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A variational inequality approach to the inelastic stress-unilateral analysis of cable-structures
Computersk Structures,Vol. 6, pp. 133-139.PergamonPress 1976.Printed inGreat Britain
A VARIATIONAL INEQUALITY APPROACH TO THE INELASTIC STRESS-UNILATERAL ANALYSIS OF CABLE-STRUCTURES P. D. PANAGIOTOPOULOS Institute of Technical Mechanics, The Technical University of Aachen, D 51 Aachen, Federal Republic of Germany (Received 18August 1975) Abstract-The present paper deals with the inelastic, stress-unilateral analysis of cable-structures undergoing large displacements. The response of the cable-structures to load- and initial strain-increments is described by a set of equations and inequalities, which are formulated compactly as variational inequalities. Two dual extremum principles-generalizations of the minimum of the potential and complementary energy-with inequalities as subsidiary conditions are derived. The numericalcalculationis preformediterativelyby using the decomposition
techniquesof the multileveloptimizationand applyingin the resulting substructuresthe algorithmsof non linear optimization.As an example the elastic,elastic-perfectlyplasticand workhardeningbehaviourof a cable-structureis examinedand some useful results from the comparisonof the solutions are obtained. INTRODUCTION
A structure, some of whose elements are not capable of transmitting compression, i.e. negative stress, is called a stress-unilateral (s-unilateral) structure. The cablestructures in this paper are considered to be stressunilateral structures. The nonnegativity inequality for the stresses of the cable-elements is taken as the principal condition in the formulation of the mathematical model. The problem is nonlinear not only because of the presence of the stress-inequalities, but also because the considered stress-strain law is non linear and the formulated theory is a large displacement theory, as usual in cable-structures. The compatibility and equilibrium equations of the structure are linearized by considering “incremental” processes caused by infinitesimal external actions added to the known situation of the structure. We suppose that any load and initial strain may be represented by a sequence of increments of properly chosen small amplitude. The behaviour of the cable-structure can be described by a set of equations and inequalities which will be formulated as variational inequalities. The stress-strain diagram may have any form. Here the calculation is performed for the cases of elastic, elasticperfectly plastic, and workhardening behaviour of the cable-elements and the corresponding results are discussed. The stress inequality for the cable-structures has been introduced in [ 1,2]. In[31 the continuous functional theory of stress unilateral cable structures and in[4,5] the elastic stress-unilateral analysis of discretized cable and membrane structures have been developed. 2. FINITE ELEMENTS DISCRETIZATION The structure is divided into n finite elements, of which u are cables. For the cables, pin-jointed bar elements are used. (The rigid body degrees of freedom of the elements are eliminated when they are unassembled, i.e. the elemental stress and strain states are defined by means of the natural stress and strain vectors [6].) Because of the large displacements and the nonlinearity of the constitutive law (Fig. 1) the developed theory is incremental. Each increment of a quantity is denoted by a dot on the symbol of the quantity. The configuration of the structure after the application of the pretension is called “initial state”. With reference to the initial state and to a fixed Cartesian
system common for the whole structure, the incremental stress-strain relations have for the ith cable the following form ii = F&i + A,,8- tii (1) ai = si t ai z 0
(2)
p = di + bi 2 0
(3)
u,yi = 0
(4)
where si, ei, s~,~,eo,i denote the stress, strain, initial stress and initial strain of the ith cable respectively. fO.iis the tangential natural flexibility matrix of the cable at the step under consideration. (- vi) is the slackness of the cable, a/(-bi) is the total stress/total slackness of the cable at the beginning of an incremental step, ui/(- yi) is the total stress/total slackness (Fig. 1) of the cable at the end of the step. Relations (l)-(4) express, that, at the end of each step either a non-negative total stress or a non-positive total slackness exists on any cable (Fig. 1). It is clear that the slackness (- u,) can be viewed as an unknown initial strain, which constitutes a reversible negative elongation. If local unloading occurs in the plastic region ABC (Fig. 1) of the ei - si diagram of a cable, hysteretic strains will appear and the new diagram EDBC is taken into account for the calculation of the flexibility matrix of the cable in the subsequent steps. If in a step qi = 0, the cable behaves like a usual pin-jointed bar element and has a positive stress. If in some step +i > 0, the cable has no tension and thus it does not contribute to the stability of the whole structure. Relation (1) represents for di = 0 the stress-strain relations of every one of the remaining n - ~7usual elements of the structure.
Fig. 1. Stress-strain law for a cable-element. 133
134
P. D. PANAGIOTOPOULOS
Relations (l)-(4) can be assembled for all the elements of the structure, as follows: e=F&t&,-ir,
(5)
c?=itii,o,
(6)
7 =Gtbro,
(7
o’y = 0.
are also introduced. Using these substititious relations (ll)-(14) can be written more compactly in the form {RUtpZO,
(20)
Relations (6)-(g) can be written equivalently under the two following inequality forms
(8)
+.’ = s’ = {S,‘, . . . ) S,‘}, Here c’ = If!,‘,. . . , i.‘}, {IA’, . . . , tivi,o,. . . , 0) = {? IO}and Fo is the tangential flexi-
UrO U’(RUt~)=O}.
{a’(?*-y)ZO,for
every ~*ZO,yZO}
(21)
and {?‘(a*-6)rO,forevery@*rO,&ZO}.
(22)
bility matrix of the unassembled elements at each incremental step. The part of a vector or a matrix extended over the cables is denoted by a bar. Index t denotes the transpose of a vector or a matrix. Further the-vectors y’ = {polo}, b’ = {l;‘lO} and the matrix F. = diag (Fo,, . . . , Fov) are introduced. For each step of the incremental process the structure behaves in a geometrically linear fashion. Thus the compatibility conditions can be written in the form
Here r* - 7 is a variation Sy of the value of r at the position of equilibrium, i.e. r* is a value in the neighbourhood of jj ; similarly the variation I?* - Cris defined. The proof, that each of the forms (21) and (22) is equivalent to the set of relations (6)-(g) is trivial: For p* =0, and r* = 29, form (21) yields eqn (8); for ST equal to vectors with zero’s in all their entries but one, where a unit is placed, form (21) yields the nonnegativity of Cr. Conversely relations (6)-(8) give 12’(7* - 7) = a’?* Z 0 for e=G’u (9) every r* z 0. A similar proof can be given for (22). Both forms are called variational inequalities. There exists a where G’ is the strain-displacement matrix of the problem duality relation between them, both in the mathematical and it is the increment of the nodal displacement vector. sense and in the physical sense. The mathematical duality Further the equations of equilibrium are linearized in each results from the fact that each of them is equivalent to the step by using the geometrical stiffness matrix[6] Kc and set of relations (6)-(8) with the difference that in (21) the variation of 9 has been taken, whereas in (22) the have the form variation of 6. The physical duality results from the fact GStKcti=R, (10) that form (21), (resp. (22)) together with relations (5), (7) (9), (10) (resp. (5), (6), (9), (10))represent all the conditions where Kc is a symmetric matrix depending on the of the problem. The physical meaning of (21), (resp. (22)) geometry and the stresses at the end of the preceding step, is, that the virtual work (resp. complementary virtual and I.tis the increment of the load vector. work) due to the variation of total slackness 9 (resp. total stresses) of the cables is non-negative in every step, over 3. VARIATIONAL INEQUALITY FORMULATlON the set of admissible variations defined by (5), (7), (9), (lo), (resp. (5), (6), (9), (10)). Using the notation introduced by OF THE PROBLEM (17)-( 19) and relation (12), the variational inequalities (21) In terms of 8, i the relations (5)-( 10)will be written as and (22), coupled with (15) and (16), can be written more KitG&+=fitGK&, (11) compactly as ~==t~=iioG'li-K~~+Ko~+BrO, (12) {(RUt CL)* (U* -U) 2 0 for every U* 2 0, U 2 0) (23) q=t+i;zo, (13) and 6+(-J (14) {U’R(U*-U)~OforeveryRU*t~~O,RUt~~O} where K,, = Fomland K = KE t KG = G&G’ + KG,and K is (24) the total stiffness of the structure. respectively. Equation (11) is written as It can be proved as before, that each of the preceding forms of the problem is equivalent to the form (20) of the KiitGK,+-R-GK&,zO (15) problem, which constitutes a “linear complementarity problem” [7]. This kind of problem appears frequently in -Ku-G&vtfitGK&rO (16) the theory of plasticity[8]. Applying for this discrete problem the results of the and variable ti as i= ti, -I-, where ti+ =(ti+~i~/2)~0 functional theory of variational inequalities [9, IO] it can and i_ = ( - li + jil/2) 2 0 componentwise. Then the vecbe concluded that the positive definiteness of matrix R tors guarantees the existence of a unique solution. The posiU’ = {li+‘,u’, p’}, (17) tive semidefiniteness of matrix R guarantees the existence of a convex set of solutions. c * = {( - GK,,b- fi - GK&)‘((GK,,b t fi t_GK&,)'((i&l; -Ko&+ii)‘} (18) 4.EXTREMUM PRINCIPLES and the symmetric matrix Suppose further that matrix R is positive semidefinite. The inequality GIL\ /K -K (19)
; (U*‘RU*- U'RU) 2 (U* - U)‘RU
(25)
135
A variational inequality approach to cable-structures
is valid, because it is equivalent to (U* - U)‘R(U* -U) 2 0,
(26)
which is satisfied for every U* and U. Forms (23) and (24) combined with (25) yield respectively: ~u*‘Ru*+~~~u*~~u~Ru+~U,U*ZO,UZO
I (27)
1 and
I
~u*Ru*z~u’RU,RU*+~
20,RU+p
20 . I (28)
solutions [ 11J. The whole set of solutions can be represented in the following manner. If U is a solution, U t E will be another solution, if inequality U t z 2 0 is valid and E satisfies the equations Re=O
and
p’c=O.
(33)
If R is positive definite, (33) yields E = 0 and the problem admits a unique solution. The difference between the two stress-increment fields corresponding to U and U t E respectively is denoted by es. Further the components of E corresponding to II+,u-, v are denoted respectively as cu+, 6.-, e,; eu =e,,- cu. in accordance with u = u+ =UL Equations (33) with eqns (12), (18), (19) yield K&‘4 t l&P” = 0.
(34)
It follows, that the solutions of the variational inequalities (23), (24) satisfy the following two quadratic minimization problems respectively:
The expression on the left hand side of eqn (34) gives the part of E, corresponding to the cables and is denoted by 4. Moreover from eqns (33) with the help of relations (5), (9), (18), (19) can be concluded that
min II,(U) = i U’RU + U’p ]U2 0
Gc, + KGe, = 0.
(29)
and min n,(U) = k U*RUIRUt p 2 0 . I
(30)
The nonnegativity of the first variation of II,(U) and IL(U) guarantees that the solution of (29) and (30) satisfies respectively the variational inequalities (23) and (24). Problems (21) and (30) constitute two quadratic minimization problems. By applying the Kuhn-Tucker theorem of nonlinear programming[ll, 121in problem (29) it can be proved again, that conditions (20) constitute the necessary and sufficient conditions of the problem. Moreover the duality theory of quadratic programming[ll] shows that the problem (30) is the dual of problem (29). The functions U, and II2 can be written in terms of I, i as
-ti’(GK&+p)+i’(a-Ko(iro-b))litb’O
Thus the cable elements have unique stress increment fields and in the remaining elements, the stress increment difference es is equilibrated by the difference of the displacement-increments. So in a structure having only cables, a cable-net for example, the stress field increments are unique. The properties of the dual problem (30) will now be discussed. It is known [ 111,that any solution of the primal problem (29) solves the dual problem (30) and the values of potential and complementary energy are equal. If U is any solution of the primal problem, and thus of the dual problem as well, the whole set of solutions of the dual problem can be given by U t E’, where E’ satisfies RE’ = 0.
I
(31)
(36)
Vector U + c’ satisfies the constraint R(Ute’)tp
min n,(i,4)=511K6+~~~Ka0+a’GKai) 1
(35)
20,
(37)
because U is a solution of the dual problem and e’ satisfies (36). From the Kuhn-Tucker conditions written for the dual problem, it results that E’ satisfies the equation
and PIG’ = 0. min U&i, v) = ; ri’ Ku + u’ G&v + u1G&h I +~b’~ttQ’~b~under:(ll)and(12)
1
. (32)
It is obvious that the functions l$ and U, represent the potential energy and the complementary energy respectively of the cable-structure. The formulated theory takes into account the loosening of the cables and for this reason the term ir enters in the expressions of II1 and II,. If v equals zero, expressions U, and HZ reduce, to the expressions of potential and complementary energy of usual structures. 5. ON THE EXISTENCEAND UNIQUENESS OF THE SOLUTION
If matrix R is positive semidefinite, the quadratic minimization problem (29) admits a convex set of
(38)
It is obvious that the subset of the solutions of the dual problem satisfying the inequality U t l’ z 0 represents exactly the set of solutions of the primal problem. If R is positive definite, then the dual problem has only one solution, the same as the primal problem. The preceding analysis gives as a result, that in the neighbourhood of the state of the structure at the end of a step there exists a set of configurations, which are in equilibrium under the same external actions. A detailed discussion of the equilibrium configurations is given in[4]. 6. NLJhiERICALCALCULATION. DECOMPOSITIONTECHNIQUES
For the numerical calculation of the problem the extremum principles (29) and (30) are used, written either in terms of vector U or in terms of the other unknown vectors of the problem[4]. We distinguish two cases according to whether or not, matrix R is positive semidefi-
P. D.
136
PANAGIOTOPOULOS
nite. In the first case the minimization problems (21) and (30) constitute convex quadratic minimization problems. For these problems the Kuhn-Tucker conditions are the necessary and sufficient conditions for a global minimum, and the fast and accurate algorithms of quadratic programming[ll, 131 can be used. Among the several numerical procedures proposed in quadratic programming, that due to Wolfe [ 11,131seems to be computationally more interesting, because it uses the Simplex algorithm of linear programming. The algorithm of Wolfe has been applied successfully in[5,6] for the elastic, stress unilateral calculation of cable-nets and is recommended for the inelastic, stress unilateral analysis as well. In most of the problems of practical interest matrix R is indeed positive semidefinite. A sufficient condition for this is matrix KG to be positive semidefinite. This happens in the case of cable nets, where the structure is discretized only by means of pin-jointed bar-elements. In the second and more general case, where matrix R is not positive semidefinite, vector U should be calculated, so as to satisfy the relations (20), which are the basic conditions of the problem. They represent the “local” Kuhn-Tucker conditions of the minimization problem (29), which is non convex in general; hence the Kuhn-Tucker conditions are necessary but not sufficient for a vector to be optimal. Thus any vector U corresponding to a local or global minimum of problem (29) satisfies the conditions (25) and accordingly it corresponds to a stable configuration of the cable structure, but the converse is not generally true. Moreover it is not possible to give a method for the construction of all the solutions of the problem, as when matrix R is positive semidefinite. For the numerical calculation of this case the algorithm of Davidon-Fletcher-Powell enriched with the modifications of Goldfarb and with the “Created Response Surface Technique” (C.R.S.T.) to accomodate the constraints, has been used[l4]. For the unconstrained problem it reduces to the algorithm of Davidon-FletcherPowell [ 141.In essence, the used algorithm belongs to the large step gradient methods or quasi-Newton methods and is coupled with a generalized projection process. The problem to be solved can be regarded as a special case of the following constrained minimization problem: {min f(x)/gi(x) Z 0 i = 1,2, . . . , r},
(39)
where f(x), g(x) are twice differentiable functions of a vector x; accordingly their gradient vectors and their Hessian matrices are available for any x. The algorithm is initiated at a feasible point x”” and continues by the method of Davidon-Fletcher-Powell, until a point xck)is reached at which one or more, say 1, of the constraints become active, i.e. ~,(x’~‘)=O i=1,2 ,...,
15n.
Then the algorithm projects in the generalized sense, V~(X’~‘) on to the set of the 1 active constraints, and let s(‘) be the unit vector in the direction of the projection. By moving from xck) along the projected gradient with a unidimensional search it is possible to find a A that yields the minimum value of f(x) and keeps X(k+l) = ,ck,+ *‘k’s@’tn the feasible region. In the case, where the constraints are linear, the optimization proceeds in large steps and yields quite easily a local minimum corresponding to a stable configuration of the structure. In most of the cable structures the number of
cables and nodes is large and accordingly the arising optimization problem has a large number of unknowns and of inequalities. For this reason a “multilevel optimization technique”[l4] suited for this kind of optimization problems will be applied. The whole system to be optimized, is decomposed into a number of subsystems. In the “first level” of the calculation every subsystem is optimized separately and in the “second level” the solutions of these subsystems are combined to yield the overall system optimum. Here the initial system, i.e. the cablestructure, is decomposed into two subsystems, i.e. substructures, a subsystem which presents only the displacement terms and corresponds to a structure resulting from the given one by transforming all the cables into bars and a subsystem, which presents only the slackness terms and corresponds to a fictitious structure. To perform the decomposition, form (31) of the potential energy is written as II@, v) = II;(u) + II:(v) t ti'GK&,
(40)
where II;(n) =; n’Kri - ti’(GK& t 0)
(41)
1 II@) = - i’Ki+i’(a2
(42)
and Ko(Co-b)).
A new variable w is introduced and the minimization problem (31) is written as min {II&i, v, w) = II’,(i) + II’;(i) t ti' G&WI+= w,ir t b 2 0). (43) The Lagrangian of (43) is n,(li,i,W)=n,(li,6,w)tp’(~-w),
(4)
where p is the vector of the Lagrange multipliers. The decomposition of the initial problem into the two subproblems can be achieved using two methods: The nonfeasible gradient controller method of Lasdon and Schoeffler [ 151 and the feasible gradient controller method of Brosilow, Lasdon, Pearson[l6]. In the nonfeasible gradient controller method the value of p is taken as constant, say p1 in the first level and the minimization problem splits into the two following subproblems (Fig. 2): min{II\(i) tdG&w-p,'w} .“.Y
(45)
min {II@) t p,‘iilir + b 2 0}, Y
(46)
and
The nonfeasible gradient controller method.
A variationalinequalityapproachto cable-structures
where problem (45) is an unconstrained minimization problem and (46) is a sign constrained minimization problem. As a result of the optimization the values of h, v, w, say ri,, v,, wr, are calculated. It is obvious that ir, = wl. The task of the “second level” is to estimate a new value of p, say p2, by means of the equation p2 = p, + k(i, - w,), where k > 0 is a properly chosen constant, and to transmit it to the “first level”. The optimization is performed again and the new values ti2, C,, w2are obtained and so on until the decreasing differences v2- w2are made small enough. It is proved in[14], that the algorithm converges in a finite number of steps if the minima exist. In the feasible gradient controller method the value of w is taken as constant, say w,, in the first level, and the minimization problem splits into the two following subproblems (Fig. 3): min {II;(i) + ri’G&w,} Ii
(47)
and min {II;(i) + p’(v V,P
wJC
+
b 2 0).
(48)
As a result of the optimization the values of ti, i, p, say u,, i,, p,, are calculated. The task of the “second level” is to estimate a new value of w, say wl, and to transmit it to the “first level”. The value w2 is given[l3] by the equation wz =
WI -
k
l3rI,(ti, i, w)
aw
>w=w,’
(49)
where k > 0 is a properly chosen constant. The optimization is performed again and yields the new values riZ,vZ,p2 and so on until the difference between consecutive values of vector w is made satisfactorily small. It is obvious that the decomposition algorithms can be applied by fixing other variables, which will give rise to substructures, with different physical meanings. The developed methods of numerical calculation and of decomposition have been applied to the inelastic analysis of the cable-structure of Fig. 4. The upper cable members (generally the carrying cables) have cross-sectional area F. = 4.00 cm’, the lower cable members (generally tension cables) have cross-sectional area F, = 1.40cm’ and the vertical cables have F, = 1.20cm’. The modulus of elasticity is common for all the cables and is equal to 1600MP/cm*. The upper and lower cables form symmet-
Fig. 3.The feasible gradient controller method,
137
Fig. 4. Dimensions and numbering of the cable-structure. ric parabolas. The common horizontal pretension in all the
upper and lower cable-members is 8.0 MP. The following two kinds of proportional loading were considered, increasing the uniform load applied to the upper cable elements by 0.01 t/m at each step. (a) symmetric loading, over the whole upper cable, ranging from 0.01 L 0.1 MP/m per unit length of the Ox axis, (b) nonsymmetric loading acting only on the half of the upper cable and ranging from 0.01 to 0.1 MP/m per unit length of the axis Ox. To simplify the problem, only the vertical displacements of the nodes are taken into account in the calculations. The solution has been obtained for an elastic, an elasticperfectly plastic, and a workhardening behaviour of the cable-elements (Fig. 5). The negative pretension stress so, is different in each cable-element and is equal to the pretension force divided by the cross-sectional area of the cable. For the decomposition the nonfeasible technique has been used, which for this problem has a faster convergence than the feasible decomposition technique. For every step of the incremental calculation, i.e. for every load increment two kinds of interation processes take place: tirst the iterations of the nonfeasible gradient controller method and second the iterations of the minimization algorithm. The whole process with Wolfe’s algorithm, which is applicable to this problem, was about two times faster than with the Davidon-Fletcher-Powell algorithm. The same example was computed without decomposition by using Wolfe’s algorithm. It was clear that the decomposition, which for large cable-structures is inevitable, was a reason of inaccuracies in the calculation. The number of iterations was considerably affected by the estimation of the starting point of the algorithm. The whole problem was programmed on a UNIVAC 1108 computer. The subroutines given in[13,14] have been used in a slightly modified form. From the variation of the tensions versus the variation of the load it is concluded that the assumption of elastic behaviour of the structure (Fig. 5a) yields the largest tensions in the cables. The elastic-perfectly plastic assumption (Fig. 5b) gives the lowest tensions in the cables. The workhardening assumption (Fig. 5c) gives tensions between the tensions of the other two cases. It is clear that the developed theory permits us to calculate the cable-structures in their “unstable region” because the possibility that a cable-element can become slack is taken into account. Accordingly the initial tension can be re-
lb.
Fig. 5. The stress-strain diagrams of the cable-elements.
138
P. D.
PANAGIOTOPOULOS
(cm)
(MP)
20
323
24 --
16 --
0 Fig. 6. Vertical displacement of the central
node (symmetric load).
O.b5
O,l(Mp/m)
Fig. 8. Tensions of cables l-2 and 7-8 vs the variation of the symmetric load.
(Mp)
(cm)
15f
I
0
Fig. 7. Verticaldisplacementsof the nodes 2.8 and 3.9 (nonsymmetric load).
duced and the resulting stress-field leads to a more economic design of the structure. It can be said that the design of cable-structures, according to the developed theory, has the same characteristics and advantages as the non elastic limit design of structures. Acknowledgements-Thispaper is a part of a research program sponsored by the “Alexander von Humboldt Stiftung” at the “Institut fiir Technische Mechanik der RWTH Aachen”. The author’s grateful thanks are due to the “Alexander von Humboldt Stiftung” for the financial support of this work and to Prof. Dr. rer. nat. G. Rieder for the continuous encouragement and support and to Mr. B. R. Witherden, B. A., M. SC. Oxon. for help in the exposition of this paper. REPERENCES
1. G. Nitsiotas, Zur Bildungund Berechnung statisch bestimmter Tragwerke mit Verbindungen aus Seilen. Bauingenieur 42, 412414 (1967). 2. G. Nitsiotas, Die Berechnung statisch umbestimmter Tragwerke mit einseitigen Bindungen. Ing. Archio. 41, 4660 (19701). 3. P. D. Panagiotopoulos, Nonlinear theory of continuous sunilateral structures. Doctor Thesis, A.U.T. (1974).
0.05
O,l(Mp/m)
Fig. 9. Tensions of cables 1-2 and 7-8 vs the variation of the nonsymmetric load.
4. P. D. Panagiotopoulos, Stress-unilateral analysis of discretized cable and membrane structures in the presence of large displacements. Ing. Archiu 45 (1975)to be published. 5. P. D. Panagiotopoulos, S-unilateraltheory of cable-structures. Proc. of the “Internat. Symposium on Cable-Structures”, Bratislaua (1975). 6. .I. Argyris, Three-dimensional anisotropic and inhomogeneous elastic media matrix analysis for small and large displacements. Ing. Archiv. 34, 33-55 (1965). 7. G. B. Dantzig and R. W. Cottle, Positive (Semi)-definite programming. In Nonlinear Programming (Edited by .I. Abadie), North-Holland, Amsterdam (1967). 8. G. Maier, Mathematical programming methods in structural analysis. In Variational Methods in Engineering. Southampton Univ. Press, Southampton (1973). 9. A. Auslender, Problbmes de minimax via l’analyse convexe et les ini?galit&svariationelles: Theorie et algorithmes. Springer Verlag (1972). 10. J. L. Lions, ContrBle des systtmes gouvemCs par des Cquations aux d&iv&espartielles. Dunod, Paris (1968). 11. H. P. Ktinzi and W. Krelle, Nichtlineare Programmierung. Springer Verlag, Berlin (1972).English translation: Nonlinear Programming, Blaisdell (1966). 12. D. Wismer, Optimization Methods for Large-scale Systems with Applications. McGraw-Hill, New York (1971).
A variational inequality approach to cable-structures 13. H. P. Kilnzi, H. Tzschach and C. Zehnder, Mathematische Optimierang. Teubner Verlag, Berlin (1966). 14. E. J. Bauman, Multilevel optimization techniques with application to travectory decomposition. Advances in Control Systems (Edited by C. T. Leondes), Vol. 6. Academic Press, New York (1968). 15. L. S. Lasdon and I. D. Schoeffler, A multilevel technique for
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optimization. Proc. “Joint Autom. Control Conf” Rensselaer Polytechn. Inst., Troy, New York (1965). 16. C. B. Brosilov, L. S. Lasdon and J. D. Pearson, Feasible optimization methods for interconnected systems. Proc. “Joint Autom. Control Conf.” Rensselaer Troy, New York (1%5).
Polytechn. Inst.
A VARIATIONAL INEQUALITY APPROACH TO THE INELASTIC STRESS-UNILATERAL ANALYSIS OF CABLE-STRUCTURES P. D. PANAGIOTOPOULOS Institute of Technical Mechanics, The Technical University of Aachen, D 51 Aachen, Federal Republic of Germany (Received 18August 1975) Abstract-The present paper deals with the inelastic, stress-unilateral analysis of cable-structures undergoing large displacements. The response of the cable-structures to load- and initial strain-increments is described by a set of equations and inequalities, which are formulated compactly as variational inequalities. Two dual extremum principles-generalizations of the minimum of the potential and complementary energy-with inequalities as subsidiary conditions are derived. The numericalcalculationis preformediterativelyby using the decomposition
techniquesof the multileveloptimizationand applyingin the resulting substructuresthe algorithmsof non linear optimization.As an example the elastic,elastic-perfectlyplasticand workhardeningbehaviourof a cable-structureis examinedand some useful results from the comparisonof the solutions are obtained. INTRODUCTION
A structure, some of whose elements are not capable of transmitting compression, i.e. negative stress, is called a stress-unilateral (s-unilateral) structure. The cablestructures in this paper are considered to be stressunilateral structures. The nonnegativity inequality for the stresses of the cable-elements is taken as the principal condition in the formulation of the mathematical model. The problem is nonlinear not only because of the presence of the stress-inequalities, but also because the considered stress-strain law is non linear and the formulated theory is a large displacement theory, as usual in cable-structures. The compatibility and equilibrium equations of the structure are linearized by considering “incremental” processes caused by infinitesimal external actions added to the known situation of the structure. We suppose that any load and initial strain may be represented by a sequence of increments of properly chosen small amplitude. The behaviour of the cable-structure can be described by a set of equations and inequalities which will be formulated as variational inequalities. The stress-strain diagram may have any form. Here the calculation is performed for the cases of elastic, elasticperfectly plastic, and workhardening behaviour of the cable-elements and the corresponding results are discussed. The stress inequality for the cable-structures has been introduced in [ 1,2]. In[31 the continuous functional theory of stress unilateral cable structures and in[4,5] the elastic stress-unilateral analysis of discretized cable and membrane structures have been developed. 2. FINITE ELEMENTS DISCRETIZATION The structure is divided into n finite elements, of which u are cables. For the cables, pin-jointed bar elements are used. (The rigid body degrees of freedom of the elements are eliminated when they are unassembled, i.e. the elemental stress and strain states are defined by means of the natural stress and strain vectors [6].) Because of the large displacements and the nonlinearity of the constitutive law (Fig. 1) the developed theory is incremental. Each increment of a quantity is denoted by a dot on the symbol of the quantity. The configuration of the structure after the application of the pretension is called “initial state”. With reference to the initial state and to a fixed Cartesian
system common for the whole structure, the incremental stress-strain relations have for the ith cable the following form ii = F&i + A,,8- tii (1) ai = si t ai z 0
(2)
p = di + bi 2 0
(3)
u,yi = 0
(4)
where si, ei, s~,~,eo,i denote the stress, strain, initial stress and initial strain of the ith cable respectively. fO.iis the tangential natural flexibility matrix of the cable at the step under consideration. (- vi) is the slackness of the cable, a/(-bi) is the total stress/total slackness of the cable at the beginning of an incremental step, ui/(- yi) is the total stress/total slackness (Fig. 1) of the cable at the end of the step. Relations (l)-(4) express, that, at the end of each step either a non-negative total stress or a non-positive total slackness exists on any cable (Fig. 1). It is clear that the slackness (- u,) can be viewed as an unknown initial strain, which constitutes a reversible negative elongation. If local unloading occurs in the plastic region ABC (Fig. 1) of the ei - si diagram of a cable, hysteretic strains will appear and the new diagram EDBC is taken into account for the calculation of the flexibility matrix of the cable in the subsequent steps. If in a step qi = 0, the cable behaves like a usual pin-jointed bar element and has a positive stress. If in some step +i > 0, the cable has no tension and thus it does not contribute to the stability of the whole structure. Relation (1) represents for di = 0 the stress-strain relations of every one of the remaining n - ~7usual elements of the structure.
Fig. 1. Stress-strain law for a cable-element. 133
134
P. D. PANAGIOTOPOULOS
Relations (l)-(4) can be assembled for all the elements of the structure, as follows: e=F&t&,-ir,
(5)
c?=itii,o,
(6)
7 =Gtbro,
(7
o’y = 0.
are also introduced. Using these substititious relations (ll)-(14) can be written more compactly in the form {RUtpZO,
(20)
Relations (6)-(g) can be written equivalently under the two following inequality forms
(8)
+.’ = s’ = {S,‘, . . . ) S,‘}, Here c’ = If!,‘,. . . , i.‘}, {IA’, . . . , tivi,o,. . . , 0) = {? IO}and Fo is the tangential flexi-
UrO U’(RUt~)=O}.
{a’(?*-y)ZO,for
every ~*ZO,yZO}
(21)
and {?‘(a*-6)rO,forevery@*rO,&ZO}.
(22)
bility matrix of the unassembled elements at each incremental step. The part of a vector or a matrix extended over the cables is denoted by a bar. Index t denotes the transpose of a vector or a matrix. Further the-vectors y’ = {polo}, b’ = {l;‘lO} and the matrix F. = diag (Fo,, . . . , Fov) are introduced. For each step of the incremental process the structure behaves in a geometrically linear fashion. Thus the compatibility conditions can be written in the form
Here r* - 7 is a variation Sy of the value of r at the position of equilibrium, i.e. r* is a value in the neighbourhood of jj ; similarly the variation I?* - Cris defined. The proof, that each of the forms (21) and (22) is equivalent to the set of relations (6)-(g) is trivial: For p* =0, and r* = 29, form (21) yields eqn (8); for ST equal to vectors with zero’s in all their entries but one, where a unit is placed, form (21) yields the nonnegativity of Cr. Conversely relations (6)-(8) give 12’(7* - 7) = a’?* Z 0 for e=G’u (9) every r* z 0. A similar proof can be given for (22). Both forms are called variational inequalities. There exists a where G’ is the strain-displacement matrix of the problem duality relation between them, both in the mathematical and it is the increment of the nodal displacement vector. sense and in the physical sense. The mathematical duality Further the equations of equilibrium are linearized in each results from the fact that each of them is equivalent to the step by using the geometrical stiffness matrix[6] Kc and set of relations (6)-(8) with the difference that in (21) the variation of 9 has been taken, whereas in (22) the have the form variation of 6. The physical duality results from the fact GStKcti=R, (10) that form (21), (resp. (22)) together with relations (5), (7) (9), (10) (resp. (5), (6), (9), (10))represent all the conditions where Kc is a symmetric matrix depending on the of the problem. The physical meaning of (21), (resp. (22)) geometry and the stresses at the end of the preceding step, is, that the virtual work (resp. complementary virtual and I.tis the increment of the load vector. work) due to the variation of total slackness 9 (resp. total stresses) of the cables is non-negative in every step, over 3. VARIATIONAL INEQUALITY FORMULATlON the set of admissible variations defined by (5), (7), (9), (lo), (resp. (5), (6), (9), (10)). Using the notation introduced by OF THE PROBLEM (17)-( 19) and relation (12), the variational inequalities (21) In terms of 8, i the relations (5)-( 10)will be written as and (22), coupled with (15) and (16), can be written more KitG&+=fitGK&, (11) compactly as ~==t~=iioG'li-K~~+Ko~+BrO, (12) {(RUt CL)* (U* -U) 2 0 for every U* 2 0, U 2 0) (23) q=t+i;zo, (13) and 6+(-J (14) {U’R(U*-U)~OforeveryRU*t~~O,RUt~~O} where K,, = Fomland K = KE t KG = G&G’ + KG,and K is (24) the total stiffness of the structure. respectively. Equation (11) is written as It can be proved as before, that each of the preceding forms of the problem is equivalent to the form (20) of the KiitGK,+-R-GK&,zO (15) problem, which constitutes a “linear complementarity problem” [7]. This kind of problem appears frequently in -Ku-G&vtfitGK&rO (16) the theory of plasticity[8]. Applying for this discrete problem the results of the and variable ti as i= ti, -I-, where ti+ =(ti+~i~/2)~0 functional theory of variational inequalities [9, IO] it can and i_ = ( - li + jil/2) 2 0 componentwise. Then the vecbe concluded that the positive definiteness of matrix R tors guarantees the existence of a unique solution. The posiU’ = {li+‘,u’, p’}, (17) tive semidefiniteness of matrix R guarantees the existence of a convex set of solutions. c * = {( - GK,,b- fi - GK&)‘((GK,,b t fi t_GK&,)'((i&l; -Ko&+ii)‘} (18) 4.EXTREMUM PRINCIPLES and the symmetric matrix Suppose further that matrix R is positive semidefinite. The inequality GIL\ /K -K (19)
; (U*‘RU*- U'RU) 2 (U* - U)‘RU
(25)
135
A variational inequality approach to cable-structures
is valid, because it is equivalent to (U* - U)‘R(U* -U) 2 0,
(26)
which is satisfied for every U* and U. Forms (23) and (24) combined with (25) yield respectively: ~u*‘Ru*+~~~u*~~u~Ru+~U,U*ZO,UZO
I (27)
1 and
I
~u*Ru*z~u’RU,RU*+~
20,RU+p
20 . I (28)
solutions [ 11J. The whole set of solutions can be represented in the following manner. If U is a solution, U t E will be another solution, if inequality U t z 2 0 is valid and E satisfies the equations Re=O
and
p’c=O.
(33)
If R is positive definite, (33) yields E = 0 and the problem admits a unique solution. The difference between the two stress-increment fields corresponding to U and U t E respectively is denoted by es. Further the components of E corresponding to II+,u-, v are denoted respectively as cu+, 6.-, e,; eu =e,,- cu. in accordance with u = u+ =UL Equations (33) with eqns (12), (18), (19) yield K&‘4 t l&P” = 0.
(34)
It follows, that the solutions of the variational inequalities (23), (24) satisfy the following two quadratic minimization problems respectively:
The expression on the left hand side of eqn (34) gives the part of E, corresponding to the cables and is denoted by 4. Moreover from eqns (33) with the help of relations (5), (9), (18), (19) can be concluded that
min II,(U) = i U’RU + U’p ]U2 0
Gc, + KGe, = 0.
(29)
and min n,(U) = k U*RUIRUt p 2 0 . I
(30)
The nonnegativity of the first variation of II,(U) and IL(U) guarantees that the solution of (29) and (30) satisfies respectively the variational inequalities (23) and (24). Problems (21) and (30) constitute two quadratic minimization problems. By applying the Kuhn-Tucker theorem of nonlinear programming[ll, 121in problem (29) it can be proved again, that conditions (20) constitute the necessary and sufficient conditions of the problem. Moreover the duality theory of quadratic programming[ll] shows that the problem (30) is the dual of problem (29). The functions U, and II2 can be written in terms of I, i as
-ti’(GK&+p)+i’(a-Ko(iro-b))litb’O
Thus the cable elements have unique stress increment fields and in the remaining elements, the stress increment difference es is equilibrated by the difference of the displacement-increments. So in a structure having only cables, a cable-net for example, the stress field increments are unique. The properties of the dual problem (30) will now be discussed. It is known [ 111,that any solution of the primal problem (29) solves the dual problem (30) and the values of potential and complementary energy are equal. If U is any solution of the primal problem, and thus of the dual problem as well, the whole set of solutions of the dual problem can be given by U t E’, where E’ satisfies RE’ = 0.
I
(31)
(36)
Vector U + c’ satisfies the constraint R(Ute’)tp
min n,(i,4)=511K6+~~~Ka0+a’GKai) 1
(35)
20,
(37)
because U is a solution of the dual problem and e’ satisfies (36). From the Kuhn-Tucker conditions written for the dual problem, it results that E’ satisfies the equation
and PIG’ = 0. min U&i, v) = ; ri’ Ku + u’ G&v + u1G&h I +~b’~ttQ’~b~under:(ll)and(12)
1
. (32)
It is obvious that the functions l$ and U, represent the potential energy and the complementary energy respectively of the cable-structure. The formulated theory takes into account the loosening of the cables and for this reason the term ir enters in the expressions of II1 and II,. If v equals zero, expressions U, and HZ reduce, to the expressions of potential and complementary energy of usual structures. 5. ON THE EXISTENCEAND UNIQUENESS OF THE SOLUTION
If matrix R is positive semidefinite, the quadratic minimization problem (29) admits a convex set of
(38)
It is obvious that the subset of the solutions of the dual problem satisfying the inequality U t l’ z 0 represents exactly the set of solutions of the primal problem. If R is positive definite, then the dual problem has only one solution, the same as the primal problem. The preceding analysis gives as a result, that in the neighbourhood of the state of the structure at the end of a step there exists a set of configurations, which are in equilibrium under the same external actions. A detailed discussion of the equilibrium configurations is given in[4]. 6. NLJhiERICALCALCULATION. DECOMPOSITIONTECHNIQUES
For the numerical calculation of the problem the extremum principles (29) and (30) are used, written either in terms of vector U or in terms of the other unknown vectors of the problem[4]. We distinguish two cases according to whether or not, matrix R is positive semidefi-
P. D.
136
PANAGIOTOPOULOS
nite. In the first case the minimization problems (21) and (30) constitute convex quadratic minimization problems. For these problems the Kuhn-Tucker conditions are the necessary and sufficient conditions for a global minimum, and the fast and accurate algorithms of quadratic programming[ll, 131 can be used. Among the several numerical procedures proposed in quadratic programming, that due to Wolfe [ 11,131seems to be computationally more interesting, because it uses the Simplex algorithm of linear programming. The algorithm of Wolfe has been applied successfully in[5,6] for the elastic, stress unilateral calculation of cable-nets and is recommended for the inelastic, stress unilateral analysis as well. In most of the problems of practical interest matrix R is indeed positive semidefinite. A sufficient condition for this is matrix KG to be positive semidefinite. This happens in the case of cable nets, where the structure is discretized only by means of pin-jointed bar-elements. In the second and more general case, where matrix R is not positive semidefinite, vector U should be calculated, so as to satisfy the relations (20), which are the basic conditions of the problem. They represent the “local” Kuhn-Tucker conditions of the minimization problem (29), which is non convex in general; hence the Kuhn-Tucker conditions are necessary but not sufficient for a vector to be optimal. Thus any vector U corresponding to a local or global minimum of problem (29) satisfies the conditions (25) and accordingly it corresponds to a stable configuration of the cable structure, but the converse is not generally true. Moreover it is not possible to give a method for the construction of all the solutions of the problem, as when matrix R is positive semidefinite. For the numerical calculation of this case the algorithm of Davidon-Fletcher-Powell enriched with the modifications of Goldfarb and with the “Created Response Surface Technique” (C.R.S.T.) to accomodate the constraints, has been used[l4]. For the unconstrained problem it reduces to the algorithm of Davidon-FletcherPowell [ 141.In essence, the used algorithm belongs to the large step gradient methods or quasi-Newton methods and is coupled with a generalized projection process. The problem to be solved can be regarded as a special case of the following constrained minimization problem: {min f(x)/gi(x) Z 0 i = 1,2, . . . , r},
(39)
where f(x), g(x) are twice differentiable functions of a vector x; accordingly their gradient vectors and their Hessian matrices are available for any x. The algorithm is initiated at a feasible point x”” and continues by the method of Davidon-Fletcher-Powell, until a point xck)is reached at which one or more, say 1, of the constraints become active, i.e. ~,(x’~‘)=O i=1,2 ,...,
15n.
Then the algorithm projects in the generalized sense, V~(X’~‘) on to the set of the 1 active constraints, and let s(‘) be the unit vector in the direction of the projection. By moving from xck) along the projected gradient with a unidimensional search it is possible to find a A that yields the minimum value of f(x) and keeps X(k+l) = ,ck,+ *‘k’s@’tn the feasible region. In the case, where the constraints are linear, the optimization proceeds in large steps and yields quite easily a local minimum corresponding to a stable configuration of the structure. In most of the cable structures the number of
cables and nodes is large and accordingly the arising optimization problem has a large number of unknowns and of inequalities. For this reason a “multilevel optimization technique”[l4] suited for this kind of optimization problems will be applied. The whole system to be optimized, is decomposed into a number of subsystems. In the “first level” of the calculation every subsystem is optimized separately and in the “second level” the solutions of these subsystems are combined to yield the overall system optimum. Here the initial system, i.e. the cablestructure, is decomposed into two subsystems, i.e. substructures, a subsystem which presents only the displacement terms and corresponds to a structure resulting from the given one by transforming all the cables into bars and a subsystem, which presents only the slackness terms and corresponds to a fictitious structure. To perform the decomposition, form (31) of the potential energy is written as II@, v) = II;(u) + II:(v) t ti'GK&,
(40)
where II;(n) =; n’Kri - ti’(GK& t 0)
(41)
1 II@) = - i’Ki+i’(a2
(42)
and Ko(Co-b)).
A new variable w is introduced and the minimization problem (31) is written as min {II&i, v, w) = II’,(i) + II’;(i) t ti' G&WI+= w,ir t b 2 0). (43) The Lagrangian of (43) is n,(li,i,W)=n,(li,6,w)tp’(~-w),
(4)
where p is the vector of the Lagrange multipliers. The decomposition of the initial problem into the two subproblems can be achieved using two methods: The nonfeasible gradient controller method of Lasdon and Schoeffler [ 151 and the feasible gradient controller method of Brosilow, Lasdon, Pearson[l6]. In the nonfeasible gradient controller method the value of p is taken as constant, say p1 in the first level and the minimization problem splits into the two following subproblems (Fig. 2): min{II\(i) tdG&w-p,'w} .“.Y
(45)
min {II@) t p,‘iilir + b 2 0}, Y
(46)
and
The nonfeasible gradient controller method.
A variationalinequalityapproachto cable-structures
where problem (45) is an unconstrained minimization problem and (46) is a sign constrained minimization problem. As a result of the optimization the values of h, v, w, say ri,, v,, wr, are calculated. It is obvious that ir, = wl. The task of the “second level” is to estimate a new value of p, say p2, by means of the equation p2 = p, + k(i, - w,), where k > 0 is a properly chosen constant, and to transmit it to the “first level”. The optimization is performed again and the new values ti2, C,, w2are obtained and so on until the decreasing differences v2- w2are made small enough. It is proved in[14], that the algorithm converges in a finite number of steps if the minima exist. In the feasible gradient controller method the value of w is taken as constant, say w,, in the first level, and the minimization problem splits into the two following subproblems (Fig. 3): min {II;(i) + ri’G&w,} Ii
(47)
and min {II;(i) + p’(v V,P
wJC
+
b 2 0).
(48)
As a result of the optimization the values of ti, i, p, say u,, i,, p,, are calculated. The task of the “second level” is to estimate a new value of w, say wl, and to transmit it to the “first level”. The value w2 is given[l3] by the equation wz =
WI -
k
l3rI,(ti, i, w)
aw
>w=w,’
(49)
where k > 0 is a properly chosen constant. The optimization is performed again and yields the new values riZ,vZ,p2 and so on until the difference between consecutive values of vector w is made satisfactorily small. It is obvious that the decomposition algorithms can be applied by fixing other variables, which will give rise to substructures, with different physical meanings. The developed methods of numerical calculation and of decomposition have been applied to the inelastic analysis of the cable-structure of Fig. 4. The upper cable members (generally the carrying cables) have cross-sectional area F. = 4.00 cm’, the lower cable members (generally tension cables) have cross-sectional area F, = 1.40cm’ and the vertical cables have F, = 1.20cm’. The modulus of elasticity is common for all the cables and is equal to 1600MP/cm*. The upper and lower cables form symmet-
Fig. 3.The feasible gradient controller method,
137
Fig. 4. Dimensions and numbering of the cable-structure. ric parabolas. The common horizontal pretension in all the
upper and lower cable-members is 8.0 MP. The following two kinds of proportional loading were considered, increasing the uniform load applied to the upper cable elements by 0.01 t/m at each step. (a) symmetric loading, over the whole upper cable, ranging from 0.01 L 0.1 MP/m per unit length of the Ox axis, (b) nonsymmetric loading acting only on the half of the upper cable and ranging from 0.01 to 0.1 MP/m per unit length of the axis Ox. To simplify the problem, only the vertical displacements of the nodes are taken into account in the calculations. The solution has been obtained for an elastic, an elasticperfectly plastic, and a workhardening behaviour of the cable-elements (Fig. 5). The negative pretension stress so, is different in each cable-element and is equal to the pretension force divided by the cross-sectional area of the cable. For the decomposition the nonfeasible technique has been used, which for this problem has a faster convergence than the feasible decomposition technique. For every step of the incremental calculation, i.e. for every load increment two kinds of interation processes take place: tirst the iterations of the nonfeasible gradient controller method and second the iterations of the minimization algorithm. The whole process with Wolfe’s algorithm, which is applicable to this problem, was about two times faster than with the Davidon-Fletcher-Powell algorithm. The same example was computed without decomposition by using Wolfe’s algorithm. It was clear that the decomposition, which for large cable-structures is inevitable, was a reason of inaccuracies in the calculation. The number of iterations was considerably affected by the estimation of the starting point of the algorithm. The whole problem was programmed on a UNIVAC 1108 computer. The subroutines given in[13,14] have been used in a slightly modified form. From the variation of the tensions versus the variation of the load it is concluded that the assumption of elastic behaviour of the structure (Fig. 5a) yields the largest tensions in the cables. The elastic-perfectly plastic assumption (Fig. 5b) gives the lowest tensions in the cables. The workhardening assumption (Fig. 5c) gives tensions between the tensions of the other two cases. It is clear that the developed theory permits us to calculate the cable-structures in their “unstable region” because the possibility that a cable-element can become slack is taken into account. Accordingly the initial tension can be re-
lb.
Fig. 5. The stress-strain diagrams of the cable-elements.
138
P. D.
PANAGIOTOPOULOS
(cm)
(MP)
20
323
24 --
16 --
0 Fig. 6. Vertical displacement of the central
node (symmetric load).
O.b5
O,l(Mp/m)
Fig. 8. Tensions of cables l-2 and 7-8 vs the variation of the symmetric load.
(Mp)
(cm)
15f
I
0
Fig. 7. Verticaldisplacementsof the nodes 2.8 and 3.9 (nonsymmetric load).
duced and the resulting stress-field leads to a more economic design of the structure. It can be said that the design of cable-structures, according to the developed theory, has the same characteristics and advantages as the non elastic limit design of structures. Acknowledgements-Thispaper is a part of a research program sponsored by the “Alexander von Humboldt Stiftung” at the “Institut fiir Technische Mechanik der RWTH Aachen”. The author’s grateful thanks are due to the “Alexander von Humboldt Stiftung” for the financial support of this work and to Prof. Dr. rer. nat. G. Rieder for the continuous encouragement and support and to Mr. B. R. Witherden, B. A., M. SC. Oxon. for help in the exposition of this paper. REPERENCES
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0.05
O,l(Mp/m)
Fig. 9. Tensions of cables 1-2 and 7-8 vs the variation of the nonsymmetric load.
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optimization. Proc. “Joint Autom. Control Conf” Rensselaer Polytechn. Inst., Troy, New York (1965). 16. C. B. Brosilov, L. S. Lasdon and J. D. Pearson, Feasible optimization methods for interconnected systems. Proc. “Joint Autom. Control Conf.” Rensselaer Troy, New York (1%5).
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