- Email: [email protected]
Determination of the collapse load of plastic structures by the use of an upper bounding algorithm
Compufers d Srrucrures Vol. 40, No. 4, pp. 1003-1008, Printed in Great Britain.
@MS-7949/91 $3.00 + 0.00 Pergamon Press plc
1991
DETERMINATION OF THE COLLAPSE LOAD OF PLASTIC STRUCTURES BY THE USE OF AN UPPER BOUNDING ALGORITHM A. V. AVDELAS Department of Civil Engineering, Aristotle University, GR-540 06 Thessaloniki, Greece (Receiued 13 June 1990) Abstract-An upper bounding algorithm is applied to the problem of the elastoplastic analysis of structures expressed as linear complementarity problems. By the use of this algorithm, the time consuming procedure of solving large quadratic optimization problems can be avoided. Applications close the paper.
are valid. Relations (1-5) are obtained when the finite
1. INTRODUCHON
As it is well known, there is an equivalence between elastoplastic analysis and linear complementarity problems [la]. It can be proved either by the use of ‘slack’ variables [l-3, 5-71 or of variational inequalities [7-lo], that these linear complementarity problems (LCP) are in turn equivalent to quadratic optimization problems. There are a number of algorithms, by different authors, that are used for the solution of these optimization problems, that is, in most cases, for the determination of the collapse load or collapse load factor. In the following, an easy to use method is implemented for the determination of the solution components that are or are not bounded without solving the LCP. The advantage of this method is that, by its use, the time-consuming procedure of solving large quadratic optimization problems or LCPs, for the different values of the loads, is avoided and in their place linear programming problems are used.
element method is used for the discretization of the structure, with the external loads applied to the nodes. Moreover, the displacements are assumed to be large, the deformations small and the existence of physical instabilizing effects is accepted. Further, the stress and strain fields of each element are described by means of ‘natural’ generalized stresses and strains [l l-161. These generalized strains and stresses are respectively not affected by the rigid body motion of the element and are self-equilibrated within each element. Further, the continuum is described by means of the nodal displacement vector u referred to a fixed Cartesian orthogonal coordinate system and to a given reference configuration. Relations (5) form a LCP [l-4]. By means of variational inequalities [&lo] and using the same notation as in [9, 17, 181, it can be proved that this LCP is equivalent to the quadratic optimization problem (primal minimization problem) n(x) = min{JI(x,)
=
ix: Mx,
+ qrx, (x, E K}.
(8)
2.1. The elastoplastic analysis problem The basic laws of incremental the assembled structure, are
elastoplasticity,
e=e,+&+c,
L 20,
for
(1)
eE=F,S
(2)
e ,=vi
(3)
P=N+W
(4)
l?
F’li=O.
For matrix M the assumption is made that it is symmetric and positive semidefinite. The former assumption is valid if and only if V = N (normality) and there exists reciprocity of the interaction of the yield modes (H = Hp. The sufficient condition for matrix M to be positive semidefinite is ensured by the positive semidefiniteness of matrices H (non-existence of softening) and KG (geometrical effects non-instabilizing). Under this assumption x?Mx is convex. Expressed in terms of the displacements it and the plastic multipliers f, (8) takes the form
(5) min{n(it, A) = fir%& + IJTHi - i’GK,,Ni
Further, the equilibrium and compatibility
equations
G$+K,b=P
(6)
b=GTli
(7) 1003
1004
A. V. AVDELAS
or
relation min{n(ti, A) = fPK,l
+ fLrHi + fe,‘K&,
-;e&e,-p%]L
>O}.
P=NrsE-(H-~r~~)i
(10)
In a similar way, under the assumption again that M is symmetric and positive semidefinite, the dual minimization problem is obtained n’(x)=min{n’(x,)
=fxTMx,]x,
eKC}.
(11)
Expressed in terms of the displacements L and the plastic multipliers f, (11) takes the form of min(n’(8,
A) = $rKu + !irHf
- LrGK,,Ni
(17)
is obtained. This relation together with (5) constitute a LCP. If matrix D = H -NrZV is symmetric (N = V, H =Hr) and positive semidefinite, then by the use again of variational inequalities, the minimization problem R(i) = min{R(&) = ii:Di,
- (Nr$qrf,
If, 2 0) (18)
is obtained. The dual of (18) is Rc(~)=min{Rc(fl)=~f~D&]Nr&‘-D~I
GO}. (19)
-(p + G&e,,) = 0, NxGrit -(H + N%,,N)i
- N%&, Q 0). (12)
From
(12), (2), (1) and (7) and by the use of problem takes the form K, = GK,,N’, the dual minimization
min{n’(P, & S) = frirKGi + !lrHi.
+ farFOF,s + ST&
x Gs+K,ri=p,N%-H3;-,
It must be noted that for the relations between the solutions of the primal [eqn (18)] and the dual [eqn (19)] problem and depending on the nature of matrix D, the various cases that have been examined for matrix M are valid. 2.2. Holonomic elastoplastic analysis Unlike the incremental elastoplastic analysis, in the holonomic case, only the final loading condition is important and not the loading history. Furthermore, after the complete unloading, there are no permanent deformations. The assumptions and relations that are valid for the structure are analogous to (l)-(7), with the difference that the Y-vector of the positive constants k is present in the relation of the yield functions [cf. eqn (4)]. Furthermore, the stress and strain vectors are finite and not incremental. The holonomic minimization problems are similar to the incremental ones. 3. THE UPPER BOUNDING ALGORITHM
x ]GS=p,N%-Hli
GO}.
(14)
When the initial assumption, that matrix M is symmetric and positive semidefinite, is valid, then the set of solutions of the primal problem (8) is convex. In this case any solution of the primal problem (l-7) or equivalently of (8), is also a solution of the dual problem (11) but not conversely [ 191. Furthermore the primal problem has a solution if and only if there exists a solution for the dual problem [6,20]. Under the assumption that matrix K is nonsingular, another family of variational inequalities is formulated, expressed with respect to the plastic multipliers n’. Through the substitutions Z = &G%-‘GK, sE=&G?(;-‘$+Ze,
- &,
(15) (16)
The above briefly presented elastoplastic analysis problems, usually end up to the problem either of studying the response of a structure to a given loading case or of finding the collapse load or collapse load (safety) factor of the structure. In the second case, one has to solve again and again a quadratic optimization problem using a time consuming procedure. The upper bounding algorithm [21] is used here in order to reduce the amount of computations needed for this purpose. The LCP w=D1+d w>o,
1.20,
(20) wrAa.0
(21)
is considered, where D is an n x n matrix and d is an n-vector. Let [21] I and L be subsets of {1,2, . . . , n } and S define the set
1005
Collapse load of plastic structures S={(l,
w)ll 3 0, w 2 O}.
(22)
In the case that matrix D is positive semidefinite, then if S # 8, there exists a partition IuL of { 1,2, . . . , n} such that 2, is boundedW, # 8, 1, is unbounded o W, = 0. Here &(resp. W,) and A, (resp. W,) denote sets of solutions of the LCP [(20)-(21)]. The steps of the algorithm determining the ZUL = { 1,2, . . . ) n}, such that ;1, is bounded, Ai, is unbounded, are as follows [21] for a positive semidefinite matrix D Algorithm
StepO: Setj=O, I,=@, L,={l,2 ,..., n}. Step 1: Solve the linear program (LP): max XjjaL,(on + d)j such that Di+d>,O,i>O. If the LP is infeasible then the LCP is also infeasible. Stop. If LP max =O, set z=Zj, then L={1,2 )..., n}\l,.Stop. If 0 < LP max < co, then set n(z) = 1, where 2 is a solution of the LP. If LP max+ co, then set n(z) = 1 + zi, where I+ z4 is feasible for all 1. > 0 and Zjis4Djcj > 0. Set J,, = Iju{ilD,l(z)
+ d, > 0,
For d, = -(NrsE - k), the Kuhn-Tucker ditions of (23) are aF
w==
=D1 +d,,
Lj+.,={1,2,...,n)\~+,.
4. APPLICATIONS
The algorithm will be next applied to one of the elastoplastic analysis problems of Sec. 2. The following assumptions are valid in addition to the ones made at the beginning of this section: (a) the material law is elastic-perfectly plastic; (b) rotations 0 are induced by the inelastic deformations concentrating in individual sections; (c) the plastic moment-rotation 0 relationship is shown in Fig. 1; (d) the moment-rotation law is reversible (holonomic). For the solution of the minimum problem, relation min{R(rl) = fr2?)1 - (NT@ - k)rA113. 2 0} has been treated [see also eqn (18)].
(23)
WZO,
ATw=O
(24)
3. 2 0.
(25)
and the LP to be solved in step 1 is max 1 (DA + d,),lDA + d, > 0, jeL,
First the simple portal frame of Fig. 2 is studied [lo, 221. The frame is of uniform cross-section EJ = 10.0 kN m2 and the plastic moment of the section is Mp = 65.73 kN m. There are eight possible critical sections of the structure. The work-hardening matrix His zero and the 16 x 16 matrix D is given by relation D = -NTZN,
(26)
where the 8 x 8 matrix Z is obtained from (15) and N is the 8 x 16 diagonal matrix of [l - 11. Further the 8 x 1 vector s6 is obtained from relation SE= K,,G%-‘p
z+oc)),
Step 2: Set j + 1+j. Step 3: Go to step 1.
120,
(27)
and k is the 16 x 1 vector of the plastic moments of the sections. For P = 78.48 kN, after the LP (25) is solved, by the use of the Simplex method, for j = l-6, the program terminates since L,, I = {a} (Table 1). That is, there is no need to solve all of the 16 LPs. The same is valid for P = 78.875 kN but forj = l-8. If P takes the value P = 78.876 kN, then no solution exists for the LPs and as a consequence the LCP is infeasible. The physical meaning of this infeasibility is that the structure has reached the collapse state, i.e. P = 78.876 kN is the collapse load. Next the structure has been solved by a computer plastic analysis program using a direct quadratic optimization algorithm [17, 18,231 with the same results. That is for P = 78.48 kN three hinges are formed and for P = 78.875 kN a fourth hinge appears. Both solutions are obtained in three pivotal steps. A further rise of the value of P causes the collapse of the structure.
M
2P 3P 2
3
4
MP
T 1.0 In 8
0
1
5
r-l
L
MP
+ Fig.
1.
Plastic moment-rotation relationship.
con-
Fig. 2.
A. V. AVDELA~
1006
Table 2. Element characteristics
Table 1 Element
6eetii
Mp (KNm)
1,2 3,4 586 7,6 9,lO 11,12
I 260 1220 1320 I 160 1300 1220
123.0 76.1 219.0 44.6 183.0
76.1
sT = [-65.73,38.247,38.247,64.738,64.738,
-65.73,
-65.73,65.73]
kN m
and Because of [24] 8=N1,
s’= [-65.73, sp=ZB
ands=sE+sP,
39.438,39.438,438.65,65.73,45.73, -65.73,
(28)
-65.73,65.73]
for P = 78.48 and 78.875 kN, respectively.
the moments vector is equal to 0.2P
30
4
40
5
t
0
@
0
P -_,
a
9
@
@
@J
11
12
+
3.0 m -+
3.0 m +
3.0 m +
3.0 m +
Fig. 3.
1.3
LoadFstw
a
Fig. 4. Rotation vs load factor, up to collapse.
3.0 m
t 3.0 m
t
kN m
1007
Collapse load of plastic structures
-
4.
--'
5.
.._... 5. ._____ 5_ .._._._,_ ____
5_
.....1. 2. I..... 10. ,...... ,O_ I,...,. ,(_ -
-I
1.0
1.1
1.2
1.3
1.4
12.
1.5
LoadFactor a
Fig. 5. Moment vs load factor, up to collapse. The frame of Fig. 3 has been analysed next under the same assumptions, with P = 137.19 kN. In Table 2 the characteristics of the elements of the structure are given. There are 24 possible critical sections of the structure. Since the work-hardening matrix H is zero, the 48 x 48 matrix D is obtained from eqn (26). The first hinge appears for a load factor a equal to 1. For CL= 1.45 collapse occurs, that is all the LPs are infeasible. The structure has been also solved by the already mentioned above computer plastic analysis program with the same results. That is the first’hinge appears for a = 1 and the last before collapse (the ninth) for a = 1.45. In Figs 4 and 5, the change of rotation 8 and of the total moment s is respectively depicted versus the load factor a.
used if the aim of the analysis is only the value of the collapse load or collapse load factor of the structure and not its total behavior under the given loads. It must be pointed out that the input data for both the upper bounding algorithm and the direct quadratic optimization one are the same since matrices D and d, are also used for the solution of the quadratic optimization problem. The usefulness of the upper bounding method presented above, is enhanced by the fact that the Simplex algorithm can solve problems with large numbers of unknowns even more so if the Karmakar algorithm is used [25]. Acknowledgemenrs-The author would like to thank Professor P. D. Panagiotopoulos of Aristotle University, Thessaloniki and Technical University, Aachen for his helpful advice.
5. CONCLUSIONS
As it has been seen, the use of the upper bounding algorithm can be very useful in the plastic analysis of structures, in the sense that it can determine the collapse load or collapse load factor of the structure without actually solving the problem i.e. the quadratic optimization problem to which it is equivalent. Instead, it solves a number of LPs at most equal to the number of elements in the Z set, but usually fewer. It must be pointed out that for the implementation of the algorithm, matrices D and dl must be formulated first. For the steps of the algorithm either a mathematic package including a linear optimization algorithm or an independent subprogram can be used. This subprogram, as it is the case here, can be incorporated in a larger and more genera1 plastic analysis program and be optionally
REFERENCES
1. G. Maier, Incremental plastic analysis in the presence of large displacements and physical instabilizing effects. Int. J. Solids Struct. I, 345-372 (1971). 2. G. Maier, Mathematical programming methods in structural analvsis. In Variational Methods in Emineering, Vol. II-(Edited by C. Brebbia and H. Totyenham). Southampton University Press, Southampton (1973). M. J. Cohn and G. Maier (Eds), Engineering plasticity by mathematical programming. Proc. NATO-AS, Waterloo, Canada 1977. Pergamon Press, New York (1977). R. W. Cottle, Numerical methods for complementarity problems in engineering and applied science. In Computing Methods in Applied Science and Engineering, 1977, I, Proceedings, IRIA, Paris (Edited by R. Glowinski and J. L. Lions). Springer, Berlin (1979).
1008
A. V. AVDELAS
5. G. Maier, Quadratic programming and theory of elastic-perfectly plastic structures. Meccanica 3, 265-273
(1968). 6. G. Maier, A matrix structural
linear elastoplasticity
theory of piece-wise with interacting yield planes.
15. J. H. Argyris, J. St Doltsinis et al., Thermomechanical response of solids at high strains-natural ;f;8ro)ach. Compt. Meths. Appl. Mech. Engng 32, I-106
Meccanica 8, 54-66 (1970). I. D. E. Grierson, A. Franchi et al., Mathematical
programming and nonlinear finite element analysis. Comput. Meths. Appl. Mech. Engng 17/18, 497-518 (1979). 8. P. D. Panagiotopoulos, Variational inequalities and multilevel optimization technique in the theory of plasticity. Comput. Struct. 8, 649650 (1978). 9. P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhluser, Boston (1985). Russian trans-
lation: Mir, Moscow (1989). 10. P. D. Paneiotoooulos. C. C. Baniotonoulos. C. C. and A. V. Avdelas, Certain propositions oh the activation of yield modes in elastoplasticity and their applications to deterministic and stochastic problems. ZAMM 64, 491-501 (1984).
Il. J. H. Argyris, Continua and discontinua. Proc. 1st Conf. Matrix Meth.
Struct.
Mech.
Wright-Patterson
F.R.,
Davton. Ohio. 1965. AFDL TR-66-80 (1966). 12. J. H. Argyris; Three-dimensional anisotropic and inhomogeneous elastic media. Matrix analysis for small and large displacements. Ing. Archiu 34, 33-55 (1965). 13. J. H. Aravris and D. W. Scharuf. Some general considerations on the natural mode* technique. Part I and Part II. Aeron J. R. Aeron. Sot. 73, 218-226, 361-368 (1969).
14. J. H. Argyris, H. Balmer er a/., Finite element methodthe natural approach. Comput. Meths. Appl. Mech. Engng 17/18, I-106 (1979).
16. P. G. Hodge Jr, Plastic Analysis of Structures. McGrawHill, New York (1959). 17. A. V. Avdelas, Application of the quadratic programming methods in the problems of the theory of plasticity (in Greek). Doctoral’ thesis, Aristotle University, Thessaloniki (1987). 18. A. V. Avdelas, A direct quadratic programming approach to the elastoplastic analysis of frames with the possibility of worksoftening. S.M.-Archives (to be published). 19. H. Kiinzi and W. Krelle, Nichtlineare Programmierung. Springer, Berlin (1962). 20. L. Collatz and W. Wetterling, Optimierungsaufgaben. Springer, Berlin (1971). 21. 0. L. Mangasarian, Simple computable bounds for solutions of linear complementarity problems and linear programs. Math. Prog. Study 25, l-12 (1985). 22. B. G. Neal, Plastic Methods of Structural Analysis. Chapman & Hall, London (1970). 23. A. V. Avdelas, The unilateral contact problem. A treatment by the use of the Keller direct method. Hel. Sot. Th. -& Appl. Mech. Proc. 1st Nat. Congr. Mech. Athens 1986. Vol. 1 DD. 207-234. T.E.E.. Athens (1987). L1 ’ 24. 0. de Donato and G. Maier, Mathematical program-
ming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity. Int. J. Numer. Meth. Engng 4, 307-329 (1972). 25. K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988).
@MS-7949/91 $3.00 + 0.00 Pergamon Press plc
1991
DETERMINATION OF THE COLLAPSE LOAD OF PLASTIC STRUCTURES BY THE USE OF AN UPPER BOUNDING ALGORITHM A. V. AVDELAS Department of Civil Engineering, Aristotle University, GR-540 06 Thessaloniki, Greece (Receiued 13 June 1990) Abstract-An upper bounding algorithm is applied to the problem of the elastoplastic analysis of structures expressed as linear complementarity problems. By the use of this algorithm, the time consuming procedure of solving large quadratic optimization problems can be avoided. Applications close the paper.
are valid. Relations (1-5) are obtained when the finite
1. INTRODUCHON
As it is well known, there is an equivalence between elastoplastic analysis and linear complementarity problems [la]. It can be proved either by the use of ‘slack’ variables [l-3, 5-71 or of variational inequalities [7-lo], that these linear complementarity problems (LCP) are in turn equivalent to quadratic optimization problems. There are a number of algorithms, by different authors, that are used for the solution of these optimization problems, that is, in most cases, for the determination of the collapse load or collapse load factor. In the following, an easy to use method is implemented for the determination of the solution components that are or are not bounded without solving the LCP. The advantage of this method is that, by its use, the time-consuming procedure of solving large quadratic optimization problems or LCPs, for the different values of the loads, is avoided and in their place linear programming problems are used.
element method is used for the discretization of the structure, with the external loads applied to the nodes. Moreover, the displacements are assumed to be large, the deformations small and the existence of physical instabilizing effects is accepted. Further, the stress and strain fields of each element are described by means of ‘natural’ generalized stresses and strains [l l-161. These generalized strains and stresses are respectively not affected by the rigid body motion of the element and are self-equilibrated within each element. Further, the continuum is described by means of the nodal displacement vector u referred to a fixed Cartesian orthogonal coordinate system and to a given reference configuration. Relations (5) form a LCP [l-4]. By means of variational inequalities [&lo] and using the same notation as in [9, 17, 181, it can be proved that this LCP is equivalent to the quadratic optimization problem (primal minimization problem) n(x) = min{JI(x,)
=
ix: Mx,
+ qrx, (x, E K}.
(8)
2.1. The elastoplastic analysis problem The basic laws of incremental the assembled structure, are
elastoplasticity,
e=e,+&+c,
L 20,
for
(1)
eE=F,S
(2)
e ,=vi
(3)
P=N+W
(4)
l?
F’li=O.
For matrix M the assumption is made that it is symmetric and positive semidefinite. The former assumption is valid if and only if V = N (normality) and there exists reciprocity of the interaction of the yield modes (H = Hp. The sufficient condition for matrix M to be positive semidefinite is ensured by the positive semidefiniteness of matrices H (non-existence of softening) and KG (geometrical effects non-instabilizing). Under this assumption x?Mx is convex. Expressed in terms of the displacements it and the plastic multipliers f, (8) takes the form
(5) min{n(it, A) = fir%& + IJTHi - i’GK,,Ni
Further, the equilibrium and compatibility
equations
G$+K,b=P
(6)
b=GTli
(7) 1003
1004
A. V. AVDELAS
or
relation min{n(ti, A) = fPK,l
+ fLrHi + fe,‘K&,
-;e&e,-p%]L
>O}.
P=NrsE-(H-~r~~)i
(10)
In a similar way, under the assumption again that M is symmetric and positive semidefinite, the dual minimization problem is obtained n’(x)=min{n’(x,)
=fxTMx,]x,
eKC}.
(11)
Expressed in terms of the displacements L and the plastic multipliers f, (11) takes the form of min(n’(8,
A) = $rKu + !irHf
- LrGK,,Ni
(17)
is obtained. This relation together with (5) constitute a LCP. If matrix D = H -NrZV is symmetric (N = V, H =Hr) and positive semidefinite, then by the use again of variational inequalities, the minimization problem R(i) = min{R(&) = ii:Di,
- (Nr$qrf,
If, 2 0) (18)
is obtained. The dual of (18) is Rc(~)=min{Rc(fl)=~f~D&]Nr&‘-D~I
GO}. (19)
-(p + G&e,,) = 0, NxGrit -(H + N%,,N)i
- N%&, Q 0). (12)
From
(12), (2), (1) and (7) and by the use of problem takes the form K, = GK,,N’, the dual minimization
min{n’(P, & S) = frirKGi + !lrHi.
+ farFOF,s + ST&
x Gs+K,ri=p,N%-H3;-,
It must be noted that for the relations between the solutions of the primal [eqn (18)] and the dual [eqn (19)] problem and depending on the nature of matrix D, the various cases that have been examined for matrix M are valid. 2.2. Holonomic elastoplastic analysis Unlike the incremental elastoplastic analysis, in the holonomic case, only the final loading condition is important and not the loading history. Furthermore, after the complete unloading, there are no permanent deformations. The assumptions and relations that are valid for the structure are analogous to (l)-(7), with the difference that the Y-vector of the positive constants k is present in the relation of the yield functions [cf. eqn (4)]. Furthermore, the stress and strain vectors are finite and not incremental. The holonomic minimization problems are similar to the incremental ones. 3. THE UPPER BOUNDING ALGORITHM
x ]GS=p,N%-Hli
GO}.
(14)
When the initial assumption, that matrix M is symmetric and positive semidefinite, is valid, then the set of solutions of the primal problem (8) is convex. In this case any solution of the primal problem (l-7) or equivalently of (8), is also a solution of the dual problem (11) but not conversely [ 191. Furthermore the primal problem has a solution if and only if there exists a solution for the dual problem [6,20]. Under the assumption that matrix K is nonsingular, another family of variational inequalities is formulated, expressed with respect to the plastic multipliers n’. Through the substitutions Z = &G%-‘GK, sE=&G?(;-‘$+Ze,
- &,
(15) (16)
The above briefly presented elastoplastic analysis problems, usually end up to the problem either of studying the response of a structure to a given loading case or of finding the collapse load or collapse load (safety) factor of the structure. In the second case, one has to solve again and again a quadratic optimization problem using a time consuming procedure. The upper bounding algorithm [21] is used here in order to reduce the amount of computations needed for this purpose. The LCP w=D1+d w>o,
1.20,
(20) wrAa.0
(21)
is considered, where D is an n x n matrix and d is an n-vector. Let [21] I and L be subsets of {1,2, . . . , n } and S define the set
1005
Collapse load of plastic structures S={(l,
w)ll 3 0, w 2 O}.
(22)
In the case that matrix D is positive semidefinite, then if S # 8, there exists a partition IuL of { 1,2, . . . , n} such that 2, is boundedW, # 8, 1, is unbounded o W, = 0. Here &(resp. W,) and A, (resp. W,) denote sets of solutions of the LCP [(20)-(21)]. The steps of the algorithm determining the ZUL = { 1,2, . . . ) n}, such that ;1, is bounded, Ai, is unbounded, are as follows [21] for a positive semidefinite matrix D Algorithm
StepO: Setj=O, I,=@, L,={l,2 ,..., n}. Step 1: Solve the linear program (LP): max XjjaL,(on + d)j such that Di+d>,O,i>O. If the LP is infeasible then the LCP is also infeasible. Stop. If LP max =O, set z=Zj, then L={1,2 )..., n}\l,.Stop. If 0 < LP max < co, then set n(z) = 1, where 2 is a solution of the LP. If LP max+ co, then set n(z) = 1 + zi, where I+ z4 is feasible for all 1. > 0 and Zjis4Djcj > 0. Set J,, = Iju{ilD,l(z)
+ d, > 0,
For d, = -(NrsE - k), the Kuhn-Tucker ditions of (23) are aF
w==
=D1 +d,,
Lj+.,={1,2,...,n)\~+,.
4. APPLICATIONS
The algorithm will be next applied to one of the elastoplastic analysis problems of Sec. 2. The following assumptions are valid in addition to the ones made at the beginning of this section: (a) the material law is elastic-perfectly plastic; (b) rotations 0 are induced by the inelastic deformations concentrating in individual sections; (c) the plastic moment-rotation 0 relationship is shown in Fig. 1; (d) the moment-rotation law is reversible (holonomic). For the solution of the minimum problem, relation min{R(rl) = fr2?)1 - (NT@ - k)rA113. 2 0} has been treated [see also eqn (18)].
(23)
WZO,
ATw=O
(24)
3. 2 0.
(25)
and the LP to be solved in step 1 is max 1 (DA + d,),lDA + d, > 0, jeL,
First the simple portal frame of Fig. 2 is studied [lo, 221. The frame is of uniform cross-section EJ = 10.0 kN m2 and the plastic moment of the section is Mp = 65.73 kN m. There are eight possible critical sections of the structure. The work-hardening matrix His zero and the 16 x 16 matrix D is given by relation D = -NTZN,
(26)
where the 8 x 8 matrix Z is obtained from (15) and N is the 8 x 16 diagonal matrix of [l - 11. Further the 8 x 1 vector s6 is obtained from relation SE= K,,G%-‘p
z+oc)),
Step 2: Set j + 1+j. Step 3: Go to step 1.
120,
(27)
and k is the 16 x 1 vector of the plastic moments of the sections. For P = 78.48 kN, after the LP (25) is solved, by the use of the Simplex method, for j = l-6, the program terminates since L,, I = {a} (Table 1). That is, there is no need to solve all of the 16 LPs. The same is valid for P = 78.875 kN but forj = l-8. If P takes the value P = 78.876 kN, then no solution exists for the LPs and as a consequence the LCP is infeasible. The physical meaning of this infeasibility is that the structure has reached the collapse state, i.e. P = 78.876 kN is the collapse load. Next the structure has been solved by a computer plastic analysis program using a direct quadratic optimization algorithm [17, 18,231 with the same results. That is for P = 78.48 kN three hinges are formed and for P = 78.875 kN a fourth hinge appears. Both solutions are obtained in three pivotal steps. A further rise of the value of P causes the collapse of the structure.
M
2P 3P 2
3
4
MP
T 1.0 In 8
0
1
5
r-l
L
MP
+ Fig.
1.
Plastic moment-rotation relationship.
con-
Fig. 2.
A. V. AVDELA~
1006
Table 2. Element characteristics
Table 1 Element
6eetii
Mp (KNm)
1,2 3,4 586 7,6 9,lO 11,12
I 260 1220 1320 I 160 1300 1220
123.0 76.1 219.0 44.6 183.0
76.1
sT = [-65.73,38.247,38.247,64.738,64.738,
-65.73,
-65.73,65.73]
kN m
and Because of [24] 8=N1,
s’= [-65.73, sp=ZB
ands=sE+sP,
39.438,39.438,438.65,65.73,45.73, -65.73,
(28)
-65.73,65.73]
for P = 78.48 and 78.875 kN, respectively.
the moments vector is equal to 0.2P
30
4
40
5
t
0
@
0
P -_,
a
9
@
@
@J
11
12
+
3.0 m -+
3.0 m +
3.0 m +
3.0 m +
Fig. 3.
1.3
LoadFstw
a
Fig. 4. Rotation vs load factor, up to collapse.
3.0 m
t 3.0 m
t
kN m
1007
Collapse load of plastic structures
-
4.
--'
5.
.._... 5. ._____ 5_ .._._._,_ ____
5_
.....1. 2. I..... 10. ,...... ,O_ I,...,. ,(_ -
-I
1.0
1.1
1.2
1.3
1.4
12.
1.5
LoadFactor a
Fig. 5. Moment vs load factor, up to collapse. The frame of Fig. 3 has been analysed next under the same assumptions, with P = 137.19 kN. In Table 2 the characteristics of the elements of the structure are given. There are 24 possible critical sections of the structure. Since the work-hardening matrix H is zero, the 48 x 48 matrix D is obtained from eqn (26). The first hinge appears for a load factor a equal to 1. For CL= 1.45 collapse occurs, that is all the LPs are infeasible. The structure has been also solved by the already mentioned above computer plastic analysis program with the same results. That is the first’hinge appears for a = 1 and the last before collapse (the ninth) for a = 1.45. In Figs 4 and 5, the change of rotation 8 and of the total moment s is respectively depicted versus the load factor a.
used if the aim of the analysis is only the value of the collapse load or collapse load factor of the structure and not its total behavior under the given loads. It must be pointed out that the input data for both the upper bounding algorithm and the direct quadratic optimization one are the same since matrices D and d, are also used for the solution of the quadratic optimization problem. The usefulness of the upper bounding method presented above, is enhanced by the fact that the Simplex algorithm can solve problems with large numbers of unknowns even more so if the Karmakar algorithm is used [25]. Acknowledgemenrs-The author would like to thank Professor P. D. Panagiotopoulos of Aristotle University, Thessaloniki and Technical University, Aachen for his helpful advice.
5. CONCLUSIONS
As it has been seen, the use of the upper bounding algorithm can be very useful in the plastic analysis of structures, in the sense that it can determine the collapse load or collapse load factor of the structure without actually solving the problem i.e. the quadratic optimization problem to which it is equivalent. Instead, it solves a number of LPs at most equal to the number of elements in the Z set, but usually fewer. It must be pointed out that for the implementation of the algorithm, matrices D and dl must be formulated first. For the steps of the algorithm either a mathematic package including a linear optimization algorithm or an independent subprogram can be used. This subprogram, as it is the case here, can be incorporated in a larger and more genera1 plastic analysis program and be optionally
REFERENCES
1. G. Maier, Incremental plastic analysis in the presence of large displacements and physical instabilizing effects. Int. J. Solids Struct. I, 345-372 (1971). 2. G. Maier, Mathematical programming methods in structural analvsis. In Variational Methods in Emineering, Vol. II-(Edited by C. Brebbia and H. Totyenham). Southampton University Press, Southampton (1973). M. J. Cohn and G. Maier (Eds), Engineering plasticity by mathematical programming. Proc. NATO-AS, Waterloo, Canada 1977. Pergamon Press, New York (1977). R. W. Cottle, Numerical methods for complementarity problems in engineering and applied science. In Computing Methods in Applied Science and Engineering, 1977, I, Proceedings, IRIA, Paris (Edited by R. Glowinski and J. L. Lions). Springer, Berlin (1979).
1008
A. V. AVDELAS
5. G. Maier, Quadratic programming and theory of elastic-perfectly plastic structures. Meccanica 3, 265-273
(1968). 6. G. Maier, A matrix structural
linear elastoplasticity
theory of piece-wise with interacting yield planes.
15. J. H. Argyris, J. St Doltsinis et al., Thermomechanical response of solids at high strains-natural ;f;8ro)ach. Compt. Meths. Appl. Mech. Engng 32, I-106
Meccanica 8, 54-66 (1970). I. D. E. Grierson, A. Franchi et al., Mathematical
programming and nonlinear finite element analysis. Comput. Meths. Appl. Mech. Engng 17/18, 497-518 (1979). 8. P. D. Panagiotopoulos, Variational inequalities and multilevel optimization technique in the theory of plasticity. Comput. Struct. 8, 649650 (1978). 9. P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhluser, Boston (1985). Russian trans-
lation: Mir, Moscow (1989). 10. P. D. Paneiotoooulos. C. C. Baniotonoulos. C. C. and A. V. Avdelas, Certain propositions oh the activation of yield modes in elastoplasticity and their applications to deterministic and stochastic problems. ZAMM 64, 491-501 (1984).
Il. J. H. Argyris, Continua and discontinua. Proc. 1st Conf. Matrix Meth.
Struct.
Mech.
Wright-Patterson
F.R.,
Davton. Ohio. 1965. AFDL TR-66-80 (1966). 12. J. H. Argyris; Three-dimensional anisotropic and inhomogeneous elastic media. Matrix analysis for small and large displacements. Ing. Archiu 34, 33-55 (1965). 13. J. H. Aravris and D. W. Scharuf. Some general considerations on the natural mode* technique. Part I and Part II. Aeron J. R. Aeron. Sot. 73, 218-226, 361-368 (1969).
14. J. H. Argyris, H. Balmer er a/., Finite element methodthe natural approach. Comput. Meths. Appl. Mech. Engng 17/18, I-106 (1979).
16. P. G. Hodge Jr, Plastic Analysis of Structures. McGrawHill, New York (1959). 17. A. V. Avdelas, Application of the quadratic programming methods in the problems of the theory of plasticity (in Greek). Doctoral’ thesis, Aristotle University, Thessaloniki (1987). 18. A. V. Avdelas, A direct quadratic programming approach to the elastoplastic analysis of frames with the possibility of worksoftening. S.M.-Archives (to be published). 19. H. Kiinzi and W. Krelle, Nichtlineare Programmierung. Springer, Berlin (1962). 20. L. Collatz and W. Wetterling, Optimierungsaufgaben. Springer, Berlin (1971). 21. 0. L. Mangasarian, Simple computable bounds for solutions of linear complementarity problems and linear programs. Math. Prog. Study 25, l-12 (1985). 22. B. G. Neal, Plastic Methods of Structural Analysis. Chapman & Hall, London (1970). 23. A. V. Avdelas, The unilateral contact problem. A treatment by the use of the Keller direct method. Hel. Sot. Th. -& Appl. Mech. Proc. 1st Nat. Congr. Mech. Athens 1986. Vol. 1 DD. 207-234. T.E.E.. Athens (1987). L1 ’ 24. 0. de Donato and G. Maier, Mathematical program-
ming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity. Int. J. Numer. Meth. Engng 4, 307-329 (1972). 25. K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988).