Bathe (1975) - FE Formulations for Large Deformation Dynamic Analysis

8/2/2019 Bathe (1975) - FE Formulations for Large Deformation Dynamic Analysis

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INT E RNATIONAL JOURNAL F OR NUM E RICA L M E T HODS IN E NG INE E RING , VOL . 9, 353-386 (1975)

FINITE ELEMENT FORMULATIONS FORL A R G E D E F O R M A T I O N D Y N A M I C A N A L Y S I SKLAUS-JURGEN BATHE*

Civil Engineering Departm ent, university of California, Berkeley, California, U . S . A .

EKKEHARD R A M M ~Universify of Sturtgarr, West Germany

EDWARD L . W I L S O N $Civil Engineering D epartme nt, Unive rsity of California, Berkeley, Ca lifornia, U.S.A

SUMMARYStarting from continuum mechanics principles, finite element incremental formulations for non-linear staticand dy namic analysis are reviewed and derived. The aim in this paper is a consistent s um mar y, com pariso n,and ev aluatio n of the form ulatio ns which have been implemented in the search for the most effective procedure.The general formulations include large displacements, large strains and material non-linearities. For specificstatic and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plasticmaterials are considered. The numerical solution of the continuum mechanics equations is achieved usingisop aram etric finite element discretization. Th e specific matrices which need be calculated in the form ulatio nsare presented an d discussed. To dem onstr ate the applicability a nd the im portan t differences in the form ulations,the solution of static and dynam ic problems involving large displacements an d large strains ar e presented.

INTRODUCTIONIn non-linear dynamic finite element analysis involving large displacements, large strains andmaterial non-linearities, it is necessary to resort to an incremental formulation of the equationsof motion. Various formulations are used in practice (see References). Some procedures aregeneral and others are restricted to account for material non-linearities only, or for large displace-ments but not for large strains, or the formulation may only be applicable to certain types ofelements. Limited results have been obtained in dynamic non-linear analysis involving largedisplacements and large strains.Currently, the general purpose non-linear finite element analysis program NONSAP is beingdeveloped at the University of California, Berkeley.z An important aspect in the developmentof the program is to assess which general finite element formulation should be implemented.In dynamic analysis numerical time integration of the finite element equations of motion isrequired. Extensive research is currently being devoted towards the development of stable andaccurate integration schemes.1.5,20.31 However, it need be realized that a proper evaluation anduse of an integration method is only possible if a consistent non-linear finite element formulationis used.

* Assistant Research Engineer.t Research Fellow of Deutsche Forschungsgemeinschaft . Form erly Visiting Scholar, University of California1Professor. Receired I8 Fehruary 19740 1975 by John Wiley & Sons, Ltd

3 5 3


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354 K.-J. BATHE, E. RAMM AND E. L . WILSONThe earliest non-linear finite element analyses were essentially based on extensions of linearanalyses and have been developed for specific applications (for a comprehensive list of References,see the books by Oden33 and Z i e n k i e ~ i c z ~ ~ ) .he procedures were primarily developed on anintuitive basis in order to obtain solutions to the specific problems considered. However, to

provide general analysis capabilities using isoparametric (and related) elements a general formula-tion need be used. The isoparametric finite element discretization procedure has proved to bevery effective in many applications, and lately it has been shown that general non-linear formula-tions based on principles of continuum mechanics can be efficiently implemented.Basically, two different approaches have been pursued in incremental non-linear finite elementanalysis. In the first, static and kinematic variables are referred to an updated configuration ineach load step. This procedure is generally called Eulerian, moving co-ordinate or updatedformulation. Murray and Wilson,28Felippa,' Yaghrnai3' Yaghmai and P O P O V , ~ ~arhoomand,'Sharifi and POPOV,~~amada,41 Stricklin and many others,38 Heifitz and Co~tantino,'~Belytschko and Hsieh6 have presented some form of this formulation.In the second approach, which is generally called Lagrangian formulation, all static andkinematic variables are referred to the initial configuration. This procedure is used by Oden,32 .33Marca1,26 Hibbitt et a1,16 Larsen," M~Namara,~'harifi and Yate~,~'tricklin and manyother^,^^.^' Haug and P0wel1.l~ survey paper of the Lagrangian formulation in static analysiswas presented by Hibbitt et al," where it is stated that additional research is required for use ofan equivalently consistent updated formulation.I t is apparent that with the different formulations available, in the development of a generalpurpose non-linear dynamic analysis program a decision need be made on the procedure to beused. An important consideration is that using any formulation based on continuum mechanicsprinciples, in which all non-linear effects are included, the same results should be obtained in theanalyses. Stricklin and many others, discussed a moving co-ordinate formulation and a Lagrang-ian formulation and pointed out that the latter is more general and computationally moreeffi~ien t.~' amada compared an Eulerian and Lagrangian formulation and predicted for a

simple truss structure a maximum difference of about 25 per cent in the displacement^.^' Dupuisand many others, analyzed arches using the Lagrangian and an updated formulation and alsocalculated a much different response by either formulation.'The purpose of this paper is to present and compare i n detail the general formulations thathave been implemented in program NONSAP, and to show their general applicability in non-linear static and dynamic analysis. The formulations are termed total Lagrangian and updatedLagrangian formulations and they are based on the work of the authors cited above. For specificsolutions in this paper, elastic, hyperelastic, and hypoelastic materials are considered.The procedures are derived from the basic principle of virtual work and are valid for non-linearmaterial behaviour, large displacement$ and large strains. I t is pointed out that, in theory, thereis no difference in the formulations. Any differences in the numerical results arise from the factthat different descriptions of material behaviour are assumed, and if the material constants aretransformed appropriately, identical numerical results are obtained. Therefore, the question ofwhich formulation should be used merely depends on the relative numerical effectiveness of themethods. In the paper specific attention is directed to the numerical efficiency of either formula-tion.To demonstrate the applicability and the important differences in the formulations, thenumerical operations required for solution are studied and a variety of sample solutions arepresented. These include the large displacement static and dynamic analysis of a cantilever,the large displacement and large strain static and dynamic analysis of a rubber-like materialand the static and dynamic, elastic and elastic-plastic large displacement analysis of arches andshells.


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LARGE DEFORMATION DYNAMIC ANALYSIS 355FORMULATION OF THE CONTINUUM MECHANICSINCREMENTAL EQUATIONS OF MOTION

Consider the motion of a body in a Cartesian co-ordinate system as shown in Figure 1. Theaim is to evaluate the equilibrium positions of the body at the discrete time points 0, A t, 2 A t,3 A t , . . .,where A t is an increment in time. Assume that the solution for the kinematic and staticvariables for all time steps from time 0 to time t, inclusive, have been solved, and that the solutionfor time t +At is required next. It is noted that the solution process for the next required equi-librium position is typical and would be applied repetitively until the complete solution pathhas been solved.

CONFIGURATIONAT TIME 0

=-0X l , t K 1 , t +A +X I

Figure 1. Motion of body in Cartesian co-ordinate system

NomenclatureI t is useful at this point to lay out the notation which will be employed.The motion of the body is considered in a fixed Cartesian co-ordinate system, Figure 1 , inwhich all kinematic and static variables are defined.The co-ordinates describing the configuration of the body at time 0 are Ox,, Ox2,Ox3, at time t

are xl ,x2, x 3 ,and at time t + A t are + A ~ l , + A ~ 2 , + A r ~ 3 ,where the left superscripts refer to theconfiguration of the body and the subscripts to the co-ordinate axes.The notation for the displacements of the body is similar to the notation for the co-ordinates;at time t the displacements are ui, = 1 , 2 , 3 and at time [ + A t the displacements are + A f u i ,i = 1 , 2 , 3 herefore

The unknown increments in the displacements from time t to t +A t are denoted asu . = + A ~ i - f ~ i : j = 1, 2, 3


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356 K . - J . BATHE, E. RAMM AND E. L . WILSONDuring motion of the body, its volume, surface area, mass density, stresses and strains arechanging continuously. The specific mass, area an d volume of the body at times 0, t and t +Atare denoted by Op, p , r + A f p : A , A , + A f Aand OV, V , + A r V espectively.Since the configuration of the body at time t + A t is not know n, the applied forces, stresses andstrains are referred to a known equilibrium configuration. In analogy to the notatio n used forco-ordinates and displacements a left superscript indicates in which configuration the quantity(body force, surface traction, stre ss,. . .) oc cur s; in addition, a left subscript indic ates with respectto which configuration the quantity is measured.The surface and body force components per unit mass at time t +A t , but measured in con-figuration t , areConsidering stresses, the Cartesian components of the Cauchy stress tensor at time t + A t aredenoted by +.si, (since Cauchy stresses are always referred t o the configuration in which theydo occur = and the Cartesian com pone nts of the 2nd Piola-Kirchhoff stresstensor corresponding t o the configuration a t time t + A t but measured in configuration at time tare denoted by r+AiS i j .Considering strains, the Cartesian components of Cauchys infinitesimal strain tensor referred

to the configuration at time t + A t are denoted by r + A r e i j ;nd the Cartesian components of theGreen-Lagrange strain tensor using the displacements from the configuration a t time t to theconfiguration at time t + A t , and referred to the configuration at time t are denoted byTh e reference configurations, w hich will be used for applied forces, Kirchhoff-Piola stressesand Green-Lagrange strains, are those at time 0 and a t t ime t .In the formulation of the governing equilibrium equations derivatives of displacements andco-ordinates need be considered. In the notation adopted, a comma denotes differentiationwith respect to the co-ordinate following, and the left time subscripts indicate the configurationin which this co-ordinate is measured ; hus, for example,

r + A : , f k , k = 1, 2, 3.

Principle of virtuul displacementsWith the notation having been explained briefly, consider again the body in Figure 1 . Sincethe solution is known at all discrete time poin ts 0, At, 2 A t , . . . , , the basic aim of the formulationis to establish an equation of virtual w ork from w hich the unkno wn static and kinematic variablesin the configuration a t time t +At can be solved. Since the is opara me tric displacement based finiteelement procedure shall be employed for numerical solution, the principle of virtual displace-ments is used to express the equilibrium of the body in the configuration at time t + A t . T h eprinciple of virtual displacements requires that

s i j6 f + A r e i j du = r+ArBs. + A t v r + Arwhere r+Ar9s the external virtual work expression,(2)

In equations (1 ) and (2) 6u, is a (virtual) variation in the curre nt displacement com pone ntsr + A r u k ,nd h f + A r e i jre the corre spond ing (virtual) variations in strains, i .e.

r + A r r + A rp r + A rf k u, d cs + A r Vr + A r B1 : ~ ; : t k 6 u k r + A f d a +t + At *6r + A r e i j = + A r u i , j + I + A r U j , i )


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LARGE DEFORMATION DYNAMIC ANALYSIS 357I t need be noted that in equation ( 1 ) and the equations to follow the sum mation convention oftensor notation is implied.Equation ( 1 ) cannot be solved directly since the configuration at time t + A t is unknown.A solution can be obtained by referring all variables to a known previously calculated equilib-rium configuration. Fo r this purpose, in principle, any on e of the already ca lculated equilibriumconfigurations co uld be used. In practice, how ever, the choice lies essentially between t w o differentformulations, namely, the total Lagrangian formulation (T.L.) and the updated Lagrangian(U.L.) formulation, which ar e presented in the following sections.Tota l Lagrangian formulationLagrangian form ulation a nd has been used a great deal in static analysis. 1 4 , 1 6 , 2 6 . 3 8at time 0 of the body. The applied forces in equa tion ( 2 ) re evaluated using

Th e formulation called here total L agrangian (T.L.) form ula tion is generally referred to asIn the formulation all variables in equatio ns ( 1 ) and (2)ar e referred t o the initial configuration

where i t is assumed that the direction a nd ma gnitud e of the forces ' + A $ L and o p ' + . ~ , f L re inde-pendent of the specific configuration at time t + A t . Loading condit ions that depend on thedeformations will be considered later.Th e volume integral of Cauc hy stresses times variations in infinitesimal strain s in equ atio n ( 1 )can be transformed t o givezs

whereto the configuration at time t + A t but measured in the configuration a t time 0,= Cartesian comp onen ts of the 2nd Piola-Kirchhoff stress tenso r corre spon ding

( 5 )and 6 r + A d ~ i j variations in the Cartesian co mp onen ts of the Green-Lagrange strain tensor in th econfiguration at time t + A t , referred to th e configuration at time 0,

r + A r6 o&ij = 64 ('+ " ; M i . j + + Ad . j. i + + A;.,., ' + A&, j )I t should be noted tha t the integral of Piola-Kirchhoff stresses times variations in the Green-Lagrange strains is defined over the initial configuration at time 0 of the body.Substituting the relations in e quatio ns (3) and (4) into eq uation s (1 ) an d ( 2 ) , he followingequilibrium equ ation for the body in the configuration at time t +At but referred to the configura-tion a t time 0 is obtained,


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LARGE DEFORMATION DYNAMIC ANALYSIS 359The increm ental stress deco mp osition used in this case is

where I t i j = Cartesian componen ts of the Cauchy stress tensor an d Sij= Cartesian compon entsof the 2nd Piola-Kirchhoff stress increment tenso r referred to the config uration a t time t . Con-sidering the strain increm ents the followingrelations hold

r E i j = t e i j + rV i jwhere

The constitutive relation between stress an d strain increm ents used now is

and equation (16)can be rewritten asJ Cijrs S&do+ t i j , q i j dv = t i j reij dv (23)V J., J

which, as equatio n ( 1 5), is a non-linear e qua tion in the incremental displacements u i .Linearization of equilibrium equations

It should be noted th at equations (1 5) and (23) are , theoretically, equivalent a nd provided theapp rop riate constitutive relations a re used, the eq uatio ns yield identical solutions. How ever,as will be seen, the finite element m atrices established for so lution ar e different.The solution of equation (15) and of equation (23) can no t be calculated directly, since they arenonlinear in the displacement increments. Approximate solutions can be obta ined by assum ingthat in equation (15) o&ij = oei j and in equation (23) l ~ i j= e i j .This means that, in addition tousing = d o e i j nd S,cij = S,eij , respectively, the incremental constitutive relations employedarean d J i j = O C i j r s O e rs (24)

r s i j = c i j r s r e r s ( 2 5 )

F I N I T E E L E M E N T S O L U T I O NIn the T.L. formulation the approximate equilibrium equation to be solved isOCijrsers oeijOdu + dSij Soq i jOdv = - I, s i j SoeijI,

whereas in the U.L. formulation the equa tion istij ,qi jdu = +Ar9 (27)


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360 K. - J . BATHE, E . R A M M A N D E . L . WILSONEquations (26) and (27) are linear equations in the incremental displacements and are used as thebasis for isoparametric finite element a n a l y ~ i s . ~ ' , ~ ~eferring to the standard procedures forassembling the structure matrices, attention need only be given to the derivation of the matricescorresponding to a single element.Finite element matrices

In the isoparametricusing element solution the co-ordinates and displacements are interpolated

k = 1 k = 1 IN t

N N'ui = k k r ~ f ; U i = 1 , ~ :k = 1 k = 1

i = 1, 2, 3 (28)

i = 1,2 , 3 (29)where 'x f is the co-ordinate of nodal point k corresponding to direction i at time t , '1.41 is definedsimilarly to 'xf, 17, is the interpolation function corresponding to nodal point k , and N is thenumber of element nodal points.43Using equations (28) and (29) to evaluate the displacement derivatives required in the integrals,equation (26) becomes, considering a single element

( d K L + 6 K N L ) ~ ' + A ' R - d F (30)where d K Lu , ~ K N L uand dF are obtained from the finite element evaluation of Jo v OCijrser ,xhoeij du, JC," S,, h,v],, Odu, and Jo v $,,oeiiOdu, respectively, i.e.

The vector ""R in equation (30) is obtained from the finite element evaluation of equation (8)in the usual way.43 In the above equations, dB, and dBN, are linear and non-linear strain-displacement transformation matrices, ,C is the incremental material property matrix, $3 is amatrix of 2nd Piola-Kirchhoff stresses, and ,$ is a vector of these stresses. All matrix elementscorrespond to the configuration at time t and are defined with respect to the configurationat time 0.Similarly, the finite element solution of equation (27), which was obtained using the U.L.formulation, results into

(34):K L + KN,)u = ' "R - Fwhere


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LARGE

and

DEFORMATION DYNAMIC ANALYSIS

:F = l, BI Z dvIn Equations (35) to (37) the elements of the linear and non-linear strain-displacement trans-formation matrices :BL and :BNL, espectively, and the elements of the incremental materialproperty matrix C correspond to and are defined with respect to the configuration at time t ,t s a matrix and Z is a vector of Cauchy stresses in the configuration at time t .It should be noted that the elements of the matrices in equations (30) to (37) are functions ofthe natural element co-ordinates and that the volume integrations are performed using a co-ordinate change from Cartesian to natural ~o - o r d i n a t e s . ~ ~able I gives the strain-displacementand stress matrices used for two-dimensional (plane stress, plane strain and axisymmetric)analysis in the U.L. and T.L. formulations. Figure 2 shows the 4 to 8 variable-number-nodeselement that has been used in the sample solution^.^

Figure 2. Two-dimensional element shown in the global x , -x2 planeDynamic analysis

In dynamic analysis, the applied body forces include inertia forces. Assuming that the massof the body considered is preserved, the mass matrix can in both formulations be evaluatedprior to the time integration using the initial configuration at time 0 as reference. Employing thestandard finite element formulation to evaluate the element mass matrix,43 the incrementalequilibrium equation for a single element in the T.L. formulation is


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362 K. - J . BATHE, E . RAMM AND E. L . WILSONTable I. Matrices used in two-dimensional analysis

~~~ ~

Total Lagrangian formulationIncremental strainsOEl = 0 U I . I + d u , , , o u l , 1 + d u 2 . 1 o u 2 .1 + t [ ( o u l . l )2 + ( o U 2 , 1 )2 1O E 2 2 = o u 2 , 2 + d u 1 , 2 O U 1 . 2+ d U 2 .2 o u 2 . 2 + f [ (o U 1 .2 ) 2 +( O U 2 .2 ) 2 10 8 1 2 = t [ o u I . 2 + 0 ~ 2 . l l + f [ d ~ 1 , 1 ou1.2+du2,1 O U 2 . 2 + d U I . Z 0 U I . l + d u 2 , 2 O U 2 , l l + t [ O U 1 . 1 O U l . 2 + 0 U 2 . 1 o u 2 . 2 1

(axisymmetric analysis)where

Linear strain-displacement transformation matrixUsing oe = dBLuwhere

oeT = [oei I 0 e 2 2 20e12 oe33l ;and

0 3 L = 03L.o f O B L 1

dBL0 =

whereN

ur = l + A ' u ~ - ' u ~ ; O X 1 = 1 , Ox; ; N = number of nodesah k = 1oh,,j =2d o x j

4 3 0 -1 0 0. " I l l O h , , , I 2 1 O h , , ,' ' ' I 1 2 O hN . 2 I 2 2 O h N , 2' . . ( I l l O h N . 2 + 1 1 2 O h , , , ) ( I 21 O h N . 2 + h 2

0


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LARGE DEFORMATION DYNAMIC ANALYSISTable I-continued

363

where N N N

Non-linear strain-displacement transformation matrix

2nd Piola-Kirchhoff stress ma trix and vector

1 -,s =$3 3

Updated Lagrangian formulation

& I 2 = [ r ~ l . 2 + ' ~ 2 . l l + t [ r ~ 1 . 1 ul.2 +'~2 .1 P2.21(axisymmetric analysis)

where au ia t x j=-

Linear strain-displacement transformation matrixUsingwhere

:BL =

re = :B,u

- 'I. 0 . . . " 0X1%

h2 0'Xh 1 0'3


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364 K . -J . BATHE, E . RAM M A N D E. L . W I LS ONTable I-continued

where' u; = ' + A ' u ~ - ' u ~ ; 'x i = f ,'.w: ; N = number of nodes1 h . j =

alxj k = 1Non-linear struin-displacement transformation matrix$ 1 . 1 0 $ 2 . 1 r h 3 . 1 ' " r h N . Ii h I . 2 i h 2 . 2 i h 3 . 2 ' ' rhN.Z

0 [ h i , , 0 1 h 2 . 1 0 lh3.1 . . . t h N . I0 i h l . 2 0 & 2 , 2 0 h 3 . 2 . . t h N , 21" 0 5 0 -3 0 . . . h " 0'XX 'X 'X

Cuuchy stress matrix and stress oector0

'712

' 7 2 20

0_ j ;0'533 1-t = Ij]and in the U.L. formulation this equation is

(:KL+:KNL)u = f + A f R - i F - M U (39)where f + A f i is a vector of the element nodal point accelerations at time t +At, and M is the elementmass matrix calculated at time 0. In integrations (38) and (39), damping effects defined by amatrix C have been ign~red.~Equilibrium iteration

It is important to realize that equations (38) and (39) are only approximations to the actualequations to be solved in each time step, i.e. equations (7) and (16), respectively. Dependingon the non-linearities in the system, the linearization of equations (15) and (23) may introduceerrors which ultimately result into solution instability. For this reason it may be necessary toiterate in each load step until, within the necessary assumptions on the variation of the materialconstants and the numerical time integration, equations (7 ) and (16) are satisfied to a requiredtolerance. The equation used in the T.L. formulation is (40):K, + KNL) A d ') - + At R- ' + % F ( ' - l ) - M + A t i j ( i ) i = 1,2 , 3 . .+A d i ) .here f + A f ( i ) - + A f u ( i - 1 )u -It should be noted that for i = 1 equation (40) corresponds to equation (38), i.e. A d 1 )= u,r + A r . . ( l ) = f + A f " > u+ A L (0)= f u, and f+AiF(o) dF.

The calculation of the acceleration approximation ' + A ' U ( i ) depends on the time integrationscheme used.The vector of nodal point forces equivalent to the element stresses, '+%F(') ,s the finite elementevaluation of j,,"+.!$;j' 8 + % ~ $ )dr, where the superscript ( i ) shows that stresses and strains areevaluated using L + A r ~ ( i ) . Since 6 L + A ; ~ i j~ ( 8 0 u i , j + 8 0 u j , i + f + A ~ u k , io u k , j + L + % u k , jouk , i ) , the


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L A R G E D E F O R M A T IO N D Y N A M IC A N A L Y S IS

nodal point forces are365

in which the matrices $Bf) andbut are defined for time t + A t and iteration (i), respectively.,S correspond to the matrices dB, and ds in Table I ,In the U . L .formulation the equation used for a single element with equilibrium iteration isin which the ith displacement and acceleration approximations are calculated as above andr+ArF S the finite element evaluation of+ A t ( i )

where :Ii:Bg)and correspond to the matrices :B, and i n Table I, respectively, but aredefined for time t +A t and iteration (i), respectively.It may be noted that the equilibrium iterations correspond to a modified Newton iterationwithin each load step.43 Table I1 summarizes the step-by-step algorithm used. For details onthe Wilson 8 and Newmark integration schemes see References 4, 5, 31.Table 11. Sum mary of step-by-step integration

Initial calculations1. Fo rm mass matrix M ; nitialize Ou, 1,ii2. Calculate the following con stants :

nitem 2 3 ;o1 < 0.01 ;Wilson 8 method : 8 2 1.37,usually 8 = 1.4, T = 8 A ta , = 615, a , = 615 a , = 2a3 = a,/@ab = At12Newmark method : 8 = 1.0,6 2 0.50, a 2 0.25 (0.5 +S), T = A fa , = l / (a A t 2 )a3 = uo u4 = - a , u 5 = -a2a b = A t ( 1 - 6 ) u , = 6 A t

in static analysis 8 = I a nd go t o A.

a4 = - a , / @a , = A t2 /6

a 5 = 1- 18

a , = l / ( a A t ) a, = 1/(2a)- 1

3. Calculate mass con tribu tion to effective stiffness matr ix : K = a,MFor each time step

A. Calculation of Displacement increment(i ) If a new stiffness matr ix is to be formed, calculate an d triangularize KK= x + K ; = LDLT

(ii) Form effective load vector : R = R +8(R - R) + M ( a , u + a , i i - F


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366 K . -J . BATHE, E. RAMM AND E. L. WILSONTable 11-continued

(iii) Solve for displacement increments using latest D, L factors:L D L T ~ = ~ + r f i

(iv) I f required, iterate for dynam ic equ ilibrium ; then initialize do) u, i = 0(a) i = i + l(b ) Calculate ( i- )st approximation to accelerat ions and displacements :t + x ( i - l ) - o u ~ i - l ~ ~ a l ~ ~ ~ a z r ~ ~+ r u -i - I ) - ~ + , , ( i - l )

(c) Calculate ( i - I)st effective out- of-b alanc e loads :I +rf i ( i - I ) = IR + O(I+ AI R - R )-M I rh(i- I - + rF(i - )

(d) Solve for ith correction to displacement increments :LDLT Au(il = I +r f i ( i - I

(e) Calculate new displacement increments :u i - 1 ) +Au("

( f ) Iteration convergence if ~ ~ A U ( ~ ) ~ ~ ~ / ~ ~ U ( ~ ) + * U I I ~to1If convergence: u = di) n d go to B ;If no convergence and i < nitem: go to (a); otherwise restart using new stiffness matrix and/or asmaller time step size.

B. Calculate new accelerations, velocities and displacementsWilson 0 method :I + A ' i i = a,u+a,~i+a, ' i i

u + t i )+ At " = r " + a 6 ( 1 + A I -I + A1u = ~ + A t ' U + a , ( ' ~ ~ ' U + 2 ' i i )

Newmark method :I + AI.. = a,u+a,'i+a,'iil + A l U = u + a 6 1 i i + a , f + A l - Ut + A ru = 'U+U

CONSTITUTIVE RELATIONSAn important aspect in the solution of non-linear problems is the calculation of the constitutivetensors, which define the stress-strain matrices in the finite element evaluations. In the iso-parametric finite element discretization it is necessary to evaluate the stress-strain matricesat the element integration points, and they are required for the calculation of the element stiffnessmatrices and stress vectors.Linear elasticity and hyp erelasticityT.L. formulation the stress-strain relations are'Elastic and hyperelastic materials are relatively easy to deal with in practical analyses. In the

dsij = d c i j r s &rs (44)where isijs the 2nd Piola-Kirchhoff stress tensor, &, is the Green-Lagrange strain tensor anddCijrss the material property tensor in the configuration at time t . The relation in equation (44)


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LARGE DEFORMATION DYNAMIC ANALYSIS 367can be written for all configurations at time 0, A t , 2 A t , . . .tutive relation equivalent to equation (44) is . In the U.L. formulation the consti-

T..=c..j 1 i j r s & rs (45)in which T ~ ~s the Cauchy stress tensor, E , , is the Almansi strain tensor and : C i j r ss the materialproperty tensor at time t.Considering linear elasticity dCijrs nd :Cijrsre both constant and defined in terms of theYoungs moduli and Poissons ratios of the material. However, it should be noted that specifyingconstant iCijrss equivalent to using a material tensor dC i j r s ,which is deformation dependent,and vice uersa ; amely the following relations exist

(46)0 0P0

d l c m n p q =7 m , i 9 x n . j i C i j r s 1 X p. r 1 x q .s

The constitutive relations in equations (44) and (45) are used in the evaluation of the elementstress matrices and stress vectors (see Table I), i.e. total 2nd Piola-Kirchhoff and Cauchy stressesare calculated directly from total Green-Lagrange and Almansi strains, respectively. However,in the calculation ofthe linear strain stiffness matrices at time t , tangent material property tensorsare required. In the T.L. and U.L. formulations the relations considered are J ij = OCijrs~ r s ndJi j = t C i j r sE , ~ , espectively, in which, for linear elasticity,

Considering hyperelasticity the stress-strain relations are derived from the strain energyfunction. 11*19*33In this study the constitutive relations defining isij nd oCijrsn terms of theGreen--Lagrange strain at time t for a rubber-like material in plane stress conditions have beend e r i ~ e d . ~ . ~herefore, to use the U.L. formulation, it is necessary to transform isij nd OCijrsto zij and C i j r s respectively, as expressed in equations (5) and (47).It is important to note that in the analysis of elastic and hyperelastic materials identicalnumerical results are obtained using the T.L. and U.L. formulations provided the materialtensors are related as given in equations (46) and (47). Also, since the material constants areindependent of the history of solution, analysis errors result only from the isoparametric finiteelement formulation and the time integration scheme, provided equilibrium iterations areperformed. Therefore, in the analysis of elastic and hyperelastic materials the analysis errorsare quite similar to those in small displacement linear elastic analysis.Hypoelasticity including elastoplasticity

For hypoelastic materials the constitutive tensors relate increments in stresses to incrementsin deformations. Since the constitutive relations depend, in general, on the stress and strainhistory, the use of a material law corresponding to the T.L. or the U.L. formulation must dependto a large degree on the possibility of performing experiments to obtain the appropriate materialconstants. In this context it should be noted that a great deal ofaddit ional research is still requiredto formulate and evaluate appropriate material constants for hypoelastic materials, in particular,for the identification of large strain behaviour. 1 9 , 2 2 , 2 3 Although the formulations presentedbelow are applicable to large strain conditions, in actual practical analysis the material law is


16/34

368 K. - J . BATHE. E . RAM M AND E . L . W I L S O Nmost likely to be defined only for small strain^.^ An important such case here included is theelastic-plastic material behaviour characterized using the flow theory, which can be used inthe analysis of large displacement but small strain problems.Using the T.L. formulation, hypoelastic material behaviour can be described using equ ation(14), i.e.

O s i j == OCijrs 08,s (14)in which oCi jrsepends on the history of the Green-Lagran ge strains and 2nd Piola-Kirchhoffstresses. Th e stresses a t time t + At are calculated using equ ation (9), i.e.

(9 )In the analysis using equatio ns (14) an d (9) it is assumed that the material tensor OCijrssevaluated in the same way as in small displacement analysis, but the stress and strain variablesof the T.L. form ulation are used to define the history of the material. A main advan tage of adop t-ing this material description is that it is relatively simple to use. Nam ely, assume th at a s ub ro uti neto calculate the material law in small displacement analysis has been written ; then the same

program would also calculate OCi j rsn large displacement analysis by simply using Green-Lagrange strains an d 2nd Piola-Kirchhoff stresses t o define the stress and strain history.Similarly to equatio n (14), in th e U.L. formulation hypoelastic material behaviour may bedescribed using equ ation (22), i.e.

f+JrS . .lJ = ;Si j+osi j

J i j = rc . .jrs r&rs (22)in which lCi j rss defined by the history of Cauchy stresses and the accumulation of the instan-taneous plastic strain increments. The constitutive relation in equ ation (22) may be m ore a ppe al-ing than the T.L. material law in equ ation (14) since physical stress com pon ents are used todefine the m aterial constants, and r ~ r s pproximated by lerscan kinematically be understood tobe the add ition of elastic and plastic strain increments, just as in small displacement analysis.Having calculated ,S i j from the relation J i j = C i j r srsr the Cauchy stresses at time t + A t a reobtained using equation (17), i.e.

(17)+ A t s . . = '.r..+ s..f I J ij r I Jand the transformation

r + A i(50)+ A r T - pi + A r x , f + A r S , . + ArI S, L I IJ I X r , j-sr rP

A third possibility is to characterize th e material behav iour using a stress rat e which is definedwith respect to the current moving co-ordinates within the time interval t to t + A t . " . ' 5 * 1 6 , 2 3The stress rate used must be invariant with respect to rigid body rotations, and one possibilityis to use the Jau m an n stress rate, which, at time t , is defined as

where D/Dt deno tes time derivative with 'x i , = 1,2 ,3, kept constant,D'r?. = c..-

lJ ' l J r S Dt ' Is


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LARGE DEFORMATION DYNAMIC ANALYSIS 369and Q p j are the Cartesian com ponents of the spin tensor,

Equations (51) to (53) need to be considered in the evaluation of the tangent stiffness matrixand in the calculation of the current stress conditions. Th e constitutive tensor relating the Jau -mann stress rate tensor z to th e incremental strain rate tensor (D/Dt),e,, is calculated in thesame way as in small displacement analysis, but using C auch y stresses to define the history of thematerial. In the evaluation of the linear strain tangent stiffness matrix, instead of equation (22),the following approximate relation may be used,D i r i j= C i j r se,s (54)

where D signifies discrete increment in, an d therefore Diers = ,e,,. Considering the calculationof Cauchy stresses at time t+ A t, it is important to use equat ions (51) to (53) in smallenough increments of time. In elastic-plastic analysis it is in any case necessary to evaluate thestress increments by numerical integration of the elastic-plastic material law times the strainincrements, and it is efficient to include eq ua tio n (51) in this integration.In this study the T.L. material description using equ ation s (14) an d (9) and the U .L. materialdescriptions given in equation s (22),(17),(50)and in equations (51) to (54) have been implemen ted.The U.L. formulations will be referred t o a s U.L. with transformation, U.L.(T), and U.L. withJau m an n stress rate, U.L. (J), respectively. The calculatio n of the elastic-plastic materia l co nst an tsand stress vectors in the T.L., U.L.(T) an d U .L.(J) form ulations is presented in de tail in Reference4.In the above enumeration it was assumed tha t the solution procedure, namely th e T.L. or U .L.formulation, is chosen according to t he definition of the constitutive tensor. Since the con stitutivetensor is dependent on stress and strain quantities, it is expedient t o use the specific incrementalfinite element form ulation, for which th e mate rial law is defined. If this is not do ne , it is necessaryto evaluate the required stress and strain quantities for the calculation of the m aterial cons tantsand transform the constitutive relations using equa tion (46) r equ ation (47).

D E F O R M A T I O N D E P E N D E N T L O A D I N GSo fa r i t has been assumed that the loads are independent of the configuration of the body.In practice, therefore, the external loads for each step can be calculated and stored o n ba ck-u pstorage before the actual time integration is carried out. However, when the structur e undergoeslarge displacements or large strains it may be necessary t o consider th e externally applied lo adsto be configuration dependent.An important type of loading, which may need to be considered as deformation dependent,is pressure loading .33In this case the loading to be used in the T.L. formulation is

and in the U.L. formulation

where n ,= component i of the norm al n in the configuration a t t ime t , an d similarly for time 0,and r + A p= surface pressure in the configuration at time t+At. Equation (55) can be written


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370 K. - J . BATHE, E. R A M M AND E. L . WILSONin the form

and similarly equation (56) becomes

where in both formulations the first integral enters the load vector and, assuming that r + A t ~ i , k=the second integral contributes to the system tangent stiffness matrix. It should be notedthat this is a non-symmetric contribution to the stiffness matrix, and is therefore, in practice,computationally inefficient to handle. Using equilibrium iterations, it appears more efficient,at least when pressures are reasonably small, to neglect the contribution of the pressure loadsto the stiffness matrix. In the iteration the loads are then evaluated asP a O x i Oni0da and -'

0

_ _ _ _, + A t ( j ) t + A t p mP a x k I + A I ( j ) I+AIda(j)nkin the T.L. and U.L. formulations, respectively, where the right superscript ( j ) ndicates theconfiguration of the iteration. It is seen that although the same approximations are involvedin both formulations, the U.L. formulation requires less numerical operations and seems morenatural to use.

SAMPLE SOLUTIONSAll solutions presented in the following have been obtained using the algorithm presented inTable 11, in which the selected parameters were to1 = 0.001, nitem = 15, 8 = 1.4, 6 = 0.50 andu = 0.25. No attempt was made to optimize the solution times by selecting the most effectiveload step increments. In the dynamic analyses, nearly always one or two equilibrium iterationswere sufficierit in each time step and the computer time used was in all analyses rather small.The time step used in an analysis is denoted by At and was selected as a reasonable fractionof the fundamental period, Tf, f the structure at time 0. In all dynamic analyses zero initialconditions on the displacements, velocities and accelerations were assumed.For the finite element discretization 4- or 8-node two-dimensional elements have beenemployed (Figure 2). The material properties given have always been assumed to be definedcorresponding to the specific formulation used for solution.Large displacement static and dynamic analysis of a cantilever

The cantilever in Figure 3 under uniformly distributed load was analyzed using the T.L. andU.L. formulations. The cantilever was idealized using five 8-node plane stress elements.Static solutions were obtained for the loading retaining its vertical direction, and for theloading remaining perpendicular t o the top and bottom surfaces of the cantilever, i.e. deformationdependent follower loading. In the finite element solution the follower loading can be definedby specifying the direction of the nodal loads to pass through two nodal points, the co-ordinatesof which are updated in each load step. In this specific analysis, the top and bottom surface nodalpoints of the cantilever have been used to define the direction of the loading. In addition to theU.L. and T.L. formulations also the U.L.(T) and U.L.(J) formulations have been used in thesolution for deformation independent loading, in order to assess the accuracy that may be


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LARGE DEFORMATION DYNAMIC ANALYSIS 37 1

L = 10 inh = l i n V . 0 2b = I in

E = I 2 lo4 lb/ in2p = 16 Ib sec2/in4

Figure 3. Cantilever under uniformly distributed loa d

expected in elastic-plastic analysis. It is seen that, t o th e precision p ossible to sh ow in Fig ure 4,th e T.L., U.L.,U.L.(T)and U.L.(J) olutions predict the sam e response (although the sam e E and vhave been used in each analysis, Figure 3) an d that excellent agreement ha s been obta ined withan an alytical solution

J290I-2EVw-ILwn

reported by H0 1de n.I~

I I I I / I

100 .o0 2 4 6 8LOAD PARAMETER K =

Figure 4. Large deflection analysis of cantilever under uniformly distributed load

Fo r the dynam ic analysis the T.L.formulation was selected. Figure 5 shows the results obtain edusing the Newm ark integration scheme. It is seen that the solution predicted using a time stepAt z Tf/42, where T is the fundamental period of the cantilever, is significantly different fromthe solution obtained with th e smaller time step At z ,/126, unless equ ilibr ium iteratio ns ar eused. An average of 4 iterations per time ste p were required. Th e analysis therefore sho ws theimportance of using equilibrium iterations in the response calculations of this structure, unlessa small time step, At , is used.It should be noted that a m ain ch aracteristic of the cantilever is that the str uctu re stiffens withincreasing displacement, which, as show n in Fig ure 5, results in a substantial decrease in ampli-tude a nd effective period of vibration. It is the stiffening of a st ru ctu re tha t can result in con-vergence difficulties in equilibrium i t e r a t i ~ n . ~


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372 K. - J . BATHE, E . RAMM AND E . L . WILSON

T /I 26

T l M E / A l ,Figure 5. Large displacement dynamic response of cantilever under uniformly distributed load, Newmark method6 = 0.50. = 0.25

Sta tic large displacement analysis of a spherical shellThe clamped shallow spherical shell in Figure 6 subjected to uniform pressure was analyzedusing a finite element idealization of eight 8-node elem ents.Figure 6 shows the load deflection curve predicted using the T.L. formu lation. In th e analysis,36 load steps with an average of about 3 to 4 equilibrium iterations in each step were used.The results are compared with a n analytical solution of Korn ishin and Isanb aeva ,2 and a finiteelement solution of Yeh.42 As shown, good agreement between the different solu tions has beenobtained. Since equilibrium iterations were performed in the present solution, the oscillatingbehaviour at the beginning of the post-buckling range in Yehs solution was not obtained.The U .L. formulation gave almost indistinguishable results to tho se of the T.L. formulatio n.

Static and dynamic large displacement analysis of a second spherical shellThe spherical shell subjected to a concentrated apex load shown in Figure 7 was analyzedfor static and dyna mic response.Figure 7 shows the static load-deflection response predicted in this study and by Stricklin3a nd M e s ~ a l l . ~ ood agreement between the different so lutions has been o btaine d. In the presentsolutions, the T.L. and U.L. formulations were used and no equilibrium iterations have beenperformed. In addition , to assess the accuracy tha t m ay be o btaine d in elastic-plastic a nalysis,the U.L.(T)and U.L.(J) formulations have been used. Figu re 8 com pares the T .L. response predic-tions with two U.L.(T) and U.L.(J) solutions. It is seen tha t the U.L.(T) and U.L.(J) solution sappro ach th e T.L. solution as the load steps become smaller.The dynamic response calculated using the W ilson 6 integration method when the apex loadis applied as a ste p load is shown in Figur e 9. It is observed tha t for this problem the differencesbetween the solutions using equilibrium iteration and not iterating for equilibrium are small.


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LARGE DEFORMATION DYNAMIC ANALYSIS

E I G H T 8 - N O D EE L E hAXIS ' TS FORIETRICIS__

~

/~

~

' W 7.-

- NALYTICAL SOLUTIONK O R N I S H I Na I S A N B A E V A [a]+ YEH' S FE S O L U T I O N [4Z]+ L S O L U TI O N W I T HE O U l L l 8 R l U M I T E R A T I O N S1

37 3

0 0 0 5 10 15 2 0 2 5DEFLECTION RATIO (W,/h)

Figure 6. Load-deflection curve fo r a shallow spherical shell

The much larger response and effective period predicted in the non-linear analysis is a resultof the softening behaviour of the structure with increasing load. It should be noted that in theanalysis of this highly non-linear shell no difficulties were encountered using the Wilson or theNewmark integration methods, and practically identical results were ~ b t a i n e d . ~Large displacement static buckling analysis of an arch

The clamped circular arch shown in Figure 10 was analyzed for buckling due to a singlestatic load using the T.L. and U.L. formulations with equilibrium iterations.Figure 10shows the calculated load-deflection curve of the arch. The differences in the displace-ments calculated using the U.L. and T.L. formulations were less than two per cent. The solutionswere obtained using 28 load steps with an average of two to three equilibrium iterations per step.The same arch was also analyzed by Mallet and Berke, who used four 'equilibrium-based'elements.24 Dupuis and many others, analyzed the arch with curved beam elements, and usedthis example to demonstrate the convergence of their Lagrangian and 'updated' formulations.'In the latter formulation only the nodal points were updated, but -not the geometry within theelements. As shown in Figure 10, the results are very sensitive to the number of elements usedand are not satisfactory. Dupuis and many others, also compared the calculated results with


22/34

374 K . - J . BATHE, E . RAMM AN D E . L . WILSON

R = 4 7 6 1 nh = 0 0 1 5 7 6 InH = 0 0859111

E = 10 x 10 l b / m 2TE N 8 N O D E E L E M E N T S v . 0 3F O R A X I S Y M M E T R I C p = 0 0 0 0 2 4 5 lb-sec2/ i$ANALYSIS

I I IT L ,U L SOLUTIONS,NOEQ UILIBRIUM ITERATIONSSTRlCKLlN [37]( T H I R T Y S H E L L E L E M E N T S ). E S C A L L [ 2 7 ] ( F I N I T E D I F F E R E N C E S O L U T IO N )

80-5d-

60s+0

W

5x 4 0aa

20

04 0 8 12 16 20D E F L E C T I O N R A T IO , W o /H

Figure 7. Load-deflection curves for spherical shell

experimental results by Gjelsvik and Bodner,I2 whose predicted buckling load is abou t 10per cent lower than that calculated by Mallet. However, i t need be realized tha t an a rch with aparameter 1 = 11.6 is already influenced by antisymmetric buckling modes, which, althoughpossible in the experiment, have not been take n acco unt of in t h e a n a l y ~ e s . ~ ~he results obtaine din this study a re therefore satisfactory.Elastic dyn amic snap buckling of a second arch

A dynam ic buckling analysis of th e circular a rch show n in F igur e 11 was carried out. T h ematerial of the arch was assumed t o be isotropic linear elastic.In the analyses the T.L. formulation was used. The uniformly distributed pressure load wasapplied as a step load. The time step to fund amental period ratio, A t / T , , was approximately 1/70.The arch is an exam ple of H umph reys analytical a nd experimental investigation, who solvedthe governing differential equation using a n analogu e compu ter. l 8 Humphreys concluded thatthe buckling load of this arch is not influenced by antisym me tric modes.Figure 12 shows the displacement response predicted in this study using the Wilson 8 integra-tion scheme. The solution obtained by H umphreys is also shown. In the Figure, the deflectionratio A defined as

average normal deflection waverage rise of arch = H / 2A =


23/34

LARGE DEFORMATION DYNAMIC ANALYSIS 375

R = 4 7 6 1 "h = 0 01576 ~nH = 0 0 8 5 9 8 ne = 10 90A : 6E = l o x lo6 I b / i n zp = 0000245 Ib-secz/on'T E N 8 N O D E E L E M E N T S Y = 0 3FO R A X I S Y M M E T R I CA N A L Y S I S

D E F L E C T I O N RATIO, W,/HFigure 8. Load-deflection curves for spherical shell

is used. The dynamic buckling of the arch occurs at that load level at which a sudden increasein the deflection ratio A is measured. Figure 12 shows that at po = 0.190 the arch oscillates abo uta position of approximately A = 0.25, and tha t a t p o = 0.200 the arch firs t snaps through , andthen oscillates abo ut a position of appro xim ately A = 2.5. Therefore, the buckling load predictedhere lies betweenp, = 0.190and po = 0.200, which is abou t five per cent lower than that predictedby Humphreys.It should be noted that for a load larger th an the buckling load, i.e. for po = 0.25, the m aximumresponse increases only little. The results for p, = 0.250 are in essential agreement withHum phreys' results, where the slightly larger response agrees with the observation tha t a smallerbuckling load w as predicted in this study. Th e discrepancies in th e results can arise from ap pro xi-mations in either analysis. Humphreys' series solution is based o n the a ssum ption of shallowness,i.e. q and w are measured vertically, and in the series solution only a finite nu m ber of terms havebeen included.It is noted th at in a practical analysis dam ping should be included an d a longer time range m aybe considered a s well.Large displacement and large strain static and dyn amic analysis of a rubber sheet with a hole

A plane stress analysis of the rubber sheet shown in Fig ure 13 was carried ou t. The purpose ofthis analysis was to test the capability of predicting static and dynam ic large stra in response.Th e material of the rubb er sheet was assum ed to be of Mooney-Rivlin type. Th e specificmaterial constants used for the hyperelastic incompressible material were C, = 25 psi, C, =


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376 K.-J. BATHE, E . RAMM AND E. L. WILSON2 5

2 0

r,9 E0aILz; 0VWJlLW

05

0 30 6 0 90 I2 0 I50 180 2 10 240T I M E /A tFigure 9. Non-linear dynamic response of spherical shell, Wilso n 0 method, 0 = 1.4

I--- LAGRANGIAN

ITERATIONS

R =1331141nh = 3/16 nb = I O m ( W I D T H )L = 34 0 nti = I09 nP = 7 3397OI = 000055 m 4E = lox 1061b/ in2

TWELVE 8 NODE ELEMENTSFO R H A L F O F A R CH = 0 2

"0 0 01 0 2 0 3 0 4VERTICAL DISPLACEMENT AT APEX w, [in]

Figure 10. Load-deflection curve for a shallow arch under concentrated load


25/34

L A R G E D E F O R M A T I O N D Y N A M I C A N A LY S IS 37 7

E = 10 x lo6 l b / i n 2ER = 67 115 i n 1v . 0 2

= 4 4 l o - 4 ~ 2i nC = 3 2 024 x 1 0 5 -&P

h = l o i nb = l o i np = I5OH 2 2871nTIME P A R A M E T E R L O A D P A R A M E T E Rr = C x i p; (Lfh X E

Figure 11 . Simply supported shallow arch

7 psi. These constants are based o n a n analytical an d experimental investigation of the rubbersheet by Iding. The finite element mesh used in the analysis is presented in Figu re 13 .Figure 14 shows the static load deflection curves for different points o n the sheet. On ly fiveequal load increments with an average of four equilibrium iterations have been used to reach th efinal load position with a displacement of more than 11 in at point B. At this stage Green-Lagrange strains of more than 4.5 are measured, see Figure 15. The results obtained are in

P bPO-- ST EPLOAD ,4 I O A t

A t = 3 3l5XIo5 sec

0 5 10 15 20 25 30 35TIME PARAMETER T - %$ X

Figure 12. Dynamic snap-through of a sha llow circular arch Wilson 0 method, 0 = 1.4


26/34

378 K.-J. BATHE, E. RAMM A N D E . L . WILSON90 Ib/ in2

1.25 s+'I in (THICKNESS)

10/3 in I 10/3in I 10/31nI I

I l l I I- 3 0 106Kl81 10 I 18 I 28- 10 0 n -Figure 13 . Finite element mesh of rubber sheet with hole

excellent agreement with those of Iding. The results of Iding have been obtained with the com-puter prog ram developed in Reference 19,but a re not given in the Reference.The dynam ic analysis was performed for the ste p load shown in Figure 16 using the Wilson 8and Newm ark integration schemes with A t / T f 2 120.Figure 16 compares the displacement response predicted using the two integration m ethods.As is seen, practically the same response was calculated using the Wilson 8 and the Newmarkmethods. In add ition, it should be noted that identical solutions have been obtain ed using eitherintegration schem e an d a n interval of stiffness reform ation of 10, 5 o r 1 time steps (see Tab le 11).This should be expected, since the solutio n is unique for th e selected time ste p A t .Elastic-plastic large displacement dynamic a nalysis of a third spherical shell

The dyn amic response of the spherical shell in Figure 17 subjected to a distributed step pressurep = 600 lb/in2 was calculated. The ma terial was assum ed to obey the von Mises yield conditionwith linear isotropic hardening. Th e purpose of this analysis was to com pare the results obtain ed


27/34

LARGE DEFORMATION DYNAMIC ANALYSIS

10

8

6

4

2

0 -

379

--

--

- -

--

0 2 4 6 8 10 12w [In1

Figure 14. Static load-deflection curve for a rubber sheet with holeusing the various non-linear large displacement form ulations available for elastic-plastic responsecalculations.Figure 17 shows the dynam ic response of the cap predicted using the N ew ma rk time integra-tion scheme in linear analysis, materially non -linear only analysis, i.e. assum ing sm all displace-ments and small strains, and com bined geometrically an d m aterially no n-linear analysis. In thefully non-linear analysis the solutions using the T.L., U.L.(T) and U.L.(J) formulations havebeen obtained. It is observed that all three formulations predict essentially the same response.The reason for obtaining almost identical solutions lies partly in tha t the mathem atical repre-sentation of the yield function is almo st the sam e in the 2nd Piola-Kirchhoff stress space aridthe Cauchy stress space. Namely, in problems of small strains bu t large rotations, such as in th e

I N I T I A LCONFIGURATION P = 150 Ib

I

+0

Figure2 4 6 8 10 12 14 16 18 20 22 [in]

5. Deformed configuration drawn to scale of rubber sheet with hole (static analysis)


28/34

380 K.-J. BATHE. E. R A M M A N D E. L . WILSON

A t - 0 0 0 l S s e c

I N T E R V AL O F S T I F F N E S SR E F O R M A T l O N = 1 - 5 , 10

Figure 16. Displacements versus time for rubber sheet with ho le. T.L. solution with equilibrium iterations

analysis of shells, the physical components of the Cauchy stress tensor in rotated (surface)co-ordinates are approximately equa l to the Cartesian c om pone nts of the 2nd Piola-Kirchhoffstress tensor.The solutions in Figure 17 demonstrate the effect of including different degrees of non-linearities. It is observed tha t the materially non-linear only so lutio n differs a great dea l from t helinear elastic response, and t hat the effect of large displacemen ts is also significant. Th e decreasein amplitude of vibration and increase in the mean deflection of the shell when non-linearitiesare taken into account should be noted.Th e response of the cap was also calculated using the Wilson 6 me thod, which gave practicallythe same result^.^A comparison of the results obtained in this study with those calculated by NagarajanZ9is given in Fig ure 18. Na gara jan used degenerate isopar am etric elements, in which i t is assumedthat the transverse normal stresses are negligibly small. This assumption affects the effectivestress patterns which control plastic loading a nd contrib utes to th e different response predictedin his study.

C O N C L U S I O N SThe objective in this paper was to review, derive and evaluate finite element formulations forgeneral non-linear sta tic and dyna mic analysis which have been implemented in the search forthe m os t effective p r ~ c e d u r e . ~he formulations have been derived from general principles ofcontin uum mechanics an d include material, large displacement a nd large strain non-linearities.Th e concep tual difference between the form ulations is the reference configuration tha t is usedfor the linearization of the incremental equ ations of motio n. In th e T.L. formulation the initialconfiguration is used as reference, where as in the U .L. formulations, th e reference configurationcorresponds to the last calculated configuration.


29/34

- 0 0 6

- 0 0 4

- 0 02

- 0E-9z+uw-1LLW0 0 0 4

0 002

0 O f

0 Of

0 IC

LARGE DEFORMATION DYNAMIC ANALYSISh = 0 4 1 , n ~ ' 6 0 0blin'

E = 10 5 lo6 lb/in2

TE N & N O D EE L E ME N T S FO R I b - s e c z / i n 4AXISYMMETRICA N A L Y S I S

P = 2 4 5

Tf = 0 5 5 x 1 0 3 s e cAt = 0 5 x lO-'sec

L I N E A R STATIC

38 1

Figure 17. Large displacement dynamic elastic-plastic analysis of spherical cap. Newmark method. 6 = 0.50. 2 = 0.15

A first important observation is that provided the constitutive tensors are defined appropri-ately, all formulations give the same numerical results. The only advantage of using one formula-tion rather than the others is its better numerical effectiveness. Theory and sample analysesshow that in small strain but large displacement analyses the differences which arise by using thesame material constants in the formulations-because, for instance, a clear definition of theconstants may not be avai lab le-can be expected to be small.With regard to the numerical operations required, Table I shows that all matrices of theformulations have corresponding patterns of zero elements, except for the linear strain-displace-ment transformation matrices. In the T.L. formulation, this matrix is full, because of the initialdisplacement effect in the linear strain terms. Therefore, the calculation of the element matricesrequires less time in the U.L. formulations.An advantage of the T.L. formulation is that the derivatives of the interpolation functionsare with respect to the initial configuration, and therefore need only be formed once, if they arestored on back-up storage for use in all load steps. However, in practice, the use of tape or disc tostore and retrieve the required derivatives in each step may be more costly than simply to re-calculate them, and, in particular, the required storage is a problem size governing factor sincesaturation of back-up storage may be reached. Auxiliary storage considerations are particularlyimportant, if a considerable amount of stress and strain history need to be stored already.It should be noted that the U.L. formulations are quite different from the moving co-ordinateformulation presented in the survey paper by Stricklin and many other^,"^ and the updated


30/34

382 K.-J. BATHE, E. RAMM AND E. L . WILSONh 1 0 4 1 , n p = 6 0 0 b / m 2

T E N 8 - NODEE L E M E N T S FORAXISYMMETRICANALYSIS \ R - 2 2 2 7 1 1 1A 1, = 0 5 Id secv 2 : 667E = 1 0 5 x 1 0 6 1 b / ~ n 2V = 0 3Uy: 24x10 lb/ in2c, : 21x lo 6 l b / m 2p = 2 4 5 ~ 1 0 - ~I b - s e c 2 / i n 40 I I I I I ct/@.t,,40 80 120 160 200

so 0 0 4z0 0 0 5VWJ 006L LWD 0 07

O o 8 C09

T L S O LU TI ONA t = A t ,S = 5 0, ~ 1 . 0 2 5

WITH E O U I L I B R I U M i T ER A T l O N S

Figure 18. Large displacement dynamic elastic-plastic analysis of spherical cap, Newmark method

formulation, which was used in the comparative stud y by Dup uis and many others. Th e incre-mental moving co-ordinate formulation surveyed by Stricklin and many other^,^*,'^,^^ wasstated to be restricted to small strains and have distinct computational disadvantages. Theseconclusions do not apply to the U.L. formulations used here. The updated formulation employedby Dupuis an d many others, in their com parative study of this formulation versus a Lagrang ianformulation did not give satisfactory results. However, using the U.L. form ulation with iso-parametric elements as presented in this paper, the results are as goo d a s those obtained usingthe T.L. formulation. The only erro rs are due to the numerical solution of the governing co n-tinuum mechanics equations.In general, using both the T.L. and the U.L. formulations equilibrium iterations should beperformed in order to ensure an accurate solution and possibly dispense with the calculation of anew non-linear stiffness matrix in each load step. If n o equilibrium iterations are carried out, th elinearization can introduce uncontrolled large errors. In the elastic and hyperelastic analysespresented here, it was possible t o calculate th e stresses in the configuration a t time t + A t directlyfrom the corresponding total strains. Therefore, the non-linear finite element eq uatio ns have beensolved exactly within the assum ptions of the tim e integration scheme an d th e convergence limitof the iteration. In pa th depend ent problems this is not possible an d to tal stresses are calculatedby addin g increments in stresses.An important consideration in path depend ent prob lems is the definition and calculation ofthe constitutive tensors, which depends on the stress an d s train history. A great deal of add ition alresearch is still required to identify various materials. Using the T.L. formulation, the effort toimplement a no n-linear constitutive relation can , in som e cases, be less tha n in the U .L. formu la-tion ;however, the appr opr iate material constants need be available. It is the ease of imp lementing


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LARGE DEFORMATION DYNAMIC ANALYSIS 38 3a new material law in a general non-linear analysis program that makes the T.L. formulationvery attractiv e.In conclusion, both the U.L. o r T.L. formulations can effectively be used in a general non-linear analysis program. It depends largely on the program design and the material constantsavailable which fo rmu lation is most effective.

A P P E N D I XN o ationThe following convention for tensor an d vector subscripts and superscripts is employed :

A left superscript denotes the time of the configuration in which the qua ntity occu rs.A left subscript can have tw o different meanings. If the qua nti ty considered is a derivative, theleft subscript deno tes the time of the configu ration, in w hich the co -or din ate is measured withrespect to w hich is differentiated. O therw ise the left subscrip t den ote s the tim e of the configura-tion in which the quantity is measured.Right lower case subscripts den ote the com pon ents of a tensor o r vector. Co mp one nts arereferred to a fixed Cartesian co-ordinate system; i,j, . . . = 1, 2, 3. Differentiation is denotedby a right lower case subscript following a co mm a, with the subscript indicating the co -ord inatewith respect to which is differentiated.

' A , ' A , +A'A = Area of body in configuration at time 0, t , t +A tdCi j rs ,Ci j rs Component of constitutive tensor at time t referred to configuration at' C i j r s ,i j r s = Com ponent of tangent constitutive tensor at time t referred to configuration

'+"dfi,::2:f, = Component of body force vector per unit mass in configuration at timeh, = Finite element interpolation function associated with nodal point k(i ) = Superscript indicating number of iteration

time 0 , tat t ime 0, tt +A t referred t o configuration at time 0, t +A t

On,, 'n i, r+"'n i = Com ponent of surface norm al in configuration at time 0 , t, t +A tf + A f p= Pressure load at time t + A tt + A f W= External virtual work expression corresponding to configuration at time

t +At, defined in equatio n (8)r , s = Na tura l element co-ordinatesisij, = Com pone nt of 2nd Piola-Kirchhoff stress tensor in config uration at timer+A:Sij Com pon ent of 2nd Piola-Kirchhoff stress tensor in config uration at timet, t +At referred t o configuration at time 0

t + A t referred t o configuration a t time t'Sij, J i j = Com pone nt of 2nd Piola-Kirchhoff stress increment at time tt, t +At = time t and t +A t, before and after time increment A t

referred t o configuration a t time 0, t +A tfiguration at tim e t , t+A t

'+Adti,:12:ti= Component of surface traction vector in configuration at time t + A t ,ui = Co m pon ent of displacement vector from initial position a t time 0 to con-ui = Increment in displacement compo nent, ui = r + A ' ~ i - ' ~ i

' u f = Displacement component of nodal point k in configuration at time tuf = Increment in ' u f

f r + A fui ,


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384 K . -J . BATHE, E. RAMM AND E. L . WILSONdui,j, = Derivative of displacement component to configuration at time t , t +At withrespect to co-ordinate Oxj

x j , x jou i , j , l + A r ~ i , j= Derivative of displacement increment with respect to co-ordinate 0xj ,I ( + A t

OV, L,+AL = Volume of body in configuration at time 0, t , t + A tI + A IOxi, x i,Ox:, x:, + A 1 ~ : = Cartesian co-ordinate of nodal point k in configuration at time 0, t , t +Atx i = Cartesian co-ordinate in configuration at time 0, t , t +At= Derivative of co-ordinate in configuration at time 0, t with respect toco-ordinate Ixj, O x j6 = Denoting variation ineij = Component of Almansi strain tensor in configuration at time t +A t , t , re-ferred to configuration at time 0

t +A t , t , referred to configuration at time 0= Component of Green-Lagrange strain tensor in configuration at time t + A t ,referred to configuration at time t (i.e. using displacements from the con-figuration at time t to the configuration at time t + A t )= Component of strain increment tensor (Green-Lagrange) referred toconfiguration at time 0, t

I + A I ~ i j , eij = Component of Green-Lagrange strain tensor in configuration at time

o ~ i j ,

oeij, eij = Linear part of strain incrementosij, ei joylij. = Non-linear part of strain incrementoeij,

p , Ip, = Specific mass of body in configuration at time 0, t , t +A tI s i j , r+As i jComponent of Cauchy stress tensor in configuration at time t , t + Atz: = Component of Jaumann stress rate tensor in configuration at time t

Inpi= Component of spin tensor in configuration at time t0

Mutr ice sdBL, B, = Linear strain-displacement matrix in configuration at time t referred to

AB,,, iBNL= Non-linear strain-displacement matrix in configuration at time t referred tooC,lC Tangent material property matrix at time t and referred to configuration

iF , :F = Vector of nodal point forces in configuration at time t (referred to configura-dKL,:KL = Linear strain stiffness matrix in configuration at time t (referred to configura-

i K N L , f K N L Non-linear strain stiffness matrix in configuration at time t (referred to

configuration at time 0, tconfiguration at time 0, tat time 0, ttion at time 0, t )tion at time 0, t )configuration at time 0, t )M = Mass matrix

r + A r R Vector of external loads in configuration at time t + A t$3, ,$ = 2nd Piola-Kirchhoff stress matrix and vector in configuration at time t andreferred to configuration at time 0T, z^ = Cauchy stress matrix and vector in configuration at time t

u, +%I= Vector of displacements at time t , t +A tu = Vector of increhental displacements at time t


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LARGE DEFORMATION DYNAMIC ANALYSIS 385REFERENCES

I . J . H. Argyris, P. C. Dunne and T. Angelopoulos, Nonlinear oscillations using the finite element technique, C o m -puter Meth. Ap pl . Mech. Engng, 2,203-250 (1973).2. K. J. Bathe and E. L. Wilson, NONSAP-A general finite element program for nonlinear dynamic analysis ofcomplex structures, Paper M3/1, Proc. 2nd Int . Conf. Stru ct. Mec h. Reactor Technology, Berlin (1973).3. K . J . Bathe, E. L. Wilson and R. H. Iding, NONSAP-A structural analysis program for static and dynamicresponse of nonlinear systems, SESM Report No. 74-3, Dept. of Civ. Engng, Univ. of California, Berkeley (1974).4. K. J. Bathe, H. Ozdemir and E. L. Wilson, Static and dynamic geometric and material nonlinear analysis, SESMReport No. 744, Dept. Civ. Engng, Univ. of California, Berkeley (1974).5 . K . J. Bathe and E. L. Wilson, Stability and accuracy analysis of direct integration methods, Int . J . Earthq. EngngStruc t . Dyn. , 1, 283-291 (1973).6. T. Belytschko and B. J. Hsieh, Nonlinear transient analysis of shells and solids of revolution by convected elements,AIAA p a p er No. 73-359, AIAA IASM EISA E 14th Struc tures, Struc t . Dyn. Materia ls Conf. , Williamsburg, Virginia(1973).7. G .A. Dupuis, H. D. Hibbitt, S . F. McNamara and P. V. Marcal, Nonlinear material and geometric behavior ofshell structures, Computers Struct., 1, 223-239 (1971).8. I . Farhoomand, Nonlinear dynamic stress analysis of two-dimensional solids, Ph .D . Dissertation, Univ. of Cali-fornia, Berkeley, 1970.9. C. A. Felippa, Refined finite element analysis of linear and nonlinear two-dimensional structures, SESM Re p o r tNo. 66-22, Dept. Civ. Engng, Univ. of California, Berkeley (1966).

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Bathe (1975) - FE Formulations for Large Deformation Dynamic Analysis

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