A general methodology for nonlinear structural analysis by mathematical programming

A general methodology for nonlinear structural analysis by mathematical programming J. A. Teixeira De Freitas Universidade T~cnica de Lisboa, Lisbon, Portugal David L l o y d

Smith

Department of Civil Engineering, Imperial College of Science and Technology, London SW7, UK

A methodology has been developed to produce a unified approach in the treatment of nonlinear structural analysis problems by combining four fundamental ingredients, namely structural discretization, substitution of structures by graphs, static-kinematic duality and mathematical programming. Graph theory is called upon to exhaust the alternative processes through which the building elements, into which the structure is discretized, can be assembled. To implement a governing system featuring symmetry, reciprocity in the constitutive relations and duality in the descriptions of equilibrium and compatibility are artificially preserved. Use is made of mathematical programming theory and algorithms to complement the resulting discrete representation with a variational interpretation and to develop numerically efficient solution procedures. Key words: elastoplasticity, directed graphs, duality, virtual work, complementarity Introduction A survey of the published work on nonlinear structural analysis reveals the existence of a multiplicity of approaches, methods and procedures designed to achieve identical objectives. Two or three decades ago, the situation in linear structural mechanics was similar. Foreseeing the advent of more efficient means of numerical implementation, efforts were then developed to establish formulations leading to a synthetic, general theory. In one such presentation, a four fundamental ingredients for unification are clearly suggested and consistently explored, namely: (1) structural discretization, to incorporate systematic procedures to formulate and solve the problem; (2) substitution of structures by graphs, to exhaust the alternative processes through which the structural building elements can be assembled; (3) static-kinematic duality (SKD) and constitutive reciprocity, to implement a governing system featuring symmetry; (4) mathematical programming, to complement the resulting discrete representation with a variational interpretation.

B2 Eng.Struct., 1984, Vol. 6, January

The present paper reports on the extension 2 of this approach into the domain of large displacement analysis of elastoplastic structures. Basic c o n c e p t s and

methodology

An engineering structure, being an orderly interconnection of substructures formed by building elements into a meaningful whole, is essentially a system. A systems approach to structural mechanics suggests a mathematical model formed by the combination of two independent sets of algebra: one - vectorial - developing from the geometric-mechanical properties and characterizing the behaviour of the constituent elements and of the substructures formed from such elements; the other Boolean - implementing the connectivity properties and thus regulating the procedure for the assemblage of the substructures to form the structure anew. Pursuing such an approach, we first resolve the structure into its simplest elements to which a specific type of connectivity can be associated, the fundamental mesh and nodal substructures suggested in Figure 1. The conditions of equilibrium between the forces applied to the selected

0141-0296/84/01052-09/$03.00 © 1984Butterworth& Co. (Publishers)Ltd

Methodology for structural analysis: J. A. T. De Freitas and D. L. Smith 0

of the structural material. The constitutive relations may then be derived by establishing the causality relations between the stress-resultants X applied to the typical element and the corresponding strain-resultants u, the components of which are shown in Figure 2. Additional, or fictitious, forces and deformations are introduced into the nonlinear elemental relationships in order to enforce artificially the essential features of the linear model, namely, a relationship of duality between the equilibrium and compatibility conditions and in the description of the static and kinematic phases of plasticity, and symmetry in the elastic causality and plastic hardening operators. As suggested in Figure 3, in most of the engineering formulations in finite-element nonlinear structural analysis the theoretical development ceases at the next stage of recovering the structural behaviour by establishing an appropriate procedure to assemble the elemental governing relations; a way is then given for the equally important aspect of searching for a convenient numerical implementation technique. In the approach suggested, the field conditions, as soon as they are derived, are processed through systems equivalence theory in order to generate the associated set of mathematical programs, which are found to represent the variational principles of nonlinear mechanics; the firstprinciples, or vectorial, and the energetic approaches are in this manner synthesized into a unified mathematical formalism which retains the advantages of each of the two complementary approaches. Finally, use is made of system qualification theory to establish conditions for the existence, uniqueness and stability of solutions, and mathematical programming algorithms are adapted to the specific physics of the problem and used to obtain numerical solutions.

El

Nodal substructure

Mesh substructure

Meshand nodal substructures

Figure I

~

L - u~

~,

"~ - ~ u i

a

x~

d~

-"

b Figure 2

--~ ;-"-- l ~

~

Deformable building elements. (a), finite element; (b),

release devices

Identification ~ of variables

Structural ~ idealization

Elemental relationship

f.

J l

Automatic L ~ Enforcement of essential features assemblage J------] of linear model

Structural idealization I I

I ~°verning t ~ system I '

implementation

I

algorithm

KKTequiva'enceH Associated theory math.programs

1 I so,u,,oo I 11 1 I

qualification

interpretation

I

uniqueness, stability

I1

I

principles

Figure 3 Traditional and present approaches. (---- --),

traditional; ( ..... ), present

substructure and the developing stress-resultants are then derived, as well as, independently, the conditions for compatibility between the strain-resultants and the displacements suffered by selected points, the movements of which are sufficient to characterize the rigid-body motion of the substructure. The description of statics and kinematics thus obtained can be exact, as no assumptions concerning the magnitude of the variables involved need be made. Next, the substructure is decomposed into its constituent finite elements which embody the mechanical properties

In the process of substitution of the actual structure by a model, three phases can be distinguished. In the first phase the relevant mechanical and topographical properties are summarized, the external actions defined and release devices used to simulate non-rigid connections linking the structural elements and the deformability of the foundation. When, as in the present case, a discrete mathematical model is to be adopted, the next phase is concerned with discretizing the structure, the objective being to identify, using rigid nodes, the constituent parts wherein stress and strain flow continuously. As the developing stresses and strains define vector fields, it is necessary to refer the graphic model to a global

-el.--

Figure 4 Third phase of structural idealization

Eng. Struct., 1984, Vol. 6, January

53

Methodology for structural analysis: J. A. T. De,Freitas and D. L. Smith

wherein Co denotes the transpose of matrix Co, if supplementary forces rr and deformations uTr:

(3) (4)

rr = QX U,r-- T6ff are added to the loading and deformation fields, respectively. In the above definitions, 4-6 the entries of the operators Q and T are nonlinear functions of the current deformations and rigid-body displacements, the latter being expressed in terms of the additional force displacements 6~r; for planar members, equations (3) and (4) reduce to:

s

m

Figure 5 Nodal and mesh connectivity

LX~Jm and

system of reference and to associate each element with a local system of reference, as shown in F i ~ r e 4 for a framed structure, where an additional rigid member is added to simulate the support offered by the foundation; such is the third phase of structural idealization. The graphic model may now be interpreted as a directed graph, that is a set of oriented branches connecting any two nodes. The graph can be disconnected using two different types of repetitive elements, the fundamental substructures: members limited by nodes - the nodal substructure - and meshes, that is connected subgraphs in every node of which there are incident exactly two members - the mesh substructure. The graph can be rebuilt either by incidence of the members upon the nodes - nodal connectivity - or by incidence of the members upon the meshes - mesh connectivity - as shown in Figure 5. Plate and shell structures can be treated similarly. 3

r ........

ui"/ =/

:_ ........

!__

i -plsL + IlLI

"

LTi

U2,n,~m

- _L/_L_-I

......

Jm

- ......

× ;7 respectively, where s = sin p and c = cos p, all intervening variables being illustrated in Figures 6-8. Summarized in Table I are the variables s and k selected to describe the static and kinematic fields of the chosen substructure; p{q} collects the indeterminate forces {displacements} and v{Y} the corresponding, or dual, displacements {forces}, It listing the displacements associated with the applied loads X.

Statics a n d k i n e m a t i c s The equilibrium and compatibility conditions are first implemented at substructure level by analysing mesh and nodal substructures of arbitrary initial topography, subject to generic static and kinematic boundary conditions. The equilibrium and compatibility conditions at structure level are obtained by processing the elemental state conditions of the mesh {nodal} substructures into which the structure is dissected, through an assemblage procedure which is designed 2 to guarantee equilibrium and compatibility of the stress- and strain-resultant {force and displacement} fields developing in the members {nodes} of the structure common to two or more incident mesh {nodal} substructures, thus satisfying the static and kinematic boundary conditions specific to the structure under analysis. The resulting Lagrangian description is:

Statics

[ :___~__E__o] [ ki]

Kinematics /C0 i - J

[sl]

(la)

s2- = k2

(lb)

The descriptions of equilibrium (equation (1 a)) and compatibility (equation (lb)), explicitly linear and valid for arbitrarily large displacements and deformations, can be expressed in dual form :

Eo C-o =

54

Eng. Struct., 1984, Vol. 6, January

(2)

\ ,4\

= X~

// ~- /

"/

J

~ 8t

/

J

q~

%

Figure 6 Definition of variables

Methodology for structural analysis: J. A. T. De Freitas and D. L. Smith

The entries of the state operators (equation (2)) are constant since they are solely dependent on the initial topography of the substructure. The linear description of statics and kinematics, ~ valid for infinitesimal displacements and deformations, is recovered by setting n and u n to zero and removing from equation (lb) the definition for 5~r, thus rendered irrelevant. Approximating second-order formulations 7-12 can be recovered~ by substituting in equations (1) the series expansion of equations (3) and (4) and neglecting the higher-order terms.

Virtual work Equations (1), consequent upon equation (2), exhibit contragradient or dual transformations; since the dual variables s and k are static and kinematic, the relationship implied in equations (1) and (2) is termed static-kinematic duality (SKD) following Munro. la The principles of virtual displacements and forces, which are often viewed as the axiomatic ingredient in linear structural mechanics, can be preserved in the nonlinear domain if they are interpreted as an energetic representation of SKD. 14 The principle of virtual displacements {forces} equation (5) {(6)}:

Sl~kl :

$2 ~ 2

(5)

(6)

I{1ASl -~- R2A$ 2

X

x;

x

f

/ b

/

u;

J

X

--X~

b

a

I

j-<

b

~p

c

Figure 9 Phases. (a), elastoplastic; (b), elastic; (e), rigid-plastic

is obtained by performing the inner product of the equilibrium {compatibility} condition (1 a) {(1 b)} with the finite increment version of the compatibility {equilibrium} condition (1 b) {(1 a)}. The principle of virtual displacements {forces} equation in linear mechanics is recovered by treating s, Ak{k, As} as infinitesimal in equation (5) {(6)}.

Constitutive relations It is assumed that the materials constituting the structural building elements follow a nonlinear elastic-plastic law relating stress- and strain-resultants. In the Lagrangian description of elasticity in equations (7) matrices F and K are block-diagonal, collecting the flexibility and stiffness matrices of the unas~embled finite elements, which are obtained by solving the elastica extended to account for axial and shear deformability effects: is

Flexibility Stiffness

uE = FX + R x

(7a)

X = Ku/r + Ru

(7b)

Collected in the nonlinear residuals Ru and R x are those terms which attempt to destroy the reciprocity of the elastic causality operators:

Figure 8 Definition of variables

I

(8a)

K -- I(

(8b)

These operators are not necessarilypositivedefinitesince their entriesdepend on the current values taken by the element stabilityfunctions,the familiardefinitionsfor which references 16-21 can be recovered by specialization of the more general relationsgiven in reference 15. The descriptionof the plasticphase in form of equations (9) is based upon Maier's matrix representation;22 however, Maier's piecewise-lineardiscretizationof the yieldingand hardening functions is not necessarily assumed in the descriptionadopted herein. Matrix N{V} collectsas columns the gradientsof the yield functions ~, {flow functions} and I] representsthe interactivework-hardening/softening between these yield modes; the plasticmultipliervector u, liststhe contribution to the plasticcomponent of the strainresultantfield up of each yield mode associatedwith the yield functions @,, and X, are the plasticcapacities(see Figure 9):

Figure 7 Definition of variables

Table

F= F

Operators appearing in equations (1)

Description Mesh Nodal

s~ X Y = 0

k u + un q

s=

{p.~.~} {-x..,x}

k2

~v=O,~.,8} tu+u..~..8}

Eo = C'o

[a a~ ao] tA ~ ~.o]

Eng. Struct., 1 9 8 4 , V o l . 6, January

55

Methodology for structural analysis: J. A. 7-. De Freitas and D. L. Smith

Statics Kinematics YieM rule

[-"

[u*l_[_l_,_l tXJ-Luoa+ [t -x,- ~ R kR:l - J ( 9(ga) b)

--V--0

cb. < 0

(9c)

Association ~ . u , = 0

(9d)

Flow rule

(9e)

u./> 0

Residual R s and Rk include those terms which otherwise would tend to destroy the assumed relationships of reciprocity and duality respectively:

[

3 1~

h

4 5 6 o n

(10a)

V= N

(10b)

between the static and kinematic phases of nonlinear, discrete plasticity; reciprocity and associated flow laws are thus not necessarily implied by the conditions of

09

0.7

o 3

/

I_

'-

/J

BS

I I X(KN)

o. 1

equations (10).

16(mm) I 20

_,_

9 ~4r~

Y2[DI2Yl+ D22y2+M2x-d2]

=

(lib) c)

( l 1 d)

(] le)

0

As illustrated by Figure 10, the governing system can be expressed in four equivalent formats, namely mesh-flexibility, mesh-stiffness, nodal-flexibility and nodal-stiffness, depending on the descriptions adopted for the equilibrium and compatibility conditions - mesh or nodal - and for the elastic causality relations - flexibility or stiffness. If the elastic {plastic} phase of the constitutive relations is deleted, the two {four} formulations of rigid-plastic {elastic} analysis are obtained. 2' 2a,24 In linear structural mechanics, the force (mesh-flexibility) method usually enjoys an advantage over the displacement (nodal-stiffness) method in so far as the size of the formulation and its numerical implementation are concerned; this advantage is not, in general, maintained when large displacements are involved. The force method resumes its competitiveness whenever a first-order nonlinear approximation to the exact static and kinematic description is sufficiently accurate, as in most practical applications, ll

I NOOcll

Stotics oncl kinernotics [ - ~

Mesh

I

Plosticity

I Stiffness

Elesticity

Figure lO Equivalent formats

56

Eng. Struct., 1984, Vol. 6, January

Y

114 1.~ 6~

l l7

I 40

I 60

I 80

1 100

(cm)

7,3

Deformation analysis

(lla)

y2 ~> 0

113

~ga 2o3 27~ 365 4o2 445

Figure 11 Examination of an irregular frame.

The general format (equations (11)) is found for the governing system after combining the equilibrium and compatibility conditions (equations (1)) with the elastoplastic constitutive relations (equations (7) and (9)); system (11) is symmetric, consequent upon equations (2), (8) and (10). This is a nonlinear complementarity problem:

(11

_l

1 2 3 4 5 6 7 8

8

M2 , --cJ

_l

- ' - 6 ~ 0 9 ~ %096- ~-6096

0,98 0,gg 109 lt2

The governing s y s t e m

[M1

Activation of

T
n = fi

Io °o_=_:__.l[!]:[,;l / =>

7 8 s m

i

I

Simple extensions of the simplex algorithm of linear programming have been developed to implement the numerical solution of the governing system of equations (1 1), which can be interpreted as a linear complementarity problem (LCP) if the entries of the matrix and right-handside vector are assumed to be known and can be treated as constants. The failure load of an elastoplastic structure subject to a proportional loading can be obtained 6 by repetitive application of Smith's adaptation of the Wolfe-Markowitz algorithm;2s the structure in its original state is increasingly loaded to failure, but the axial force distribution is assumed to maintain the constant values found when attaining collapse in the previous iteration - three to four iterations being generally sufficient to guarantee convergence. The same algorithm can be applied if the plastic hinge formation sequence is to be followed in order to obtain the correct static and kinematic configurations developing along an equilibrium path, as illustrated in Figure 11 for the irregular frame shown therein, the cross-sectional properties of which can be found in reference 26. Instead of iterating on the elastoplastic failure load, convergence is now required at each basic solution (BS), which defines the activation of a yield mode, or plastic hinge. The plastic hinges are numbered according to their order of formation, and X is the load factor applied to the loading shown in Figure 11. For the example under consideration, it was found that plastic unstressing occurs at plastic hinges 4, 5, 6 and 7 at the instant plastic straining commences at plastic hinge 8. The frame collapses owing to mobilization of the combined mechanism involving plastic hinges 1, 2, 3 and 8; this fortuitously coincides with the mechanism predicted by plastic limit analysis which, however, overestimates the failure load by 11%.

Perturbation analysis Equations (11) can be replaced by an equivalent infinite sequence of linear complementarity problems 14by taking

Methodologyfor structural analysis: J. A. 7".De Freitas and D. L. Smith finite increments of each intervening set of variables, say v, and equating terms of the same order in a power series expansion on an arbitrary perturbation parameter e: vi+l = vl+ Av

(12a)

A v = ~ v(n) en n=l

(12b)

n!

The format of the nth-order component of the resulting infinite sequence is structurally identical to equations (11), but now vfn) replaces the generic variable v and the entries of the intervening arrays define the state of stress and strain existing in the structure prior to the implementation of the t'mite increment. Solutions of the resulting sequence of linear complementarity problems may be recombined using equations (12).

Extremum principles There is an equivalence between each of these linear complementarity problems and the pair of the primaldual nth-order quadratic programs of equations (13) and (14) obtained by enforcing the Karush-Kuhn-Tucker equivalence conditions: Primal

Min z(x, y) = ½~Dy + ~~Cx -~ ~c (13) subject to D y + Mx{~} d

Dual

Max w (x, y) = -- ½~Dy -- ½~Cx + ~d subject to I~y -- Cx = c,

Y2 >t 0

(14)

The physical interpretation of equations (13) and (14) reveals that the primal {dual} constraints represent static {kinematic} admissibility, while the primal {dual} objective function identifies with the nth-order term in the series expansion of the finite incremental potential co-energy {energy} of the system. In other words, equations (13) and (14) represent the incremental principles of minimum potential co-energy and minimum potential energy of nonlinear mechanics, 14 which when specialized to infinitesimal displacements and deformations reduce to the HaarKarman and Kachanov-Hodge principles, respectively. If the elastic {plastic} phase of the constitutive relations is deleted, equations (13) and (14) reduce to the minimum potential energy and co-energy principles of rigid-plastic {elastic} analysis.2a' 24 The role of the complementary energy principles in kinematically nonlinear mechanics has been to date a polemic issue, as is apparent in the works by Z u b o v , 27 Fraeijs de Veubeke ~ and Koiter29 devoted to this subject. It is worth stressing that it is the preservation of SKD and reciprocity in the constitutive relations, and the processing through mathematical programming theory of the resulting (exact) symmetric governing system, that enable the extension into the nonlinear domain of the dual role of energy and complementary energy demonstrated by Westergaard 3° and Argyris31 for linear behaviour. The minimum potential energy theorems proposed in references 32-35 can be obtained if in the nodal-stiffness description of equation (14) the finite increment on the generic (kinematic) variable Av is replaced by its first-order variation, which according to equation (12b) is defined by: Av = lim - ¢~0

e

Qualification of solutions Sufficient conditions for a kinematically nonlinear solution to exist and to be unique are obtained by applying to the programs of equations (13) and (14) the mathematical programming theorems on duality and uniqueness, a6 Multiple solutions are characterized by applying the theorem on multiplicitya7 to the composite form of these programs in order to guarantee that such solutions are simultaneously statically and kinematically admissible. Stability criteria are established by processing the formulation through the stability postulate of Drucker. 3a The solution qualification statements thus obtained 2'a9 involve all state variables, namely generalized displacements and their associated loads, as well as strain- and stressresultants, since the procedure is applied to each of the four equivalent formulations of elastoplastic structural analysis: such statements contain, therefore, those proposed by Corradi4° and MaierY who first suggested this method of approach, working on tangent stiffness formulations.

Numerical implementation The perturbation method, applied to the analysis of elastoplastic structures which sustain large displacements and deformations, invests the mathematical formulation with considerable linearity. The infinite sequence of linear complementarity prob-. lems associated with equations (13) and (14) can then be solved by an algorithm41 which is a simple extension of the simplex method. It is the virtue of such an algorithm that it provides a regular pattern for successive improvements of the solution. Multiple plastic unstressing is automatically implemented through a routine that establishes a priority sequence among the several yield modes which may show a tendelacy to be de-activated. The solution procedures developed in elastomechanics which are designed to detect and solve for limit and branching points can be readily adapted to the specific physics of elastoplastic behaviour.~ This non-iterative, f'mite increment algorithm also incorporates a procedure designed to maximise the steplength, which is adjusted either to reveal the activation of a new yield mode or to minimize the accumulation of numerical error resulting from the truncation of the series expansion of equation (12b).

Illustrative examples As an illustration of plastic unstressing, consider the single-bay two-storey frame shown in Figure 12. Horne,42 using a first-order nonlinear formulation,6 has found that the frame collapses owing to overall instability after the formation of the third plastic hinge. The authors' solution, shown in Figure 13, differs most from Home's from the moment critical section 2 becomes active (BS6), well into the post-buckling phase. At this stage Home identifies plastic unstressing at critical section 1 1, followed by the successive activation of sections 10 and 8, the sway mechanism being then mobilized with sections 6, 10 and 12 unstressing simultaneously. When yielding started at section 2 the authors found that plastic unstressing occurred simultaneously at critical sections 6 and 11. The activation of sections 10 and 8, consecutive in Home's solution, was separated by the

Eng. Struct., 1984, Vol. 6, January

57

Methodology for structural analysis: J. A. T. De Freitas and D. L. Smith

reactivation of section 6; and this section, upon final activation of section 8, again unstressed together with sections 10 and 12 to mobilize the same sway mechanism found by Horne. As an illustration of the effect of geometric imperfections in the response of shallow domes, consider the elastic space truss shown in Figure 14. The structure was analysed 43 under the two loading conditions summarized in Table 2. Plotted in Figures 14 and 16 are the vertical displacements of node 1 for each load case, the graphs in Figures 15 and 1 7 representing the radial displacements of the nodes surrounding the centre node. Figure 12 Examination of single-bay two-storey frame

Table 2 Components of the applied loads Direction

X

Y

Z

Nodes

1-7

1-7

1

2-7

Loading 1 Loading 2

0 0

0 0

k k

2k 0

0.5(

/

0.2 ~,

o •

//

Home Deform=ion anolysis Perturbation analysis

~

\

~o 2.0 I

I

I

0.04

O.08

0.12

6/L

B

C (3,4,6,7)

Figure 13 Comparison of results

I

<

-0.6

A

I

-0.4

I

1

-0.2 5 (mm)

Figure 15 Radial displacement of nodes 2-7 under loading 1

2 < 1 Lg

6 (cm) i

___.L

-0.2

_#

J

0.2

J

0.4 8(cm)

.1__

0.6

_L

I

J

0.8 i

Figure 14 Vertical displacement of node 1 under loading 1

58

Eng. Struct., 1984, Vol. 6, January

Figure 16 Vertical displacement of node 1 under loading 2

Methodology for structural analysis: £ A. 7". De Freitas and D. L. Smith

C (3,/..6.7)~

10&%lEA

the incremental complementary ~ariational principles of minimum potential energy and co-energy. Conditions for the existence, uniqueness and stability of elastoplastic solutions in the presence of finite displacements and deformations are obtained by processing the governing system and associated quadratic programs through system qualification theory.

Acknowledgements This research was sponsored by the National (Portuguese) Institute of Scientific Research (INIC) through the Mechanics and Structural Engineering Centre (CMEST), Technical University o f Lisbon. Figure 17 Radial displacement of nodes 2-7 under loading 2

References The response o f the perfect structure is characterized by the A-labelled curves, the results obtained being in agreement with those provided by Hangai; 44 the 'snapthrough' phenomenon, typical in the response of shallow arches and domes, is well illustrated by load condition 2 in F/gum 16. The effect of an initial curvature in the truss members was analysed next. The B-labelled curves represent the equilibrium paths when each truss member has an initial transverse mid-span deflection o f 0.1% of its length. Under load condition 1 such imperfections reduce the load-carrying capacity o f the truss by as much as 80%, their effect being only marginal under load condition 2. The C-labelled curves represent the response o f the truss with imperfect members when imperfections in the topography of the structure are introduced; the z coordinates of nodes 2 and 5 are now assumed to be 1.8 cm instead of 2.0 cm. The symmetry in the displacement mode is lost, with drastic changes in the response of the truss being exposed in the vicinity of the bifurcation point of the perfect system.

Conclusions A systematic and consistent procedure has been described for formulating and solving the general problem of nonlinear analysis of structures which can be discretized into a finite number o f repetitive building elements. The alternative processes through which such elements can be assembled are exhausted by interpreting the structure as a directed graph. A governing system featuring symmetry is obtained by preserving reciprocity in the constitutive relations and duality in the descriptions of equilibrium and compatibility, from which the virtual work concepts are derived. The Lagrangian governing system is replaced by an infinite sequence o f linear complementarity problems, which can be solved by an algorithm that is a simple extension of the simplex method used in linear programming. It has been shown that there is a direct equivalence between each linear complementarity problem in the infinite sequence and a dual pair of symmetric quadratic programs. Such a sequence of quadratic programs forms a discrete variational approach to the structural problem, and the dual nature of these programs has been shown to represent

1 Smith, D. L. 'Plastic limit analysis and synthesis of structures by linear programming', PhD Thesis, University of London, 1974 2 De Freitas, J. A. T. 'The elastoplastie analysis of planar frames for large displacements by mathematical programming', PhD Thesis, University of London, 1979 3 Da Fonseca, A. M. A. 'Plastic analysis and synthesis of plates and shells by mathematical programming', PhD Thesis, University of London, 1980 4 De Freitas, J. A. T. 'An~lise el~stica de estmturas articuladas espaciais sujeitas a grandes deslocamentos', CTE 41, CMEST, 1981 5 De Freitas, J. A. T. 'Statics and kinematics for space frames subject to large displacements', CTE 45, CMEST, 1981 6 De Freitas, J. A. T. and Smith D. L. 'Elastoplastic analysis of planar structures for large displacements' (in press) 7 Denke, P. H. 'Nonlinear and thermal effects on elastic vibrations', Report SM-30426, Douglas Aircraft Company, 1960 8 Argyris, J. H. 'Recent advances in matrix methods of structural analysis', in 'Progress of aeronautical services' (ed. Kuchemann, D. and Sterne, L. H. G.), Macmillan, London, 1964 9 Haisler,W. E. et al. 'Development and evaluation of solution procedures for geometdcaUy nonlinear structural analysis', J. AIAA. 1982, 10, 264 10 Oliveira,E. R. A. 'A method of fictitious forces for the geemetrically nonlinear analysis of structures', ]4th Int. Cong. Theoretical and Applied Mechanics (ed. Koiter, W. T.), 1974 11 Smith, D. L. 'First-order large displacement elastoplastie analysis of frames using the generalized force concept', in 'Engineering plasticity by mathematical programming' (ed. Cohn, M. Z. and Maier, G.), Pergamon Press, Oxford, 1977 12 Kohnke, P. C. 'Large deflection analysis of frame structures by fictitious forces',Int. £ Numer. Meth. Engrg., 1979, 12, 13 Munro, J. and Smith, D. L. 'Linear programming duality in plastic analysis and synthesis', Int. Syrup. on Computer Aided Design, University of Warwick, 1972 14 De Freitas, J. A. T. and Smith, D. L. 'The potential energy and co-energy theorems in large displacements elastoplastic analysis', CTE 42, CMEST, 1981 15 De Freitas, J. A. T. and Smith, D. L. 'A finite element representation of elastic beam-columns' (in press) 16 Manderla, H. 'Die Berechnung der SekundarSpannungen, welche in einfachen fachwerk infolge starrer Knotonverbindungen auftreten', Allgemeine 8auzeitung, 1880 17 Berry, A. 'The calculation of stresses in aeroplane spars', ~ans. Roy. Aero. See., 1916, 1 18 James, B.W. 'Principal effects of axial loads on moment distribution analysis of rigid structures', NACA TN 534, Washington DC, 1935 19 Jennings, A. 'Frame analysis including change of goemetry', Proc. ASCE, J. Struet. Div., 1968, 94,627 20 Mallet, R. H. and Marcal, R. V. 'Finite element analysis of nonlinear structures', Proc. ASCE, £ Struet. Div., 1965, 94, 2081 21 Powell, G. H. 'Theory of nonlinear structures', Proc. ASCE, £ Struct. Div., 1969, 95, 2687

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