Arch and Vault - Engineering Class Home Pages
Arch and Vault - Engineering Class Home Pages
Arch and Vault - Engineering Class Home Pages
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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