ANALYSIS OF A HYPERBOLIC PARABOLOID ,,

by

Carlos Alberto Asturias

Theai1 submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SCIENCE

in

Structural Engineering

December 17, 1965

Blacksburg, Virginia ·

-2-

TABLE OF CONTENTS

I. INTRODUCTION •••••••••••••••••••••••••••••••••••

11. OBJECTIVE ••••••••••••••••••••••••••••••••••••••

Ill. HISTORICAL BACKGROUND • •••••••••••••••••••••••••

Page

5

8

9

IV. NOTATION ••••••••••••••••••••••••••••••••••••••• 10

v. VI.

VII.

VIII.

IX.

METHOD OF ANALYSIS • ••••••••••••••••••••••••••••

FORMULATION OF THE METHOD

THEORETICAL INVESTIGATION

• •••••••••••••••••••••

• •••••••••••••••••••••

A.

B.

DEVELOPMENT OF EQUATIONS • ••••••••••••••

ILLUSTRATIVE EXAMPLE • ••••••••••••••••••

12

15

24

24

37

EXPERIMENTAL WORK • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 50

A. DESCRIPTION OF THE MODEL ••••••••••••••• 50

B. TEST SET UP •••••••••••••••••••••••••••• 53

TEST RESULTS AND CONCLUSIONS • •••••••••••••••••• 61

X. BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••• 65

XI. ACKNOWLEDGMENT • •••••••••••••••••••••••••••••••• 67

XII. VITA • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 68

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

I.

II.

III.

IV.

v. VI.

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LIST OF FIGURES

HYPERBOLIC PARABOLOID TYPES •••••••••••••

DIFFERENTIAL ELEMENT .................... AXES .................................... DIMENSIONS ••••••••••••••••••••••••••••••

STRESSES ., • 0 •••••••••••••••••••••••••••••

STRESSES ••••••••••••••••••••••••••• ll ••••

Page

7

13

36

36

44

45

FIGURE VII. BENDING MOMENTS ••••••••••••••••••••••••• 47

FIGURE VIII. FORM AND DETAILS •••••••••••••••••••••••• 51

FIGURE

FIGURE

FIG•JRE

FIGURE

IX. WALLS AND DETAILS ••••••••••••••••••••••• : 54

Xo MODEL ···~······················•••o••••• 55

XI. LOCATION OP DIAL GAGES

XII. FRAME WORK AND LOADING

••e••••••••••••e•o

e••••••••e•••••••o

56

57

FIGURE XIII. CRACICING PATTERN •••••••••••••••••••••••• 64

TABLE I.

TABLE 11.

-4-

LIST•OF TABLES

EXPERIMENTAL DATA • ••••••••••••••••••••••

COMPILATION OF RESULTS •••••••••••••••••••

Page

59

62

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I. INTRODUCTION

The Hyperbolic Paraboloid is a surface which may be

generated by the translation of a concave parabola along the

path of a normal convex parabola. It may also be generated

by the combined translation and rotation of a straight line.

a fact that permits the construction of such a surface using

straight line elements.

Because of its structural properties. simplicity of

construction. economy. and the wide number of shapes that can

be obtained from cOfllbinations of parabolas of different

curvatures and from sections with principal axes rotated or

translated. the Hyperbolic Paraboloid has become a popular form

for roof shells.

A simplified anaJysis (Membrane Theory) has been presented

by the Portland Cement Association (1) for the use of designers»

but the application is limited to shells with stiffener edge

members (edge beams). For shells not meeting this limitation

the mathematical solution is rather involved.

One method of analysis (Shallow Shell Theory) requires

the solution of simultaneous eighth order differential equations»

which is s tedious job when done by hand. The availability of

the Electronic Digital Computer has msde it aasier to solve such

problems and has opened this subject to more intensiva inveati• gation by many researchers.

-6-

There are other techniques of shell analysis, but the

Membrane Theory and Shallow Shell Theory are the ones most

widely used in engineering design. Both theories are based

primarily on the general theory of thin shells with some individual

assumptions.

The Hyperbolic Paraboloid can also be considered in two

different ways according to the shape of its edges and according

to its radii of curvature.

The edges can be straight or curved (see Fig. I). One

of the popular straight edge types is the "Umbrella" form. A

coamon curved edge type is the "Saddle" form. The basic form

is the same in all cases; differences in appearance are produced

by rotation of the horizontal axes about the vertical axis. The

analyses are basically the same; they differ only because of the

rotation of axes.

The curvatures may be large or small. The analysis is

the same in either case, but small curvatures produce flat shell•

and introduce large bending stresses. With larger curvatures

bending stresses may usually be neglected.

-7-

' ..- _...-

I I I I I

' I \I ,.-..- .-JI'"

'I ,,,,,.,...-~-- Curved Eda•

Straight Edge

Fig. i Types of Hyperbolic Paraboloids

-8-

II. OBJECTIVE

The principal objective of this thesis was to find a

comprehensive method of analysis of the Hyperbolic Paraboloid

with curved edges and verify its reliability by comparing the

analytical and experimental results of displacements of a thin

shell model (l/8 inch thick) of a plaster saddle (36" X 36"

square) without edge beams and supported along two opposite

edges.

The analysis was based on the Shallow Shell Theory and

represent• a combination of the techniques used by Apeland (2)

and Billington (3). This is explained in detail in Section VI.

The details of the experimental work are given in Section VIII.

Finally, a comparison of the results is discussed in Section IX.

In general good agreement was found between the analytical

computations and experimental measurements of displacements.

-9-

III. HISTORIC.AL BACKGROUND

The firet approach to the analysis of thin ehells wae

the formulation of the "Membrane Analogy" in 1826 by Lame and

Clapeyron (4). The General Elastic Theory of thin shell• wae

developed by A. E. H. Love (5) in 1888.

A major development of thin shell theory occurred in

the early 1940'• when Marguerre (6) in 1938, Vlaeov (7) in 1944

and Reiesner (8) in 1947 developed the original formulation of

the Shallow Shell Theory.

Revieions by Nazarev (9) in 1949 and Munro (10) ·in

1961 made it more workable. Their results are incorporated

comprehensively in a recent presentation by Billington (3) in · ' ·

1965. There is of course a great amount of work.presented by

Wang (11) Timoshenko (12) and Flugge (13) with illustrations of ·

•pplication to particular structures.

It is an interesting fact that during the development of

these theories the lack of an assured technique for accurate

analysis did not deter the construction of many expressive shell

structures. The need for adequate predictions of the structural

action of Hyperbolic Paraboloids persists today so that auch work -

needs to be done to improve and extend theoretical snalyees.

t::\., b

c,.,c,

.:x.~, J!

l('X,'J)

/\,,fl;. ... , ... An

D

£

F

M ••• Mg,

MIJ'lt' Mx~

N:n, H'!

N1tlf • Hy ..

Qx., Qy

~ .. , ~,,

Pi

-10-

IV. NOTATION

Coefficients which convert the parameter•

to lengths.

Spans dimensions

Rises of shel 1

Thickness of shell

Imaginary term

Shape parameters

Radius of Curvature

Displacement components, longitudinal,

tangential and radial

Rectangular carteaian coordinates

Surface function of shell

Real constants of integration

Flexural rigidity

Modulus of elasticity

- Airy stress function

Flexural momenta

Twisting momenta

Normal stress resultants

Shear stress resultants

Transverse shear resultants

Effective tran1ver1e edge atre1s reaultants

Real part

- l l-

U, V 7 vv nisplacement components in the

directions (x., '-(., Z)

Functional parts of the characteristic

equation

Shell parameter

Ratio of shape parameters

Shell parameter

/ " Non-dimensional coordinates

Stress-displacement function

Root of the characteristic equation

Poisson's ratio

Qi + ,;)z -=>><-~ .;;J~'I.

~~ - .;;>'l ~~ ~~~z 'Vi ( "Vz)

'14("V4)

"~.

-12-

V. METIIOD OF ANALYSIS

The method of analysis presented and used in this

paper, is based on the Shallow Shell Theory and is a combination

of methods outlined by Apeland (2) and Billington (3) and using

the notation of Fig. 11.

This method is a general one which is easily understood

and can be used for the analysis of any shell structure.

The assumptions in the analysis of the behavior of

thin shells are:

l. Deflections under load are small enough so that changes

in geometry will not affect the equilibrium of the

system.

2. Points in lines normal to the middle surface remain

on lines normal to the middle surface after

deformation.

J. Deformation due to radial shear can be neglected.

Based on these assumptions, the general theory of

Thin Shella, as formulated and presented by Billington (3),

appears in equation form aa follows:

-13-

..... ........... y

........ , y

Fig. II Differential Element

-14-

(j)

_Where a •• f ti. and a,.r, , the Q!t term in the firat Eq. vanilhea

aa do the Q. tera in the second and the N,,'l and the "11Jx tenna in

the third.

(2)

M • _ o (..1.. ;> e. + ~ ~Cl,, + JJ (.l. t.16, + 9>c Ola1 )] " a • .w" 011a, ~ r a, ~d.IJ a.a, ~ [ I ;e., Ox ~it ( \ 9)eh. 9., ~Cb)] M, - O o, ~ + a .. a, ~ .. }' a" ~ + a;a, ~,

M ·-M • Det-,Ji) Q ~9c + ..!. ig' _ Q" ~- ~ ~') ., !• 2 \""Ot ~!t Q" 411• a11a, Q.ly Ga0J 9>.i.IJC

.When we recall that

9a & J!:. Ji- :;) (13' r... Q•:w"

and

-15-

VI. FORMUIATION OF THE METHOD

The preceding equations of general thin 1hell theory

are u1ed a1 a ba1i1 of the Shallow Shell Theory with the

following aaaumption1:

1. The 1lope of the shell la small compared with

1ome reference plane •

. 2. The curvature of the surface is small.

3. The shell boundaries are 1uch that the aurface load•

are carried primarily b1 the in-plane 1treaa

reaultantl

4. The change• ln curvatur~ of the 1urface are amall.

Applying theae a1aumptiona to Equation• l and 2, the

following equation• are obtained:

1 ~Na. + _1 ~MIJ"' + Qx ~ a, a-., px - 0

1 9'My 1 ~"'"'' Cit' ~ + ~ ~-" + Pc.t • ..!...~ ... -1.~'I + ~""" 1~'°'~. a,., ~ a':I a.a1;1 r" r'Mt

0 l &)

+~:+pz. - 0

• 0

•

-16-

r , ~I..(. u::r ( l dll" "' ) ] v . -·· ··- - - + - - - --t .-:i. • .. :~ r,. ~ ~ ~·~ r'i

f1 ~hr t.J (l ~LL o:r)] "'[a':t ~.>y - -r-; + ~ ·a,.-~; .. - (";

h / -~ <:hr + _1_ ~v.. _ 2u::r ) G \ u,. ;)o... a&J .'.:>o.':S r,.,,

(. M;i:& -D -.\, i ..::>'o.:r \

·' ,u-=-i ·.-··---i) t..~-1. -._: ._ ..... ~

~ . I ~·._.:: . ·. .·.-:',. '. M>'. - u (a,..• Q>d,. • -t· /' (:';)-, -~~:;,)

M~ j ~ D (L-jA) ( ~"°'J £:~.~)

Combining EQs. (5) ~iv~s

'Zul ) r,.,

Equations (7) are differentiated twice and combined to give

1

~·

Whet>. I,, , r_, and rx, are assumed constant over the surface

1

The stress resultants may be expressed in terms of a stress

function, F, as

c -; )

(6)

( 7)

( 8)

(9)

-17-

(10)

By sub.;titutio;1 n;- Eqs. JO into Eq. 8 a g~m.•ralizeci form is

obtained.

(11)

Pr.oper substitutions of Eqs. 4 into the third of Eq. 3

&ives

2_L_ a .. a~ 0 (l2)

Sub6titution of the stress function F into Eq. 12 gives

'VJ+w- - -fr -v: F - 15- - ~ ( ;~/p.,.a,..dw.,.-+ * /p'J a!j a.),) (B)

Which can be written as 4 i T.

V CJJ - O V" F • I f (p) (13a)

Rewriting Equation i1 gives V 4 F +I'. (1-f") V~c.U • f (p)

Operating V4 on Eq. lla and \'. (1- ~") 'V~ on Eq.13a gives

(J4)

E.h (l- )'~)

l::q. 14 OP.com» s \78 F + l'Z. E ~ 'V:F : "J4f (f>) 4- \I-. (I-}>,_) \l;, f' ( p) (i4 .,.)

-18-

Thia theory of Shallow Shells ia nov applied to the

particular problem of the hyperbolic paraboloid with curved

~, ... The equation of a Translational Shell i• C.rteaian

Coordinate• ia given by the equation

in which

if { >O hyperbolic paraboloid

~ • 0 parabolic cylinder

~ ~ 0 elliptic paraholoid

With the middle surface of the shell defined (geometry)

(1r;)

and the loading condition already specified, proper subatitution·

of these two conditions into Eqs. (11) end (13) give•

which in turu becomes· ( g TfA cfZ(J.) )

v4 F - la.a Eh -- - r -aJl' u ;>~'Z.

( Oi'F a'F ) () V"w + "1.-z 9);i' - r a!l2 Introduction of a atreea•diaplac-.nt fu.octtoa.

defined -.Y'

= 0 (st•)

'II fz (SJ.)

-19-

and

F=

in which 11 ;,1! ;;;>'

v l • :;,)"1 -1 ~· Then eubetituting Eq. 16 into Eq. llb

e • D

The condition• of Eq. 13 must be met for the

ahel l with the edge conditions along ~"' '!: !' being moment

free aupport on wall• rigid in the vertical direction but

offering negligible reeietance in the horizontal direction.

(16)

(17)

The equation that baa been found to eatiefy th••• requirement• 1a in th.£ form

¢•I, Cn e P~!f cos .kx. (1&) n•t

Introduction of iq. 14 into Eq. 13 gives

[ (-A2 + .<1 p1)4~+ ci.(-4".2 -r.< .. p1) 2 ] Cn e"A"cosAx. •O in which

p.Jm+r and

(. • 2 ..<. • ~n-:rr-=:;::=:===-~--. ~4do Q~~(1- }'1>t•s/h)1

eubetitution of lqa. 20 and 21 into Eq. 19 yields. m4 + t4 (t m + 1+t)2 = o

(19)

cio)

(21)

(22)

-20-

The eolutlon of Equation 22 yields the four root•

R-4 • : ( O.s ~ i. f.>s ) R.a • : ( d-'l ~ .: "''l )

"""')· ~ [ Aa e ''"' + At e f1~ + A, er~~" + A.4 e f.i11""' + Arie fwl"it-+ /;..'- e~·~•+ P.1ePYNs,... f\~cE'w'-•] cos A')(

('U)

-21-

All the unknown atreaaea and diaplacement• in the ahell are

given in terms of the atre•• - displacement function

which are obtained by aubstitution of Eq. (15a) into Eq1.

10, 3 and 6• yielding

NK• 1:1

., ... ~h Ir. vi. ( di¢) • d.!j s ~ ay• M.,'f • ~ .. & h ~ v' ( dz,p)

dKa .. ct a)C.• t-h!

,;lZF .. - E h k a v; ( d'-"' ) a---

a~z chc.0)'1

('25)

0-· -D (v~> D v• ( ac;6) ~"

::. O>>C a,. -0 (~)=

C>t;j D v•(~) e. • Q .. -+ O>M'-'11

~ ~~~ G., + ~14

~

-22-

The aolution of the homogeneous equation givea:

A, eC•·+i.t...><11 + A. e<~·-'r.-.>cti

rj (i<.'J) • A,eC~a+{~•>'4>+ t.,4 eC-'a-<f.Ja)<l- cos A'I. .

• 1',e-W.+C:f.11l~ .. ~. e(.»1_,,.,,)...,

... ti..1e-C..lt•'~'l"'-+ "' e c~.- '-,.,·l•

The first row of Eq. 26 may be expressed aa

[ ~. (cos f->•4- +~sin ~.q,) + "'" ( c.os ~4'- ~sin f.>• 4') J e-·+

vbich can be written a1

[ (1'. + t>..) cos ~ c¥ .._ ~ (A. - ~oz.) sin {J.>14' J e 0 •+

now, adding

So

A,+ A-a. ... 'la}+ A, - At. '2.t b,

'ZA, ... 'l (a+Lb)

A1= (a+&:b)

Then the equation reduce• to: + t.., e p,,.. + A,. e fc.,..

'2f2 cos /...x

where 2ll mun• two ti•• the rul part, and the imaginary part

ia dropped

(26)

-2'.\-

Particular Integral of Surface Load

The uaual way to obtnin the particular integral ii by u1ing a

Trigonometric Serie• expanaion. For a uniform 1urface load

a Fourier coaine 1erie1 i1 uaed a1 follov1

p1 • ~ f:>n C.OS Ax.

~ (x.~) •I.. ~ .... cos. l..x ....

giving the following 1olution

vi ~ + cl. \7; ¢ • P;, ( A & ;. al A 4 ) ¢" c:os /.. -x. = f... c; s ..<:x.

frOll which

where

and

('l7)

(2')

(!9)

~)

-24-

VII. THEORETICAL INVESTIGATION

A. Developwent of Equations

Following the development of the general method of

anAlysis of a hyperbolic parabolid shell, ~ specific

problem was treated to show the application of the method.

The shell was square (a • b) with free edges without

edge beams , a long !f = t T and with continuous support

-,. + .2... ,,., - ?. giving the boundary conditions

N.,., = 0

N"''S = 0

The load was assumed uniformily distributed over the projected

area (horizontal). The dimensions choaen were:

a= b::. ?,(011

giving the following values

).. ... 0.0872 nm \.oo

d .. '2.'?>b ~- o.~

-25-

Solving for the first boundary condition using

iq. 25 N~lj = '-h \r.. Vt' ( ¢', x" )

Homogeneous solution:

Calling 'Zh '-: .. = C,

and noting that

then

cos /...x

[ ~· ~,ef•'P + ~·tt.~ePa.iP l

+ ,<2~ :+ P, 1 /:i..~ er~·~-+ ~ ... t ta.,e~~

(a• ib)(c.osf.>.tp+ ~s.1nf.11 t¥)ed1 V'

+( c + id )(c.os f>-..4' + L sil'\ ~q,) edto/

+ (e"" l..f)(c.osf.)lo/- l.sin ~14') e-~.c.p

+Cc;-+ l.h)(cos ~ .. o/- i. sin {J;Lo/) e-..li.4>

I 8,.(a+i.b)(cos f.7•<P+ L~in ~1tp) ed14J 1 + p,2 (c~ l.d)(c..os ~v.4~ ~sin ~t.cp)e.,).,.~

11

~~t toS~X + e\'l (e*::f)(ws ~·Vt- l.5in r.>14') e.,)1+

"+ W (~· Lh) Cc.os f->•<i'- is1n f>,.4) e ...i • ., J ~

-26-

(a+ i.b)( A~ 4- J. ~ p.'l) (cos~''-" + ~sin ~·4') e~•-4-'

-+(c + i.d)(A4 + ,(}({')(cos ~·4' + isiO ~14') ed&4

+ (e ... (f)(1-.\. ,(~ P.') (cos ~·l¥- (sih ~,q.,)e~'.,.

+ (~+ t.h)(A4+ A~ fi') (cosf.<J- - lSIO ~tp) e-dt

becauae of aymmetry of loading and ge011etry we have that

a • •· b • f, g • c, d • h

Pn • d.n + L ~" Then the equation ia:

(a+i:lo)(.<.4 .... A14'" (d-1+~ ~,)z )(cos. ~14' + i sin ('>.q,) ed•4'

(c ... i.d)(~4 +Ard (c:h"' I'. f'~}')( CO'S ~i.tf + 'sin fi~tl') e01.o/

(o+ i'.b)(A4 + A'-t (~1 + i~,)'J( c.o• f1•4' - .: sin~· tP) e~•'f'

(c.+ i.d)(J.4 + -'¢ (d.a • ~ (>1tf)(c.os f''Lq.-i.sin ~t.41-Je-Ja.,,

cos J.x.

(a+ .:b)(~('\1'\'-+ i.s\n~1'\')• o cosf•o/+a\".s1'n~14' +~bcos(l,1<.f

- bStOf'1 o/

(a - &.b)(co•~·~ - \.sin f>•'¥) ~ a cos~·++ o\. •in~·~""' i.bc.c&~·~

- bs1n ~·4

-27-

[acos~,tp (A4 + ,< 1 (..>,t-~t)) - b sen (J,.'f'(/./t+ /...w.(d..,1-(!>,z))

+ a "n ~·4> (- .( 'l 2 c;J., ~· ) + b c.os ~· +(- ,<. • '2~, e,,. ) ] ed•+

+ [()CD~ <4•4 (J.4+ ).z (o),z-~,z)) - bs1ri ~1+(A'°'+ /..1'(.d,•- ~11})

- Q s1'n ~' ~ (- "• 2J., ~·) - b cos f.>• 'Y (-A 'I. '2d• ~'} J e-.l•'f-

becauae ot aynaetry it will be worked only for the tare

(a + lb)

e-J•+ • cosh c:J., 'I- + sin hot:)., 4'

~•'lo• c.osh di qi - S\n h.J1 o/

calling

( ,( 4 + ,( r.~ ( oi:),t _ ~,z)) • )(,

( - ,(I 2 ol1 ~·) .... X?.

(J.,.+ t,tif (ci>&'- ~t·}) • X;

(-/..2'2d..~,) 111 ~4

Q { C.OS ~·~ CDShd1<P >(, + ~1n r;14'9inhd1C¥ )(,_)

+ b ( c.os ~1ct- c.o-;hd,u.. )(-z. - s\n ~.ct- '51'nhd•t.\1 'IC,) t-t ,~ • 'l c.

+ c (eos ('nl}'{.O!.h*~· x~ +si n~4$inh~~ X4 )

+d (co~ r;~~ c.osh .Ji.4, )(4- sm (:kq. s'nhdz<V'1Cs

/...x. (11)

-28-

Particular •olution:

fl a L <;/,,.. COS kx "

Sub•titution into Eq. 17 yield• ~" co~"-..

using Iq. 2S

w,, . t: h ~t va- ( ¢',11:~) fr1· ~ ~~

"'"!\ • 1z .. (\'l)(\-jJT.) €4 h .. A4 (-t4 +4)

N1t~ • ~ c:.os ~'X I~ €'i \e: 4

ii"" c ~ cos t.. x

l + e."' 2T. 4

D

f'z

(Jt)

-29-

Solving for the second boundary condition froa Eq. 25:

Mlj'f = - 0 V 4 ( <J.y, + f ?, xx)

Homogeneous solution:

then

a"p -9~1 -

-30-

(a+ ib)(cos ~4> + ~sin~·"' )(-1.sp.•+ e.• ~ ~ 1,1 _ ~ J. 1 (',') e.»1+

•(c+ ~d}(<o~~lf'+ ~11iti~'4')(-A"P'\"i' f'..4+JJA.1-Ji1' .. ~•) ecJa4' M~, • -0 os~X

+(e+ ~f)(cos f>•«.¥- i.sin p,."-')(-A'P, 1 ... ~.•,..~At- r1.·w·) .~+ -"('i + \h)(cos f2>•'4- - ~~•n ('>,4)(-A • R '~ f,\ ~}. .. _~A 1~ ") e·.J"+

cal ltns

f.<'tir+ f. 4 + }' J."-~ CP,?.),

~I.'~'+ f• •+}'A'- }A /.11({1).

X.,s -/..11(~,.,,- [.!-") +~,4-ro~.'.J,&+ (?>. 4,. 1.'JA(l-d1•-+p.') >C"&• -/..' (•~- ~) +-d..•- c:; ~~~ .. ·+{it. 4_., ;.1..c (:-<d,'+ ~·} "'' ... -A 1(-'Zci.~ )- 4f.' • .J11 + 4 ~·'.a, - i' /' (-'2.>.P.,.) X.. _ A1(-Z~ ~) -4~·~ +-'~1al, -1.'JI-(-~~ fl.>.)

MYS a -4 D

a ( CO'i ~·'f coshc/.1 c.11 X 1 -+ si"" ~· 'f> ~'"~'c.¥ X5) + b(cos~1't'coshd1c.f )(!> - s\n ~'~ !t1n~1'fX1)

+ c (co~ ~"1<.pc.oshdz~ )(, .._ s\n ~1lp1a\nhc)"~X4)

+ d (c~ f.>'t t.\r cosh .J14' )(4 -9io ~..:c.V~\nhcl"'~ ><J

-31·

Parti~ular •olution

where

operating ';;/ 4 on f (x.~) a"d r'- yields

Mlo\11 • (---1 ) ( P• ) ('!_A) ( ..'... ~) cos /...')(. I +- £ 4 12z. t 'Z I."

4

(S4)

-32·

Solving for the thlrd boundary condltlon from

Eq. 25: ~,. - o [ v" (p,.,) + ('- ~) v 4 ( 9', ""IJ )]

Homogeneou• •olutlon:

then the et1va~ion is:

fl.,_. - D

(a.&. i b)(cos f».r¥ +'$in foul.JI)~~"+~.,-(\-~) A'"P. -(1- ~).<1 f.') ~.,,. + (c + Ld)(eo~ @,«11 .. ~s1n~"tV)(-e,A' ... ~7- (,-}')(J.'"{i+J.1t(i"))e~1.,.

- ( e + i.f) (cos ~''¥- i.~1·n ~·t¥X-P. I."'+ f.7-(\- ,.)(~"~ + .<''f."))e...)'"' - ('i+&.h)(eo,~~ - ts in ~~)(-~t""" ~7-(,-Ja)(.<~+~'fl")) e·.l&

.<x.

-33-

(-f. A•+ (f' -(1-}')(A'f. ~J.tf.i;));. .

C:-f1A'+ ~ 7-(• -/' )(J.'fr +t<'fti;)). )(,. -A~1 +.l,7-'Zl..l~ ~·' .. is; ~'(l.4 - 7~~.

')(t= -.<".Je+ .-,,' - th~~ e.: +sr;.: ~-7~-&

)(!a:r -~'~ - 7.l,"~ • S'i ~"..J,4 - 21 (',. •..). • + f.-.1

' C.4 ,tS It 1.2...f I 41 ')("·-A {'>L- 7d.\ , ... ._sc;r,.. ~. -21r4~' + 1 ...

~ (co~~·+ ~1nh~>. 4' X1 + s\n (°?>.1Vcosh~1q.. ')(s)

.._ b( cos ~·'f' !>1nh-l1'f' 'X1 - sin~.~ cosh~1cp X,)

""c ( coc ~r. 4' "l inh~hq.,")(? + s1°n 0>.o/~hc:)z.~>'4)

""d (c°"~t4 s1nhdz~ X..- s•n~t4 coshw .. 'f-X,)

-34-

Solving for the fourth boundary condition from

Eq. 25: N"'!I = - I; h iz1. v; (¢, ,,,, )

Homogeneoua aolution:

anc1

(Asf. + &,( p.i).

( -<'ei 1- ~ ,<. f./)

(c~ id)(co~f.>i.o/+ Lsin~·~) X 1 ed1L'fo

(e ~ q')( co-. {!»1 'f' - l 5\0 f.i1 lV) )(~ e~•'i-

(ca+ '-h)(c.os ~-z4' - ~ ~\o ~<¥) ~ e-._.1.._

a( ccx ~·IYS\nn.Jiq- )(, + ~•n @.~cosh.J,'f>><s)

+ b (co~ ~,q. ~1nh.l,4 x,. - S\n fi•"I' eoshdi~ X.1) HMJ s. -4C1. nA°"

-4- c ( cos@c ~ 5\nhc:l~~ Xi+ s10 ~ ... ~ eoshdz~X..,.) . ~ .

-35-

With the boundary conditions stated along the~-~ axis, the

principal roots of the characteristic equation are computed

for the parameters of the shell. The particular integral for

the type of load ie computed, and the two solutions, Homogeneou1

and Particular, are added together to determine the equations

which will give the constants of integration. For the ca1e of

uniform load and &)'llllletric geoa1etry the problem reduces to

four eimultaneoua equations.

. "''

The para .. tar of. the .•hell (Fia• .. :"111 and IV) are:

a. ba ~ io.

P· o.15

£111 3~ iO"' p.s.~

h11 Ye" in.

c ... c"ll. 9 in.

-36-

Fig. 111 Axea

Fig. IV Dimenaiona

.37.

B. Illustrative Example

A thin shell plaster model of a "saddle type" hyperbolic

paraboloid was analyzed using the relations developed in

Section VIJA. The complete analysis ls outlined step by step

in this section. The dimensions are given and the boundary

conditions are the eame •• those specified in Section VII.

Step 1

Dimensions and Loads:

L• Q • b • !»6 in.

C,• Ca • t 9 in.

h ' . • "i '"· C( • J. p.s. L

Step 2

Tbe shell constants: (see Fig. IV)

~ • o. 0872. ~

2 .

d· \'Z(•- ;•) ( \£~) = 'Z. ~b

'l..( ~ • O.l

Step 3

From Eq. (18a) c:ilo • 10. 07b

f'a. 9. 9t'i&

ola a 0.0099

-38-

Step 4

Along the edge ~=- + Q. 1:

~.A,• \").700 0

d, A~• o. 01s;~

p.,, ,(~ • tt:?.&000

fl>. A~ r: l. C:,600

Step 5

Trigonometric functiona:

sin ~· lJ' • o.o cos ~· o/. -l.O sin ~"': 1. 0

cos ~ .. o/. o.o sinh<=>.~· I\ \6'1.bO

cosh~. o/• I\ i6f.60

cosh d-1.\.\' • L oo Trigonometric and hyperbolic functiona:

sin ~,...,.;nh.i1'-¥• O

sin~wcosh.i,~- o

c~ (!>. 41 s\nh~,4' • - \\., l6'l.. 6

co~ ~·~ co•hd,~ • - 11., i b'Z. 6

$1.n ~4' ~inhd1.Vf ..

sin ~'I} CO!>n1::h.~ ..

c°' ~a "f cosh .J& ~ a

~ ~'Y ~1nhcJi1t4' ..

.O'lO

i.00

o.o 0.0

-39-

Step 6

Shell edge conditions:

~"" • ~ .. , hOCllO'lf'HU9 ·~ ~ .. " pc11rhe41\Qr • 0

M1tit,. M'.i'Jh + MIJ'Jp - o I?.! • iz,h - 0

- 0

Step 7

From the first boundary condition (lCf· !tl)

Homogeneous solution

x, ~ o. o 874 + 0.0811 ( 10. 07';' - 9. 9~1 )

~,. 0.0&1~+ o.on•( 0.0099'- o.CJ,iJ&')

>'-1 • - 'Z 1l 0.0&7 1" i0.07'-,. C),q?,

><4• - 2 ~ o.o&7z1t0.oon it o.99'8

•O. OIOG>

a-0.007'i

.:r-!.'50,

=-o.oooli

'ZC. (-1'1.'Z.&a + l6 74-4.s;b + .ooooo!.+..00011?) cosA'll

C,• 2E.hlE~ 'ZC· ,,,ooow.'1

ParticvJ•~ solvtlon (Ml\· ~~)

;- 22. & COi /..:x.

-J,, '100, 000" ~ "2, \10~ ooo b +. 59"C ~ 19.8d 111 +22.& ('7)

-40-

From the aecond boundary condition

Homogeneeu• aolution

a) Determine the values of v X X X "' , t ~ ~, 4

X,• - J.'1 (d,"-f>i1t)+fJ.14 - G~•'d,1 + ~4 + ~/.. 1- fAl.1(d11·,,.'J ><a• -/.1 (.-.'- ~,'} +d.r.4 -fD~:~&· + ~4+ ~A'- }A "-•(.;i,'-~1)

Xs.• -.<•(-ta\,~) - 4f.>."cl.,1 + 413,•ei., -}"Jl.(-'l~. @i)

x •• _,(.•f.t~~.)- 4~~-' + 4f.>i1~ - ,<.·,,. (-z~.,..J

0 • Eh' • 'too l'Z(•- ,,.,

. from ~· !>"!>

-4D• -'Zo,ooo

(- 88~10"° a - 57-8 ><. 10' b - 1!67 c + a9'2.4 d) CoS A-x

Particular solution

from '"1· ~ - 2<0. 00 cos AX

-88~ 1010a - ?>7.8••o'b - \17.t7tc ... 59'Z.4d • +'l6.oo

• -s• 4~.o • 0.9&l

.. - \700

• .o~a

(~&)

From the third boundary condition

Homogeneous solution

a) Determine X, 't Xt ., X,. and X"-

x, I: - A "cl1 +. ~.7- 21 cl It; f->17 + fS'=> ~I 'J f.-1 4 - 7 ~1,WI

X~· -/..cf.>, - 7<d, •f.>, + SS r;, .,...l, 4 - 21 ~·" ..); ... ~ f.>.1

><--i • - ""~. + .J ... 7 - '2\ .,J~ ~ ... + 3? ~" f.>,. Lt - 7 f:>t. '.J l

"A.. ' ' 4 a.." & (:!_., )(4 • -/... l'·- 7.,J, ~'I + S'5 ~ ~&. - '211 .. 1. ~ ... + 1-•

10 10 • 74.G ~ 10 a ~ 1~7'1C. 10 b + l.O c 1- .o"9d =0

• 1.000

(19)

-42-

From the fourth boundary condition

Homogeneous solution

a) Determine X,, )(, T ')(.,. and X4,

x.. .( ... ~ ... I. (~.~- !»f.i.~·)

>'-s• -1.'f-• + /.. ( •. [?,.~,z ... ~.')

><i • A'\J•+ /..(~:- 3~1d&)

X4• -,< '(?>t + ~ ( ~{'>w..J.,• + p, .. ')

from kq· 3CO

o.orfc:.-.1o'b "° .o&7c + .002'; cl

Step It

Solution of 1imultaneous equations.

-16.2 -.1o"a + ~.II 1t 109 b +o.~q, c + 19.&d

- 89.o "' 1010 o "!.78 I< IOIOb - \'7."l"Z c ~ !>9'2. 4d

-74.'-x 1cf 0- + Ho1.oo ~1010b + 1.00 c + o.o~tdl

-1.9 "iO~Q - O.Oil,K\04b + ·0~7 c + O.OO'Z-d

giving·

_., a .. - o.49& 'K 10

bs - 0.0021 I( 10 _.,

Cw + o.o2!>4

d .. - I. O.'l.

• \.70

- \.04

- -.012"

- 0·087

- o.

-+'li.1

• + '2.6.00

= 0.

• o.

-43-

The cons~ants of integration Eqs. 41 were substituted

into the equ4tions for the stresses and their values obtained

(Figs. 5, 6, 7). Equations for displacements were obtained

from the geometric considerations (see reference 3).

·A computer program was then written to obtain values

for stresses and displacements of the shell surface at intervals

of a/12 by by/12 in the X and Y directions. The results for

the model to be teated were obtained and are presented in

graphical form (Fig. V, VI, and VII).

The equations for the deflections are as follows where

U, V and W represent the displacements in the X, Y, and Z

directions.

• Cl

CJ t

..J

"' II .... c GI

"

31.2

0

+;. 0

0

0

0

crown

-·

crt1 w n

Nvx

-44-

----~ I_,.,,,,,,,,-

~

1s;• au pported edcte

Fig. V Stresses

~

v ~

M&

., • ~ -

cu 'Oft ,, • • " ~ ~

·~· MJ.

QI t!.• ~

.... L • ... ~

" u o• 0

y .. o

Nvv -=-

y.o

~Lt

-45-

0 y .. 6

' \

0

\ ' U4.4 '(. 6

H.2

Fig. VI Stresses

0 10.4 Yal2

'" ~ \

' C S~0.6

YsU

'1 ..,. "G

" u u ... ~

+/n 0 Y1 l&

0 Yrl8

Attention Patron:

Page _____ omitted from

numbering

46

-47-

45• 146.4 IU..? 7!t,.!.

so"

Q) Q) ". "" ' , ~

Cll

b Q) 111 " ... .c .. " ~

" o• :ft/n 0 0 0

y, 0 '<• 6 lalZ Y:14

H xx -=

•;.. crown

0

1.o\& QI 0 c

!t6 ~ 0

" !IJ ~

7~ • ' ~

QI ... Cl 0

Sa mp I "I $uppod:eai ed~e

Mvv =--= Fig. VII Moments

-48-

Diaplacement in the Z direction

1.o (cos ~·4' cosht:.t14' )(, + sin f?i• 4' s1nh'*'1u~ Xa

+ 2to ( ros ~,...i. i:"!j;h .J, 'J.. x. - s1i, f·"~ ,.iohJ1~ X1

W = 'l CO$,()(, + '2c (cosf.>t~ cosh.h.~ '){'I+ s10 ~.-.. "1inh.lzcp')(4

+ '2d (co,f.>s."- cos\i •• h~ x.\ - -sin ~-4' t.1nh-~~x~

Di1placement in the X direction

from ~· 2?

u = tz-z ul + ~) 'VSi,1( _ <· +, )( ~~ ~ ), 'll. _ ")(. ~"¢ J

U,. (•-+4)v9<,"X. u". -(1+}')(vt </.),-x.

u,,. -rxQ·~

(4'2)

Substituting Eq. (25) for

o (c.o,f>.~c.osh.l14- X, -4- s'n ~~sinh.),~ 'X-.)

+ b (to~~·"" cosh~1'1-')( 5 - sin ~,4 s1nh.;;)1't X,) u, • 8tr.,

+c (tos~'-" c.osn~'"' Xt + 5in ~'"'° s1nn.)tc.f> X.)

+d ( cos('1'4- c.osheh.'#- X"- 9\n ~t'4-s'"heh.14- X1

a(c.osP,,.4'cosh.J1c.+-'><; + sln~•'-1-sinh.,),4' Y.~)

+b (co!>~·~ Coshd1t+X~..:. ~1n ~'"' s1'nh~1~ )(.') (43) U7 =-4.olc, (•+f') •1n.kx.

+ c (cos f->1"' cosn ..li ')(.~ + slo ~&&4- sinh.Ji.4~)

l+ d (cos ~t4 cosh.Ji. X~ - sin~,"' '~h.J-z.c.vX~)

a (cos~'"' c.osh.J14> X, • + sin{?l1'4- s1.nhc11~ x,:) + b( cos~·~ cosh.1 • ...- ><3" - sin fa·~ sinhJu~ X," )

+ c ( c.os ~z.IJ. c.oshJ,4 x; + s'1n r.>i.4 slnhdt~ x;) 4- d ( C.OS ~Ci- c.osh~htf1'X~ - Sin ~'l.4 -J\nhJz.1'- 'J.."i.•) J

Displacements in tbe Y direction

from i;q. 2~

V • I-ls r-(Ht)(vr•9),~ - (H· ).l)(v'l4 ~), ~ -v ~ 't'9']

-50-

Vil I. KXPERIMENTAL WORK

In order to verify the analytical results obtained from

the equations developed in Section VIIB, a plaster model of a

thin-shell hyperbolic paraboloid was constructed and loaded

with a simulated uniform load (over the horizontal projection).

Displacements were ~easured at several points.

A. Description of the model.

The steps involved in the construction of the model

were as follows:

l. With the dimensions of the shell established the

first step was to build a wooden form (see Fig. VIII)

for the frame, 5/8" plywood pieces were cut with

the desired edge curvatures and fastened together

with wood screws. For the deck, thin strips of

wood 1/4" wide by 3/4" deep were nailed close together

at 45° from the axes to form the surface. They

were then sanded and varnished to provide a smooth,

non-bonding surface. Thin guide strips were put

along the edges and along the principal axes as a

reference to get the required thickness.

2. The model was cast on the form and screeded with a

straight edge parallel to the straight line elements

of the su1.·face to give the necessary thickness. The

-51-

'W 00 D Ii H fO IU1

Fig. VIII Form and Details

-52-

thickness was essentially uniform, although it

did have some variations up to+ 'I& inch and - l/&4 ';nch

3. The material used for the model was Ultracal 30

plaster manufactured by the U. S. Gypsum Company.

Because of its rapid setting properties it was

necessary to place it in a very ehort period of

time after mixing (approximately four minutes).

The mortar consisted of a mix of 60% sand and 40%

Ultracal by weight. The percentage of water was

15% of total weight.

Small batches of plaster were made continuously

by one person while the mixes were placed continuously

on the surface by a second person. Placing was done

so that new plaster was always placed in contact

with a previous batch before it began to set.

This technique required some practice before it was

successfully used to produce a good shell.

After curing in a moist room for three to five

days, the shell was removed from the form and

placed upon two edge walls of plaster previously

prepared.

-53-

B. Test set-up

The vertical walls on which the model was mounted

were placed a long two opposite edges ( ')( • '! ~) • The walls were

set on the parallel edges of two tables to provide working space

under the model. A good bearing surface for the shell was

provi.ded by packing mortar between the contact· surfaces. The

shell was not bonded to the walls but was free to rotate and

translate horizontally at the top of the walls. (see Fig. IX).

In order to provide lateral stability, one wall was

holted and braced to remain fixed. The other wall was set on

transverse rollers so to provide no resistance to horizontal

displacements.

Hangers consisting of 6" X 6" X 1/4" plywood pans were

hung by wire to the surface of the shell (Fig. X). They were

spaced on a 6" X 6" horizontal rectangular grid pattern. The

pans were positioned at different levels to provide more

working space and minimum of interference while loading.

A framework of "Dexion" light gage steel was constructed

around the model to support eleven dial gages for displacement

measurements (see Fig. XI).

I I I

-54-

~ y4 11 (free ib rotOJte). Detail 'Z. Det~i l 1.

Fig. IX Walls and Details

-55-

Fig. X Model

1 ' 2. s. 4. r;, c;.

7, 8

9, &0 1 II

-56-

hori1onta\ di~plQcement in 1ne x direction

lon~itudincil displacement 1n 1hc ~ dirr"chon

verhcal dls~Qcement 1n 1kc z dtr~.On

Fig. XI Location of Dial Gagea

-57-

Fig . ~I I Frame Work and Loading

-58-

C. Test procedure

After initial dial gage readings were recorded, ·the

hanger pans were loaded with small steel blocks weighing

approximately 0.6 pounds each. To avoid large a>ncentrated

loads, all pans were loaded and unloaded at the same rate

of one block increments. The dial gages were read after each

load ircrement and dP.crement. The data is recorded in Table I.

The load was increased to 28.1 pounds per square

foot and reduced to zero for four cycles. When the load

was thenincreased to 33.6 pounds per square foot the shell

cracked and collapsed.

Glo.G-1:

No 0 '2.44 4-BS 7.3'Z

I O.l74 0.171 o.1•t O.IC.4

'? q,090!. o.oca f 0.09, o. uc. !> 0.017 o.ou Q.08.'l o.oqi 4 0.087 o. ocu o.oq" o.o•u s. •. 0 s;" ft .ft t;I( a.or.11 o.r.1:.1 ~ o.044 0.041 0.037 o.O~I

7 0.4G4 o.Ar.2 o.4r.o 0-4'i' a 0.050 0.010 0.079 o.ou --, 0.'100 Q."'\UI o.~01 0.'10"L 10 o.~ ... ., o. '!.,0 o.3&b o.iso 11 0.900 o.~O! o.qo4 o.qo'

I .o gc;(O 0.101 0.107 o. W2. 1 • 9901 I.Olli I. 112. \.'109 J .o87Z .oti'B ,og'Z? .o9H ~ .04'S5 .04'0 • Ml'i .ol_~ r; .4100 .4CllO ,4q40 • 4qi;r;

' • 'i'Z61:i • l;')IO . 'IZ'3C. . 'ii e.1 7 .41qc; .4q11 ,4qo4 .4i-i4 • .ouo . •'Z'i3 • O'Z"lO .019"& IJ • qoeo .'toiO .t.t~'i ,qoli'i --10 •''>IC\ • H&G. .1EWi .'3710 II .qo'o • qoCi'Z • qoJ'Z .401.,

~

l O A 0 ( p.s. ·f) lst. C"'de.

'·'' i'l. 'lo l4.64 17.0 s 19.i;7 '2l. C3" 14.40 '26.'14 29.'H O.l'i9 o.1r;o 0-14& 0.140 Q.11'1' 0.130 0.1'29 o. l'ZG O.i'Z4 0.117 0.1~' 0.11, o.1i:;z o. ''"' 0.1"72 0.119 O. llC> o.1cn 0.097 0.09, 0.107 0.117 o.11r:; 0.129 1.130 Q.1~4. 0.1 !" n.107 0.114 o.IZJ lt.l~I O.l~.6. 0.139 0.14 ~ 0.147 0.1Ci4 "' ftt..:E. 11,0C,P. lt.OCiO ft,04.:7 ..... ~]9. ~.040 11.0117 n.o!_~ ""~~ a.oi& o.O'ZE. 0.001 O·OOC, o.oo.5 o.oo~ o.oo I -o.oolf -~.007 0.41j& 0.44, 0.441 o.4!»' t\.A.U> n.4~0 o 4U o. 4'Zf. I'). 4'1.7 0.011 o. 0 tr; Q.070 o.o IS 0.010 o.oOB 0.001 o .ooc; 0.001 o.Qo4 0 .1104. 0.904 0 -90'i 0.QOCj a.CJ oc; o -~oi. o.QO(. D."l'lb O.l>7B o.1c;,4 o.3i:;; o.H<O o.31q a.Jl 7 o. ~I '2 0-!0g 0.10; 0.901 o.qoq o."'l o.'314 o.91s; 0·911'3 0.917 0.91 e, 0-~1~

'Znd C'4c:le. o .u 8 O.ltZ Q.\'l6 O.l!IZ. 0.134 0.1!19 0.1~97 0.1404 0. \.C.I \

\.'JOO 1.nz 1.44'- 1.c;o1 \.(Dl\4 \.(DlG. 1.743 1.840 l.'106 ,\046 ,1\09 .1140 ,1118 • l'Z4'2 .1'190 • l'l'Z.~ .1180 • 1+4 i;

-'-~-'-'8- .. 92..§~_ ~Ul_L ... !l_?OQ_ L...0.L40 _o t'lC ... oo•'i .00\7 .0010 .4910 • 4'-c;o • 4 lj"U) • 4'iOI -~4.'10 .44~1i • 44'2& .4411 .4400

• lljl 'lr::; .lj096 .'i0% • 'iO'ZG .4q73 .~}Z • 4'to'- • 4<t~3 .4&40 .4f.Sf, ,4"7Ao .47Cjlf .4"'100 .41:.'-0 -411i'i .4c; 0 .4!80 .4')10 .01ia~ .016, . Ol'iO .Oii~ .oon .OOlj? .oo4o • 0010 -.oollj(, .oio4j_ ~Q40 .Gto~q .'\04)~ __ .~o?>ti .'tD~i ,qo37 .'103& .003'7 • 36~~-- . 3'i70 • 3i; II • ~i;o .1380 . nos; • ?l"Z40 • ?II\, .3091i .9947 .8990 • f,t)6'Z ,f)C)"D • 994'i .M1fi • tl9'ZI .41900 • 8S I! r,

., . . ,, ... ·0'· 1"~ Z 'I ,

Table I. Experimental Data

19.C-,1 9.7~

0.1;£> 0.1,1 o.1i:;' o.o,'ll 0.1'2.7 0.10, 0.1~1 0.10~ II nl•A n 01.4. 0 .001 Q.O'l.C. o.Li.~i; !l./.t.l;Q

0.012 o. a-i; ll.Q04 0.,01 O·!!>O a.no o.q1" 0-~0]

0.1~7 o. \I c; 1.bl5'J , . .,.,a .11qs ... ll6B .0100 .0100 .44·13 • .4q72 • 4q20 • 'H?IO • <4'i7 • .q.77 • 001\% • 0110

• Clt04 • q,~!) .31C\O • 1'iOO • &q40 • gqgo

"

-·--0

0.170 -0.090 --0.01~ -·-o.o~j-o~ a .o44 0.4,7 \).0;1 o.901 o.?.97 o.~071

.o 87 3,

o9'Z7! • 0 8"-C7 .0 'iO\ .4~1'2.

• 'i'!> '2 .4'1~

.Ot4

.1:u2~ ,3a,..o • qo~o

' '-" '° '

.--~----------------------~--·~-----

GAGl LO A. 0 ( "' p.'5.t I ~- C';icle.

No 0 C)."T b ICJ.'i'2 zq:zti !~.Cj2 9./C. 0 l • OC\4'i f.147 a. 12~ It. l'i'Z O. I 'Z'O 1. I 17 Ii .O~G7

2 .9SOO J. 'ZIO 1.1)10 '· f,J 0 1.!J<JO 1. ~o.o n'l.h" , • 0870 .1oc;o • l'Z 41 .1Ac;o .1400 • It 0 0 • 08"9. 4 .0410 .0?140 , 0 lbO .oo~o .0120 • O?i'Z. 0 .o i;oo c; .4901 • s;o oo • 4'i0 .4401 .M-?_ ~L ____ ,__.4~10

'" • 'SZCJO . "114 .Aqs .a~Ci .4~3 . c;14 • s:;?> 21 7 .41q4 .4'b~ .4<0_7 • it.,.£) • 41:b .41:.7 . 47'-0 8 • ozoo . o 1qo • 007 - • l'VI c, noc.. .0'2.0 . 0 "Z '!>I g • 907"o .qoog • qooo .iqq; --~..L_ .,___,_ti9c;; • Cf\ O'i 10 . ,qio .~C..4 • 1401 ~!i.l~ ~fol -~-~ . "?.q \I 11 .qo"o • totOt> • ec;,11 .s904 • e,~4~ .~q~ .C\ObO

4-tt-i C<jc:\e. I .OCl(;,4- • \'i8 • 1?>4 • IG:.3 • l'?.7 • I c;. C. .oqso 2 .9900 f.310 1-6~0 J..:.6~o l.'390 1 • .llo .q430 3 • O!.'C> • IO 1)0 , t'Z.~0 .t4So I .CiGO • 1'2.0 • n P."To 4. • 04GO • o~ljO . o \lt..o (\~l) .nl~"'" A 11 "S: 11n

~ .A"IOO • '700 • 4 c;1 AAi:; .ol?.n .c;oo • .d.CllO

'" .ljlOo • "i'lOI • CiO \ , AAo .440 . "i l !;. • s;-:i,7

7 ,4'2.00 .4~7 .4"14 431 • "iOI • ..LJ.~ .42'-t' 8 .01oo .011 .00% .ooo • 00!1 . ozo O""

9 ,q100 .qo1 .11\oo .qo .4:to4 • '10'- ci4o7 lo .1ctOI .~{,4 . \4o .31t !ZO • ~c;1 ll.O!nn

" • C\0'0 .qo I .900 .egg .{l,~Q .M°" -a nc.

(continuation) Table I. Experimental Data

I O" 0

'

-6 I -

IX. TEST RESULTS AND CONCLUSIONS

The deflection measurements from the test data are

summarized in.Table II and the average values are compared

with the computed values based on the equations of Section VII.

The differences between experimental and theoretical

values varied from 4% to 9% which does not appear to be un-

reasonable for this test set up.

In the development of the analysis, the assumption

was made that the longitudinal edges (~ 2 i ~) were supported

over deep slender walls that provided no hori~ontal restraint.

In the model the walls were not as slender as intended, and so

there were surely some restraints not accounted for in the

assumptions. This is noted for example by the readings on

dials 'l and & , whose changes are smaller than those

predicted by theory for loading. Also, upon unloading, the

displacements measured by dials 17 '3 1 , 4 1 'i and t, 6 did not

reduce to zero.

In noting the failure crack pattern (Fig. Xlll)it is

seen that the cracks occurred where the tensile stresses

were m~imum.

The shell did not collapse immediately although the

cracks were complete. An attempt was made to reduce the load>

.but within av ery few seconds the structure collapsed completely.

T ti 5 T L 0 A OS TE'ST ITME.OIP.ETICA.L ~V£~A~E VA.LU~S

l,t. c9cle 'Znd. Cyc:le 3th CIJde 41'1~de POSITION f9.tr.s.f ~9.'2p.if '29.'2 f'·f 29.? r.s.f ~Oll.1tot4TAL.

1-4

2 - c; .oao -- .Oto'!. .00'7 .08'Z .07q(..

3-6 .010 .010 . Oii .00<3 .010 ,QI I

7-~ .014 .Ol'i .014 .Ole;) .o l'l"r . 0 \'le;

'llSP.TICAI...

9

10 ,09'7 .oe>~ .oe,\- -.07e .oe~ .oa4o

u .01<=, -- .ol~t; . Cl_t;b_ - .ooz .0\~ .ol'2S

Table II. Compilation of Results

OIFf'<ltENC.E

.6o'24

.001

.oo~

.ooZ

.0007

%

~.?.%

9Yo

2'1%

2Yo

r:.,/:;

I O' N I

-63-

The computed maximum tensile stress at the failure load

was 425 psi while the average ot tensile test specimens was 450

psi giving more evi.dence to show the validity of the theoretical

solution.

It was intended to place a number of SR-4 electric strain

gages over the surface to measure strains that could be compared

with the computed values discussed in Section VU.&, but it was

felt that this should not be attempted until a number of shells

had been tested to failure for deflection measurements. It is

recom.nended that this course of action be followed in future

studies.

During the testing, there were some sounds··of cracking

which were traced to 'the breaking of the small residual bond

between the shell and the packing mortar on the walls. The

shell showed no evidence of lack of structural integrity until

the full load ·of 33.6 pounds per square foot was applied.

When cracking started it appeared to start near all

four corners and developed up almost directly to the center

(see Fig. XIII). Because of the small holes where the hanger

wires pierced the shell and because of minor variations of

thickness the.cracks did not develop along straight lines but

in slightly irregular lines.

-64-

er'l<:kt wich !"'\ado the ~+rudure +o c:ollopse.

Fi&· XIII Cracking Patt~rn

-65-

X. BIBLIOGRAPHY

1. Portland Cement Association, "Elementary Analysis of

Hyperbolic Paraboloid Shells," 1960.

2. Ape land, K .• "Stress Analysis of Translational Shells"

Proceedings, American Society of Civil Engineers, Paper

No. 2743, February, 1961.

3. Billington, D., "Thin Shell Concrete Structures"

McGraw-~Ull Book Company, 1965.

4. l.ame and Clayperon , "Membrane Analogy"

1882.

5. Love, A. E. H., "A Treatise on the Mathematical Theory of

Elasticity," 4th ed., Dover Publications, Inc., New York, 1944.

6. Marguerre, K., "Zur Theorie der gekrummten Platte grosser

Formanderung," Proceedings of the 5th International Congress

of Applied Mechanics, 1938, pp. 93-101.

7. Vlasov, V. Z., "Basic Differential Equations in General Theory

of Elastic Shells," (English Trans. in NACA TM 1241, 1951)

8. Reissner, E., "The Flexural Theory of Shallow Spherical Shells,"

J. Math. Phys., Vol. 25, 1946, and Vol. 25, 1947.

9. Nazarova A. A., "On the Theory of their Shallow Shells,"

Prikl. Mat. Mech., Vol. 13, pp. 547-550, 1949 (English Trans.

in NACA TM 1241, 1951).

-66-

10. Munro, J., "The Linear Analysis of Thin Shallow Shells,"

Proc. Inst. Civil Engrs~ Vol. 19, pp. 291-306, July, 1961.

11. Wang, Chi-Teh., "Applied Elasticity," McGraw-Hill Book

Company, New York, 1953.

12. Timoshenko, S. P., and Gere, J.M., "Theory of Elastic

Stability," 2d ed., McGraw-Hill Book Company, New York,

1961.

13. Flugge ~ W. , "Stresses in Shel ls," Springer-Verlag OGH,

Berlin, 1960.

-67-

XI. ACKNOWLEDGMENT

The author wishes to express his sincere appreciation

to his major advisor, Dr. George A. Gray, Professor, Department

of Civil Engineering, Virginia Polytechnic Institute, for his

assistance, advice and encouragment in preparation of this work,

and also for his guidance and help over the past year.

Also, the author wishes to express his greatfulness to

Dr. Richard M. Barker, Assistant Professor, Department of

Civil Engineering, Virginia Polytechnic Institute, for his

teaching and assistance during the studies of the author at

Virginia Polytechnic I~stitute.

To the United States and Guatemala Government for the

scholarship that made possible his studies at Virginia Polytechnic,

and finally to the Department of Civil Engineering at Virginia

Polytechnic Institute for the opportunity that he has to increase

his knowledge in his chosen field and for financing the experimental

part of this work.

ABSTRACT

Hyperbolic Paraboloid Analysis

Because of its structural properties, simplicity of construction, economy, and the wide number of shapes that can be obtained from combinations of parabolas of different curvatures and from sections with principal axes rotated or translated, the Hyperbolic Paraboloid has become a popular form for roof shells.

In this thesis a comprehensive method os presented for the analysis of a hyperbolic paraboloid surface with curved edges. Its reliability is verified by comparing the analytical and experimental results of displacements of a thin shell plaster model without edge beams and supported along two opposite edges. The model tested was 36" X 36" X l /8" thick.

The an~lysis is based on the Shallow Shell Theory and represents a combination of known techniques. In general good agreement was found between the analytical computations and the experimental measurements of displacements.