- - ' - -'Aj. -'-. -S.L

AA IN 1.' W OT $tj~t

HAIFA,

II

If ~ .l~ I ratioq, It. . - j ~ ~ S~1 I

*(~lATSI to j N --.. on s .emj I* I Ns t

T, chnion Resiearch and Dev2oprment z' --- 1 1,% itGA;I1n

haifa, Israel

3 flEPO'T TITLE

-)N 'Mr BucLfl:B ANID PSIBUCYLIM, OF rCW-M2 AltCi t; uiD) HR1'.

4OE$CRhPTIVE NOTES (7YP* oftor.5ot and 2flcfusve dtoo)

S AUTH4OR(S) fMagt n"me. Ila t ntme. IisfiaIl)

Dyrn L. Clive

6 EPORT OATE 7. TOT&~N a- 0A4 ' 17b to OF PrV,

Ang~ust 1971 V"i 15

8A COVJTRACT OR GRANT NO0 90 orIGINA M0.3 n!:aopy "Sim"E's)

F 61052-69-0-0040h PPOJCCT N~O TAB iep:rt No. 133

C 5b 0114CR r P OWC~I ,,1 o1h,rnuent- a IhL# r',.v b. meqiadChia report)

10 A VA IL AIILITY 'LIITAT1ON1 NOTIC.ES

T',hi! liocument has been approved for puh.Lsh releaso nr vaciu;

: .tribution is unlimited,

II ~1P -,%Fk~y NoTes 12 5P04S~jRlNG MWtTAieP-.IV1TY

___________ jAir Farce Of'f -'.ce 3ciei t 1fic hisuearcL.

________ 1400. *Wiionl1: 1aa2220)Ilk ABSTRACT

This paper treats the buckling and postbuckling oehavit,- or circulararches and rings under conztant directional pressure. Nlew erict andAn rxmte solations are iven for the linearized ei~renval' e 1lrlem.

Apymptoti - ana'yses of erry rnatbuCkline, t'iuior :-!rr, : >--* thetheory of Koiter and the foralism of Budienckg ai:4 :!utch-L.'on- j 3t-buckling behavior is shown to te stable, and unf I-ec d by, Ly -.:onor inextensibility.

CD RI 147"

Security Classification _______ ______ ______

14 LINK A LINK 0 LINK CKYWRSROLE WT ROLE ITW ROLE WT

1. Buckling and Postbuckling.

2. Circular Arahes and Ringt.

3. Imperfection sensitivity.

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SecuritV Classification

F61052-69-C-0040

SR 5 AUGUST 1971

SCIENTIFIC I ,LPORr No. S.

ON THE BUCKLING AND POSTBUCKLING OF CIRCULAR ARCHES

AND RINGS.

by

CLIVE L. DYMDepartment of Aeronautical Engineering

Technion- Israel Institute of Technology,Kaifa, Israel.

T.A.E. REPORT No.133

DDC

IV6_ ..-- |

C

This document has been approved for public release and sale, its distributionis unlimited.

The researth reported in this document has been sponsored in part by the AirForce Office of Scientific Research, through the European office of AerospaceResearch, United States Air Force, under Contract F61052-69-C-0040.

* On leave from Carnegie-.1ellon University, Pittsburgh, ra., U.S.A.

_, a %'ir lidm ],

" " ""'" ':r ,, " >-,., - ,. - , -, -- " L.

TABLE OF CONTENTS

PAGE NO.

ABSTRACT I

NOMENCLATURE II - IiI

1. Introduction 1 - 2

2. General Equations 3 - 5

3. Linearized (Buckling)Solutions 6- 9

4. Postbuckling: The Budiansky-Hutchinson Approach 10 - 16

5. Postbuckling: Kolter's Approach - 23

6. ODiscssinTadcncuin 24 6

REFEPENCES 27 - 28

TABLE 1 28

FIGURE 1

I , -

ABSTRACT

This paper treats the buckling and postbuckling behavior of circular

arches and rings under constant directional pressure. New exact and

approximate solutions are given for the linearized eigenvalue problem.

IL is clearly demonstrated that the assumption of inextensibility is

quite reasonable for the asymmetric buckling of steeper arches and of

rings. Asymptotic aaalyses of early postbuckling behavior are given,

based on the theory of Koiter and the formalism of Budiansky and

Hutchinson. The postbuckling behavior is shown to be stable, and un-

affected by the assumption of :.nextensibility.

I'

- II -

I"

NOMENCLATURE

a, b, a', b' perturbation solution parameters defined by Eqs. (40) and

(37)

A, Ai displacement coefficientr, dimensionless

A displacement coefficient, dimensional*

A displacement coefficient (=

e, ai extensional strain

E Young's modulus

h arch thickness

H thickness parameter

moment of inertia

K, K curvature, dimensionlessI

L i differential operators

M, 14 imoment rosulta,.1, dimensionless

N, 1. stress resultant, dimensionless

F, P potential energy functionalsm

q load per unit length of arch

Q extensibility parameter defined by Eq. (16)

R arch radius

Si nonlinear extensional strain

v, V i tangential displacement, dimensionless

w, w tangential displacement, dice-nsionless

I'

•- -- .

half anigle of arch vertex

6 first order variationk

C porturbiitiOfl paramfeter

arch coordinate

arch l0ading,dimensionions

Xr Xc rotation of norm~al to arch centerlinles? dimensionless

-- 1

1. INTRODUCTION

This paper treats the buckling and the postbuckling behavior of

pinned arches, under constant directional ('d. ad") pressure, where the

arch radius of curvature is a constant. A linearized treatment of this

problem has been given by Chwalla and Kollbrunner [1]*, who assumed that

the arch was inextensible during buckling. Kaxmel [2] has shown that

extensibility did not have a great effect on the buckling loads, a result

confirmed by the present study. Also, Levy [3] and Carrier (4] have

given "closed form" solutions for the postbuckling of inextensional

circular rings in terms of (implicit) elliptical integrals. It is also

worth noting the recent contribution of Singer and BBc_ , w 0

discuss the relationship of semi-circular arches under dead pressure to

tbat of completn rings, pointing c'it the need to account for rigid body

displacements.

The present work treats the linearized buckling problem both exactly,

within the framework of ring theory, accounting for extensibility, and

anproxiataty: nxtndina Trv'l approximation for iings to arches of varios

vertex angles. A qualitative me.asure of the midplane extension is also

provided from the new exact solution.

Two asymptotic analyses of the early postbuckling behavior are presented,

both based on the theory advanced by Koiter (6]. The first makes use of the

formalism expounded by Budiansky and Hutchinson [7,8,9], and is carried

N I:umbers in brackets refer to entries in the list of References.

- . . --

I

22

out to include terms in the displacement expansion of order e , where c

is the small buckling deflection parameter. It is shown that the post-

buckling behavior of the arch following asymmetric buckling is stable,

and it is not appreciately effected, qualitatively or quantitatively, by

the inextensibility assumption. Further, it is shown that the inclusion

of the second order terms in the displacements has no appreciable effect

on the quantitative outcome.

Finally an anlysis based directly on Koiter's approach is presented,

2also accurate in the displacement expansion to order £ 2D. he- reits are

identical, for ihi pv o d using the Budiansky-

Hutchinson formalism, and serve to confirm the idea that (at least for

*simple" problems) accurate quantitative information can be obtained r-sing

only the buckling mode shapes in the postbuckling analysis. Koiter has

previously considered the elastica [6, and a two-bar frame [101 using only

the buckling mode, while Haftka and Nachbar [li] have conpared results of

a "one-term" solution with an exact solution, and .te comparison was quite

good.

is

-3-

2. GENERAL EQUATIONS

The equations used to describe the arch behavior are those of ring

theory, which are also a one-dimensional subset of the nonlinear shell

equations of Sander 1 " sonle3s form thEy aprear as follows.

e stress and moment resultants are

1 2

M=- f (lb)dO

where e Lnd X are the linear in-plane strain and the rotation,

respectively, which when expressed in terms of a tangential displacement

v and a radial displacement w take the form

dv dwe = - w, = + V (2)

The displacements and the circumferential ixch coordinate have been

nondimensionalized with respect to the arch radius R, while the stress

and moment resultants have been rendereI dimensionless by dividing by

(EI/R2) and (EI/R), respectively. This results in the thickness ratio

being introduced in equation (la), i.e.,

H = 1 (h)2 (3)12 R

Further, a dimensionless applied loading X wili. be defined as

3A = ,-4)El

where q is the load (uniform here) per unit length of arch. The geometry

-4- t

is picttired in Figure 1.

Now the (dimensionless) total potential e.aergy of the transversely

1o~ipa ewritten down as

P = 2R (U+V)El

(e2+2dO + f () do- 2A wdo (5)

-C H d -CL

In eiuation (5) the first integral represents the (linear and nonlinear)

utrain ener v - dle surface extension, the second integral is the

bending contribution to the energy, while the £inr l term is the potential

of the applied load.

From equation (5) one can deduce the potential energy corresponding

to the linearized buckling problem (See Eqns. (47)of Section 5), deroted

here as P X

X 1 a 2 a dx1 2 a 2P 2 =- e.d + - f -) d - X f Xd.6)

-i -CL -CL

The subscript one will, throughout this paper, denote a buckling dis-

placement or stress quantity. The Eular-Lagrange equations of the quadratic

functional (6) are straightforwardly obtained as

LIwI + L2v I - 0 (7a)

L3w1 + L 1v - 0 (7b)

where the Li are the differential operators

- U

--

d3 d

de2

L w ( + H) - + XH (8)2 d2

d4 d 23 4 a2

it is immediately of intereat to novs that equations (7) car be uncou;pled

to yield a single equation in v. , or the identical equation in w1 alone.

Fuithermore, that-single equation is identically that given by Chwalla and

Kollbrunnor[11, which they derived by assuming inextensibility in the buckling

state, i.e., eI - 0. That equation is, obtained here with no assumption

regarding the extensibility;

(LL - L 2 VI Vli + (2 + X)vI U + ' (9)

3 2-L 13 1 12 1+~ 1 -

The prime superscripts in equation (9) correspond to d6fferentiation with

respect to ti.e arch coord!iate. The question of inextensibility will be

raised again in the seuel.

For completeness the natural (in terms of stress resultants) and geo-

metric boundary conditions for the linearized prozlem are also noted

Either N1 - = 0 or vI is pre.-cribed

da1Either Ti- - kxl = 0 or wI is prescribed (IC)

~dw 1

Either M 0 or d- is prescribed

The boundary conditions are applied at t a, where 2a is the

arch vertex angle (Figure 1). Results for the complete ring are ol taxn-

ed by setting a = 15].2

f -6-

3. LIEARIZED (BUCKLlIG) SOLUTIOUS

A solution to t.c eigenvalue problem represented by the differential

equation (9) has been given by Chwalla and Kollbrunner (11 for a simply

supported arch, using the boundary conditions (reflecting the inextensibility

conditicn)

V= v1 = = 0 at = 11)

Some of the resulting eigenvalues, corresponding to a solution that is

asymmetric with respect to € = 0, are shown in Table I.

An approximate solutlon for a semi-circular arch (- v/2 < 7/2) has

been given by Levy [31, and recently it has been generalized by Babcock and

Singer [5] to properly account for the vanishing of the displacement v1 at

the arch ends. This approximation, which assvimes an incompressible arch, is

here generalized for arbitrary vertex angle 2%, to take the form:

OIAWI = A sin -, V - ( + cos-) (12)

It is easily verified that the qolution (12) satisfies the inextensible pin

conditions at i = ± a. Then the linearized poLe,.tial ancrgy (6? can be

easily converted into a Rayleigh Quotient to obtain (inextensible) approximate

critical loads. i.e.,

a dXJ P1- I ) dO

A = '13)

Substititn of the assumptions (12) into the quotient (13) yields

-7-

2? 2 2

2 2= c.14)

cr 22+ (--!) 2

2a

Some numerical values of A are given in Table 1. The agreement is

obvious'.y quite good. :.uso, it is noted bere that for the case ,A =1/2,

i.e., a complete ring, the mode shape represented by Eqs. (12) has been

shown to be essentially identical to that obtained by Ch,ialla and Kollbrunner

in their annlysis [5].

Finally, in an attempt to assess the effect of the inextensibility

assumption on the buckling load (aee the work of Kamel [2]) and to lay the

groundwork for the study of the effects of inextensibility on the post-

buckling behavior, an ecct solution to the differential equations (7) ana

the boundary conditions (10) is now given. The asymetric solution to

equations (7) is

Ew1 A1 s'n y r 2 sinO + Y coso

(15)

Av1 =o + (QA3 - A2)cosO +A sinO

where

1+ H{A -1) (16)S= - ( - 1)

and the Ai are constants, determined to within an arbitrary factor by

satisfaction of boundary coisditions.

• This solution is not valid for A = 1. However for 0 < 4 /2, thi is

of no concern here, as A > 3.

-8,,

Using the results (US), the exact in-plane strain and rotation formulae

(for buckling) can be written as:

e= (1- Q) A3 siniz (17a)

E 1f- AcoO s (1 + Q)A 3cos (17b)

The eigenvalue- (buckling loads) are obtained after satisfying the following

boundaryI conditions:

V 0, - 0 at 4. (18)

The resulting transcendental equation from which the buckling load is calculated

is found to be:

2(1 + Q) sin ct (19)( + 2Q - AQ)sinacosa + a(l - X)

For the sake of comparison, the eigenvalue equation obtained by Chwalla and

Kollbrunner is also given.

rX tan Nra = 2 sin2a (20)(3 - X)sinacosa + a(l - X)

It is seen that Ecs, '19) and (20) are identical when Q = 1, which corresponds

to inextensional deformation (Eq. (17a)). Numerical values for the buckling

pressures given by Eq. (19) are presented in Table 1, and again the agreement

is excellent.

It is worth noting at this juncture a few featurus of the extensional

strain eE resulting from the exact solution. First, considering the co-

efficient on the right hand side of Eq. (17a), and the definition of Q (Eq.

(1,5), it can be seen that

9-

2H(A - 1) H 1.1 - H( - 1)(21)

rropo'tional to the square of the thickness-to-radius of the arch, and

is thus quite small.

Secondly, it can be observed from Eq. (17a) that the extensional

strain is an odd function of the arch coordinate, i.e.,

e -€1 = -e () (22)

Also, it turns out that there exists another uncoupling of the exact

differential equations (7) that yields the result

2Ed e E+ e = (2:)

Thus, in addition to the odd flnction represented by the sinusoid al-

ready obtained, there is another sclution to the equations for -Which

W and eE(C) are even functions of € , tUe latter being a cosinusoid.1 1

This solution is not considered at all in the present work as attention is

restricted o rly to arches steep enough for asjnetric(biturcation) buckling

to take place.

...I m~ epl!Iwimlm =

-10-

4. POSTBUCKLING: TIE BUDIANSKY-HUTCUINSON APPROACH

In this sectir. the early postbuckling behavior of the ar,- will be

obtained, foliowing the asymptotic analysis outlined in the very lucid

exposition of B' Ciansky [9j. As with the buckling analysis, only asym-

metri- buckling of steeper arches is considered here. For very shallow

arches symmetric buckling can take place at lower pressures [133. how-

ever this snap type of buckling cannot be analysed with the Koiter theory.

The starting point is the virtual work (or variational) statement of the

full nonlinear problem:

f [; 6e + NX 6x+M 6K- Xw]dO = 0 (24)

In addition to the nomenclature defined in the previous section, the

curvature K has been introduced in Eq. (24). Then

H = K , - (.!5)dO

The variational equation (24) must hold fcr any kinezatically consistent

.. d-- .ncc... vr 4 AI-4 nn 5v and 6w. with corresponding strain

field variations le, 6X and 6K.

The following expansions are now introduced:

v = XV 0 cv 1 + c 2v2 +....

w= Xw 0+w 1 + C 2 w2 +

e = )e + Ce1 + C2 e2 +....

S= X° + £X1 + CX 2 +..K 1 2

31

N = 0-o + eN I + C2 N2 + ...

.M 1 + C2 (26)

Here c is a smn&1l parameter such that when C - 0, then X ) Xcr ,

the biturcation buckling pressure. The curvature and moment expansions

begin with terms of order £ to denote that the p.e-buckling state

(zeroth order terms in Eqs. (26)) Is a membrane stae. In fact, as it

is easily demonstrated that a nonlinear membrane state cannot exist for

this problem, the quartity x° shall be henceforth taken as zero, and

thus only a linear membrane pre-buckling state'is considered.

For the "unvariedr quantities in Eq. (24) the expansions (26) are

substituted. After rearrangement by increasing powers of e , the following

new variational equation results:

af [- 1Se - A6w ]d

a.cf [N16e + MI6K- X 1*X]dO

+ C f rN A. ' -(7-a . ..- 2 ... . .. ' I' 1 -^2P ,,,,O , (27)

a+ 3- [N 3

6 e + M36 K + (N1 X2 + N2X No 0X3 )6X]dO

+........= 0.

The pre-buckling state is defined by the vanishing of the zeroth

order variation. After a simple calculation that state is found to be

- 12 -

N A, e =-H, w = 11, v 0 0 (25)0 0 0 0

With the vanishing of the zeroth order term in Eq. (27), an c may

be divided through, and thus the buckling state itself is now found to

be, recalling that e - o implies A - A cr:

fI 511 6e + MS6K - 'crXl6X]dO = 0 (29)

The Euler-Lagrange equations of this variational equation are;

dN1 dM1

d- + A crX1 0 0 (30a)

!~ d g,dx1

Expressed in terms of displacements, Eqs. (30) are identical to the

linearized equations (7? given earlier.

It was pointed out earlier that the set of displacement variations

constantly alluded to need only be kinematically consistent, and since

the buckling displacements most surely are such a set, they can be

appropriately Iart-cd i-n Eq. !291 to yield the Rayleigh Quotient

(Budiansky's "energy equation") for the critical load,

f(Nl1e 1 + MI1K )dOcr 2(~e= - (31)

(I) 2dO

The same substitution in the more complete variational statement (27)

must be equally valid, and noting the quotient (31), it is found that

Ic -13-

(A -2

(X cr A) !~d

CL

2+ f (N2e1 + M 2K1 + N - XXX]dQ (32)-ai

2 O 2+ C f [rT3e1 + "3N ! + N +yIX2 + N2 X1 - k.lx 3 di 0-Oi

At this point the relationship between (A - A cr appears tci be taking

shape. However, it appears that equations of the third order will have to

be solved in order to obtain the roefficient of C2 above, In order to

avoid this difficulty, an orthogonality condition is introduced as follows

f (?I e + M d = 0; =-2,3,4 ... (33)th

It follows from Eas. (33) and (29), since the m order mode shapes must

be kinematically admissible, that

fX= ; 2,3,4 (34)-ai

in order to appiy the orthogonaiir.y condi-Lion i331 tuiid ,L6 ion ie

(34) successfully, the following consequenccs of constitutive linearity

and geometric nonlinearity must 3e noted:

* The precise meaning of this crthogonality condition has not beenestablishel ar is the subject of some discussion7 see Reissner[14].

:t - 14 -

2 21 2N2e = 2 1 1 N 2 = ll 2 +2 1 )

N3 e1 = N 3S I = N1 S3 = N1 (e 3 + X IX2 ); (35)

M 2K I M 1K2 ; MI3K1 = 1K3 ;

In Eas. (35), S. has been used t: denote the total nonlinear strain of

order c , which is made up in part of lower order terms due to the

geometric nonlinearity. Then notings Eqs. (35), (33) and (34), the

variational equation (32) can be put into the very simple form:

X = A ;- a' + b'e +.... (36)cr

where 2f N X1 d

at = 2 -a (37a)f X 1 dO5

OL 2]df [2NIXIx 2 + N2x1 ]dO

b' -a a 2 (37b)

f xi de

Since

a N -S -(e +-(8H -1 H1 2 H 2 =H 2 x 1)(8

the result (36) can be written in its final form:

A 2l + a + bc +.... (39)cr

where 2fe 1 da ' 3 -cca = j- =-(40a)Acr 2HXcr a 2

- 1

Q 2 1 4f[2ex\ 2 X+e -X xP:

b _b 1 Q _ -- 1 1_2 21 1(40b)

cr Hcr f~d

Recall that according to the %.Leory, .it a 3 0, then the stracture will

be in an unstable state in the cotbucki.ng range, and it will also be im-

perfection sensitive. If a = 0, '. , 0, then t-e poitbuckling stability

and the imperfection sensitivity va.y as the arithmctic !-gl of b.

In the present result it app.2:s at first glance that the existence of

a stable or unstable equilibrium statE in the pcstbuckling range depends on

whether or not inextensicility is assumed. For from equation (40a), ii

e 1() is non-zero, then the coefficient a might be. However, recalling

Eq. (221, it is clear th&t the coefficient in question will always vanLsh

for asymmetric buckling! Thus the exclusion (or inclusion) of the arch

extensibility can at best cnly influence the quantitat.ve nature of the post-

buckling equilibrium state, and not its qualitative nature.

It has already been shown that for asymmetric bucklrng the buckl tng

extensibility eI is quite small in magn:tude, with no appreciabis effect

on the buckling load. With this in mind, as well as the discussion of the

previous paragraph, it seems quite reasonable to stipuletc Lhat

e 0 and e 2 =0 (41)

In this instance the coefficient b reduces toct 4f xld;

b L (42)cr a 2

I ds

- 16 -

Using the excellent approximate solution (12), with its critical load

defined b Eq. (14), it is then a straightforward matter to calculate

the postbuckling relationship:

2 2_[1+3 (--l) + 2- -1) 21CL 2 a (12 a 2 .CA .2

- cr 2) 2 2 -- j (43)[2 +(2- - 1)

(In writing this result cognizance has been taken of the fact that the co-

efficient A in Eq. (12) was nondimensionalixed with respect to the radius*

R, i.e. A = , and of the definition (3)).

It is clear from Eq. (43' that X > Xct, and thus the postbucklingbehavior is always stable, and that the arch (or ring if a = 2) will not

be sensitive to imperfections :in its" behavior.

- 17 -

5. POSTBUCKLING: OITER'S APPROACH

In order to examine the stability of the bifurcation poin% and of

the postbuckling behavior using Koiter's approach directly, the total

potential energy is first divided into the prebuckling contributions,

and into the terms due to the buckling and prebuckling behavior. If

the prebuckling behavior is again linearized, the "expansions" (44) are

now introduced, i.e., let

e = Ae + e0

w = Aw + w (44)

0

X= X

Substituting these expansions into the energy functional (5) yields

2R (U + V) 2 XE =" f (Xe 2 - 2A2wo)do

-a.

+ (27,s a - 2.A--iwjd1'H 0 0

1 2- 2 2 14-aX

I- (e7 +Xe X 2+ eX2 + I 4]do

+ f (dX)2 do (45)(do

In view of the prebuckling results obtained earlier, the total potential

energy due to deformation beyond the prebuckling state can be written ao

- 18 -

p 2RA A A

- (U + V) +2aiHA 2 = P1 + P2 +P 3 + P4 (46)

where

P = - 2 f (e + w)do (47a)

X i a 2

P 3 f eX do (47c)-a

X 1 2L 41= 1 a 2

P4 iH xdo (47d)

aNow, by the definition of e(4¢) it is clear that

X dvA~ =- f (- w + O DE2(a) - v(-Q (48)

Hence, for a supported arch, P X 0. Thus the potential enezgy change1~

during and after buckling is simply

PX = PA + PA PA (49)2 3 4

where PN, PX and PA are given by equations (47b, c,d).

'2 3 4eutos

Recall that the behavior of interest takes place at loads very close to

A k A the cinod. Thus, to & - . -th.......

functionals (47) can be expanded in Taylor series about the point A = Acr

As P3 ' p4 do not (for this problem) explicitly depend on A, they appear

only as constants, while

A AA +l) dPA2

P1 +(- A)-d- .

2 r A=Acr cr cr

=P 2 + ( - Acr)P 2 +.. (50)

- 19 -

where1a a a

P 2 = e 2+ () 2 d o (51a)

2 acr-OL -aL

pc =- f X2d (51b)-a

Then the total energy change is

P A ;2 + (X - X ) p' + P + P (52)A1

where P3 = P3 and P4 = P4

Now the following displacement and strain expansions are introduced:

2W = Cw + C w2 +

V = CV I + C v 2 +(53)

2e = Ce + C e2 + ...

X = EX1 + £2X2 +

where £ is the same small parameter used previously. Introduction of the

4expansions (53) into the energy functionals yields, to order c

a 1 a, a dX1 5P C4 f ~~ed - X I X d + rc 12 - do2 _a cr-a -a

3 1 a a a dx dX23 2. - c u-+2c f e e2dO - do + do) (54a)

H1 41 Xc _, 2lX f+ 4 1 (e 2 + 2ele3)do - Xv f (X2 + 2XiX3)d

if-a 2 13-a 2 13

a dx 2 2 dX. dX3+ f(-) + 2 jo-- do

do

'1~ -20-

6- =2c f x - 2 (54b)P 2 2 I xld-- f[X2 + 2XIX 3]d(4

-CL t

ac -cc

3 1 2 4 12P 3 = C- H elXl10 + C if f [ex I1 + 2e xiX2])d (540)

P 4 = 4 C r4id

-CL

Then the same orthogonality conditions applied in the Budiansky-

Hutchinson formalism are applied again (written entirely in terms of

kinematic quantities) to tne functtonals (54). These functionals are then

greatly simplified and take the form:

2[1 a e2¢ 2 Ct X 1€+P2 = e ( _i X +r f W d

2 H 1 1 -CL d

(55a)

+4 I e 2d¢ f X2 dO + fL 2d 2H ea2 y cr _Ci 2 -ad

p=_ 2 X~d,- 4 x~cc (Sb-CL -CL

i C d- (55c)

L3 1 CL

U 4 1 1

+ C 4 1 [e2XI + 2elXiX2]do

S4 1_ X 4 do 05d)4 C T, d4l

Note that the term of order c 2 in P2 is simply the Rayleigh

quotient, and thus it vanishes identically. Further, the term of order

4.~£ in P2 will be multiplied by (A - Acr . according to equation (52).

- 21 -

S.nce it in expe:,ted that for this problem that (A - Xcr) will be of

order £2, the te= of order C4 in P2 may be deleted as being of

higher order. Thus the final form of the energy change is:

2 a 2 31a 2C .X( -A cr) f X do + 3 FI e1 2 de-G -a

14 a 2 1 4+ e f [e 2X1 + 2e 1 X1 X2 + -- X] do (56)

-ci

.2 2 a dX2+ -fed -cr I 4d + f (=-,do

H-_a -a -a d

The resemblance between the present approach and the Budiansky-Hutchin3on

approach is now more visible. However, it appears that there will be some

difference due to the term

1 a2 2 dX d 2-,2 [W2,V2 ] f 2do - Acr 2 + d

in the expression of order C4 in Eq. (56). Terms o this type do appear

consistently in Koiter's original analysis (see, for example, Section 3' of

Ref. (6].), although they do not appear in analyses using the Budiansky-

Hutchinson formalism (see, for example, Ref. [91.). For the present work,

under the well established assumption of inextensibility, the terms

P 2 [w2,v2] will be shown to yield no contribution to the result, and the

results will be identical to those already obtained.

The cecond order displacements, v2 and w2, are obtained as the

solution of the variational equation

~- 22 -

ae2 a ., a OX2d66 { f e~d- ac x.d, + f (-) dIt-CE cr za .W-_ - -a

a 2 1 4+ le X2 + 2,1Yl +-11de 0 (58)

at 2 1-'T2 4

In carryir.; out this variation all terms of first order are of course

considered as knwzn or given quantities. If inextensibility of the arch

is asoswed, e= 2= 0, then the vatiationa1 L.-ation is justI 2

2d1( (2x f X2d = 0 (59)6(I (v ¢:-a

Thus the variational equation leads to set of homogeneous cquati,as that

can have only a trivial solution . :.ence, for thc inextcnsible arch, it

*l is possible to take as solutions for the second order displacements the

trivial ones, i.e.,

w2 =v 2 =o (60)

Hence the energy functional reduces to (recalling eI = 0 again):

2 a 2 4 a- 2 (A - X v2 d ' 4 1. f d. (61)cr d 411-

Using the approximate solution (12), with A = A /R, it is a . ,ht-

forward natter tu cc;ipute the energy char.e 1, asX C 3 2 * 5 2 2

S -[ + '.-- 1)1(X - X) + ( ) -- - - )"- -)2 '2 cr R 4 2

(62)

* A solution linearly deprkndent or, th- buc:zlin s-.0ution i; tu}cl out by

the orthogonality condition.

4'

A ,Jl l .. ,.. .

23-

At tLe c.litical lcad, ' = > *-t is st .h- 1 . and th's iC-cr'

bifurcation (critical) state is iLself stable. The inltLal postbuckling

load-deflection relationship is determined by the condit4 n

dp "

'63= c *3)

d(cA )

from which it follows that

2I + 31- -1) 1) 21

1= r + (12) - (,2 (64)IT2T2 + (r - 1)

C1

This is exactly the result (43) obtained previously.

* This is an approximation to 6P = 0

- 24 -

6. DISCUSSION AND CONCLUSIONS

The present paper has considered the buckling and postbuckling

behavior of circular arches (and rings) under constant directional

loading. For arches steep enough so that asymmetric; bifurcation

buckling is critical, it has been demonstrated that the arch inextensi-

bility is a negligible quantity in magnitude, in its effect on the buckling

load, and in its effect on initial postbuckling behavior. In addition an

approximate solution for the linearized eigenvalue problem was presented

that duplicated almost precisely the buckling pressures obtained from the

exact solution.

The numerical results for the eigenvalue problem as solved by Chwalla

and Kollbrunner and as given by the present approximate and exact sclutions

have been given in Table 1. It is worth pointing out that the approximate

(inextensible? eigenvalues are always greater than or equal to Chwalla and

Kollbrunner's (inextensible) results, in accord with Rayleigh's Principle.

The results when the extensibility is accounted for are always less than the

inextensible results. This is in direct contrast to the analysis of Knmel

[2] which indicated that the buckling pressures would increase if extensi-

bility were accounted for As it has already been pointed out, the

* Singer and Babcock (5] have already pointed out that K nmel's result was

also erroneous in arriving at the coefficient X = 4 for constantor

j directional pressure, assuming inextensible deformations.

- 25 -

differential eauation solved by Chwalla and Kollbrunner, assuming in-

extensibility, is identical to that solved here, without that assumption.

The difference is in the boundary conditions, where the inextensibility

assumption provides less freedom (greater stiffness). Hence it is quite

proper that the inextensible eigenvalues be upper bounds to the (freer)

extensible eigenvalues.

It is also worth pointing out that the results given in Table 1 for

the Chwalla-Kollbrunner eigenvalues are not precisely those given in the

original paper (1]. Rather they were calculated (using a digital computer)

directly from the transcendental equation (20). There ari some dis-

crepancies, although they are very small. For example:

Chwalla and Kollbrunner [I] Chwalla and KollbrunnerE . (20) - Digital Computer

8.725 8.7273

a M 3.265 3.2712

It is easily verified that the original Chwalla and Kollbrunner results satisfy

Eq. (20) to within the precision of limited-accuracy trigonometric tables and

slide rule accuracy that must have governed calculations in the early 1930's.

The postbuckling behavior for the constant-directionally arches ard rings

has been shown to be quite stable, and thus imperfection sensitivity is not

an issue here. Also it has been shown, under the reasonable assumption of in-

extensible buckling and posthuckling deformation, that the second order dis-

placement solution does not change the results that would be obtained if

only first order (buckling) displacementswere used in the calculations.

- 26 -

Finally it has been showm that, ror the special circumstances of

this problem, the approach of Koiter as originally given and the variation

proposed by Budiansky and Hutchinson lead to identical results.

ACKNOWLEDGEMEUTS

The author is grateful for the partial financial support offered by

Carnegie-Mellon University that made his visit to the Technion possible.

It is also a great pleasure to thank Professor Josef Singer for

arranging the visit, and Professors Singer and Menachem Baruch for many

helpful and informative discussions and critiques.

V-27-

REFERENCES

1. Chwalla, E., and Kollbrunner, C.F., " Beitrage sun Knickproblen des

Bogantragers und des Raimen", Stahlbau, Vol. 11, No. 10, May 1938,

pp. 73-78.

2. K)aMel, G., "Der Einfluss der angsdehnung auf de elastische Stabilitat

geschlossener Kreisringe", Acta Maohanica, Vol. 4, 1967, pp.34-42.

3. Levy, M., "Memoire sur un nouvea cas integrable du problem* de l'elastque

et 1'une des ses applications", Journal de Mathematiques pures et

Appliquees, Liouville, Series 3, Vol. 10, 1884, pp.5-42.

4. Carrier, G.F., " On the Buckling of Elastic Rings', Journal of Mathematics

and Physics, Vol. 26, 1947, pp. 94-103.

5. Singer, J., and Babcock, C.D., " On the Buckling of Rings tnder Constant

Directional and Centrally Directed Pressiure", Journal of Applied Mechanics,

Vol. 37, No. 1, March 1970, pp. 215-218.

6. Koiter, W.T., " Over de stabiliteit van het .elastisch evernvicht" ("On

the Stability of Elastic Equilibrium". Thesis, Delft, N.J. Paris,

Amsterda!m; 1945. Enalish trns1ation issued a, ?$5A 70k F-10:833,1967.

7. Budiansky, B., and Hlutchinson, J.W., "Dynaic )auckling of Imperfection -

Sensitive Structures", Proceedings of the X! International Congress of

Aplied Mechanics, H. Gortler, Ed., Springer Verlag, New-York,1964,pp.636-651.

8. Budiansky, B., "Dynamic Buckling of Elastic Structures: Criteria and

Estimates," Proceedings, Internatioz" Confererce on Dynamic Stability o!

Structures, Pergamon, flew York, 1966, pp.83-06.

C

- 28 -

9. Budiansky, B., "Post-Buckling Behavior of Cylinders in Torsion", Theory

of Thin Shells, F.L. Niordson, Ed., Springer Verlag, Berlin, 1969.

10. Koiter, W.T., " Post-Buckling Analysis of a Simple Two-Bar Frame",

Recent Progress in Applied Mechanics, Ali~ruist and Wiksell, Stockholm,

1966, pp.337-354.

11. Haftka, R., and Nachbar, W., "Post-Buckling Analysxs of an Elastically-

Restrained Column," Intmenational Journal of Solids and Structures,

Vol. 6, No. 7, 1970, pp.1433-1449,

12. Sanders, J.L., "Non-Linear Theories for Thin Shells", Quarterly of

Applied Mathematics, Vol. 21, No. 1 , 1963, pp. 21-36.

13. Schreyer, H.L., and Masur, E.F., "Buckling of Shallow Arches", Journal

of the Engineering Mechanics Division, ASCE, Vol. 92, No. EM4, August

1966, pp.1-19.

14. Reissner, E., " A Note on Postbuckling Behavior of Pressurized Shallow

Spherical Shells", Journal of Applied Mechanics, Vol. 37, No. 2, June

1970, pp.533-534.

15. Roorda, J., "The Buckling Behavior of Imperfect Structural Systems",

Journal of Mechanics and Physics of solids, Voi. 13, 1965, pp.267-280.

-29-

TABLE 1: ASYM14ETRIC BUCKLING PRESSURES FOR CIRCULAR ARCHES; 1 = qR3 /EI.

Solution of Chwalla Present Present Exactand Follbrunner Approximation Result Eq. (19)Eq. (20) Eq. (14) (h/R = 1/10)

- 99.979598 99.979598 99.97954110

63.967766 63.967766 63.967676

35.941318 35.941320 35.9411516

15.859006 15.859031 15.858590

1 3.271245 3.272727 3.269233

k____

3t6

qR3-NIFORM

w

tI Eb3;UNIFORM12

EA= Ebb ; UNIFORM

FIGURE, 1 ARCH GEOMETRY AND LOADING