Analysis of Slackened Skeletal Structures by Mathematical Programming

PERGAMON

Computers and Structures 69 (1998) 639±654

Analysis of slackened skeletal structures by mathematical programming A. Gawe° cki *, M. S. Kuczma, P. KruÈger Institute of Structural Engineering, Poznan University of Technology, ul. Piotrowo 5, 60-965, Poznan, Poland Received 15 July 1997; accepted 17 November 1997

Abstract The paper deals with the holonomic behavior of slackened-elastic±plastic (SEP) skeletal structures (beams, frames, trusses) using the quadratic programming formulations. The considerations are restricted to `frictionless' quasi-static processes, small strains and piecewise linear approximations of the yield and clearance surfaces. The results of numerical experiments for several illustrative examples are presented, including the unilateral contact problem between an elastic-perfectly plastic beam and an elastic±perfectly plastic foundation. In the analysis the discrete FEM-oriented mathematical model [A. Gawe° cki, Elasto-plasticity of slackened systems, Arch. Mech. 1992;44:363± 390 was employed. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Engineering plasticity; Unilateral contact; Finite elements; Mathematical programming

1. Notation Matrix notation is used throughout the paper. Bold face upper-case letters indicate matrices, and bold face lower-case letters stand for column vectors. u p E s l c k l f g C CT E

n  1 column vector of generalized nodal displacements n  1 column vector of generalized nodal loads m  1 column vector of generalized nodal strains m  1 column vector of generalized nodal stresses s  1 column vector of plastic strain multipliers r  1 column vector of stress multipliers s  1 column vector of plastic moduli r  1 column vector of clearance moduli s  1 column vector of yield condition planes r  1 column vector of contact condition planes m  n matrix of kinematic compatibility n  m matrix of equilibrium m  m matrix of elasticity

* Author to whom correspondence should be addressed.

H N M Z K n m r s

s  s matrix of linear plastic hardening m  s matrix of normals to the yield hyperpolyhedron m  r matrix of normals to the clearance hyperpolyhedron m  m matrix of distortion in¯uence m  s matrix of sti€ness number of components of displacement or load vectors number of components of strain or stress vectors number of planes of the clearance hyperpolyhedron number of planes of the yield hyperpolyhedron.

2. Introduction The problem of slackened structures (i.e. structures with gaps at structural joints) belongs to the mechanics of systems with unilateral constraints. The general theory of such structures has been given in ref. [1], introducing the following fundamental assumptions (see refs [2, 3]):

0045-7949/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 7 ) 0 0 1 6 3 - 7

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. the system consists of deformable (elastic±plastic) structural elements, and indestructible and undeformable connecting elements of very small dimensions; . the constrained motion (due to the presence of gaps) between the structural and connecting elements is permitted; displacements and gaps are small enough so that the kinematically linear theory can be used; . all friction e€ects are neglected; . only quasi-static processes are considered; . the ideal structure (i.e. structure without gaps) is kinematically stable. The presence of gaps at joints induces some additional concentrated `clearance strains' which correspond to relative generalized displacements (longitudinal and transverse translations, rotations) between the elements and corresponding connecting plates. The clearance strains can vary within the `clearance region' de®ned in the clearance strain space. The clearance region is bounded by the so-called clearance surface corresponding to the locking locus in the Prager theory of ideal locking materials, cf. ref. [4]. Thus, the deformations of slackened structures result from elastic, plastic and clearance strains. The solution uniqueness is guaranteed if the clearance region is strictly convex. Absence of friction forces leads to the normality rule for stresses. In other words the vector of generalized stresses is orthogonal to the clearance surface. The behavior of slackened-elastic±plastic (SEP) structures includes a lot of interesting e€ects, in particular when plastic deformations occur, leading to many conceptual and numerical diculties. On a macro scale the behavior is qualitatively similar to that of the locking-elastic±plastic continuum. The non-convexity of a ®rst yielding (elastic) region in the external load space, clearance strains, non-uniqueness and the strongly nonlinear response (even in the elastic range) appear to be characteristic features of slackened systems. It can be shown that in the case of perfect plastic structural elements the presence of gaps does not a€ect the ultimate limit load [5]. However, this state is usually preceded by the so called `sublimit' states when mixed, clearance-plastic ¯ow mechanisms develop. The ultimate limit load is attained step-wise, and the sublimit states correspond to lower values of the load multiplier. From the theoretical and numerical analysis it follows that there exists a certain gap distribution which corresponds to the maximum elastic strength. Furthermore, the limit load can be reached on a purely nonlinear elastic way. By introducing the potential function we can formulate the problem of SEP structures as a saddle point (min± max) problem or in the form of dual extremum principles, as the quadratic programming (QP) problems [2, 3]. The numerical procedure we used in solving the QP problem in this paper is based on the

Karush±Kuhn±Tucker necessary optimality conditions for this problem. The linear complementarity formulation for the problem of the SEP structures was proposed in Ref. [6]. It should be pointed out that the theory of slackened systems covers all the `frictionless' cases of unilateral constraints. The main purpose of this paper is to present the results of numerical analysis for some typical examples of slackened systems, using the holonomic quadratic programing formulation [2]. The mathematical model is discussed in Section 2, and some aspects of the quadratic programming problem which were used in developing a computer code are enclosed in the Appendix. The numerical results are illustrated in Section 3. The examples include particular cases of slackened structures made of the elastic±plastic material, for which a one-parameter yield condition and a one-parameter contact condition at structural joints are considered. The mathematical model is formulated in terms of the discrete matrix description proposed by G. Maier and his co-workers for elastic±plastic structures, (see, ref. [7]). It should be mentioned that a similar problem to that presented here was considered by L. Corradi and G. Maier in 1969, [8]. Those authors constructed a FEM-oriented model of elastic-locking systems and included numerical results for illustrative plane strain problems. According to the Prager's concept [4] the C±M theory describes materials like foam rubber which contains small cavities and behaves elastically up to certain critical strains for which locking behavior begins. However, that theory does not cover plastic deformations, and the elastic e€ects are introduced into the clearance region. It is clearly seen that from the physical point of view the C±M problem is essentially di€erent from the problem of slackenedelastic±plastic structures. Nevertheless, the sense of matrix descriptions in both problems is similar. 3. Mathematical description of slackened-elastic±plastic structures 3.1. The mathematical model of slackened-elastic±plastic structures In this section we will recall the basic relations of the ®nite dimensional model of slackened-elastic±plastic systems [2]. A given SEP system is treated as an assemblage of elastic±plastic structural members and ideally rigid, undeformable connecting elements of very small dimensions. In the interior of each connecting element a certain point called `node' is distinguished, and a load p can be applied only at the nodes. Kinematics of the system is described by a vector u that collects displacement components of the connecting elements. Let us denote by s the vector of (gener-

A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

alized) stresses and by the vector of (generalized) strains, which have to be de®ned consistently with the equation of virtual work: puT ˆ sT E

…1†

where `T ' denotes the transpose. The general mathematical model of SEP structures is described by the following matrix relations: 1: Cu ÿ E ˆ 0;

4: EE ˆ Eÿ1 s;

…5†

DL  sT E_ L ˆ 0;

T s_ T EL ˆ c_ l …6†

Eq. (6) say that the clearance work W L is non-negative, whereas clearance `dissipation' DL is always zero. Relations similar to Eq. (4) are in force for functions f and l_ , so we have

T

7: l_ r0; T 8: l_
f ˆ 0;

9: g ˆ MT EL ÿ lR0;

T l_ f_ ˆ 0:

10: s ˆ Mc c; 11: cr0; …2†

In Eq. (2) C is the kinematic compatibility matrix for an `ideal' structure, i.e. structure without clearances. Since the problem is considered in the framework of small deformations, the standard additive strain decomposition is employed in which the true values of clearance E L , elastic E E and plastic E P components (2)3 are introduced, cf. ref. [9]. Matrix Eq. (2)4 represents the generalized Hooke's law where E denotes the strictly positive-de®nite elasticity matrix. According to G. Maier's description the piecewise linear approximations of the yield and contact conditions are used in the form of linear matrix inequalities (2)5, (2)9, respectively; N and M are rectangular matrices that collect the external normals to all the sides of yield and clearance polyhedrons. Relations (2)6 and (2)10 express the normality rule for plastic strain rates and stresses, respectively. The linear plastic hardening is described by matrix H; l_ and c are the plastic strain rate and stress multipliers vectors, respectively. Generalized yield limits are collected in vector k. The vector l represents clearance moduli (i.e. limit values of gaps). A dot (
) denotes di€erentiation with respect to time. In our considerations here we assume that the ideal structure is kinematically stable which corresponds to the condition det…CT C† 6ˆ 0:

T cT g_ ˆ c_ g ˆ 0:

W L  sT EL ˆ cT lr0;

5: f ˆ NT s ÿ Hl l ÿ kR0; _ 6: E_ P ˆ Nl ;

12: c
g ˆ 0:

The complementarity conditions (2)9,11,12 express the well-known conceptual assumptions then c ˆ 0; g < 0; _ then c ˆ 0; …4† g ˆ 0 and g < 0 then c ˆ 0: g ˆ 0 and g_ ˆ 0 From (4) and c_ Tgc c+c cTgÇ =0 we have the following orthogonality conditions

Using orthogonality conditions (2)12 and (5) one obtains the following relations for stresses and clearance strains:

2: CT s ÿ p ˆ 0; 3: E ˆ E L ‡ E E ‡ EP ;

T

641

…3†

A proof of the requirement (3) is given in the Appendix.

…7†

Upon substituting in Eq. (7) the function f de®ned in Eq. (2)5 we obtain the condition of material stability T s_ T E_ P ˆ l_ Hl_ r0:

…8†

It is clearly seen from Eq. (8) that when H is a diagonal matrix with positive (non-negative) entries, then the material is stable. In a more general case, according to W. Prager's concept of kinematic plastic hardening, matrix H may be expressed as H ˆ hNT N;

…9†

where h is a non-negative scalar. Upon adopting the Prager hardening rule (9) and assuming l(0) = 0, we can derive from the plasticity conditions (2)5±8 the known formula for the plastic dissipation T D  …s s ÿ Ds sT †E_ P ˆ l_ kr0:

…10†

In Eq. (10)Ds s=hEEP is the back-stress. From the mathematical model Eq. (2) one can also derive the relationship p ˆ CT s ˆ CT EEE ˆ CT E‰Cu ÿ …EEL ‡ EP †Š ˆ Ku ÿ CT E…EEL ‡ E P †;

…11†

where K = CTEC is the sti€ness matrix of the linear elastic structure which is, under condition (3), strictly positive de®nite. Further, using Eqs. (11) and (2)1±4 we obtain the following split of the stress vector [10], s ˆ E…Cu ÿ ED † ˆ ECKÿ1 p ‡ ECKÿ1 CT EEED ÿ EEED ˆ se ‡ ZEED

…12†

where Z ˆ ECKÿ1 CT E ÿ E;

ED ˆ EL ‡ EP :

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Subscript e relates to the linear elastic system without distortions and subjected to load p. Z is a square singular symmetric and semi-negative de®nite matrix of distortion in¯uence (cf. ref. [7]). As can easily be shown the matrices Z and C are orthogonal to each other, i.e. ZC = 0 and CTZ = 0. The ®rst identity means that the kinematically admissible strains induce no stresses, while the other case leads to the conclusion that the stresses ZEED induced by the distortion strains are in equilibrium with zero-valued external loads. It should be mentioned that the fundamental relations (2) enable us to describe arbitrary boundary conditions for the analyzed structural elements. Any change of the type of structure a€ecting the degree of statical indeterminacy can be taken into account in a simple way by a proper choice of the components of the clearance moduli vector l}. The particular case of l = 0 corresponds to the ideal structure with bilateral constraints at all connections. 3.2. Dual extremum principles for the holonomic model of SEP structures The holonomic behavior of the system takes place if the ®nal states of stresses and strains do not depend on the deformation history. It corresponds to the response of a system made of a nonlinear elastic material. In the case of slackened-elastic±plastic systems the convexity of the clearance region is additionally required. The holonomic behavior is true for proportional loading if local plastic unloading does not occur. The mathematical model (2) describes also the holonomic behavior if the rate-quantities are replaced by their ®nal values. For the holonomic case, which is treated in this paper, the mathematical model leads to the dual extremum principles which can be derived in the following way. Making use of equality relations of Eq. (2) we can write (2)2, (2)5 and (2)9 in the form T

ÿf ˆ Hl l ÿ N Mc c ‡ kr0; CT Mc c ÿ p ˆ0; T

l ‡ MT CuÿMT Eÿ1 Mc c ÿ lR0; g ˆ ÿM Nl or in the following table form (cf. ref. [11]) l

u

c T

Fl ˆ ÿf ˆ

H

0

ÿN M

Fu ˆ

0

0

CT M

ÿMT N

MT C

ÿG

Fc ˆ g ˆ

r0

1 k

r0

ÿp ˆ 0 ÿl

R0

r0 …13†

where G = MT Eÿ1M. Of course, relations (13) have to be supplemented with the remaining conditions of

Eq. (2) l T f ˆ 0;

cT g ˆ 0:

…14†

Note that the system (13) is symmetric as matrices H and G are positive semi-de®nite. Hence, relations (13) may be treated as the indicated therein partial derivatives of a potential F(l l, u, c), which is convex with respect to l, u and concave with respect to c. By integrating Eq. (13) we obtain 1 1 F…l l; u; c† ˆ lT Hl l ÿ cT Gc c ÿ lT NT Mc c ‡ uT CT Mc c 2 2 T T T ‡ l k ÿ c l ÿ u p: Now the solution of the problem consists in ®nding the saddle point {l l8, u8, c8} such that F…l l ; u ; c † ˆ min max F…l l; u; c† lu

c

is subject to the constraints lr0;

lT Fl ˆ 0;

cr0;

cT Fc ˆ 0:

The saddle point problem can be transformed into two dual extremum principles by applying the Legendre transformation, i.e. F 0 ˆ F ÿ cT Fc ˆ) min jF; cR0;

lr0;

F 0 0 ˆ F ÿ lT Fl ÿ uT Fu ˆ) max jFl r0;

Fu ˆ 0

The explicit forms of the dual extremum principles are 1 1 F 0 …l l; EE ; u† ˆ lT Hl l ‡ E TE ‡ lT k ÿ uT p ˆ) min 2 2

…15†

subject to the constraints MT(Cu ÿ Nl lÿEEE ÿ l R 0, l r 0, and 1 1 00 F …l l; c† ˆ ÿ lT Hl l ÿ cT Gc c ÿ cT l ˆ) max 2 2

…16†

subject to the constraints Hl lÿNTMc c+kr 0, CTMf fÿp = 0, f r0. Note that in the particular case with no slackened strains the problems (15) and (16) reduce to the principle of minimum of potential energy and the principle of maximum of complementary energy, respectively. Our numerical experiments are restricted to solving the primal problem (15) because it provides the complete information on strain and displacement states of the structure. The standard form of this QP problem and the numerical algorithm we used are presented in the Appendix. 3.3. Original structure problem If the system is kinematically unstable due to the presence of gaps, it is necessary, ®rst before the true

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Fig. 1. Mechanical models of plane slackened connections: (a) general case; (b) beam connection; (c) truss connection.

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

Fig. 2. Beam with rotation constraints under concentrated force P: (a) beam, load, and rotation gaps; (b) P ÿ D diagram.

analysis begins, to ®nd a non-zero sti€ness `original' structure that can carry the prescribed loads with reference load vector p0. The structural system is then treated as consisting of undeformable elements, so the corresponding extremum principles take the form of dual linear programming problems (LPPs), [2, 5]: F 0 …u† ˆ ÿuT p0 ˆ) min jMT Cu ÿ lR0; F 0 0 …c† ˆ ÿc cT l ˆ) max jCT Mc c ÿ p0 ˆ 0;

…17† cr0:

…18†

We remark that among the three possible situations in LP programs, the case of an unbounded region of constraints corresponds here to a structural mechanism which is unable to sustain any loading. It may also be noted that a good approximate solution of (11) can be obtained by assuming a ``small'' elastic resistance within the clearance region, cf. ref. [12].

3.4. Sublimit and limit load problems In the case of slackened±perfectly plastic (SP) structures, the formulation of the limit load problem is much more complex than that of `regular' perfectly plastic structures. From the mathematical model for SP structures one obtains the following LPPs, [5]: T _ ˆl_ k ˆ) min jMTa …Cu_ ÿ Nl_ †R0; F 0 …l_ ; u† u_ T p ˆ1; l_ r0;

…19†

00

F …m; ca † ˆm ˆ) max j ÿ NT Ma cTa ‡ kr0; CT Ma cTa ÿ mp ˆ0;

cr0:

…20†

where m denotes the load multiplier and subscript a indicates the active submatrices determined from the solution of the original structure problem. Solutions of both problems allows us to determine the kinematic

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645

Fig. 3. Portal frame with rotation constraints: (a) frame, load and rotation gaps; (b) P ÿ D diagram and modi®cation of structure type.

quantities (uÇ , E_ L, E_ P) and statical ones (s s, m). In gen-

4. Examples

eral, these solutions are di€erent from those for the reference ideal structure without clearances, viz. the load multiplier appears to be less than that for the ideal structure and a clearance-plastic ¯ow mechanism develops. This situation corresponds to the sublimit state. A further problem consists of the determination of the state when the clearance-plastic mechanism stops due to the appearance of new contact at slackened connections. The solution of this problem provides a new matrix partition into active and passive parts. Then the problems (19) and (20) are solved again. The ultimate limit load is reached if the clearance strain rate vector becomes equal to zero, i.e. E_ L=0. It can be shown that the presence of gaps does not a€ect the ultimate limit load.

4.1. Introduction Examples presented herein concern plane skeletal slackened structures with one-parameter contact conditions, i.e. beams and frames with rotation constraints and trusses and support elements with longitudinal gaps. Since Clapeyron's postulates are not satis®ed, a nonlinear behavior of slackened structures, even in the pure elastic range, can be expected. The analyzed structures are made of an elastic±perfectly plastic material with Young's modulus E = 205 GPa and yield limit sY=300 MPa. The general mechanical model of a plane slackened connection is illustrated in Fig. 1(a). Two bars (structural elements) are joined by bolts attached to the connecting element (plate). Owing to the presence of gaps between bolts and holes drilled in the end (rigid) parts

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

problem are the vectors of kinematical quantities (displacements of connecting plates; clearance and plastic strains). The QP procedure was developed on the basis of an algorithm given in ref. [13]. The numerical algorithm used allows us to determine the state of the SEP system at a given load level just in one step. But in order to determine the deformation paths and to get more information on the history of the deformation process, the QP problem must be solved for a sucient number of load levels.

4.2. Simple beam with non-symmetric rotation constraints at both supports

Fig. 4. Two storey slackened frame: (a) frame, loads, rotation gaps; (b) bending moment diagram at limit load state.

of the structural elements, a relative motion of the bar elements and connecting plate may occur. The relative displacements here, play a role of concentrated generalized clearance strains. Thus, as in the theory of plastic structures, the slackened connections can be treated as generalized clearance hinges. The slackened connections considered in the examples can be regarded as particular cases of the general mechanical model. The generalized clearance hinges with rotation and longitudinal gaps together with their symbolic indications are shown in Fig. 1(b) and (c), respectively. In both cases the components of clearance strains have to satisfy the following inequality: ‡ ÿlÿ i RELi Rli ;

‡ lÿ i ; li r0:

All the calculations have been carried out by means of our own computer code which solves the primal quadratic programming problem (15). Unknowns of the

In example 1, which is of instructive character, we consider a simple beam subjected to vertical concentrated load P acting at the midspan point. The beam, load, rotation gaps, moment of inertia of the cross section I and yield moment MY are presented in Fig. 2(a). The ®nal results are illustrated in the P ÿ D diagram (Fig. 2(b)). The load increases from 0 up to limit load PY=60 kN. Segments AB and BC correspond to the elastic range. Note that the nonlinear response of the beam in this range is due to the unilateral constraints, at the left support the contact starts for P = 20.05 kN. At point C the plastic deformation at the midspan point occurs and the sti€ness of the beam decreases. Segment DE corresponds to the sublimit state where the clearance-plastic mechanism develops. At point E the new contact at the right support appears and the kinematical mechanism converts again into a structure. For the further increase of loading the accumulation of elastic±plastic deformations in the beam can be observed up to point F where the ultimate limit load is reached. It should be mentioned that the displacement and deformations of the beam are small.

4.3. Elastic±perfectly plastic portal frame with rotation constraints The portal frame with pin-ended columns is loaded by horizontal force P. The clearance hinges with di€erent moduli are introduced at nodes 2 and 3 [cf. Fig. 3(a)]. The P ÿ D diagram and the changing type of the structure are presented in Fig. 3(b). At the initial stage the deformations are induced only by the clearance strains, and segment AB corresponds to the sublimit state where the clearance mechanism develops. Further deformations up to point D are clearance-elastic and elastic. At point D the plastic hinge at critical Section 3 forms, and segment DE corresponds to elastic±plastic deformations. The nonlinear behavior of the structure can be seen.

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647

Fig. 5. Diagram of load multiplier m vs weighted displacement d and the actual type of the structure for di€erent loads multiplier and Young's moduli.

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654 Table 2 Slackened truss: displacements of nodes Node No

Displacements Horizontal [m]

1 2 3 4 5

0.000000 0.010293 0.000003 0.021535 0.000001

Vertical [m] ÿ0.000000 0.021955 0.000002 ÿ0.029216 ÿ0.000001

4.5. Slackened truss The truss is composed of six elastic±perfectly plastic bars with longitudinal gaps (cf. Fig. 6). The load and clearance modulus vectors are assumed as follows: P ˆ‰P1 ; P2 ; P3 ; P4 ŠT ˆ ‰200; 300; 150; 150ŠT …kN ‡ ÿ ‡ ÿ ‡ T l ˆ‰lÿ 1 ; l1 ; l2 ; l2 ; . . . ; l6 ; l6 Š

Fig. 6. Six-bar slackened truss under proportional loads.

4.4. Two-storey slackened-elastic-perfectly plastic frame The frame, loads, rotation gaps and other data are shown in Fig. 4(a). The loads increase proportionally with parameter m up to the limit state when m = mY=0.727. The bending moment diagram at this yield point load is presented in Fig. 4(b). The deformations of the frame are characterized by m ÿ d diagram, where d denotes the weighted displacement with dimension of energy, d = SiP0iui. The in¯uence of Young's modulus on the frame's behaviour is demonstrated for three values of Young's modulus in Fig. 5, wherein the associated types of the frame for the increasing load multiplier are also shown. The solid line indicates the step-wise m ÿ d relation for the limit load problem of the slackened±perfectly plastic frame. The limit load problem for a similar frame of slightly di€erent dimensions was considered in ref. [14]. It should be mentioned that the change of frame dimensions leads to quantitative and also qualitative changes in step-wise m ÿ d diagrams.

ˆ‰1; 2; 1; 0; 1; 5; 2; 1; 2; 2; 3:4; 2ŠT
10ÿ2 …m† The ®nal values of generalized stresses and strains are gathered in Table 1, while the horizontal and vertical displacements of nodes are listed in Table 2. 4.6. Unilateral contact of elastic±plastic beam and elastic-perfectly plastic foundation The mathematical model of slackened systems can be used to analyze a considerably wide class of problems with `frictionless' unilateral constraints. Let us consider an elastic-perfectly plastic beam clamped at the left support and simply supported at the point located 1.0 m from its right end, see Fig. 7(a). Additionally, there is the elastic±plastic foundation, situated 0.5 cm below the beam. So, for suciently large de¯ections of the beam, some contact zone between the beam and foundation can occur. The beam is loaded with its dead load p = 2.0 kN/m and concentrated force P which is gradually increased from 0 to 360 kN. The foundation is characterized by elastic sti€ness coecient c = 6000 kN/m3 and yield limit sY=0.20 MPa.

Table 1 Slackened truss: generalized stresses (normal forces) and strains (bar elongations) Bar No 1 2 3 4 5 6

Generalized stresses (normal force) [kN] 20.000005 ÿ299.999870 ÿ59.999913 ÿ18.504120 ÿ104.705117 93.995780

Generalized strains (elongation or shortening) EP [m] EL [m]

EE [m]

0.000000 ÿ0.016419 0.000000 0.000000 0.000000 0.000000

0.000293 ÿ0.007317 ÿ0.001171 ÿ0.000451 ÿ0.001532 0.002293

0.010000 ÿ0.000000 ÿ0.050000 ÿ0.010000 ÿ0.020000 0.034000

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649

Fig. 7. Elastic±plastic beam with unilateral contact conditions: (a) beam, loads; (b) mechanical model of the system.

The unilateral contact between the beam and the foundation can be modeled by a number of vertical truss elements with vertical gaps. The clearance moduli for i-th truss element have been assumed as follows, + lÿ i =0.005 m and li is a suciently large number. Then only the compression forces in the contact plane can occur, and the requirement of unilateral contact condition is satis®ed. The system is divided into 24 beam elements and 25 truss elements, its mechanical model is diagrammatically illustrated in Fig. 7(b). P ÿ D diagram for the beam and some comments on the structure behavior are displayed in Fig. 8. The diagrams of bending moment and de¯ection of the beam and the distribution of the foundation pressure are shown in Fig. 9, for load multiplier m = 0.36 that corresponds to the load level at which ®rst yielding occurs in the foundation. Finally, to make a comparison we also analyzed the test problem considered already in ref. [15]. It is an elastic±perfectly plastic beam freely resting on an elastic±perfectly plastic foundation and loaded as shown

in Fig. 10, where the distributions of bending moment, reaction of the foundation and displacement are also included. The results obtained from our formulation are presented in the right column of Fig. 10, while those of ref. [15] in the left column. The characteristic behavior of the system and good agreement of both series of results can be observed. A similar unilateral contact problem to that presented here has been analyzed in ref. [6] where the elastic±plastic behavior with linear hardening both for the beam and the foundation is assumed and the case of non-proportional loading is treated. Therein the incremental formulation in the form of a linear complementarity problem (LCP) has been used. In such a case the problem is actually treated as a series of holonomic problems.

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

Fig. 8. P ÿ D diagram for the beam-foundation system.

5. Concluding remarks In the paper a general, uni®ed, ®nite dimensional mathematical framework for slackened-elastic±plastic structural systems was presented. The problem is formulated in the form of two quadratic programming programs, which in the special cases of the `original' structure problem and the sublimit and limit problems reduce to linear programming programs. The included numerical results for a number of test problems have an illustrative character of the impact of clearances on the global response of the structure. They show that this in¯uence may be substantial. In the numerical experiments it was assumed that the plastic and the clearance e€ects were concentrated (lumped) at the nodes of ®nite element mesh. This conceptual idealization, as concerns plastic e€ects, seems to be admissible for the elastic perfect plastic behavior, but leads to some inconsistency of the ®nite element model when plastic hardening is accounted for. Some numerical examples for the case of kinematic plastic hardening together with the ®nite element forms of the matrices H, C and others are provided in ref. [6], where the present pro-

blem is solved as a linear complementarity problem. These numerical results we obtained by solving the primal problem (15), using an available QP algorithm [13]. In this paper it was not the authors' aim to test and compare the eciency of various QP algorithms for ®nding a solution to the present QP problem. Such an analysis for the elastic±plastic systems is made by Maier et al. [15]. We are under way to develop other QP algorithms for our problem, with emphasis laid on iterative schemes. Yet, the main diculty in employing iterative algorithms is the fact that the matrix governing the problem is only semi-positive de®nite.

Acknowledgements The work was supported by the Polish Committee of Scienti®c Research (KBN) under Grant no. 8T11F01710.

A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

Fig. 9. Plots of bending moment and de¯ection of the beam, and of support reaction for load multiplier m = 0.36.

651

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

Fig. 10. Results for the test problem considered in [13] (left column), and obtained with our formulation (right column).

A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

Appendix A

R ˆ CT C: We observe now that for any matrix C the following holds (for a proof see ref. [17]).

A.1. On kinematic stability of structures In this appendix we substantiate the condition of kinematic stability of the ideal structure, Eq. (3). In common circumstances, kinematic stability of a structure is assured if the sti€ness matrix of the structure is non-singular, i.e. det K  det…CT EC† 6ˆ 0; where C, dim C = m  n, m r n, is the matrix of kinematic compatibility in which the boundary conditions are taken into account. However, in a general case, any criterion of kinematic stability (called also geometric invariability) need not be dependent on the material of a structure and should also be relevant to structures made of ideally rigid components. For such structures it is impossible to determine the standard sti€ness matrix K. To consider this question more speci®cally, let us ®rst recall that a structure is said to be kinematically stable if any non-zero component of generalized displacement vector of the structure is associated with at least one non-zero component of generalized strain vector (cf. ref. [16]). The following theorem holds. Theorem. Let C be an m  n matrix with m rn, which de®nes the relation between the displacement vector u and the strain vector , Cu ˆ E : Then, the condition of kinematic stability is det…CT C† 6ˆ 0:

…A1†

Proof. We notice ®rst that from the fact that zerovalued strains imply zero-valued displacements it follows that the only solution to the system Cu ˆ 0 must be the trivial one, u = 0. This situation is possible if the rank of matrix C is equal to the number of components of vector u, i.e. rank C ˆ n:

653

…A2†

In the other case in which rank C < n the structure constitutes a mechanism. In order to show that the condition (A1) is equivalent to the requirement (A2) we introduce the n  n Gram matrix R of matrix C, de®ned as follows

rank C ˆ rank R: But, rank R = n if and only if det R$ 0. In other words, when det R $ 0 then rank C = n. q A.2. Description of the numerical algorithm for the QP problem We present here brie¯y the main steps of the numerical algorithm [13] which we adapted for solving the quadratic programming problem (15). A general quadratic programming problem, as given in (15) and (16), may be formulated as follows: ®nd x8 $ RN such that 1 F…x † ˆ xT Qx ‡ qT x 2

…B1†

subject to A 0 xRb 0 ; A 0 0 x ˆb 0 0 ; x 0 r0; where dim Q = N  N, dim q = N, dim A' = lgN, dim b' = lg, dim A0 = lhN, dim b0 = lh, dim x' = N', with N, N', lg, lh being some integers. The vector of unknowns x is split into two parts; the ®rst part contains components xj, j = 1,. . ., Nu which are unrestricted in sign, whilst the second one, denoted in (B.1) by x', has the components which should not be negative, i.e. xj r 0, j = Nu+1,. . .,N. By direct comparison of the problems (15) and (B.1) we have 2 3 2 3 2 3 0 u ÿp 4 5 4 5 4 Q :ˆ E ; x :ˆ E E ; q :ˆ 0 5; H l k A 0 :ˆ ‰MT C ÿ I

NŠ;

b 0 :ˆ l;

x 0 :ˆ l;

with N = n + m + s, lg=r, lh=0, N' = r where n, m, r, s are de®ned in the notation. Further, in order to eliminate the inequality constraints let us introduce a vector of nonnegative slack variables, dim y = lg, and denote

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A. Gawe° cki et al. / Computers and Structures 69 (1998) 639±654

      x q ~ :ˆ Q 0 ; ~ ; Q q:‡ ; y 0 0 0  0  0  I b ~ :ˆ A A : ; ~ b :ˆ 00 A0 0 0 b ~ :ˆ x

Finally, for the modi®ed quadratic problem we have the Lagrangian function ~x ÿ ~ ~x ‡ ~ ~T Q~ ~ ‡ wT …A~ ~ ~ w; f† ˆ x L…x; qT x b† ÿ fT x and the resulting Karush-Kuhn-Tucker conditions are ~ x ˆ~ A~ b; ~T

~ x ÿ A w ‡ f ˆ~q; ÿQ~ x~ j r0; wj ˆ0;

j ˆ Nu ‡ 1; . . . ; Nu ‡ lg ; j ˆ 1; 2; . . . ; Nu ;

wj r0;

j ˆ Nu ‡ 1; . . . ; Nu ‡ lg ;

wj x~ j ˆ0;

j ˆ 1; 2; . . . ; N ‡ lg :

Vectors w, dim w=lg+lh, and f, dim f = N + lg, play the role of Lagrange multipliers. It is worth emphasizing that, under the assumption of semi-de®niteness of the matrix Q, the system (B2) constitutes the necessary and sucient optimality conditions for the quadratic program (B1). The solution algorithm determines a solution to the system (B2) in three phases. Phase I consists in ®nding a basic solution to the arti®cial problem de®ned by (B2)1,2,3 in which p = 0 is assumed. Then by making use of the multiplier p the solution is further modi®ed in order that the orthogonality condition (B2)6 be satis®edÐPhase II. Within Phase III the obtained basic solution is ®nally processed by the simplex algorithm which maintains the conditions (B2)5,6. References [1] Gawe° cki A. Elastic±plastic Skeletal Structures with Clearances. In: Ser `Rozprawy', No 185, (in Polish). Poznan University of Technology Press, Poznan, 1987.

[2] Gawe° cki A. Elasto-plasticity of slackened systems. Arch Mech 1992;44:363±90. [3] Gawe° cki A. Mechanics of slackened systems. Comp Assisted Mech Engng Sci 1994;1:3±25. [4] Prager W. On ideal locking materials. Trans Soc Rheology 1957;1:169±75. [5] Gawe° cki A. On plastic ¯ow mechanisms in the perfectly plastic- slackened structures. Arch Mech 1988;40:653±63. [6] Gawe° cki A, Kuczma MS. Elastic±plastic unilateral contact problems for slackened systems. J Comp Appl Math 1995;63:313±23. [7] Maier G. Mathematical programming methods in analysis of elastic±plastic structures, (in Polish). Arch InzÇ La° d 1975;31:387±411. [8] Corradi L, Maier G. A matrix theory of elastic-locking structures. Meccanica 1969;4(4):298±313. [9] Genna FA. A nonlinear inequality, ®nite element approach to the direct computation of shakedown load safety factors. Int J Mech Sci 1988;30:769±89. [10] KoÈnig. Shakedown of Elastic±plastic Structures. PWNElsevier, Warszawa, 1987. [11] Borkowski A. Analysis of Skeletal Structural Systems in the Elastic and Elastic±plastic Range. PWN-Elsevier, Warszawa, 1988. [12] Gawe° cki A, JaninÂska B. Computer analysis of slackened elastic±plastic beams under non-proportional loads. Computer Assisted Mech Engng Sci 1995;2:25±40. [13] Kre° glewski T, Rogowski T, RuszczynÂski A, Szymanowski J. Optimization Methods in the FORTRAN Language (in Polish). PWN, Warszawa, 1984. [14] Gawe° cki A, KruÈger P, Slackened systems under variable loads, In: MroÂz Z, Weichert D, Dorosz S, editors, Proc EUROMECH 298, Inelastic behaviour of structures under variable loads, 1992. pp 399-417, Kluwer Academic Publishers, Dordrecht: 1995. [15] Maier G, Grierson DE, Best MJ. Mathematical programming methods for deformation analysis at plastic collapse. Computers & Structures 1977;7:599±612. [16] Fra° ckiewicz H, Kinematic criteria of geometric nonrigidity of discrete Lattice. Bull Ac Pol Sci Ser Sci Tech 1969;5(7). [17] Mostowski A, Stark M. Elements of Higher Algebra (in Polish). PWN, Warszawa, 1965.