SHEAR DEFLECTIOIN IN 4PB TESTS

A.C. Pronk & M. Huurman

Delft University of Technology,

Delft, ZH, The Netherlands

Introduction

Problem

Solution

Comparison

Conclusions

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Introduction of problem

When in a 4PB test a beam is bended not

only a deflection due to bending will occur

but also a deflection due to the shear force

will be present.

If the ratio of height H over the length L is

small (slender beam) e.g. < 1/10 the

deflection due to shear can be neglected

However, in our practice the ratio is higher

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Problem

In the analytical approach (beam theory)

the actual beam is replaced by a single

line (1D) with no real geometrical

dimensions.

As a consequence an assumption has to

be made for the shear force distribution

over a cross section of the beam

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Possible Solution:

Determine the real 3D solution by Finite

Element Calculations (FEM).

Introduced new (minor) problems:

1) Finite Element Calculations are not exact.

2) For bending a beam in FEM you have to

touch (grab) the beam.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

External touching the material

introduces constraints e.g. stress

concentrations

The following slides are made by Rien

Huurman and presented already at the

1 st 4PB Workshop in Delft

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

EXAMPLES, 4PB

2D static elastic ⇒ AA

ABAQUS

Rien Huurman 1 st 4PB Workshop

In case of a 3D model the effects of

grabbing the beam becomes even

more visible.

Due to this touching the beam is

‘hampered’ in its ‘freedom’

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

EXAMPLES, 4PB

3D static elastic ⇒ AA

ABAQUS

Why different behaviour at different clamps

Rien Huurman 1 st 4PB Workshop

EXAMPLES, 4PB

Bending >

longitudinal stress.

3D static elastic ⇒ AA

Material >

contract/expand

perpendicularly.

Clamp blocks this >

Extra system

stiffness

Rien Huurman 1 st 4PB Workshop

So, the problem to be solved is:

How to bend a 3D beam without

touching or grabbing the beam

which will result in constraints

Solution: Apply a shear load

distribution over the cross section

at the inner supports.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

But you don’t know on forehand the

shear load distribution over a cross

section.

Solution: Make use of the St Venant

principle

The form of the shear load

distribution “far” away from the

source loading is “independent” of

the form of the source loading.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Iterative Solution Procedure

1.Apply a homogenous shear load

distribution at the inner supports.

2.Determine the shear load distribution

halfway the beam (L/2)

3.Use this distribution as the new input.

4.Repeat step 2 and 3.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

1 st Step

2 nd Step

Deviations

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

25.0

Height [mm]

16.7

8.3

0.0

-8.3

-16.7

mid clamp response

1st response at clamp

2nd response at clamp

-25.0

0.00 0.05 0.10 0.15

Shear stress [MPa]

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Deflection [m m ]

Applied shear #2 Applied shear #1 Appliead shear #1 - Applied shear #2

0.1

0

-0.1

-0.2

-0.3

-0.4

0 50 100 150 200 250

Distance to middle of beam [mm]

0.00045

0.00035

0.00025

0.00015

0.00005

-0.00005

Difference [m m ]

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Cowper 1966 (10+10μ)/(12+11μ) 0,8517

Timoshenko 1974 5/6 0,8333

Timoshenko 1922 (5+5μ)/(6+5μ) 0,8710

Olson 1935 (20+20μ)/(24+15μ) 0,9231

Pickett 1945 24,612(1+μ)/(29,538+5,942μ+64,077μ 2 ) 0,8419

Tanji 1972 (6+12μ+6μ 2 )/(7+12μ+4μ 2 ) 0,9354

Pai 1999 5/(6+(μ/(1+μ)) 2 (H/B) 4 [1-

(90/π 4 )Σ{Tanh(nπB/H)/(n 5 (π(B/H))}])

Hutchinson 2001 5(1+μ)/(6+5μ-(μ/(1+μ)) 2 (H/B) 4 [1-

(90/π 4 )Σ{Tanh(nπB/H)/(n 5 (π(B/H))}])

ABACUS 2009 Comparison analytical solution and finite element

calculations

0,8269

0,8763

0,859

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

CONCLUSIONS

Using an internal shear loading over the

cross section at the clamp it is possible to bend a

beam without touching or grabbing it. In this way

unwanted constraints are avoided.

Application of the Saint-Venant principle enables

the determination of the unknown shear force

distribution at the clamps of a 4PB test.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

CONCLUSIONS

This is done by replacing the homogenous shear

force distribution (per element) at the clamp in the

first calculation by the calculated distribution half

way the inner and outer clamp in the second

calculation.

Equalizing the analytical solution for the

deflection profile and the deflection calculated by

FEM leads to a value of 0,859 for the shear

correction coefficient. This is close to the value of

0,85 used in the 1D calculations in ABACUS.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

CONCLUSIONS

The method described in this paper opens the

possibility to obtain in a numerical way the

correct expression for the shear correction

coefficient.

2 nd European Workshop on 4PB, 24-25 September 2009,

University of Minho, Guimarães, Portugal

Thank you for listening

Any Questions

2 nd European 4PB Workshop, 26/27-09-2009, Univ. of Minho

Ad & Rien ; Delft University of Technology