dr. cw engmeer, f 0 boa; j2

Elastoplastic dynamic analysis by the

DR-BEM in modal co-ordinates

D.P.N. Kontoni

Dr. CW Engmeer, f 0 Boa; J2<S7, GTZ-g^^O f afra Greece

ABSTRACT

A dynamic analysis of elastoplastic problems by the Dual Reciprocity BoundaryElement Method (DR-BEM), incorporating the modal co-ordinatetransformation technique, is presented. The proposed methodology combines theadvantages of the DR-BEM with the efficiency of the modal superpositiontechnique. The DR-BEM utilizes the simple elastostatic fundamental solutionand reduces the number of unknowns only to the boundary, causing aconsiderable reduction in the size of the problem. Interior cells, which arerequired to take care of the elastoplasticiry, can be restricted only to thoseportions of the domain expected to become elastoplastic. The use of the modalsuperposition technique, which is more economical than the direct timeintegration solution, is investigated. Numerical results are presented to illustratethe use and check the accuracy of the present methodology.

INTRODUCTION

The Boundary Element Method (BEM) has been employed for the analysis ofmaterially nonlinear problems under static and quasi-static loadings [1,2] andrecently under transient dynamic loadings [3-15]. Ahmad [3] and Ahmad andBanerjee [4] treated the dynamic elastoplastic problem in incremental form bythe direct BEM of the initial stress type in conjunction with the elastodynamicfundamental solution. Kontoni and Beskos [5-8] presented general and completedirect BEM formulations both in current and incremental form, based on theinitial strain or stress approach, for the transient dynamic inelastic problemutilizing the elastodynamic [5,6,8] or the elastostatic [7,8] fundamental solutionand provided solution procedures for the resulting boundary element methods.Burczynski and Adamczyk [9] and Panzeca et al [10] have also presentedboundary integral formulations for dynamic elastoplastic analysis employing theelastodynamic fundamental solution. The use of the elastodynamic fundamentalsolution eliminates the inertial term and makes possible the handling ofproblems with infinite or semi-infinite domains. However, the computational

Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

192 Boundary Element Technology

cost is very high due to the complicated tensors involved. The use of theelastostatic fundamental solution creates an inertial domain integral in theformulation, but the simplicity of the static fundamental solution leads tocomputational advantages. However, this choice restricts the method to finitedomains. The presence of domain integrals due to inelastic and inertial terms inthis formulation requires an interior discretization of the domain in addition tothe boundary one and the resulting method is called by Kontoni and Beskos [7]Domain Boundary Element Method (D-BEM). Carrer and Telles [11] furtherdeveloped and employed the Domain Boundary Element Method to solvesuccessfully dynamic elastoplastic problems.

The inertial domain integral can be further transformed into boundaryintegrals by approximating the accelerations within the domain and the resultingintegral equation forms the basis for "the Dual Reciprocity Boundary ElementMethod (DR-BEM) for the dynamic analysis of inelastic problems" which waspresented by Kontoni [12-14] and Kontoni and Beskos [13,15]. The DR-BEMutilizes the simple elastostatic fundamental solution, gives accurate results andis computationally efficient because the number of unknowns is reduced only tothe boundary. Interior cells, in addition to boundary elements, are required totake care of the inelastic integrals. However, these cells can be restricted only tothose portions of the domain expected to become inelastic. Moreover, thenumber of unknowns in the resultant algebraic systems depends only on theboundary discretization, resulting in a considerable reduction in the size of theproblem. In References 12-15 the direct time integration method was utilized inthe solution procedure.

In the present work the dynamic analysis of elastoplastic problems by theDual Reciprocity Boundary Element Method (DR-BEM), incorporating themodal co-ordinate transformation technique, is presented. The proposedmethodology combines the advantages of the DR-BEM with the efficiency of themodal superposition technique. The use of the modal superposition technique,which is more economical than the direct time integration solution, isinvestigated. Numerical results are presented to illustrate the use and check theaccuracy of the present methodology.

BOUNDARY INTEGRAL FORMULATION

Consider an inelastic body of volume Q enclosed by a surface F whichexperiences a motion under the influence of dynamic loading, within the contextof small displacement and small strain theory. When the elastostatic (Kelvin's)fundamental solution is adopted, the reciprocal theorem or the weighted residualstatement with the aid of the equilibrium, kinematical and constitutive equationsprovides the integral equations of the problem [7,12-15]. Following Nardini andBrebbia [16,17] the inertial domain integral in these integral equations can befurther transformed into boundary integrals by approximating the accelerationsu« within the domain. Thus, the accelerations u-(x,t) can be expressed by a sum

i 1 1of m co-ordinate functions f (x) multiplied by a- (t), where a- (t) are unknown

time dependent functions, i.e.,


Boundary Element Technology 193

with summation on k = 1 to m implied. The inertial domain integral can bereduced to equivalent boundary integrals, using the usual reciprocity relationshipand the integral equations of the problem can take the form [7,12-15]

o r

/y/x,s)n = or o

(initial stress approach)

(2)

with c-{E) = ( 1/2)6-- whenever the F boundary is smooth, while 6-- is the

Kronecker's delta. Here and x are points in the domain, while capital lettersE and X denote points on the boundary F. In the above equations, u- and bj arethe components of the displacement and body force vectors, respectively, p- are

the components of the traction vector, oVr are the components of the initial

stress tensor and o is the density of the body. Expressions for Kelvin'sfundamental solution U-- and the associated tensors E-j.; and T-- in two- and

v **J vk

three- dimensions can be found elsewhere [1,2]. Furthermore, y^ is thek

displacement solution and TJ ., are the corresponding tractions of the "pseudo-state" elastostatic problem

in an unbounded domain. In References 12-18 and in this work a simple class ofco-ordinate functions f (X) = R(A ,X) where R is the distance from the pointA% where the function is applied to a field point X, was successfully employed.

The above integral expression (2) forms the basis for the "Dual ReciprocityBoundary Element Method (DR-BEM) for the dynamic analysis of inelasticproblems" [12-15] for which a numerical solution procedure can be derived.

Stresses at interior points can be determined by the DR-BEM using theproper integral equations in a pointwise fashion [14,15]. The stresses at boundarypoints can be directly calculated from the constitutive equation, the boundarytractions and the directional derivatives of the boundary displacements withoutinvolving any integration.



SOLUTION PROCEDURE

In order to solve the previously derived integral equations numerically theboundary of the domain must be discretized into a number of boundary elementsand that part of the domain where inelastic behaviour is expected needs to bediscretized into interior cells. The integral equation (2) under the assumption ofzero body forces, and after the spatial discretizations and the pertinentintegrations, can be assembled into the form

[H\(U(t)} - (G](P(t)} + Q([H\M - [G]M )%(!)} - [Q]fo = 0

Applying equation (1) to every nodal point and choosing a linearly independentset of functions, their number being equal to the number of nodal points, onereceives

(U(t)} = [F]{a(t)} and (a(f)} = [E\(U(t)} &

Using this substitution for (a(t)}, equation (4) yields

[M] {#(;)}

where [H], [G], and [Q] are the usual matrices of static inelasticity and the massmatrix [M] is given by

[M] -p([H]M - [G]M)[E] (7)

Taking into account the boundary conditions of the problem ,i.e.,

uy = i/y (prescribed) on F^ , py = py (prescribed) on 1"^ ; F = Fj U

equation (6) can be written as [12]

where (x(t)} is the vector of the unknown nodal boundary tractions anddisplacements. In order to be able to obtain the solution for displacements andtractions, a distinction between the two types of boundaries has to be made.The variables at F-. will be denoted by the subscript 1 and at F^ by 2.Rearranging and partitioning the global matrices into the correspondingsubmatrices yields [12]

t/2(f) + 222 #2(0 = "2(0

where Ugft) is the vector of the unknown nodal boundary displacements.(In the absence of support excitations one has Uj(t) = U-j(t) = 0 )



In References 12-15 equation (10) was solved by a direct time integrationmethod. The Houbolt time integration method was found to be the most suitablebecause it introduces artificial (numerical) damping for the high frequencies ofthe problem, which is equivalent to truncating the influence of higher modes inthe response, (this fact has proved to be quite helpful to the DR-BEM [12-18]),and approximates the acceleration components in terms of the displacementcomponents by using a four-point backward-difference formula. The followingguideline was found [13-15] to be suitable for the selection of the time step : At<. 2 T /100 , where T, is the fundamental period for linear undamped vibration.

Nardini and Brebbia [18] decoupled the system response into independentmodes and solved the transient elastodynamic problem using modalsuperposition.

In the present work, the main objective is to extend the modal co-ordinatetransformation technique, known as the modal superposition method in linearanalysis, to solve inelastic dynamic problems. The formulation presented hererequires only a single evaluation of the modal spectrum of the initial elastic stateof the structure.

The evaluation of the natural frequencies and modes of the initial elasticstate of the problem can be deduced from equation (10) by settingand the external influences to zero, obtaining

[/2W + Z22 #2(f) = 0

This is the case of the free vibration problem in which damping is neglected. Thesolution to equation (11) can be postulated to be of the form

-f) (12)

where $~ is a vector of order N , where N^ is the order of vector L t), t is thetime variable, t a time constant and co a constant identified to represent thefrequency of vibration (rad/sec) of the vector <J>2 . Substituting equation (12)into equation (11), the generalized eigenproblem is obtained which can befurther reduced to a standard one, as follows

/ / _ 12 2 2 ' ^

The eigenproblem in (13) yields the N eigensolutions (co (j.), i= 1,2,...,N.The vector Q*' called the ith-mode shape vector and co^ is the correspondingfrequency of vibration (rad/sec). It should be emphasized that equation (11) issatisfied using any of the N^ displacement solutions 4^ sinco t-t ), i = 1,2,...,N.It is important to point out that matrix %22 is non-symmetric and therefore,some of its eigenvalues will be complex. Consequently, care should be taken inthe choice of the appropriate eigenvalue solution algorithm. The non-symmetricmatrix %22 can be transformed into Hessenberg form and the eigenvalues of thetransformed matrix can be calculated by the Q-R algorithm [17-19]. Alternatively,



Kontoni, Partridge and Brebbia [20] presented an eigenvalue solution procedurewhich avoids the complex eigenvalues associated with the non-symmetric BEMmatrix and which is at the same time very easy to implement. In this work, theeigenvalues are calculated by the above Q-R algorithm, while the eigenvectorsare determined in a way similar to that proposed by Kontoni et al [20].

The U2(t) can be represented in the basis spanned by the eigenvectors as

where X-(t) are called modal contributions and are yet unknown functions oftime, are the modal shapes and p is the number of modes considered. The"approximately equal sign" in equation (14) expresses the fact that anapproximation to the solution of equation (10) is obtained because p is less thanthe number of the unknown nodal boundary displacements of the problem. Itshould be noted that it is a usual practice to employ only the first p eigenmodesas in many practical problems high frequencies contribute insignificantly in theoverall response. However, care should be exercised to include a sufficientnumber of modal components in the analysis, otherwise poor results may beobtained. Substituting equation (14) into equation (10), one obtains

with F(t) representing the right-hand side of equation (10).

Since matrix %22 is non-symmetric, one has to work with two bases [18],one corresponding to the original eigenvalue problem (13) and the othercorresponding to its adjoint (transposed) problem, i.e.

T -with %22 being the transpose of %22 and $2 the alternative basis. The set of

eigenvalues %/ is the same for both the eigenvalue problems (13) and (16), whileonly the eigenvectors are different.

Following Nardini and Brebbia [18] and pre-multiplying expression (15) by

Tvector <j>2 • of the dual basis, the system of equations (15) can be decoupled, due

to the orhogonality of $2 and $2 with respect to matrix %22 and the identitymatrix, into the form

/ T

or



2i 2i 2f *2i

(i = 1, 2,-,p)

which represents p independent one-degree-of-freedom systems that can besolved by a time integration algorithm. In this work, the Houbolt timeintegration method was found to be the most suitable time integration algorithm.

The solution to the original problem, namely L t), is then achieved usingequation (14). Thus, the required computations in solving equation (10) can bereduced significantly using the concepts of modal superposition.

Certainly, one should recognize that the actual (instantaneous) frequenciesand vibration mode shapes are continuously changing during a nonlinearresponse. Specifically, the elastoplasticity will cause variations in the naturalstructural frequency spectrum. However, the methodology employed in this workit is effective because employs only the initial elastic eigenvalues andeigenvectors throughout the analysis. The non-linearity in material behaviourenters in the term |L)(t)}, which need to be calculated in each time step. ( Theresponse must be transferred from the generalized coordinates to the naturalcoordinates in each time step in order to evaluate the inelastic term.)

Although the above analysis procedure it is usually called "modalsuperposition", it is more appropriate to look at the method simply as aco-ordinate transformation. The basic step in this solution is a change of basisfrom the N unknown nodal boundary displacements to p modal generalizeddisplacements, where p is much less than N^, prior to the step-by-step solution.Naturally, the efficiency of the modal superposition method becomes moreevident when it is applied to large structural systems.

In elastic analysis, p depends on the structure considered and on the spatialdistribution and frequency content of the loading. In elastoplastic analysis,however, p may have to be considerably larger than in elastic analysis. Otherwise,if p is too small, the response ttyt) cannot be represented adequately by theselected p eigenvectors c^ . Since the accuracy of the predicted responsedepends entirely on the quality of the transformation vectors used, goodboundary element discretization should be utilized in order to calculate theeigenvalues and eigenvectors with the necessary accuracy. The advantage ofmodal superposition is essentially that since frequencies and mode shapes havebeen obtained, a variety of response history analyses can be carried out withrelatively small additional cost. Moreover, in the case of inelastic analysis, theinelastic material constants ( e.g., a ) as well as the interior cell discretizationcan be easily modified and the analysis can be repeated with relatively small cost.



The unknown tractions are determined at each time step from equation (9)and thus, the vector (x(t)} can be used for the calculation of the stresses at thattime step. Stresses can be calculated at each time step as follows [13-15]

where vector |n(t)} represents the elastic dynamic stress component and matrix[S ] reduces to matrix [S] of static inelasticity for density g = 0.

Since the initial stress distribution in the interior cells is not known a priori,the Standard Initial Stress (SIS) technique [1,11] for inviscid plasticity has beenemployed in this work for the calculation of (aW(t)} at each time step. Thus,equation (10) for direct integration or eqn (18) for modal superposition is solvedtwice at each time step, first with (L t)} = { 0 } and next with the calculated{L2)(t)}. (The latter solution's values are used for the time marching.)

NUMERICAL EXAMPLE

Simply-supported deep beamConsider a simply supported deep beam under plane stress conditions with anelastic perfectly plastic material obeying the Von Mises yield criterion which issubmitted to a suddenly applied uniform loading p(t) = p H(t-O) as shown inFigure 1. The numerical data are L = 24. , h = 6. , E = 100. , v = 0.333 , g = 1.50, a = 0.16 and p = 0.75 p^ = 0.015 where p^ = 2 a h /L is the staticcollapse load. This problem was analysed by the proposed DR-BEM whichutilizes the Modal Superposition technique (DR-BEM & M.S.). In order to verifythe results obtained by the present DR-BEM in Modal Co-ordinates, thisproblem was also analysed by the DR-BEM which uses Direct Integration(DR-BEM & D.I.) and by the FEM program NONSAP [21]. Figure 2 shows thediscretization utilized by the DR-BEM, using linear boundary elements andlinear triangular interior cells. It should be noted that no internal points areincluded as additional collocation points (additional degrees of freedom) and thenumber of the DR-BEM collocation points is equal to the number of boundarynodes. The FEM-discretizations utilized (employing 4-node and 8-nodeisoparametric elements) are shown in Figure 3. Figure 4 depicts the time historyof the vertical displacement of point A (L/2, h/2) as obtained by the DR-BEMand Modal Superposition (DR-BEM & M.S.), including p = 1, 2,..., 14 initialelastic modes in the analysis. It may be seen that the best elastoplastic results areobtained when 9 p £ 14. Figure 5 portrays the time history of the verticaldisplacement of point A (L/2, h/2) as obtained by the DR-BEM with ModalSuperposition (including p = 9 modes and p=14 modes), the DR-BEM withDirect Integration and the FEM with Direct Integration. The elastoplastic resultsobtained by the DR-BEM & M.S. with p= 14 modes are within plotting accuracywith the DR-BEM & D.I. results. The DR-BEM analyses were performed by theHoubolt time integration scheme and for the elastoplastic analyses the StandardInitial Stress (SIS) technique was employed. The FEM analyses were performedby the Wilson 9 time integration scheme. The time step used by the DR-BEMwas At = 0.56 , while the FEM elastoplastic analysis required smaller time steps,i.e., At = 0.28 and At = 0.14 were used for FEM-meshes a and b respectively. (Itis worth mentioning that the elastic dynamic analysis by both versions of theDR-BEM is obtained without interior discretization into cells.)



Th

i .

P

, A

L z:

L

p(t) a

O

t

Figure 1 : Geometry, loading and material description of the simply supporteddeep beam.

AA

A

A

A

<]i A

L/

Figure 2 : Boundary element and interior cell discretization of the beam.

i

1 / 2

mesh a(with 4-node elements)

mesh b(with 8-node elements)

Figure 3 : Finite Element discretizations of the simply supported deep beam.



ELASTIC S INELASTIC DYNAMIC ANALYSIS OF A SIMPLY SUPPORTED BEAM0.000.

number of nodes included :1,2, 3. 4, 5. 6.7,8, a 10,11.12.13.14

14 modes, dt=0.56 ] TIME

Figure 4 : Elastic and elastoplastic dynamic vertical displacement of point A ofthe simply supported deep beam by the DR-BEM in Modal Co-ordinates.

ELASTIC & INELASTIC DYNAMIC ANALYSIS OF A SIMPLY SUPPORTED BEAM0.010 ,

oCL

zLU.LUCJ_o_C/D

CELU

§ §18 8TIME

[ DR-BEM S M.S. : 9 Modes, 14 Modes, dt=0.56 3[ FEM (mesh a. dt=0.2B). .. . . . . FEM (mesh b. dt=0.14) ]

[ DR-BEM S D.I.

Figure 5 : Elastic and elastoplastic dynamic vertical displacement of point A ofthe simply supported deep beam by the DR-BEM and the FEM.



CONCLUSIONS

The Dual Reciprocity Boundary Element Method (DR-BEM) for the transientdynamic analysis of inelastic structures, incorporating the modal co-ordinatetransformation technique, is presented and applied to elastoplastic dynamicproblems. The proposed methodology combines the advantages of the DR-BEMwith the efficiency of the modal superposition technique. The DR-BEM utilizesthe simple elastostatic fundamental solution and reduces the number ofunknowns only to the boundary, causing a considerable reduction in the size ofthe problem. Interior cells, which are required to take care of the elastoplasticity,can be restricted only to those portions of the domain expected to becomeelastoplastic. The modal superposition technique is an efficient alternative to thedirect time integration solution. A solution procedure for the present DR-BEMis provided. The numerical results presented indicate that the DR-BEM inconjunction with the modal co-ordinate transformation method is capable ofpredicting the response an elastoplastic structure subjected to dynamic loadings.

REFERENCES

1. Telles, J.C.F. The Boundary Element Method Applied to InelasticProblems, Lecture Notes in Engineering, Vol. 1, Springer-Verlag, Berlin,1983,

2. Mukherjee, S. Boundary Element Methods in Creep and Fracture, AppliedScience Publishers, London, 1982.

3. Ahmad, S. Linear and Nonlinear Dynamic Analysis by Boundary ElementMethod, Ph.D. Thesis, State University of New York at Buffalo, 1986.

4. Ahmad, S. and Banerjee, P.K. inelastic Transient Dynamic Analysis ofThree- Dimensional Problems by BEM' Int. J. Num. Meth. Engng, Vol. 29,pp. 371-390, 1990.

5. Kontoni, D.P.N. and Beskos, D.E. inelastic Dynamic Analysis by theBoundary Element Method', in Boundary Elements IX (Ed. Brebbia, C.A.,Wendland, W.L. and Kuhn, G.), Vol. 2, pp. 335-351, Proceedings of the 9thInternational Conference on Boundary Element Methods in Engineering,Stuttgart, Germany, 1987. Springer-Verlag, Berlin, 1987.

6. Kontoni, D.P.N. and Beskos, D.E. 'Boundary Element Formulation forDynamic Analysis of Nonlinear Systems' Engineering Analysis, Vol. 5, No.3, pp. 114-125, 1988.

7. Kontoni, D.P.N. and Beskos, D.E. 'BEM Dynamic Analysis of MateriallyNonlinear Problems', in Boundary Elements X (Ed. Brebbia, C.A.), Vol. 3.,pp. 119-132, Proceedings of the 10th International Conference onBoundary Element Methods in Engineering, Southampton, England, 1988.Springer- Verlag, Berlin, 1988.

8. Kontoni, D.P.N. and Beskos, D.E. 'Dynamic Response of Nonlinear Systemsby Boundary Element Methods', Vol. 1, pp. 175-184, Proceedings of the 2ndNational Congress of Mechanics, Athens, Greece, 1989.

9. Burczynski, T. and Adamczyk, T. 'Analysis of Nonlinear Systems in terms ofthe Boundary Element Method', Mechanics and Computer, Vol. 7, pp. 149-164, 1988.



10. Panzeca, T., Polizzotto, C. and Zito, M. 'A BEM Formulation of theDynamic Elastic-Plastic Structural Problem via Variational Principle', inBoundary Elements in Mechanical and Electrical Engineering (Ed. C.A.Brebbia and A. Chaudouet-Miranda), pp. 193-204, Proceedings of theInternational Boundary Element Symposium, Nice, France, 1990. CMP andSpringer-Verlag, Southampton and Berlin, 1990.

11. Carrer, J.A.M. and Telles, J.C.F. Transient Dynamic Elastoplastic Analysisby the Boundary Element Method', in Boundary Element Technology VI(Ed. Brebbia, C.A.), pp. 265-277, Proceedings of the 6th InternationalConference on Boundary Element Technology, Southampton, England,1991. CMP and Elsevier, Southampton and London, 1991.

12. Kontoni, D.P.N. The Dual Reciprocity Boundary Element Method for theTransient Dynamic Analysis of Elastoplastic Problems', in BoundaryElement Technology VII (Ed. Brebbia, C.A. and Ingber, M.S.), pp. 653-669,Proceedings of the 7th International Conference on Boundary ElementTechnology, Albuquerque, U.S.A.,1992. CMP (Computational MechanicsPublications) and Elsevier, Southampton and London, 1992.

13. Kontoni, D.P.N. and Beskos, D.E. 'Applications of the DR-BEM in InelasticDynamic Problems', in Boundary Elements XIV (Ed. Brebbia, C.A.,Dominguez, J. and Paris, F.), Vol. 2, pp. 259-273, Proceedings of the 14thInternational Conference on Boundary Element Methods - BEM 14,Seville, Spain, 1992. CMP and Elsevier, Southampton and London, 1992.

14. Kontoni, D.P.N. Dynamic Elastoplastic Analysis by the Boundary ElementMethod, Doctoral Dissertation, University of Patras, Patras, Greece, 1992(in Greek).

15. Kontoni, D.P.N. and Beskos, D.E. Transient Dynamic Elastoplastic Analysisby the Dual-Reciprocity BEM', Engineering Analysis with BoundaryElements, 1993. (in press)

16. Nardini, D. and Brebbia, C.A. Transient Dynamic Analysis by the BoundaryElement Method', in Boundary Elements (Ed. Brebbia, C.A., Futagami, T.and Tanaka M.), pp. 719-730, Proceedings of the Fifth InternationalConference, Hiroshima, Japan, 1983. Springer-Verlag, Berlin, 1983.

17. Nardini, D. and Brebbia, C.A. 'Boundary Integral Formulation of MassMatrices for Dynamic Analysis' Chapter 7, Topics in Boundary ElementResearch, Ed. Brebbia, C.A., Vol 2, pp. 191-208, Springer-Verlag, Berlin andNew York, 1985.

18. Nardini, D. and Brebbia, C.A. Transient Boundary Element ElastodynamicsUsing the Dual Reciprocity Method and Modal Superposition', in BoundaryElements VIII (Ed. Tanaka, M. and Brebbia, C.A.), Vol. 1, pp. 435-443,Proceedings of the 8th International Conference on Boundary Elements,Tokyo, Japan, 1986. Springer-Verlag, Berlin, 1986.

19. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. NumericalRecipes - The Art of Scientific Computing, Cambridge University Press,Cambridge and New York, 1990.

20. Kontoni, D.P.N., Partridge, P.W. and Brebbia, C.A. 'The Dual ReciprocityBoundary Element Method for the Eigenvalue Analysis of HelmholtzProblems', Advances in Engineering Software and Workstations, Vol.13(1), pp. 2-16, 1991.

21. Bathe, K.J., Wilson, E.L. and Iding, R. NONSAP - A Structural AnalysisProgram for Static and Dynamic Response of Nonlinear Systems, ReportNo. UCSESM 74-3, University of California, Berkeley, 1974.


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