Arch and vault structures Prof Schierle 1

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1 Arch and vault Arch and vault structures Prof Schierle 1

2 Funicular vs. load Load type Funicular 1+2 Single point load Triangle 3+4 Two point loads Trapezoid 5+6 Uniform load Parabola 7+8 Mixed load Gothic curve 9+10 Self weight Catenary Radial load Circular Arch and vault structures Prof Schierle 2

3 11 12 Load and form 1 Polar polygon of parabolic cable 2 Parabolic funicular cable under uniform load 3 Polar polygon of parabolic funicular arch 4 Parabolic funicular arch under uniform load 5 Polar polygon of asymmetrically loaded cable 6 Funicular cable under asymmetric load 7 Polar polygon of asymmetrically loaded arch 8 Arch funicular under asymmetric load 9 Global moment of horizontal couple M = H d 10 Arch bending due to funicular offset M=Fe F=archforce e = arch offset from funicular line 11 Variable arch depth (optimal span/depth = 5) 12 Arch force vs. arch depth (rise) Arch and vault structures Prof Schierle 3

4 Arch hinges 1 Fixed-end arch 2 Fixed-end arch bend under temperature change 3 Fixed-end arch footing subject to overturn moment 4 Fixed-end arch bend under uneven settlements 5 Two-hinge arch 6 Two-hinge arch, bend under temperature variation 7 Two-hinge arch footing without overturn moment 8 Two-hinge arch, bend under uneven settlements 9 Three-hinge arch 10 Three-hinge arch, free to move under temperature change without secondary bending stress 11 Three-hinge arch foundation, with vertical and horizontal loads 12 Three-hinge arch, free to move under uneven settlement without secondary bending stress Arch and vault structures Prof Schierle 4

5 Exhibit hall Klagenfurt, Austria (1966) Architect: O Loider Engineer: Timber construction contractor The 96x75m hall 3-hinge wood arches span 96 m Arches of twin I-beam cross-sections, spaced 6.8 m, are crescent-shaped to fit the funicular pressure line for various loads to minimize bending stress. 1 Axon 2 Wind racing detail 3 Arch crescent profile 4 Arch cross-section A B C D E Glue-lam twin arches, 2-16x100 to 187 cm Arch flanges, 16x41cm glue-lam Roof purlins, 8x22cm solid wood L-shaped purlins, 2 8x22cm, brace arches Wind bracing, 8x8 cm Arch and vault structures Prof Schierle 5

6 Storage hall Walsum, Germany Engineer: Bauabteilung Brüninghof The circular hall of 94.6 m diameter is 20.8 m high Eight radial 3-hinge glue-lam arches span 94.6 m A concrete tension ring/wall resists the lateral arch thrust Arches are crescent-shaped to fit the funicular pressure line for various loads to minimize bending stress. 1 Roof framing plan 2 Cross-section 3 Hinge support 4 Arch bracing detail A Glue-lam arch, 20x cm, crescent shaped B Glue-lam beams, 8-16/16-70 cm, based on span C Arch bracing, 8x16cm D Steel hinge E Concrete tension ring Arch and vault structures Prof Schierle 6

7 Wood arch design Assume: Glue-lam arches, spaced 16, three-hinged for ease of transportation and to avoid settlement stress. Available dimensions: ¾ laminations; 3 1/8, 5 1/8, 6 3/4, 8 3/4, 10 3/4 wide). Based on case studies, use conservative allowable buckling stress: F c = 200 psi Code live load of 20 psf, reduced 40% to 12 psf for tributary area > 600 sq. ft. Loads: LL = 12 psf DL = 18 psf = 30 psf Arch and vault structures Prof Schierle 7

8 Distributed load w = 30 psf x16 /1000 w = 0.48 klf Global moment M = w L 2 /8 = 0.48 x / 8 M = 600 k Horizontal reaction H = M/d = 600 / 20 H = 30 k Vertical reaction R= wl/2= 0.48x100/2 R = 24 k Arch compression (max.) C= (H 2 +R 2 ) 1/2 =( ) 1/2 C = 38 k Cross section area A= C/F c = 38/0.2 ksi A = 190 in 2 Glue-lam depth (try 51/8 wide glue-lam) t =A/width =190/5.125= 37 Use 50 boards of ¾ t = 37.5 C H R Check slenderness ratio L/t= 100 x12 /37.5 L/t = 32, OK Arch and vault structures Prof Schierle 8

9 Note Arch slenderness L/t is usually about 30 to 40; hence 32 is OK; while the 5 1/8 arch width is braced against buckling by roof diaphragm. Graphic Method Draw equilibrium vector at support. starting with computed vertical reaction Draw a line parallel to support tangent and a horizontal line Measure length of unknown vectors: horizontal vector is horizontal reaction sloping vector is is max. arch force. Arch and vault structures Prof Schierle 9

10 Wood arch details 1 Two-hinge arch 2 Three-hinge arch 3 Crown hinge concealed 4 Crown hinge exposed 5 Base hinge concealed 6 Base hinge exposed 7 Base moment joint concealed 8 Base moment joint exposed Arch and vault structures Prof Schierle 10

Bus Station Chur, Switzerland (1992) Architect: Richard Brosi / Robert

station, the bus station connects ski resorts.

Inclined 16 steel arches span a 164 platform.

11 Bus Station Chur, Switzerland (1992) Architect: Richard Brosi / Robert Obrist Engineer: Toscano / Ove Arup (Peter Rice) Located over a train station, the bus station connects ski resorts. The glass roof provides scenic mountain views. Inclined 16 steel arches span a 164 platform. Radial strands resist lateral thrust and buckling. Arches are suspended from outrigger masts. Arch/strut triangles resist lateral load. Arch and vault structures Prof Schierle 11

12 Assume: Arch span L = 50 m / 0,3048 L ~ 164 Arch rise d ~ 30 Arch spacing e = (7.5 m/2) / e = 12.3 Arch outside =406 mm / 25.4 ~ 16 Arch wall thickness t ~ ¼ Arch inside diameter i = 15.5 Allowable steel stress F a =0.6x50 ksi F a = 30 ksi Allowable strand stress F a = 210/3 F a = 70 ks LL = 1.6 kpa x 0.145x144 in 2 /ft 2 LL = 33 psf DL (estimate) DL = 27 psf LL+DL = 60 psf Uniform arch load w = 60 psf x 12.3 / 1000 w = 0.74 klf Global moment M = w L 2 /8 = 0.74 x /8 M = 2488 k Horizontal reaction H =M / d = 2488 / 30 H = 83 k Vertical reaction R = w L /2 = 0.74 x 164 /2 R = 61 k Max arch compression C = (H 2 + R 2 ) 1/2 = ( ) 1/2 C = 103 k Arch and vault structures Prof Schierle 12

13 Max arch compression (from last slide) C = 103 k Check allowable buckling stress Radius of gyration r = ( 2 + i 2 ) 1/2 /4 = ( ) 1/2 /4 r = 5.48 Unbraced length (between strands) KL = 1.1 x164 / 7 KL = 26 Slenderness ratio KL/r = 26 x 12 / 5.48 KL/r = 57 Allowable buckling stress (AISC table) F a =23 ksi Arch cross section A = ( 2 - i 2 )/4= ( )/4 A = 12 in 2 A = r 2 r 2 =A/ = 2r =2(A/ ) 1/2 Max. arch stress f a = C/A= 103 / 12 f a = 8.6 ksi Check f a F a 8.6<23, OK Max strand force T ~ H = 83 T~ 83 k Required metallic strand area A m = T/F a = 83 / 70 A m = 1.2 in 2 Gross strand area (70% metallic) A g = A m /07 = 1.2/0.7 A g = 1.7 in 2 Strand size = 2(A/ ) 1/2 = 2(1.7/ ) 1/2 = 1.47 Use 1 ½ Arch and vault structures Prof Schierle 13

14 Arch and vault structures Prof Schierle 14

15 Vault cross sections 1 Cylindrical vault 2 Rib vault 3 Inverted cylindrical vault 4 Folded vault 5 Undulated vault 6 Corrugated vault Vault compositions Some vault compositions generate cross vaults with intersections that provide implied ribs for improved buckling resistance. Arch and vault structures Prof Schierle 15

Exposition hall, Turin (1947-49) Engineer: Pierre Luigi

units to resist buckling, are joined by site-cast

The wire mesh ferro-cement units integrate natural

16 Exposition hall, Turin ( ) Engineer: Pierre Luigi Nervi The 75/94 m concrete vault of prefab Ferro-cement units to resist buckling, are joined by site-cast concrete. The wire mesh ferro-cement units integrate natural lighting. A Ferro-cement unit B Site-cast concrete rib C Skylight Arch and vault structures Prof Schierle 16

Garden Festival Hall, Liverpool Architect/Engineer: Ove Arup This 78/62 m project was designed for a dual purpose: Central focal point for the festival - and afterwards Sports center with pool, a

17 Garden Festival Hall, Liverpool Architect/Engineer: Ove Arup This 78/62 m project was designed for a dual purpose: Central focal point for the festival - and afterwards Sports center with pool, a multipurpose hall squash courts, a gymnasium, and related facilities Structure: Three-hinge truss arches, spaced 3m provides the flexibility required for both programs Steel pylons support gravity load and lateral thrust The vault has translucent 2 cm polycarbonate panels The round endings are glad with corrugated aluminum Arch and vault structures Prof Schierle 17

18 Airship hanger, Orly airport, France Engineer: Eugène Freyssinet The first of two hangers, build in 1915 was the first reinforced concrete vault. Each vault spans 80m, is 300m long and 56m high. The parabolic cross-section fits the funicular pressure line for uniform load distributed horizontally. To resist buckling under unbalanced load, the vaults Consist of ribs of required depth without great dead load. The 6 cm concrete ribs are 7.5 m wide, and vary in depth from 5.4 m at the base to 3 m on top. Skylights are integrated with the ribs. Palace Ctesiphon (531 AD) The ancient Palce Ctesiphon (Mesopotamien plain) has a brick vault of 80 ft span (about 1/3 of Fryssinet s vault). The vault cross section approximates the parabolic funicular pressure line for minimal bending stress. Arch and vault structures Prof Schierle 18

19 IBM traveling exhibit Architect: Renzo Piano Engineer: Ove Arup/Peter Rice The design objective for this traveling exhibit pavilion was light weight and ease of assembly and disassembly. The 10x50 m pavilion was on exhibit in major European cities. Translucent polycarbonate pyramids for natural daylight are supported by two sets of glue-lam arches on the inside and outside Aluminum joints link arch segments The three-hinge vault allows thermal change without secondary stress A base platform adjusts for local sites Arch and vault structures Prof Schierle 19

20 Architect: Minoru Yamasaki; Engineer: Roberts & Schaefer Architect: Camelot Maily Zehrfuss Engineer: Nicholas Esquillan Air terminal St. Louis CNIT exhibit hall Paris (At 600 ft span the longest span structure in the world) Alternate design by Nervi Arch and vault structures Prof Schierle 20

21 Design great arches Arch and vault structures Prof Schierle 21

Suspended high-rise. Suspended high-rise Copyright G G Schierle, press Esc to end, for next, for previous slide 1

Suspended high-rise. Suspended high-rise Copyright G G Schierle, press Esc to end, for next, for previous slide 1 Suspended high-rise Suspended high-rise Copyright G G Schierle, 2001-06 press Esc to end, for next, for previous slide 1 Suspended high-rise 1 Gravity load path 2 Differential deflection 3 Prestress to