artigo powell

8/12/2019 Artigo Powell

1/9

2686 December, 1969Experiments, Bul/etin 96, Iowa Engineering Experiment Stati on, Feb rua ry 19, 1930.

10.Mctcalf, L., and Eddy , H. P ., American Sewerage Practice, McGraw Hil l Company, New York1916, pp.481-534. 11. Rowe, R. R., Co ncrete Pipe in Highway Construction, Technical Memorandum, AmericanConcrete Pipe Association, January I, 1962.12. Spangler, M. G., The Supporti ng S trength of Rigid Pipe Culverts, Bul/etin 112. lowa Engi.neering Experiment Station, February 8, 1933. ~13.Spangler, M. G., Analy si s o f Loads and Supporting Strengths , and Principies of Desi gn fo r ,~Highway Culverts, Proceedings, Highway Research Board, 1946, pp. 189-214.,:14. Timrners, J. H., Load Study of Flexible Pipes Unde r High F ill, Bul/etin 125. Highway

Research Board, 1956, pp. 1-11.

.,~

.~:

6943 December, 1969 ST12

Journal of theSTRUCTURAL DIVISION

Proceedings of the American Society of Civil Engineers

i : l:;~ 1 :

THEORYOF NONLINEARELASTICSTRUCTURESByGraham H. Powell,l M. ASCE

INTRODUCTION

i

General theoretical formulations and computational techniques for theanalysis of linear elastic structures have been very thoroughly studied inrecent years. The theory canbe elegantly and concisely expressed in standardmatrix form. Efficient solution procedures for both the force and displace-ment methods are well established. Nosuch standardzaton exists, however,for the analysis ofnonlnear elastic structures, as evidenced by recent papers(1,2,3,4,5,6).2 Indeed, it is still not clear what constitutes a consistent theo-retical formulation of the problem.In this paper, an attempt is made to separate, physically, the differenttypes of nonlinearity which are associated with large displacements of elasticstructures, to present a theoretical formulation in which these types ofnon-linearity can be easily identified, and to examine a number ofproblems as-sociated with solution techniques. Thedisplacement methodofanalysis appliedto discrete element systems is usedthroughout, andit s assumed that stransare small although displacements may be large. A member of a plane rigid

frame s selected to illustrate the theory. This member is selected becausethe results can be expressed compactly, andyetthe features of more complexstructural elements are present. In particular, both flexural effects and axialstran effects are present, and therefore, the principIes developed canbeextended to finite elements for plate bending, as well as for plane and three-dimensional stress.As the theory is developed, two alternative formulations are presented inNote.-Discusslon open until May 1, 1970. To extend the closing date one month, a

written request must be filed with the Executive Secretary, ASCE. This paper is partof the copyrighted Journal of the Structural Division, Proceedings of the AmericanSociety of Civil Engineers, Vol. 95, No. ST12, December, 1969. Manuscript was sub-rntted for review for possible publication on February 19, 1969.

1Assist. ProL Civ. Engrg. , Univ. of Calif., Berkeley, Calif.2 Numerais in parentheses refer to corresponding items in the Appendix.- References.

2687


2/9

December, 1969 ST 12 , Iow a Engineering Experiment Station, February 19, 1930.1. P., American Sewerage Practice, McGraw i Company, New Yor k,: Pipe in Highway Construction, Technical Memorandum , Americanm, January I, 1962.Supporting Strength of Rigid Pipe Culverts , Bulletin 112, lowa Engi-on, February 8, 1933,ysis of Loads and Supporting Strengths, and Principies of Design foreedings, Highway Research Board, 1946, pp. 189-214.Study of Flexible Pipes Under High Fill Bulletin 125 H h J,I-II. Ig way

6943 December, 1969 ST 12

Journal of theSTRUCTURAL DIVISIO

Proceedings of the American Society of Civil Engineers

THEORY OF NONLINEAR ELASTIC STRUCTURESBy Graham H. Powe ll ; ' M. ASCE

INTRODUCTIONGeneral theoretical formulations and computational techniques for the

analysis of linear elastic structures have been very thoroughly studied nrecent years. The theory can be elegantly and concisely expressed in standardmatrix formo Efficient solution procedures for both the force and displace-ment methods are well established. No such standardization exists, however,for the analyss of nonlinear elastic structures, as evidenced by recent papers(1,2,3,4,5,6).2 Indeed, t is still not clear what constitutes a consistent theo-


3/9

r2688 December, 1969parallel, one of a traditional engineering type and one in general matrixform. The development proceeds as follows:

1. Basic definitions and equations are presented.2. Geometric compatibili ty relationships between member deformationsand jont displacements are examined.3. Equilibrium equations for members and joints are developed.4. The essential equations of the problem are identif ied.5. The role of the tangent stiffness in analysis is described, and expres ,

sons for tangent stiffness are derived.In this development, member deformations and joint displacements are

separated, in order to present a theory which is applicable when rigid bodyrotatons of the members are large. A more direct formulation, in whichmember deformations are not expl'cttly examined, is considered at the endof the paper. This type of formulation has commonly been used, but it sargued herein that t is not as satisfactory as the less direct formulation.

DEFINITIONS ANDBASIC RELATIONSHIPSStructure. - The structure consis ts of a number of rigid body [onts linked

together by deformable structural members. In ths study the members are

S IFIG. 1.-;\1EMBER ACTIONS AND DEFORMATIONS

initially straight, and their elast c properties are assumed to be concentratedalong ther axes. External loads are applied only at the joints.Differential Element.-Each member is made up of an infinite number of

differential elements. The deformations of all differential elements are as-sumed to be small, and the relationships between actions and deformationsare aSSumed to be linear. In general, differential elements will be elementsof material wth stress and stran related by Hooke's law. However, an ele-ment of a plane frame member is more simply taken as an elemental lengthof beam, with stress and strain resultants related by

CJ1=EIE1 and CJ2 = E2 E2 ............ ............. . . . . . . . . . . (1)(2)

ST 12 NONLINEAR ELASTIC STRUCTURES 2689in which CJ1= bending moment; CJ2 = axial force; E1 = EI; E = Young's modu-lus; I = second moment of area of cross section; E2 = EA; A = area of crosssection; E 1 = curvature of member axis; and E 2 = axial strain.

Member Actions and Deformations .-ln order to develop a theory which isapplicable when member rigid body rotations are large, the deformations andrigid body displacements of the members are separated. Fig. 1 shows a framemember supported so that its chord undergoes no rigid body displacement.The member generalized deformations, v1, v2 and v3, as shown, and the corre-sponding actions, 51, S2 and S3 are selected for the derivation of the theory.

The member deformations are assumed to be small. That is, v1 and v2 aresmalI n compartson with unty , and v i, v~, and vs/ L are negligible in com-parison with unity. Members which may undergo large flexural deformationsmust be subdivided into shorter members. Large axial deformations are notpermitted.

In order for the behavior of the complete structure to be defined uniquelyn terms of its joint displacements, the deformations of alI differential ele-

r :' : :~ ; ;

R rlY L xFIG. 2.-DISPLACEMENTS OF DIFFERENTIALELEMENT

FIG. 3.-JOINT LOADSAND DISPLACEMENTS

ments in a member must be defined uniquely in terms of the member gener-alized deformations. Transverse displacements, w, are therefore constrainedto vary cubically with the distance along the member chord, and longitudinaldisplacements, u, to vary linearly (Fig. 2). The following linear transformationis then consistent wth the small deformations assumption:tl = [ L U ? - : P .p) : J l : : \ ........... (3)L P 3 _ p2 )oin which the deformed length, L is not significantly different from the originallength, Lo.

The differential element deformations are given byd2w 1 1E 1 = d x2 = L 6p - 4)v1 + L 6p - 2)v2 .. ...... (4)

~t


4/9

2690 December, 1969which is a linear function ofthe displacements, and

du 1 (dW)2E2 dx + 2 dx4p + l)v, + (3p' - 2p)v,) (

........... (5)E 3 + . {(3p2 _L 2in which the quadratic term introduces the effects oflarge displacements.More generally, if {v} = T, then

Ei = (6e)Joint Loads and Displacements .-The displacements of any joint consist oftranslations, r1 and r2 along the fixed coordinate directions, X and Y, and arotation rs, as in Fig. 3. The joint loads are the corresponding forces Rl 1 R2and R3.Geometric Compatibility.-For a vanishingly small change of geometry ofthe structure about any deformed or undeformed configuration, a linear ds-placement transformation can bewritten as

{dv} = [c] {dr} (7)in which {dv} matrix ofdeformation increments for a single member or anygroup of members; {dr} matrix of displacement increments at one or morejoints; and [c ] = displacement transformation matrix which is instantaneouslycorrect inthe particular configuration of the structure. For the single memberin Fig. 4, the transformation is well known, as follows:

r;d v [ _ l c 1 s -c ; jd v l - i L L L r;c 5 -c drsL O L T dr ......... (8)4dvs -c -5 O C 5 drsr,

in which 5 = sn ; and c = cos.For large displacements, the transformation is not, of course, linear. Forpurposes of computation, the nonlinear transformation for a frame member

ST 2

(6b)(6c)

(6d)

ST 12 NONLINEAR ELASTIC STRUCTURES 2691displacing from an initial state, O,to a new state, A, can be formulated asfollows (Fig. 5):

= Xo + r4 - r1 9a) 9b ) 9c )

YA = Yo + rs - r2 v3 = (x3 t + y 3 t 1 / 2 - Lo .

e o = tan-1 ~9d)

O tan-1 (~) ge)................

More

v1 = 00 + r3 - e A ... .. ... .. . . . . . ... ... . 9f)v2 e o + r6 - e A ................. 9g)generally, as a structure deforms from state Oto state A, it moves

s t '{6

~n ~r4

r5

~/-y \.: ~rlC x

-r

Yo

r l l XA _ FIG. 5.-INITIAL AND FINALSTATES OF MEMBERFIG. 4.-MEMBER ARBITRARIL YORIENTED IN SPACE

through an infinite number of intermediate states, and the transformation canbewritten as

{rA}{VA} J [cr] {dr} ........................... (10)oin which {VA} = matrx of member deformations in state A; {rA} matrix ofjoint displacements in state A; and [c r] instantaneously correct displace-ment transformation matrix at any intermediate state.Eqs. 9a-9g are smple to use in computation for a frame member, and itshouldgenerally be possible to establish similar direct relationships for otherstructural elements. However, it may be convenient to evaluate member de-ormations bynumerical integration of Eq. 10. For example, the form

N{VA} = ~ ~ [cn] {rA} ........................ (11)

n=lcan be used, in which {r A} is divided into N equal parts and [ cn ] is the instan-

~. , . ; . . . .. , . . :, . . .

. . . . . . . . .. . . t t


5/9

~2692 December, 1969 ST 12taneously correct displacement transformation at the beginning or middle ofeach parto

EQUILIBRIUMJoint Equilibrium.-All jonts and members must be in equilibrium in the

deformed configuration of the structure. The equilibrium equations are con-veniently established by application of the Virtual Displacements PrincipIe.

For an infinitesimal imaginary displacement, {dr}, about the deformedstate, A, the member deformations, {dV}, are given by{dv} [CA] {dr} (12)

in which [c ] is the instantaneously correct displacement transformationmatrix in state A. The virtual work equation is then

{dj:} T { RA } = {dv} T { SA } ... .. .. .. .. .. ... .. ..... (13)in which {RA} isthe matrix of [ont loads in state A; and {SA} isthe matrix ofmember actions in state A. Therefore, Irom Eq. 12, and because {dr} isarbitrary

{RA} = [cAl T {SA} (14)The force transformation matrtx, [c A ] T , is dependent on the displacementsof the structure.

Member Equilibrium.-The chain of equations linking joint displacements{rA} and loads { R A } in state A is completed byestablishing a re lationship be-tween member actions {SA} and deformations {VA}

For a single frame member, an infinitesimal imaginary displacement {dv}about its deformed posi tion leads, by differentiation of Eqs. 4 and 5, to imag-inary differential element deformations dEand dEz given by

1 1dE1 L 6p - 4)dv1 + L 6p - 2)dvz . . . . . . . . . . . . . . . . . (15)- dv { }and dE z T + 3p2 - 4p + l)v1 + 3pZ - 2p)vzx { 3pZ - 4p + 1)dv1 + 3pZ - 2P)dv2} (16)

More generally, the following equation is obtained by differentiation of Eq. 6:dEi {v} + {v}T


6/9

2694 December, 1969fication, possibly wth no great loss in accuracy, can be achieved byassigningconstant values to Yi

ESSENTIALEQUATIONSThe essential equations which must be satisfied can be summarized asfollows: {rA}

{V A} = J [an ]{dr} .o{SA l = [k s] {V A }{R A } = [ a A F {S A l .

Nonlinearity is present because the displacement transformation is nonlinear(Eq. 10), the members behave nonlinearly (Eq. 21), and the equilibrium equa-tons must be formulated in the deformed configuration (Eq. 14).

In computational algorithms, one or more of these nonlinearites may beignored, and approxrnate results obtaned. For example, for finite elementanalyses in which the elements are smal l, a commonassumption might bethatthe member nonlinearity, represented by Eq. 21, canbe ignored. The influenceof this particular assumption onthe solution technique is noted subsequently.TANGENTSTIFFNESS

General.-In exact computational algorithms for large nonlinear struc-tures, the non-linear problem is most commonly solved bya Newton type of{R } F IR ST D IS PL AC EM EN T E ST IM AT E

S EC ON D D IS PL AC EM EN T E ST IM AT EE X A CT S O L UT IO N

TANGENTAT O

BS E C ON D U N B AL A NC E DFORCE

F IR S T U N BA L AN C EDFORCE{ r}

O

FIG. 6.-DIAGRAMMATIC ILLUSTRATION OF NEWTON PROCEDUREmethod in a series of linear steps. The procedure is well known andphysicallyreasonable, being one in which a soluton which satisfies compatibility is sue-cessively corrected until it also satisfies equilibrium. The basie method isillustrated diagramatically in Fig. 6.

For this method to be applied, tangent stiffnesses for the structure arerequired. In this secton , expressions for the tangent stiffness matrix of asingle member are developed. The tangent stiffness of the complete structureis then considered in the following section.

It should be emphasized that, except for stability investigations, the tangent

ST 12

(10)(21)(14)

NONLINEARELA$TIC STRUCTURES 2695ST 125tiffness s largely just a tool which is a part ofsome soluton techniques. Thefundamental equations are Eqs. 10, 21 and 14, and if a solution can be Ioundwhich satisfies these equations, by any technique whatsoever, this is a validsolution.Member Tangent Stiffness in Member Coordinates.-In any state , A, anequation of the orrn

{ d S } [ tA ] {d v}s sought, in whieh [ IA] is the member tangent stiffness in state A. This ma-trix is obtained most directly for a frame member by differentiating Eq. 20with respect to {v} . The result is

........................... ..... (24)

as, \ 21 O j r lS2 ) 2 2 1 2 o dV 2

dS s o ~ dv s[( L(12,: - 3v,u + vj) ~) ( -L(3: - , + 3,:) _ - )140 + 15 280 30

+ EA .t . ( L v~ - 3 v) v2 + 12 vV 5)symme rrc 140 + 15 4v) 3~ v2 )

4 V 2 3 ~ V i )o

l d V i . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)dV 2 d vswhieh ean bewritten as

{zs} [[ti] + [ 1 2 V A ) ] ] {d v} [ tA ] {d v} (26)More generally, the tangent stiffness is obtained by differentiating Eq. 19.This differentiation is conveniently, and instruetively, carried out as follows.Two states are eonsidered, namely {V A }, {S A} and {V A + to v} , {S A + toS} . lneachof these states a virtual displacement is imposed andan equilibrium equa-tion in the form of Eq. 19 is developed. The first equation is then subtractedfrom the seeond togive anequation for { toS}. Both { toS} and { to v} are then madeto tend to zero, 50that high order terms in {to v } can be ignored. The resultean then be arranged as

{dS} = Ir T e, Ldp . {dv}o+ If T e, Yi Ldp . {dv}o

. .i .. ;1~ 4 .. . .. . . ,. , . ; ; : ,',


7/9

fr;;' ~I:: i ' 2696 December, 1969

+ L t< b i> T E iY i< a i> Ld p {d v }o1+ L f < b i > T r, z, r, < b i> L d p . {d v }

+ L t< bi> T ai L d p . {d v }oThe first term on the rght-hand side of this equation is the ordinary linearstiffness of the member. The remaining terms depend on the deformationsor the member, and may be collectively termed the intal stran= stiffnessofthe member. The tangent stiffness matrix will always be symmetrical, andnumerical evaluation of the integraIs is possible.

Initial St re ss S tiffne ss .-If , in Eq. 25, the flexural deformations, vl and v 2,are made equal to zero, the result is

[t 2 v A )~ -~

: 115 30

EA I _ a. ~ ..................... (28)30 15O O

2 -1 o5 30 : J1 2 . . . . . . . . . . . . . . . . . . . . . . . . . (29)sL I 30 TIO Oor [ 1 2 V A )This matrix nowcorresponds tothe last term onthe right-hand side of Eq. 27.It is appropriately termed the initial stress stiffness matrix of themember,following the terminology used by Turner, Martin and others (6,5). Afurthercontribution tothe structure tangent stiffness, more appropriately identified asa geometric stiffness, followingtheterminology of Argyris (1), is considcredin the next secton,

It can be seen that the true member tangent stiffness is not n general ob-tained byadding the ordinary linear stiffness to the ntal stress= stiffness.Nevertheless, because convergence can be obtaned by a Newtontype of methodwithout the use of an exact tangent, itmay be satisfactory n practice to useths approximation to the true tangent stiffness. For stability investigationsof such structures as axially loaded, geometrically perfect columns, the as-sumption that the flexural deformations are initially zero is, however, correct.

It may also be noted that the derivations of Eqs. 23 and 27 are identical inprincipIe to the use of the strain energy expressionsSi = .E . -

Vi ......................... . . . . . . . . . . (30)

ST 12 NONLINEARELASTICSTRUCTURES 26971l 2Uand tij = - ,-,- , ,. . . . . . . . . . (31)uVi uVj

nwhich U = strain energy; and tij = term of tangent stiffness matrix.STRUCTURETANGENTSTIFFNESS

M em be r Ta nge nt Stiffness in St r u c t u re C o o r d i n a te s .-An equation of theform

{dR} = [ T A ) {d r } (32)s sought, in which [T A 1 is the tangent stiffness ofa singIe member or a groupofmembers in the fixed coordinates, X, Y. The following equations can bewritten:

{ d S } = [t A 1 {d v } (26){d v } = [ cA l {d r } (33)

and {dR} = [ C A ] T { d S } + [ d c F{ S A } (34)Eq. 34 is obtained by differentiation of Eq. 14, in which the force transforma-tion matrix, [ C A ] T , is a funct on of the joint dispIacements, {r } . Eqs. 26 and33 can be substituted into Eq. 34 to give

{dR} = [ c A F [ tA ) [ c A l {d r } + [ d c lT {SA I t (35) [ T l) {d r } + [d c F { S A } Matrix [ T l ] is the member tangent stiffness transformed into structure co-ordinates, and accounts for the changes in magnitude of the member end forcesas the structure deforms. Term [d c F{S A } accounts, essentially, for thechanges indirection of the member end forces as the structure deforms. Thissecond term must be rearranged as

[d c ]T {SA I = [T 2) { d r } (36)inwhich matrix [T 2) can be termed the geometric stiffness.Two methods of making this rearrangement are considered herein. Thefirst method, which is essentially that used byArgyris (1), involves differen-tation of the force transformation matrix, [cA) T, and s exact but does notappear to guarantee that a symmetric stiffness matrix will resulto The secondmethod follows more cIosely the derivations ofthe prevous sectons, but re-quires that some terms be ignored a prior i.

Method 1 D iffe re nti al io n o f F or ce T ra ns fo rm ati on .- Th e matrtx, c A F ,for a frame member s a function of the intial geometry of the structure andthe joint displacements, rl through re (Fig. 4). Therefore

T _ \ [ S C J Td c ) - / . J Ilr i d r i , (37)t=1

and [ dc JT { SA } [ S C J T J Ilr i lS A } d r jz = 1

. . . . . . . . . . . . . . . . . . .. (38)

.., : : v- ... 1- /.:


8/9

2698 December, 1969 ST 1.Eq. 38 can now be arranged in the form of Eq. 36, in which column iof [T2l i[8 e/ er il T {SA} If the multiplication sequence is reordered, matrix [T2l Cbe expressed in the more convenient form:

[T2l = SI [T2] + S2[T22] + S3[T23lin which column iof [T2j 1 is column j of [ e/ r i J T.For a planeframe member, matrices [le/8ri]T can be evaluated from Eq.8, using .

[~JT 8riThe result is

[8eJT ~ + [~JT u:88 8ri si. 8ri-2se (e2 _ S2) O 2se (S2 _ e2) O

2se O (S2 - c2) -2se O[T21l [T22] i 2 I O O O O

symmetric -2se (e2 _ S2) O2se O

O

S2 -se O _S2 se Oe2 O se _e2 O

O O O Oand [T23] = b I O I .............ymmetric S2 -sec2 O

Oin which s = sns, and e = cose.

For more complex structural elements, it should generally be possible tocarry out the corresponding manipulations. The member tangent sttifness instructure coordinates is then given by

[TA) = [TI l + [T 2l (43)However, by this method there appears to be no guarantee that [ T

21 will al-ways be symmetric.

If the manipulations required to develop Eqs. 41 and 42 are examined, it isfound that for [T 21] and T22] there are contributions of similar orders ofmagnitude from the differentiations with respect to both e and L. That is, theinfluence of the change in mangitude of the end shears, SI + S2)/ L, as themember changes length is of the same order of magnitude as the influence ofthe changes in direction of those shears as the member rotates. However, ithas already been assumed that change of member length is negligible, and,therefore, for consistency [ T21) and [ T221should both be ignored. Onthe otherhand, the terms in [T23] result solely from differentiation with respect to e,and are not negligible.

Method 2, Use of Nonl inear Deformation-DisPlaeement Relat ionship.-As-

(41)

(42)

ST 12 NONLINEAR ELASTIC STRUCTURES 2699surne that for moderately small changes of [oint displacernent { ~r } about somedeforrned postton of the structure , nonlinear relationships between changeof rnember deforrnation and change of joint displacement can be written, andthat these relationships can be approximated bythe linear and quadratic termsonlyas

t.Vi = {t.r} + ~ {~rF [eil {t.r} (44)For an infinitesimal increase, {d~r}, in {~r}, such as will be mposed in

applying the virtual displacements principIe, the following equation is obtainedby differentiating Eq. 44:d~Vi = {d~r} + {~rF [eil {dt.r} (45)

Consider, now, the equilibrium between the [oint loads and the member actionsin two adjacent states {r}, {R}, {S} and {r + t.r}, {R + t.R}, {S}, in which{S}remains constant. In the first state, the virtual work equation is

{d~ tF {R} ~ dt.Vi Sil,6 r=o {dt. t}T L : Si T (46)In the second state, the virtual work equation is

{dt.rF {R + t.R} = ~ dt.Vi Si,6r=M ~

= {dt.rF L : Si [T + [eiV {~r}l)Therefore, if Eq. 46 is subtracted from Eq. 47 the result s

{t.R} ~ Si [ ei1T] {~ r } , (48)For a frarne mem ber, it is found that there is no simple and direct way todetermine [e il for the flexural effects, i= 1 and i= 2. However, if tts as-sumed, a priori, that the geometric stiffness effects associated with theflexural actions SI and S2 are zero, then this problem can be ignored. Thequadratic term inthe equation for ~ v3 s then L ~e)2, and therefore matrix [e31is readi1y written as

[e31 T ...............................in which L1/2 -coseTherefore, matrix (T21 becomes S3[ e3], in which [e31 is found to be identicalto matrix [T231 in Eq. 42.Correct Tangent Sli ffness for Use in Large tnepiacements Analysis.-For any rnathernatically idealized structure, the most reliable and rapid con-vergence can be expected by a Newton type of method if the tangent stiffnesswhich is used is exact for the idealized structure. As noted previously, acommon assumpton in the analyss of finite element systems is that the mem-ber acton deformation relationship of Eqs. 21 and 23 can be linearized as

(47)

. . . . . . . . . . . . 1 1

,'\. \~:.:

(49)50 )


9/9

2700 December, 1969 ST {S A } \ J 1[k1l {VA} i J T s, JJlp . {VA}o . (51)

If this assumption is made, it becomes an integral part of the mathematical .idealization. It follows immediately that the member tangent stiffness, from f ,Eqs. 26 and 27 is ~1 . {.{dS} = [t 1 {di} ~T tt, Ldp . {dd ......... (52)That is, the initial strain stiffness is implicitly ignored, and the only add-tional contribution to the structure tangent stiffness must be the geometricstiffness. If a more elaborate form of the tangent stiffness is used, it wiUactually be less exact, and may be expected to lead to less rapid convergence.

DIRECT FORMULATIONIn the preceding the ory , mem ber deformations we re de liberately separated

from member rigid body displacements, in order to develop a general theorypermitting large rigid body rotations. If the [oint displacements are such thatthe member rigid body rotatons are known not to be large, then direct non-linear relationships between the differential element deformations and jointdisplacements can be written as

Ei = {r } + ~ {r } T T lt ; {r } (53)for which appropriate matrices < f[ i? and h ; can be determined. Expressionsfor the secant and tangent stiffnesses direct ly in structure coordinates can nowbe derived, by the same methods and with similar form as Eqs. 23 and 27.

If the joint displacements are not moderately small, however, the re lation-ship represented by Eq. 53 may not be sufficiently accurate. In th s case,either higher order terms must be included in Eq. 53 or the theory of theprevious sections must be applied. Even if Eq. 53 is sufficiently accurate, thedirect formulation does not require substantially less computational or com-puter coding effort toapply than the indirect formulation. Fu rthe r , the indirectformulat ion has the advantage that the initial strain and the geometricnonlinear effects are distinctly separated, so that the initial strain effectcan be ignored if the initialassumptions require it. It appears, therefore, thatthe indirect formulation should be preferable in all cases.

CONCLUS10NSIn this paper the three major sources of nonlinearity in elastic structures

undergoing large displacements have been distinguished, and the essentialequations of the problem have been stated. It has been shown that the non-lineari ty of the structure results from the nonlinearity of the transformationsrelating member deformations to joint displacements, from the necessity offormulating the joint equilibrium equations in the displaced configuration ofthe structure, and from the nonlinear act ion-deformation relationships of thestructural members. The nonlinearities orginating within the structural mern-

NONLINEAR ELAST'IC STRUCTURES 2701ST 12bers result in turn from the nonlinear transformations relating differentialelement deformations to member deformations, and from the necessity offormulating the member equilibrium equations in the deformed configurationsof the members. The nonlinearity which originates within the members hasbeen termed the initial strain effect herein, and that which originates outsidethe members has been termed the geometric effect.Expressions for the secant and tangent stiffne sses of members and struc-tnr es have been presented in a cornpact Ior m, which can be extended to f initeelement systems of a variely of types. The importance o selecting consistenttangent stiffnesses for solutions by methods of Newton type has been empha-sized, and it has been shown that the initial strain erfcct may have to beignored.Of course, a statement o theory does not constitute a solution. Consider-able work remains to be carried out to select efficient computational algorithmsand to determine the relative importance of the different types of nonlinearity.

APPENDN.-REFERENCES

I .A rgyris, J. H., Kelsey, S., and Kamel, H .. Matrix Methods in Structural Analysis. Agardo-graph 72, Pergamon Press, New York, 1964 ,pp. 1-164.2.Connor, J. J., Logcher. R. D., and Chan. S. c., N onlinear Analysis of Elas ti c F ramed Struc-urres. .Iournal o/ lhe Structural ivision, ASCE. Vol. 94, No. ST6, Proc. Paper 6011, June 1968,pp.1525-1548.3. Felippa, C. A., R efined Finite Elemenl Analysis of Linear and Nonlinear Two-DimensionalStrur.tures, SES M Repor/66-22. University of California, Berkeley, 1966.4_Malleu, R. H., and Marcal, P. V., Finite Element Analysis of Nonlinear Structur es. .Iournal 0 1ihe Structura Division, ASCE, Vol. 94, No. ST9, Proc. Paper 6115, SepL. 1968, pp. 2081--2105.

5. Martin, H_ c., O n lhe De riva tion of Stiffness Matrices for the Analysis of Large Deflection andStability Problerns , Pruceedings. Conference on Matrix Methods in Structur al Mechanics,Wright Patterson Air Force Bas e, Ohio, October, 1965, pp. 697- 716.6. Turner, M. J., Martin, 1-1.c., and Weikel, R. C .. Further Development and Application of theStiffness Method. Agardograph 72. Pcrgamon Pr ess. New Yor k , 1964, pp. 203- 266.

~~.

? r: \

~ .~~

artigo powell

Documents