- Email: [email protected]
Method of fictitious system for evaluation of frame shakedown displacements
Compurm 12 Srrrcrwes Vol. 50, No. 4. pp. 563467. 1994 Copyri&t Q 1994 Elwicr Sc*lre Ltd Printed in Great Britain. All ri&ts -cd 004s7%9/w 56.00 + 0.00
Pergamon
METHOD
OF FICTITIOUS SYSTEM FOR EVALUATION FRAME SHAKEDOWN DISPLACEMENTS
OF
J. ATKOCIONASand A. NORKUS Department of Structural Mechanics, Vilnius Technical University, Vilnius, Lithuania (Received 20 July 1992)
Abstract-An ideal elastic-plastic framed bendable system with prescribed physical parameters is considered. Actual residual displacements depend on the history of loading. Since the repeated-variable loading is characterized only by bounds of its variation, the only possibility is to determine the variation bounds of residual displacements. The problem is formulated as a problem of nonlinear mathematical programming. A physical interpretation of the developed mathematical model of the problem is presented, namely, the displacements of a considered structure are covered by displacements of a fictitious system, conforming to the holonomic law.
1. FORMUWTtONOF THE PROBLEM An elastic-plastic bendable frame with prescribed physical parameters under a repeated-variable loading (RVL), ensuring the shakedown state and characterized by bounds of variation F,, F,, is to be considered. The problem of estimating residual displacements for elastic-plastic structures has been investigated in many works [l-6]. The problem of determining the variation bounds of residual displacements II, is formulated as a problem of nonlinear mathematical programming. The objective function depends on plastic strains, conditions of statical and kinematical admissibility for residual stresses and strains estimating the energetic bounds of shakedown states are restrictions in this problem. Energetic bounds are defined by the minimum value a* of elastic potential confirming to the residual stresses
produced more accurately, these come under the restrictions of the problem to be considered. Applying this, the upper and lower bounds of the actual displacements are determined more accurately by solving the problem of mathematical optimization. 2. ENERGETICBOUNDSOF SHAKEDOWN The parameter a * is obtained by solving the problem of quadratic mathematical programming [7,8]. Static formulation
Find min
$M,~]M,
subject to
[AIM, = 0,
= O!*
(1)
Pl(Mr+Me,mJdM,>
a = OSM,‘p]M,
[@l(W+Me,mim)~Wt*
and the maximum value of the total energy dissipation LYmXfor all possible states of shakedown when
(2)
Kinematic formulation F, G F(t) G Fs,.
Find
At the case of unloading D =A%&,.
Here M and 1. are the vectors of limit stresses and plasticity multipliers for linear yield conditions
(3)
[Qrl(M,+M,)GM, respectively. The minimum principle of elastic potential and compatibility equations corresponding to it enable the above-mentioned energetic values to be
(4) 563
J. ATK~&%NAS and A. NORKLJS
564
Here [D] is the n size flexibility matrix, [A] is algebraic operator of equilibrium equations (m lines n columns), and
Here
[6] = -[@][B]r Cm,= Mo- I@lK,mm 1
By using the influence matrix [a], extreme vectors from n components of the elastic moments M,,_, M,,,,, are in the linear relationship with FNP, F,. For the elimination of some unknown values, these mathematical models are reconstructed. The algorithm of reconstruction is based on the fact that the mathematical model of the problem make up a dual pair of mathematical programming problems. First the equilibrium equations
Cm,= M - I@IKnin . Taking into account the above-mentioned expressions the reconstructed mathematical model of the problem (1) and (2) obtains the following form. Find min subject to
[AIM, = 0 are eliminated in the mathematical model (1) and (2). For this purpose these expressions are to be expressed in the form
Thus the problem, which is dual to the problem (5), (6), has the following form max
[[A’], [A”]]M, = 0. Here [A’] is quadratic matrix of size (m x m), which has the inverse matrix [A’]-‘. The vector M, is parted according to the matrices [A’] and [A”] to the vectors M’I and M”,9 where
IkM:
)‘@bK’
-&(C_
- [d]M:‘)
- aL(C,in - [Q,IW )} subject to
@]M: + [&lr(,Im, + A,,,,) = 0
M; = -[A’]-‘[A”]M:‘. By using the matrix [B] = [[A”]r([A’]r)-‘, -m, sidual moments M, are expressed in the form
03) re-
The strains equations
M, = - [B]?M;. According to the structure of the vector M,, the flexibility matrix [D] can be expressed in the form
PI=
[
1
[D'l [D”].
Then the elastic potential of residual stresses is
(7)
are obtained
M:‘=
from the compatibility
-@]-‘pi]?,
where
By putting the obtained expression into the objective function of the problem (7), (8) the following analysis problem is obtained. Find max
{ -o.s~TtW
+~&x,,(PlMm, - Ma)
The yield conditions
subject to 42= Am.y+ L,
A~.~B 0, A,,,~,, 3 0. (10)
Here also can be expressed in the form
[&Y’ G C,, [~IW’ d C,,,,m .
[n] = [6]@][B]? From the other hand it is easy to find that
[fJl= Pmwl~
Evaluation of frame shakedown displacements because M, = [Gl4 where
El = (bIM[Dl-’ - PI-WIT. The vector 1* is determined by solving the problem (9), (10) then
565
Here [r] is a configuration matrix of a system, Z is a set of’indexes for the cross-sections to be considered (i = 1,2,. . . , z). In addition to the development of the vector M,, the vector of elastic stresses M: E {M:}’ (i E I) and the matrix [a*] for the yield conditions in the form of equalities are to be determined. Then by using the vector w, which is of a smaller size than M,,, the equations
M,* = [GIla*, then a* = 0.5. I*yfa]rl* and the minimum value of energy dissipation, ensuring the reach of a shakedown state D Ill,”=A+‘M 0 is obtained. If D = 0, the system is in an elastic state. The minimum energy potential principle of residual stresses enables the actual M,*(t) for the final state just before a plastic failure to be determined without consideration of the loading history and, at the same time, that of the places of structure, where the rates of plastic strains are equal to zero. Thus the following proposition is true [9]. If all possible histories of loading lead to the only state of residual stresses M,* under the minimum value of the elastic potential a*, the upper bound of energy dissipation 6,_ for the actual processes of shakedown is obtained by solving the problem (9) (IO). A fictitious elastic-plastic system is created for the purpose of using the possibilities of this proposition for an analysis of residual displacements of structure at shakedown. In this system which has a strict mathematical basis, as it will be shown later, all possible histories of loading
are valid. It means that in a fictitious structure unloading does not appear for all the possible histories of loading
C. The previously the newly vector l*
problem (9), (lo), taking into account the obtained vectors MamaX, M,:,, as well as developed vector M is to be solved. The and the value 6_
= PSI,
are finally obtained. It is evident that the final state just before a plastic failure of the fictitious system is actually considered in this case, and the number of components of the vector I* not equal to zero is not greater than the degree of statical indetermination k. The algorithm which follows from the mathematical optimization models for elastic-plastic systems [lo] is valid for the states that are close to a cyclic plastic failure. The stress state does not depend on the history of loading if D,, = L?,, .
For elastic-plastic structures, equations of residual strains
compatibility
W’, = P,IW lead to the only stress state M,1, which is characterized by the parameter a *. The ‘appearance’ of the M,* for all the histories of loading is obtained by fulfilling the following three stages of the proposed algorithm. A. By solving the problem (9), (10) for the primary structure, which is characterized by the vector M under a prescribed M,,,, M,,, the parameter a* and vector M,* are to be obtained. B. On the basis of the vector M: obtained, a new vector of limit stresses M, = ]YI&
and the minimum principle for residual stresses are in correspondence ([l 11). Thus the algorithm for determining LI,,, can be analysed by solving these equations and taking into account M,=M,*. For linear conditions
the plastic strains are 8, = [O *]ra
and compatibility form
equations finally obtained have a
P,l~ = PrIM,.
(12)
is to be developed, using the rule %, =
maxPhIoM3+ Mcii.mar,,,b 1, i E1.
The matrix [B,] has k lines and z columns. Now it is that a fictitious elastic-plastic system,
(11) obvious
566
J.
ATKOEIUNAS and A. NORKUS
characterized by the vector a,,, confirm to the compatibility equations (12) for the stresses M,*, when a* = O.SM,“[D]M,*. The vectors of basic solutions from the equations systems (12) confirm to trajectories of a plastic deformation that lead to (quasi) determinable systems (a, for which has the largest values). The actual vector I* among these basic vectors is the one which confirm to the maximum value of energy dissipation 0
0
,
o
1 1.10
1 .os s
This value is fully identical with the value obtained by solving the algorithm stage C.
Fig. 1. A two-storeyecl frame. 4. NUMERICALEXAMPLE A two-storeyed frame (Fig. 1) under freely variable loads
3. MATHEMATICAL
MODEL OF EVALUATION SHAKEDOWN DISPLACEMENTS
OQ V,
The theory of mathematical programming enables the problem of w,~,_, u,~,~~, determining when %,in/
g
h,inf
G
hi
Cf 1 g
4i_nq
s
url,,sup 9
to be solved as an optimization problem, using the influence matrix [HI] for residual displacements find z%i*l= (
subject to
m
(13)
[@*]([G*]A + M:) S l@
(14)
3.20,
(15)
QWJ, %,in/
i=l,2
,...,
>
;rqn+]A
nTf,1Gb,,
(16)
The unknown value in the (13)-(16) problem is the vector 1. Let us to analyse the mathematical model (13x16) of the problem. The restrictions (14) and (16) ensure the satisfaction of the minimum elastic potential for residual stresses of a structure under consideration (the equality
-0.25F
< H < 0.25F
is considered. The limit bending moments and the inertia moments are the following: for columns - M,, , I; for beams - 2M,, 2&. The u,, for the upper beam under the various values of the safety coefficients s is to be determined. The extreme elastic bending moments for F = 1 are the following: me,,
&,,,
= (0.2457, 0.3327, 0.4168, 0.4462, O., 0.0428, 0.0197, 0.0721, 0.1153, 0.0918, O., 0.7496)r = { -0.1153, -0.0721, -0.0197, -0.0428, -0.5965, -0.4462, -0.4168, -0.3327, -0.2457, -0.7496, -0.8421, -0.0918jr.
For the parameter of limit load F. = 2.5128M,,/L the solution of the problem (9), (10) yields a* = O.O152M~L/EJ, D,,,i,= 0.4008MiL/EJ,
M,+ = (0.0819, 0.1638, -0.0476, -0.1213, -0.1213, 0.1213,0.0476, -0.1638, -0.0819, -0.1162,0.1162, 0.1162}=. Then the vector R = {0.6993, l., l., l., 1.6204,
[G*]L = M,’ I., l., l., 0.6993, 2., 2., 2.}= is satisfied). It is easy to find that a potential value smaller than a* for a statically admissible M, does not exist. That is why the restrictions (14) and (16) will always be satisfied as equalities [while at the same time the restrictions (15) can be satisfied as inequalities]. An optimal solution of the problem (13)-(16) can be obtained not only for the basic vectors 1 because s,, is the same for all i in the restrictions (15). But all this ensures the reliability in the determination of the bounds u,~,~, u,,,,.
was obtained by using the formula (11). By solving the problem, (9) (10) for the second time and by using the previously developed vector l@, the upper bound of the energy dissipation a,, = 0.7271 MiL/EZ was determined. The vector M,+ = {0.6174, 0.8361, 1.0476, 1.1213, - 1.4991, - 1.1210, -1.0476, -0.8361, -0.6174, -1.8838, -2.1162, 1.8838)‘. By solving the problem (13x16) u,,,~ = O.O838M, L*/EZwas determined. The solutions
Evaluation of frame shakedown displacements for other values of the coefficients s are shown in the form of circles in the Fig. 1. The results of a solution,
based on the use of the widely known Koiter’s for determining Dmx are also shown there graphically [ 1,6].
inequality
5. CONCLUSIONS
A mathematical model of the problem (13)-(16) is evident in its mathematical and physical interpretation, namely, the displacements of an actual structure under RVL are covered by the displacements of a fictitious system. Such a method of analysing the displacements for the only prescribed Fsup, ii,,, could be useful. REFERENCES
1. J. A. Kiinig, Shakedown of Elastic-Plastic Structures. PWN, Warsaw (1968). 2. A. R. S. Ponter, An upper bound to the small displacements of elastic-perfectly plastic structures. J. appl. Mech. 39, 959-963 (1972).
561
3. E. Vitiello, Upper bounds to plastic strains in shakedown of structures subjected to cyclic loads. Meccanica 7, 205-213 (1972). 4. M. Capurso, A displacement bounding principle in
shakedown of structures subjected to cyclic loads. Inr. J. Solids Struct. 10, 17-92 (1974).
5. J. Atkditinas, Mathematical model of analysis problem of elastic-plastic system at shakedown. Prikladnaja Mechanika 21, 79-85 (1985) (in Russian). 6. J. AtkoGnas, A. Borkowski and J. A. Kdnig, Improved bounds for displacements at shakedown. Comput. Meth. appl. Mech. Engng 28, 365-376 (1981). 7. G. Maier, Quadratic programming and theory of elas-
tic- lastic structures. Meccanica 7, 4 (1968). 8. A. 8 yras and J. AtkoEiiinas, Mathematical model for the analysis of elastic-plastic structures under repeatedvariable-loading. Me&. Res. comm. 11, 353-360<1984). 9. J. AtkoEitinas, Extmmal principle and analysis problem for shakedown systems under variable loadings. Vilnius, Lithuanian Institute of information, No. 2628 LI (1991) (in Russian). 10. J. AtkoEitinas, Algorithm of design for elastic-plastic systems under repeated-variable loading. Sopro;. material. i teor. sooruz, Kieu, 45,75-77 (1984) (in Russian). 11. N. I. Rezuchov, Fundamentals of Elasticity Plasticity and Creep Theories. Visshaja shkola, Moscow (1968) (in Russian).
Pergamon
METHOD
OF FICTITIOUS SYSTEM FOR EVALUATION FRAME SHAKEDOWN DISPLACEMENTS
OF
J. ATKOCIONASand A. NORKUS Department of Structural Mechanics, Vilnius Technical University, Vilnius, Lithuania (Received 20 July 1992)
Abstract-An ideal elastic-plastic framed bendable system with prescribed physical parameters is considered. Actual residual displacements depend on the history of loading. Since the repeated-variable loading is characterized only by bounds of its variation, the only possibility is to determine the variation bounds of residual displacements. The problem is formulated as a problem of nonlinear mathematical programming. A physical interpretation of the developed mathematical model of the problem is presented, namely, the displacements of a considered structure are covered by displacements of a fictitious system, conforming to the holonomic law.
1. FORMUWTtONOF THE PROBLEM An elastic-plastic bendable frame with prescribed physical parameters under a repeated-variable loading (RVL), ensuring the shakedown state and characterized by bounds of variation F,, F,, is to be considered. The problem of estimating residual displacements for elastic-plastic structures has been investigated in many works [l-6]. The problem of determining the variation bounds of residual displacements II, is formulated as a problem of nonlinear mathematical programming. The objective function depends on plastic strains, conditions of statical and kinematical admissibility for residual stresses and strains estimating the energetic bounds of shakedown states are restrictions in this problem. Energetic bounds are defined by the minimum value a* of elastic potential confirming to the residual stresses
produced more accurately, these come under the restrictions of the problem to be considered. Applying this, the upper and lower bounds of the actual displacements are determined more accurately by solving the problem of mathematical optimization. 2. ENERGETICBOUNDSOF SHAKEDOWN The parameter a * is obtained by solving the problem of quadratic mathematical programming [7,8]. Static formulation
Find min
$M,~]M,
subject to
[AIM, = 0,
= O!*
(1)
Pl(Mr+Me,mJdM,>
a = OSM,‘p]M,
[@l(W+Me,mim)~Wt*
and the maximum value of the total energy dissipation LYmXfor all possible states of shakedown when
(2)
Kinematic formulation F, G F(t) G Fs,.
Find
At the case of unloading D =A%&,.
Here M and 1. are the vectors of limit stresses and plasticity multipliers for linear yield conditions
(3)
[Qrl(M,+M,)GM, respectively. The minimum principle of elastic potential and compatibility equations corresponding to it enable the above-mentioned energetic values to be
(4) 563
J. ATK~&%NAS and A. NORKLJS
564
Here [D] is the n size flexibility matrix, [A] is algebraic operator of equilibrium equations (m lines n columns), and
Here
[6] = -[@][B]r Cm,= Mo- I@lK,mm 1
By using the influence matrix [a], extreme vectors from n components of the elastic moments M,,_, M,,,,, are in the linear relationship with FNP, F,. For the elimination of some unknown values, these mathematical models are reconstructed. The algorithm of reconstruction is based on the fact that the mathematical model of the problem make up a dual pair of mathematical programming problems. First the equilibrium equations
Cm,= M - I@IKnin . Taking into account the above-mentioned expressions the reconstructed mathematical model of the problem (1) and (2) obtains the following form. Find min subject to
[AIM, = 0 are eliminated in the mathematical model (1) and (2). For this purpose these expressions are to be expressed in the form
Thus the problem, which is dual to the problem (5), (6), has the following form max
[[A’], [A”]]M, = 0. Here [A’] is quadratic matrix of size (m x m), which has the inverse matrix [A’]-‘. The vector M, is parted according to the matrices [A’] and [A”] to the vectors M’I and M”,9 where
IkM:
)‘@bK’
-&(C_
- [d]M:‘)
- aL(C,in - [Q,IW )} subject to
@]M: + [&lr(,Im, + A,,,,) = 0
M; = -[A’]-‘[A”]M:‘. By using the matrix [B] = [[A”]r([A’]r)-‘, -m, sidual moments M, are expressed in the form
03) re-
The strains equations
M, = - [B]?M;. According to the structure of the vector M,, the flexibility matrix [D] can be expressed in the form
PI=
[
1
[D'l [D”].
Then the elastic potential of residual stresses is
(7)
are obtained
M:‘=
from the compatibility
-@]-‘pi]?,
where
By putting the obtained expression into the objective function of the problem (7), (8) the following analysis problem is obtained. Find max
{ -o.s~TtW
+~&x,,(PlMm, - Ma)
The yield conditions
subject to 42= Am.y+ L,
A~.~B 0, A,,,~,, 3 0. (10)
Here also can be expressed in the form
[&Y’ G C,, [~IW’ d C,,,,m .
[n] = [6]@][B]? From the other hand it is easy to find that
[fJl= Pmwl~
Evaluation of frame shakedown displacements because M, = [Gl4 where
El = (bIM[Dl-’ - PI-WIT. The vector 1* is determined by solving the problem (9), (10) then
565
Here [r] is a configuration matrix of a system, Z is a set of’indexes for the cross-sections to be considered (i = 1,2,. . . , z). In addition to the development of the vector M,, the vector of elastic stresses M: E {M:}’ (i E I) and the matrix [a*] for the yield conditions in the form of equalities are to be determined. Then by using the vector w, which is of a smaller size than M,,, the equations
M,* = [GIla*, then a* = 0.5. I*yfa]rl* and the minimum value of energy dissipation, ensuring the reach of a shakedown state D Ill,”=A+‘M 0 is obtained. If D = 0, the system is in an elastic state. The minimum energy potential principle of residual stresses enables the actual M,*(t) for the final state just before a plastic failure to be determined without consideration of the loading history and, at the same time, that of the places of structure, where the rates of plastic strains are equal to zero. Thus the following proposition is true [9]. If all possible histories of loading lead to the only state of residual stresses M,* under the minimum value of the elastic potential a*, the upper bound of energy dissipation 6,_ for the actual processes of shakedown is obtained by solving the problem (9) (IO). A fictitious elastic-plastic system is created for the purpose of using the possibilities of this proposition for an analysis of residual displacements of structure at shakedown. In this system which has a strict mathematical basis, as it will be shown later, all possible histories of loading
are valid. It means that in a fictitious structure unloading does not appear for all the possible histories of loading
C. The previously the newly vector l*
problem (9), (lo), taking into account the obtained vectors MamaX, M,:,, as well as developed vector M is to be solved. The and the value 6_
= PSI,
are finally obtained. It is evident that the final state just before a plastic failure of the fictitious system is actually considered in this case, and the number of components of the vector I* not equal to zero is not greater than the degree of statical indetermination k. The algorithm which follows from the mathematical optimization models for elastic-plastic systems [lo] is valid for the states that are close to a cyclic plastic failure. The stress state does not depend on the history of loading if D,, = L?,, .
For elastic-plastic structures, equations of residual strains
compatibility
W’, = P,IW lead to the only stress state M,1, which is characterized by the parameter a *. The ‘appearance’ of the M,* for all the histories of loading is obtained by fulfilling the following three stages of the proposed algorithm. A. By solving the problem (9), (10) for the primary structure, which is characterized by the vector M under a prescribed M,,,, M,,, the parameter a* and vector M,* are to be obtained. B. On the basis of the vector M: obtained, a new vector of limit stresses M, = ]YI&
and the minimum principle for residual stresses are in correspondence ([l 11). Thus the algorithm for determining LI,,, can be analysed by solving these equations and taking into account M,=M,*. For linear conditions
the plastic strains are 8, = [O *]ra
and compatibility form
equations finally obtained have a
P,l~ = PrIM,.
(12)
is to be developed, using the rule %, =
maxPhIoM3+ Mcii.mar,,,b 1, i E1.
The matrix [B,] has k lines and z columns. Now it is that a fictitious elastic-plastic system,
(11) obvious
566
J.
ATKOEIUNAS and A. NORKUS
characterized by the vector a,,, confirm to the compatibility equations (12) for the stresses M,*, when a* = O.SM,“[D]M,*. The vectors of basic solutions from the equations systems (12) confirm to trajectories of a plastic deformation that lead to (quasi) determinable systems (a, for which has the largest values). The actual vector I* among these basic vectors is the one which confirm to the maximum value of energy dissipation 0
0
,
o
1 1.10
1 .os s
This value is fully identical with the value obtained by solving the algorithm stage C.
Fig. 1. A two-storeyecl frame. 4. NUMERICALEXAMPLE A two-storeyed frame (Fig. 1) under freely variable loads
3. MATHEMATICAL
MODEL OF EVALUATION SHAKEDOWN DISPLACEMENTS
OQ V,
The theory of mathematical programming enables the problem of w,~,_, u,~,~~, determining when %,in/
g
h,inf
G
hi
Cf 1 g
4i_nq
s
url,,sup 9
to be solved as an optimization problem, using the influence matrix [HI] for residual displacements find z%i*l= (
subject to
m
(13)
[@*]([G*]A + M:) S l@
(14)
3.20,
(15)
QWJ, %,in/
i=l,2
,...,
>
;rqn+]A
nTf,1Gb,,
(16)
The unknown value in the (13)-(16) problem is the vector 1. Let us to analyse the mathematical model (13x16) of the problem. The restrictions (14) and (16) ensure the satisfaction of the minimum elastic potential for residual stresses of a structure under consideration (the equality
-0.25F
< H < 0.25F
is considered. The limit bending moments and the inertia moments are the following: for columns - M,, , I; for beams - 2M,, 2&. The u,, for the upper beam under the various values of the safety coefficients s is to be determined. The extreme elastic bending moments for F = 1 are the following: me,,
&,,,
= (0.2457, 0.3327, 0.4168, 0.4462, O., 0.0428, 0.0197, 0.0721, 0.1153, 0.0918, O., 0.7496)r = { -0.1153, -0.0721, -0.0197, -0.0428, -0.5965, -0.4462, -0.4168, -0.3327, -0.2457, -0.7496, -0.8421, -0.0918jr.
For the parameter of limit load F. = 2.5128M,,/L the solution of the problem (9), (10) yields a* = O.O152M~L/EJ, D,,,i,= 0.4008MiL/EJ,
M,+ = (0.0819, 0.1638, -0.0476, -0.1213, -0.1213, 0.1213,0.0476, -0.1638, -0.0819, -0.1162,0.1162, 0.1162}=. Then the vector R = {0.6993, l., l., l., 1.6204,
[G*]L = M,’ I., l., l., 0.6993, 2., 2., 2.}= is satisfied). It is easy to find that a potential value smaller than a* for a statically admissible M, does not exist. That is why the restrictions (14) and (16) will always be satisfied as equalities [while at the same time the restrictions (15) can be satisfied as inequalities]. An optimal solution of the problem (13)-(16) can be obtained not only for the basic vectors 1 because s,, is the same for all i in the restrictions (15). But all this ensures the reliability in the determination of the bounds u,~,~, u,,,,.
was obtained by using the formula (11). By solving the problem, (9) (10) for the second time and by using the previously developed vector l@, the upper bound of the energy dissipation a,, = 0.7271 MiL/EZ was determined. The vector M,+ = {0.6174, 0.8361, 1.0476, 1.1213, - 1.4991, - 1.1210, -1.0476, -0.8361, -0.6174, -1.8838, -2.1162, 1.8838)‘. By solving the problem (13x16) u,,,~ = O.O838M, L*/EZwas determined. The solutions
Evaluation of frame shakedown displacements for other values of the coefficients s are shown in the form of circles in the Fig. 1. The results of a solution,
based on the use of the widely known Koiter’s for determining Dmx are also shown there graphically [ 1,6].
inequality
5. CONCLUSIONS
A mathematical model of the problem (13)-(16) is evident in its mathematical and physical interpretation, namely, the displacements of an actual structure under RVL are covered by the displacements of a fictitious system. Such a method of analysing the displacements for the only prescribed Fsup, ii,,, could be useful. REFERENCES
1. J. A. Kiinig, Shakedown of Elastic-Plastic Structures. PWN, Warsaw (1968). 2. A. R. S. Ponter, An upper bound to the small displacements of elastic-perfectly plastic structures. J. appl. Mech. 39, 959-963 (1972).
561
3. E. Vitiello, Upper bounds to plastic strains in shakedown of structures subjected to cyclic loads. Meccanica 7, 205-213 (1972). 4. M. Capurso, A displacement bounding principle in
shakedown of structures subjected to cyclic loads. Inr. J. Solids Struct. 10, 17-92 (1974).
5. J. Atkditinas, Mathematical model of analysis problem of elastic-plastic system at shakedown. Prikladnaja Mechanika 21, 79-85 (1985) (in Russian). 6. J. AtkoGnas, A. Borkowski and J. A. Kdnig, Improved bounds for displacements at shakedown. Comput. Meth. appl. Mech. Engng 28, 365-376 (1981). 7. G. Maier, Quadratic programming and theory of elas-
tic- lastic structures. Meccanica 7, 4 (1968). 8. A. 8 yras and J. AtkoEiiinas, Mathematical model for the analysis of elastic-plastic structures under repeatedvariable-loading. Me&. Res. comm. 11, 353-360<1984). 9. J. AtkoEitinas, Extmmal principle and analysis problem for shakedown systems under variable loadings. Vilnius, Lithuanian Institute of information, No. 2628 LI (1991) (in Russian). 10. J. AtkoEitinas, Algorithm of design for elastic-plastic systems under repeated-variable loading. Sopro;. material. i teor. sooruz, Kieu, 45,75-77 (1984) (in Russian). 11. N. I. Rezuchov, Fundamentals of Elasticity Plasticity and Creep Theories. Visshaja shkola, Moscow (1968) (in Russian).