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<strong>Arch</strong> <strong>and</strong> <strong>Vault</strong><br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 1


Tension Compression<br />

Funicular vs. load<br />

Load type Funicular<br />

1+2 Single point load Triangle<br />

3+4 Two point loads Trapezoid<br />

5+6 Uniform load Parabola<br />

7+8 Mixed load Gothic arc<br />

9+10 Self weight Catenary<br />

11+12 Radial load Circular<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 2


11<br />

12<br />

Load <strong>and</strong> form<br />

1 Polar polygon of parabolic cable<br />

2 Parabolic funicular cable under uniform load<br />

3 Polar polygon of parabolic funicular arch<br />

4 Parabolic funicular arch under uniform load<br />

5 Polar polygon of asymmetrically loaded cable<br />

6 Funicular cable under asymmetric load<br />

7 Polar polygon of asymmetrically loaded arch<br />

8 <strong>Arch</strong> funicular under asymmetric load<br />

9 Global moment of horizontal couple M = H d<br />

10 <strong>Arch</strong> bending due to funicular offset<br />

M=Fe<br />

F=archforce<br />

e = arch offset from funicular line<br />

11 Variable arch depth (optimal span/depth = 5)<br />

12 <strong>Arch</strong> force vs. arch depth (rise)<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 3


<strong>Arch</strong> hinges<br />

1 Fixed-end arch<br />

2 Fixed-end arch bend under temperature change<br />

3 Fixed-end arch footing subject to overturn moment<br />

4 Fixed-end arch bend under uneven settlements<br />

5 Two-hinge arch<br />

6 Two-hinge arch, bend under temperature variation<br />

7 Two-hinge arch footing without overturn moment<br />

8 Two-hinge arch, bend under uneven settlements<br />

9 Three-hinge arch<br />

10 Three-hinge arch, free to move under temperature<br />

change without secondary bending stress<br />

11 Three-hinge arch foundation, with vertical <strong>and</strong><br />

horizontal loads<br />

12 Three-hinge arch, free to move under uneven<br />

settlement without secondary bending stress<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 4


Wood arch details<br />

1 Two-hinge arch<br />

2 Three-hinge arch<br />

3 Crown hinge concealed<br />

4 Crown hinge exposed<br />

5 Base hinge concealed<br />

6 Base hinge exposed<br />

7 Base moment joint concealed<br />

8 Base moment joint exposed<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 5


Wood arch design<br />

Assume:<br />

Three-hinge glue-lam arches at 16’<br />

Available dimensions: (¾” boards;<br />

3 1/8”, 5 1/8”, 6 3/4”, 8 3/4”, 10 3/4” wide).<br />

Based on case studies, use conservative<br />

allowable buckling stress: F c’= 200 psi<br />

Loads:<br />

LL = 12 psf (60% of 20 psf for trib. area>600 ft 2 )<br />

DL = 18 psf<br />

� = 30 psf<br />

<strong>Arch</strong> load w = 30 psf x16’/1000 w = 0.48 klf<br />

Reactions R = 0.48 x100’/2 R = 24 k<br />

Graphic Method<br />

• Draw equilibrium vector at support.<br />

starting with vertical reaction<br />

• Draw C & H vectors<br />

• Measure vector lengths:<br />

• C = max. arch force<br />

H = horizontal reaction<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 6


C<br />

H<br />

R<br />

Numeric method:<br />

<strong>Arch</strong> load (from last slide) w = 0.48 klf<br />

Global moment<br />

M = w L2 /8 = 0.48 x 1002 / 8 M = 600 k’<br />

Horizontal reaction<br />

H = M/d = 600 / 20 H = 30 k<br />

Vertical reaction<br />

R= wL/2= 0.48x100/2 R = 24 k<br />

<strong>Arch</strong> compression (max.)<br />

C= (H2 +R2 ) 1/2 =(302 +242 ) 1/2 C = 38 k<br />

Cross section area<br />

A= C/Fc’= 38/0.2 ksi A = 190 in2 Glue-lam depth (try 51/8“ wide glue-lam)<br />

t =A/width =190/5.125= 37<br />

Use 50 boards of ¾” t = 37.5”<br />

Check slenderness ratio<br />

L/t= 100’x12”/37.5 L/t = 32, OK<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 7


e = 22’<br />

L = 315’<br />

d = 78’’<br />

Exhibit hall Klagenfurt, Austria<br />

<strong>Arch</strong>itect: O Loider<br />

Engineer: Timber construction contractor<br />

The 96x75m hall has 3-hinge wood arches, crescent to fit<br />

the funicular pressure line to minimize bending stress.<br />

Span L = 315’<br />

Depth d= 78’<br />

Spaced at e= 22’<br />

1 Axon<br />

2 Wind racing detail<br />

3 <strong>Arch</strong> crescent profile<br />

4 <strong>Arch</strong>, A = 5824-8608 cm2 (903-1334 in2 )<br />

A Glue-lam twin arches, 2 x 16x100 to 187 cm<br />

B <strong>Arch</strong> flanges, 4 x 16x41cm glue-lam<br />

C Roof purlins, 8x22cm solid wood<br />

D L-shaped purlins, 2 – 8x22cm, brace arches<br />

E Wind bracing, 8x8 cm<br />

Load = 30 psf LL + 20 psf DL = 50 psf<br />

<strong>Arch</strong> load w = 50 psf x22’/1000 w = 1.1 klf<br />

H = wL2 /(8d) = 1.1 x 3152 /(8x78) H = 175 k<br />

R = wL/2 =1.1 x 315/2 R = 173 k<br />

C = (H2 +R2 ) 1/2 = (1752 +1732 ) 1/2 C = 246 k<br />

Max. stress f = 246k x1000#/902 f = 273 psi<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 8


Storage hall Walsum, Germany<br />

Engineer: Bauabteilung Brüninghof<br />

The circular hall of 94.6 m (310’) diameter <strong>and</strong> 20.8 m (68’)<br />

<strong>Arch</strong> height features eight radial 3-hinge glue-lam arches.<br />

A concrete tension ring / wall resists the lateral arch thrust.<br />

1 Roof framing plan<br />

2 Cross-section, arch span L = 310’, rise d = 68’<br />

3 Hinge support<br />

4 <strong>Arch</strong> bracing detail<br />

A Glue-lam arches, 20x140-226 cm (7.9”x55-89”)<br />

B Glue-lam beams, 8-16/16-70 cm, based on span<br />

C <strong>Arch</strong> bracing, 8x16cm<br />

D Steel hinge<br />

E Concrete tension ring<br />

Load = 12 psf LL + 20 psf DL = 32 psf<br />

Circumference C = � ø = 3.1416 x310’ C = 974’<br />

Max. arch spacing e = 974’/16 e = 61’<br />

Max. arch load w = 32 psf x61’/1000 w = 1.95 klf<br />

M = wL 2 /24 = 1.95x310 2 /24 M = 7808 k’<br />

H = M/d = 7808k’/ 68’ H = 115 k<br />

R = wL/2 = 1.95/2 klf x310’/2 R = 151 k<br />

Max. arch compression C = (115 2+ 151 2 ) 1/2 C = 170 k<br />

Max. arch stress<br />

f = C/A = 170k x 1000# / (7.9”x55”) f = 391 psi<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 9


Bus Station Chur, Switzerl<strong>and</strong> (1992)<br />

<strong>Arch</strong>itect: Richard Brosi / Robert Obrist<br />

Engineer: Toscano / Ove Arup (Peter Rice)<br />

Located over a train station, the bus station<br />

connects ski resorts.<br />

The glass roof provides scenic mountain views.<br />

Inclined 16” steel arches span a 164’ platform.<br />

Radial str<strong>and</strong>s resist lateral thrust <strong>and</strong> buckling.<br />

<strong>Arch</strong>es are suspended from outrigger masts.<br />

<strong>Arch</strong>/strut triangles resist lateral load.<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 10


Assume:<br />

<strong>Arch</strong> span L = 50 m / 0,3048 L ~ 164’<br />

<strong>Arch</strong> rise d ~ 30’<br />

<strong>Arch</strong> spacing e = (7.5 m/2) / 0.3048 e = 12.3’<br />

<strong>Arch</strong> outside � =406 mm / 25.4 � ~ 16”<br />

<strong>Arch</strong> wall thickness t ~ ¼”<br />

<strong>Arch</strong> inside diameter �i = 15.5”<br />

Allowable steel stress F a =0.6x50 ksi F a = 30 ksi<br />

Allowable str<strong>and</strong> stress F a = 210/3 F a = 70 ks<br />

LL = 1.6 kPa x 0.145x144 in 2 /ft 2 LL = 33 psf<br />

DL (estimate) DL = 27 psf<br />

� LL+DL � = 60 psf<br />

Uniform arch load<br />

w = 60 psf x 12.3’ / 1000 w = 0.74 klf<br />

Global moment<br />

M = w L 2 /8 = 0.74 x 164 2 /8 M = 2488 k’<br />

Horizontal reaction<br />

H =M / d = 2488 / 30 H = 83 k<br />

Vertical reaction<br />

R = w L /2 = 0.74 x 164 /2 R = 61 k<br />

Max arch compression<br />

C = (H 2 + R 2 ) 1/2 = (83 2 + 61 2 ) 1/2 C = 103 k<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 11


A = �r 2 r 2 =A/� �= 2r =2(A/�) 1/2<br />

Max arch compression (from last slide) C = 103 k<br />

Check allowable buckling stress<br />

Radius of gyration<br />

r = (� 2 + �i 2 ) 1/2 /4 = (16 2 + 15 2 ) 1/2 /4 r = 5.48”<br />

Unbraced length (between str<strong>and</strong>s)<br />

KL = 1.1 x164 / 7 KL = 26’<br />

Slenderness ratio<br />

KL/r = 26’ x 12” / 5.48” KL/r = 57<br />

Allowable buckling stress (AISC table) F a=23 ksi<br />

<strong>Arch</strong> cross section<br />

A =�(� 2 -�i 2 )/4= �(16 2 -15.5 2 )/4 A = 12 in 2<br />

Max. arch stress<br />

fa= C/A= 103 / 12<br />

Check fa≤ Fa Max str<strong>and</strong> force<br />

fa = 8.6 ksi<br />

8.6


<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 13


Olympic stadium Athens<br />

<strong>Arch</strong>itect/Engineer: Santiago Calatrava<br />

Length section<br />

Cross section<br />

Support arch<br />

Suspenders<br />

Tension chord<br />

Rafters<br />

Deformation simulation<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 14


<strong>Vault</strong> cross sections<br />

1 Cylindrical vault<br />

2 Rib vault<br />

3 Inverted cylindrical vault<br />

4 Folded vault<br />

5 Undulated vault<br />

6 Corrugated vault<br />

<strong>Vault</strong> compositions<br />

Some vault compositions generate<br />

cross vaults with intersections that<br />

provide implied ribs for improved<br />

buckling resistance.<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 15


Exposition hall, Turin (1947-49)<br />

Engineer: Pierre Luigi Nervi<br />

The 75/94 m concrete vault of prefab Ferro-cement units<br />

to resist buckling, are joined by site-cast concrete. The<br />

wire mesh ferro-cement units integrate natural lighting.<br />

A Ferro-cement unit<br />

B Site-cast concrete rib<br />

C Skylight<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 16


d=25’<br />

L=200’<br />

8.2’<br />

Exposition hall, Turin (1947-49)<br />

Engineer: Pierre Luigi Nervi<br />

Assume:<br />

Span L = 200’<br />

Depth d = 25’<br />

DL = 38 psf<br />

LL = 12 psf<br />

∑ = 50 psf<br />

Uniform load per rib<br />

w = 50 x 8.2’/1000 w = 0.41 klf<br />

Global moment<br />

M = wL 2 /8 = 0.41x200 2 /8 M = 2050 k’<br />

Horizontal reaction<br />

H = M/d = 2050/25 H = 82 k<br />

Veritcal reaction<br />

R = wL/2 = 0.41x200/2 R = 41 k<br />

Rib compression<br />

C = (H 2 +R 2 ) 1/2 = (82 2 +41 2 ) 1/2 C = 92 k<br />

Rib cross sections<br />

A = 3”x10” + 4”x8” A = 62 in 2<br />

Average concrete stress<br />

f = 92kx1000#/62 f = 1484 psi<br />

Note: rib rebars resist bending <strong>and</strong> compression<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 17


Neue Messe (New Fair) Leipzig<br />

<strong>Arch</strong>itect: GMP Hamburg<br />

Engineer: Polonyi Köln with Haringer München<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 18


Neue Messe Leipzig<br />

Truss arch Detail<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 19


Garden Festival Hall, Liverpool<br />

<strong>Arch</strong>itect/Engineer: Ove Arup<br />

This 78/62 m project was designed for a dual purpose:<br />

• Central focal point for the festival - <strong>and</strong> afterwards<br />

• Sports center with pool, a multipurpose hall<br />

squash courts, a gymnasium, <strong>and</strong> related facilities<br />

Structure:<br />

• Three-hinge truss arches, spaced 3m<br />

• provides the flexibility required for both programs<br />

• Steel pylons support gravity load <strong>and</strong> lateral thrust<br />

• The vault has translucent 2 cm polycarbonate panels<br />

• The round endings are glad with corrugated aluminum<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 20


Airship hanger, Orly airport, France<br />

Engineer: Eugène Freyssinet<br />

The first of two hangers, build in 1915 was the first<br />

reinforced concrete vault.<br />

Each vault spans 80m, is 300m long <strong>and</strong> 56m high.<br />

The parabolic cross-section fits the funicular pressure<br />

line for uniform load distributed horizontally.<br />

To resist buckling under unbalanced load, the vaults<br />

Consist of ribs of required depth without great dead load.<br />

The 6 cm concrete ribs are 7.5 m wide, <strong>and</strong> vary<br />

in depth from 5.4 m at the base to 3 m on top.<br />

Skylights are integrated with the ribs.<br />

Palace Ctesiphon (531 AD)<br />

The ancient Palce Ctesiphon (Mesopotamien plain) has a<br />

brick vault of 80 ft span (about 1/3 of Fryssinet’s vault).<br />

The vault cross section approximates the parabolic<br />

funicular pressure line for minimal bending stress.<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 21


IBM traveling exhibit<br />

<strong>Arch</strong>itect: Renzo Piano<br />

Engineer: Ove Arup/Peter Rice<br />

The design objective for this traveling<br />

exhibit pavilion was light weight <strong>and</strong><br />

ease of assembly <strong>and</strong> disassembly.<br />

The 10x50 m pavilion was on exhibit in<br />

major European cities.<br />

• Translucent polycarbonate pyramids<br />

for natural daylight are supported by<br />

two sets of glue-lam arches on the<br />

inside <strong>and</strong> outside<br />

• Aluminum joints link arch segments<br />

• The three-hinge vault allows thermal<br />

change without secondary stress<br />

• A base platform adjusts for local sites<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 22


Air terminal St. Louis<br />

<strong>Arch</strong>itect: Minoru Yamasaki;<br />

Engineer: Roberts & Schaefer<br />

CNIT exhibit hall Paris<br />

(At 600 ft span the longest span structure in the world)<br />

Engineer: Nicholas Esquillan<br />

Alternate design by Nervi<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 23


t<br />

d<br />

Static model<br />

<strong>Arch</strong>es are subject to axial stress <strong>and</strong> bending stress<br />

Therefore:<br />

• Em = Eo must be satisfied<br />

• Detail dimensions must be in geometric scale<br />

Force scale<br />

S f = P m/P o<br />

S f = S g 2<br />

Assume:<br />

E m = E o = 1,600,000 psi<br />

<strong>Arch</strong> size 5.125” x 37.5”<br />

Geometric scale S g = 1:25<br />

Force scale<br />

S f = S g 2 = 1/25 2 S f = 1:625<br />

Model arch size<br />

t = 5.125/25 t = 0.2”<br />

d = 37.5/25 d = 1.5”<br />

Load on original arch<br />

P o = 30 psf x 100’ x 16’ P o = 48,000 #<br />

P m = P o/S f = 48,000/625 P m = 76.8 #<br />

Use 24 cups at 76.8/24 P m= 3.2 #/cup<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 24


Berlin Ostbahnhof<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 25


Dresden HBF<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 26


Norman Foster:<br />

Reichstag Dome Berlin<br />

radial arches<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 27


Galleria Milano<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 28


Van Brienenoordbrug Bridge near Rotterdam<br />

<strong>Arch</strong> <strong>and</strong> vault Copyright © G G Schierle, 2010 press Esc to end, � for next, � for previous slide ‹#›


St. Louis Gateway <strong>Arch</strong>, 1968<br />

Parabolic, 192 m high<br />

<strong>Arch</strong>itect: Eero Saarinen<br />

Engineer: Hanskarl B<strong>and</strong>el<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 30


Antonio Gaudi: Casa Milà parabolic arches<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 31


<strong>Arch</strong>es at Campeche Mexico<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 32


Bad Lippspringe Castle Germany<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 33


<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 34


Design models<br />

Design model<br />

Test model �<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 35


Presentation<br />

of<br />

test results<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 36


Design great<br />

<strong>Arch</strong>es <strong>and</strong> <strong>Vault</strong>s<br />

<strong>Arch</strong> <strong>and</strong> vault structures Copyright Prof Schierle 2012 37

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