Inelastic analysis of suspension structures by nonlinear programming

COMPUTER METHODS fN APPLIED MECHANICS AND ENGINEERXNG 5 (19755 127-143 0 NORTH-HOLLAND PUBLISHLNG COMPANY

INELASTIC ANALYSIS OF SUSPENSION STRUCTURES BY NONLINEAR PROGRAMMING

Received 4 June 1974 The cable behavior which includes loosening and @astic yielding is described by a multilinear force-elongation law. A simpLe analytical re~rese~tat~on of this type of Law is used and shown to be easily adjusted in order to allow for irreversible plastic strains. The determination of the structural response to given externaf actions in the presence of combined geometrical and physical noniinearities is shown to be amenable to the minimization of a nanlinear energy function subject to sign-constraints for some variables, whereas the nodal displacements represent unconstrained variables. A mathematical programming technique which solves this problem efficiently is discussed and used in studying the comparative behavior of sample tension structures (up to fatiure) in the domain of large configuration CfiaRg&

1. Introduction Geometric nonlinearity represents an essential feature of suspension structures. Under the customary assumption of linear elastic behavior of individual cable members, the total potential energy is a highly nonlinear ~no~quad~tic) function of the free nodal displacements, and the configuration changes due to loads can be determined by the unconstrained minimization of this function. This energy approach to the analysis of geometrically nonlinear but physically linear discrete tension structures was developed by Buchholdt [ 1,2]. Alternative approaches involve iterative solution of systems of nonlinear equations by the Newton-Raphson iteration scheme or by modi~~ations thereof (see e.g. [ 3]), or step-by-step “historical” procedures of solving a sequence of individually linearized problems for small load increments (see e.g. f4]). Combined geometric and physical nonlinearities must be allowed for when structural responses to exceptional loads are to be studied in order to assess ultimate strength or serviceability. In fact, in these cases the relationships between tensile force and elongation of individual cable members, even in the absence of transversal loads, becomes strongly nonlinear, since it must cover, in genera both foosening and plastic tensile strains, “Inelastic” cable behavior including either one or the other or both of these circumstances has been described as an approximation by means of nonlinear equations, On this basis, analysis methods centered an Newton-Raphson-type iterations were developed by Greenberg [ 51, Jonatowski and Birnstiel 161, et al.; procedures resting on energy minimization were studied by Murray and Willems [ 7 I _ In this paper the nonlinear behavior of individual cables is depicted by a multilinear law, which is able to describe, with satisfactory accuracy and simplicity, both slackening and inelastic yielding simultaneously. The analytical representation of these laws implies merely linear equations, * Presented

at the International

Conference on Tension Roof Structures, London, April 1974.

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R. Contra et al., Inelastic analysis of suspension structures by nonlinear programming

sign constraints and a “complementary” or orthogonality relation (i.e. it exhibits the mathematical structure which is known in recent literature as a “linear complementarity problem”). If the irreversible nature of plastic deformation has to be taken into account, the above law needs to be only slightly adjusted in some parameters, and preserves all its analytical features. On this basis the determination of the geometry change due to loads and imposed strains (such as thermal strains) becomes amenable to the minimization of a non-quadratic energy function of the displacements and of the parameters which govern the deviations from linearity in the cable law. The latter variables are nonnegative and their sign requirements are the only constraints in the optimization process. The degree of nonlinea~ty of the objective function is the same as in elastic analysis in the presence of geometric nonlinearity alone. The type of mathematical optimization problem thus formulated is dealt with in an abundant literature, e.g. [ 8,9]. The algorithm employed here in working out illustrative examples exploits the linear nature of the constraints and turns out to be fairly efficient [ lo]. The extremum property stated can be interpreted as an extension of the potential energy theorem f 111. Both perfectly plastic and work-hardening yielding behaviors are envisaged in the sample structures considered. The irreversibility of inelastic tensile strains in taken into account in passing from one loading stage to the subsequent one. Clearly, the nonholonomic nature of plastic strains could be fully allowed for only by means of incremental analysis which would imply solving a quadratic programming problem at each step [ 121.

2. Multilinear idealization of nonlinear behavior of cables 2, I. Let Qi denote the change of the tension force in a generic cable member (index i) initially subject to a force Qi; let qi indicate the corresponding elongation change. The multilinear Qi versus qi plot of fig. 1a can be subdivided into the plots of figs. 1b and 1c. The former relates Qi to the member elastic stiffness Ei, the “elastic” part ei of qi; the latter relates Qi to “inelastic” or “corrective” part fi of qi : qi=Ei-‘Qi+

(1)

fi.

Fig. 1. Multilinear idealization of the force-elongation

law for a cable member.

R. Contro et al., Inelastic analysisof suspensionstructures by nonlinearprogramming

129

The nonnegative constants K,, K,, K, define the ordinates of the angular points in the plot of fig. 1a. The nonnegative constants H,,, iFi,, If, define the slopes of the three branches of the multilinear in fig. 1c as shown there, Let $, X/ (i = 1,2,3) indicate auxiliary variables, the latter being measures of the various deviations from linearity as shown in fig. le. With these symbols, the Qi versus fi relationship depicted in fig. lc can be analytically described as follows: &=-h~+h;+X:,

s; = -Qi

-

(2) (Kf +Hfh',)

,

(34

q;=Qi-(K;+H;X',),

(3bl

‘p; = Qi - (K;+H;X;),

(3c)

$20, +

(4) (I= 1,2,3,

0

(5)

Consider a value Qi > Kk of the cable force. By eq. (3c), pf < 0; hence, because of eq. (6), ??, = 0; by eqs. (3b), (3c), (4) and (S), neither Xi nor hi may be zero, and therefore, because of eq. (6), both ‘pi and & must be zero, Thus, eqs. (3b) and (3~) define uniquely A’, and hi, which in turn define by eq. (2) the inelastic elongation fi . Similar paths of reasoning justify the above stated equivalence between the graphical representation of fig. lc and the analytical relation set eqs. (2) -(6). 2.2. By properly choosing the constants K and H, eqs. (2)~(6) provide a piecewise linear approximation of the real behavior of cable members along a broad range of situations. The corner marked by index j = 1 corresponds to the onset of loosening. Therefore

K:=(ji,

Hi,=0

(7)

are obvious, mandatory assumptions. If Yi denotes the yield limit of the cable, clearly Hi,

= Yi -

Q, .

(8)

Some best-fitting criterion can be adopted to select K”, and Hi,:Ii', = 0 would represent a “perfectly plastic” stage before rupture; positive Hi, and Hi, characterize “hardening” stages; for H',=a3relations and variables with index i = 3 become immaterial and the corresponding plot is a trilateral one. If an additional linear branch were introduced in order to improve the accuracy of the idealization, index 1 would run over 1,2,3,4, but the essential mathematical features of eqs. (2)-(6)

would remain unaltered, Clearly these are the same for any number of branches in the multilateral and consist of the linearity of all relations except eqs. (61, which require that, in each pair of ~o~es~onding variables X and ip, at least one must vanish. 2.3. The above laws, Qi versus & (and hence Qi versus qj>! describe reversible deformations (i.e. with fully recoverable, nonhysteretic strains) and, in fact, were used first in “holonomic” plasticity theory. Negative elongations of slack cable members are clearly reversible. However, if strain reversal (“lo& unloading”) occurs after some positive fi does decrease in the reality, as Qi decreases, then, in order to describe correctly the subsequent deformation process, irreversib~ity must be allowed for. This can be done in the context of the above piecewise-linear description of cable member behavior, simply by modifying slightly these laws.

a)

b)

Fig. 2. Typic& cksnges of member taw from one step to the subsequent one in the multistage procedure.

Fig. 2 illustrates the required modifications for two typical cases. Dashed axes and plots and primed quantities refer to the preceding stressing stage, which has led to the current tension Qi’ at the end of a plastic yielding process. The Qi versus fj laws to be used for describing the subsequent stage which might involve unloading are shown in solid lines. In fig. 2a it is assumed KfZi < Q:’ < Kts5. In this case the new K constant9 5~ consider are

K’, = fj; + Q;' ;

K”, = 0;

K’, = Kri3 - Q;* ,

(91

whereas the H constants remain unaltered. In the case of fig<2b, where K’,, < K’,j < Qi*+ the new K constants are indicated in the same figure. However, it is clearly possibie to interpret the new law alternatively as a trilateral characterized by the constants:

R. Contra et al., Inehtic

131

analysis of suspension structures by nonlinear programming

3. Analysis of tension structures as a problem in nonlinear programming with sign constraints only 3. f. Consider an equil~b~um configuration 2 of a tension structure whose deformability is govern{ by n nodal degrees of freedom and which consists of yyzcable members (i.e. cables not indi~dually subjected to loads, but ending at “nodes”’ where loads may be active). Let the vectors p = {a’, . . . Fn}, @ = {Q, ,.. Drn} define the-given external forces and the known internal forces respectively, present in the static situation C. The problem to be discussed below is the determination of the structural response to additional loads described by vector P = {PI...P, and to imposed elongations (e.g. due to thermal strains or turnbuckle operations) defined by vector D = {D1 . .. D,, I. The static configuration change E + IS due to the above external actions is characterized by the nodal displacements u = {ur .. . un} referred to the same Cartesian reference frame as for loads, the additional elongations 4 = (4 1 . . . q,), and the additional internal forces Q = IQ1 . .. Q,]. Clearly, th e nodal further elongations t = {tr . . . fm} are nonlinear functions of the displacements U, whose magnitude is expected to be generally large. Specifically, consider the 6th cable member. As for the two nodes between which it is inserted, let a prime and a double prime indicate the node corresponding to the lower number and to the higher number, respectively, in the numeration assumed for the nodes. Let Zibe the cable length in the situation z1, and r~; its direction cosines in z’, the index h running over the set of reference axes (2 for plane, 3 for space st~ctur~s). With this notation we may write fi = [

F (lp.; + 24: - U;)*]lR - zi

(i

= 1 ... m)

,

(11)

t=q+D=t(u).

(12)

Eq. (11) confers a precise meaning to the symbolic functional dependence t(u) in the geometric compatibility eq. ( 12). The virtual work equation @+Q)%=@+P)%

(13)

imposed for any Fu provides the equilibrium equations for the new configuration

M(g+Q)=P+P,

IX

(14)

where the entries of the matrix M = ~~~~~ are first derivatives, calculated in X, of the geometric relations eq. ( 11): at,

Mki=

( ) au

(k=l...n,

i=l...m).

(15)

k

The nonlinear behavior of each cable will be described by multilinear Qi versus qi laws as in sect. 2. Define the vectors

and the diagonal matrices

H’= [HfH;H;],

i= 1,2,... m;

E= [E,E,

.*. E,]

)

and construct the diagonal matrices H=

[H* HZ **. H” ]

)

(order 3 ~11X 3 pn) ,

Iv=

[n n 1. a]

(order m

X

3 m) .

By means of these symbols, all relation sets eqs. (1) to (6), for i = 1 &..m, can be condensed into the foilowing single set of matrix relations, after having substituted eq. (2) into eq. f 1):

cp=NtQ-HA-K,

(17)

9%

(1%

=O.

3.2. The matrix relations expressing the assemblage geometric compatibility

in the configuration change ( 121, the equilib~um of the final configuration ( 141, and the individual deformability laws of all members f 16)-( 19) fully govern the response to the external actions P and D superimposed on the initial static situation z. The set of dl these relations can be easily shown to represent the Kuhn-Tucker conditions of the following nonlinear programming problem [8- 111: minw(z/, &I =$t’(u)Et(u)

+$ sf(H+N”EN)

+(iZ-Ef))‘t(~)-(ij+p)t*+I;i’t),+I)’ENi,

1- L,fNfEt(zr) (201

subject to: k>O”

(21)

Under weak restrictions (“contraint qu~i~catiun~*~ certainly fuh?iQed in the present context, the “local” Kuhn-Tucker conditions of the above mi~rni2~t~on problem are 181:

(22)

R. Cantro et al., Inelastic anulysis

ofsuspension structures by nonlinear programming

133

When the derivatives are calculated, eqs. (22) are readily seen through trivial substitutions to become fully equivalent to the whole set of the governing equations formulated in sect. 3.1. The objective function eq. (20) is not convex in general; hence, the Kuhn-Tucker conditions are necessary, but in general not also sufficient for a vector be optimal in problem eqs. (20) -(21). Therefore any vectors u*, A* which correspond to the global or a local minimum of w over the feasible domain 3, > o must satisfy the Kuhn-Tucker conditions and hence do represent a solution to the structural problem formulated in sect. 3.1. However, there may be vectors u**, 1** which satisfy those conditions and hence solve the structural problem, but do not correspond to a minimum of w. It can be readily shown that the former (starred) vectors define stable (in the small) equilibrium configurations. Since most numerical minimization techniques, including that adopted in this paper, lead to some (at least local) minimum, it can be concluded that the optimization process of solving eqs. (20) - (2 1) provides a stable configuration C of the suspended structures under the given external actions. 3.3. If the behavior of all members is assumed as linear (i.e. if the number of all X variables becomes zero, or, alternately, all constants H become -), eqs. (20) -(21) reduce to a formulation of the potential energy principle in the presence of geometric nonlinearities. However, in contrast to this principle, the more general extremum property expressed by eqs. (20) -(21) for multilinear behavior represents a way of enforcing via optimization not only equilibrium but also some features of the constitutive laws (i.e. Q,4 o, cprI = 0). Moreover, the objective function o (20) is readily seen to represent the total potential energy for a solution, not for any feasible vector [ 111 If the geometric relation t(u) is linear, and configurations 2 and 2 are regarded as coincident in writing the equilibrium equation (14) (first-order theory), the extremum property expressed by eqs. (20)-(21) reduce to that established by Maier in holonomic plasticity.

4. Multistage procedure taking account of permanent strains and loosening 4.1. The conjuration change 2 + Z under consideration is now assumed to be preceded by another one which led to the static situation % starting from an initial situation X, with all cables elastic and causing plastic yielding in some cables. For the analysis of the structural response to the preceding (say proportional) loading process o + P, suppose that use has been made of the tension-elongation laws of fig. 1 which are plotted with dashed lines in fig. 2. In order to determine the final static situation ZI, the same multilinear Qi - qi laws could be adopted in analysJng by a single optimization the configuration change provoked by the total loading process P + P. Clearly this might lead to serious errors if significant strain reversals happen to occur in reality during the second loading stage in cables which were plastically deformed during the first one. In fact, the irreversible nature of these deformations would be totally ignored in this way. Alternatively one can perform the analysis of the confi~ration change and, subsequently, the analysis of the structural response to the second loading stage P alone, starting from the known situation z, but assuming the constitutive laws indicated in fig. 2 with solid lines for cables which exhibit plastic strains in %. This adjustment of Qj versus qi relations is easily done in the analytica description of multilinear laws, as pointed out in sect. 2.3, and does not alter the essential feature

134

R. Conlro et al., lnelastie analysis of suspensian stluclures by nonhear

progmmning

of the mathematical problem. It is worth stressing that it would not be so if a “smooth” representation, by means of nonlinear equations, were adopted. The above “multistage” procedure of analysing the structure for each finite loading step in sequence, by adjusting the laws of the yielded cabIes at the beginning of each step, allows for the irreversibility of plastic strains at a large extent, probably satisfactory in practice if the loading steps are proportional and not too large. However, a sequence yielding-unloading of a cable within a single stage is still, clearly, a possible cause of errors, which can only be reduced by decreasing the step amplitudes of the individual ~ro~o~ion~ loading processes. 4.2. Consider the case in which some cable, say the i-th, is slack in the static situation :, i.e. it exhibits a corrective negative elongation fi = -Ai (Ai > 0 being the excess of its stressless straight length over the distance between the nodes at its ends). This occurrence can be dealt with as follows. The slackness of member i can be removed through. a fictitious turnbuckle operation which shortens its length by Aj, Then the usual Qi versus& law of fig. 1 can be adopted by assuming in it Kf = 0, and the present analysis method can be applied, with the only measure included among the given external actions is the imposed positive elongation Dt = Ai in order to compensate for the fictitious shortening operated before the analysis, as mentioned above.

5. An outline of the algorithm employed Problem (20) -(21) can be regarded as a special case of the folIowing general line&y constrained optimization problem: min F(x), subject to: Ax 2 b ,

(23)

x

where: F denotes a differentiable function of the vector X, so that its gradient vector g(xi) is available for any xi: A is a given matrix, b a given vector. It is assumed that the feasible domain defined by the constraints in (23) is bounded; this condition can be always complied with by adopting, if necessary, fictitious constraints non-active in the solution. The present problem can be cast into form (23) obeying the above condition by setting:

r-s

11 .Jl4

X=

3

A=

.I

Tl’ I

b=

i j

0.

s

1

(241

i

where: I represents the identity matrix of order equal to the number of components of X; S and 1 are positive vectors of large arbitrary components. The ~go~thm employed has been the accelerated conjugate direction method of Best and Ritter, which is basically a development of 2outendijk”s procedure of feasible directions. This method is described in detail in [ 131. The initial point is any feasible vector x0. The typical step, starting from a vector xi which is reached through the preceding computations and fails to satisfy an optimality (statio~arity) test

R. Contra et al., Inelastic analysisof suspension structures by nonlinear programming

135

can be briefly outlined as follows: (1) the new direction si = yi - xi is found b y solving the linear programming problem : min g/@--xi),

subject to:

Y

Ay>

b;

Djy= ci ,

(25)

where: Di is a matrix formed on the basis of differences between suitable previously calculated gradients; ci coincides withgi if it is not o, and the relevant equation represents the conjugation conditicn; yi is optimal vector of (25). (2) a linear search along sj: min “(Xi f QliSi), CY

subject to

0 G CY~ < 1

(26)

is performed to determine aj on the basis of an approximation (quadratic or cubic) of w(a). (3) one sets Xi+%= xi + QiSj and computes Dj+lt c~+~. Improvements pointed out in [ 131 are: ( 1’) accelerated direction finding procedure, which replaces at some Xi the normal procedure ( 1) and determines si as a linear combination of previous gradients through the optimal vector of the dual to program (25); (2’) dete~ination of “optimal step size” through special procedures due to authors cited in E13 I instead of (2) for some xi. Alternative “ad hoc” algorithms for solving problems (23) are available in the recent literature but their efficiency in the present peculiar context has still to be examined.

6. Appli~tio~

and computational

remarks

6.1. The method developed in the preceding sections has been applied to the analysis of the lensshaped, plane roof structures whose geometric characteristics in the unloaded pretensioned state are defined in figs. 3 and 4 along with the node and member numeration.

Fig. 3. Initial configuration and numbering for the first sample structure.

136

R. Contro et al,, Inelastic analysis of suspension structures by nonlinear programming

Fig. 4. Initial configuration and numbering for the second sample structure.

In both cases the nodes are placed on parabolae whose vertices are in elevation 5 m above or below the level of the relevant anchorages, the distance of which (span) being 100 m. The cross section areas are 20 cm2 in the sagging main cable, 10 cm2 in the hogging main cable. For all vertical links (both struts and ties) a common cross section of 12 cm2 has been chosen. The elastic moduli were assumed to be E = 1,700,OOOkg cme2 for the main cables (apparent material elastic constant), E = 2,100,OOO kg cmm2 for ties and struts. The following idealizations has been adopted for the behavior of individual cable members: (a) sagging cable elements: loosening has been not envisaged (Hf =-); in tension, three alternatives are taken into consideration for comparison purposes: perfectly plastic yielding (Hi= 0); strainhardening with Hi = E/3; unlimitedly linear elasticity (Hi = -); (b) hogging cable elements: loosening is allowed for (Hf = 0), whereas inelastic strains in tension were ruled out; (c) vertical linkage elements were supposed linear elastic in all static conditions. The above assumptions, based on obvious engineering judgement and conjectures on the responses to vertical loads, keep the number of the sign-constrained variables h reasonably small (precisely = 20 for both structures). However, it can be readily checked at the end of the analysis whether assumptions of this type are fulfilled or not, and in the negative case they can be properly relaxed for a new calculation. These checks always give positive results in the present examples. 6.2. The tension structure with intersecting cable (fig. 3) was analysed starting from the unloaded state with the rather high pre-tension Q= 80 ton, Q being the common horizontal component of the pre-tension in all main cable members. The following two kinds of proportional loading were considered, increasing by 2 ton at each step the load applied to each loaded node:

R. Contra et al., inelastic analysis of suspension structures by nonlinear programming

137

(a) symmetric loading, formed by equal vertical forces acting on all upper nodes and ranging from 2 to 14 ton at each node; (b) nonsymmet~c loading consisting of equal vertical forces, from 6 up to 18 ton each, on upper nodes 13, 14 = 3,4,5, and from 3 up to 9 on the central node 6. The lens-shaped roof of fig. 4 under nonsymmetric loading conditions of type (b) alone, starting from an unloaded state with c= 40 ton. (Similar data were assumed in [ 1] for illustrating elastic analysis via potential energy minimization.) Some of the equilibrium configurations obtained are presented in figs. 5 to 8. For the sake of clearness in these graphs the initial (only prestressed) con~guration is indicated in solid lines with ordinates amplified by a factor 4 with respect to the abscissae; this provides the basis for the vertical displacements which are amplified by a factor 8 with respect to the abscissae. The horizontal displacements are not shown, as they are generally far smaller. In each figure two deformed configurations involving inelastic strains are represented, one with heavy dashed lines, the other with heavy dash-dot lines. Cable members undergoing loosening (Xi,> 0) or plastic strains (Xi > 0) are marked by asterisks. For comparison purposes the configuration obtained under linear elastic idealization of cable behavior for the latter (heavier) loading condition is plotted with light dash-dot lines. Figs. 10 and 11 show how the vertical displacement of the lower central node and two meaning ful cable forces (of members 1 and 11) vary as loads increase. Again, the co~esponding values obtained under linear elastic assumptions for cable behavior are represented by similar light lines. It is quite apparent from these diagrams that, at least for the structures and the loading ranges considered, physical nonlinearities cause nonlinear overall responses (such those reflected by

Fig. 5. Vertical displacements under symmetric loads; perfectly plastic behavior.

R. Contro et al., Inelastic analysis of suspension structures by nonlinear programming

138

6

displ

Fig. 6. Vertical displacements under non-symmetric

loads; perfectly plastic behavior.

Fig. 7. Vertical displacements under symmetric loads; hardening behavior.

R. Contra et al., Inelastic analysisof suspension structures by nonlinear programming

6 =*ot H = E/3

Fig. 8. Vertical displacements under non-symmetric

141

I

-6

I

loads; hardening behavior.

; /

I

!

1.

d.9d.

Fig. 9. Vertical displacements under non-symmetric

loads, hardening behavior, for the second structure (fig. 4).

139

140

R. Contra et al., inelastic analysis of suspensim

structures by nonlinear programming

Fig. 10. Central displacement and cable forces versus symmetric load intensity.

d~~~~~~n-~oad relations) far more directly and effectively than the intrinsic geometric nunfinearity. Moreover, the linear-elastic (H = -) and perfectly plastic assumptions seem to provide upper and lower bounds for meaningful quantities corresponding to more realistic workhardening idealizations (such as that with H = E/3 considered herein). Assuming 0.03 as rupture tensile strain, ultimate loads were easily determined on the basis of the inelastic analysis performed and marked by dots in figs. 10 and Il. 6.3. All computations of the present examples were carried out using a computer code centered on the subroutine FCDPAK. This was written by M. .T.Best as an implementation of the nonlinear programming algorithm described in [ 131and briefly outlined here in sect. 5. The structure of fig. 3 led to minimization problems involving 32 free variables (displacements u) and 20 sign~onstr~ned variables (inelastic deformation parameters X). The numbers of iterations in the minimization process are indicated in fig, 12, It can be observed that the iteration number increases roughly by a factor 2 in passing from unconstrained minimization in 32 u variables (linear elastic hypothesis H = -) to sign-constrained minimization in 52 variables (32 of which being free). On the contrary both the iteration number and the running time turned out

R. Contro et al., Inelastic analysisof suspension structures by nonlinear programming

Fig. 11. Central displacement and cable forces versus non-symmetric

load intensity.

Symmetric loads

68

15

99

100

100

100

100

0

162

166

193

177

221

222

221

$E

137

166

141

173

167

174

176

8

10

12

14

16

18

98

99

100

100

100

100

100

166

169

202

171

171

171

171

162

165

169

137

175

171

168

Non-symmetric

loads

P (L/joint)

6

H

0 SE

Fig. 12. Numbers of iterations for tension structure of fig. 3. G’= 80 ton; variables: 32 u; 20 h.

141

142

R. Contro et al., Inelastic analysis of suspension structures by nonlinear programming

to be little affected both by the number of X variables which are nonzero in the solution and by the tolerances adopted in checking the fulfilment of Kuhn-Tucker conditions for termination. The storage occupation in a UNIVAC 1108 computer was 18 R for the instructions and 32 for the data. 6.4. The multistage method described in sect. 4 was applied to the structure of fig. 3 with H = E/3 and 0 = 80 ton for the following loading paths: (a) from the only prestressed state to a symmetric load distribution of 8 ton on each upper node; addition of 6 ton after adjustment of the law for yielded cables; (b) symmetric loading of 8 ton as in (a), followed by a nonsymmetric loading stage of additional 8tononnodes13,14z3,4,5and4tononnode6. (c) proportional symmetric loading up to 10 ton each upper node and subsequent unloading up to P = 0.01 ton after constitutive law adjustment. In case (a), performed for check purposes, exactly the same plural state was found as in the single-step analysis performed earlier, as expected. In fact in this case no local unloading occured in the latter stage. In case (b) significant local unloading phenomena did occur, thus justifying the recourse to the multistage procedure. In the load cycle case (c) the pe~anen~y deformed con~guration shown in fig. 13 was determined. All cases seem to indicate the feasibility and the efficiency of the proposed procedure of adjusting the constitutive laws of yielded cables.

Fig. 13. Vertical displacements under symmetric loads and residual displacements after unloading.

R. Contra et al., Inelastic analysisof suspension structures by nonlinear programming

143

Acknowledgements This paper is part of a research program sponsored by the National (Italian) Research Committee (CNR, Gruppo Costruzioni Metalliche). The support under CNR grant is gratefully acknowledged. The authors wish to thank Dr. M. I. Best of Waterloo University, for the optimization subroutine used in the examples and Mrs. L. Binda for her assistance in the computational work.

References [l] H.A. Buchholdt, A nonlinear deformation theory applied to two-dimensional pretensioned cable assemblies, Proc. Inst. Civ. Engrs. 42 (1969) 129-141. [2] H.A. Buchholdt and B.R. McMillan, Iterative methods for the solution of pretensioned cable structures and pinjointed assemblies having significant geometrical displacements, IASS Pacific Symposium on Tension Structures and Space Frames (Tokyo and Kyoto, Oct. 1971) pp. 3,7,1-3,7,1X [3] H. M#llmann, Analysis of hanging roofs using the displacement method, Acta Polytechnica Scandinavica C168 (1971) l-49. [4] G. Maier and 0. De Donato, Elastic analysis of plane pretensioned systems, IASS Pacific Symposium on Tension Structures and Space Frames (Tokyo and Kyoto, Oct. 1971) pp. 3,2,1-3,2,12. [S] D.P. Greenberg, Inelastic analysis of suspension roof-structures, J. Struct. Div. A.S.C.E. 96 (1970) 905-930. ]6] J. J. Jonatowski and C. Birnstiel, Inelastic stiffned suspension space structures, J. Struct. Div. A.S.C.E. 96 (1970) 1143-l 166. [7] T.A. Murray and N. Willems, Analysis of inelastic suspension structures, 3. Struct. Div. A.S.C.E. 97 (1971) 2791-2806. 181 W.S. Zangwill, Nonlinear programming (Prentice-Hall, Englewood Cliffs, N.J., 1969). [9] G. Zoutendijk, Nonlinear programming, computational methods, in: J. Abadie (ed.), Nonlinear programming (NorthHolland, 1970). [lo] M.J. Best, A feasible conjugate direction method to solve linearly constrained minimization problems, Research Report CORR 79-12 (University of Waterloo, Canada, 1973). [ 1 l] R. Contro and G. Maier, Energy approach to the inelastic analysis of suspension structures, Techn. Rep. N.16, ISTC (Politecnico, Milan, July 1973). [ 121 G. Maier, Increments plastic analysis in the presence of large displacements and physical instab~izing effects, Int. 3. Solids Structures 7 (1971) 345-372. [ 131 M.J. Best and K. Ritter, An accelerated conjugate direction method to solve linearly constrained minimization problems, Research Report CORR 73-16 (University of Waterloo, Canada, 1973). [ 14) R. Fletcher, An efficient, globally convergent, algorithm for unconstrained and linearly constrained optimization probiems, 7th Int. MathematicaJ Programming Symposium (The Hague, 1970), T.P. 431 Atomic Energy Research Establishment (Harwell, Dec. 1970). [IS] P.E. Gill and W. Murray, Quasi-Newton methods for linearly constrained optimization, NPL Report NAC 32 (National Physical Laboratory, May 1973).