subspace Identification for Structural Parameter Estimation. Proc. of SPIE,

Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, 2009. 7292.

[14] Catarina, J. M., Delgado, P., Lopes dos Santos and J.L. Martins de Carvalho. Numerical Algorithm for Recursive

Subspace Identification. Proc. of the 37 th IEEE Conference on Decision & Control, Tampa, Florida USA, 1998, 1848-1849.

[15] Weng, J. H. Application of subspace identification in system identification and structural damage detection. Ph.D.

Thesis, National Taiwan University, 2010.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

振动台叠层剪切箱改进

孙海峰

1 , 景立平

1 , 王宁伟

2 1

, 孟宪春

(1. 中国地震局工程力学研究所哈尔滨 150080;2. 沈阳建筑大学沈阳 110168)

摘要 : 振动台模型试验是研究地震作用下土 - 结动力相互作用的重要方法 , 而土箱的性质将直

接影响到试验结果的准确性。本文在总结国内外研制土箱经验的基础上 , 研制出了刚度可调的叠层

剪切箱。通过三层三跨地下结构全粘土试验证明 : 剪切箱具有针对不同试验中所选择的模型土种类

的不同而改变剪切箱的刚度的功能 , 很好的解决了在进行单一水平地震荷载输入试验时的模型箱边

界效应问题 , 能够满足不同模型土试验中对降低模型箱边界效应的要求。

关键词 : 叠层剪切箱振动台试验边界效应刚度土 - 结动力相互作用

DEVELOPMENT OF MULTIFUNCTIONAL LAMINAR SHEAR CONTAINER FOR

SHAKING TABLE TEST

Sun Haifeng 1 , Jing Liping 1 , Wang Ningwei 2 , Meng Xianchun 1

1: Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China;

2: Shenyang Jianzhu University, Shenyang 110168, China

Abstract:Shaking table test is an important method to study on the problem of the soil-structure dynamic

interaction. The property of the soil container directly affects the accuracy of the result. A laminar shear

container was designed for shaking table test. And a shaking table test on soil-underground structure dynamic

interaction which structure lay in clay was conducted. The results of the test show that the soil container

eliminated the boundary effect when the dynamic load was applied in only one horizontal direction. Meanwhile,

the stiffness of the soil container could be changed according to the change of the model soil, which is

applicable to decrease the boundary effect.

Key words: Laminar shear container, Shaking table test, Boundary effect, Stiffness, Soil-structure dynamic

interaction.

一、引言

地震作用下 , 土 - 结构相互作用是一个非常复杂的问题 , 它涉及到地震工程、岩土工程和结构工

程等多个领域。目前 , 研究此问题主要有三种方法 : 原型观测 , 模型试验和数值模拟。其中 , 原型

观测方法由于观测技术及地震的准确遇见性差 , 目前地震原型观测的数据还很难采集。而数值模拟

方法由于参数选取和计算方法的不同 , 其获得的结果间也往往有很大的差异。因此 , 模型试验是目

前进行土 - 结构相互作用研究的最有效方法 [1,2] 。由于振动台试验具有几何相似比可以更大 , 可在较

短的时间内进行多次模型试验以消除一些随机因数的影响 , 还可以进行二维、三维振动的模拟的特

点 , 它已成为研究土 - 结构相互作用的最重要方法。而模型箱的性质将直接影响到实验结果的准确性。

本文在总结国内外研制模型箱经验的基础上 , 设计并制作了刚度可调的、振动台试验能叠层剪切箱 ,

并通过三层三跨地下结构全粘土振动台试验验证了本剪切箱在解决边界效应问题上良好的性能。

基金项目 : 中国地震局公益性行业科研专项课题 (2008419022)

作者简介 : 孙海峰 (1981-), 男 , 博士研究生 , 主要从事岩土工程研究工作 . E-mail:sunhaifeng3639@163.com。

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二、模型土箱研究现状

要进行土 - 结相互作用研究 , 就必须要有盛土的容器。由于土箱的容积及振动台各种技术参数的

限制 , 必须对土箱的边界进行处理来模拟地基土的半无限性 [3] 。常用的土箱大概有三种 : 刚性土箱、

[4-6]

圆筒形柔性土箱和叠层剪切土箱。其中 , 刚性土箱是通过在箱内壁上贴柔性材料来吸收抵达侧边

界的能量 , 但是 , 柔性材料设置的过柔将会导致土体发生的是弯曲变形而不是剪切变形 , 太刚则会

导致反射波太强。圆筒形柔性土箱 , 由美国的 Meymand(1998) [7] 博士首先设计的。这种容器包括一

块围成圆筒形的橡胶膜 , 上端由钢圆环固定 , 下端固定在基底钢板上。它允许容器内的模型土发生

多方向平动的剪切变形 , 橡胶膜外包纤维带或钢丝提供径向刚度。柔性容器的外包纤维带的间距对

试验结果的影响很大 , 太小则成了刚性容器 , 太大则在振动时土体向外膨胀 , 导致土体约束压力的

释放 , 同时土层可能发生弯曲变形。目前 , 国内外最常用的是叠层剪切土箱。这种土箱是由多层刚

性框架叠放在一起 , 每两层框架通过刚性滚珠相连来实现层间的剪切运动。Whitman 和

Lambe(1981) [8] 最早研制出了一种叠环式模型箱。Matsuda(1988) [9] 研制了该模型土箱并最早完成了饱

和砂土振动台试验研究。国内 , 伍小平等 (2002) [10] 最早研制了钢制矩形层状剪切变形土箱 ; 黄春霞

(2006) [11] 研制了 3.0 m×1.5 m×1.8 m( 长 × 宽 × 高 ) 的由 15 层长方形钢框架组成的剪切土箱 ; 史晓军

(2009) [12] 研制了可在相互垂直的两个方向上分别模拟土体在水平地震作用下的层状剪切变形 ; 高博

(2009) [13] 用轴承和端部弹性约束系统分别取代常见的滚珠和侧向刚性约束系统的方法对叠层剪切

模型箱进行技术改进。陈国兴 (2010) [14] 研制了一个 15 层叠层方钢管框架并辅之以双侧面钢板约束的

叠层剪切型模型土箱。这些模型箱有一个共同的特点 : 为了限制剪切箱垂直方向的变形及平面扭转

变形 , 同时又能为箱体提供恢复力 , 设计者们在剪切箱的两端设置了固定的钢板。但是 , 每次试验

中 , 剪切箱内所盛装的模型土的刚度并不相同 , 当模型土的刚度较低时 , 土与剪切箱的刚度比就会

降低 , 在进行振动台试验时将会在箱壁上产生反射波 , 影响试验结果的准确性。因此 , 有必要研制

可以根据试验要求改变箱体刚度的叠层剪切箱。

三、剪切箱的设计与动力性能

3.1 构造要求

(1) 箱体牢固、安全可靠。

(2) 减小剪切箱的边界效应。

(3) 剪切箱的尺寸应小于振动台台面几何尺寸 , 剪切箱在盛装模型土后的总质量应小于振动

台的最大承载能力。

(4) 箱体每层框架满足刚度要求。

3.2 剪切箱设计方案

根据中国地震局工程力学研究所大型振动台的几何尺寸和承载能力 , 模型箱主体尺寸设计为

3.70m( 纵向 )×2.40m( 横向 )×1.70m( 高度 ), 底座尺寸为 4.18m( 纵向 )×2.82m( 横向 )×0.12m

( 厚度 ), 为增加底座强度 , 防止在吊装过程中出现危险情况 , 在底座设置了加强梁。剪切箱采用

15 层方形钢管框架叠合而成 , 每层钢框架由四根方形钢管焊接而成 , 方形钢管截面尺寸为

100mm×100mm, 壁厚 3mm。除最上一层框架外 , 每层框架四条边的两侧分别焊接两片具有 V 形凹

槽的 200mm×80mm×10mm 不锈钢垫板 , 每个凹槽内放置钢滚珠 8 个 , 形成可以自由滑动的支承点。

在垂直于振动方向的每个横向侧面 , 设计了三种尺寸的五块钢板 , 尺寸分别为 :

240mmx1696mm、490mmx1696mm、840mmx1696mm, 这样就可以根据试验要求选择所要使用钢板

的数量 , 同时经过计算 , 合理布孔 , 使每种尺寸的钢板可以随意移动位置 , 这样就可以提高钢板使

用的灵活性 , 满足改变刚度的要求。这些钢板通过用螺栓将其固定于横向侧面。箱体侧面如图 1。

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图 1 二维叠层剪切箱

图 2 二维剪切箱模型图

剪切箱纵向两侧分别安装两根圆形钢立柱 , 立柱上安装轴承 , 钢管立柱与箱底座通过 4 个螺栓

相连 , 用两根方钢管连接纵向两侧立柱以限制土箱垂直运动及平面扭转运动。模型箱内壁衬 2mm

厚的特制橡胶垫以防止剪切箱内的土和水的漏出。用螺栓将剪切箱底座固定在振动台台面上。

3.3 剪切箱的动力性能

为了使模型土能够模拟原型地基土的半无限性 , 在剪切箱与模型土组成的体系中 , 剪切箱的刚

度应小于模型土的刚度 , 以保证土体在振动过程中起控制作用 , 因此 , 剪切箱的频率应远离模型土

的频率 ; 剪切箱的阻尼应小于模型土的阻尼 , 才能使模型土与原型地基土有相同的运动规律。

采用扫频的方法测试侧面装有三块钢板时空箱的基频 , 测得值为 3.8Hz, 同样 , 采用扫频的方

法获得了装满粘土的土箱 - 模型土整体的基频为 11.8Hz。

应用有限元软件 ABAQUS [15,16] 建立了三维剪切箱模型并进行了振型分析 , 模型如图 2 所示。采

用实体单元模拟方型方钢 , 用壳单元模拟两侧的钢板 , 层与层之间通过钢板垫相连 , 并可以自由滑

动。由此计算出剪切箱在激振方向上的基频为 3.6Hz。同时 , 又建立了土体与地下结构三维有限元

计算模型 , 土体采用摩尔 - 库伦模型 , 地下结构采用完全弹性模型 , 模型底边界采用固定边界条件 ,

侧边界采用自由场边界条件 , 对模型提取一阶频率 , 值为 12.7Hz。计算值与振动台试验得出的剪切

箱和剪切箱中装满粘土的基频基本一致 , 相互证明了结果的可靠性。

根据自由振动衰减法 , 使剪切箱产生一定位移后释放 , 获得剪切箱的阻尼比为 4.51%。地震动

作用下土体的阻尼比一般在 5%-25% 之间 , 因此 , 在振动台模型试验中土箱的阻尼不会给模型土体

的地震反应带来不良影响。

四、全粘土试验剪切箱边界效应测试

Atop1

Aup1

Aup2

A1-4

粘土

A1-3

A1-2

A10

A1-1

Abo1

图 3 传感器布置图

为了检测剪切箱边界效应情况 , 进行了地下结构二维全粘土振动台试验。试验采用哈尔滨粘土

作为模型地基土 , 土层厚为 1.5 m., 土体分层夯实 , 在试验前采用白噪声激振振动台台面 , 使土体

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密实 , 预振后 2 天再开始试验。土层中布置加速度传感器如图 3。通过对比试验中同一深度处模型

地基土各测点地震动特性 , 可以得出剪切箱边界效应的程度。由于地下结构会对周围土体的震动响

应造成影响 , 因此选择埋深为 1.13m 的 A3、A4、A5 和埋深为 0.53m 的 A8、A9、A10 两组数据来

分析剪切箱的边界效应。

0.2

Acceleration(g)

0.1

0.0

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Time(s)

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0 5 10 15 20 25

Time(s)

0.006

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0.004

0.002

Amplitude

0.004

0.002

Amplitude

0.004

0.002

0.000

0 10 20 30 40 50

Frequency

(a) 输入 0.2gEl-centro 波时埋深 1.13m 处沿振动方向各测点加速度时程及傅氏谱

0 .6

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0 5 10 15 20 25

Time

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0 5 10 15 20 25

Time

0.025

0.020

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0.015

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0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50

Frequency

(b) 输入 0.6gEl-centro 波时埋深 1.13m 处沿振动方向各测点加速度时程及傅氏谱

图 4 输入 El-centro 波时埋深 1.13m 处沿振动方向各测点加速度时程及傅氏谱

0 .3

0.2

0 .3

0.2

0 .3

0.2

A10

Acceleration(g)

0.1

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Acceleration(g)

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0 5 10 15 20 25

Time

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0 5 10 15 20 25

Time

0.008

0.006

Amplitude

0.004

Amplitude

0.004

Amplitude

0.004

0.002

0.000

0 10 20 30 40 50

Frequency

0.000

0 10 20 30 40 50

Frequency

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0 10 20 30 40 50

Frequency

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(a) 输入 0.2gEl-centro 波时埋深 0.53m 处沿振动方向各测点加速度时程及傅氏谱

Acceleration(g)

0 .8

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Time

Acceleration(g)

0 .8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

A10

0 5 10 15 20 25

Time

0.04

0.03

Amplitude

0.02

Amplitude

0.02

Amplitude

0.02

0.01

0.00

0 10 20 30 40 50

Frequency

0.00

0 10 20 30 40 50

Frequency

0.00

0 10 20 30 40 50

Frequency

(b) 输入 0.6gEl-centro 波时埋深 0.53m 处沿振动方向各测点加速度时程及傅氏谱

图 5 输入 El-centro 波时埋深 0.53m 处沿振动方向各测点加速度时程及傅氏谱

限于篇幅 , 这里只选取 El-centro 波 , 幅值分别为 0.2g 和 0.6g 的结果对剪切箱的边界效应进行

分析。同一深度的 A3、A4 和 A5 各点加速度的时程及傅里叶频率 - 幅值谱如图 4(a) 和图 4(b)。从图

中可以看出 :0.2g 和 0.6g 输入时 ,A3、A4 和 A5 各点的加速度时程和傅里叶频率 - 幅值谱重合的非

常好。同一深度处的 A8、A9 和 A10 各点加速度的时程如图 5(a) 和图 5(b)。从图中可以看出 :A8、

A9 和 A10 各点加速度的时程和傅里叶频率幅值谱重合的也非常好 , 只是 A8 的最大加速度值高于

A10, 原因是地震波在地下结构处产生的反射波与入射波在 A8 处叠加 , 使得 A8 的峰值大于 A10。

表 1 列出了各点的最大加速度绝对值和同一深度处各点的加速度峰值与距离土层中心点最近点的加

速度峰值的相对误差。从表中的数据不难看出 , 剪切箱性能良好 , 很好的解决了模型箱的边界效应

问题。

表 1 模型地基同一深度处各测点的有效峰值加速度及相对误差

输入峰值 /(g) A3/(g) A4/(g) A5/(g)

A3−

A3 × 100%

A4−

A3 × 100%

A5−

A3 × 100%

0.2 0.23 0.23 0.21 0.00 0.00% 2.00%

0.6 0.5 0.46 0.5 0.00% 4.00% 0.00%

输入峰值 /(g) A8/(g) A9/(g) A10/(g)

A8−

A8 × 100%

A9−

A8 × 100%

A10 − A8 × 100%

0.2 0.25 0.26 0.28 0.00% 1.00% 3.00%

0.6 0.68 0.66 0.63 0.00% 2.00% 5.00%

五、结论

模型试验是研究地震作用下土 - 结相互作用的重要方法 , 而土箱的性质将直接影响到试验结果的

准确性。本文在总结国内外研制土箱经验的基础上 , 在国内首次设计并制作了刚度可调的叠层剪切

箱。通过三层三跨地下结构全粘土试验证明了 : 本剪切箱很好的解决了在进行单一水平地震荷载输

入试验时的模型箱边界效应问题。同时 , 本剪切箱可以根据试验中所选择的模型土种类的不同而改

变剪切箱的刚度 , 以满足试验对降低模型箱边界效应的要求 , 为今后进行各类不同场地条件下土 -

结相互作用试验提供了性能良好的试验箱。

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参考文献

[1] 刘晶波 , 李彬 . 地铁地下结构抗震分析及设计中的几个关键问题 . 土木工程学报 , 2006, 39(6): 106-110.

[2] 何海健 , 刘维宁 , 王霆 . 地下铁道抗震研究的现状与探讨 . 中国安全科学学报 , 2005, 15(8): 3-7.

[3] 廖振鹏 . 工程波动理论导论 ( 第二版 ), 北京 : 科学出版社 , 2004.

[4] 杨林德 , 季倩倩 , 郑永来 , 杨超 . 地铁车站结构振动台试验中模型箱设计的研究 . 岩土工程学报 , 2004, 26(1):

75-78.

[5] 陈国兴 , 庄海洋 , 程绍革 , 杜修力 , 李亮 . 土 - 地铁隧道动力相互作用的大型振动台试验 : 试验方案设计 . 地震工

程与工程振动 , 2006, 26(6): 178-183.

[6] 景立平 , 姚运生 , 郑志华 . 高速铁路路基粉土地震模拟振动台试验研究 . 地震工程与工程振动 , 2006, 27(6):

160-165.

[7] Meymand, P. Shaking table scale model test of nonlinear building- pile- superstructure interaction in soft clay, Berkeley:

University of California Berkeley, 1998.

[8] Whitman, R. V., Lambe, P. C. and Kutter, B. L. Initial results from a stacked ring apparatus for simulation of a soil profile.

In: Proceedings of the International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil

Dynamics, St. Louis, USA: [s. n.], 1981, 1, 105–1110.

[9] Matsuda, T. and Goto, Y. Studies on experimental technique of shaking table test for geotechnical problem. Proceedings

of the 9th World Conference Earthquake Engineering, Tokyo, Japan, 1998: 837-842.

[10] 伍小平 , 孙利民 , 胡世德 , 范立础 . 振动台试验用层状剪切变形土箱的研制 . 同济大学学报 , 2002, 30(7):

781-785.

[11] 黄春霞 , 张鸿儒 , 隋志龙 . 大型叠层剪切变形模型箱的研制 . 岩石力学与工程学报 , 2006, 25(10): 2128-2134.

[12] 史晓军 , 陈隽 , 李杰 . 层状双向剪切模型箱的设计及振动台试验验证 . 地下空间与工程学报 , 2009, 5(2):

254-261.

[13] 高博 , 张鸿儒 . 堆叠式剪切模型箱的改进 . 岩石力学与工程学报 , 2009, 28( 增 2): 4021-4026.

[14] 陈国兴 , 王志华 , 左熹 , 杜修力 , 韩晓健 . 振动台试验叠层剪切型土箱的研制 , 岩土工程学报 , 2010, 32(1): 89-98.

[15] 庄茁 . 基于 ABAQUS 的有限元分析和应用 , 北京 : 清华大学出版社 , 2009.

[16] 石亦平 , 周玉蓉等 . ABAQUS 有限元分析实例详解 , 北京 : 机械工业出版社 , 2006.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

锈裂混凝土裂缝形态的试验研究

赵羽习余江金伟良

( 浙江大学结构工程研究所 , 浙江杭州 310027)

摘要 : 钢筋锈蚀引起的混凝土结构开裂被认为是钢筋混凝土结构耐久性失效的主要原因。在锈

胀开裂过程中 , 裂缝开展的模式如何 , 是影响锈裂模型预测结果的重要因素。本文对人工气候环境

下加速劣化达两年的混凝土试块进行切片研究 , 利用数码显微镜观测了锈胀裂缝的开展情况 , 包括

不同半径位置处对应的环向裂缝宽度和混凝土保护层表面的裂缝宽度 , 由此分析随锈蚀增长裂缝开

展的过程 , 并建立了锈胀裂缝开展模型。

关键词 : 混凝土钢筋锈蚀裂缝

EXPERIMENTAL STUDY ON CRACK MODE IN REINFORCED CONCRETE

STRUCTURES WITH REBAR CORROSION

Y.X. Zhao, J. Yu and W.L. Jin

Institute of Structural Engineering, Zhejiang University, Hangzhou 310027, China

Abstract: Steel corrosion is one of the most predominant deterioration mechanisms in reinforced concrete

structures. In corrosion-induced cracking process, the crack mode is a major factor that affects the prediction of

the cracking models. A reinforced concrete specimen which has deteriorated in artificial environment for two

years was investigated in this study. The crack mode was observed by digital microscope, including the

circumferential crack width at different radius and the crack width on the surface of the concrete cover. The

pattern of crack propagation was discussed based on the observed results and a crack propagation model was

established.

Keywords: Concrete, rebar, corrosion, crack.

一、前言

钢筋锈蚀引起的混凝土结构开裂被认为是钢筋混凝土结构耐久性失效的主要原因。由于钢筋锈

蚀产物体积膨胀引起的钢筋 / 混凝土界面锈胀力会导致混凝土保护层受拉而开裂。一旦混凝土保护层

中出现裂缝 , 环境中的氯盐等有害介质就会通过裂缝直接侵入到混凝土内部接触到钢筋 , 从而导致

钢筋锈蚀大大加剧 , 甚至造成混凝土保护层的剥落 , 最终导致混凝土结构失效。因此 , 混凝土结构

锈胀开裂的研究对混凝土结构的使用性能评估和剩余寿命的预测具有相当重要的意义。对于混凝土

保护层表面锈胀开裂后的情况 , 国内外很多学者通过试验进行了研究 [1-6] 。由于混凝土保护层表面裂

缝的形态能够直观地反映表面锈裂后混凝土破坏的过程 , 因此 , 这些研究集中于对保护层表面裂缝

宽度的观测这一方面 , 通过试验数据建立了表面裂缝宽度与钢筋锈蚀量的关系。但是 , 在锈胀开裂

的理论计算和有限元分析中 , 为了更加真实地模拟出锈胀开裂的过程 , 必须对混凝土内部锈胀裂缝

的形态进行把握 , 这就使得仅仅直观地观测表面裂缝宽度是不足的 , 还需对锈胀裂缝的内部形态进

行研究。

有鉴于上述情况 , 本文中 , 对人工环境中劣化达两年的钢筋混凝土试块切片 , 利用数码显微镜

基金项目 : 国家自然科学基金项目 (50808157); 浙江省钱江人才计划项目 (2010R10099)

作者简介 : 赵羽习 (1973-), 女 , 浙江杭州人 , 教授 , 博导 , 从事结构工程的教学科研工作。Email:yxzhao@zju.edu.cn

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进行观察 , 基于观测数据分析锈胀裂缝的开展情况。研究成果可用于混凝土锈裂过程的理论分析和

数值模拟 , 对于准确预测混凝土表面锈胀开裂具有重要的意义。

二、试验方案

2.1 混凝土试块

本次试验采用的试块为杭州湾跨海大桥建造时现场浇筑的高强混凝土试块 , 按照与浇筑大桥钢

筋混凝土构件同样的规格技术和搅拌工艺浇筑。试块的尺寸为 150mm×150mm×300mm。顶部均匀

配置三根带肋钢筋 , 钢筋直径为 16mm, 混凝土保护层厚度为 20mm, 如图 1 所示。试块的配合比

见表 1。边长 150mm 的立方体试块 28 天抗压强度值是 56.0MPa。

3φ16

150

300

150

图 1 混凝土试块细节图

表 1 混凝土试块的配合比 (Kg/m 3 )

水泥磨细矿渣粉煤灰砂骨料水水胶比减水剂防腐剂

126 168 126 735 1068 145 0.345 5.04 8.4

为了确保在加速劣化过程中氯离子的单向扩散 , 得到不均匀锈蚀的情况 , 试块的四周表面和底

面用环氧树脂和聚氨酯底漆做了表面处理 , 因此氯离子只能通过试块的顶面渗入混凝土中。

2.2 试件加速劣化

试块在浙江大学混凝土结构耐久性实验室内的步入式人工气候实验箱中进行加速劣化。经历交

替的干湿循环过程 , 每个试验周期持续 3 天 , 包括喷洒 3.35% 氯化钠溶液 4 小时 , 剩余的时间在 40

o C 下恒温干燥以完成一次循环。本试块在此人工气候环境中经历 2 年的时间。

2.3 样品制备

由于试块已经开裂 , 部分位置出现了宽度较大的裂缝 , 为了避免混凝土试块在后续的切割、制

样过程中出现进一步的人为的损伤 , 采取环氧树脂浸渍的方法 , 用低粘度的环氧树脂密封试块 , 填

入表面开裂的裂缝中 , 确保了试块的完整性。

混凝土试块中样本的切取如图 2 所示。采用 Ф355mm 的混凝土切割机将试块沿钢筋方向切割成

长条状。按照其相对位置分别被命名为 L( 左部 ),M( 中部 ) 和 R( 右部 )。切下的长条状试样继

续用环氧树脂侵入密封。然后采用精密切割机 SYJ-200 将条状试样进行切片 , 切片的厚度大约为

1cm, 切片的命名按照编号 1,2,3,4…… 进行。例如图 2 中 M-14 就代表试块中部条状试样中第

14 个切片。在观测之前将切割好的钢筋混凝土切片用不同规格的砂纸进行打磨抛光。并将切片保存

在相对湿度低于 30% 的环境中 , 以防止钢筋进一步锈蚀。

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L 组 M 组 R 组

M-

R-

切割线

-4

钢筋

图 2 样品制备图

2.4 样本观测

待观测样本中的钢筋具有不同的锈蚀程度 , 对应的锈胀裂缝的数目和宽度也有所不同。在 R 组

试件中 ,R-1 至 R-12 的试样裂缝切割后状态良好 , 而在 R-13 至 R-20 的试样中 , 观察到不同程度的

被破坏的裂缝。这是由于锈胀裂缝未开展到保护层表面 , 或者裂缝虽然开展到保护层表面 , 但非常

微小 , 环氧不能很好地渗入其中 , 导致锈胀裂缝在切割扰动的作用下 , 裂缝长度继续扩展 , 裂缝边

缘也出现了一定程度的损坏 , 不能进行裂缝宽度的观测。对于 M 组试样 , 存在与 R-13 至 R-20 试样

相似的情况。对于 L 组试样 , 裂缝分布与 R 组相似。在本次研究中 , 着重对 R-1 至 R-12 的试样进

行裂缝开展情况的研究。

在研究过程中 , 为了对钢筋和混凝土的不同部位分别进行观测 , 采取塑料薄膜划线分区的方式

来进行研究。首先 , 在一张塑料透明薄膜上确定一个中心点 , 过中心点每隔 30 度刻线 , 间隔 3mm

刻同心圆环。然后将做好的薄膜贴在试样之上 , 让薄膜的中心点对准钢筋的中心位置。如图 3 所示。

这样钢筋被分成了 12 个区 , 混凝土也相应地被分区。然后利用数码显微镜对钢筋的每个区和混凝

土中有裂缝的区域进行观测分析。数码显微镜有多级放大倍数 , 与计算机相连后 , 直接采集彩色显示

的数字图片 , 可以清楚的分辨出锈蚀产物层和裂缝。利用显微观测的颜色和表观形态判断锈蚀层和

裂缝的位置和分布范围 , 利用数码显微镜附带软件的测量功能进行量测。对于铁锈层 , 直接利用软

件的面积测量功能分别测量 12 个区的铁锈面积。对于裂缝 , 借助圆环和射线的分区 , 按 1mm 间隔

测量同一半径 R i 下所有裂缝的宽度 , 如图 4 所示。

w s1

w 1,20

w 1i

w 2i

w s2

R i

图 3 样本观测方法示意图

图 4 裂宽观测示意图

将同一半径下测得的所有的裂缝宽度进行叠加 , 得到该半径对应的环向裂缝总宽度 :

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∑

W i = w ki

k = 1

(1)

式中 : W i 表示任一半径 R i(mm) 处对应的环向裂缝总宽度 (mm), w ki 为单根裂缝在半径 R i 处

的裂缝宽度 (mm),j 为半径 R 处裂缝的数量。

根据测量计算得到的环向裂缝总宽度 W , 可建立其与裂缝所在半径位置 R 的关系 :

W = f R )

(2)

另外 , 对试样 R-1 至 R-12 的混凝土保护层表面的裂缝宽度进行了测量。同样 , 将试样顶面和

侧面各条表面裂缝的宽度叠加 , 得到混凝土表面总裂缝宽度 W :

∑

( i

W s = w sk

(3)

式中 : W s 表示混凝土表面裂缝总宽度 (mm), w sk 为单根裂缝在保护层表面处的裂缝宽度 (mm),

j 为保护层表面处的裂缝的数量。

三、试验结果与分析

试块在劣化环境中经历两年后 , 钢筋锈蚀已经较严重。锈胀裂缝在试块表面已经非常明显并且

在试块的端部可以清晰的看到锈斑。各样品的锈蚀程度各不相同。从钢筋所处的位置来看 , 角区试

样 L、R 的钢筋锈蚀程度大于中间试样 M; 对于同一试样 , 钢筋端部的锈蚀程度明显大于中间部位。

下面 , 分别从裂缝形态、混凝土保护层表面裂缝宽度和铁锈分布三方面来进行分析讨论。

3.1 裂缝形态

3.1.1 裂缝宽度观测结果

对于待研究样本 , 将 12 个区的锈蚀产物面积进行叠加 , 然后假定铁锈膨胀率为 2, 可以计算出

各个样本的锈蚀率 ρ。将同一半径下测得的所有的裂缝宽度进行叠加 , 可得到该半径对应的环向裂

缝总宽度。建立环向裂缝总宽度与裂缝所在半径位置的关系 , 采取下述线性方程来描述裂缝总宽度

的变化 :

W = a( R − R)

(4)

i i +

式中 : W 表示环向裂缝总宽度 (mm), R 表示混凝土中任一点所在的半径位置 (mm),R 为

钢筋半径 (mm),a、b 是描述裂缝宽度分布特点的参数。在图 5 所示的裂缝宽度简化模型中 ,a 为

裂缝宽度变化系数 , 而 b 则是钢筋表面的裂宽系数。

W s

b θ=arctana

3.0

2.5

2.0

W I

/mm

1.5

1.0

0.5

0.0

0 5 10 15 20 25

(R I

-R)/mm

图 5 锈胀裂缝宽度简化模型

图 6 试样 R-1 裂缝宽度随半径变化图

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典型试样 (R-1) 混凝土保护层内的锈胀裂缝宽度随半径变化的测试结果如图 6 示。根据测试

数据回归得到的裂缝宽度变化系数 a 和钢筋表面裂宽系数 b 列于表 2 中。由于混凝土为不均质材料 ,

产生的裂缝宽度波动较大 , 即使在相邻的裂缝位置处 , 裂缝宽度也可能有较大的变化。因此 , 模型

拟合的离散性较大。可以看出 , 对于任一试样 , 离保护层表面越近处 , 锈胀裂缝宽度越大。在钢筋

锈蚀率较小的情况下 , 裂缝宽度值比较小 , 裂缝内外宽度的变化梯度不大。随着钢筋锈蚀率的增长 ,

一方面 , 钢筋表面位置处的裂缝宽度逐渐增大 , 另一方面 , 裂缝宽度内外变化梯度也越来越明显。

3.1.2 裂宽模型讨论

表 2 裂宽模型中参数回归值

试样 ρ a b

R-1 6.07% 0.035 1.507

R-2 7.07% 0.052 1.495

R-3 5.89% 0.039 0.823

R-4 4.71% 0.017 1.163

R-5 3.50% 0.016 0.924

R-6 1.16% 0.011 0.345

R-7 0.55% 0.004 0.378

R-8 0.75% 0.007 0.557

R-9 0.71% 0.006 0.595

R-10 2.34% 0.008 0.451

R-11 2.78% 0.015 0.268

R-12 2.20% 0.018 0.209

(1) 裂缝宽度变化系数 a

在裂缝宽度模型中 ,a 反应的是裂缝宽度沿径向的变化情况 , 图 7 为裂缝宽度变化系数 a 与钢

筋锈蚀率的关系。可以看出 , 随着锈蚀的增长 ,a 的值呈现不断增大的趋势 , 可采用线性模型近似

拟合 :

a = 0.67857ρ

− 0.00234 ( R 2 = 0.850 ) (5)

式中 , ρ 为钢筋锈蚀率。

观察图 7 发现 , 当钢筋锈蚀率为 0.34% 时 , 拟合线与 x 轴相交 , 裂缝宽度变化系数 a 的值为 0,

即此时裂缝内外宽度是一致的。对于锈蚀率小于 0.34% 的情况 , 参数 a 则小于 0, 即裂缝外部的宽

度小于内部的宽度。由此可以推断 ,0.34% 即为混凝土保护层表面锈裂时刻的钢筋锈蚀率值。在钢

筋锈蚀率小于 0.34% 时 , 裂缝未开展到混凝土保护层表面 , 裂缝形态呈内宽外窄的特点。而当裂缝

开展到混凝土保护层表面时刻 , 锈胀裂缝宽度内外一致 ; 之后随着钢筋锈蚀率的增大 , 外部裂缝宽

度逐渐大于内部裂缝宽度 , 并且内外裂缝宽度的变化梯度越来越明显。

为了验证本模型在表面开裂前的正确性 , 试验研究中 , 对于裂缝尚未开展到混凝土保护层表面

的内裂情况 , 也进行了观察。但是 , 由于裂缝多数在切割过程中发生了损坏 , 因此 , 不能进行定量

的裂宽观测研究。通过对某些状态稍好的内裂裂缝观察发现 , 裂宽沿径向几乎保持一致 , 只在裂缝

端部区域发生变化 , 最终在裂缝尖端处 , 裂缝宽度等于 0。对于此种实际情况 , 裂宽测量点的分布

形式应该为 : 靠近钢筋处 , 裂宽几乎相等 , 呈水平分布 , 只在裂缝端部呈下降趋势 , 并在尖端位置

变为 0。若用直线进行拟合 , 参数 a 应小于 0, 与图 7 所示的情况保持一致。

(2) 钢筋表面的裂宽系数 b

从图 5 所示的锈胀裂缝宽度简化模型中可以看出 ,b 反映的是钢筋表面裂缝的宽度。图 8 为系

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数 b 与钢筋锈蚀率的关系。可以看出 , 随着锈蚀的增长 , 钢筋表面的裂缝宽度呈现不断增大的趋势。

即随着锈蚀率的增长 , 钢筋表面裂缝宽度不断增大 , 可采用线性模型进行拟合 :

2 =

b = 20.93ρ ( R 0.612 ) (6)

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.00 0.02 0.04 0.06 0.08

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.00 0.02 0.04 0.06 0.08

图 7 裂缝宽度变化系数 a 与钢筋锈蚀率 ρ 的关系

图 8 参数 b 与钢筋锈蚀率 ρ 的关系

(3) 环向裂缝总宽度 W i

将系数 a 与 b 模型拟合的结果代入 W 的表达式中 , 即将公式 (5) 和 (6) 代入公式 (4), 即

可得到环向裂缝总宽度 W 与钢筋锈蚀率的关系 :

W = [ 0.67857( R − R)

+ 20.93] ρ − 0.00234( R − R)

(7)

根据公式 (7), 可以计算在不同钢筋锈蚀率情况下不同半径位置处的锈胀裂缝的宽度。

对于混凝土保护层表面位置处的情况 , 即 R i − R = 20mm

时 , 由式 (7) 可知环向裂缝总宽度为 :.

W = 34.5014ρ

− 0.0468

(7’)

对于钢筋表面位置处的情况 , 即 R i = R 时 , 由式 (7) 可知环向裂缝总宽度为 :

W = 20.93ρ

(7”)

若以参数 a 讨论中得到的表面开裂时刻的钢筋锈蚀率 0.34% 代入式 (7’) 和式 (7’’) 中进行研

究 , 可以算得此时的钢筋表面裂缝宽度和混凝土表面裂缝宽度均为 0 .071mm

。

w 1,1

3.0

2.5

2.0

公式 (9)

公式 (7’)

w 1,2

W S

/mm

1.5

1.0

0.5

0.0

0.00 0.02 0.04 0.06 0.08

图 9 文献 [13] 中的裂宽计算几何模型

图 10 保护层表面裂缝宽度与钢筋锈蚀率 ρ 的关系

Molina [7] 对锈胀裂缝宽度进行了理论分析研究 , 假定断裂能在裂缝处完全释放 , 并且不考虑钢

筋周围混凝土开裂时受外围混凝土的约束作用 , 将裂缝宽度的问题简化为简单的几何学问题 , 如图

9 所示 , 可得到钢筋表面裂缝宽度与钢筋锈蚀深度之间的关系 :

W i = 2π ( n −1)δ

(8)

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式中 ,n 为锈蚀产物体积膨胀率 , 本文中取 n=2,δ 为钢筋锈蚀深度 (mm)。以本文中得到的表面

开裂时刻的钢筋锈蚀率为例 , 钢筋锈蚀深度 δ = R − 1−

ρ ⋅ R = 13.6μm

; 此时 Molina 的理论模型结果

为 W i = 0.085mm

, 略大于试验数据拟合得到的结果。这是由于在理论模型中 , 由几何关系直接计算

得到表面裂缝宽度 , 未考虑钢筋周围混凝土开裂时受外围混凝土的约束作用无法自由开展的因素 ,

从而导致计算结果偏大。

3.2 保护层表面裂缝宽度与钢筋锈蚀率的关系

对裂宽模型的讨论 , 着重于对裂缝开展的内在机理的研究 , 对于锈胀开裂的理论研究或有限元

分析 , 具有相当重要的意义。而对保护层表面裂缝宽度的研究 , 着重于直观的观测 , 以便在实际工

程中通过观测表面裂缝宽度的发展情况来推断钢筋锈蚀的程度。

3.2.1 试验结果与分析

本试验研究中 , 对混凝土保护层表面的裂缝宽度进行了测量 , 采用线性模型对保护层表面总裂

宽 W s 与钢筋锈蚀率的关系进行拟合 , 结果如下 :

W s = 39.14ρ

− 0.04904 ( R 2 = 0.82 ) (9)

公式 (7’) 给出了混凝土表面位置处的环向裂缝总宽度 , 其值略小于上述保护层表面裂缝宽度

公式 (9) 的值 , 如图 10 所示。这是由于在实际情况中 , 裂缝并非严格地沿径向发展 , 这就导致在

R i -R=20mm 处测得的环向裂缝总宽度与在混凝土保护层表面测得的裂缝宽度不相等。由图 4 所示的

裂缝情况也可以看出 , 裂缝 1 在 R i -R=20mm 处的环向裂缝宽度小于其在混凝土保护层表面的宽度值。

3.2.2 混凝土表面开裂时的裂宽

在以往混凝土保护层锈裂过程的研究中 , 混凝土保护层开裂时刻钢筋锈蚀率的预测是非常被关

注的 , 其通常被定义为混凝土保护层表面裂缝宽度为 0 时的钢筋锈蚀率 [2-5] 。研究人员通过混凝土

结构的锈裂试验 , 建立混凝土表面裂缝宽度与钢筋锈蚀率之间的关系 , 然后令混凝土表面裂缝宽度

为 0 来预测表面开裂时刻的钢筋锈蚀率。

但是 , 基于 3.1.2 中对裂缝宽度变化系数 a 的讨论 , 在保护层表面开裂的瞬间 , 由于混凝土的

脆性特性 , 裂缝尖端的宽度迅速由 0 变为一定值 , 锈胀裂缝内外宽度保持一致。因此 , 令表面裂缝

宽度等于 0 来计算此时的钢筋锈蚀率是不合理的 ; 而是应该令表面开裂时刻的裂宽等于一定值来进

行计算。

在 3.1.2 的第三部分中 , 通过公式 (7’) 计算得到表面开裂时刻的钢筋表面裂缝宽度为 0.071mm,

将此值代入公式 (9) 中 , 得到钢筋锈蚀率 ρ =0.31%, 即为混凝土保护层表面开裂的临界锈蚀率 ,

与前述系数 a 讨论中得到的临界锈蚀率基本保持一致。

基于上述分析 , 笔者认为 , 在锈裂预测过程中 , 根据锈胀裂缝宽度与钢筋锈蚀率的经验公式 ,

宜采用一定的表面临界锈裂裂宽来进行表面锈裂的预测 , 结合试验的情况 , 建议取值为 0.05-0.1mm。

3.2.3 混凝土表面裂宽预测模型对比

在以往的研究工作中 , 一些研究人员也根据其试验数据建立了表面裂缝宽度与钢筋锈蚀之间的

[8]

关系。其中 , 王深、Rodriguez [2] 建立了裂缝宽度与钢筋锈蚀率之间的线性模型 , 与本文的研究结

果是一致的。另外 ,Vidal [3] [5]

建立了裂缝宽度和钢筋锈蚀截面损失面积之间的线性模型 , 夏晋建立

了裂缝宽度与钢筋锈蚀率之间的对数模型。本次研究中 , 采用已有的其他非线性模型对表面裂缝宽

度测试数据分别进行拟合 , 并与前述线性模型拟合的结果进行比较如下。

(1)Vidal 的模型

Vidal [3] 建立了裂缝宽度和钢筋锈蚀截面损失面积的关系如下 :

W = .0575( ΔA S − ΔA

)

(10)

0 S0

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式中 , w 为锈胀裂缝宽度 (mm), Δ A S 为钢筋截面损失面积 (mm 2 ), Δ A S 0 为混凝土表面出现第一

条裂缝的时刻的钢筋截面损失面积 (mm 2 )。 Δ A S 和 Δ A S 0 可分别由下式计算 :

ΔA 2αδ

α 2 S = ( d − δ )

(11)

ΔA

= A [1 − (1 − 0.001α (7.53 9.32c

/ d)

/ ) ]

(12)

S 0 S

+ d

式中 ,α 为不均匀锈蚀系数 , 对于均匀锈蚀的情况 ,α =2。 A S 为钢筋的初始横截面面积 (mm 2 )。δ

为钢筋锈蚀深度 ( μ m )。

将公式 (11) 和 (12) 代入公式 (10) 中 , 整理可得到表面裂缝宽度与钢筋锈蚀率之间关系的

表达式 , 可以发现 ,Vidal 的模型采用二次多项式形式对锈胀裂缝宽度与钢筋锈蚀率之间的关系进行

了拟合。

若采用二次多项式形式对本次试验数据进行锈胀裂缝宽度与钢筋锈蚀率关系拟合 , 结果如下 :

W s = 0.49458 + 1.4731ρ + 435ρ

( R 2 = 0. 89 ) (13)

按照公式 (13) 的拟合结果 , 钢筋锈蚀率为 0 时 , 表面裂缝宽度约为 0.49mm, 与实际情况不

符 , 是不合理的。

分析还发现 , 在公式 (13) 中 , ρ 2

的系数越小 , 拟合线越趋向于直线。随着 ρ 的系数的减小 ,

锈蚀率等于 0 时的裂缝宽度值也不断减小 , 越来越接近实际的情况。当 ρ 的系数等于 0 时 , 拟合线

退化成线性直线 , 即为本文中建立的线性模型情况。

3.0

2.5

2.0

W S

/mm

1.5

1.0

W s

/mm

1.5

1.0

0.5

0.0

0.00 0.02 0.04 0.06 0.08

0.0

0.00 0.02 0.04 0.06 0.08

图 11 二次多项式模型拟合结果

图 12 对数模型拟合结果

(2) 夏晋的模型

采用夏晋使用的对数模型进行拟合时 , 结果如下 :

W = ln(0.6255 + 114.56ρ)

( R 2 = 0. 70 ) (14)

对模型的决定系数 R 2 进行比较发现 , 本文采取的线性模型的决定系数优于对数模型 , 因此 , 线

性模型比较适合用来描述表面裂缝宽度的变化情况。

本节中 , 对锈胀裂缝的形态进行了观测研究 , 建立了环向裂缝总宽度以及保护层表面裂缝宽度

与钢筋锈蚀率的关系。但需要注意的是 , 由于试件种类有限 , 本试验只对钢筋直径为 16mm、混凝

土保护层厚度为 20mm 的情况进行了研究 , 因此 , 本文建立公式的主要目的是用来反映裂缝宽度线

性变化的规律 , 而对于公式中系数的取值 , 仍需针对不同的类型的试件分别进行考虑。

四、锈胀裂缝开展模型

从 3.1 的分析中可知 , 锈胀裂缝宽度的变化情况可用线性模型进行模拟 , 可以采用图 13 所示模

型来描述随着钢筋锈蚀裂缝扩展的情况。

对于能够开展到保护层表面的外裂裂缝 , 表面开裂前 , 靠近钢筋处的裂宽保持一致 , 只在裂缝

端部区域发生变化 , 在裂缝尖端处裂缝宽度等于 0。在表面开裂时刻 , 裂缝内外宽度相等。表面开

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裂以后 , 裂缝形状呈梯形 , 裂缝外部宽度大于内部宽度。随着锈蚀的增长 , 钢筋表面裂缝宽度增大 ,

内外宽度变化趋势越来越明显。

而对于两钢筋之间内裂裂缝 , 在裂缝贯通之前 , 裂缝形态与外裂裂缝相似 , 靠近钢筋处宽度保

持一致 , 尖端处宽度变为 0; 裂缝贯通时刻 , 裂缝宽度保持一致 , 之后 , 随着锈蚀的增大 , 裂缝宽

度不断增大 , 但裂缝各处的宽度基本保持一致。

混凝土

钢筋

(a) 表面开裂前 (b) 表面开裂时刻 (c) 表面开裂后

图 13 裂缝开展过程模型

五、结论

本文对人工气候环境下加速劣化的混凝土试块进行切片研究 , 观测了锈胀裂缝在不同半径位置

处对应的环向裂缝宽度 , 以及混凝土保护层表面的裂缝宽度 , 得到了如下结论 :

(1) 在混凝土保护层开裂之前 , 裂缝呈现内宽外窄的特点 , 靠近钢筋处的裂宽保持一致 , 只

在裂缝端部区域发生变化 , 在裂缝的尖端处裂宽为 0; 随着锈蚀的增长 , 裂缝长度不断扩展 , 钢筋

表面的裂宽也不断增大 ; 一旦裂缝开展到保护层表面 , 则内外宽度一致 ; 之后 , 随着锈蚀的不断发

展 , 一方面 , 钢筋表面的裂缝宽度不断增大 , 另一方面 , 外部裂宽开始大于内部裂缝 , 并且裂缝内

外宽度变化的梯度也越来越大。

(2) 混凝土保护层表面裂缝总宽度与钢筋锈蚀率呈线性关系 ; 使用经验公式进行表面锈裂预

测时 , 保护层表面开裂临界时刻的表面裂缝宽度应取为一定值 , 而不应取为 0, 建议取值为

0.05-0.1mm。

参考文献

[ 1 ] Alonso, C., Andrade, C., Rodriguez, J. and Diez, J. M. Factors controlling cracking of concrete affected by

reinforcement corrosion. Materials and Structures. 1998, 31: 435-441.

[ 2 ] Rodriguez, J., Ortega, L. M., Casal, J., and Diez, J. M. Corrosion of reinforcement and service life of concrete

structures. Proc., 7th Int. Conf. on Durability of Building Materials and Components, London, 1996, 1: 117-126.

[ 3 ] Vidal, T., Castel, A. and Francois. R. Analyzing crack width to predict corrosion in reinforced concrete. Cement and

Concrete Research, 2004, 34: 165-174.

[ 4 ] Zhang, R., Castel, A. and Francois, R. Concrete cover cracking with reinforcement corrosion of RC beam during

chloride-induced corrosion process. Cement and Concrete Research. 2010, 40: 415-425.

[ 5 ] 夏晋 . 锈蚀钢筋混凝土结构力学性能研究 . 杭州 : 浙江大学 , 2010.

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predictive models. Research Report No.275.05.2009.

[ 7 ] Molina, F.J., Alonso, C. and Andrade, C. Cover cracking as a function of bar corrosion: part 2-Numerical model.

Materials and Structures. 1993, 26(163): 535-548.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

构造柱与蒸压加气混凝土砌块组合墙体抗震性能的试验研究

1 1

凌沛春赵均

1,3

周炳章吴会阁苗启松

1 1

朱玉玉涂军

(1. 北京工业大学建筑工程学院 , 北京 100124;2. 北京市建筑设计研究院 , 北京 100045;

3. 石家庄经济学院工程学院 , 河北石家庄 050031)

摘要 : 本文通过对两片设置构造柱的蒸压加气混凝土砌块墙体足尺试件的拟静力水平加载试

验 , 研究了这种蒸压加气混凝土砌块组合墙的破坏过程与特征 , 实测了在水平荷载下墙体试件的滞

回曲线及承载力、变形能力等性能指标 , 并比较了洞口对墙体性能的影响。试验结果表明 , 两端设

置构造柱的蒸压加气混凝土砌块墙具有较高的抗震抗剪承载力和开裂后的承载能力储备 , 墙体开裂

前变形小 , 而后期有较好的极限变形能力 , 在反复循环加载下耗能能力较好。因此 , 这种组合墙体

能够满足地震区层数不多的砌体建筑的抗震要求。

关键词 : 蒸压加气混凝土砌体结构构造柱抗震性能

EXPERIMENTAL STUDY ON SEISMIC BEHAVIOR OF AUTOCLAVED AERATED

CONCRETE BLOCK COMPOSITE WALLS WITH STRUCTURAL COLUMNS

Peichun Ling 1 , Jun Zhao 1 , Bingzhang Zhou 2 , Huige Wu 1,3 , Qisong Miao 2 , Yuyu Zhu 1 , Jun Tu 1

1 College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China;

2 Beijing Institute of Architectural Design, Beijing 100045, China; 3 College of Engineering, Shijiazhuang

University of Economics, Shijiazhuang 050031, China

Abstract: Two full-scale specimens of autoclaved aerated concrete (AAC) composite walls with structural

column were tested under laterally pseudo-static loading to investigate their performance, including their failure

mode, seismic load-carrying capacity, deformation capacity and hysteretic behavior, etc, The effect of the wall’s

opening on its seismic behavior was also studied. It is shown that AAC composite walls with structural columns

have very small lateral deformation before cracking, good seismic shear capacity and reasonable safety margin

after cracking. They also have adequate energy-dissipation ability under repeated cyclic loading and

deformation capacity. So AAC composite walls with structural columns can satisfy seismic requirements for

multi-storey low-rise buildings in earthquake zones.

Keywords: Autoclaved aerated concrete, masonry structure, structural column, seismic behavior.

一、前言

随着粘土砖被逐步禁用 , 亟需有新的承重材料作为其替代物 , 用于多层砌体结构。蒸压加气混

凝土 , 作为有着保温、隔热吸声、耐火等良好物理性能的轻质材料 [1] , 在建筑中通常主要用作填充

墙、围护墙等非承重用途。近些年来国内外已开始研究将其作为承重材料 [2~4] , 形成新型结构体系 ,

基金项目 : 北京市建设委员会科技开发项目 “ 新农村低成本、低能耗墙体成套技术研究 ”

作者简介 : 凌沛春 (1983-), 男 , 山东临沂人 , 硕士研究生 , 从事混凝土结构和砌体结构抗震研究。

赵均 (1954-), 男 , 北京市人 , 教授 , 从事混凝土结构、砌体结构和结构抗震的教学、科研工作。

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以解决建造层数不多的砌体建筑的需要 , 实现保温节能与承重一体化。然而 , 蒸压加气混凝土自身

强度不高 , 加之砌筑墙体的整体性较差 , 如何确保其抗震安全 , 成为了这种墙体在地震区应用推广

的关键。为此 , 北京工业大学和北京市建筑设计研究院合作进行了这种墙体的抗震研究。

设置钢筋混凝土构造柱已被实践证明对于一般砌体结构抗震有重要作用 , 故本文在蒸压加气混

凝土砌块墙体中采用这种措施 , 形成组合墙体 , 通过拟静力试验 , 研究其抗震性能。考虑到以往试

验多采用真实墙体的缩尺模型试件 , 为使试验结果更贴近实际 , 本文进行的是足尺试件的大型试验。

二、试验概况

2.1 试件设计

本文试验共包括两片墙体 , 均为两端带构造柱的蒸压加气混凝土砌块墙 , 分别编号为 WJ-1 和

WJ-2。参考实际工程中的有关墙体尺寸 , 取试件的墙体长度和高度 ( 含构造柱和用以加载及模拟圈

梁的墙顶梁 ) 分别为 4230mm 和 2990mm, 墙体厚度 200mm。WJ-1 为无洞口的整体墙 ,WJ-2 为带洞

口的墙体 , 即在 WJ-1 基础上开设 1800mm×1200mm 的窗洞。为了模拟墙体底端的边界条件 , 底端采

用截面为 400mm×450mm 底梁与墙体连接。墙体采用蒸压砂加气混凝土砌块 ( 尺寸为

600mm×240mm×200mm) 砌筑 , 其强度等级为 A2.5; 砂浆采用加气混凝土砌块专用砌筑砂浆 , 其强

度等级为 M5.0; 构造柱和圈梁的混凝土强度设计等级均为 C20。构造柱截面尺寸为 200×200mm, 钢

筋为 4Φ8。试件的立面尺寸见图 1, 剖面尺寸及配筋见图 2。

(a)WJ-1

图 1 试件立面图

(b)WJ-2

图 2 试件剖面

图 3 加载装置示意图

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2.2 加载装置及加载方案

加载装置的竖向系统由龙门架、加载梁、分配梁和 2 个垂直千斤顶组成。水平荷载由支承于反

力墙的水平拉压千斤顶来施加。正式试验开始后 , 先在试件顶部施加竖向荷载 , 一次性加至每个千

斤顶 100kN, 并保持不变 ; 然后在试件顶部施加水平低周反复荷载。在加载过程中采用力和位移混

合控制 : 墙体开裂前 , 由水平力的进行控制 , 开裂后 , 由加载点处水平位移控制 ; 每级荷载反复循

环 1 次 ; 荷载下降至最大荷载的 85% 为试件破坏准则。试验加载装置见图 3。

2.3 量测方案

本文试验的主要量测内容有 : 墙顶竖向荷载 , 墙顶水平加载点处的水平荷载及位移 , 构造柱纵

筋及 WJ-2 窗洞下墙体水平钢筋的钢筋应变 , 底梁的水平位移和转动等。所有数据均采用 IMP 数据采

集系统由计算机全程监控采集。

此外 , 还用观测的方法 , 全程观测和记录在加载过程中试件的外部特征。

三、试验结果及分析

3.1 试件破坏过程及特征

1、试件 WJ-1

墙体试件在开裂之前刚度较大 , 侧移很小 , 实测荷载 - 位移曲线基本成直线 , 墙体处于弹性状

态。第 4 级加载时 , 墙体左右侧分别出现斜裂缝。此后 , 新裂缝不断出现 , 形成交叉斜裂缝 , 同时

构造柱水平裂缝陆续出现。墙体裂缝相互交叉、张开闭合、宽度增大 , 把砌块分割成许多小块 , 并

不断脱落 , 最终导致墙体中部逐渐被压碎 , 试件承载力下降。加载至第 17 循环 , 终止试验。

本试件墙体中间部位的裂缝数量多 , 间距密 , 破坏严重 ; 两侧构造柱从下到上出现大量贯通的

水平裂缝 , 且墙体四角处均有斜裂缝延伸到构造柱 ; 圈梁上除两条角部延伸的斜裂缝外 , 其他处基

本完好。整体破坏形状见图 4(a)。

(a)WJ-1

(b)WJ-2

图 4 试件正面破坏图

2、试件 WJ-2

墙体试件在开裂之前实测荷载 - 位移曲线基本呈线性变化 , 墙体基本处于弹性阶段。第 4 级加

载时 , 墙体左右两侧窗间墙分别出现第一条斜裂缝 , 随后发展为贯穿各自墙肢 , 在窗洞两侧的墙肢

形成了 “ 八 ” 形主裂缝。荷载达到极限荷载的约 90% 时 , 右窗间墙形成交叉形裂缝 , 随后 , 左窗间

墙形成交叉形裂缝。此后 , 随着加载级数的增加 , 窗间墙被相互交叉的裂缝将墙体分割、压碎 , 并

出现 “ 掉块 ” 现象 , 试件承载力下降。

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本试件由于有窗洞口的存在 , 墙体开裂荷载较小。从整体上看 , 窗间墙的裂缝多而密 , 破坏严

重 , 窗下墙裂缝稀而少 ; 构造柱沿高度出现多条的水平裂缝以及少数几条由墙体延伸而来的斜裂缝 ,

破坏不严重 ; 窗过梁位于洞口范围的部分出现两条竖向裂缝 , 但未贯穿到梁顶 , 破坏不明显 ; 圈梁

有两条墙体延伸的竖向裂缝 , 基本完好。整体破坏形状见图 4(b)。

从上述两片墙体的破坏过程和破坏形态可见 , 有无洞口墙体均在墙体中间高度形成交叉密集的

剪切斜裂缝 , 破坏严重 ; 专用砂浆粘结良好 , 沿水平灰缝出现的裂缝很少。同时 , 两片墙体的构造

柱均有多条水平裂缝 , 表明单片砌体墙的构造柱和圈梁构成弱框架 , 发挥了对墙体的约束作用 , 并

直接抗剪抗弯。

3.2 承载力及变形分析

各试件的开裂荷载、极限荷载、破坏荷载及其对应的位移、位移角的实测值如表 1 所示。表中 :

F ci 、F ui 、F di 分别为各试件开裂荷载、极限荷载和破坏荷载 , 均取正反向加载的平均值。△ ci 、△ ui 、

△ di 分别为各试件加载点处在开裂、极限荷载和破坏时的位移 , 均取正反向加载的平均值。

表 1 承载力、位移、位移角实测值

试件

开裂荷载时极限荷载时破坏荷载时

编号 F ci /kN △ ci /mm θ ci F ui /kN △ ui /mm θ ui F di /kN △ di /mm θ di

WJ-1 153.79 1.57 1/1815 184.96 5.64 1/504 157.77 14.00 1/203

WJ-2 75.45 1.45 1/1965 112.95 5.69 1/500 92.41 10.15 1/280

(1) 试件 WJ-2 开裂荷载和极限荷载分别为 WJ-1 开裂荷载和极限荷载的 49% 和 61%, 说明洞口

的存在降低了墙体的承载力。

(2)WJ-1、WJ-2 开裂荷载分别占各自极限荷载的 83%、67%, 其所占比例的大小反映了从墙

体裂缝出现宏观裂缝至达到最大承载力的过程长短。这说明墙体在开裂后 , 其承载力均有不同程度

的增长 , 但开设窗洞试件的开裂后承载力百分比提高较多 , 这是因为开洞墙体的构造柱所提供的抗

剪承载力占其总承载力比重大于不开洞的墙体。

(3) 试件位移角能反映构件在各阶段的变形能力。墙体开裂时位移角在 1/1800~1/2000, 极限

承载力时位移角在 1/500 左右 , 破坏时 , 位移角在 1/200~1/280 左右。说明两端设置构造柱的蒸压加

气混凝土砌块墙 , 在开裂前刚度较大 , 变形较小 , 可满足正常使用 ; 而在强震下达到极限荷载和破

坏荷载时 , 位移角较大 , 变形能力较好。

3.3 滞回曲线及骨架曲线分析

两试件在水平加载点处的荷载 - 位移滞回曲线及其骨架曲线分别见图 5 和图 6。

从图 5 可以看到 , 各试件滞回曲线具有一些共性特征 : 试件在开裂前 , 滞回曲线为基本重合的

狭长滞回环 , 滞回面积极小 , 试件处于弹性工作阶段。试件开裂后 , 随着墙体裂缝的发展 , 滞回环

面积增加 , 并开始向位移轴倾斜 , 试件进入非弹性工作阶段。试件达到极限荷载后 , 承载力出现下

降 , 而滞回环面积继续增大 , 并较快地向位移轴倾斜。

观察图 6 的骨架曲线 , 并结合对墙体破坏过程的观测 , 可以大致将其受力过程分为 3 个阶段 : 第

一阶段为从开始加载到墙体出现第一条裂缝前。试件基本上处于弹性阶段。第二阶段为从墙体出现

第一条裂缝到试件达到极限承载力。在此阶段 , 随着裂缝不断发展及新裂缝的陆续出现 , 墙体的刚

度不断下降 , 水平承载力继续提高 , 曲线明显弯曲 , 直到达到极限荷载。第三阶段为从极限荷载下

降至试件破坏。试件裂缝进一步发展 , 墙体斜向反复受拉受压 , 损伤不断加重 , 但由于构造柱的有

效约束作用及其直接提供的抗力 , 使承载力下降较为缓慢 , 试件表现出了良好的延性 , 有效地防止

了脆性破坏和倒塌。相比之下 , 试件 WJ-2 后期骨架曲线的下降速度比 WJ-1 快一些。

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3.4 滞回耗能分析

耗能是指结构或构件在地震作用下发生塑性变形、吸收能量的能力。滞回环所包含的面积反映

了该循环结构或构件的耗能情况。本文采用等效粘滞阻尼系数 h e 描述墙体在试验中所耗散能量的多

少。表 2 列出了各试件在开裂荷载循环、极限荷载循环及破坏荷载循环 ( 即下降到不低于 85% 极限荷

载时 ) 所确定的等效粘滞阻尼系数 h e 。

荷载 /kN

200

100

位移 /mm

-20 -10-100

0 10 20

-200

-300

荷载 /kN

200

100

位移 /mm

-20 -10-100

0 10 20

-200

(a)WJ-1

图 5 试件水平荷载 - 位移滞回曲线

(b)WJ-2

荷载 /kN

200

100

位移 /mm

-20 -10-100

0 10 20

-200

-300

荷载 /kN

200

100

位移 /mm

-20 -10-100

0 10 20

-200

(a)WJ-1

图 6 试件骨架曲线

(b)WJ-2

表 2 试件等效粘滞阻尼系数 h e

试件编号开裂荷载循环极限荷载循环破坏荷载循环

WJ-1 0.065 0.097 0.141

WJ-2 0.072 0.111 0.130

由表 2 可见 , 各试件从开裂状态经过极限状态直至破坏荷载状态 , 随着墙体变形的增加 , 其等

效粘滞阻尼系数都随之增大。这反映了随着加载过程的进行 , 墙体的裂缝不断发展 , 数量增加 , 裂

缝间的滑动摩擦以及蒸压加气混凝土砌块的不断压碎破坏 , 使其耗能不断增多。尤其到破坏时 , 两

个试件的等效粘滞阻尼系数 h e 都较大 , 其中不开洞试件 WJ-1 的耗能能力则更好一些 , 且与文献 [5] 中

构造柱设置形式相同的缩尺试件等效粘滞阻尼系数 h e 试验实测值较为接近。

四、结论

(1) 墙体破坏过程及破坏形态的特征是 , 先在墙体 ( 或窗间墙 ) 中部产生剪切斜裂缝 , 然后

形成交叉斜裂缝 , 墙体逐渐被裂缝分割成许多小块 , 并导致砌体不断被压碎、承载力下降而破坏。

(2) 墙体两端的构造柱能有效约束内部砌块砌体 , 增强墙体的整体性 , 防止砌体的滑移、错

位 , 减缓砌体酥碎现象的产生 , 加之其直接提供的抗力 , 显著提高组合墙体的水平承载力和极限变

形能力 , 避免墙体突然倒塌。

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(3) 设置构造柱的蒸压加气混凝土组合墙体 , 在开裂前变形较小 , 可满足正常使用 ; 而在强

震下达到极限荷载和破坏荷载时 , 可以有良好的变形能力。因此 , 这种组合墙体能够满足地震区层

数不多的砌体建筑的抗震要求。

(4) 设置构造柱的蒸压加气混凝土组合墙体在反复加载循环下耗能能力较好。

参考文献

[1] Narayanan, N. and Ramamurthy, K. Structure and Properties of Aerated Concrete: a review. Cement

and Concrete Composites, 2000, 22(5): 321-329.

[2] Trunk, B., Schober, G., Helbling, A. K. and Wittmann, F. H. et al. Fracture Mechanics Parameters of

Autoclaved Aerated Concrete. Cement and Concrete Research, 1999, 29: 855–859.

[3] 于敬海 , 苏志德 , 田淑明 , 等 . 蒸压砂加气混凝土承重墙抗震性能研究 . 低温建筑技术 , 2007,

(04): 38-40.

[4] 赵成文 , 刘雅楠 , 周培厚 , 等 . 不同砌筑方式蒸压加气混凝土砌块砌体抗压强度研究 . 新型建筑

材料 , 2010, (01): 32-35.

[5] 蒋伟 . 新型轻质混凝土承重砌体抗震性能试验研究 , 天津大学硕士学位论文 , 2010, 天津 .

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

SEISMIC RESISTANT DESIGN AND ANALYSIS OF

VERTICAL BOUNDARY ELEMENTS IN STEEL PLATE SHEAR WALLS

Keh-Chyuan Tsai 1 , Chao-Hsien Li 2 , Jing-Tang Chang 1 and Chih-Han Lin 2

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan.

2 National Center for Research on Earthquake Engineering, Taipei, Taiwan.

ABSTRACT

This paper describes the recent experimental researches on the steel plate shear wall (SPSW) at National Center

for Research on Earthquake Engineering (NCREE). In addition, the design implications learned from the test

results are presented. In 2007, the cyclic tests of four full-scale two-story narrow SPSWs confirmed that the

yielding of the first story column can be confined in the bottom of the column when the proposed capacity

design principle is followed. Test results also suggest that, when the column hinging is allowed at a location

slightly above the column base, the column size can be cost-effectively reduced without comprising the seismic

performance of the SPSW. In 2009, the cyclic test of a reduced scaled coupled SPSW (C-SPSW) substructure

was conducted at NCREE. The specimen was the 2/5-scaled substructure of the lowest two-and-half-story of a

6-story C-SPSW prototype building. The C-SPSW specimen consisted of two SPSWs connected together by the

coupling beams. In addition to a constant vertical force representing the gravity load effects, cyclic increasing

displacements and the associated overturning moments were applied at the boundary of the specimen using the

Multi-Axial Testing System (MATS) in NCREE. Test results suggested that the proposed design method, which

aims to limit the hinging of the bottom column within the bottom quarter column height, could be a choice of

design in the practice. The test results confirm that the effect of the coupling beams in reducing the axial forces

in the inner boundary columns. The relationship between the coupling beam rotation and the story drift is

explored based on the test data. Test results suggested that the cyclic responses of the C-SPSW specimen can be

satisfactorily predicted using the general purposes frame response software incorporating the strip model.

KEYWORDS

Steel plate shear wall, seismic design of steel structure, boundary element.

INTRODUCTION

Steel plate shear wall (SPSW) has seen increased usage in North America and Asia in recent years. An SPSW is

composed of a structural frame and infill steel plates. The beams and columns surrounding the infill plates are

named boundary beams and boundary columns, respectively. SPSW can effectively resist horizontal earthquake

forces by allowing the development of diagonal tension field action after the infill plates buckle in shear. The

input energy is then dissipated through the cyclic yielding of the infill plates in tension. The boundary elements

in an SPSW must to be designed to anchor the plastic tension field action of the infill plates. Recently, many

researches (Vian and Bruneau, 2005; Berman and Bruneau, 2008; Qu and Bruneau, 2010) focused on the

capacity designs for the boundary elements. In 2007, the objective of the cyclic tests of four 2-story narrow

SPSWs at National Center for Research on Earthquake Engineering (NCREE) was to verify the capacity design

method for the boundary column proposed by Tsai et al. (2010). The test results have confirmed the

effectiveness of Tsai et al.’s method in limiting the plastic hinge formation on the boundary column at the

column base.

The infilling the steel plate into the structural frame would conflict with the architectural demand for the

doorways. In a tall and slender SPSW, the significant overturning moments will result in high axial forces in the

boundary columns. Moreover, the tall SPSWs likely behave in the flexure-dominated deformation mode under

the lateral forces. It would cause the steel plates in the top stories of the wall to fail in developing the plastic

tension field action. The coupled steel plate shear wall (C-SPSW) is considered as a solution to the application

of SPSW in the high-rise buildings. In order to investigate the seismic behavior and design of C-SPSW, cyclic

test of a recued-scaled C-SPSW substructure was conducted at NCREE in 2009 using the brand-new

Multi-Axial Testing System (MATS) facility. In addition, a new design philosophy of boundary column was

examined based on the test results of the C-SPSW.

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CYCLIC TESTS OF FOUR TWO-STORY NARROW SPSWS

Column capacity design philosophy

The main objective of this experiment is to verify the effectiveness of the column capacity design proposed by

Tsai et al. (2010). The objective of the design method was to limit the hinging of the column to the column base.

Figure 1 illustrates the philosophy of the proposed method. The action on the boundary elements of an SPSW

subjected to lateral forces can be considered as the superposition of two parts. One part is only due to frame

sway (see Figure 1a-left) while the other is due to panel forces (see Figure 1a-middle). For properly portioned

SPSWs, the infill plates would develop the plastic tension field prior to the yielding of the columns. The yield

panel forces acting on the boundary elements can be considered as distributed loads with a tension field angle

α (AISC, 2005a) from the vertical. The horizontal yield panel force acting on the 1 st story columns is denoted as

ω ch1 .

The top and bottom halves of Figure 1a demonstrate two types of the bending moment distribution patterns on

the compressed column. The bending moment from the frame sway action can be assumed as linearly

distributed (Figure 1a-left). The quadratic- distributed moment (Figure 1a-middle) can be estimated by assuming

that the column has fixed ends and the column is subjected to a distributed load ω ch1 . If the column strength is

sufficient to resist enough swaying moment, as shown in the right top of Figure 1a, the maximum superposed

moment will locate at the bottom end, which implies that the plastic hinge will form at the bottom end (Figure

1b-top). Tsai et al. (2010) derived the equation for the column moment demand based on this kind of moment

distribution (right top of Figure 1a). If the column is smaller, as shown in the bottom half of Figure 1, the plastic

hinge would form above the bottom end.

Experimental program

Figure 1 Philosophies of the design methods for the bottom column of SPSW

Four 2-story 2.14-meter wide by 6.5-meter tall narrow SPSWs were cyclically tested to a 5% rad. roof drift at

NCREE. All four specimens had the same center-to-center line dimensions. All the infill steel plates were 2.6

mm thick low yield strength (LYS) steel with a measured yield stress, f yp = 195MPa. All boundary elements

were made of A572GR50 steel. The key differences in the specimens were the size of the boundary elements

and the use of restrainers. The aspect ratio of the infill plates ranged from 0.62 to 0.64, which was less than the

lower limit (0.8) specified by 2005 AISC seismic provisions (AISC 2005a). Figure 2 shows the test setup.

Increasing cyclic horizontal displacements were applied on the top level of the specimens through two actuators.

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Figure 2 Test setup of cyclic tests of four 2-story narrow SPSWs

RBS

Figure 3 Details of Specimen N

Figure 4 Details of Specimen RS

This paper focuses on the cyclic performances of three of the four specimens. The details of the specimens and

the sizes of the boundary elements are provided in Figures 3 and 4. Specimens N and S are typical unstiffened

SPSWs; Specimen RS is a restrained SPSW (R-SPSW) (Tsai et al., 2010). The boundary elements are classified

into four categories and abbreviated as: the top beam (TB), the middle beam (MB), the bottom beam (BB) and

the column (C). All beam-to-column connections are welded moment connections. Specimen N was normal,

named from the fact that the column complied with the proposed capacity design method (Tsai et al., 2010). The

boundary elements in Specimen S were smaller than those in Specimen N. The column in Specimen S did not

satisfy the proposed capacity design method. Specimen RS was identical to Specimen S except it adopted two

pairs of restrainers for each story. As the restrainers can reduce the horizontal panel force effect on the boundary

column, the boundary columns in Specimen RS conformed to the proposed capacity design method (Tsai et al.,

2010). Figure 5 shows the predicted plastic mechanism of the three specimens based on the proposed capacity

design method. As the boundary column strength in Specimen S was insufficient, it was expected that a plastic

hinge would form within the column height.

Experimental results and learned implications

Figure 6 shows the lateral force versus story drift relationships of Specimens N, RS and S. In general, all

specimens remained elastic before a drift of 0.3%. The boundary elements slightly yielded at this stage. Plastic

hinges on the boundary elements were observed at the boundary element ends at a roof drift level of about 1.0%

radian. As the roof drifts exceeded 2.5% radians, flange or web local buckling occurred gradually at the plastic

hinge zones. Nevertheless, the load-carrying capacity did not decrease significantly.

Figure 7 shows the overall views of the specimens after the tests. Based on the flaking of the whitewashes, the

locations of the plastic hinges agree with the predictions shown in Figure 5. As shown in Figure 8, the plastic

hinges developed at the first story column bottom ends can be recognized from the flaking of the whitewashes.

Two types of plastic hinges can be observed in the columns. One is the plastic hinge developed when the

column is compressed. This kind of plastic hinge can be observed on the column web shown in the photo

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(Figure 8c) as a ring shape. This is due to the occurrence of web local buckling at the center of the ring. The

other type of plastic zone develops when the column is tensioned. In this case, uniform web yielding would be

evident as shown in the photo (Figure 8c) at the column bottom end. As there is no plastic hinge zone forming

within the column height for Specimens N and RS, these two types of plastic zones overlapped at the column

bottom ends (Figures 8a and 8b). On the other hand, in Specimen S, the ring-shape zone was evidently above

the column bottom end where uniform yielding zone developed (Figure 8c). The distance from the bottom end

to the ring center is about two times the column depth. It clearly indicated that, in Specimen S, the plastic hinges

formed away from the bottom ends when the columns were compressed.

Test results confirmed that the proposed capacity design is effective in limiting the plastic hinge formation to the

columnbase. However, the cyclic responses shown in Figure 5 indicate that Specimen S possessed satisfactory

ductility and load carrying capacity. This suggested that the cyclic performance of an SPSW might not be

significantly deteriorated when it has plastic hinges slightly above the column bases. To achieve a more

economical SPSW, a slight under-design for the column as illustrated in the case of Specimen S should be

acceptable.

Story Shear (kN)

Base Shear (kN)

1500

1000

500

-500

-1000

-1500

1500

1000

500

-500

-1000

-1500

1500

1000

500

-500

-1000

-1500

Figure 5 Predicated plastic mechanism

-6 -4 -2 0 2 4 6

Story Drift (% Rad.)

-6 -4 -2 0 2 4 6

Roof Drift (% Rad.)

-6 -4 -2 0 2 4 6

Story Drift (% Rad.)

-6 -4 -2 0 2 4 6

Roof Drift (% Rad.)

Figure 6 F-D relationships

-6 -4 -2 0 2 4 6

Story Drift (% Rad.)

-6 -4 -2 0 2 4 6

Roof Drift (% Rad.)

CYCLIC TEST OF A REDUCED-SCALE C-SPSW SUBSTRUCTURE

Specimen design

(a) Specimen N (b) Specimen RS (c) Specimen S

Figure 7 Specimens after tests

(a) Specimen N (b) Specimen RS (c) Specimen S

Figure 8 Plastic zones on the columns

A prototype 6-story coupled steel plate shear wall (C-SPSW) building (see Figure 9) was designed based on the

model U.S. standards (AISC, 2005a and 2005b). The site is located at the east zone of the Chiayi City in Taiwan.

A 2-D strip model (Thorburn et al., 1983) for the C-SPSW was developed using the structural analysis program

SAP2000 to check that the infill plates and the other frame elements remain elastic under the code prescribed

LRFD load combinations (AISC 2005b). The seismic load was calculated according to Taiwan seismic building

code (ABRI, 2002).

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9 10 9

Figure 9 (a) Floor plan and (b) elevation of the prototype C-SPSW building

The bottom boundary column of the prototype 6-story C-SPSW was selected based on a new column capacity

design philosophy (Chang, 2009): the plastic hinge on the bottom column is allowed to form above the bottom

end but shall be limited within the bottom quarter column height as shown in the bottom half of Figure 1. This

design philosophy would result in a more economical design results than the previously proposed design method

(Tsai et al., 2010), which limits the plastic hinge formation at the column base. The coupling beams of the

6-story C-SPSW prototype were designed as shear links. By extending the current provisions (AISC 2005a) for

the link beam of the eccentrically braced frame (EBF) to the design of the coupling beams, the coupling beam

section was selected in accordance with the relationship e < 1.6M p /V p , where M p and V p are the plastic flexural

and shear strength of the coupling beam, respectively.

Using the 6-story C-SPSW prototype as a basis, a 40% scale specimen was constructed as the bottom 10.5-m

high substructure of the original C-SPSW. The region of the substructure contains the lowest two and half

stories of the original structure (shown in Figure 10-left). The infill plates were 3.5-mm thick low yield strength

(LYS) steel with a measured yield stress f yp = 220MPa. All the boundary elements and coupling beams were

made of A572GR50 steel. The design results of the test specimen are shown in Figure 11. The Reduced Beam

Section (RBS) was employed at the ends of the boundary beams. The spacing of the stiffener plates in the

coupling beam webs was determined based on the rotational demands of the coupling beams predicted from a

finite element analysis. A shell model of the 2/5-scaled C-SPSW specimens was developed using the

commercially available software ABAQUS/Standard. A nonlinear pushover analysis up to a 5% rad. roof drift

was conducted to obtain the rotational demands of the coupling wall.

Figure 10 Test setup of cyclic test of reduced-scaled

C-SPSW substructure

Figure 11 Schematic of the C-SPSW

substructure specimen

Experimental Program

The specimen was tested using the Multi-Axial Testing System (MATS) at the NCREE. In order to simulate the

effects of the upper substructure acting on the bottom two-and-half story substructure, as illustrated in Figure 10,

the specimen was subjected to the cyclic lateral forces, F H , the cyclic overturning moments, M OT , and a constant

1400 kN vertical force, P V , which represents the gravity load. The specimen was set upside down in the MATS.

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The base beam of the specimen was mounted on the cross beam of MATS. The top boundary of each SPSW was

connected with a transfer beam. The column top ends were pin-connected with the transfer beam ends; and the

top steel plate was welded to the transfer beam using the fishplate connection. The mid-span of the transfer

beam was pin-supported on the platen. A lateral support system was constructed on the platen and the reaction

frame (A-frame) of MATS.

The actuator system applied forces on the platen. Two horizontal actuators were employed to apply the lateral

displacement on the specimen through the platen. Two cycles of 0.1%, 0.2%, 0.3%, 0.5%, 0.75%, 1.0%, 1.5%,

2.0%, 3.0%, 4.0% and 5.0% rad. roof drifts were imposed sequentially on the specimen. The vertical actuators

can be classified into 3 categories: (1) two rows of vertical pancake type actuators pushing the bottom of the

platen. The difference in the applied forces between the two rows of actuators induced the overturning moment

effects on the platen; (2) two built-in hold down actuators which were mounted on the A-frame and pushing the

top face of the platen; and (3) two additional actuators which were anchored between the platen and the cross

beam. The two additional actuators provided a constant 1960 kN vertical compressive force on the platen. It

should be noticed that the pancake type actuators must be always in contact with the platen in compression. The

hold down actuators and additional actuators provided vertical compressive forces on the platen to insure the six

pancake type actuators were always in compression.

The resultant forces of these vertical actuators applied a constant vertical force (P V = 1400 kN). The relationship

between the resultant moment M OT of these vertical actuators and the lateral force F H applied by horizontal

actuators is: M OT = F H × (2.51 m). The relationship is calculated based on the inverted triangle vertical

distribution of the lateral forces, which was determined from the current Taiwan’s seismic code (ABRI, 2002).

Test results and design implications

Figure 12 illustrates the force versus displacement relationship of the specimen. The C-SPSW specimen

exhibited an excellently ductile behavior and dissipated significant amount of hysteresis energy during the

cyclic loading test. The lateral and vertical load-carrying capacities did not notably deteriorate when the overall

drift of the specimen reached 5% rad. An unanticipated fracture occurred at the early stage of the test. The weld

between the fishplate and the transfer beam at the 3 rd story of the southern SPSW were totally torn off at 1% rad.

roof drift. However, that fracture did not significantly deteriorate the load-carrying capacity of the specimen.

Thus, the test went on until the two cycles of 5% rad. roof drift had been completed.

story shear (kN)

1500

1000

500

-500

-1000

-1500

-6 -4 -2 0 2 4 6

total drift (% rad.)

-6 -4 -2 0 2 4 6

3F drift (% rad.)

-6 -4 -2 0 2 4 6

2F drift (% rad.)

Figure 12 Force versus displacement relationships

Test

ANA

-6 -4 -2 0 2 4 6

1F drift (% rad.)

After the test, as shown in Figure 13a, the flaking of the whitewashes on the specimen showed that the specimen

had developed the plastic mechanism. The coupling beams developed shear plastic hinges (Figure 13b). The

flexural plastic hinges formed at the ends of the boundary beams (Figure 13c) and near the column bases

(Figures 13d, 13e and 13f).

Figure 13d shows the flaking of the whitewashes on the bottom (first story) columns of the northern SPSW after

the test. As shown in Figure 13e, the plastic zone on the outer bottom column spread over a range from 40 to

100mm measured from the column base. On the other hand, the plastic zone on the inner bottom column

concentrated at the column base (Figure 13f). The plastic zone on the outer column was wider than that on the

inner column. In addition, the plastic zone on the outer column was located at a higher position than that on the

inner column. The difference in the distribution of the plastic zone between the outer and inner columns could

be attributed to that the axial force in the inner column was smaller than that in the outer column because of the

coupling beam effect illustrated in Figure 14. Hence, the reduced flexural strength of the outer column was

smaller than that of the inner column.

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Figure 13 Deformed shapes of the specimen after test: (a) overall; (b) the 3 rd floor coupling beam; (c) the 2 nd

floor boundary beam; (d) the 1 st story of the southern SPSW; the plastic zones on the (e) outer and (f) inner

bottom columns

Figure 15 shows the relative deflections of the first story columns in northern SPSW when the lateral forces

approached zero during the first cycle of the various drift levels. As shown in the top right of Figure 15, the

relative deflection is defined as the difference between the absolute deflection and the reference line, which is

the straight line from the top end to the bottom end of the column. The relative deflection can be utilized to

estimate the inward flexural deformation of the column induced by the horizontal tension field forces of the

infill plate. The column deflections measured when the lateral forces approaches zero in the various drift levels

could represent the residual deflection after the specimen has experienced such level of drift. As shown in

Figure 15, it can be found that residual “pull-in” deformations on the outer column were much larger than those

on the inner column. This should be attributed to that the axial forces in the outer column were much higher,

thus, the flexural strength of the outer column is smaller due to the axial-flexural interaction effect. However,

form the Figure 15, it can be found that the maximum residual deflection of the outer column was about 10 mm

(= h 1 /200) after the specimen had suffered a 2.1% rad. first story drift. The 1/200 of the story height could serve

as the deflection index limiting the development of large secondary forces in the columns (Ellingwood, 2003).

Moreover, the past test (Tsai et al., 2006) has shown that peak story drift of a well-designed SPSW specimen

during the collapse prevention level (2/50 hazard level) earthquake was about 2.0 to 2.5% rad. The test result

suggested that, even if the plastic zones on the bottom columns had spread over the bottom half of the column

height, the residual “pull-in” column deflection would not cause a significant secondary effect

Figure 16 shows the shear deformation of the second floor (2F) coupling beam, γ 2CB , versus the average drift,

θ 1+2 , of the 1 st and 2 nd story relationship. The slope of the line composed of the data at the various level peak

drifts had a sudden change as the story drift reached about 0.75% rad. It indicates that the plastic shear hinge

developed at this drift level. From the data at the peak drifts after the shear yielding, it can be found that the

plastic shear deformation of coupling beam is close to the inelastic story drift. It suggested that, for seismic

design purpose, the plastic rotational demand of the coupling beam at the bottom of a C-SPSW can be estimated

as the design story drift. The design of the stiffeners in the coupling beam could be based on the estimated

rotational demand.

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2.0

Northern SPSW

outer 1F col. inner 1F col.

column height (m)

1.5

1.0

0.5

0.0

0 10 20 -20 -10 0

residual relative deflection (mm)

(in the 1 st cycle)

near the zero force pts after

the [peak 1F story drifts ]

in the 1 st cycle

of the various drift levels

[0.67, -0.64 ]

[1.36, -1.35 ]

[2.11, -2.14 ]

[2.85, -2.91 ]

[3.44, -3.56 ]

Figure 14 Effects of the coupling beam on

reducing the axial forces in the inner boundary

columns

Figure 15 Residual relative deflections of the bottom

columns in the northern SPSW in the various drift

levels

2F coupling beam

shear deformation, γ 2CB

(% rad.)

-1

-2

-3

-4

-5

1.2

peak (elastic)

0.8 pos. peak (plastic)

0.6 neg. peak (plastic)

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5

-1 -0.5 0 0.5 1

average drift of the 1 st and 2 nd stories, θ 1+2

(% rad.)

RL of the positive

plastic drift peaks

= 1.14 1+2 − 0.398

R 2 = 0.996

RL of the negative

plastic drift peaks

= 1.06 1+2 + 0.343

R 2 = 0.998

elastic cyclic response

plastic cyclic response

regression line (elastic)

regression line (plastic)

Figure 16 2F coupling beam shear deformation (γ 2CB ) versus the average drift (θ 1+2 ) of the 1 st and 2 nd story

relationship

Figure 17 Strip model for the C-SPSW

specimen

2F coupling beam

shear deformation, γ 2CB

(% rad.)

ANA

Test

0 1 2 3 4

average drift of the 1 st and 2 nd stories

θ 1+2

(% rad.)

Figure 18 Comparison of experimental and analytical

γ 2CB -θ 1+2 relationships

Analytical Simulation

In this research, as shown in figure 17, a strip model (Shishkin et al., 2010) for the C-SPSW specimen was

constructed using the nonlinear frame analysis program PISA3D (Lin et al. 2008). The tension field actions of

the infill plates at the 1 st , 2 nd and 3 rd story were represented by twelve, eleven and eight truss elements,

respectively. The orientation of the tensile strips was determined from the tension field angle prescribed by the

modern U.S. building codes (AISC, 2005a). In addition, a diagonal compression strut (Shishkin et al., 2010)

was employed to represent the compressive stiffness and strength of each infill plate. The columns were

modeled using the PISA3D Fiber Beam-Column element. The boundary beams, coupling beams and the transfer

beams were represented by using the PISA3D Beam-Column element. The reduced beam section (RBS) regions

were simplified using a prismatic beam segment with the same cross sectional properties as those of the most

reduced section. All beam-to-column joints were modeled using rigid end offsets with a rotational spring to

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epresenting the flexibility of the panel zone. Special constraints were set to represent the pin-connections

between the transfer beams and the boundary columns. The platen in the MATS was modeled by a rigid frame

consisting of a rigid beam and two rigid columns representing the half height of the platen. Except the rigid

frame presenting the platen was modeled using elastic material, the material properties of the elements

representing the infill plates, beams and columns were bilinear, fitting the coupon test results. Nonlinear

pushover analysis up to 5% rad. roof drift was carried out on the strip model. The constant 1400 kN vertical

force P V , the lateral displacement control with a load pattern composing of a horizontal force F H and a

overturning moment, M OT , were applied on the node at the mid-span of the rigid beam representing the platen.

Figure 12 shows that the analytical pushover results agree well with the experimental hysteresis loop. Figure 18

shows that the analytical relationship between the shear deformation of the 2F coupling beam and the average

story drift of the 1 st and 2 nd stories quite match the test results.

CONCLUSIONS

The test results of the Specimen S in 2007 and the reduced-scaled C-SPSW specimen in 2009 showed that the

plastic hinge formation slightly above the column base would not result in significant deterioration of strength

and ductility. This suggests that the proposed column capacity design, which aims at limiting the plastic hinge

on the bottom column within bottom quarter column height, could be a choice of design in the practice. The test

results suggested that the rotational demand of the coupling beam at the lowest level of a C-SPSW can be

estimated as the design story drift. The strip model can predict the overall and local responses of the C-SPSW

specimen very well. Further studies on the seismic design and behaviours of the C-SPSW can be conducted

using strip models.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support provided by the Taiwan National Science Council and

NCREE. Technical supported provided by NCREE is very much appreciated.

REFERENCES

American Institute of Steel Construction (AISC). (2005a). Seismic Provisions for Structural Steel, Chicago,

Illinois.

American Institute of Steel Construction (AISC). (2005b). Specification for Structural Steel Buildings, Chicago,

Illinois.

Architectural Building Research Institute (ABRI). (2002). Seismic Force Requirements for Building Structures,

Taipei, Taiwan.

Berman, J.W and Bruneau, M. (2008). “Capacity Design of Vertical Boundary Elements in Steel Plate Shear

Walls”, Engrg. J., AISC, 15(1), 55-71.

Chang, J.T. (2009). “Seismic Responses of Multi Story Coupled Steel Plate Shear Walls System”, Master Thesis,

Dept. of Civil Engrg., National Taiwan Univ., Taipei, Taiwan. (In Chinese)

Ellingwood, B. (2003). “Specification for Structural Steel Buildings”, Engineering Journal Article. AISC.

Lin, B.Z., Chuang, M.C. and Tsai, K.C. (2009). “Object-oriented Development and Application of a Nonlinear

Structural Analysis Framework”, Advances in Engineering Software, 40(1), 66-82.

Qu, B. and Bruneau, M. (2010). “Behavior of Vertical Boundary Elements in Steel Plate Shear Walls”, Engrg. J.,

AISC, 17(2), 109-122.

Shishkin, J.J., River, R.G., Grondin, G.Y., (2009). “Analysis of Steel Plate Shear Walls Using the Modified

Strip Model”, J. of Struct. Engrg., ASCE, 135(11), 1357-1366.

Thorburn, L.J., Kulak, G.L. and Montgomery C.J. (1983). “Analysis of Steel Plate Shear Walls”, Structural

Engineering Report No. 107, Dept. of Civil Engrg., Univ. of Alberta, Edmonton, Alberta, Canada.

Tsai, K.C., Li, C.H., Lin, C.H., Tsai, C.Y. and Yu, Y.J. (2010). “Cyclic Tests of Four Two-Story Narrow Steel

Plate Shear Walls. Part 1: Analytical Studies and Specimen Design”, Earthquake Engrg. & Struct. Dyn.,

39(7), 775-799.

Tsai, K.C., Lin, C.H., Lin, Y.C., Hsieh, W.D and Qu, B. (2006). “Substructural Hybrid Tests of A Full Scale

2-Story Steel Plate Shear Wall”, Technical Report NCREE-06-017, National Center for Research on

Earthquake Engineering, Taipei, Taiwan. (In Chinese)

Vian, D. and Bruneau, M. (2005) “Steel Plate Shear Walls for Seismic Design and Retrofit of Building

Structures”, Technical Report MCEER-05-0010, Multidisciplinary Center for Earthquake Engineering

Research, Buffalo, New York.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

APPLICATION OF SINGULAR SPECTRUM ANALYSIS TO HEALTH

MONITORING OF BRIDGE STRUCTURE

Shu-Hsien Chao 1 , Chin-Hsiung Loh 2 , Jiang-Huang Weng 1 , Pei-Yang Lin 3 , Chia-Hui Chen 4

Post-Doctoral Fellow, Department of Civil Engineering,

National Taiwan University, Taipei, Taiwan, Email: d92521005@ntu.edu.tw

Professor, Department of Civil Engineering,

National Taiwan University, Taipei, Taiwan, Email: lohc0220@ntu.edu.tw

Research Fellow, National Center for Earthquake Engineering, Taipei, Taiwan.

Research Assistant, Department of Civil Engineering,

National Taiwan University, Taipei, Taiwan

ABSTRACT

Singular spectrum analysis (SSA) is a novel technique and has proven to be a powerful tool for time data series

analysis. It takes the singular value decomposition of Hankel matrix embedded by the analyzed time data series

and decomposes the data to several simple, independent and identifiable components. Basic capability of SSA

includes finding trend, extracting periodic component, smoothing time series and de-noising of time series. It

has already been widely applied to process climatic, meteorological, geophysical and economic data. In this

study, singular spectrum analysis is applied to process velocity data from the scouring test of a four span bridge

model. It is shown that coupling level between 1 st and 2 nd singular values in singular spectrum analysis can be

used to monitor scouring extent, scouring location and settlement occurrence of the bridge pier. It provides a

useful feature of time series and a powerful indicator for structural health monitoring and early warning. The

analysis algorithm is simple and suitable for real structural health monitoring (SHM) application.

KEYWORDS

Singular spectrum analysis, bridge, scouring, structural health monitoring.

INTRODUCTION

Every year several typhoons strike Taiwan from June to September and bring heavy rains and strong wind.

Because of special terrain in Taiwan, water level of river rises suddenly and flow rate is very high during

typhoon strike. The heavy flow threats bridge safety. In the past, there were many bridge collapses during

typhoon strike. For example, on August, 2009, Typhoon Morakot caused more than 10 bridges broken in

southern Taiwan and caused many economic and life loss. For this reason, it is very important to develop an

early warning system for bridge to provide information and send early warning message before it collapses.

A main reason for bridge collapse is scouring. Scouring results from erosive action of flowing water, which

excavating and carrying away material from bed and banks of stream. This phenomenon will result in bearing

capacity loss of bridge pier foundation. Once foundation bearing capacity is insufficient to sustain bridge weight,

bridge pier may displace and bridge deck may collapse. So scouring extent of bridge pier foundation is

important information about structural safety and should be monitored directly during heavy flow. However,

scouring extent monitoring of bridge pier foundation is very difficult because sensor should be installed under

soil around bridge foundation, and they should be protected appropriately to ensure their functionality during

heavy flow, which may bring rock and sand. These make sensor set-up and maintenance difficult. Another

indirect way is to monitor vibration response of bridge deck. Sensor only needs to be installed on bridge deck,

and it can be maintained easily and regularly. Foundation capacity loss causes structural dynamic property

change, which this change can be monitored from structural vibration signal. So vibration based approach seems

to be a solution to monitor bridge pier foundation scouring extent and structural safety.

In the past, many vibration based method were developed to monitor structural safety. A common structural

monitoring approach is modal analysis, which using system identification technique to identify structural modal

frequency, modal damping ratio and mode shape from vibration data [1]. Damage can be detected by modal

frequency change [2], and damage location can be detected from mode shape information. Modal analysis is a

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asic and efficient tool for structural health monitoring, but some drawbacks make it not easy to be applied to

real monitoring system. A main reason is that system identification computation work is heavy and may produce

so called ‘redundant mode’, and analysis result depends on several parameters of system identification algorithm.

So the modal frequency, especially the modal damping, is difficult to be identified accurately. Besides, modal

frequency is not sensitive to the damage especially for large and complex structural system. For these reasons,

many other research used non-model based or signal based analysis approach to monitor structural condition and

detect damage, such as using auto-regressive model (AR) or auto-regressive moving average model (ARMA) [3,

4].

Singular spectrum analysis (SSA) is a novel technique of time series analysis. It is a model-free and

non-parametric analysis method. The birth of SSA is usually associated with publication of several papers by

Broomhead and King (1986) and Broomhead et al. (1987) [5~7]. Since then, the technique has attracted a lot of

attention and applied to several areas. SSA procedure mainly includes two stages: decomposition and

reconstruction. It takes singular value decomposition of trajectory (or Hankel) matrix embedded by analyzed

time series and decomposes it into several simple, independent and identifiable components. SSA algorithm is

simple but a lot of probabilistic and statistical elements are employed in it, such as principal component analysis.

Basic capabilities of SSA include finding trend, extracting periodic component, smoothing time series and

de-noising of time series. It has become a standard tool in analysis of climatic, meteorological, geophysical and

economic time series. Recently some research shown similarity between the analysis results of SSA, Fast

Fourier Transform (FFT) and Wavelet transform (WT) [8, 9]. However, it remains a unique analysis tool and a

powerful technique to analyze and realize data structure. Until to now, issue about SSA application to structural

health monitoring is rarely discussed. Simple algorithm, model free and successful application in many different

areas are main reasons for why we study singular spectrum analysis and apply it to structural health monitoring

issue.

In this study, singular spectrum analysis is used to analyze vibration data derived from scouring test of a four

span bridge model. In next section, set-up and measurement of the bridge scouring test will be introduced. Then

basic SSA algorithm and analysis result of test data will be discussed. We will show that singular spectrum

analysis can also be applied successfully to structural health monitoring issue.

TEST SET-UP AND BASIC ANALYIS WORK

Figure 1 Sketch of the bridge test model and set-up

In order to realize scouring effect on bridge structure dynamic characteristics, a four span bridge model with

simply supported girder on each pier was constructed to across a watercourse of width about 4.3 meters in the

hydraulic lab. The bridge pier foundation is a hollow transparent acrylic tube with 1.5cm thickness intensified

by steel plate on top and bottom of the tube and steel bar inside the tube. The total weight of each pier is about

25kg. The bridge deck is a steel plate with thickness 2cm. Two bridge decks of middle span are 47×100cm. Two

side spans are 47×125cm. Their weight is 75kg and 100kg respectively. Simple rubber spacer with thickness

2mm and diameter 1.5cm is used as the bridge bearing between deck and pier. Sketch and dimension of this

bridge model are shown in Figure 1. The bridge pier foundations are embedded in sand with depth of 30 cm. To

focus major scouring phenomenon on one single bridge pier, major running stream water was guided and

focuses scouring effect mainly on pier #3. Photos of bridge test are shown in Figure 2(a). Total run time of this

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test is 180 minutes. Water flows to the bridge pier at 10 minutes after test starts, and flow condition is stable

until 30 minutes after test starts. After 130 minutes, bridge pier #3 gradually displaces and descends resulting

from scouring around pier #3 foundation. Figure 2(b) shows picture of the bridge final state with descending

pier and deck.

( a ) During the test ( b ) After the test

Figure 2 Pictures of bridge model during the test and after the test

There are three monitoring targets in this experiment. First is imbedded depth of each pier. Two cameras were

installed inside each bridge pier to observe scouring phenomena and imbedded depth. Figure 3 shows the

imbedded depth time history of each bridge pier from camera video image. Another monitoring target is

dynamic response of the bridge. 12 velocity sensors are installed on the bridge deck to collect the bridge

vibration signal in transverse direction (along stream line). VSE-15D sensor is used which is a servo velocity

meter produced by Tokyo Sokushin Co., Ltd. This sensor is very sensitive to detect low level vibration motion

and its linear range (0.2Hz~70Hz) is suitable for SHM applications. It can be used to measure acceleration or

velocity response. In this test, velocity response data of the bridge during scouring process are collected. Figure

4(a) shows velocity time history collected from sensor at node #9. Third monitoring target is displacement of

bridge pier #3. A Laser displacement sensor was installed to measure vertical displacement of pier #3 during the

test. Figure 4(b) shows displacement response time history at top of pier #3. Data acquisition system collected

all velocity and displacement responses of the bridge model from velocity and displacement sensors with

sampling rate 200 Hz.

Imbedded Depth (cm)

Pier 1

Pier 2

Pier 3

50 100 150

Time (min)

Figure 3 Imbedded depth of each pier derived from camera image

In real situation, scouring depth extent of real bridge pier foundation and settlement of bridge pier are very

difficult to be monitored. In contrast, vibration response of bridge deck can be monitored simply by installing

sensor on bridge deck. So objective of analysis work of this test will focus on how to detect, quantify and locate

damage from velocity response measurement before bridge collapses. In this study, analysis work will focus on

three issues:

1. How to detect scouring extent?

2. How to detect scouring location?

3. How to detect settlement occurrence of bridge pier before bridge collapse?

In the next section, basic algorithm of singular spectrum analysis will be introduced. Then it will be applied to

analyze velocity response data of this test. It will be shown that it can be used to solve three issues listed above.

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0.5

Node 9

Pier 3

Velocity (cm/s)

Vertical Displacement (cm)

-0.5

50 100 150

Time (min)

-2

50 100 150

Time (min)

( a ) Horizontal Velocity response at Node 9 ( b ) Vertical displacement at top of Pier 3

Figure 4 Velocity data and displacement data

SINGULAR SPECTRUM ANALYSIS

This section briefly describes calculation and properties of basic SSA, mainly based on the book by Golyandina

et al. (2001) [10]. Basic SSA consists four steps: embedding, singular value decomposition, grouping and

diagonal averaging. First two steps are called decomposition stage, and final two steps are called reconstruction

stages.

1 st Step: Embedding

Consider a real-valued discrete time series x[n] with size N (n = 1 ~ N) where x[n] = x(nΔt) and Δt is sample

time interval. Let L be an integer and 1 < L < N. Then we embed L-trajectory matrix of the time series x as:

⎡ x[1]

x[2]

L x[

⎤

⎢

⎥

⎢

x[2]

x[3]

L x[

K + 1]

X =

⎥

(1)

⎢ M M O M ⎥

⎢

⎥

⎣x[

L + 1] L x[

⎦

Where K = N – L + 1. Apparently embedding procedure maps a time series with length N to L time delay series

with length K.

2 nd Step: Singular Value Decomposition

Second step of SSA is singular value decomposition (SVD) of trajectory matrix X as:

X = USV

(2)

Where U is an L×L unitary matrix, S is an L×K diagonal matrix with nonnegative real numbers on diagonal, and

V* (the conjugate transpose of V) is an K×K real or complex unitary matrix. Diagonal entries of S are known as

singular values of X. L columns of U and K columns of V are called left singular vectors and right singular

vectors of X, respectively. We denote U = [u 1 , u 2 , …, u L ], V = [v 1 , v 2 , …, v K ] and S=[diag(σ 1 , σ 2 , …, σ L ), 0]. By

using singular value decomposition, it is possible to write X as a sum of L elementary matrices X 1 ~ X L (if L<

K):

X = X + X + L+

1 2

σ1 u1v1

+ σ

2u2v

+ L+

σ L

uLvL

(3)

T T

u1p1

+ u2p2

+ L+

uLpL

Where p 1 ~ p L are L principal components of L time delay series with length K.

3 rd Step: Grouping

The grouping procedure partitions L elementary matrices into several subsets as:

X = X1 + X2

+ L + X

(4)

The question about how to group depends on objectives of SSA. For example, if one wants to de-noise time

series x[n] we can group elementary matrices into two subsets as:

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X = L +

(5a)

+ X2

+ Xd

= Xd

+ Xd+

+ XL

L +

(5b)

Generally elementary matrices with small singular value are noise term. Selection of d depends on signal to

noise ratio and singular value distribution. More information about how to group elementary matrices can refer

to reference [10].

4 th Step: Diagonal Averaging

After 3 rd step, trajectory matrix is divided into several elementary matrices. Then original time series x can be

divided into several elementary time series x 1 ~x D as:

x = x1 + x2

+ L+

x D

(6)

Where each elementary time series can be derived by diagonal averaging of each elementary matrix as:

x i

= d. a.{

Xi}

(7)

We denotes d.a.{.} as diagonal averaging operator.

ANALYSIS RESULTS AND DISCUSSIONS

Recently some research showed that precursors of volcanic explosion can be discovered by singular spectrum

analysis of seismic data [11]. Ratio of two singular values, which quantifying coupling level between two

singular values in singular spectrum analysis, was used as a possible indicator of volcano instability. Evidently

coupling level of singular value in singular spectrum analysis is an important feature of data series and may

provide available information for structural health monitoring issue. In this study we only focus on coupling

level between 1 st and 2 nd singular values because they represent most principal components in analyzed time

series. Instead of using ratio between 1 st and 2 nd singular value, we define eigenvalue ratio difference (EVRD) to

quantify coupling level between 1 st and 2 nd singular value as:

2 2

σ1

−σ

EVRD =

(8)

∑

Where σ is the singular value in singular spectrum analysis, L is the row size of trajectory matrix. The name

“eigenvalue” is used because non-zero singular values of X are square roots of non-zero eigenvalues of X T X or

XX T . The eigenvalue ratio difference of a time series depends only on the selection of trajectory row size L. The

relationship between eigenvalue ratio difference and L are called the eigenvalue ratio difference structure in this

study. For a long time series, it is more appropriate to conduct singular spectrum analysis by moving widow or

recursive concept to trace the time variation of the eigenvalue ratio difference structure [12].

Figure 5 shows the time variation of eigenvalue ratio difference structure of the bridge model velocity response

at node #9. The widow length is 5sec. It shows the eigenvalue ratio difference structure varies with time. At first

120 minutes, the variation resulting from the scouring of the bridge pier foundation. While bridge pier

settlement occurs, the eigenvalue ratio difference become very large no matter what trajectory matrix row size is

used. This means that the first eigenvalue and second eigenvalue become decoupling. It is similar to the concept

in the system identification algorithm. The poles or the eigenvalues of the system matrix will not be in pair for

unstable system.

In real application, it is very time consuming while doing singular value decomposition of trajectory matrix with

many different row size L. It is more suitable to evaluate eigenvalue ratio difference under a certain size

trajectory matrix. Figure 6 shows the time variation of eigenvalue ratio difference by using two different sizes of

trajectory matrices (L = 8 and L = 100). For small L, the eigenvalue ratio difference result can be used to

monitor bridge pier foundation scouring and settlement. The analysis result is sensitive to bridge characteristic

change. For large L, the eigenvalue ratio difference only can be used to monitor bridge pier settlement. This

analysis result focuses on the limit state of the bridge structure. The issue about which one is better depends on

the objective of structural health monitoring. Figure 7 shows the comparison between eigenvalue ratio

difference result and vertical displacement measurement at top of pier #3. It is proved that the eigenvalue ratio

difference accurately provide occurring time information of bridge pier settlement. It is a very useful feature of

time series and a powerful indicator for structural health monitoring and early warning.

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Eigen-Value Ratio Difference (%)

100

50 100 150

Time (min)

Figure 5 Eigen-Value Ratio Difference Structure of velocity response at Node 9 ( 5sec window length )

L = 8

L = 100

100

Eigen-Value Ratio Difference (%)

50 100 150

Time (min)

Eigen-Value Ratio Difference (%)

50 100 150

Time (min)

Figure 6 Eigenvalue ratio difference of velocity response at Node 9

Displacement (cm)

EVRD (%)

100

120 130 140 150 160 170

Time (min)

Figure 7 Comparison of displacement response of pier 3 and eigenvalue ratio difference of velocity response at

Node 9

L = 8

100

EVRD (%)

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Node 9

Node 6

Node 3

50 100 150

Time (min)

Figure 8 Comparison of eigenvalue ratio difference of velocity response at Node 9, Node 6 and Node 3, which

are un-biased to zero at t=0 ( L = 8 )

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In order to detect the damage location of the bridge, we analyze the velocity data of node #9, # 6 and #3, which

are close to pier 3, pier 2 and pier 1 respectively. Figure 8 shows the comparison of eigenvalue ratio difference

of the velocity data of node #9, #6 and #3 respectively, which were un-biased to zero at t=0. The velocity data of

the node near the damage location represent larger eigenvalue ratio difference change than the node far from the

damage location. This phenomenon is coincident with the damage detection method which using two-stage

auto-regressive model [1]. It shows that the feature of the data near the damage location change more great than

the data far from the damage location, and this feature can be extracted from the data series by using eigenvalue

ratio difference in singular spectrum analysis.

CONCLUSIONS

This paper introduces singular spectrum analysis and applies it for structural health monitoring of bridge

structure. A bridge model test data, which is designed to clarify scouring effect on bridge dynamic properties, is

used to verify applicability of singular spectrum analysis. We use eigenvalue ratio difference to quantify the

coupling level between 1 st and 2 nd singular values in singular spectrum analysis. It is proven that eigenvalue

ratio difference extracted from the bridge velocity data can be used to monitor scouring extent of the bridge,

scouring location and settlement occurrence of the bridge pier. It is a very useful feature of time series and a

powerful indicator for structural health monitoring and early warning. The analysis algorithm is simple and

suitable for real structural health monitoring (SHM) application. This study is a preliminary work about

application of singular spectrum analysis to structural health monitoring. The mathematical explanation of

eigenvalue ratio difference structure, relationship between modal property and singular spectrum analysis result

still need to be studied in the future.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support provided by National Science Council (NSC) under

grant number NSC 98-2625-M-002-018-MY3. Experimental supports from National Center for Research on

Earthquake Engineering (NCREE) are also acknowledged.

REFERENCES

[1] Weng, J.H., Loh, C.H., Lynch, J.P., Lu, K.C., Lin, P.Y., Wang, Y. (2008). “Output-only modal identification

of a cable-stayed bridge using wireless monitoring systems”, Engineering Structure, 30(7), 1820-1830.

[2] Loh, C.H., Weng, J.H., Liu, Y.C., Lin, P.Y., Huang, S.K. (2011). “ Structural damage diagnosis based on

on-line recursive stochastic subspace identification”, Smart Materials and Structures, 20(5), 1-10..

[3] Sohn, H., Farrar, C.R. (2001). “Damage diagnosis using time series analysis of vibration signal”, Smart

Materials and Structures, 10 (3), 446-451.

[4] Zheng, H., Mita, A. (2007). “Two-stage damage diagnosis based on the distance between ARM A models

and pre-whitening filters”, Smart Materials and Structures, 16(5), 1829-1836.

[5] Broomhead, D.S., King, G.P. (1986). “Extracting qualitative dynamics from experimental data”, Physica D:

Nonlinear Phenomena, 20 (2-3), 217–236.

[6] Broomhead, D.S., King, G.P. (1986), “On the qualitative analysis of experimental dynamical systems”, In S.

Sarkar, editor, Nonlinear Phenomena and Chaos, pages 113–144. Adam Hilger, Bristol.

[7] Broomhead, D.S., Jones, R., King, G.P., Pike, E.R. (1987), “Singular system analysis with application to

dynamical systems”, In E.R. Pike and L.A. Lugaito, editors, Chaos, Noise and Fractals, 15–27. IOP

Publishing, Bristol.

[8] Bozzo, E., Carniel, R., Fasino, D. (2010), “Relationship between Singular Spectrum Analysis and Fourier

analysis: Theory and application to the monitoring of volcanic activity”, Computers and Mathematics with

Applications, 60 (3), 812-820.

[9] Zhao, X., Ye, B. (2009), “Similarity of signal processing effect between Hankel matrix-based SVD and

wavelet transform and its mechanism analysis”, Mechanical Systems and Signal Processing, 23 (4),

1062-1075.

[10] Golyandina, N., Nekrutkin, V., Zhigljavsky, A. (2001), “Analysis of time series structure: SSA and related

techniques”, Monographs on statistics and applied probability, vol.90. Boca Raton: Chapman & Hall, CRC.

[11] Carniel, R., Barazza, F., Tárraga, M., Ortiz, R. (2006), “On the singular values decoupling in the singular

spectrum analysis of volcanic tremor at Stromboli”, Natural Hazards and Earth System Science, 6 (6),

903-909.

[12] Haavisto, O. (2010), “Detection and analysis of oscillations in a mineral flotation circuit”, Control

Engineering Practice, 18(1), 23-30.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

连续钢 - 砼组合梁桥剪力连接件力学性能的有限元分析

曹鸿猷

陈志军

朱宏平

杨宏印

( 华中科技大学土木工程与力学学院 , 武汉湖北 430074)

Email: hpzhu@mail.hust.edu.cn, 86-27-87542631

摘要 : 武汉二七长江大桥非通航孔采用了 6×90m 等跨钢 - 混凝土组合连续梁结构 , 采用顶推施

工工艺使钢结构部分先行就位 , 桥面板在钢结构顶推就位后分区段分批铺设 , 通过桥面板预先开孔 ,

在开孔区域后浇混凝土与钢梁剪力连接件结合。这种设计构想很好的解决了江滩桥梁施工的大吨位

吊装难题 , 但钢 - 混凝土结合面相对集中 , 剪力连接件的类型、布置范围、布置间距以及受力性能将

极大的影响桥梁结构的整体力学性能。本文以该桥为工程背景 , 采取有限元分析软件 ANSYS 中

COMBIN39 单元模拟剪力钉 , 考虑群钉效应计算分析剪力钉的剪力和滑移 , 在多种荷载组合下对关

键受力区域剪力连接件的布置方式和力学性能进行分析和比较。根据设计图纸上剪力钉的分布情

况 , 模型中剪力钉近似于沿桥长方向均匀布置 , 且每排钉的间距为 1m, 布置在托板的中线和边缘

线上。对于剪力连接件在各支座和各跨跨中区域布置的疏密不同以及剪力刚度不同 , 模型中通过调

整相应部分的 COMBIN39 单元的具体参数以实现精确的模拟。分析计算结果表明 , 在各种荷载组

合下各控制截面剪力连接件承受的剪力和滑移均在合理范围内 , 且中间支座处和跨中处剪力连接件

承受的剪力和滑移都很小 , 边支座的剪力连接件的剪力值和滑移相对较大 ,1/4 跨处剪力连接件承

受的剪力和滑移最大 , 尤其是边跨中距中间支座 1/4 跨处剪力钉的剪力和滑移值需要重点关注 ; 另

外 , 原设计中每个栓钉的剪力值均小于的 67%, 滑移量均小于 0.2mm, 表明柔性剪力连接件整体工

作性能良好 , 可以应用于连续钢 - 混凝土组合梁桥的结合面。

关键词 : 连续钢 - 砼组合梁桥剪力连接件滑移柔性连接件

FEM ANALYSIS OF SHEAR CONNECTOR BEHAVIOR OF CONTINUOUS

STEEL-CONCRETE COMPOSITE GIRDER BRIDGE

Cao Hong You 1 Chen Zhi Jun 1 Zhu Hong Ping 1 Yang Hong Yin 2

School of Civil Engineering & Mechanics, Huazhong University of Science & Technology,

Wuhan 430074, CHINA

Abstract: The non-navigable bridge of Erqi Yangtze River Bridge in Wuhan, China , is a continuous

steel-concrete composite box girder bridge with a span of 6×90m. In this bridge, steel box girder with shear

connectors is seated first by incremental launching method (ILM), after which precast bridge decks with

openings are laid in batches and in sections, concrete are then poured into the openings to bond the steel box

girder and precast bridge decks and thus make the composite bridge. By this method, the problem of bridge

construction with large tonnage hoisting over rivers is solved very well. Nevertheless, in this kind of bridge, the

overall bridge behaviors are significantly dependent on the bond behaviors of the interfaces between the slabs

and the steel girder which have a great bearing on the mechanical performance of shear connector, as a result,

the type, layout (distribution and distance), and mechanical characteristics of shear connector are critical issues

in the design of the bridge. With Erqi Yangtze River Bridge as engineering background, this paper presents the

项目基金 : 国家自然科学基金 (51078164) 国家杰出青年科学基金资助项目 (50925828)

作者简介 : 曹鸿猷 (1986-), 男 , 湖北黄冈人 , 博士生 , 从事桥梁结构数值仿真、车桥耦合动力学研究。

陈志军 (1969-), 男 , 湖北咸宁人 , 副教授 , 大跨度桥梁分析理论与方法、高速铁路桥梁动力行为 ,

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朱宏平 (1965-), 男 , 湖北新洲人 , 教授 , 博导 , 主要从事结构抗震、损伤识别及健康检测研究 (E-mail: hpzhu@mail.hust.edu.cn)

esults of a numerical investigation on the effects of the above issues using an FEM model established in

commercial finite element(FE) software ANSYS with nonlinear spring element COMBIN39 to model the

bond-slip behavior of shear connectors, with special attention to explore the group effects of shear connectors

and different load combinations. In the FEM model, shear connectors were laid out uniformly along the center

line and edge line of the bridge respectively with a center-to-center spacing of 1 mm; the stiffness and layout of

shear connector can be accurately modeled by appropriately defining parameters of element COMBIN39.

According to the numerical results, all of the shear forces and slips on the different control sections are in a

reasonable range for Erqi Yangtze River Bridge under different load combinations. Shear force and slip are

small near the mid-bearing and at mid-span cross sections, relatively larger at the cross sections near the

end-bearing, and maximal at the cross sections near the 1/4 span of the bridge, specially the 1/4 span next to the

mid-bearing at the end-span, where additional caution should be taken in the design of shear connectors . The

results also show that, based on the design of the Erqi Yangtze River Bridge, all shear forces in all the shear

connectors are less than 69% of their load-bearing capacity with the slips less than 0.2 mm; therefore, it can be

said that the flexible shear connectors used in the bridge work well and thus provides an appropriate choice in

designing continuous composite beams.

Keywords: Continuous composite beams, shear connectors, slip, flexible connectors.

一、前言

钢 - 混凝土组合梁具有结构高度小、自重轻、承载力高、刚度大、节省支模工序和模板、减少现

场湿作业量、施工速度快、综合效益好等显著优点 , 作为重要的横向承重构件之一可以广泛用于地

下结构、高层结构和桥梁结构等 [1] 。钢 - 混凝土组合结构的优势在于组合作用 , 但组合作用的大小是

由组合梁中的关键部件 — 剪切连接件决定的。早期剪力连接件一般设计成能抵抗所有作用其上的荷

载 ; 在过去的二十年中慢慢出现了在某些荷载情况下允许部分剪力连接件破坏的设计形式 , 这种形

式虽然降低了结构的强度和刚度 , 但其经济性和整个结构的延性却有较大提高。实际工程中连接形

式的选择需要考虑结构性能的要求、施工条件 , 同时要充分考虑结合面的受力特点。

目前研究钢 - 混组合梁柔性剪力连接件的方法主要分为两种 : 现场实验法和数值仿真法。现场实

验虽然能够得到最可靠的数据 , 然而实验花费极高、现场采集数据困难 , 某些情况下甚至难以实现 ,

这使越来越多的研究者采用数值分析和计算机手段来解决这一问题 [2] 。许多学者已经成功的应用有

限元方法研究了钢 - 混组合梁的性能 , 一些研究者从工程设计者实用性的角度出发 , 采用超级单元法 ,

将钢梁、混凝土板两种不同构件放在一个单元中考虑 ( 如 ,Salari MR [3] 、Sebastian WM [4] 、Pi [5] et al),

这种特殊的单元能够计算组合梁两种不同构件间通过剪力连接件的非线性工作特性 , 然而不可避免

的带有一些假设 , 难以较精确的模拟单个剪力钉的工作性能 , 但建模、计算效率较高 , 为工程设计

人员提供了一种新的较为准确、有效的计算方法。另一些研究者则利用实体、板壳及连接单元建立

精细化的有限元模型对结合梁的力学性能进行分析。El-Lobody and Lam [6] 通过 ABAQUS 软件包 ,

建立有限元三维模型 , 通过人为判断混凝土压应力及剪力连接件的剪力成功求得组合梁的极限承载

力。2002 年 Baskar K , Shanmugam NE [7] 等也使用通用有限元软件建立精细化的有限元组合梁模型 ,

研究结构在负弯矩和剪力作用下的非线性特性 , 结果表明 , 合适的有限元模型能够准确的预测组合

梁的极限承载力。2007 年 Qiu WL , Jiang M [8] 等通过按时间调整的等效弹性模量法研究了收缩徐变

对组合梁剪力连接件粘结滑移特性 , 并与实验数据吻合较好。近年一些学者开始对组合梁剪力连接

件的性能进行参数化研究 [9,10] , 也都使用有限元方法 , 不难看出 , 有限元方法在组合梁分析中的可

靠性、精确性越来越受人们认可。可以看出他们中一些研究者把重点放在基于实验数据的有限元模

型表述与校正上 , 受到实验模型的制约 , 所研究对象一般是小的结构构件 ; 一些研究者则对复杂的

施工过程模拟不详细 , 长期作用下收缩徐变关系考虑不仔细、甚至忽略 , 这些都限制着有限元方法

在组合梁设计设计中更加深入的应用。

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武汉二七长江大桥非通航孔采用了 6×90m 等跨钢 - 混凝土结合连续梁结构 , 用顶推施工工艺使

钢结构部分先行就位 , 桥面板在钢结构顶推就位后分区段分批铺设 , 通过桥面板预先开孔 , 在开孔

区域后浇混凝土与钢梁剪力连接件结合。本文以该桥为工程背景 , 采取有限元分析软件 ANSYS 建

立详细的三维有限元模型 , 考虑混凝土桥面板、剪力连接件、钢梁各自材料的非线性特征 ; 利用

COMBIN39 单元模拟剪力钉的非线性性能 , 并考虑群钉效应计算分析剪力钉的剪力和滑移 , 在多种

荷载组合下对关键受力区域剪力连接件的布置方式和力学性能进行分析和比较。

二、有限元模型

2.1 单元类型及模型建立

有限元模型中 , 混凝土桥面板采用 ANSYS 中的空间实体单元 SOLID65, 钢箱梁及其横隔板和

托板采用 SHELL63 单元模拟 , 梁内撑杆和预应力筋则采用空间杆单元 LINK8 模拟 , 采用非线性弹

簧单元 COMBIN39 模拟剪力连接件。

SOLID65 单元用于含钢筋或不含钢筋的三维实体模型。该实体模型可具有拉裂与压碎的性能。

在混凝土的应用方面 , 可用单元的实体性能来模拟混凝土 , 而用加筋性能来模拟钢筋的作用。

SHELL63 是薄壳单元 , 包含弯曲和薄膜效应 , 忽略横向剪切变形 , 具有 4 个节点 , 每个节点 6 个自

由度 , 可以承受平面内荷载和法向荷载 ; 弹塑性、蠕变、应力刚化、大变形、大应变等特性均考虑

其中。LINK8 单元是有着广泛的工程应用的杆单元 , 比如可以用来模拟 : 桁架、缆索、连杆、弹簧

等等。这种三维杆单元是杆轴方向的拉压单元 , 每个节点具有三个自由度 : 沿节点坐标系 X、Y、Z

方向的平动。

COMBIN39 是一个具有非线性功能的单向单元 , 可对此单元输入广义的力 - 变形曲线。该单

元可用于任何分析之中 , 在一维、二维和三维的应用中 , 本单元都有轴向或扭转功能。此单元仅当

每个节点有两个或者三个自由度的时候 , 才可以具有大位移的功能。

预应力混凝土桥面板采用实体力筋法来模拟 , 预应力的模拟方法采用降温法 , 可以设定不同位

置的预应力索具有不同的预应力分布。在建模过程中 , 对预应力筋进行适当简化 , 将三根合为一根 ,

相应修正预应力筋的截面积和预应力值。

如图 1 所示 , 钢 - 砼组合梁在混凝土板与钢梁交界面处通过布置不同分布形式的剪力栓钉 ( 图

1.a) 来传递桥面板与钢梁间的剪力。通常在实际计算中 , 认为桥面板与钢梁间没有滑移错动 , 在对

组合梁按照桥面板、钢梁按不同材料特性分开建模时往往忽略剪力连接件 , 混凝土材料单元与钢材

料单元在交界面处公用节点 ( 图 1.b)。本文考虑桥面板与钢梁间剪力连接件的滑移 , 在混凝土桥面

板单元与钢梁单元间加入 COMBIN39 非线性弹簧单元来更加接近真实情况的模拟剪力连接件的作

用 ; 认为两种单元竖向相对位移为 0, 故在交界面耦合两种单元相应节点的竖向位移 ; 通过设置

COMBIN39 单元特性 , 来模型剪力连接件在桥面板相对于钢梁在纵、横桥向发生滑移时的工作性能

( 图 1.c)。

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图 1 剪力连接件有限元模型示意图

2.2 材料模型

2.2.1 混凝土材料

C50 混凝土初始弹性模量 3.45×10 5 MPa, 泊松比 0.2, 线膨胀系数 1.0×10 -5 , 其本构关系采用

E.Hognestad 建议的混凝土轴向受压应力 —— 应变曲线模型 , 强度上升段为二次抛物线 , 强度下降段

为斜直线 , 表达式 [11] :

⎧ f ⎡

2 εc / εy − ( εc / εy)

⎤

⎪ ⎣

⎦

= ⎨

⎪ f ⎡

1−0.15 ( ε

−εy) /( ε −ε

) ⎤

⎣ cu y

⎩

⎦

( εc

≤ εy)

( ε < ε ≤ε

)

c c cu

式中 : f c

为混凝土轴心抗压强度 ; ε y

相应于 f

时的应变 , 取 ε

=0.002; ε cu

为极限压应变 , 取

(1)

=0.0038。

根据《公路钢筋混凝土及预应力混凝土桥涵设计规范》(JTG D62—2004) 提供的混凝土收缩徐

变计算公式 , 对于特定环境下的某一构件 , 收缩徐变 ε

与徐变系数 φ 均为混凝土龄期 t 的单值函数 ,

其具体表达形式如下 :

ε () t = k ⋅

φ()

1 2

T + k2

⋅ h

⎛

⎞

⋅⎜ ⎟

+ T

⎝

⎠

0.3

(2)

(3)

式中 , T = t− t , 定义为计算龄期 ;k 1 ,k 2 ,k 3 ,k 4 为根据环境条件、混凝土标号与构件尺寸由

规范提供的公式计算得到的常数。

混凝土收缩在 ANSYS 中可通过温度荷载来实现。由式 (2) 计算得到相应龄期的混凝土收缩应变 ,

在把收缩应变换算成该龄期混凝土单元相应节点上的温度荷载。

ANSYS 提供了很多蠕变 ( 徐变 ) 准则 , 分析中选用 C6=1 的显式方程 :

C2 C3

−C4 / T

& =

1σ

(4)

εcr C t e

为了能够与规范提供的式 (3) 相容 , 需要在每个微小的时间间隔内 , 式 (4) 分别取适当的系数

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(C 1 -C 4 ) 近似拟合式 (3)。根据式 (3) 有 :

为了实现式 (3) 和 (4) 相容 , 只需满足下式 :

−0.7

dφ

⎛ T ⎞ k

= 0.3k

⎜ ⎟

dt k T k T

3 2

⎝ 4

+ ⎠ ( 4

+ )

σ dφ

= C1σ

T e

Edt

C2 C3

−C4 / T

(5)

(6)

如果取 :C2=1,C3=0.3,C4=0, 则有如下关系 :

C ( T)

0.3kk

3 4

1 1.3

ET ⋅ ⋅ ( k4

+ T)

(7)

因此 , 在 ANSYS 计算分析混凝徐变时 , 在每一步时间步长内 , 时变系数 C 1 按式 (7) 取值 , 同时 ,

时间步长的设置必须遵循一条原则 : 在徐变系数变化较快的时段 , 必须设置较小的时间步长 , 以保

证计算精度。

2.2.2 徐变方程验证

为了验证上述方法拟合规范公式的可靠性 , 分别取正方形截面立柱模型和 T 形截面立柱模型 ,

柱底固结 , 顶面施加均布竖向荷载 , 计算徐变系数 , 并与规范计算结果相比较如图 2 所示。结果表

明 , 该方法计算徐变系数值结果与规范值误差在 1% 以内 , 且误差不具有时间累加效应 , 因此 , 采

用该方法进行混凝土徐变过程分析是可靠的。

图 2 徐变系数值对比

2.2.3 钢梁、预应力钢筋

钢梁、预应力钢筋的应力应变关系采用理想弹塑性模型 , 钢材弹性模量 Es=2×10 5 MPa, 泊松比

0.3, 线膨胀系数 1.2×10 -5 , 钢梁屈服强度 σ

=370Mpa, 预应力钢筋 σ

=1860Mpa。其应力 σ

——

应变 ε

关系表达式为 :

⎧⎪ ε

= ⎨ ⎪ ⎩σ

(0 ≤ ε ≤ ε )

( ε > ε )

(8)

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2.2.4 剪力钉

根据中国《钢结构设计规范》(GB500172003), 栓钉抗剪设计承载力为

N = 0.43A E f ≤ 0.7Aγ

(9)

v s c c s

由推出试验得到的荷载 — 滑移曲线是反映剪力件工作性能的特征曲线 , 模型中采用 J.W.Fisher

建议的剪力 — 滑移曲线 [12] :

Q=Q u (1-e -0.71Δ ) 0.4 (10)

上述两式 (9)、(10) 中字母含义如表 (1) 所示 :

表 1 参数含义表

E c 混凝土的弹性模量 γ 栓钉材料抗拉强度最小值与屈服强度之比

f c 混凝土轴心抗压强度设计值 Δ 栓钉滑移量

A s 圆柱头焊钉 ( 栓钉 ) 钉杆截面面积 Q 栓钉连接件工作受力

f 圆柱头焊钉 ( 栓钉 ) 抗拉强度设计值 Q u 栓钉连接件极限承载力

当栓钉材料性能等级为 4.6 级时 , 取 f =215N/mm 2 ,γ=1.67。

2.3 荷载工况

根据桥涵设计规范 , 考虑桥梁运营过程中恒载 , 活载 , 温度荷载三种主要荷载形式。计算剪力

连接件内力和滑移时 , 考虑如下六种荷载工况 :

(1) 恒载 ;

(2) 恒载 + 活载 ( 车道荷载加人群荷载 ) 满布 ;

(3) 组合 (2)+ 整体升温 ;

(4) 组合 (2)+ 整体降温 ;

(5) 组合 (2)+ 梯度升温 ;

(6) 组合 (2)+ 梯度降温。

三、有限元分析结果

二七长江大桥非通航孔深水区采用了 6×90m 等跨钢 - 混凝土结合连续梁新结构 , 采用顶推法施

工新工艺 , 在钢结构顶推就位后 , 再分区段分批铺设桥面板。标准桥面板与钢梁连接处断面如下图

3 所示。

图 3.a 结合梁标准横断面图

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图 3.b 桥面板与钢梁连接处标准断面图

3.1 结构计算控制截面及剪力连接件布置

由于结构和荷载对称 , 故应力和滑移亦具有对称性 , 取半桥计算结果进行分析 , 剪力钉按照设

计图纸布置方式设置 , 取 13 个控制截面 , 其编号及位置示意图见图 4。

图 4 控制截面编号及位置示意图

根据设计院提供的图纸 , 设计所采用的是栓钉连接件。在有限元分析中 , 由于模型的限制 , 对

剪力钉的参数和布置做了一定简化。根据图纸上钉的分布情况 , 模型中钉近似于沿桥长方向均匀布

置 , 且每排钉的间距约为 1m, 布置在顶板的中线和边缘线上。对于图纸上钉在各支座和各跨跨中段

布置的疏密不同情形 , 模型中相应部分的 COMBIN39 单元的具体参数则做出相应的制定 , 以实现

精确的模拟。由此 , 模型中的每个 COMBIN39 单元实际对应数个栓钉 , 其抗剪承载力考虑乘以 0.9

的群钉折减系数。对于图纸上钉在各支座和各跨跨中段布置的疏密不同情形 , 各截面剪力钉的布置

及承载力情况如下表。如图 4 中 S1 截面的抗剪承载力计算如下 : 0.9×73.7×17×11.35=12798KN。

表 2 示出每个截面剪力钉的布置及承载力 ( 由于模型的适当简化 , 剪力钉个数将出现小数 )。

表 2 剪力钉布置与承载力

截面每根栓钉抗剪承载力 /KN 截面实际栓钉数 / 个截面抗剪承载力 /KN

S1 73.7 17× 11.35 12798

Q1L 73.7 6× 10.85 4318

M1 73.7 6× 10.85 4318

Q1R 73.7 6× 10.85 4318

S2 73.7 17× 12.71 14332

Q2L 73.7 6× 9.53 3793

M2 73.7 6× 9.53 3793

Q2R 73.7 6× 9.53 3793

S3 73.7 17× 12.71 14332

Q3L 73.7 6× 8.94 3558

M3 73.7 6× 8.94 3558

Q3R 73.7 6× 8.94 3558

S4 73.7 17× 12.71 14332

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3.2 制截面剪力连接件剪力及滑移分析

设计中 , 预制桥面板采用 C50 混凝土 , 张拉横向预应力 ; 为了减小收缩徐变的影响 , 存放 6 个

月以上后才安装使用。为研究混凝土收缩徐变对剪力连接件性能的影响 , 建立了不考虑收缩徐变及

考虑 1800 天收缩徐变效应两种有限元模型。

3.2.1 剪力连接件滑移分析

图 5 不考虑徐变时不同控制截面剪力钉滑移量

图 6 考虑徐变时不同控制截面剪力钉滑移量

由图 5、图 6 可知收缩徐变对剪力钉滑移量的影响较小 , 主要因为在铺设桥面板时 , 桥面板已

预制完成并存放 6 各月以上 ; 说明该措施能够有效减小混凝土桥面板收缩徐变在运营阶段对剪力连

接件的影响。最大滑移量发生在荷载工况四时 , 边跨边支座处 , 剪力钉滑移量达 0.19mm; 各荷载工

况作用下 , 边跨靠近中支座 1/4 截面 (Q1R 截面 ) 处剪力钉在各种荷载工况作用下剪力钉滑移量均

处于相对较高值 , 最大值达 0.17mm, 最小值为 0.14mm。除边跨边支座 (S1 截面 ) 外 , 有较大滑移

量的截面均是每跨的 1/4 截面处 , 因为设计图纸中这些截面位置剪力钉的布置方式与跨中截面相同 ,

这些截面与跨中截面不同 , 弯、剪组合受力状态相对明显 , 剪力钉布置形式宜于跨中截面有所差别 ;

支座截面虽然剪力较大 , 但剪力钉布置数量较多 , 与跨中截面一样 , 剪力钉滑移量很小。与降温荷

载相比 , 升温荷载对剪力钉的滑移量影响更大 , 特别是梯度升温荷载。

3.2.2 剪力连接件剪力分析

图 7 不考虑徐变时单个栓钉剪力值图 8 考虑徐变时单个栓钉剪力值

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图 7、图 8 分别给出了不考虑和考虑混凝土收缩徐变时 , 关键截面处剪力钉的单钉剪力值。与

剪力钉滑移量相同 , 收缩徐变对剪力钉剪力值影响也较小。除边跨边支座 (S1 截面 ) 外 , 各墩顶截

面及跨中截面处剪力钉的剪力值均很小 , 各种荷载工况作用下 , 最大剪力值均发生在边跨靠近中墩

1/4 截面 (Q1R 截面 ) 处 , 最大剪力值为 49.10KN, 小于其承载强度的 67%, 对应位置的剪力钉滑

[13]

移量小于 0.2mm。根据王连广研究得出 :“ 栓钉连接件的变形能力必须满足当承载力达到最终承

载力的 60% 时 , 滑移量不应超过 0.4mm, 达到最终承载力时 , 滑移不应大于 3~5mm”, 表明柔性剪

力连接件整体工作性能良好 , 可以应用于连续钢 - 混凝土组合梁桥的结合面。

四、结语

利用 ANSYS 建立武汉二七长江大桥非通航孔钢 - 混凝土结合连续梁桥的有限元模型 , 考虑了钢

梁、混凝土桥面板、预应力钢筋的材料非线性特性 , 特别针对混凝土的缩徐变特性做了专门的分析 ,

用 COMBIN39 单元模拟该桥的柔性剪力连接件 , 研究在多种荷载工况作用下柔性剪力连接件的工

作性能。计算结果表明 , 在剪力钉抗剪滑移分析中 , 混凝土的收缩徐变对剪力钉的剪力和滑移的影

响很小 ; 在各种荷载组合下各控制截面剪力连接件承受的剪力和滑移均在合理范围内 , 实际上每个

栓钉的应力值均小于其承载强度的 67%, 滑移量均小于 0.2mm, 且中间支座处和跨中处剪力连接件

承受的剪力和滑移都很小 , 边支座的剪力连接件的剪力值和滑移相对较大 ,1/4 跨处剪力连接件承

受的剪力和滑移最大 , 且边跨中距中间支座 1/4 跨处剪力钉的应力和滑移最大。总体来说柔性剪力

连接件整体工作性能良好 , 可以应用于连续钢 - 混凝土组合梁桥的结合面。

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[10] El-Lobody, E. and Young, B. Performance of shear connection in composite beams with profiled steel sheeting. Journal

of Constructional Steel Research, 2006, 62(7): 682–94.

[11] 顾祥林 . 混凝土结构基本原理 , 上海 : 同济大学出版社 , 2004.

[12] 聂建国 . 钢 - 混凝土组合梁结构 : 试验、理论与应用 , 北京 : 科学出版社 , 2005.

[13] 王广连 . 钢与混凝土组合结构理论与计算 , 北京 : 科学出版社 , 2005.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

A STUDY ON DAMAGE ASSESSMENT OF THE SCOURED BRIDGES

C. Y. Wang 1 and Y. C. Sung 2

Department of Civil Engineering & Graduate Institute of Civil and Disaster Prevention Engineering,

National Taipei University of Technology, Taiwan. Email:581ce013@gmail.com

Department of Civil Engineering & Graduate Institute of Civil and Disaster Prevention Engineering,

National Taipei University of Technology, Taiwan. Email: sungyc@ntut.edu.tw

ABSTRACT

In the past decades, the heavy rain frequently caused the scoured bridges collapse during typhoon period in

Taiwan. To get a better warming system, this study first concluded three fundamental parameters including the

water level, the scouring depth of the bridge foundation and the flood velocity sensitive to the failure of the

bridges and proposed an analysis process in determining the safety level. In this paper, various possible failure

modes of the pile-foundation and caisson-foundation were classified and employed as the check-point of the

governed issue during the analysis. Through the 3D finite element software of MIDAS-GTS, the stresses

concerned during various situations of scoured riverbed are able to be determined and, accordingly, the most

possible failure mode of the bridge foundation can be detected through the choice of minimum ratio of capacity

to stress. As a result, different kinds of safety level represented by the surfaces composed of the parameters

could benefit the bridge engineers a good making decision on management of the scoured bridges.

KEYWORDS

Flood-resistant assessment, Bridge management, Early warning system

INTRODUCTION

Taiwan is located in south-east of Pacific Ocean. Strict flood owing to typhoon causes great bridge damages

during flood season from May to October each year. The severe flood frequently accelerated the scouring of

riverbed and resulted in foundation scouring of cross-river bridge (Melville et al. 2000). In 2000, the Kaopeng

Birdge in southern Taiwan collapsed beyond expectation few days after the Typhoon Bilis. With strong winds

and torrential rain pounded Taiwan in August, 2009, the Typhoon Morakot yielded dozens of villages deluged

with floodwaters and caused lots of people death as well as more than 60 bridges collapsed.

The bridge damage caused by scouring could impact livelihood activities in ordinarily, emergency relief and

material transportation in post-disaster. Therefore, the engineers took more effort finding good solutions to

mitigate the threat of bridge scouring. After the typhoon Morkaot, the high-tech monitoring had been

implemented generally on the existing bridges in Taiwan. An application of Fiber Bragg Grating (FBG) sensors

(Lin et al. 2004) and micro-electro-mechanical (MEM) system (Lin et al. 2010) had been applied practically. By

the wireless Zigbee network transmission, the real time signals of scouring depth, water level, flow velocity, etc.

are already able to be detected easily and served as the input data of safety assessment of bridge scouring.

This paper intends to establish a system of damage assessment of bridge scouring based on finite element

method considering the interaction between soil and structure. Two proposes are able to be achieved: (1)

Identification of the failure modes of scoured bridges, (2) Safety evaluation according to the signal of sensitive

parameters detected by bridge real-time monitoring device to build up a warning system for bridge management.

FUNDAMENTAL FACTORS TO STRUCTURAL SAFETY OF SCOURED BRIDGE

The fundamental factors sensitive to the structural safety of the scoured bridge are summarized as the following:

(1) Load-carrying Capacity of Foundation Soil

A. Pile Foundation

Based on soil mechanics, the ultimate load-carrying capacity of a pile foundation includes the bearing force

supported by the tip of pile Qp( tonf ) together with the frictional force resisting by the pile shaft Qs( tonf )

and can be expressed by the equation as follow:

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The allowable load-carrying capacity

Q a is:

Qu= Qp+ Qs

(1)

= Qu

/ FS

(2)

FS is the safety factor adopted as three in this study.

The more the scouring depth of the pile is, the less the contribution of Qs( tonf ) becomes. Accordingly,

the decrease of Q a goes on the way of soil removing due to scouring until the resulting Q a is less than

the external loading from superstructure when a critical scouring depth D b is reached.

B. Caisson Foundation

According to the foundation design code in Taiwan (CPAMI 2001), the ultimate load-carrying capacity of

soil pressure q ( tonf / m ) and the allowable one q ( tonf / m ) can be expressed by Eqs 3 and 4,

respectively.

qu = αCNc + r2DfNq + 0.5βr2BN

(3)

q = ( q − rD )/ FS + r D

(4)

a u 1 f 2 f

Where, r ( tonf / m ) is the unit weight of soil under the bottom surface of caisson, B( m ) is the caisson

width, Df

( m ) is embedded depth, N c 、 N q 、 N r are the load-carrying indices, α and β are the

influence indeices sensitive to the foundation shape and are specified in the design code, FS is the safety

factor adopted as three in this study. The ultimate load-carrying capacity Q ( tonf / m ) as well as the

allowable load-carrying capacity Q ( tonf / m ) is able to be obtained by the product of q or q and

contact area between the tip of caisson and the bearing soil.

The more the scouring depth of the caisson is, the less the Q ( tonf / m ) becomes. Accordingly, the

decrease of Q a goes on the way of soil removing due to scouring until the resulting Q a is insufficient to

carry the external loading from superstructure when a critical scouring depth D b is reached.

(2) Strength of Foundation Body

A. Interaction between Axial force and Bending Moment

The bending moment M associated with the axial force P of the foundation body varies with the change of

scouring depth. To assure the structural safety of the scoured foundation, the establishment of the

interaction curve between axial force and bending moment (P-M curve) is necessary for determination of

safety factor with respect to different scouring status. As the scouring becomes serious, the resulting stress

point represented by a pair of (P, M) usually approaches closer to the boundary of the P-M curve diagram.

And as can be seen in Figure 1, the failure mechanism of bending moment triggers with the resulting stress

point locating beyond the P-M curve when a critical scouring depth D M is reached.

B. Interaction between Axial force and Shear Strength

According to the Taiwan Specification for Highway Bridges (MOTC 2009), the shear strength capacity of

the foundation body mainly contributed by concrete and transverse reinforcement can be expressed as:

φV = φ( V + V )

(5)

n c s

Vc = 0.53(1 + F)

fcAe

(6)

π Af

h yhD

Vs = ≤ 2.12 fc A

(7)

2 a

where Vn( kgf ) is the nominal shear strength of foundation body, Vc( kgf ) and Vs( kgf ) are the shear strength

of concrete and transverse reinforcement, respectively, Ah( cm ) and a ( cm)

are the cross sectional area and

the space of transverse reinforcement, respectively, Ae( cm ) and Acm ( ) are the effective and gross cross

sectional area of the foundation body; F is the modification factor related to the interaction between shear

strength and axial force, D( cm)

is pile diameter; φ is the reduction factor adopted as 0.85 in this study.

As the scouring becomes serious, the resulting shear force appears more significant. When a critical

scouring depth D V is reached, the resulting shear force is found to be larger than the shear strength obtained

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y Eqs 5 6 and 7 and the failure mechanism of shear is regarded as governing.

(3) Debris or Gravel Impact Forces on Bridge

Figure 1 Pile moment failure mode

The debris accumulation yields the impact force to the foundation, substructure and even superstructure of

the bridges (NCHRP 1999). It is essential to take the impact force induced by debris accumulation into

account for the assessment of flood-resistant capacity. Since few researches focused on this issue could be

available, the determination of such kind of impact force on the bridges so far is immature of reveal in the

bridge design code. A design guideline for highway bridges considering debris flow (MOTC 1999) is

referred here. As can be seen in Eq 8, the impact force of gravel P ( tonf ) on the bridges is taken into

account as an equivalent static force applying at the top of the water level to the structural members on the

upstream side.

Where U ( m / sec) is the averaged flow velocity; ( m)

(4) Limitations on Deformation of Foundation

1.24 2

P = 48U

(8)

R is the averaged gravel diameter.

The scouring of bridge foundation must have enlarged the lateral deformation of foundation or substructure.

To keep a qualified functionality of bridge service, a certain degree of limitations on deformation is

sufficient and necessary.

A. Pile Foundation

According to the Seismic Design Specification for Highway Bridges (MOTC 2008), a stability index Qs

Eq 9 is adopted to restrict the relative displacement of the piers due to significant earthquake. If Qs

experiencing of being larger than 0.25, the significant geometric nonlinearity of bridge structure usually

takes place with high potential and the nonlinear dynamic analysis should be carried out to confirm the

structural safety.

PdΔ0

= (9)

Vl d c

Where Pd( tonf ) is the pier axial force due to dead load, Δ0( m)

is the relative deformation between top and

bottom of the pier, Vd( tonf ) is the shear force in the pier of plastic status, lc( m ) is the pier height.

In order to have a higher safety, the limitations on deformation of foundation should be more rigorous than

those of substructure. A conservative value 0.2 of Q s is taken into account and the allowable relative

deformation of the pile foundation can be defined by Eq 10:

Δ allowable = 0.2

(10)

Where Δ allowable( m)

is the allowable relative deformation between top and first inflection point of the pile,

Mu( tonf − m)

is the ultimate bending moment strength of pile member.

The P-M curve of pile member can be used to investigate if the deformation of pile is qualified. As can be

seen in Figure 2, the current status of pile force including the axial force ( P i ) and bending moment ( M i )

can be obtained in FEM solution and denoted by point A. By inserting a horizontal line passing point A to

find point B at the intersection of the line and P-M curve, the ultimate bending moment Mu( tonf − m)

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corresponding to the current axial force P i is able to be recognized as the ordinate value of point B on the

P-M curve. Consequently, the allowable relative deformation of pile at check point can be obtained by Eq

10 and the bridge functionally can be investigated accordingly.

Figure 2 Determination of the ultimate bending moment Mu( tonf − m)

w.r.t. the current axial force P i

B. Limitation of Caisson Foundation

According to the structural foundation design code in Taiwan, the deformation at the design surface of

caisson is not allowed to be greater than both of 1% of caisson width and 5cm. In addition to that, the

allowable rotation is limited within 0.005rad.

(5) Passive Soil Pressure on Caisson Foundation

The passive soil pressure distributing on caisson at down-stream side gives a resistance of either sliding or

overturning related to bridge scouring and can be expressed as:

σ = σ z K + 2C K

(11)

p p p

cos φ

K p

sin( φ− δ)sin( φ+

α)

(12)

cos δ[1 −

]

cosδcosα

In which, σ p( tonf / m ) and σ z( tonf / m ) are the passive pressure and the effective vertical pressure at where

distant to ground with depth z, K p is coefficient of passive pressure, Ctonf ( / m ) and φ (degree) are the

parameter of cohesion and the friction angle in soil, respectively, δ (degree) is the inclination of ground

with respect to horizontal axis.

If the resulting horizontal pressure σ h( tonf / m ) of the soil elements at down-stream side of caisson is

found to reach the soil passive pressure in Eq 11, it means that the soil is approaching to the ultimate state

and will soon be lack of sufficient resistance to sliding or overturning of caisson.

DAMAGE ASSESSMENT OF SCOURED BRIDGE

The proposed procedure of damage assessment of the scoured bridges is listed as follows:

(1) Establish the FEM model including the bridge substructure, foundation and soil layers.

(2) Evaluate the proper material properties of soil layers based on the boring data and the design code of

foundation.

(3) Adopt the suitable material properties of substructure and foundation of the existing bridges.

(4) Choose the safety factor FS i considered.

(5) Specify the water level H i

(6) Specify the flow velocity V i .

(7) Set the soil layers without scouring.

(8) Apply the water pressure on the substructure beyond the top of soil layers and the impact force of debris

flow or gravel at the top of water level. According to the Taiwan Specification for Highway Bridges

(MOTC 2009), water pressure can be expressed as:

P = 52.5KV

(13)

Where, P( kgf / m ) is water pressure, K is the influence indeices sensitive to the pier shape and

V( m/sec)

is flow velocity.

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(9) Perform FEM analysis to check if the load-carrying capacity of foundation soil is insufficient to sustain the

load from superstructure. If it happens, the critical depth D b of load-carrying can be determined

correspondingly.

(10) Check if failure of foundation body occurs within the scouring depth of D b .

A. Pile Foundation

I. Get the axial force P and the bending moment M of pile member for each step of soil-removing

within the scouring depth of Db. Find the critical depth of bending moment Dm at where the stress

point denoted by the pair of P and M locates beyond the P-M curve interaction.

II. Get the shear force of pile member for each step of soil-removing within the scouring depth of Db.

Find the critical depth of shear DV at where the shear force obtained is greater than the shear capacity

of pile.

III. Capture axial force and bending moment of pile member and use Eq 10 to check if the relative

displacement between the pile top and the first inflection point of pile is qualified. Find the critical

depth of unqualified deformation D d at where the relative displacement exceeds the allowable value.

B. Caisson Foundation

I. Get the displacement of the caisson member for each step of soil-removing within the scouring depth

of Db. Find the critical depth of unqualified deformation Dd at where the displacement exceeds the

allowable value specified in the design code.

II. Get the horizontal stress in soil element around the caisson member on down-stream side for each

step of soil-removing within the scouring depth of Db. Find the critical depth of excessive passive soil

pressure deformation Dh at where the soil is approaching to the ultimate state and will soon be lack of

sufficient resistance to sliding or overturning of caisson.

(11) Take the minimum of {D b , D m , D v , D d } for pile foundation and the minimum of {D b , D d , D h } for caisson

foundation as the governing scouring depth D G of bridge souring with respect to the flow velocity V i and

water level H i .

(12) Make a decrease of soil layer considering the scouring and return to step (7).

(13) Make an increase Δ V to the flow velocity V i and return to step (6).

(14) Make an increase Δ H to the water level H i and return to step (5).

(15) Make a choice of different FS and return to step (4).

(16) Construct the diagram represented by the relationship between the converged sets of (V i , H i , D G , FS i ).

(17) Do the nonlinear regressive analysis of the converged sets of (V i , H i , D G , FS i ) to establish the prediction

model. The rapid evaluation of FS and damage model of the scoured bridge can then be fulfilled by the

model through the actual data including the flow velocity V, the water level H, the scouring depth D

observed by the monitoring instrument in site as input.

CASE STUDY

CASE 1 Failure analysis of Shuan-Yuan bridge

The Typhoon Morakot pounded Taiwan with strong winds and torrential rain in August, 2009. The Shuan-Yuan

Bridge crossing the Kao-Pin river in southern Taiwan is one of the collapsed bridges caused at that time. Based

on the proposed assessment method, the failure of this bridge is analyzed and discussed.

Description of the bridge

The Shuan-Yuan Bridge was originally constructed in 1975 with two-lane traffic and supplemented to four-lane

traffic in 1981. It has a total length of 2082.8 m and a net deck width of 19 m . The basic structure unit of the bridge

is composed of six PCI girders as superstructure in each span and single pier as substructure supported by six

piles as foundation. The total length of the piles varies from 30 m to 43 m . Two different sizes of pile diameter,

0.9 m and 0.76 m , are constructed in the top 10 m and below, respectively. To increase the flood-resistance of each

foundation, four additional piles with a diameter of 0.9 m and a length of 50 m were retrofitted beneath the

enlarged footing in 2002.

Description of the analytical model

A 3D FEM model as shown in Figure 3 is established according to engineering drawings of original as well as

retrofitting design. Three kinds of element were used in this case study: (1) The triangular solid element to

simulate the soil layers with Mohr-Coulomb constitutive model and the pier together with pile cap with elastic

constitutive model, (2) The elastic beam element to simulate pile, (3) The interface element to simulate the

interaction between pile and surrounding soil. Based on the Japanese design specifications of highway bridge

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(Japan Road Association 1996), the important parameters of cohesive coefficientc ( tonf / m ) and shear-resistance

angle φ (degree) are able to be estimated by Eqs 14 and 15 through the inspected SPT-N value of each soil

layers. The geotechnical parameters for the analysis are listed in Table 1.

After the collapse of this bridge, an investigation report (CECI 2009) pointed out that the maximum flow

velocity of 10m/sec. and the highest water level of 6m were observed during the Typhoon Morakot.

C= (0.6~1) N

(14)

φ = 15 + 15N

≤ 45 o

(15)

Where N is the inspected SPT-N value of the soil layer.

Figure 3 D Finite Element model of the Shuan-Yuan Bridge

Table 1 Geotechnical design parameters for analysis of the Shuan-Yuan Bridge

Geotechnical layer Depth (m) SPT-N C (tonf/m 2 ) φ Unit weight (tonf/m 3 )

Silt 5 3.6 2.2 15 1.6

Sandy loam (I) 5.75 11 0 27 1.8

Sandy loam(II) 6 12 0 28 1.8

Clay loam(I) 8.25 9 5.4 26 1.9

Clay loam(II) 12 13 7.8 28 1.9

Sandy loam 13 19 0 31 1.8

Results and discussions

Based on the data actually observed including the maximum flow velocity of 10m/sec., the highest water level

of 6m and the maximum scouring depth of 16.23 m during the Typhoon Morakot, this analysis is performed and

discussed as follows:

(1) The results obtained show that the pile loses sufficient load-carrying capacity when the soil layers are taken

off to the situation of 31 m distant from the bottom of pile cap. In the other words, the critical scouring depth

related to load-carrying capacity is D b =31 m .

(2) The resulting stress point represented by a pair of (P, M) of the pile goes outside the envelope of the P-M

curve diagram when the soil layers are taken off to the situation of 10 m distant from the bottom of pile cap

as shown in Figure 4. The critical scouring depth related to interaction between axial force and bending

moment is D m =10 m .

(3) The resulting shear force of the pile exceeds the shear strength when the soil layers are taken off to the

situation of 11 m distant from the bottom of pile cap. The critical scouring depth related to interaction

between axial force and shear strength is D m =11 m .

(4) The deformation of pile is found to always satisfy the limitations, it means that the critical depth related to

unqualified deformation D d is greater than D m =10 m .

(5) The analytical governing scouring depth D G =10 m is lower than the actual scoured depth 16.23 m . As a result,

the insufficient bending capacity of pile body occurs first and the damage of the foundation reveals

reasonably.

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Figure 4 Pile moment excess P-M curve in souring depth arrived 10 meter

Table 2 Result of failure analysis of Shuan-Yuan bridge

Scouring depth Relative

Failure mode

(under pile cap displacement

Relative rotation

bottom surface)

Plie bearing capacity 31 m 1.9 cm 0.0027 rad

Moment strength capacity 10 m 1.35 cm 0.0019 rad

Shear strength capacity 11 m 1.38 cm 0.0019 rad

Bridge functionality > 31 m < 1.9 cm < 0.0027 rad

CASE 2 Bridge management of flood-resistance for the existing Dai-Ja-Hsi bridge

Description of the bridge

Located in middle Taiwan, the Da-Jia-Hsi Bridge has the continuum box girder with varying span from 40 m to

45 m as superstructure and single pier with 32 m height as substructure and supported by 8-1.5 m φ piles with 20 m

as foundation. The geotechnical design parameters for analysis are listed in Table 2.

Table 2 Geotechnical design parameters for the Da-Jia-Si bridge analysis

Geotechnical layer Depth (m) SPT-N C(tonf/m 2 ) φ Unit weight(tonf/m 3 )

Sandy Gravel 19 50 0 42 2.1

Sandy Loam 1.5 50 5 38 1.9

Sandy Gravel 7.5 50 0 42 2.1

Since the bridge locates at a high potential zone of scouring, some advanced monitoring instruments had been

implemented to the bridge to evaluate the structural safety against scouring as shown in Figure 5. The real-time

information including the scouring depth (SD), the water level (H) and the flow velocity (V) can be monitored

and, through the 3G transmission network, transferred to the head office at National Center for Research on

Earthquake Engineering (NCREE), Taipei. Associated with the damage assessment system developed, the

structural safety of the bridge subject to flood is able to be evaluated rapidly and served as the important

pre-warning to catastrophic disaster as well as decision making in emergency.

Figure 5 Monitoring system implemented at

the Da-Jia-Hsi Bridge

Figure 6 Finite Element Model of the Da-Jia-Hsi

Bridge

Numerical analysis

Similar to the simulation adopted in the Shuan-Yuan Bridge, the 3D finite element model is established as

shown in Figure 6. According to the hydraulic analysis, the maximum flow velocity at the bridge site is 3.45

m/sec. in estimation with respect to the flood with return period of 200 years. The highest water level considered

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is taken as 8.3 m that is just the elevation of the top of embankment. Three different levels of safety factor FS=1.5,

FS=1.2 and FS=1.0, are taken into account for analysis corresponding to the performance levels of safe,

operational and near collapse, respectively. By varying the possible values of FS, V, H and SD, the database of

the relationships between FS, V, H and SD as can be seen in Figures 7 to 9 can be obtained and applicable to the

bridge management.

Application of bridge stability assessment

The availability of Figures (7) to (9) is somewhat indirect to the practical management of the bridge by the

engineers. To enhance their convenience in practical use, the regression analysis of the relationships between FS,

V, H and SD was made to reveal the function of SD in terms of V and H with respect to FS=1.0, FS=1.2 and

FS=1.5, respectively. The function employed is written as:

b1 d1

SD1.0 = ⎡

⎣a1H + cV

+ e1 ( H × V ) + f ⎤

1⎦

exp( g1)

+ h1

for FS=1.0 (16)

b2 d2

SD1.2 = ⎡

⎣a2H + c2V + e2 ( H × V ) + f ⎤

2 ⎦ exp( g2)

+ h2

for FS=1.2 (17)

b3 d3

SD1.5 = ⎡

⎣a3H + c3V + e3 ( H × V ) + f ⎤

3⎦exp( g3)

+ h3

for FS=1.5 (18)

Where ai

~ h i are the unknown parameters to be determined in regressive expression.

The Nelder-MEAD simplex method (Jeffrey C et al. 1998) is quite efficient and was employed for the

optimization solution of Eqs 16 to 18. Table 3 gives the analyzed results of these parameters. The corresponding

R-squared value of three regression results were 0.985, 0.9801 and 0.9796, respectively, and it seems good of

precision obtained.

A set of (V, H, SD) can be detected by the monitoring instruments simultaneously. Through the input of the

monitored V and H into Eqs 15 to 17, the critical scouring depths of SD 1.0 , SD 1.2 , and SD 1.5 are able to be

found immediately and compared with the actual SD monitored. As a result, there are three different situations

will be acquired.

(1) If SD ≦ SD 1.5 , the bridge is regarded as safe level.

(2) If SD 1.5 < SD ≦ SD 1.2 , the bridge is regarded as operational level and a successive attention should be paid

on the bridge if there is any irregular conditions found.

(3) If SD 1.2 < SD < SD 1.0 , the bridge is regarded as near collapse and a necessary inspection and even repairing

or retrofitting work should be done. If necessary, the close of bridge has to be taken into account as well.

(4) If SD 1.0 < SD, the bridge is regarded as unsafe and the repairing or retrofitting work should be done as

soon as possible. The close of bridge has to be carried out.

Figure 7 Relationship between

water level, flow velocity and

scouring depth for near collapse

condition (FS=1)

Figure 8 Relationship between

water level, flow velocity and

scouring depth for operational

condition (FS=1.2)

Figure 9 Relationship between

water level, flow velocity and

scouring depth for safe condition

(FS=1.5)

Table 3 The parameters and R-squared values of three regression results

a i b i c i d i e i f i g i h i R 2

SD 1.0 0.157 0.363 7.787 0.011 -0.011 -2.425 -0.931 -5.414 0.985

SD 1.2 0.097 0.429 7.420 0.009 -0.009 -2.425 -0.973 -5.452 0.9801

SD 1.3 0.092 0.422 6.949 0.010 -0.009 -2.244 -0.948 -4.940 0.9796

CONCLUSIONS

This paper aims at building up a health diagnosis system of flood-resistant capacity assessment for the existing

bridges. The damage assessment procedure was proposed. A case study on the Shuan-Yuan Bridge collapsed in

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the Typhoon Morakot was analyzed and discussed through the proposed method. The reasons to why the bridge

do collapse is able to be obtained. Furthermore, the practical application of the proposed method to the

Da-Jia-Hsi Bridge was conducted and, accordingly, a rapid evaluation system of flood-resistance for the bridge

was established with association with the implementation of the monitoring instruments in site. The results

obtained benefit the bridge management of flood-resistance and the effective strategy to disaster prevention

could be adopted.

Base on results obtained, some conclusions can be drawn as follows:

(1) The water level, the flow velocity and the scouring depth induced by flood give a significant contribution of

dynamic loading as well as support boundary to the bridge structure and can be regarded as the governing

parameters to the flood-resistance of the existing bridges. Since they can be captured already by the

advanced monitoring instruments and, accordingly, served as the input of the health diagnosis system of

bridge scouring. Theoretically, these parameters are not independent. The interaction of them depends on

the riverbed, the quantity as well as velocity of water flow, etc. However, it is difficult to distinguish them

one by one exactly because of the complicated varying situations between them during flood. In order to

evaluate quantitatively their influences on structural safety of the scoured bridge, this paper regarded them

as independent and performed the analysis assuming that one parameter is a variable while the other two are

constants to have corresponding the safety factor of the bridge.

(2) The fundamental entries sensitive to the failure mode of the scoured foundation were summarized as (1)

load-carrying capacity of foundation soil, (2) bending moment capacity as well as shear capacity of

foundation body, (3) deformation of foundation and (4) passive soil pressure of caisson. Through the

proposed method, the possible damage pattern of the scoured bridge can be identified and the corresponding

set of the governing parameters can be determined.

(3) The Shuan-Yuan Bridge was reported its collapse in the Typhoon Morakot with the action of the highest

water level H=6 m , the maximum water velocity V=10m/sec. and the maximum scouring depth of

SD=16.23 m . The analysis result shows that the governing scouring depth D G =10 m is lower than the actual

scoured depth SD=16.23 m so that this bridge loses sufficient bending capacity to sustain the structure during

Typhoon. The conclusion is consistent with the reality.

(4) Associated with the monitoring instruments implemented in the Da-Jia-Hsi Bridge, the damage assessment

system developed can evaluate rapidly the structural safety of the bridge subject to flood and serve as the

important pre-warning to catastrophic disaster as well as decision making in emergency.

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Bridges”, Taiwan.

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Design Guideline ”, Taiwan.

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Bridge”, Taiwan.

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Research Program, Washington, D.C.

Jeffrey, C.L., James, A.R., Margaret, H.W., and Paul, E.W. (1998). “Convergence properties of the nelder-mead

simplex method in low dimensions, SIAM Journal of Optimization”, 9(1), 112-147.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

天幕钢结构屋盖风洞试验与风振分析

朱红飞陈务军何艳丽董石麟

( 上海交通大学空间结构研究中心 , 上海 200030)

摘要 : 本文以上海漕宝路田林地块天幕钢结构工程为背景。首先进行了风洞试验 , 得到该屋盖

结构的平均风压系数 , 并对其特征进行了分析。在此基础上 , 采用频域内截断模态法计算结构的风

振位移响应 , 给出了位移风振系数 , 为工程提供设计依据。

关键词 : 天幕钢结构屋盖风洞试验平均风压系数频域法位移风振系数

WIND TUNNEL TEST AND WIND-INDUCED STRUCTURAL DYNAMIC

RESPONSE ANALYSIS FOR A STEEL VELARIUM ROOF

H. F. Zhu W. J. Chen Y. L. He S. L. Dong

Space Structures Research Centre-SSRC, Shanghai Jiao Tong University, Shanghai 200030, China

Abstract: This paper take the steel velarium structure project at Tianlin district in Caobao road of Shanghai as

the background. First, wind tunnel test is carried out, its process is presented in the paper and the characteristics

of the mean wind pressure coefficients obtained by the test are analyzed. On the basis of the experimental results,

the wind-induced displacement response is evaluated by using the truncated modes method in frequency domain.

Finally,displacement wind vibration coefficients are presented,which provide a basis for the engineering

design.

Keywords: Steel Velarium Roof, wind tunnel test, mean wind pressure coefficient, frequency domain method,

displacement wind vibration coefficient.

一、工程简介

本工程位于上海漕宝路田林地块 , 天幕钢结构屋盖 ( 见图 1) 采用网架体系 , 上表面覆盖采光

板 , 下表面吊顶采用膜结构帷幕。屋面网架平面呈椭圆形 , 长轴 156m, 短轴 62.4m。网架支点最大

跨度 42m, 网架厚度 0.6~3.0m, 网格宽度 2.75m。整个屋盖由 12 根钢管柱支承 , 每根柱子在柱顶

按树形结构分为四个小柱。这 12 根柱子分别起于四个混凝土塔楼的屋面 ( 柱底标高 50.40m)。

天幕钢结构屋盖 , 体型复杂 , 屋盖敞开 , 上下表面均受到风压作用。本项目位于漕宝路和沪闵

高架交叉口 , 周边建筑物分布密集 , 且多为高层结构 , 使得建筑环境也颇为复杂。不同于结构风工

程中一般屋盖结构 , 天幕屋盖仅由 12 根钢结构柱支承 , 同时 12 根柱子又分别位于四个高 50.40m

的混凝土塔楼上 , 使得该屋盖结构的风振响应更加复杂。

-636-

图 1 建筑效果图

二、风洞试验

考虑到建筑形体的复杂性和周边环境的影响 , 设计风荷载通过风洞试验确定。

2.1. 模型制作

本试验取建筑物周边 300m 范围内建立刚性模型 , 采用 1:200 的几何缩尺比 , 该缩尺比能保证风

洞试验阻塞度的要求 , 同时能得到比较高信噪比的试验数据。试验模型主要采用有机玻璃和 ABS 板

两种材料 , 这两种材料均具有质量轻、强度高、高温下可塑性好 , 非常适合制作表面复杂的刚性试

验模型。图 2 为安置在风洞中的试验模型 , 图中结构蓝色部分为天幕钢结构部分。

图 2 试验模型

2.2. 测点布置

天幕屋盖上下表面对应布置测点 , 共布置测点 514 个 , 其中上下表面各 257 个测点。如图所示 :

图 3 测点布置

图 4 试验风向角

-637-

-1.35 -1.23 -0.74

-1.56 -1.39 -0.72

-1.34 -1.39

-1.72 -1.51 -0.71

-1.1

-1.29

-0.98

-1.14

-0.77

-0.62

-0.72

-0.94

-0.66

-0.89

-0.55

-0.51

-0.69

-0.81

-0.49

-0.85

-0.38

-0.39

-0.61

-0.68

-0.34

-0.59

-0.3

-0.27

-0.24

-0.2

-0.17

-0.09

-0.15

-0.05

-0.07

-0.06

-0.1

-0.08

-0.06

-0.04

-0.01

-0.03

-0.01

-0.27 -0.13 -0.06 0.08 0.02 0.09 0.03 0.08 -0.22 -0.2 -0.33

-0.34 -0.03 0.04 0.05 0.09 0.18 0.12 0.09 0.08 -0.05 -0.15 -0.21 -0.31 -0.3

-0.39 -0.26 0.01 0.01 0.04 0.11 0.13 0.18 0.16 0.09 0.06 -0.01 -0.15 -0.21 -0.32 -0.32 -0.3

-0.48 -0.43 -0.18 -0.18 -0.12 -0.05 0.11 0.17 0.06 0.13 0.04 0.06 -0.14 -0.14 -0.29 -0.2 -0.36 -0.37 -0.39 -0.22

-1.06

-0.7

-1.54

-1.08

-1.11

-0.47 -0.41 -0.26 0.01 0.02 0 0.07 0.14 0.09 0.17 0.14 0.01 -0.08 -0.26 -0.36 -0.49 -0.43 -0.49 -0.4

-0.76

-0.57

-0.67

-1.1

-0.74

-0.8

-0.5 -0.17 -0.15 0.03 -0.06 0.03 -0.02 -0.01 -0.03 0.07 -0.18 -0.3 -0.28 -0.28

-0.61

-0.52

-0.53

-0.91

-0.6

-0.77

-0.45

-0.42

-0.41

-0.77

-0.39

-0.54

-0.56 -0.32 -0.12 0 0.03 0.04 0.07 0.07 -0.14 -0.23 -0.26

-0.59

-0.5

-0.32

-0.28

-0.43

-0.32

-0.35

-0.21

-0.18

-0.21

-0.39

-0.12

-0.31

-0.15

-0.17

-0.05

-0.18

-0.08

-0.26

-0.06

-0.34

-0.09

0.03

-0.04

-0.01

-0.06 -0.05 0.01 -0.02 0.04

-0.25

0.03

-0.31

-0.08

-0.24

-0.07

-0.18

-0.09

-0.13

-0.25

-0.21

-0.04

-0.08

-0.09

-0.29 -0.11 -0.08 0 -0.08 -0.04 -0.13 -0.09 -0.31 -0.28 -0.36

-0.34 -0.09 -0.05 -0.07 -0.03 0.04 -0.02 -0.06 -0.07 -0.16 -0.21 -0.27 -0.32 -0.29

-0.35 -0.29 -0.07 -0.08 -0.05 -0.02 -0.05 0.02 0 -0.05 -0.05 -0.1 -0.2 -0.24 -0.29 -0.29 -0.28

-0.48 -0.46 -0.24 -0.21 -0.18 -0.14 0 0.01 -0.11 -0.02 -0.1 -0.1 -0.23 -0.17 -0.29 -0.15 -0.3 -0.31 -0.34 -0.21

-0.47 -0.38 -0.24 0 -0.03 -0.08 -0.04 0.03 -0.02 0.06 0.04 -0.11 -0.18 -0.26 -0.31 -0.41 -0.37 -0.42 -0.38

-0.47 -0.17 -0.16 -0.02 -0.11 -0.04 -0.07 -0.06 -0.1 -0.04 -0.2 -0.28 -0.22 -0.21

-0.5 -0.23 -0.05 0.02 0.03 0.05 0.05 0.04 -0.12 -0.16 -0.2

-0.43

-0.61

-0.43

-0.17

-0.3

-0.37

-0.17

-0.09

-0.24

-0.27

-0.07

-0.12

-0.15

-0.07

-0.09

-0.07

0.02 0.01 0.05 0 0.06

-0.15

-0.13

-0.22

-0.07

-0.05

-0.14

-0.16

-0.01

-0.03

-0.02

0.03

0.1

-0.06

0.15

-0.1

0.01

-0.17

0.12

-0.07

-0.13

-0.11

-0.1

0.04

-0.19

0.07

-0.09

-0.23

-0.16

-0.12

-0.14

-0.2

-0.16

-0.07

-0.13

-0.14

-0.22

-0.06

-0.12

0.04

-0.16

0.15

-0.25

-0.14

-0.24

-0.26

-0.22

-0.21

-0.27

-0.36

-0.22

-0.3

-0.03

-0.21

-0.26

-0.12

-0.23

-0.31

-0.18

-0.26

-0.18

-0.21

-0.14

-0.26

-0.27

-0.13

-0.26

-0.21

-0.16

-0.22

-0.16

-0.14

0.03

-0.21

-0.23

-0.21

-0.3

-0.23

-0.22

-0.32

-0.19

-0.42

-0.21

-0.41

-0.23

-0.39

-0.23

-0.24

-0.38

-0.21

-0.12

-0.27

-0.1

-0.25

-0.16

-0.35

-0.2

-0.16

-0.35

-0.22

-0.42

-0.38

-0.39

-0.32

-0.38

-0.33

-0.34

-0.28

-0.18

-0.19

-0.33 -0.23

-0.3

-0.37 -0.27

-0.24

0.2 0.18 0.18

0.17 0.2 0.2

0.22 0.32

0.27 0.23

0.22 0.23 0.26

0.12

0.24

0 0.25 0.23

-0.22

0.1

0.22

-0.02

0.17

-0.17

0.26

0.15

0.1

-0.12

0.07

0.09

0.05

0.17

0.23

-0.05

0.02

-0.01

-0.11

-0.03

-0.14

0.02

-0.13

0.39

-0.13

-0.21

-0.37

-0.5

-0.1

-0.39

-0.17

-0.29

-0.08

-0.28

-0.08

-0.14

0.01

-0.3

0.01

-0.18

-0.28

-0.16

-0.15

-0.16

-0.26

-0.17

-0.15

-0.14

-0.16

-0.32

-0.15

-0.28

-0.12

-0.13

-0.14

-0.27

-0.31

-0.27

-0.07

-0.11

-0.05

-0.28

-0.11

-0.28

-0.12

-0.17 -0.14 -0.12 -0.1 -0.12

-0.07 -0.1 -0.06 -0.17 -0.14 -0.13 -0.1 -0.08 -0.22 -0.12 -0.16

0.33 -0.42 -0.37 -0.27 -0.11 -0.15 -0.16 -0.13 -0.12 -0.19 -0.2 -0.17 -0.12 -0.17

0.22 0.21 -0.04 -0.16 -0.2 -0.19 -0.15 -0.14 -0.1 -0.11 -0.1 -0.13 -0.16 -0.13 -0.1 -0.15 -0.08

0.21 0.33 0.15 0.08 0.12 -0.29 -0.29 -0.11 -0.23 -0.17 -0.21 -0.19 -0.05 -0.22 -0.21 -0.18 -0.18 -0.15 -0.2

0.18

-0.32

0.34

0.21

0.05

-0.07

0.17 0.22 -0.01 -0.01 -0.2 -0.18 -0.24 -0.21 -0.04 -0.17 -0.11 -0.24 -0.18 -0.27 -0.25 -0.27 -0.25 -0.15 -0.24

0.3

0.05

-0.12

0.22

0.25

0.01

0.28 -0.23 -0.22 -0.27 -0.15 -0.19 -0.23 -0.13 -0.25 -0.24 -0.16 -0.3 -0.24 -0.23

-0.08

0.07

-0.17

-0.06

0.21

-0.13

0.33

-0.16

-0.1

-0.29

-0.38

-0.16 -0.31 -0.13 -0.12 -0.04 -0.11 -0.12 -0.08 -0.08 -0.14 -0.13

-0.14

-0.22

-0.18

-0.23

-0.13

-0.19

-0.12

-0.01

-0.22

-0.02

-0.08

-0.03

-0.2

-0.14

-0.15

-0.13

-0.2

-0.14

-0.13 -0.12 -0.1 -0.09 -0.07

-0.11

-0.12

-0.11

-0.08

-0.09

-0.11

-0.02

-0.06

-0.01

-0.09

-0.12 -0.1 -0.1 -0.07 -0.08

-0.01 -0.12 -0.03 -0.13 -0.12 -0.11 -0.1 -0.06 -0.2 -0.08 -0.14

0.41 -0.54 -0.44 -0.34 -0.08 -0.1 -0.12 -0.1 -0.1 -0.12 -0.16 -0.11 -0.05 -0.16

0.3 0.32 0.01 -0.14 -0.19 -0.16 -0.11 -0.08 -0.05 -0.06 -0.05 -0.06 -0.09 -0.08 -0.05 -0.12 -0.03

0.27 0.38 0.2 0.13 0.18 -0.23 -0.24 -0.07 -0.19 -0.12 -0.15 -0.14 0 -0.17 -0.18 -0.15 -0.15 -0.13 -0.18

0.18 0.2 -0.01 0.06 -0.1 -0.11 -0.2 -0.16 0.02 -0.12 -0.06 -0.18 -0.14 -0.23 -0.21 -0.23 -0.23 -0.15 -0.23

0.2 -0.13 -0.09 -0.17 -0.19 -0.16 -0.18 -0.07 -0.19 -0.19 -0.16 -0.27 -0.21 -0.21

-0.15 -0.15 -0.01 -0.11 -0.03 -0.09 -0.1 -0.08 -0.09 -0.13 -0.11

-0.13 -0.11 -0.09 -0.08 -0.08

-0.12

-0.35

-0.17

-0.27

-0.26

-0.29

-0.27

-0.12

-0.27

-0.1

-0.09

-0.17

-0.12

-0.17

-0.08

0.11

-0.12

0.1

-0.14

-0.29

-0.22

-0.25

-0.15

-0.28

-0.24

-0.19

-0.13

-0.3

-0.24

-0.21

-0.13

-0.32

-0.16

-0.3

-0.09

-0.18

-0.14

-0.16

-0.21

-0.22

-0.24

-0.12

-0.14

-0.22

-0.23

-0.13

-0.05

-0.24

-0.16

-0.07

-0.28

-0.24

-0.26

-0.04

-0.18

-0.08

-0.2

-0.3

-0.16

-0.24

-0.22

-0.32

-0.15

-0.24

-0.25

-0.14

-0.28

-0.14

-0.29

-0.08

-0.05

0.04

-0.19

-0.11

-0.06

0.02

-0.23

-0.11

-0.26

-0.11

-0.26

-0.05

-0.2

-0.31

-0.07

-0.2

-0.32

-0.08

-0.2

-0.1

-0.19

-0.16

-0.23

-0.03

-0.34

-0.14

-0.25

-0.02

-0.32

-0.29

-0.03

-0.68

-0.28

-0.02

-0.67

-0.38

0.16

-0.3

-0.36

0.17

-0.27

-0.16

-0.2

2.3 试验工况

风平行于结构长轴方向为 0° 风向角 , 风向角按照逆时针方向每间隔 10° 增加一个风向角 , 共

计 36 个风向角工况 , 图 4 为模型方位与试验风向角。

2.4 求解方法

本文所有系数和风压值的正负号意义如下 : 正号表示风压沿结构表面法向向内 , 即对表面产生

压力 ; 负号表示风压沿结构表面法向向外 , 即对结构表面产生吸力。

采用无量纲风压系数来描述结构表面的风压

p − p p − p

= =

p 1

− ps

ρU

i s i s

其中 C pi 为 i 点的风压系数 ,p i 为测点 i 处的压力 ,p s 为参考点静压 ,p t 为参考点总压 ,ρ 为空

气密度 ,U r 为参考点风速。

体型系数可由测点的风压系数计算得到 :

(1)

⎛Zi

⎞

μsi

= C

pimean ⎜

⎟

⎝ ⎠

2α

(2)

其中 μ 为测点 i 的体型系数 ,C pimean 表示 C pi 的均值 , 称为风压系数均值 , Z 为测点 i 所处的

高度 ,α 为地貌粗糙度指数 ,C 类地貌 α 取 0.22。

三、风洞试验结果

根据上述风洞试验模型及计算方法 , 取项目所在地的 50 年一遇基本风压为 0.55kN/m 2 ,C 类地

貌 , 测得 36 个风向角下的每个测点的平均风压系数 , 图 5、图 6、图 7、图 8 分别给出了 0°、20

°、70°、90° 风向角下屋盖的测点平均风压系数等值线图 :

-1.58 -1.33 -0.71 0.01 0.04 0.03 0.01 0.03

(a)0° 上表面

(b)0° 下表面

图 5 0° 风向角下天幕钢结构屋盖测点平均风压系数等值线图

-1.78 -1.41 -0.7 0 -0.01 -0.07 -0.11 -0.11

(a)20° 上表面

(b)20° 下表面

图 6 20° 风向角下天幕钢结构屋盖测点平均风压系数等值线图

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-1.18 -0.75 -0.7 -0.76 -0.86 -0.89 -0.84 -1.01 -1.16 -1.01 -1.07 -1.29 -1.18 -1.24 -1.26 -1.22 -1.02 -1.06 -0.85 -0.91 -0.91 -0.83 -0.69 -0.77 -0.82 -0.69

-0.55 -0.77

-0.7

-0.69 -0.88 -0.69

-0.57

-0.58

-0.59

-0.47

-0.45

-0.35

-0.3

-0.33

-0.32

-0.31

-0.3

-0.37

-0.38

-0.35

-0.32

-0.36

-0.33

-0.3

-0.31

-0.36

-0.33

-0.31

-0.32

-0.42

-0.44

-0.42

-0.37

-0.4

-0.27

-0.33

-0.41

-0.37

-0.41

-0.3

-0.32

-0.37

-0.45

-0.33

-0.37

-0.36

-0.38

-0.47

-0.39

-0.34

-0.44

-0.52 -0.49 -0.53 -0.64 -0.73 -0.7 -0.72 -0.69 -0.58 -0.54 -0.52

-0.42

-0.52

-0.43

-0.66 -0.6 -0.69 -0.74 -0.89 -1 -0.97 -1.21 -1.43 -1.3 -1.29 -1.5 -1.34 -1.33 -1.23 -1.07 -0.75 -0.8 -0.53 -0.56 -0.5 -0.47 -0.35 -0.47 -0.48 -0.4

-0.9

-1.37

-0.37

-0.35

-0.4

-0.31

-0.43

-0.28

-0.38

-0.42

-0.44

-0.42

-0.33

-0.48

-0.27

-0.28

-0.5

-0.43

-0.3

-0.29

-0.43

-0.46

-0.51

-0.34

-0.53

-0.44

-0.42

-0.32

-0.34

-0.61

-0.54

-0.32

-0.36

-0.32

-0.68

-0.34

-0.49

-0.45

-0.41

-0.37

-0.76

-0.45

-0.46

-0.49 -0.44 -0.47 -0.52 -0.62 -0.62 -0.73 -0.77 -0.84 -0.8 -0.83

-0.67

-0.64

-0.53

-0.51

-0.55

-0.65

-0.63

-0.59

-0.61

-0.64 -0.64 -0.68 -0.78 -0.76 -0.77 -0.83 -0.86 -0.91 -1 -0.92 -0.96 -0.98 -0.85

-0.69

-0.89

-0.97 -0.79 -0.89 -0.74 -0.92 -0.99 -1.04 -1.03 -1.13 -0.99 -1.12 -1.09 -0.84 -0.83

-0.79

-0.97

-0.62

-1.6

-0.89

-0.95

-0.59

-1.13

-0.68

-1.7

-0.62

-1.25

-1.21 -0.9 -0.66 -0.57 -0.57 -0.57 -0.49 -0.51 -0.53 -0.54 -0.55

-0.49

-1.43

-0.28

-1.1

-0.35

-2.41

-0.52

-1.61

-0.28

-1.01

-0.28 -0.32 -0.27 -0.17 -0.2

-0.29

-2.2

-0.34

-0.26

-1.6

-0.59

-0.15

-1.44

-0.43

-0.06

-0.01

-0.24

-2.12

-0.22

-0.13

-0.15

-1.81

-1.25

-0.13

-0.41

0.08

-1.7

-0.07

0.05

-0.17

-0.72

0.04

-0.12

-0.23

0.17

-0.86

-0.94

-1.4

-0.66 -0.7 -0.74 -0.82 -0.89 -1.01 -1.15 -1.05 -1.04 -1.06 -1.06 -1.01 -0.97 -0.96 -0.94 -0.92 -0.85

-0.89 -0.94 -0.97 -0.8 -0.83 -0.93 -0.99 -1.04 -1.08 -1.03 -1.08 -1.21 -1.18 -1.13 -1.1 -1.14 -0.98 -1 -0.9

-0.66 -0.74 -0.81 -0.98 -0.94 -0.99 -0.95 -0.91 -0.85 -0.85 -0.67 -0.66 -0.63 -0.51

-0.69 -0.77 -0.92 -0.97 -1.12 -1.31 -1.38 -1.3 -1.21 -1.14 -1.05 -0.88 -0.74 -0.69 -0.64 -0.54 -0.5

-0.9 -1.04 -1.12 -0.96 -1.02 -1.1 -1.21 -1.32 -1.27 -1.19 -1.14 -1.2 -1.03 -0.89 -0.84 -0.82 -0.62 -0.65 -0.52

-1.01 -0.89 -1.02 -0.85 -1.08 -1.24 -1.23 -1.17 -1.21 -0.97 -1.01 -0.87 -0.61 -0.54

-0.9

-0.27

-1.18

-1.17 -0.92 -0.69 -0.68 -0.66 -0.66 -0.54 -0.52 -0.38 -0.37 -0.35

-0.25

-1.13

-0.06

-0.9

-0.18

-1.52

-0.28

-1.08

-0.2

-0.8

-0.31 -0.37 -0.3 -0.21 -0.21

-0.19

-1.5

-0.24

-0.12

-1.02

-0.51

-0.13

-1.05

-0.47

-0.08

0.02

-0.16

-1.66

-0.25

-0.15

-0.19

-1.49

-0.87

-0.13

-0.09

-0.27

0.08

-1.54

-0.06

0.03

-0.08

-0.46

0.09

-0.29

0.17

-0.41

-0.46

-0.42

-0.38

-0.26

-0.37

-0.38

-0.47 -0.4

-0.45

-0.34 -0.42 -0.35

-0.3

-0.47

-0.51

-0.68 -0.6

-0.72

-0.67 -0.69 -0.72

-0.13 -0.07

-0.04

-0.13 -0.12 -0.04 -0.06 0.02 -0.15 -0.2 0.02 -0.22 -0.21 -0.09 -0.22 -0.14 -0.06 -0.19 0 -0.1 -0.18 -0.24 -0.3 -0.2 -0.21 -0.23 -0.05 -0.18

-0.3 -0.25

-0.21

-0.14

-0.06

-0.09

-0.05

0.05

-0.28

-0.22

-0.14

-0.29

-0.23

-0.2

-0.24

-0.19

-0.13

-0.19

-0.26

-0.14

-0.26

-0.18

-0.1

-0.26

-0.23

-0.22

-0.18

-0.04

-0.25

-0.03

-0.16

-0.23

-0.25

-0.2

-0.01

-0.18

-0.23

-0.42

-0.22

-0.17

-0.21

-0.14

-0.21

-0.15

-0.22

-0.11

-0.18

-0.3

-0.17

-0.17 -0.24 -0.19 -0.22 -0.17 -0.11 -0.04 0.14 -0.32 -0.13 -0.12

-0.34

-0.27

-0.06

-0.27

-0.4

-0.13

-0.04 -0.3 -0.25 -0.24 -0.13 -0.14 -0.1 -0.02 0.15 0.04 -0.18 -0.01 0.08 -0.25

-0.15

-0.05 -0.09 -0.14 -0.17 -0.18 -0.18 -0.17 -0.14 -0.06 0.01 0.1 0.03 -0.07 0 0 -0.24 -0.13

-0.19 -0.15 -0.19 -0.06 -0.19 -0.16 -0.24 -0.22 -0.06 -0.17 -0.06 -0.23 -0.33 -0.38 -0.3 -0.32 -0.48 -0.32 -0.33

-0.24

-0.29 -0.3 -0.23 -0.3 -0.28 -0.25 -0.29 -0.17 -0.38 -0.6 -0.36 -0.49 -0.41 -0.42

-0.49

-0.45

-0.56

-0.45

-0.52

-0.13

-0.34

-0.38

-0.47

-0.64

-0.29

-0.16

-0.41 -0.41 -0.23 -0.21 -0.1 -0.14 -0.3 -0.42 -0.28 -0.35 0.02

-0.16

-0.31

-0.5

-0.52

-0.21

-0.57

-0.19

-0.35

-0.44

-0.36

-0.29 -0.23 -0.46 -0.56 -0.48

-0.43

-0.29

-0.75

-0.44

-0.59

-0.51

-0.45

-0.71

-0.4

-0.53

-0.32

-0.7

-0.96

-0.68

-0.96

-0.85

-1.05

-0.52

-0.44

-0.31

-0.85

-0.6

-1.12

-1.49

-0.94

-0.69

-0.42

-1.44

-0.47

-1.19

-0.98

-0.78

-0.24

-0.18

-0.15

-0.25

-0.2

-0.22

-0.2

-0.17

-0.15

-0.21

-0.22

-0.17

-0.24

-0.21

-0.19

-0.25

-0.21

-0.19

-0.17

-0.09

-0.22

-0.15

-0.18

-0.23

-0.24

-0.22

-0.17

-0.2

-0.26

-0.45

-0.23

-0.17

-0.29

-0.2

-0.24

-0.2

-0.28

-0.18

-0.24

-0.26

-0.24

-0.21 -0.23 -0.25 -0.19 -0.16 -0.13 -0.12 -0.09 -0.19 -0.19 -0.2

-0.27

-0.19

-0.13

-0.21

-0.2

-0.23

-0.13

-0.21

-0.22 -0.29 -0.25 -0.24 -0.15 -0.17 -0.15 -0.14 -0.16 -0.18 -0.21 -0.19 -0.17 -0.23

-0.17 -0.24 -0.18 -0.19 -0.22 -0.21 -0.21 -0.18 -0.13 -0.13 -0.16 -0.18 -0.19 -0.18 -0.14 -0.24 -0.22

-0.36 -0.3 -0.24 -0.24 -0.05 -0.27 -0.25 -0.03 -0.25 -0.24 -0.13 -0.26 -0.22 -0.26 -0.26 -0.08 -0.22 -0.28 -0.26 -0.27 -0.25 -0.31 -0.32 -0.09 -0.23

-0.26 -0.23 -0.26 -0.14 -0.25 -0.18 -0.26 -0.26 -0.11 -0.27 -0.22 -0.37 -0.3 -0.37 -0.35 -0.39 -0.41 -0.26 -0.4

-0.36

-0.36 -0.38 -0.3 -0.32 -0.32 -0.32 -0.4 -0.32 -0.44 -0.43 -0.33 -0.4 -0.36 -0.41

-0.32

-0.71

-0.35

-0.75

-0.39

-0.17

-0.21

-0.55

-0.39

-0.67

-0.39

-0.29

-0.45 -0.48 -0.3 -0.32 -0.37 -0.44 -0.45 -0.36 -0.23 -0.3 -0.27

-0.24

-0.4

-0.43

-0.5

-0.39

-0.75

-0.23

-0.33

-0.4

-0.51

-0.62 -0.65 -0.57 -0.54 -0.52

-0.46

-0.51

-0.44

-0.42

-0.74

-0.71

-0.5

-0.84

-0.83

-0.6

-0.38

-0.47

-0.88

-0.8

-0.74

-0.73

-0.92

-0.84

-0.6

-0.3

-1.01

-0.66

-1.12

-0.81

-0.73

-0.62

-0.59

-0.99

-0.51

-0.7

-0.69

-0.7

-0.24

-0.22

-0.21

-0.08 -0.68

-0.14 -0.72 -0.32

-0.31

-0.24

-0.3

-0.17 -0.51

-0.17 -0.75 -0.35

-1.49 -1.06 -0.81 -0.39 -0.44 -0.51 -0.55 -0.58

-0.38 -0.17 -0.15 -0.23 -0.2 -0.14 -0.03 0.07

(a)70° 上表面

(b)70° 下表面

图 7 70° 风向角下天幕钢结构屋盖测点平均风压系数等值线图

-0.85 -0.89 -0.76 -0.45 -0.49 -0.51 -0.5 -0.49

-0.44 -0.22 -0.3 -0.2 -0.17 -0.13 -0.09 -0.1

(a)90° 上表面

图 8 90° 风向角下天幕钢结构屋盖测点平均风压系数等值线图

图 5(a) 为 0° 风向角 ( 气流从左向右流动 ) 下天幕屋盖的平均外压 ( 上表面所承受的风压 )

系数。由于周边建筑环境的影响 , 平均外压相对建筑中轴线没有对称分布。屋盖中间表现为风压 ,

最大值 0.18, 其他均表现为风吸。绝对值最大的平均负风压系数发生在迎风面檐口 , 为 -1.58 左右 ,

沿着气流方向 , 平均风压系数的绝对值减小。背风面平均负风压系数在 -0.3~-0.4 左右。图 5(b) 为

0° 风向角下平均内压系数 , 与外压相似 , 也不呈对称分布。屋盖中间和背风面表现为风吸 , 檐口

部分平均负风压系数的绝对值比中间大。不同的是 , 下表面迎风端表现为风压 , 平均风压系数最大

值达 +0.39, 这是由于下部混凝土塔楼的存在造成向下的气流受阻 , 产生向上的气动力。和平均外压

系数相比 , 平局内压系数绝对值相对较小。

图 6(a)、(b) 分别为 20° 风向角下天幕屋盖的平均外压系数和平均内压系数 , 基本和 0° 风

向角相似。其平均外压基本表现为风压 , 绝对值最大值出现在迎风面檐口 , 达 -1.78。平均内压在迎

风端最大值为 +0.38。

图 7(a)、(b) 分别为 70° 风向角下天幕屋盖的平均外压系数和平均内压系数。平均外压全部

表现负压 , 绝对值最大值出现在迎风面檐口 , 达 -2.41。和 0°、20° 风向角不同 ,70° 风向角下平

均内压也全部表现为负值 , 迎风面檐口绝对值最大值为 -1.49。90° 风向角 ( 图 8(a)、(b)) 和 70

° 风向角基本相似。

其他风向角均与上诉情况相似。

综上所诉 , 天幕钢结构屋盖的外压一般表现为风吸 , 最大值发生在迎风面檐口处。平均内压系

数会受内部结构的影响 , 表现为风压或者风吸 , 但其绝对值一般小于外压系数。最不利风向角位于

60°~90° 之间。

四、风振响应及风振系数计算方法

4.1. 风振响应采用频域分析法

(b)90° 下表面

网架脉动风振有限元动力方程可写成如下形式 :

[ M ]{ u&& } + [ C]{ u& } + [ K]{ u} = P( t)

(3)

u 分别为结

式中 :[M]、[C]、[K] 分别为网壳的总体质量矩阵、阻尼矩阵和刚度矩阵 ,{ u&& } 、{ u& } 、{ }

构的位移、速度、加速度矢量 ,{ P()

t } 为时变的荷载矢量。

本文水平脉动风速功率谱采用工程界广泛采用的 Davenport 谱 , 把式 (3) 振型分解之后 , 得到下

式 :

-639-

式中 :[ φ ] 为归一化模态矩阵 ,{ }

{ u} [ φ]{ q}

q 为广义坐标向量。

= (4)

则结构在脉动风作用下的运动方程可以化为关于广义坐标的独立振动方程的组合 :

{} q 2ξ

[ ]{} q [ ] 2

{} q { F}

式中 :{ F } 为相应的广义动力荷载向量。

{ F} [ φ] { P}

{ F } 的空间 - 频率谱密度矩阵可表示为 :

+ Ω + Ω = (5)

∗T

[ SP

] = S { P} ⋅ { P}

= S φ ⋅ F ⋅ F ⋅ φ

[ φ] [ D][ RC][ D][ φ] Svf ( w) [ Q] Svf

( w)

式中 : [ Q] = [ φ] T

[ D][ R ][ D][ φ]

, [ ]

= (6)

( ) ([ ] { } { } [ ])

= ⋅ = ⋅ (7)

D 是对角矩阵 , 其对角线元素为 dii = Ai μziμf μrw0

μ , w 0

为建筑

物所在地区的基本风压 , μ 为结构体形系数 , μ 为风压高度变化系数 , μ 为峰值因子 , μ 为带保

证因子的脉动风压系数 , R c

为脉动风荷载的相干系数。

在独立模态空间下 , 振动方程 (6) 的稳态频响函数为 :

j ( w) =

2 2

( j = 1, 2, L , m)

(8)

w − w + 2ξ

w w⋅i

因此系统输入与输出的关系式为 :

{ q} = [ H]{ F}

(9)

则广义位移响应的谱密度矩阵为 :

⎣

⎡Sq⎦ ⎤ = S( {}{} q ⋅ q ) = S( [ H][ SF]

⎡⎣H⎤⎦) = [ H][ Q] ⎡⎣H⎤⎦

⋅ Svf

( w)

(10)

式中 : ⎡H

⎤ H 的共轭矩阵。

⎣ ⎦

为 [ ]

进而可求得物理坐标空间下 , 网架的位移响应谱密度矩阵 :

那么 , 结构位移响应的均方值为 :

( φ φ ) [ φ] ⎡ ⎤

q [ φ]

[ ] [ ] {}{} [ ]

S = S ⋅ q ⋅ q = ⎣S

⎦ (11)

+∞

{ σ

u} ([ u]

)

= ∫ diag S dω

(12)

−∞

均方响应反应了随机过程偏离均值的程度 , 结构在总风力下的设计位移可表示为 :

u = u + μ⋅ σ ⋅∗ sign u

(13)

{} { } { u} ({ })

式中 : 符号 ⋅∗ 表示矩阵的相应元素相乘而不是矩阵相乘。

4.1. 风振系数的定义

脉动风引起的结构响应其性质是动力的 , 与静力分析相比复杂的多。因此为了工程设计简便易

行 , 人们总是希望将结构的动力响应用静力响应来表示 , 这一思想可通过风振系数来实现。现行规

范中 , 结构的风振系数定义为 :“ 总风荷载的概率统计值与静风荷载的概率统计值的比值 ”, 此处的

总风荷载包括平均风荷载和脉动风荷载两部分。从风致振动的节点位移出发 , 此处引入节点位移风

振系数。

节点位移风振系数可定义为 :

ui + μσ ⋅

⋅sign( ui)

βU

= = (14)

本文的风振系数以此定义为准。

五、风振系数分析

根据荷载规范基本风压取为 0.55kN/m 2 ,C 类地貌。结构的体型系数根据前面的风洞试验结果 ,

提供了 36 种风向角下的测点体型系数。采用频率内的摸态分析法对此结构进行风振响应分析 , 由

于此类结构是频率密集性结构 , 需要考虑多振型参与并且应该考虑各阶振型相互之间的耦合。本文

-640-

对此结构考虑了前 30 阶模态。

计算结构的频率时 , 考虑了恒载作为附加质量 , 根据结构设计要求取恒载上下表面各为

0.3kN/m 2 , 且不考虑活荷载产生的附加质量。根据模态对整个结构在脉动风作用下应变能的贡献多

少来定义模态对结构风振响应的贡献 , 网架结构的前 30 阶模态的自振频率和前 30 阶模态应变能列

于表 1 中。这里能量系数是指第 j 阶模态贡献的能量与系统总能量的比值 , 即 Ej

/ E 。

用在频域内的振型分解法 , 结构在脉动风下的位移响应可表示为

q { ϕ }

是第 j 阶广义位移 , j

其中 , j

是第 j 阶模态向量。

结构系统总的应变能用模态方式表示为

{ x } = ∑ q { ϕ }

(15)

E E k q

1 2

j 2 j j

其中 , E E

为系统总能量 ,

j 为第 j 阶模态贡献的能量。

5.1. 模型选取

= = (16)

计算模型取天幕钢结构屋盖和下部钢结构柱 ( 见图 6), 这里不考虑下部混凝土部分的影响。

5.2. 自振频率及其对系统的能量贡献

图 9 计算模型

表 1 前 30 阶模态的自振频率及其对系统能量贡献

振型阶数频率 (hz) 能量贡献振型阶数频率 (hz) 能量贡献振型阶数频率 (hz) 能量贡献

1 1.3944 0.0076 11 4.0146 0.0002 21 5.4087 0.0000

2 1.9637 0.0030 12 4.0708 0.0001 22 5.5243 0.0001

3 2.4191 0.1561 13 4.2245 0.0000 23 5.5524 0.0001

4 2.6120 0.0149 14 4.4927 0.0277 24 5.7041 0.0000

5 2.6805 0.0005 15 4.5584 0.0130 25 5.8931 0.0001

6 2.7527 0.0305 16 4.6357 0.0000 26 5.9033 0.0002

7 3.0406 0.0328 17 4.8118 0.0000 27 6.2023 0.0000

8 3.2453 0.0002 18 5.0597 0.0000 28 6.3306 0.0015

9 3.7571 0.0006 19 5.1136 0.0091 29 6.3956 0.0005

10 3.9696 0.0062 20 5.2813 0.0001 30 6.5835 0.0000

六、风振系数结果

根据上述的风振响应分析方法和风振系数的定义 , 选取了 0°、20°、70°、90°、110°、160

°、180°、200°、250°、270°、290°、340° 这 12 种风向角 , 计算得到这 12 个风向下的节点

位移风振系数。图 7 为在 0°、20°、70°、90° 风向角下的风振系数的分区域表示 :

-641-

0° 风向角屋盖上表面位移风振系数 0° 风向角屋盖下表面位移风振系数

20° 风向角屋盖上表面位移风振系数 20° 风向角屋盖下表面位移风振系数

70° 风向角屋盖上表面位移风振系数 70° 风向角屋盖下表面位移风振系数

七、结语

90° 风向角屋盖上表面位移风振系数 90° 风向角屋盖下表面位移风振系数

图 10 四种风向角下天幕钢结构屋盖位移风振系数

根据风振系数统计结果 , 基本都大于 1.6, 局部达到 3.0+。和一般大跨度屋盖结构类似 , 该屋

盖檐口风振系数分布大于中间部分 , 屋盖下表面的风振系数和上表面的相差不大 , 和封闭结构相比 ,

敞开结构屋盖下表面风振系数偏大 , 在工程设计时需要注意。建议在结构设计时 : 方法一 , 屋盖外

延可取平均风振系数 2.3~2.5, 中间部分可取平均风振系数 1.8~2.0; 方法二 , 按整个屋盖在平均值范

围内取统一值 2.2; 方法三 , 直接用计算所得的节点位移风振系数。

参考文献

[1] 何艳丽 , 董石麟 , 龚景海 . 空间网格结构频域风振响应分析模态补偿法 . 工程力学 , 2002. 19(4): 1-6.

[2] 黄本才 , 汪丛军 . 结构抗风分析原理及应用 , 上海 : 同济大学出版社 , 2008.

[3] 任涛 . 单层索网体育场罩蓬结构分析及施工模拟研究 , 上海 : 上海交通大学 , 2008.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

考虑剪切影响的钢筋混凝土柱荷载 - 变形特性分析

张勤 , 贡金鑫

( 大连理工大学建设工程学部 , 辽宁大连 116024)

摘要 : 准确计算钢筋混凝土柱在地震荷载作用下的荷载 - 变形曲线 , 是进行钢筋混凝土结构抗

震性能分析的基础。传统的荷载 - 变形特性分析中 , 通常忽略剪力的影响 , 这不能准确反映弯剪荷载

作用下构件的荷载 - 变形特性。为建立包含剪力影响的柱荷载 - 变形关系 , 在弯曲理论分析方法基础

上 , 通过对现有试验数据的分析 , 提出了一种模拟轴向和水平荷载作用下柱荷载 - 变形曲线的简化计

算方法。计算结果与试验滞回曲线的包络线吻合较好。

关键词 : 钢筋混凝土柱荷载 - 变形特性剪切性能

LOAD-DEFORMATION ANALYSIS OF REINFORCED CONCRETE COLUMNS

INCLUDING SHEAR EFFECTS

Q. ZHANG 1 and J. X. GONG 1

1 Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

Abstract: Accurate calculation on lateral load-deformation relationship of reinforced concrete columns under

seismic load is the foundation of seismic performance analysis of reinforced concrete structure. In traditional

lateral load-deformation relations analysis, the effects of shear behavior is usually ignored due to the shear

behavior of cracked reinforced concrete section is not well understood, and the lateral load- deformation

properties of reinforced concrete members under combined flexural and shear load can not be accurately

described. In order to predict the lateral load-deformation relations of reinforced concrete columns including

shear effects, a simplified calculation method is proposed on basis of flexure theory and existing test data

analysis in this paper. The analytical results are compared with the experimental data and consistent agreement

is achieved.

Keywords: Reinforced concrete, column, lateral load-deformation properties, shear behavior.

一、前言

近年来 , 全球地震灾害频发 , 人们对于建筑抗震的要求与意识也逐渐提高。震害调查及试验研

究表明 , 钢筋混凝土柱作为混凝土结构的主要承重构件 , 在强烈地震作用下易发生破坏 , 这会给人

们的生命财产造成巨大损失 , 因而其在强烈地震作用下的受力性能 , 一直倍受关注 [1-3] 。钢筋混凝土

柱在单调荷载作用下的荷载 - 变形关系可用来建立恢复力模型及进行抗震性能评估 , 所以准确模拟地

震荷载作用下柱的荷载 - 变形特性具有重要意义。

在传统的柱荷载 - 变形曲线分析中 , 主要侧重对柱弯曲特性的模拟 , 往往忽略剪切对柱荷载 - 变形

特性的影响 , 这主要是由于钢筋混凝土柱通常设计为以弯曲破坏为主 , 剪力影响影响较小 ; 同时由

于混凝土开裂后的剪切特性比较复杂 , 目前还没有统一的理论来阐述这一机理 , 比较难以模拟剪力

对柱荷载 - 变形特性的影响。而实际上 , 钢筋混凝土柱在地震荷载作用下会出现弯曲、弯剪以及剪切

三种破坏模式 , 当剪切破坏起控制作用时 , 则剪力对柱荷载 - 变形特性的影响则不能忽略。另外 , 由

收稿日期 :2010-

基金项目 : 国家自然科学基金重点资助项目 ( 90815027 )

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作者简介 :* 张勤 (1983-), 男 , 安徽庐江人 , 博士研究生 , 主要从事钢筋混凝土结构抗震研究 (E- mail:

zhangqin8190@163.com .);

于柱端钢筋滑移引起的滑移变形影响也需要包含在柱荷载 - 变形特性分析中。

Setzler 和 Sezen [4] 考虑弯曲、滑移和剪切变形的组合建立包含剪切影响的荷载 - 变形关系 ;

Mostafaei 和 Kabeyasawa [5] 考虑轴力 - 剪力 - 弯矩相互作用 , 应用修正压力场理论来描述剪切特性 , 建

[6]

立了包含轴力 - 剪力 - 弯矩相互作用的荷载 - 变形关系 ; 魏巍巍和贡金鑫结合修正压力场理论和弯曲

理论建立了双轴应力的截面分析模型 , 再根据平衡条件确定了柱水平荷载与总体变形间的关系。这

些方法从不同角度考虑了剪切对柱荷载 - 变形特性的影响 , 分析过程均较复杂且存在大量的计算 , 不

易被工程技术人员所掌握。鉴于此 , 本文通过分析已有的钢筋混凝土柱试验结果 , 并结合柱的受力

的特点 , 建立包含剪切及滑移影响的经验修正系数 , 通过修正按传统弯曲理论建立的柱荷载 - 弯曲变

形曲线来得到包含剪切影响的荷载 - 变形全曲线。

二、荷载 - 弯曲变形特性

2.1. 材料本构关系

(1) 钢筋应力 - 应变关系采用 Esmaeily & Xiao [7] 的三线段强化模型 , 并认为钢筋受压时的本构

关系与受拉时相同。

(2) 考虑箍筋对核心混凝土的约束作用 , 保护层和核心混凝土分别采用文献 [8] 中的 Marder 等

的无约束混凝土模型和 Saatcioglu-Razvi 的约束混凝土模型。

2.2. 荷载 - 弯曲变形计算

钢筋混凝土柱在单调荷载作用下的荷载 - 弯曲变形关系 , 可采用经典的弯曲理论求得弯矩 - 曲率

关系 , 再利用塑性铰模型建立曲率与柱顶位移的关系式 , 进而得到荷载 - 弯曲变形关系。分析中假定

截面保持平截面 , 并不考虑混凝土的抗拉作用。柱截面轴力和弯矩的平衡关系可表示为 :

∫ ∫ c ∫ s

(1)

A Ac

N = σ dA = σ dA + σ dA

∫ ∫ c ∫ s

(2)

A Ac

M = σ xdA = σ xdA + σ xdA

式中 : A c

、 A s

分别为纵向钢筋和混凝土的面积 ; σc()

ε 、 σs()

ε 为应变为 ε 时混凝土和钢筋的应力 ,

可根据材料本构关系求得 , 其中的材料应变 ε 可根据平截面假定由初始压应变 ε

和曲率 ϕ 计算得到。

通过编制的程序 , 逐步增大曲率 ϕ , 并使之满足上述平衡方程式即可得到固定轴力作用下 , 侧

向荷载单调增加的柱截面弯矩 - 曲率关系 , 再根据塑性铰模型 , 即可建立柱荷载 - 变形关系。本文分

析中 , 柱构件的塑性铰长度按 Priestley & Park [9] 的模型计算。

三、考虑剪力影响的荷载 - 变形特性

钢筋混凝土柱的总侧向变形主要由弯曲变形、剪切变形和滑移变形组成 [4] 。对于弯曲破坏柱 ,

总侧向变形以弯曲变形为主 , 剪切和滑移变形分量所占比例较小 , 有时甚至可以忽略 ; 而对于弯剪

破坏柱 , 总侧向变形中的剪切变形分量所占的比例较大 , 特别是在纵筋屈服后 , 因而导致按弯曲理

论计算的曲线明显高于试验曲线 , 不能反映柱后期变形阶段的真实特性。本文分析主要针对弯曲、

弯剪破坏柱 , 对于纵筋屈服前就发生剪切破坏的柱不是本文关注的重点。分析中全部柱以弯剪破坏

看待 , 弯曲起主要作用时考虑剪切修正的系数小 , 剪切作用明显时修正的系数大。曲线修正分两个

阶段进行 , 在柱纵筋屈服前后分别给出相应的经验公式。

3.1. 纵筋屈服前的变形修正

纵筋屈服前可认为柱处于弹性反应阶段 , 该阶段柱的总侧向变形 Δ

和弯曲变形 Δ

之间的关系

可采用系数 k Δ

( χ ) 来修正。系数 k Δ

( χ ) 考虑了剪切及滑移变形的影响并与反映柱基本特征的参数 χ 有

关 , 可通过柱实际屈服位移 Δ

和计算的屈服位移 Δ

之间的关系来建立 , 即 :

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Δry

Δ = k Δ

( χ)

Δ = Δ

T f f

(3)

式中 : Δ fy

为按弯曲理论计算的柱屈服位移 , 对应于截面最外侧纵向受拉钢筋首次屈服时的位移 ; Δ ry

为柱实际屈服位移 , 对应于弯矩为 M 时试验骨架曲线上的实测位移 ( M 为按弯曲理论计算的屈服

弯矩 , 对应于截面最外侧纵向受拉钢筋首次屈服时的弯矩 )。

本文选取了 34 根钢筋混凝土柱试件进行分析 , 各试件尺寸及相关参数见表 1, 其中试件尺寸及

保护层厚度的单位均为 mm, 强度单位为 MPa, 力单位为 kN; 试件破坏模式划分以 PEER 结构抗震

[10]

性能试验数据库使用手册的报道为准。考虑剪跨比、轴压比、箍筋配筋率、纵筋配筋率、纵筋屈

服强度这些主要因素对钢筋混凝土柱剪切和滑移变形的影响 , 参数 χ 采用以下形式 :

( ρ ) 1.5

χ=

( n + 0.01) ρ λ

0.2 0.1 2

(4)

式中 : λ 为柱剪跨比 ; n 为柱轴压比 ; ρ c

为箍筋配筋率 ; ρ s

、 f y

分别为纵筋配筋率和屈服强度。结

合试验数据 , 对试验柱屈服位移比值 ( Δ ry

/ Δ

) 与参数 χ 之间的关系进行回归分析 , 可得到如下统

计公式 :

式中 : 各符号意义同式 (4)。

k Δ

Δry

( χ ) = = 0.147χ+

表 1 试件尺寸及相关参数

纵筋参数

箍筋参数

试件

试件混凝土

轴压力 / 保护层

屈服极限

屈服

名称

尺寸强度

配筋率配箍率轴压比厚度

强度强度

强度

Ang-U3 400×400×1600 23.6 427 670 0.0151 320 0.0133 1435/0.380 24.5

Ang-U4 400×400×1600 25.0 427 670 0.0151 280 0.0101 840/0.210 22.5

Gill-S2 550×550×1200 41.4 375 635.6 0.0179 316 0.0111 2680/0.214 38.0

Tanaka-U1 400×400×1600 25.6 474 721 0.0157 333 0.0138 819/0.200 40.0

Tanaka-U2 400×400×1600 25.6 474 721 0.0157 333 0.0138 819/0.200 40.0

Tanaka-U3 400×400×1600 25.6 474 721 0.0157 333 0.0184 819/0.200 40.0

Tanaka-U4 400×400×1600 25.6 474 721 0.0157 333 0.0184 819/0.200 40.0

Tanaka-U5 550×550×1650 32.0 511 675 0.0125 325 0.009 968/0.100 40.0

Tanaka-U6 550×550×1650 32.0 511 675 0.0125 325 0.009 968/0.100 40.0

Atalay-3S1 305×305×1676 29.2 367 578 0.0163 363 0.0081 267/0.098 32.0

Atalay-5S1 305×305×1676 29.4 429 657 0.0163 392 0.0081 534/0.195 32.0

Atalay-6S1 305×305×1676 31.8 429 657 0.0163 392 0.0048 534/0.181 32.0

Atalay-9 305×305×1676 33.3 363 563 0.0163 392 0.0081 801/0.259 32.0

Atalay-10 305×305×1676 32.4 363 563 0.0163 392 0.0048 801/0.266 32.0

Atalay-11 305×305×1676 31.0 363 563 0.0163 373 0.0081 801/0.278 32.0

Atalay-11 305×305×1676 31.8 363 563 0.0163 373 0.0048 801/0.271 32.0

Saatcioglu-U1 350×350×1000 43.6 430 - 0.0320 470 0.0035 0/0.000 22.5

Saatcioglu-U3 350×350×1000 34.8 430 - 0.0320 470 0.0071 600/0.141 22.5

Saatcioglu-U4 350×350×1000 32.0 438 - 0.0320 470 0.0106 600/0.153 22.5

Matamoros-C520N 203×203×610 48.3 586 739 0.0193 407 0.0159 285/0.143 38.3

Matamoros-C520S 203×203×610 48.3 587 740 0.0193 408 0.0159 285/0.143 39.0

(5)

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Matamoros-C540N 203×203×610 38.1 572 729 0.0193 514 0.0122 569/0.362 20.7

Matamoros-C540S 203×203×610 38.1 573 730 0.0193 515 0.0121 569/0.362 20.7

Nagasaka-1063 200×200×300 21.6 371 541 0.0133 344 0.0080 147/0.170 12.0

Nagasaka-1932 200×200×300 21.0 371 541 0.0133 344 0.0139 294/0.350 12.0

Ohue-2D16RS 200×200×400 32.0 369 - 0.0201 316 0.0066 183/0.140 11.0

Ohue-4D13RS 200×200×400 29.9 370 - 0.0265 316 0.0067 183/0.150 12.5

Ono-CA025C 200×200×300 25.8 361 533 0.0213 426 0.0104 265/0.257 19.0

Sezen-No.1 457×457×1473 21.1 434 645 0.0247 476 0.0020 667/0.151 65.1

Sezen-No.4 457×457×1473 21.8 434 645 0.0247 476 0.0020 667/0.146 65.1

Lynn-3CLH18 457×457×1473 26.9 331 496 0.0303 400 0.0008 503/0.089 38.1

Lynn-2CLH18 457×457×1473 33.1 331 496 0.0194 400 0.0008 503/0.073 38.1

Lynn-3CMH18 457×457×1473 27.6 331 496 0.0303 400 0.0008 1512/0.262 38.1

Lynn-3CMD12 457×457×1473 27.6 331 496 0.0303 400 0.0017 1512/0.262 38.1

3.2. 纵筋屈服后的荷载修正

纵筋屈服后柱进入弹塑性反应阶段 , 该阶段通过建立柱水平荷载 F

的修正系数 kF

( μ

Δ ) 来考虑剪

切及滑移的影响 , 即在相同的变形下 , 柱的水平荷载按下式计算 :

F = k ( μ ) F

(6)

Tf F Δ f

式中 : F f

为变形为 Δ

时按弯曲理论计算的柱的水平荷载 ; F Tf

为变形为 Δ

=Δ

时考虑剪切及滑移影

响计算的柱的水平荷载 ; kF

( μ

Δ ) 为根据试验结果确定的荷载修正系数 , 即相同变形下柱的实际水平

荷载与按弯曲理论计算的水平荷载之比 , 与柱的相对位移 μ

( μ Δ

定义为柱屈服后的位移 Δ

与实际

屈服位移 Δ

的比值 ) 有关。

通过对纵筋屈服后柱试验骨架曲线与所计算的荷载 - 变形曲线的分析 , 荷载修正系数 k ( μ ) 可写

成以下表达形式 :

⎡

1.5 1.5

a( n+ b)

( ρ

sf )

⎤

a( n+ b)

( ρsfy)

⎡ π⎤

( μ

Δ)

= cos ⎢

( μ 1) ( 1) 0,

0.7 Δ

− ⎥

0.7

− ∈

( )

⎢ 2 ⎥

(7)

⎢ ρcfty λ ⎥ ρcf

⎣ ⎦

⎣

⎦

式中 : f ty

为箍筋屈服强度 ; a 、 b 为经验系数 , 经对试验数据的优化分析得 a = 0.252 , b = 0.197 。

3.3. 计算结果与试验结果比较

图 1、2 分别给出了部分弯曲和弯剪破坏柱的荷载 - 变形曲线的计算结果及试验结果比较。图中

曲线 1 是按弯曲理论计算的荷载 - 弯曲变形曲线 , 这一曲线未考虑剪切及纵筋滑移的影响 , 因而不能

完全反映柱的真实受力特性 , 特别是对于纵筋屈服后发生剪切破坏的柱 ; 曲线 2 是在曲线 1 基础上

采用式 (12) 和式 (16) 修正后的荷载 - 变形曲线 , 反映的是柱侧向荷载与总侧向变形之间的关系 ,

包含了弯曲、剪切及纵筋滑移对柱抗震性能不同程度的影响。

由图 1 可看出 , 对于发生弯曲破坏的柱 , 曲线 1 和曲线 2 比较接近 , 这表明剪力对弯曲破坏柱

的荷载 - 变形特性影响较小 , 有时甚至可以忽略。而图 2 中曲线 1 和曲线 2 相差较大 , 特别是在纵筋

屈服后 , 未考虑剪力的影响的荷载 - 变形曲线 ( 曲线 1) 明显高于试验滞回曲线 , 而考虑剪力影响的

荷载 - 变形曲线 ( 曲线 2) 与试验结果吻合较好 , 这表明对弯剪破坏柱荷载 - 变形特性进行分析时需考

虑剪力的影响。采用本文方法可以非常简便地模拟包含剪力影响的柱荷载 - 变形特性。

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Ang-U3

Lateral load/kN

200

150

100

Tanaka-U1

Lateral load/kN

200

150

100

-60 -40 -20 0 20 40 60

Displacement/mm

-50

-100

-90 -60 -30 0 30 60 90 120 150

-50

Displacement/mm

-100

-150

-200

Test curve

predicted curve 1

predicted curve 2(Modified)

-150

-200

Test curve

predicted curve 1

predicted curve 2(Modified)

Atalay-5S1

Lateral load/kN

Atalay-9

Lateral load/kN

-60 -40 -20 0 20 40 60

-20

Displacement/mm

-40

-60 -40 -20 0 20 40 60

-20

Displacement/mm

-40

-60

-80

Test curve

predicted curve 1

predicted curve 2(Modified)

-60

-80

Test curve

predicted curve 1

predicted curve 2(Modified)

图 1 弯曲破坏柱荷载 - 变形曲线计算结果与试验结果比较

Nagasaka-1932

Lateral load/kN

150

100

Ohue-2D16RS

Lateral load/kN

120

-15 -10 -5 0 5 10 15

Displacement/mm

-50

-20 -15 -10 -5 0 5 10 15 20

Displacement/mm

-40

-100

-150

Test curve

predicted curve 1

predicted curve 2(Modified)

-80

-120

Test curve

predicted curve 1

predicted curve 2(Modified)

Sezen-No.1

Lateral load/kN

300

200

100

-80 -60 -40 -20 0 20 40 60 80

-100

Displacement/mm

Lynn-3CMH18

Lateral load/kN

400

300

200

100

-40 -30 -20 -10 0 10 20 30 40 50

Displacement/mm

-100

-200

-300

Test curve

predicted curve 1

predicted curve 2(Modified)

-200

-300

-400

Test curve

predicted curve 1

predicted curve 2(Modified)

图 2 弯剪破坏柱荷载 - 变形曲线计算结果与试验结果比较

四、结语

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地震荷载作用下 , 钢筋混凝土构件会因服役期过长或配箍不足而发生包含剪切影响的破坏 , 此

时需考虑剪力对构件荷载 - 变形特性的影响。本文在截面分析的基础上 , 结合现有的柱反复荷载试验

数据 , 提出了一种考虑剪力影响的柱荷载 - 变形特性分析方法 , 以该方法建立的荷载 - 变形曲线与试

验滞回曲线包络线基本吻合 , 可用于地震荷载下柱荷载 - 变形性能的分析。

参考文献

[1] 李乔 , 赵世春 , 何川 , 等 . 汶川大地震工程震害分析 , 西南交通大学出版社 , 2008.

[2] Sezen, H. Seismic Behavior and Modeling of Reinforced Concrete Building Columns, University of California, Berkeley,

2000, Berkeley.

[3] 贡金鑫 , 王雪婷 , 张勤 . 从汶川地震灾害看现行国内外桥梁抗震设计方法 . 大连理工大学学报 , 2009, 49(5):

739-747.

[4] Setzler, E. J. and Sezen, H. Model for the Lateral Behavior of Reinforced Concrete Columns Including Shear

Deformations. Earthquake Spectra, 2008, 24(2): 493-511.

[5] Mostafaei, H, Kabeyasawa, T. Axial-Shear-Flexure Interaction Approach for Reinforced Concrete Columns. ACI

Structural Journal, 2007, 104(2): 218-226.

[6] 魏巍巍 . 钢筋混凝土结构基于修正压力场理论的承载力和变形研究 . 大连理工大学博士学位论文 , 2011, 大连 .

[7] Esmaeily-GH, A. and Xiao, Y. Seismic Behavior of Bridge Columns Subjected to Various Loading Patterns, Pacific

Earthquake Engineering Research Center, University of California, Berkeley, 2002, Berkeley.

[8] 贡金鑫 , 魏巍巍 , 赵尚传 . 现代混凝土结构基本理论及应用 , 中国建筑工业出版社 , 2009, 北京 .

[9] Prietley, M. J. N. and Park, R. Strength and Ductility of Concrete Bridge Columns under Seismic Loading. ACI

Structural Journal, 1987, 84(8): 61-76.

[10] Berry, M., Parrish, M. and Eberhard, M. PEER Structural Performance Database User’s Manual (Version 1.0),

University of California, Berkeley, 2004, Berkeley.

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The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

SYSTEMATIC CATEGORIZATION OF STRUCTURAL COMPONENTS IN

STONECUTTERS BRIDGE

K. C. Lin 1 , X. W. Ye 1 , Y. Q. Ni 1 and K. Y. Wong 2

Department of Civil and Structural Engineering,

The Hong Kong Polytechnic University, Hong Kong. Email: ceyqni@polyu.edu.hk

Bridge & Structures Division, Highways Department,

The Government of The Hong Kong Special Administrative Region, Hong Kong.

ABSTRACT

This paper presents a systematic categorization of structural components in Stonecutters Bridge (SCB). Based

on the corresponding locations and dominated load resistances, the structural components in SCB are

categorized into five groups. Various structure types, component types and elements within each group are

identified and categorized. Each structural element is assigned with a unique tag-name, which provides the

information of global coordinate and local coordinate. The categorization results are well organized into a

database, in which the tag-name setups a linkage between the element locations in a 3D bridge model and other

element information, such as design drawings, physical properties, structural health monitoring (SHM) data, and

inspection\maintenance records. A bridge inventory system is proposed by integration of the database and the

3D bridge model. The sophisticated systematic categorization work will facilitate in searching and locating

structural elements conveniently and accurately during the execution of structural condition rating and bridge

inspection and maintenance.

KEYWORDS

Systematic categorization, database, structural health monitoring, bridge management system, bridge condition

rating.

INTRODUCTION

Long-span bridges play important roles in the transportation system of Hong Kong with a special

geomorphology of mountains and isolated islands. By spanning natural barriers, these bridges serve as vital

links in the transportation system with one of the heaviest traffic in the world. The smooth operation of bridges

is a necessary support to the local prosperous economy. Any damage or traffic posting on the bridges will

inevitably interrupt the public transportation system carrying daily necessities of life and imports and exports of

the world’s busiest seaport. Bridge management system (BMS) is a well-accepted tool in the management of

bridge operation, with the aid of several systems including bridge structural health monitoring (SHM) system,

and bridge condition rating (BCR) system. A high effective BMS is necessary in order to minimize the

interruption of traffic due to bridge inspection and maintenance activities. One requirement of the BMS is to

provide necessary information to the engineers to quickly locate the object of inspection and maintenance on

site. Therefore, systematic categorization of bridge structural components should be carried out under this

consideration. The categorization also provides a basis for the development of BCR (Wong, 2006). Bridge load

rating in BCR should be performed at different load effects including tension, compression, bending, etc.; load

rating of a bridge is governed by a component having the lowest rating; factors in load rating codes depend on

load effects of components considered (AASHTO, 1994; 2003). As specified in BS5400 Part 3, limit states in

flexure, shear, and torsion should be considered in the design and evaluation of beams; compression is the

dominated load effect in compression members; and tension strength governs the capacity of members subjected

to axial tension (BSI, 2006). Therefore, it is necessary to carrying out a systematic investigation of the

components according to their load effects under in-service condition.

Methods for systematic categorization of bridge structures for different purposes have been available in

literature. Geier and Wenzel (2002) classified bridge structures according to their conditions evaluated from

dynamic tests. Three classes of structures were identified, and the classification results provided a basis for

prioritizing bridge maintenance actions required. Cheng et al. (2009) categorized a reinforced concrete bridge

into different structure categories and associate elements. Possible damage patterns for each bridge element

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under earthquake were defined. Maintenance activities corresponding to each element were suggested and

subdivided into post-earthquake repair and rehabilitation. Ramamoorthy (1999) separated an armored vehicle

launched bridge into different components according to their functions in the structure. These components were

classified into either functional components or structural components. Likely damage scenarios of each

component were defined according to the Federal Highway Administration (FHWA) manual (Hartle et al., 1990).

Typical damage mechanisms were identified as battle damage, fatigue cracking, large deformation, plastic

cracking, missing components, environmental damage and separation of components.

The aim of this paper is to develop a method of systematic categorization for the purpose of facilitating bridge

management and bridge condition rating. Both the locations and load effects of structural components are

considered in the categorization. The locations of the components will provide necessary information for bridge

inspection and maintenance, while the consideration of load effects provide a basis for executing bridge

condition rating under each category of components. A bridge inventory system is proposed based on the

categorization work.

METHOD OF SYSTEMATIC CATEGORIZATION OF BRIDGE STRUCTURE

According to global locations and dominated load effects under in-service condition, the structural components

of a typical cable-stayed bridge can be systematically categorized into five groups, namely, stay cable group

(SCG), vertical structure group (VSG), horizontal structure group (HSG), articulation group (ATG), and

foundation group (FDG). Each group is divided into several structure types; each structure type is further

divided into several component types; finally, different elements within each component type can be identified

and categorized. The bridge elements include pipes, plates, bars, strand wires, walls, corbels, webs, tendons,

dampers, buffers, bearings, movement joints, pile caps, piles, and others. Each structural element is assigned

with a unique tag-name, which provides four types of information, namely, (i) group name, (ii) structure type,

(iii) component type or segment which serves as local coordinate, and (iv) element. The categorization results

are stored in a database, as illustrated in Figure 1 and Figure 2.

Figure 1 General principle of component categorization

Figure 2 Sketch of component group

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The database has five groups of tables aiming at the five categorization groups, respectively. In the storage of

each categorization group, a table is sub-divided into several partitioned tables according to global coordinates,

such as back span and main span. Therefore, the global and local coordinates of an element can be retrieved

from the database given the tag-name of target element.

The load effects of an element vary widely based on its location and functions in the bridge structure. Typical

load effects include tension, compression, shear, bending, and torsion. Possible load effects for each bridge

element within each group are presented in Table 1.

Table 1 Dominated load effects of structure components

Group Name

Load Effects

Tension

Stay Cable Group

Shear

Compression

Vertical Structure Group

Shear

Bending

Shear

Horizontal Structure Group Tension

Compression

Torsion

Articulation Group

Shear

Compression

Foundation Group

Shear

SYSTEMATIC CATEGORIZATION OF STONECUTTERS BRIDGE

Stonecutters Bridge (SCB) in Hong Kong, with a main span stretching 1,018 m between the 298 m high tapering

towers, is the second longest spanning cable-stayed bridge in the world carrying dual three-lane highway traffics.

The structural configuration of the bridge is illustrated in Figure 3. After completed in 2009, it serves as a key

part of transportation network linking the Hong Kong International Airport to the urban areas.

Concrete

Steel

Concrete

West Tower

East Tower

Backspan

Mainspan

Backspan

P4W

P3W

P2W P1W

P1E P2E P3E P4E

North Side

South Side

Figure 3 Structural configuration of Stonecutters Bridge

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According to the proposed systematic categorization method, the structural components of SCB are categorized

into five groups. The Stay Cable Group refers to the stay cables and their attachments to decks and towers,

which is composed of the following 4 component types, namely, Stay Tower Anchor, Stay Deck Anchor, Stay

Cable Strand, and Stay Cable Dampers. Elements within each component type are identified in Table 2. The

Vertical Structure Group refers to the towers and piers, which are composed of the following structure types,

namely, piers P1W, P2W, P3W, P4W, P1E, P2E, P3E, P4E, West Tower (WT) concrete part and composite part,

East Tower (ET) concrete part and composite part, and Cable Anchor Box at each towers. Each pier and tower is

divided into several segments from bottom to top. Each segment has elements of inner walls, outer walls, and

quadrates; The Horizontal Structure Group refers to the bridge-deck structure, which is composed of the

following structure types: Longitudinal Steel Box-girder, Cross Steel Box-girder, Longitudinal Hybrid

Box-girder, Longitudinal Concrete Box-girder, Cross Concrete Box-girder, and Concrete Cross Heads. Elements

within each structure type are identified in Table 3. The Articulation Group refers to the structure components of

Hydraulic Buffers, Lateral Bearings, and Movement Joints. These components are identified as the elements.

The Foundation Group refers to the foundation structure, which is composed of two structure types, namely, Pile

Cap and Bored Pile. Pile Cap has elements of quadrates, while Bored Pile has elements of piles. The

categorization results of SCB are summarized in Table 4.

Component Type

(a) Stay Tower Anchor

(b) Stay Deck Anchor

(d) Stay Cable Dampers

Table 2 Elements within stay cable group

List of Element

Anchorage Tube, Anchorage Bearing Plate, Flange Plate, Internal

Guide, Girder Side Socket, Neoprene Sleeve, Sear Bar, Stainless Steel

Clamp, Stainless Steel Baseplate, and Stainless Steel Gusset Plates

Guide Pipe, Guide Pipe Extension, Guide Pipe Slice, Tower Side

Socket, Anchor Plate, Web Plate, Flat Bar, Flange Plate, Seal Plate,

Interior Stiffener Plate, and Exterior Stiffener Plate

High-Density Polyethylene Tube, Filament Tape, and Galvanized Wire

Internal Damper Housing, Vertical Hydraulic Damper, Horizontal

Hydraulic Damper, Stopper Device, Cable Clamp, Protection Sheet,

Damper Bracket, Rubber Damper, and Backup Material

Table 3 Elements within horizontal structure group

Component Type

List of Element

Deck Plate, Nose Plate, Edge Plate, Bottom Plate, Outer Central Web,

(a) Longitudinal Steel

Inner Central Web, Inclined Inner Web, Vertical Inner Web, Diaphragm,

Box-girder

Trough, and T-Stiffener

(b) Cross Steel Box-girder Web Plate, Flange Plate, Diaphragms, and T-Stiffeners

Strand Prestressing Tendons, Shear Connectors , and Prestressing Bars

Box-girder

(d) Longitudinal Concrete

Box-girder

Top Slab, Bottom Slab, Inclined Inner Slab, Outer Web, Central Web,

Inner Web, External Tendons and Internal Tendons

(e) Cross Concrete Top Slab, Bottom Slab, Inclined Inner Slab, Outer Web, Central Web,

Box-girder

Inner Web, and Transverse Prestressing Tendon

(f) Concrete Cross Heads Bottom Slab, Inclined Slab, Outer Web, and Central Web

Table 4 Summary of systematic categorization work

Group Name Number of Structure Type Number of Element

Stay Cable Group 4 6,889

Vertical Structure Group 16 1,447

Horizontal Structure Group 6 12,316

Articulation Group 3 18

Foundation Group 2 148

The results of systematic categorization of structural components are stored in a relational database IBM DB2.

The database also maintains other information of the structural components such as design drawings, design

physical properties, photos from visual inspection, maintenance records, monitoring data from SHM system,

condition rating from BCR system. The unique tag-name of a bridge element serves as the linkage of all the

information. An inventory system is proposed to retrieve, update, and display these data in a user-friendly

interface.

The inventory system comprises a user-friendly interface, a database, and a 3D model. The running of this

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system begins with the input at the interface by users; the interface automatically transforms the input into

Structured Query Language (SQL) command which can be recognized by the database; the SQL command is

transferred to the database; the database retrieves data records according to the command; the retrieving result is

transferred back to the interface; the interface calls the ANSYS software and display the retrieving results in

3D-model. The logical design of the inventory system is illustrated in Figure 4.

In the interface, a navigation tree is provided, in which each categorization group and element has a check box.

Check boxes control whether to show up the corresponding components or not in the 3D model. Users can pick

up target elements in the visualized 3D model by zooming in/out, moving, and mouse-clicking. In addition,

elements can be extracted from the database by running a SELECT command by giving its tag-name. In general,

four methods of input are supported in the interface: (a) select the targeted tag-names/No. from the drop-down

list box, or keyboard input these values; (b) mouse click the targeted structural components in the 3D models; (c)

write the Structured Query Language (SQL) command in the database; and (d) select the targeted tag-names/No.

from the retrieving result. Once picked up, the target element is highlighted in the model, and its associate

information is singled out and displayed, following the tag-name that links 3D model to other information. The

output in the user interface includes tag-names, physical properties, maintenance records, monitoring data, and

other information of structural components. The output can be displayed in four manners: (a) display the

retrieving data records in tables; (b) store the retrieving data records in Text/Excel format; (c) display the

retrieving data records in figures; and (d) display the structural components in 3D models.

Figure 4 Logical design of inventory system

The inventory system integrates information of bridge structural components, and displays this information in a

visualized 3D model. It provides comprehensive information of the bridge structure in a user-friendly interface.

Bridge manager can retrieve and update information from this system without soaking within abundant and

tenebrous design drawings and database. Thus, it will greatly facilitate bridge management activities.

CONCLUSIONS

A systematic categorization of bridge structural components for the purpose of facilitating bridge management

and bridge condition rating is proposed in this paper. Structural components of cables-stayed bridges are

categorized into five groups according to locations as well as dominated load effects under in-service condition.

The method is applied to the categorization of structural components in cable-stayed Stonecutters Bridge (SCB).

The categorization results and other information of structural components are well-organized in a relational

database. An inventory system is proposed to retrieve, update, and display these data in a visualized 3D model.

It will facilitate front-line engineers in carrying out bridge inspection and maintenance activities required.

ACKNOWLEDGMENTS

The work described in this paper is part of a collaborative research project “Development of structural health

rating system for inspection and maintenance of Stonecutters Bridge under in-service condition” (Project No.

K-ZB52) which is jointly supported by the HKSAR Highways Department and HKPolyU. The authors

gratefully acknowledge the advice, cooperation and financial support from the HKSAR Highway Department.

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REFERENCES

AASHTO (1994). Manual for Condition Evaluation of Bridges, Washington, D.C., USA.

AASHTO (2003). Manual for Condition Evaluation and Load and Resistance Factor Rating of Highway

Bridges, Washington, D.C., USA.

BSI (2006). BS5400: Steel, Concrete and Composite Bridges, Part 3: Code of Practice for Design of Steel

Bridges, British Standards Institution, London, England.

Cheng, M.Y., Wu, Y.W., Chen, S.J. and Weng, M.C. (2009). “Economic evaluation model for post-earthquake

bridge repair/rehabilitation: Taiwan case studies”, Automation in Construction, 18(2), 204-218.

Geier, R. and Wenzel, H. (2002). “Bridge classification based upon ambient vibration monitoring”, Proceedings

of the First European Conference on Structural Health Monitoring, Paris, 981-988.

Hartle, R.A., Amrhein, W.J., Wilson, K.E., Baughman, D.R. and Tkacs, J.J. (1990). Bridge Inspector’s Training

Manual 90, FHWA, Washington, D.C.

Ramamoorthy, V. (1999). Development of a Decision Support System for Assessment of Mobile Bridges, Master

Thesis, West Virginia University, Morgantown, West Virginia, USA.

Wong, K.Y. (2006). “Criticality and vulnerability analyses of Tsing Ma Bridge”, Proceedings of the

International Conference on Bridge Engineering – Challenges in the 21st Century, Hong Kong, China.

(CD-ROM).

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固顶型变壁厚钢储罐在不均匀沉降下的稳定性能

王震雷翔赵阳谢新宇

( 浙江大学建筑工程学院 , 浙江杭州 310058)

摘要 : 立式钢储罐是一类典型的金属薄壳结构 , 其对罐周不均匀沉降十分敏感。以往的研究大

多基于理想化的谐波沉降 , 但对于高度非线性的薄壁壳体 , 通过谐波沉降叠加得到实际沉降下的结

构显然是不合适的。本文将储罐结构的实际沉降分为两类 —— 整体不均匀沉降和局部不均匀沉降 ,

研究工程中典型几何尺寸的固顶型钢储罐在不均匀沉降下的非线性响应和稳定性能。研究表明 , 在

整体不均匀沉降下 , 罐壁上部首先出现剪压屈曲 , 屈曲后基本不能继续承载。在局部不均匀沉降下 ,

储罐的结构响应与沉降周向跨度有关 , 跨度较小时 , 较小的沉降即能使罐壁产生跳跃屈曲 , 同时较

大的屈曲变形说明局部不均匀沉降对于储罐稳定性较为不利。

关键词 : 变壁厚固顶罐圆柱壳不均匀沉降非线性分析稳定性

STABILITY BEHAVIOR OF FIXED-ROOF TAPERED-WALL STEEL TANKS UNDER

DIFFERENTIAL SETTLEMENT

WANG Zhen 1 , LEI Xiang 1 , ZHAO Yang 1 and XIE Xinyu 1

1 College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

Abstract: Vertical cylindrical steel tanks are sensitive to differential settlement beneath the tank wall. Most

previous studies were based on idealized harmonic settlement. However, for thin shell structures of high

nonlinear behavior, it is obviously inappropriate to obtain the results under real settlement by simple summation

of harmonic solutions. Real settlement of steel tanks can be grouped into two types - global differential

settlement and local differential settlement. This paper examines the nonlinear response and stability behavior of

fixed-roof steel tanks under both types of settlement. It is shown that, for tanks under global differential

settlement, shear buckling occurs at the upper part of the tank wall, and cannot carry more for post-buckling

behavior; while for tanks under local differential settlement, the structural response is related to the degree of

localization. At highly localized settlement, local snap-through buckling occurs at the tank wall, and the

corresponding large deformation of the tank wall shows that the localized settlement is adverse to the stability

behavior of the tank.

Keywords: Tapered-wall fixed-roof tanks, cylindrical shells, differential settlement, nonlinear analysis, stability.

一、前言

钢储罐广泛应用于石油、化工、电力等诸多工业领域 , 是金属薄壳结构应用于实际工程的典型

例子。建造在软土地基上的大型钢储罐会产生各种沉降变形 , 包括整体均匀沉降、整体平面倾斜沉

降、罐周不均匀沉降以及底板的碟形沉降等 , 其中罐周不均匀沉降对结构的影响最为不利 [1,2] 。

目前对不均匀沉降下钢储罐性能的研究主要可分为两类 , 一类是基于壳体理论的解析解 , 如文

献 [3] 采用薄膜理论、修正 Donnell 理论和无延伸理论给出了储罐应力和变形的理论解 ; 另一类则是

利用有限元方法 , 如文献 [4] 对均匀壁厚及变壁厚的储罐进行了有限元分析 , 并与理论解进行了比较。

基金项目 : 国家自然科学基金资助项目 (50778159)

作者简介 : 王震 (1985-), 男 , 浙江台州人 , 博士研究生。

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这些研究都局限于单一谐波沉降下储罐结构的线性行为 , 文献 [5] 则对谐波沉降下的储罐结构进行了

非线性分析 , 而文献 [6] 对一假定的局部不均匀沉降下的储罐结构进行了几何非线性分析 , 指出了以

往研究中存在的问题。但是 , 储罐结构的实际不均匀沉降并不是以单一的谐波形式出现的 , 而研究

谐波沉降的主要理由在于通过对不同谐波沉降下解的叠加得到实际沉降下的结果。简单的叠加显然

只对结构的线性行为适用 , 对非线性程度不高的情形误差也许还不太大 , 但钢储罐这类薄壳结构的

结构响应表现出很强的非线性 [5,7] , 因此对谐波沉降的分析无法真正了解储罐在不均匀沉降下的结构

性能。

本文将储罐结构的实测沉降分为两类 —— 整体不均匀沉降和局部不均匀沉降 , 以实际工程中典

型几何尺寸的变壁厚固顶罐为分析对象 , 通过非线性有限元分析系统考察固顶罐在不均匀沉降下的

稳定性能。

二、沉降形式与分析模型

5.1. 典型沉降形式

根据对钢储罐罐周实测沉降数据的分析 [8] , 可分为整体不均匀沉降和局部不均匀沉降。整体不

均匀沉降按 n=1 谐波项的大小又可分成三种情况 , 本文分别以 G1、G4、G7 罐沉降所示为例 , 并令

沉降峰值 S max =1.0 获得沉降模型 ( 如图 1 所示 ), 分别对应整体倾斜、整体倾斜为主和局部不均匀

特征较显著的整体不均匀沉降。对于代表局部不均匀沉降的 G8 罐沉降采用相同方法得到 G8 沉降模

型 ( 如图 2 所示 )。

0.02

沉降值 (mm)

0.01

0.00

0 60 120 180 240 300 360

-0.01

罐底周向展开角 (°)

沉降值 (mm)

0.01

0.00

0 60 120 180 240 300 360

-0.01

罐底周向展开角 (°)

-0.02

Smax=tav

-0.02

Smax=tav

(a) G1 沉降模型

(b) G4 沉降模型

0.02 Smax=tav

0.02

Smax=tav

沉降值 (mm)

0.01

0.00

0 60 120 180 240 300 360

-0.01

罐底周向展开角 (°)

-0.02

沉降值 (mm)

0.01

0.00

0 60 120 180 240 300 360

-0.01

罐底周向展开角 (°)

-0.02

图 1 整体不均匀沉降模型

5.2. 分析模型

根据某实际工程的大型钢储罐 , 选取结构几何尺寸典型的固顶罐为分析对象。储罐结构尺寸如

图 3 所示 , 高径比 H/R=0.57, 平均壁厚 av =0.0181m, 径厚比 R/ av =1423, 顶盖中心高 16.24m、

坡度 3.6°、厚度 0.007m( 本文模型中忽略顶盖 )。

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采用通用有限元软件 ABAQUS 对结构进行非线性分析。由于储罐属于薄壁结构 , 径厚比较大 ,

故分析中只考虑几何非线性。采用 4 节点薄壳单元 S4R5, 通过弧长法跟踪沉降 - 位移全过程。结构

有限元模型如图 3 所示 , 采用柱坐标 x-y-z(x- 径向 ,y- 切向 ,z- 竖向 ), 忽略顶盖模型 , 用约束顶边

径向和切向线位移来替换模拟固顶罐作用 , 同时约束罐底边径向和切向线位移 , 在竖向施加相应的

沉降模型。储罐结构的材料参数 : 钢材弹性模量 E=2.09GPa, 泊松比 v=0.3。

2.12

2.5

H=14.62

2.5

3.6°

0.009

0.01

0.013

0.018

0.031

0.033

R=25.75

图 3 实例固顶罐结构尺寸及有限元模型

三、整体不均匀沉降下的非线性分析

图 4 给出了实例储罐在三种沉降模型 G1、G4、G7 下的典型沉降 - 位移曲线 , 纵坐标是罐周最

大沉降点的相对沉降值 , 横坐标是罐壁最大变形点的相对径向位移。储罐在罐周整体不均匀沉降作

用下 , 开始结构响应基本呈线性 , 且罐壁变形也较小 , 当沉降到达特征点 A、B、C( 对应 G1、G4、

G7 的临界沉降值 ) 时罐壁发生屈曲。局部屈曲后 , 储罐承受的最大沉降值基本不变 , 而罐壁变形不

断发展 , 因此可将罐壁屈曲对应的临界沉降值作为结构的极限沉降值。对比这三种沉降下的临界沉

降值 , 可知 G1 沉降 ( 整体倾斜沉降 ) 对应的临界沉降值 (A 点 ,S max,cr = 0.76 av ) 明显高于 G4 和

G7 沉降的结果 (B 点 ,S max,cr = 0.28 av ;C 点 ,S max,cr = 0.21 av )。这表明整体不均匀沉降的局部不

均匀特征较显著时 , 结构的极限沉降值较低 , 即整体不均匀沉降中的局部化特征对储罐结构的稳定

性十分不利 , 此时很小的沉降值即能使罐壁产生屈曲变形。

0.8

Smax/tav

0.6

0.4

0.2

0.0

0.0 0.5 1.0 1.5 2.0 2.5

Ux/tav

图 4 整体不均匀沉降下储罐的沉降 - 位移曲线

图 5 给出了储罐在不同沉降模型下的屈曲变形图。三者的共同表现均是在罐壁上部出现了较多

的斜向波状凹陷变形。分析结构屈曲时的罐壁应力变化可知 , 由于顶盖对罐顶的径向和切向作用 ,

使得不均匀沉降作用下的变壁厚储罐在罐壁上部产生较大的剪切应力 , 随着沉降增加 , 罐壁剪切应

力不断增大 , 而罐壁上剪切应力最大的区域首先发生罐壁凹陷变形 ; 凹陷变形的产生使得罐壁上相

应位置上出现了剪切应力和轴压应力的集中 , 并且随着变形发展应力值也不断提高。因此可知罐壁

屈曲是该两种应力共同作用的剪压屈曲 , 其屈曲变形则为斜向凹陷。

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(a) G1 沉降 (b) G4 沉降 (c) G7 沉降

图 5 不同沉降下的屈曲变形图

四、局部不均匀沉降下的非线性分析

G8 沉降 ( 如图 2) 下储罐的非线性分析表明 , 结构的非线性响应类似于整体不均匀沉降中 G7

沉降模型的结果。为更充分体现局部不均匀沉降的影响 , 本文将 G8 沉降模型中周向角在 298°~343

° 之外的沉降值缩小 10 倍 , 并对周向角在 298°~343° 之间的沉降值进行谐波化处理 , 形成修正的

G8-45° 沉降模型 ( 如图 6)。

沉降值 (mm)

0.02

Smax=tav

0.01

0.00

0 60 120 180 240 300 360

-0.01

罐底周向展开角 (°)

Smax/tav

0.8

0.6

0.4

0.2 A

0.0

0 3 6 9 12 15 18 21 24 27 30

Ux/tav

0.10

0.05

-0.02

图 6 修正的 G8-45° 沉降模型

0.00

0 2 4 6 8

图 7 修正的 G8-45° 沉降下储罐的沉降 - 位移曲线

图 7 和图 8 分别给出了储罐的沉降 - 位移曲线和结构变形发展图。结构初始变形表现出与罐底沉

降相对应的罐壁波状变形 ; 当沉降峰值达到约 0.10 av 时 ( 图 7 中的 A 点 ), 罐壁上部产生斜向凹陷

变形并迅速加深扩大 ( 图 8a— 图 8b), 这在沉降 - 位移曲线上表现为跳跃屈曲。分析结构屈曲时罐壁

应力变化可知 , 罐壁初始凹陷是由集中剪应力引起的剪压变形 , 随后迅速转变为轴压变形。当轴压

凹陷屈曲发生之后 , 应力随着变形的发展得以重新分布使得应力集中得以扩散 —— 原来轴压应力很

大的屈曲区域的应力开始减小 , 而变形周围区域的轴压应力开始增大 —— 从而导致储罐沉降能力在

发生罐壁跳跃屈曲后还能继续上升。这在沉降 - 位移曲线上表现为跳跃屈曲后的强化表现 , 结构变形

发展则表现为凹陷变形不断扩大加深最终形成 “V” 形轴压屈曲变形 ( 图 8c— 图 8d)。

与整体不均匀沉降相比 , 典型局部沉降作用下储罐表现的比较敏感同时罐壁局部轴压屈曲后结

构表现出明显的后屈曲强度 , 往往能继续加载至数倍壁厚同时产生显著的罐壁变形。因此在储罐设

计时应充分利用结构的后屈曲强度 , 可以将整体失稳破坏作为结构极限承载力状态 , 以罐壁变形控

制作为结构的正常使用极限状态。

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(a) A 点 (b) B 点 (c) C 点 (d) D 点

五、局部沉降周向角的影响

图 8 修正的 G8-45° 沉降下储罐的变形发展图

局部不均匀沉降存在一个沉降周向角跨度的问题 , 本节分析不同周向角 α 对储罐结构性能的影

响。局部不均匀沉降模型均采用完整的正弦谐波形式 , 如当 α=60°, 沉降模型记为 sina-60°。

图 9 给出了不同周向角局部不均匀沉降下储罐的沉降 - 位移曲线。当 α 较大时 ( 如图 9a), 结构

响应与局部化较明显的整体不均匀沉降的结果类似 , 主要呈现剪压屈曲 ; 当 α 较小时 ( 如图 9b),

结构表现为轴压跳跃屈曲 , 随着周向角的减小 , 结构跳跃屈曲产生的变形 ( 径向位移 ) 趋于明显 ;

当 α 更小 ( α =15°, 图 9c) 时 , 结构表现出稳定的后屈曲性能。

Smax/tav

0.25

0.20

0.15

0.10

0.05

sina_120°

sina_90°

0.00

0 1 2 3 4 5

Ux/tav

Smax/tav

0.15

0.12

0.09

0.06

0.03

0.00

0 2 4 6 8 10 12

Ux/tav

sina_60°

sina_45°

(a) (b) (c)

Smax/tav

0.30

0.25

0.20

0.15

0.10

0.05

图 9 不同周向角沉降下的沉降 - 位移曲线

0.00

0 3 6 9 12 15 18

Ux/tav

sina_30°

sina_15°

图 10a、b、c 分别给出了周向角 α 与结构屈曲临界沉降值 ( 取初始屈曲时沉降峰值 )、对应屈曲

点的临界变形值和结构敏感度的关系曲线 ; 其中结构沉降敏感度指罐底承受某一沉降时的罐壁变形

量 , 这里取临界变形值和临界沉降值的比值。由图 10a 可知 : 当结构表现为跳跃屈曲时 , 临界沉降

值随着 α 的减小而减小 ; 当 α 很小 , 结构临界沉降值有显著提高。由图 10b 可知 : 当 α 较大时 , 结

构剪压屈曲对应的临界变形很小 ; 当 α 较小结构呈轴压屈曲时 , 随着 α 的减小临界变形值不断增大。

综上 , 沉降周向角 α 越小则对结构稳定性越不利 , 即储罐对不均匀沉降越敏感 ( 如图 10c)。

0.20

Settlement

Deformation

Sensitivity

0.18

2.5

0.16

2.0

Smax/tav

0.14

0.12

0.10

0.08

0 40 80 120

(a)

Ux/tav

1.5

1.0

0.5

0.0

0 40 80 120

Central angle (° )

(b)

Ux/Smax

0 40 80 120

(c)

图 10 周向角 α 与临界沉降值、临界变形值和敏感度的关系图

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六、结论

本文通过非线性有限元分析系统考察了固顶型钢储罐在不均匀沉降下的非线性响应和稳定性

能 , 得到以下主要结论 :

(1) 在整体不均匀沉降下 , 储罐表现为罐壁上部的剪压屈曲 , 屈曲位置对应于罐底沉降变化

最迅速的区域。局部屈曲后 , 储罐承受的最大沉降值基本不变 , 而罐壁变形不断增大。可将屈曲临

界沉降值作为储罐极限沉降值 , 即将罐壁上部发生剪压屈曲视为储罐结构失效的标志。

(2) 固顶储罐发生整体倾斜沉降时 , 其剪压屈曲沉降值和对应的罐壁最大剪应力明显高于其

他类型的整体不均匀沉降结果 , 说明整体不均匀沉降中的局部化特征对储罐结构的稳定性十分不

利 , 此时很小的沉降值即能时罐壁产生屈曲变形。

(3) 在局部不均匀沉降作用下 , 储罐的结构响应与沉降周向跨度有关。当跨度较大时 , 结构

响应与局部化较明显的整体不均匀沉降的结果类似 ; 跨度较小时 , 结构表现出沉降峰值对应的罐壁

上部的轴压跳跃屈曲 , 屈曲沉降值较小而罐壁变形较大 , 并有着明显的后屈曲强度。在储罐设计时

应充分利用结构的后屈曲强度 , 可以将整体失稳破坏作为结构极限承载力状态 , 以罐壁变形控制作

为结构的正常使用极限状态。

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Parallel Session –

Geotechnical II

The 5th Cross-strait Conference on Structural and Geotechnical Engineering (SGE-5)

Hong Kong, China, 13-15 July 2011

广州市中心局部区域地下结构对地下水影响的初步分析

曹洪

骆冠勇

( 1 华南理工大学亚热带建筑科学国家重点实验室 , 广州 510640 )

摘要 : 本文以广州城区作为研究对象 , 通过对城区内的地质水文情况和一个典型区域的地下结

构物分析调查 , 利用城区区域渗流有限元分析程序 , 分析了城区内成行成片的地下结构对地下水的

水头和流速这两个方面影响。分析结果表明广州城区内现有的地下阻水结构物对地下水的影响具有

以下两个特点 :1) 结构物对水头大小的影响不大 , 但其扰动范围广。由于结构物之间存在透水缝

隙 , 绝大部分区域的扰动量级在 ±0.5m 之内 ; 同时由于大量的阻水体成片成行分布 ,

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