\10:~ PfliG,';lATIC FOLDED ROOF GTflUCTURE ·~=~===============================

~ ~~ by FLDRENCIO DEL POZO and AVELINO SAMAATIN

SUM'AAAY

The dP.sign of a prestressed concr ete roof is described . A special linear ela stic analysis of non-prismatic folded structures has been a pplied . -­The obtained r esults have been compar ed to the r esults deducted f r om a -sm:tll sc3le model test. Conclusions abou t the E·fficency of this typ8 of •, t:--~ Jcturcs arr; shm· n.

>¥Prof . Dr . Ingeniero de Caminos . Laboratorio Central ** Dr . Ingeniero dn Caminos . r.~inisterio de Obras Pt:iblica

- 1 -

lrltr oduction .

Among the different sport ha l ls built at the Barcelona Football Club , t he Hod<ey Ice Pa bellon presents several interesting features .

The rectangular planform of this building is 44 . 00 x 66 . 00 m. approximat£ l y , and severa l structural solutions for its roof have been co~sidered. -One of the studied solutions , a non-prismatic foldP.d plate is here comment ed with some deta i l .

Design descr iption

The prestressed concrete roof is formed by nine identical inner non-prism~ tic folded plate structures. with the following dimensions : 44 . ,5 m. span and 6 m. width . Each of these shells are simply supported on t r a nsversal -end gables, a nd along its longitudina l edges the con tinuity exists . The two outer non-prismatic folded plate structur es , are identical to t he inner one except the lateral edges, that are cantilevers of 1. 25 m. span (Figure 1).

The shell thickness is 0 , 12 m. and each non- prismatic folded pla te is for­med by four triangles . Due to analysis and constr uction consider·ations , -­the thickness along the intersection of t wo triangles have been i.ncre.ased. according to figure 2 (section A-A) .

The pr estr essed cable have been designed as shown in figure 3 in order to car ry the dead and live loads . The main co~sideration in i ts des ign was -the longitudinal bending str esses . The ~nall transversal flexura l str esses have been res i sted by means of ordinary steel reinforcement (figure 3) .

Analysis

During the selection of the method of the r oof a nalysis the possibili t y -to use a srnall easily ·a vailable digital computer has been an importan t -­consideration .

- 2 -

PLAN

PLAN

4 SIGHT X l..l> ------------------------ ---7 ---------------------A

ELEVATION

>O<

u. 0 0 0::

w :I: 1-

~ IJ... I 0 ~ N > z z .,.._ .,.._ 0 0 J: J: 1-

1-(!) <!> z

u If) If) IJ... w w If) a

N

w 0:: ::J \.!)

IJ...

- 4 -

\J1 PRESTRESSED CABLES

'.

LONGITUDINAL SECTION PRESTRESSED CABLES

TRANSVERSAL SECTION

FIGURE 3.- ROOF REINFORCEMENT

ORDINARY STEEL REINFORCEMENT

For that reason ; ver y general numerical methods as Finite El ements , wer e a voided . Among the s pecific methods of analysis of non- pr ismatic fo l ded pl<'lte structures , the method given in the reference ( 1) . wich is an exte~ sion of t he clasical Yitzhaki ana lysis is developed . The main steps o f -the co:nputations a r e not r epa:1 tGd here but only the introducted changes -to the original presentation. namely :

1) Us e the Flexibility or Forct% method during the transversa l anal ysis . that appea r s to be more effici ent than the previous used Stiffness method .

::: ) Apply l;he same computer soubrout ine (conjugate beam proper ties) to the computation of the longitudina l bend ing moments (M ) and the deflexion of each plate (~·;) . L

3) In the secondar y analysis . that takes into account the effects of f i ct2:_ cions r B:.1ctions applied to t he shell, the fo l lowing change was considered : Use as reaction representation a parabolic distr ibuted loads between each b .:o adjacent stations insteads of concentrated punctua l loads .

4) Into the cornpu ter program genera l boundar y conditions hav e been consi­dered .

The computer progr am has been written i n FORTRAN IV f or an IBM- 1130/ 32K computer , using overlay technique .

In order to test the computer calculations the prismatic folded pl ate of t he figure 4, has been a nalysed using the ordina ry folded pla te t heory ( 2 ) L. P . P . (3 terms of the Fourier series) a nd the general non- prismatic fol ded plate theory L. P . N. P . (number of s t ations = 8 ) .

The resul ts obtained from the tv:o methods have agreed satisfactorily for two types of transversal boundary conditions . I n the tabl e I these r esults a re shown .

- 6 -

-.:J

I

L~1 = 0.340 em=0.340

~f =0.120 em:0.172

6

~f = 0.120 ~m=0.172

8 e1 = 0.120 em=O.I72

'

t I 25 f t + ~ 8 X 150 ~ ~ ~ ~ l SPAN= 44.34

E = 1.31 X 106

\)=0

UNifORM LOAD

UNITS: METERS TONS .

FIGURE 4._CROSS SECTION OF THE PRISMATIC FOLDED PLATE STRUCTURE OF THE EXAMPLE

TABLE I

COMPARISON EXAMPLE - RESULTS

A) HYPOTESIS: FREE EDGE 2 .

VERTICAL DISPLACEMENTS (mm , f +) CENTER TRANSVERSAL SECTION (MIDSPAN )

Edge LPP LPNP

188 176

2 156 150

3 127 119

4

130 12 1

5 128 119

6 135 125

7 137 127

8 140 130

9 141 132

HORIZONTAL DISPLACEMENTS (mm, _,..+) CENTER TRANSVERSAL SECTION (MIDSPAN)

Edge

LPP

LPNP

2 3 4 5 6

62 . 2 62 . 2 15. 5 10 . 6 8 . 0 - 2 . 1

62 . 5 62 . 5 13 .6 10 . 2 7 . 0 - 2 . 6

7 8

2 . 0 -1 . 9

1 . 6 - 2 . 2

9

2 .2

0 . 1

10

14-1 1J2

10

D

0

LONGITUDINAL BENDING STRESSES (Kg ./cm2, «1}.) CENTEf~ TRJ.\NSVERSAL SECTION (MIOSPAN)

Edge 2 3 4 5 6 7 8 9 10 LPP 16 . 1 58. 3 -1 15 . 8 90. 9 - 96. 6 105 . 3 - 102 . 0 108 . 4 - 106 . 2 108 . 7 LPNP 12 . 6 59 . 2 - 113 . 0 87 . 2 - 93 . 1 101 . 6 - 98 . 3 104 . 9 -102 . 7 105 . 1

TRANSVERSAL BENDING MaAENTS (mt/ml ,(D?+) CENTER TRANSVERSAL SECT ION (MIOSPAN)

Edge 1 2 3 4 5 6 7 8 9 10 LPP o. - 0 . 19 - 2 . 60 0 .98 -1 . 12 0 . 35 - 0 . 31 - 0 . 07 - 0 . 11 - 0 . 18 LPNP 0 . - 0 . 23 - 2 . 53 1. 00 - 1. 03 0 . 38 - 0 .22 - 0 . 03 - 0 . 02 - 0 . 13

SHEAR STRESSES AT EDGES (t/ml , + 'll<~ ) SUPPORTS

Edge 2 3 4 5 6 7 8 9 10

LPP 0 11J3 1 1. 61 - 3 . 12 - 3 . 98 - 2 . 49 - 1 . 92 - 0 . 80 - 0 .42 0 LPNP 0 10 . 85 1 . 5 1 - 2 . 96 - 3 . 95 - 2 .5 0 - 1 . 93 - 0 . 80 -0.42 0

- 8 -

TABLE I (Contd)

B) HYPOTESIS: VERTICALY SUPPORTED EDGE 2 .

VERTICAL OISPLACEMENTS (mm , l +) CENTER TRANSVERSAL SECTION. ( :.nDSP NJ)

Edge 1 2 3 4 5 6 7 B 9 10 LPP -45 0 52 96 123 137 143 145 145 145 LPNP -40 0 48 88 113 127 133 135 136 135

HORIZONTAL DI SPLACEMENTS (mm. __,.,. +) CENTER TRANSVERSAL SECTION ( MIOSPAN )

Edge 2 3 4 5 6 7 8 9 10 LPP -41 .4 -41 .4 42 .3 - 27 .5 15 . 8 - 7 .4 2.3 -0.4 -0.2 0 LPNP - 37. 5 - 37.5 38. 9 - 25 . 8 14.9 - 7 . 0 2 .3 - 0 .5 - 0 .2 0

LONGITUDINAL BENDING STRESSES (Kg . /cm2 , ...... ~) CENTER TRAN~JVERSAL SECTION (MIOSPAN)

Edge 1 2 3 4 5 6 7 8 9 10

LPP 29 .2 - 3 . 5 -42 .7 75 .9 - 95 . 0 -106 .3 -109 . 5 110. 8 110.4 110.3

LPNP 27 .3 - 3 .5 -40 .4 72 .9 - 91 .5 -1 02 .5 -105 . 8 107. 2 - 106 .8 105.7

TRANSVERSAL BENOING Ma.1ENTS ( mt/ml , + (a)) CENT ER TRANSVERSAL SECTION (MIOSPAN)

Edge 1 2 3 4 5 6 7 8 9 10 LPP o. - 0 . 19 0 . 19 0 . 71 0 . 34 0 .21 o.oo - 0. 11 - 0 .16 -0.19 LPNP 0 . - 0 .23 0.20 0 .69 0 .38 0 .25 0 . 06 -0. 05 - 0.09 -0. 12

SHEAR STRESSES AT EDGES ( t/ml , + • or.!) SUPPORTS

Edge 1 2 3 4 5 6 7 8 9 10 LPP o. 4 .47 -4.95 1. 87 -1 .73 0 . 29 - 0 .27 -0. 05 0. 01 o. LPNP o. 3 .60 -4. 00 1.62 - 1.58 0 . 31 - 0 .25 - 0 . 03 -0. 02 0 .

- 9 -

From the table I , t he number of stations (ne = 8) has been considered suf

ficiente from the •pra ctical accuracy point of view. Nevertheless in the analysis of the roof this number was increased to ne = 10 .

The data used in the analysis of the actual non-prismatic folded plate -­r oof a r e given in t he table II .

In the table III the computer results of this analysis a re presented .

Scale model test

In order to obtain additional information about the s hell structural beha

viour, exper imental analysis was considered . The model test \'.'as carried out at the Laborator io Central de Ensayos de \ateriales de ~~drid .

The model scale was 1/24, large enough in order to r eproduce without spe­cial difficulties the actual thickness of the shell . In fact the model - ­shell thickness v!as r educed to 5 mm .

The construction of the model has been carried out accor ding to the follo wing steps :

1) M accurate wood model was built. From this previous model the corres-­ponding plaster formworks ( molds) were manufactured .

2 ) The selected model material was a mix of polyester r esin a nd cement .

This mixture has been experimented at the labor atory Central for a long -time and has presented the following elastic constants :

Young modulus E M

Poisson Ratio l)M

- 2 73500 Kp. cm

0 . 32

3) The mix \'.•as poured into the above mentioned formworks . After a partial mater ial polymerization , the mol ds wer e taken out and the model was ther­mosetted in an O\'!en at 50Q C during a wee!<.

In the photographs 1 and 2 the finished model is shown .

4) Following the usual experimental practice sever al muterial samples we­

r e prepared simultaneously to the model construction . These s pecimens \'le-

- 10 -

TABLE II

DATA OF THE NON- PRISMATIC FOLDED PLATE ROOF

(Notation is given in the appendix A. )

1 . 1: Gener al data .

I = 9 N = '1J A = 10 E = 13 10000 t / rrl2. .

1. 2 . Plate longitudina l data .

1. 3

L ( ~) = 44 • 34 m. (.l (t)= 4 . 44 g

;"-~(• )= -4 .44 Q

Plate 1.

i = 1 , 2 , ••• 9 .

i = 2 , 5 , 6 , 9 . i = 3 , 4 , 7 , 8 .

JL(o(f)= 1. 25 v(?( , 1) o. (~ ( ,; 1 1) = o.34 e (•1( , 1) "' ·I t

·.X. = 1 , 2 , ... 10

Plates 2 , 5 , 6 a nd 9 .

i) V (1 , i) ~

( 1 ' i) h ( 1 ' = 0 . 150 = ..:. 0 . 240 e = 0 .299 ~

m h (2 , i) = 0 .445 V (2 , i) = ..: 0 . 710 e (2 , i) = 0 . 265

m h (3 , i) = 0 . 745 (3 , i) + 1. 190 (3 , i) ~· 0 . 233 V e

m h (4 , i) 1. 050 (4 , i ) + 1. 680 ( 4 , i) = 0 . 203 = V = - e

m h (5 , i) 1. 345 V (5 , i ) +

( 4 , i ) = - 2 . 160 e = 0 . 182 ~

m h (6 , i) = 1. 645 V (6 , i ) = ..: 2 . 630 e (4 , i ) = 0 . 165

~ m

h (7 , i) - 1. 960 V (7 , i ) = ..: 3 . 120 e (4 , i) = 0 . 152 m

(8 , i) v ( 8 , i) ~

( 4 , i) h = 2 . 250 = ..: 3 . 610 e = 0 . 145 ~

m h (9 , i) = 2 . 550 V (9 , i) = ..: 4 . 100 e ( 4 ' i) = 0 . 140

m (10 , i) V (10 , i)

~

(4 , i) h = 2 . 850 = ..: 4 . 600 e = 0 . 137 m

and ef (~ , i) = 0 . 120

where.,(,= 1 , 2 , .. 10 i = 2 , 5 , 6 a nd 9 .

The sign + corresponds t o i = 2 , a nd 6 and thr

s ign - to i = 5 and 9 in the values of v (·X , i )

- 11 -

= _0 . 34

TABLE II (Cont . )

Plates 3 . 4 7 and 8 .

Let be i = 4 and 8 or and 7. and i

1= 2 and 6 or 5 and9 th~

h ( i 1"" ) :: h ( i 1 1 11- 0( )

v (i , o<) = v(i1

, 11- o<)

e ( i , oZ ) = e ( i 1

, 11- o<.) m m e~( ~ . <x. J = ~ i 1, 11- o< J

o( = 1 ' 2' . . • 10

1.4 . Loads

Uniform distributed loads in plates .

- 2 P (1) = -300 kp m - 2

P ( i) = -570 kp m i = 2, 3 1 • • • 9 .

- 12 -

~A---~: .. ~ _ )~~I._

RESULTS ;Y7 '/:£;.·~ ! 20.:- ?_,: ~-- !. ··• :; ··?~:u ,~: --. _ _.:~ "· :..;

ROOF \1 o+~ ':"o·- i '· .-iv('r i ·· 1 -!·>c ., .. ,. e, ,:: .; .. : ;'-.. • .J ,. (, .. • •: • -- • ._ .. • • (.# • • • -'- __ .. L - •

3 .. !_)·!, _: Jf.:.~:~~: : .::.--:., ( · ... :~:··~. ) .

~v. .. .-\.:_.;j o~- i0·-~_; 2 . EJa ;: 3. _.t;:,:_! ; l . ill~f ~ ~- . ~1:: ,_: .~ . • r"' - ' r-~;· ,:u j •

,.- . -~ 1 -;.. :._) J _:. ' • 2:·· ~~~ ~1 .

- --·-- - ---- - -- ~----- - ---- - --- - ---- - ----2 -17.50 -14.76 33.69 42.40 -10.58 -55.1 S 6.56 50 .41

3 -4.35 -20.42 63.77 31 . 52 -0.79 -35.0 !.: 8.93 32.17

4 21.25 - 21.85 73. 94 15 . 90 9.77 -13.4 8 8 .75 13:.24

~ 5 47. 37 - 21.88 66. 04 6.51 19 .11 -1.8 3 7.21 3 . 8 1 w I

6 66.38 -17.93 44.58 -2.34 26.40 5. 7 7 5.03 - 3 . 0 1

1 73.61 -11.44 15.21 -16.01 30.28 14.8 9 2.50 -13.11

8 63.79 - 0 . 84 -17. 86 -40.20 29.10 32.2 7 - 0 .12 -33.08

9 37.13 3.19 -44. 86 -6 5.33 2 1. 00 47.7 9 - 2 . 87 -52.16

• e • / •

TADL·~ III . (Cont. )

10 ·c-rr·· -)T ·r · L ·-.;. .. ~··>·~"' s·"Tr '' ., .,., ( i'G/C}T2) _: .!.. i ~ ~: .• !. · L-e·. ):__·. y '.•::. u )•J .:·J::> --~ ..... •

srf_l_~.r IO:! P!Ar":.!i 1. ?L:1.T~ .J.•. ;~< l'l~E J !:>L-~'-~~ t .. PL!i. .. .'E 5 ._ EJ~.2_E 6 ._ ?I .. .':.' i~ 7 . PIA':._.r~ !":~ ~!_j_ f!.~ lE 9. - - -- -=-·~-:- :-.. ·-..... ...,.. - ...,_. - ------ .. -- --·~ -------..- - ·-----·-

2 S1 -8.05 -0.15 2. 53 13.14 -10.44 -37.1 3 -3.06 21.47 -8.40 S2 - 0.18 2.61 11.76 -10.57 -37.14 -2.9 2 21.27 -8 . 53 -36.86

3 S1 1.23 -3.53 -6.23 34.23 -44.84 24.8 6 -45.50 59.77 -52.48 S2 -3.28 -6.26 33.45 -44.65 25.14 -45.1 t 59.73 -52.25 32.87

4 Sl 13.10 - 3.52 -17.78 51.82 -72.63 69.2 5 -80.64 88-52 -87.04 S2 -3.10 -17.85 51.q4 -72.51 69.59 -80.7 '! 88.57 -86.92 80.48

-5 Sl 23.96 - 2.12 -27.62 61.12 -84.35 89 . 4 J -98.42 101.85 -103.21 S2 -1.66 -27 •. 69 61. 88 -84.31 89.78 -98.4~ 101.94 -103.18 101.57

6 s 1 28.92 1.94 -34.74 60.64 -80.81 91.7 2 -99.79 99.37 -102. 8 1 ..... S2 2.31 -34.79 61. 66 -80. 85 92.01 -99.7 s 99.47 -102. 84 103.99 +=> I

7 Sl 29.95 4.59 -34. 93 47.14 -62-46 78.8 4 -84.45 78.26 -85.91 S2 4.68 -34.92 48.15 -62.62 78.97 -84.3 2 78.35 - 86.05 90.59

8 S1 24.31 6.01 -26. 95 15.81 -29.12 51.6 8 -50.86 31.04 -50.88 S2 5.70 -26.90 16. 57 -29.42 51.52 -50.6 5 31.1l -51.12 61.58

9 S1 13.55 4.58 -14. 60 -28 .31 9.76 16.3 3 -8.29 -37.19 -1. 20 S2 3 . 92 -14.59 -28. 55 9.92 15.70 -8.4 3 -37.19 - 7.06 22.59

·--·- .. _ .. -- - ·- - - -----~ ...... ~- ·~· · -~-- -· ... / .

?ft~:.~ EI ~~.::1~ . )

r· n • ··r·• ~ '' T • • '') - · ,., ' ' I ' ·• .· ( r j-.·• ) :.!"- \! . ....; , :. .. .; ... -: . .J .. ... -~.!. ...: t•!L . J .!.,; \··.:..1

.!.L •

s~.~.·;..T:::mr i.DGE 1. _ifoJG J 2. EOiih: 3. E.1nE 4. :&JtLi! ·~ --.,~

EJ)rl 11 t) __....::.::..!.. .: .!_ ~on.3_. 7. :.!D0J<J 3. -~).rj_~ 9. lfl)t}3 I U.

2 o. oo -0 . 23 1. 80 - 0 . 64 -0 . 86 1.8 0 - 0 . 04 -0 . 29 - 0 . 59 1.95

3 o. oo - 0 . 23 1. 8 1 - 0 . 46 -1.19 2 . 9 6 - ·0 . 4 1 -0 . 16 - 0 . 9 7 3 . 00

4 o. oo -0.23 1. 37 -0.31 - 0.72 1.8 9 -0. 20 - 0. 30 - 0 . 59 1.75

5 o. oo - 0 . 23 0 . 63 0 . 02 -0 . 04 0 . 6 5 o. oo - 0 .18 -0 .1 8 0 .45

6 o.oo -0 . 23 -o. 29 0 . 68 0 . 40 - o.o 1 - 0.14 0 - 51 -0.15 - 0 . 22

7 o. oo -0 . 23 -1. 22 1.63 0 . 53 - 0 . 31 - 0 . 6 3 1.80 - 0 . 50 - 0 . 3 1

8 o. oo -0 . 23 -1. 79 2. 38 0 . 5 1 - 0 .2 6 -1. 04 3 . 0 1 -0. 82 - 0 .13 . ..... m 9 o. oo - 0 . 23 -1 . ss 1.40 0 .71 - 0 .4 1 -0 . 68 1.92 - 0 . 43 - 0 . 25

I

.. . I .

'i'A3L3 EI. (Cont . )

D ..;; F 1 E C '!! I 0 :~ 3 (10- 3 !!) •

S:::'ATIOiZ PLA'~· i 1. ?LA'.':!: 2. FL ;.;~ 3. FLA.• .... :3 Ll . ?~u·':_ ' :; ~ . P:.:\. , :;_: , • FL..'.':.'l:; 7 . FL.:..~!."..:1 ~t . ;:·~· ~~.'~·~ :y. - --- - - - - · - ... _ .., __ - -- -·-··- - --- --- ----

2 - 7.14 -3 . 79 13 . 55 -20.71 26. 44 - 27.47 29.77 -30 . 64 32.43

3 - 15 . 00 - 7. 96 26 . 59 -40.26 55 . 10 -58.9 2 !>ts . 20 -59 . 75 67 . 67

4 - 22 . 18 - 11. 75 37.83 -56 . 33 77 . 26 -82.7 3 81.99 - 83 . 92 93 . 47

5 -27.02 -14.28 45 . 55 - 66.19 86.33 - 93 .3 e 97 . 33 -99 . 34 104 . 50 . -Gl 6 -28 . 25 -14.98 47 . 95 -67.51 83 .01 -90 . 67 100 .85 -1 02 . 67 100 .94 I

1 - 25 . 61 - 13.57 43.49 -58. 80 69 . 52 - 76 . 6 9 90.18 -91:.60 85.17

8 -19.34 -1 0 -23 31 - 74 -40.56 48. 96 - 54.5 6 65. 16 - 66.05 60.59

9 -10 .37 - 5. 48 15 . 20 -17 . 74 24.83 - 27.9 {: 31 . 12 -31. 47 31.06

re -tested under simple bendi.ng s tress e s in order to obtain the vnlues of E and ') .

~.1 M

5 ) The model test stresses '-'lere rrEa3ured during t he testing at 126 points, according to the figur es 5 and 6 . These points corresponding to 5 2 t r ian gular ros ettes and 7/l. l ongitudinal strain- gages .

6 ) Thr· cons idered loading condition ,:,·a:> the sel f - \·:eight and '-'!as simu] u ted

by fn8::'"t ns extensional rubber tubes sepnratec! in plan 6 . ;?.!5 ems . By appliC£ tions nf three hydraulic jod<s over a horizontal strong steel frame the above ment ioned rubber t ub e:, '-';Grr:: under tension producing thn desired si mula'!:cd dood load in the model .

Photographs 3 and 4 s hm·:s t he loa ding technique used .

7) The shell actual bounda ry conditions wer e repre: ented into the rnodel r1ccorclino t o phot ograph 4 and its importance was ovident during t he tes -!::ing .

8) The distributed d ead load consider ed in t he prototype was - :?

p = 0 . 055 k p C.'ll -p

The model u cting load in each rubber s pring i s 1 · k p$ , and the equi valent

distributed load PM is

p = M --~

6 . 25""

- 2 0 . 256 kp cm

And the stress r a tio i s

c; r_ (f t-1

The deflection ratio is

f _ P_

f M t:: ~\ L M

f) I f

PI"\

e-r Le E M

~

It \'1as assumed E p

13G ClOD kp cm - <.:.

The abov e formula a r e va lid onl y if J f :: }J M . Some P.Xp8rience and ~gineer­lfi ng j udgement clre necesary in order t o ob tain an ae.corated model r esults interpretation .

- 17-

_. CX>

I

I

I .---~

r-I=

I-

I= ~

I-

I= I=

,___

~

~ .I ~L-

I e ROSE:TTE:S i o STRAI N-GAUGE:S

I I I I I ! ! ! I

i I = H

I i I . 1--

I ! f-~ .

i F= k I F=

i i f-

1--

I I ~ 413.

4 48

23 37 147 i 60 F= k '"' 22 36 Tsv ~ 35

14 46

I. 140 58 62 .

[ 20 j33 44 7 I-f-l ! 041

32 I i F= H i I F=

. i . 119 Ill ' 43 1$5

1--f7 f- I-! 18

30 t 42 )5' 29 41 61·· . i z 52

17 27 • 4 0 .51 15 HI: 26 :19 r-~ . 25 50

2 r 4 r 4 T "' r ·9 I

FIGURE 5.-MODEL I NNER FA CE.-SITUATION OF STRAI N-GAUGES AND ROSETTES

I I .... ~ I

I ! I i

(

\ \ ! l I

I J

I I i 1 I I I

I r ''" .....

\.0 \ I ! I _\ 7!1 100

.811 zo ' 10

87 ; gg ,. I

~67 I 86 98 ,97 tOll

I 117 4 85 101

66 ]74 84

~ )8~ 96 ' I~ 7 16

\ I 9!5

~ i

/_ I

94 I l I

t !.f. ' IIIZ j ll3 c 0 8 lli

\ An 81 4 92 ,105 114 122

65 70 91 ... 79 4 104

I 11) ~121 14 78 4 90 102 4

'!'u 77 19 101 Ill 1 ] 1 49

31 24 1 ' 4

2

E TTE:S • ROS

o STRAIN - GA UGE:S

FI GURE 6.- MODE L OUTER FACE.- SI TUATION OF STRAI N- GAUGES AND ROSETTES

The . model str ess \'ter e obtained from the str..i. : l~ by t he follm·1ing formuloe

-r ~"' ----

) l.. A- ), M

( M l l - -·- (~ L I"\ + ;) M (- 1 '1 ~ 1 - /} l

K

•acre E., ,, J C: 2 M G. h(~ f 3M a re the stroins measured by the rosettes of a!! gle 4SQ . In the figures 7 and 8 the prototype str esses a r c shmm ond i11 the figure 9 the normal displaccments a r e also given.

Conclus ions

From the exp<:rience ga ined from the desian of this type of s tructur es ( 3 ) s ome provis ional conclus ions can be dravm :

1) The non-pri::;matic folded plate roof r epresents a very s imple structure f rom t he design point of viev! .

2 ) The presented method of numerical nna lysis . very specialised for this type of strucbJres. seems to be vGry sui table from the econoroic;:.~ l and -­compu t:ttion point of vie1·: . The accuracy of t he obtained r esuJ. t ;; o re enough for practical purpo~es .

3 ) The construction and testing of the c l astic model of thi s type of -s tructures does not offer any n ev1 difficul ty with respect to another s hell structures r ather it seems offer more construct ional advantages due to its geometric simplicity .

4) A comparative s tudy of the results from t he numerical ana lys is a nd m£ dd test shows a quite good agreemen t beto.·!een the:n. Some sm<'tll differcm­ccs of these r esults can be expl~ined due to the sensitivity of the model t o the a ccuratel y r epresentation of the existing boundary conditions .

- _() -

I) ~

I

.

I \

Q HORIZONTAL TRASOUCTER

QS) VERTICAL

Q

I

"'\(

I

..1

I

I

I

I .....,

I

9

0 0 I

I

I

I

' T

~ I

T <;>'

y l 9

6 J

I

I

~ ± .

+ I I .

± t I

I

9

0 I

I

I

I . i . I

I

I . I

I . I

I . I . I

9

FIGURE 7.-MODEL OUTER FACE - S ITUA TION OF DEFLECTION TRASDUCTERS

r I H

I _

l i\

-I

STR E SSES ( kp .cm -Z) STRESSE S { kp . cm - 2 )

POINT' DIRECTION 11DIRECTION 21DIRECTION 3

(0"~ ) ( (j2) ( ""()

POINT 'DIRECTIO N 11 DIRECTION 21 DIRECTION 3

(0"1) (0"2) (""()

95 29 -29 33 11 5 - -

-12 -65 - - ---- -

-~ 9 10 47 -----·- ··- ·- -- . - - -

-~-1-- ---~- 0 8 I 39 108 --- -- -- - - o- --·--

74 22 -14 40 - ------ -- - -. - -----10 14 -5 -3 -- --- -- -- .. ·- ----24 -56 I -44

---- -30 - 2! ---- - -- -

L ---~~--17 . 48 -- -30 -4 -4 40

- - - -- - - 1- -

-22 72 ·- -

;~ 34 -5 -4 72 » ·--· --- - - ·

3 -25 I -55 I -- ··-·

83 - - - ··-- --16 -8 -!4 -26 I -58

- · ·-· .. 17 -10 8 _ .. ___ 18 66 13 3 13 -- - ----- ---

~ 9

··---0 -60 I -37

- -- - ------ ---- -I -48 - 18

- - - -- - -- - - -2 57 18 9 24 L -12 I 18 -- -- -- - - - -

76 - 19 -- --- - - - -28 -23 ~~------ - . --·-- -7 57 21 - - - - - ------ -

~ -35 58 -26 -23 13 0 - - - - ·- - - - ---j i 3 27 13 -17 -2 ~~9 4 i8 -4

- -- - -- --- 19 MO_ 30 - __j

IL_ __ 2 - - - . - -- - -- -- - --

29 25 -2 -3 61 0 -8 -2 - ·- - . --- -- - -- ·- - -. 30 22 41 -6 62 I 2 5

-- - -- ·-- - - -~I 68 63 I I -3 6

--32 86

F IGU RE 8._ PROTOTYPE STRESSES._ INNER FACE (kp cm2)

\) ;..)

STRESSES ( kp.cm 2

) STRESSES ( k P~ C"'-2 )

PO INT DIRECTION 1 DIRECTION 2 DIRECTION 3 POINT DIRECTION 1 DIRECTION 2 DIRECTION 3

t er, l ( (j2) ( "( ) t er,> ( CT2

l ( ""C)

64 -31 -33 16 i-- . -

65 7 - 13 5

-78 !---

95 -97 -66 4 7 7 67 - 12 2 8

~6 50 97 -97

68 5 98 4 3 0 -2

69 -18 -7 2 99 52 1-- ----

_ 70 -18 --26 71 4 -20

100 -lOO 1-- ---101 -5 -- - --~

72 -11 102 -30 -19 0 - -73 -18 103 - 17

- 74 ---· ·--12 104 -55 -27 -4 - - -75 -3 15 7 105 2 32 -10 -- - -- - -·--- - -~-- - - --- ·-76 21 106 3 1 -- 77

- -~ - ---6

-~-- - 1--·- - -- -- -78 6 -12 8

-!-- 107 4 6 -- ---- -. ·-

108 -74 - - 1----- -- -· - -79 53 -4 7 - - 1- ·-_ 80 -30 - --- ---

------ - -- - -109 - 11 -26 - 3

1-- - 1-- --f-~ f--110 _ 16 ·-

81 11 - 4 6 Ill -I --82 12 112 - 5 I I - -83 20 113 - 10

-- - --84 -73 - -85 25 37 5

r-- -- 1--- ----- -86 4 33 0 -

11 4 14 23 - 4 - -- - !-- -·-115 17

c- -- - '-·--116 21

i--87 46 117 -1 5 - -----·-88 5 36 - 5

!- - -- ---_8~ -9 -

,___j_l8 - 9 - 14 - 16 - 1---- -119 16 - -

90 5 4 5 13 - ---- ----·--- - -1

~I -26 - - - -------

120 - 11 -18 - 17 1-·

121 - 4 0 9 -12 1-- - - -

92 -20 2 2 -7 122 -7 10 - 11 - -·----

9 3 3 3 ------- L- -

FIGURE 9._ PROTOTYPE STRESSES._ OUTER FACE ( kp .cm-2

)

3

"---- - 2

~

_6_1

l l I

I

3

6

1-6 ~

...<'-3

I l

01

+ I

"-;"_7

l

11

11

·-a

j_

o' 0-1 l 1- 4

?= ~

-9

).- 12

~ -13

. 13

i 12

- 10

,.I. ~

5 j_

I . I 2 ? I C(o Q HORIZONTAL TRASOUCTER

0 VERTICAL FIGURE 10._ PROTOTYPE DISPLACEMENTS (mm. )

Q2 J

....J

-L j

I

j

~ I 1

C( -1

u

'6 ....... N

~

i (/)

__ _.,_

~ z "Q u 0

1-u w (/)

,<t 'lo

..<. I

'<{ "'

ts

11'---___:_--~­t<.

. ., .. ,

N - ­r<

0 c w 1-<{ .....1 Cl.

w ::r:: 1-

1.1.. 0

z 0 1-

z LL. w 0

I ....:

J w 0:: :::> 0

z LL. <{

...J .....1 Cl.

- ~~6 -

w 1-<( ..J a..

w 1-<( ..J a.. w :I: I-

I&.. 0

z 0 1-u w tJ')

..J <( tJ')

er w > tJ')

z <( er I-

I

~ w er :::> 1.-' r;:

EDGEn

~

/ /

e , ---------------

/ /

~/

/ /

/

; /

; ;

;

TRANSVERSAL SECTION

F

PLAN

Fl GURE 13 .- SIGN CONVENTIONS

- 27 -

- EDGEn•1

V

_.,..- PLATE i

PIIO'IOGIAra 1.- Hoilel t..... f••· lttuai.eo ef •wd.a ..... _. roeett.u.

- 21-

P80'IOGUh 2.- Kodel oat•• faee. stnatf.oa ef suata ........ ne•tt••·

- 29 -

- 30 -

- 31 -

.l\cknowledl]ments .

Ti 11.1 <lc>sistancc of thn l ate I nneniaro rJe Caminos J . L . Mar t ln i n the nume d .cal an<> lysis is r ecognised . The nmde}_ analysis have been carried uu L by t:fte staff of t he LBborator:i.u Cen l:r al. under thP d :irncti.on of the Or .

,J. Moreno Torres , Inonnicr:J cJc r::umin::n . The! collaborn t:i.nn o f Dr . ,J .

l~lartinez , I ngeniero de Caminos in wri. ttin~ thn comrutcr [lr ogrnm is nlso a cl<nowlt::ugeci .

Aibliograph~ .

( 1) "LOI'!G NON- Fl!-lHi·AATIC FOLDED :3TF1UCTIJ::IES"

C. D. ,Johnson a nd T. Lee . ,Journ . !-:i !: r ue . Div . A ~.i CE . ,June 1 , 9[:;[1

( 2 ) " A SU~lVEY ON FOLDEO PLATL:: SFHJCTlFiE 3"

A. Snrnnrt:tn a nd J . Martinez . I . P, . S . ··j . Colloquium of Madrid . Sep t . Oct . 1. 969 .

(3) " TRIBUNE FDn THE DOG R/\CING STADI UM OF MADniD"

F . del Pozo , J . A. Torro_jn r:mcf F1. L6pez Palanco . nctober 1 . 962 World Conf8rence on Shell SLl'ucturcs helrl CJt Si'ln Franci sco . Na­tionnl Ac<'!demy of Scie nce , Ptlblication 1 . 18'7 - 1. 96-11 .

- 32 -

Appendix A. -

Notations Computer Program Description .

1. Input Data 1. 1. General Data

I = number of pla tes N = number of edges (N = I+ 1) A = number of statio ns E = Modulus of elasticity

1 . 2 . Plate data

1. 2 . 1. Longitudinal (fig 101 i = i dentificati on nu1nber of each plate L(i) =span of the plate nD.i p(i) =angle of t he plate nO.i

1. 2 . 2 . Tr ansversal (fig 11)

,-x_ = station or t r ansver sal section a t position ,">( For a""lch plate n9 i at t he sta tiono( is given :

hG~. i) = horizontal projection v(~,i) = ver tica l projection e ~((>(, ~) = slab thickness ( flexura l behaviour) eM (•-<, ~) = plate thid~neSS ( me:nbane beln Vi our) S(o<,i.) = area z (o<>. ) == strengh modulus

Note . - If the section ()( of the plate nO. i is rectangular , then.

S(ot,L ) :: ~('-'<1 1.) 1 , ,1

(.X1

1. )

2 (o<, ~-)::.-!- .S{_<><, ~ ) l(o<, L) b I

were /J l.. l ] 2: R { <><, ~ ) : [ er ( "", t) -r 1-t- ( .x, t) _

1. 3 . Load definition

1. 3 . 1 . Self weight Q'{t.}w: unit density of the plate~ m . i

1. 3 . 2 . Prestr essing (I

Y{<><,;)= axial force acting at the station of the plate n9 i ;-:(0(1t)= bending mom:.. nt acting at the station of the plate "n9 i

- 33

1. 3 . 3 . Uniform distributed loading

p(i) = vcrti e<l l uniform distributed loadina for uni t horizon tal proj ected c:tr' ln of the nla Le n n .i .

1. 3 . 4 . Unifor m distribu ted l oadinn a l ong edges

q( n) = vertical uni fan~ rlist ributed l oading for unit of l ength o f cdgr~ nn. n .

1. 3 . 5 . Concent r ated punctua l loads

R(.x;•0= punctua l load acting at the sta t i on of tho edge n .

1.4 . Boundar y condit i ons .

n = edge number

Y\ e(11)= boundary condi t i on hUJn)= boundary condition

ni<,,U,) of the edge n .

cod e for

code for t he rotation a l ong the edge N

the hor i zo nta l displi:lGC81nent

Note . - I f the bou nda ry condition code i s equa l 0 or 1 m(.!'.ans unconstrained or contr a ined dis placement res pective l y .

2 . Computer results (fi g 1~ )

tfi:(o<lt) =

Qk (c< ,c) =

rotation along end k (k = 1 , 2 ) of t he plate i at the s tation C>(

r educed shear per uni t leng th of t he end I< (I< = 1, :~ ) of the pl £ te i at the sta t i on 'X.

h1(~_. f\ ) = transversa l bendi ng momen t per uni t l nngth o f the edge n at the s tation o(

Mt.(o<, ,t.,)= l ongi t udina l bcndi']:JiTIOrnent at t he station of th8 pltd;J i asuming its structurul bch;'lviour as a s i mply s upporter! be::1:n.

tJ (IX I..)= longitudina l axia l force at the station of t he plate i. L ' ·

S/c<, L)= normal stress acting a l ong end k (k =1. ;2 ) of the pla te i ond due to M ( o<. . i) .

Tt~--<.~) = total shear acting a lona t he end k(k = 1 .2 ) o f the plate i a t ­the s t a tion

tll(<X',L) = beam dis pl a c ement of the plate i at the secti on

)._/o<,tJ= norrn~ l displa cement of t :1e end k ( k = 1 ,2 ) o f the plc1tr. i at the section

- 34 ·-