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Optimum design for work-hardening adaptation
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 12 (1977) 129-144 0 NORTH-HOLLAND PUBLISHING COMPANY
OPTIMUM DESIGN FOR WORK-HARDENING ADAPTATION * C. POLIZZOTTO, C. MAZZARELLA
and T. PANZECA
Facoltcidi Architettura, Universitridi Palermo, ViaMaqueda, 17.5. 90133 Palermo, Italy Manuscript received 23 December 1976 The finite element-linear programming approach and the work-hardening adaptation criterion are used to formulate a general theory of optimum design of rigid-work-hardening structures subjected to loads which vary statically within given limits. Self-weight, as well as some technological constraints, can be introduced into the framework of the optimization problem. The optimality conditions are discussed with the aid of geometrical descriptions as well, and a comparison is made with the standard limit design. Numerical applications are given for a plane truss and a plane frame with axial force-bending moment interaction.
Introduction The notion of Bauschinger adaptation is due to Prager [ 1 ] - [ 31. It is a special kind of shakedown in which the adaptation is produced only by (plastic) deformations occuring in a rigidkinematically hardening structure which is subjected to statically varying loadings. Some other contributions have followed Prager’s work. Polizzotto [41 has given a generalization of Prager’s results to a broader class of hardening rules, as well as an inadaptation theorem analogous to that of Koiter for elastic-plastic inadaptation [5 I. Second-order geometric effects and temperature cycles have been accounted for by KBnig and Maier [61, who have also given bounding criteria for plastic deformations and displacements. Finally, Polizzotto and Mazzarella [ 71 have adopted the work-hardening adaptation criterion for a formulation of the optimum design problem. The present paper considers the topic treated in [ 71, discussing a number of aspects more fully and introducing aspects that were disregarded before. The structure to be designed is a general structure conceived as a discrete model. This has a fixed node layout, and its finite elements have dimensions, sizes and physical properties to be determined in such a way that the resulting design satisfies the two following conditions: a) The cost is minimum. b) There is safety as regards quasistatic loads which vary with an unknown time sequence but remain in the interior of a given loading domain. To this goal some simplifying hypotheses are introduced. They are: 1) The typical finite element has an idealized piecewise linear rigid-plastic behavior. In any stage the yield surface is a polyhedron whose faces can only translate as a result of the work-hardening [41, [71, [81*
2) The work-hardening rule has symmetry with respect to the yielding modes [ 81. 3) The pEastic resistances, i.e. the distances from the origin of the faces of the virgin yield polyhedron, are linearly dependent on the design variables. * Friendly dedicated to Professor Ugo Fuxa.
C. Polizotto,
130
C. Maazzarellaand T. Panzera. Optinutm design fbr work-hardening adaptation
4) The cost is a linear function of the design variables. 5) A continuous spectrum of elements is available. 6) Geometrical effects on equilibrium equations are ignored. In the present context safet.~ means capability of the structure to adapt itself to a purely rigid state, generally after an initial period during which plastic dissipation occurs. A sufficient criterion for ruork-hardening adaptation, suitable for the present application, is that formulated in [4]. An optimized structure, designed to be safe in the sense explained above, can however suffer from dangers like fatigue failure or displacements which are too large and so become unserviceable. The amount of plastic dissipation energy involved in a real adaptation process, though limited, could in fact be excessive in comparison with some fatigue and service restrictions. Then an enlarged safety notion should be used as a basis for the optimization, but this will be the proposition of future work. The present paper only considers possible limitations on dimension and size changes of the virgin yield domain as a result of the work-hardening adaptation process.
1. Preliminary
propositions
The structure is acted upon by variable loads which, for the sake of simplicity, are considered to be applied directly on the nodes of the given layout. These loads, which are described by the vector f, are the sum of service, fixed, and self-weight loads, respectively p, pf and pw , i.e.
Let the vector u describe the (infinitesimal) displacements of the nodes, and let the vectors and u describe the (generalized) strains and stresses of the elements, respectively. After these definitions the compatibility and equilibrium equations are readily written as
E
E=CU,
(3)
C’a =f,
(3)
where C is the compatibility matrix, which depends only on the given node layout. The conditions of conformity, namely the conditions dictated by the plasticity laws, are written using a matrix description which is now becoming usual in structural analysis and synthesis (see e.g. [41, [6]-[8]). With re f erence to a single element of the structure, say the hth, these conditions are the following: ;h
,j/hih
,
+h
= (Nh)t&
_
Hh
kh
__ kh I? =
@GO,
ih>O,
(+h)tjh=o.
($‘)+=o
1)
2, . . . n .
(4)
’
Fig. 1 is an illustration of the meaning of eqs. (4). The superposed dot means derivative with respect to time t, N* is the matrix of unit external normals to the faces of the yield polyhedron, while Hh is the work-hardening matrix (by hypothesis symmetric and positive semidefinite).
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
Fig. 1. Sketch of yield surface in two dimensions: i: = N,i, 6, = @*= 0, @3,...@6< 0.
+ N,& + . .. + N,i,
= N,& + N,&;
131
i,, & > 0, i3 = ... i6 = 0;
Moreover k” is the vector of plastic resistances, and $* is the vector of plastic potentials. There is plastic yielding in the element, that is kh + o, only if(i) some components (at least one) of the plustic activation coefficient vector ih are nonzero, and (ii) the corresponding components of the vector +h, as well as those of the vector 9” = (Nh)‘ih - Hh ih, are zero. Relations analogous to eqs. (4) can be written for the overall structure by introducing supervectors and supermatrices which respectively collect all the pertaining vector and matrix quantities, for instance: N = [Nl, N2, ... N”]
H=
)
d = [(a’)’ , (u2)$ ... (a”)fl , Then the conformity i=Ni, +
pP,IP, ...H”J ,
k’= [(k’)‘,
(k2)‘,... (k”)‘] .
conditions for the overall structure are
+=N’a-HA-k i>o,
+ti=o,
&=o
*
(5)
It is worth noting that the two orthogonality conditions of eqs. (5) are equivalent to the analogous conditions of eqs. (4) written for all n elements because the relevant variables are all sign-constrained.
C. Polizzotto, C. Mazzarella and T. Panzera, Optimum design ftir work-hardenirzg adaptatiotl
132
2. Design variables A design is completely described by a number of nonnegative (geometrical or physical) parameters called design variables. These are collected in a design vector x > o. The design x = o is the weakest design, that is the design obtained using a material cover which is the lightest, taking into consideration given construction requirements. When there are no such requirements, the design x = o identifies with the uncovered layout. The cost g, the plastic resistances k and the self-weight p, are design-dependent variables. This dependence is generally nonlinear, but quite frequently it is supposed to be linear. Though this linear idealization causes of course some inaccuracy, solutions which are more easily obtained are however expected to give sufficient information about the nonlinear problem. As a result the following linear relations hold: g=gO
+dx,
(6)
k = k” + Vx , Pw =pw0
(7)
+wx,
03)
where the superscript ’ refers to the weakest (or initial) design. We observe: i) The cost gradient vector c has nonnegative components and depends on the geometric and physical properties of the elements as well as on the way the design variables are defined. ii) The matrix V, with nonnegative entries only, depends on the piecewise linearization of the element yield domains. It characterizes the way the element yield polyhedra expand when the design variables increase. Particular expansion modes are the free expansion mode (the faces of each polyhedron translate each other independently), and the uniform expamiorz mode (the yield domain remains geometrically similar to itself during the expansion). In any case the matrix V must be defined in such a way that the expanded polyhedra described by eq. (7) and the given matrix N could be considered as good piecewise linear approximations of the true yield domains of all the available elements. In the sequel the column vectors of V will be supposed to be unit vectors [71, [9], [ 101. iii) The self-weight matrix W depends on the given layout, as well as on some physical and geometrical properties of the elements [ 7 I , [ 9 I, [ 10 1.
3. The statical adaptation
criterion
In 141 two work-hardening adaptation criteria were given, one of statical type, the other of kinematical type. Both could be used as basic tools in the present optimization problem, but to achieve this goal the first seems to be more convenient than the second. The statical adaptation criterion named above is based on the notion of statically admissible yield surface. This is a subsequelzt yield surface (namely a surface obtained from the virgin one by changes which comply with the specific work-hardening law) such that for every loading condition there is an equilibrated stress distribution which is not in the exterior of the same yield surface.
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
133
Let the set of loading conditions be described by the homogeneous vector function
f(T)=P(T)+PF >
PF =pf +Pw
(9)
)
where r is a vector parameter belonging to an r-dimensional loading domain R. Now a statically admissible yield surface is defined taking a vector 1,> 0 (hardening intensity vector) such that the following conditions are met (see eqs. (3) and (5)): C%(t)-p(t)-pF=o,
N’s(r)-H)c-k
VTER.
(10)
The statical adaptation criterion says that is such a vector A exists, the structure possesses adaptation ability for all the loading conditions mp(r) +pF, whatever m is, but 0 < m < 1. It is easy to prove the following theorem on the statically admissible yield surfaces: THEOREM. Zf a yield surface is statically admissible with respect to a set of loading conditions, it is also statically admissible with respect to the loading conditions of the convex hull [ 111 of this set. Proof
Equilibrium and conformity (10) are by hypothesis and in particular in two certain points pi, f2 of it. Depending equations and on convexity of the yield domain, equilibrium be met also in the interior of the segment with end points rl {r=(l
-Q)ti
+(Yt2,
met in the set of loading conditions, on linearity of the equilibrium and conformity are easily shown to and f2, i.e. the segment
vol: o
to which the following loading set corresponds:
{f(T)=(l
-a)f(rl)+orf(t2),
Va:
o
Since every point of the convex hull belongs to at least one such segment, the theorem is proved. A general smooth loading domain is always supposed to be piecewise linearized. Then the convex hull of a (piecewise linear or linearized) loading domain R is an r-dimensional polyhedron of, say, p vertices. In the sequel we shall define the relevant loading conditions of a given loading domain R to be the smallest discrete set of loading conditions whose convex hull is also the convex hull of R. In other words, the relevant loading conditions are just the p vertices of the convex hull. In the frequent case of p- 1 alternative loading conditions, each consisting of loads which vary proportionally to a single nonnegative parameter, i;e. fi=ripi+pF,
O
1,
i= 1,2 ,... p-l,
there are p relevant loading conditions, that is the p- 1 points ri = 1, 7,. = 0, i # i (i, j = 1, 2, . . . p-l), as well as the origin r1 = T* = .. . = rP_ i = 0 . By virtue of the theorem above, to apply the statical adaptation criterion to a given loading domain R, it is sufficient to write the conditions (10) only for the p relevant loading conditions ofR.
134
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
4. The optimum design problem
Let the following loadings: fi=pi+pF,
ViEP,
(11)
withP= (1, 2, . .. p}, be the set of relevant loading conditions of a given (piecewise linearized) loading domain R. Taking into account all the hypotheses and positions of the previous paragraphs, eqs. (6)-Q) and (lo), (1 l), we write the following minimization problem: c*ai-
N’oi - Hk-
Min cP=c’x: i
wx-f;
I
X20,
=o, Vx -k”<
Vi E
o ,
120,
P
ViEP
,
(12)
I
wheref,? =pi+pr+pt, i E P, are the loadings which should be applied upon the weakest design, and the objective function Q differs from the cost g (6) by a constant term only. Following known classical methods [ 1 I 1, [ 121, the optimality conditions are now deduced, that is the necessary and sufficient conditions in order that the vectors x*, L* , tstY, Vi E P, and some other vectors zi,!, k;, Vi E P, satisfy the optimization problem (12). These conditions are written after some manipulations as follows (a being an arbitrary positive constant): C*a,! - Wx’-f:
=o,
ViEP
(equilibrium) ,
(13)
(conformity)
(14)
+fI = N’af I - Hn* - Vx* - k” , 6,: =Nif +T < 0 ) i;>o, Nif-Cti,?=o,
+Tif=O,
,
ViEP
ViEP
(compatibility)
,
(15)
JI* = V’i;; - W’&;; - (ye., &* = vx* (uniformity) +**x* = 0
$20,
x*20,
X* =Hi;
, h’ =H1*
xX0,
l.*>o,
if*‘&* =o
,
(16)
(adaptation) ,
(17)
(resultant mechanism) .
(18)
1
I
Eqs. (13)-( 18) describe how the optimum design - considered to be in the limit state where adaptation has occurred - behaves under the action of the p loading conditions. This behavior has the following characteristics: (a) Each of the p loading conditions produces in the structure a compatible strain field and an
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design jbr work-hardening adaptation
135
equilibrated
stress field which are conforming, that is they correspond to each other through the plasticity laws (13)-( 15). (b) The resultunt mechanism, that is the sum of the mechanisms respectively produced under the action of the p loading conditions, satisfies uniformity and adup tu tion requirements (16)-( 18). As it has been explained in [9], [ 101, eqs. (16) have been named “uniformity conditions” because they are substantially equivalent to the “uniform dissipation energy principle” of Drucker and Shield [ 131. Moreover, a physical analogy can be envisaged on the basis of the formal similarity between eq. (14) and eq. (16). By virtue of this analogy the optimum design x* can be viewed as the result of a growing process in which theplastic resistance vector Ak” = I/X* is linked to the resultant plastic strain rate intensity vector 5: through the uniformity conditions (16): just as plastic geformation is a result of the yielding process, in which the plastic strain rate vector Ef = NL; is linked to the stress vector Go!through the conformity conditions (14).
Fig. 2. Sketch of growing surfqce and hardening hypercone in two dimensions: Ak = Y,x,+ V,x, =V,x,;h=H,h,+H,h,=H,h,; x1 = 0,x, > 0; hI = 0, A2 > 0; rl,1 < 0, $2 = o;x1<0, x2 = 0.
The above analogy can also be described by using geometrical language [9], [ 101. In fact, in the i-space (see fig. 2) a (convex polyhedral) growing surface exists which has a,many faces as there are design variables and whose (external unit) normals are the column vectors of the matrix V. The distances from the origin of these faces in the case where the self-weight is ignored (i.e. if W = 0) are measured by the components of the vector (YC;in other words they are proportional to the components of the cost gradient vector c. In the more general case where the self-weight is present (i.e. if W # 0) these distances are measured by the vector CYC = 01c+ W'irg , and the modified distance aZj of the jth face is generally greater but even less than the simple distance aci. Then the presence of the self-weight produces translations of the faces of the growing surface, with consequent cost raising (or lowering) effects. Finally, we see that the vector Ak’ = VX* , which describes the plastic resistance increments of the optimum design, satisfies the following growing law: (i! it is displayed along the external generalized normal to_the growing surface at the point where AR touches this surfaces; (ii) it is zero in the case where A, does not touch the surface.
136
C. Polizzotto,
C. Mazzarella
and T. Panzeca,
Optimum
design for work-hardening
adaptation
The adaptation conditions (17) were discussed in 141. They give information about the yield domain of the optimum design in the limit state where work-hardening adaptation has been reached. In fact, they link the resultant mechanism &i, i;T given by eq. (18) with the ~z~orkhardening intensity vector I*, and therefore with the hardening displacement vector h* = H A*, whose components are the translations of the faces of the yield domain from the virgin state to the limit state above. These adaptation conditions can receive adequate description in the i-space, where a (polyhedral) adaptation hvpercone is defined. Its vertex is the origin, and the external normals of i!s faces are described by the column vectors of the hardening matrix H (see fig. 2). The vector Ai cannot be external to this cone: when it lies along a generatrix of the cone, the yector h* = HA* is perpendicular to the conical surface along the activated generatrix, but when LI; is internal to the cone, h” = o. The resultant mechanism, given by eq. (1 S), is a definition of the so-called Fotllkes mechanism
[91, [lOI. In concluding this paragraph, we observe that the primal problem given previously, can be reformulated in a dual form. Following known methods [ll], [12], wehave I
i Max q = (Y-’ c
(<,fio)‘zii - (k’)’
ii):
iEP
Of course, the optimum = c
((fo)‘;;
- Ctii = o ,
vi, - W’liR H&Go, i,
;
&X*
.
Nii
Vi E P
ii ) li,
iEP
\
(YCGO,
Vi E P
$20,
= c
values of the two objective
-
= c
1.
(19)
iii
iEP
functions
eq. (12).
coincide,
I
i.e.
- (P)’ i;).
iEP
5. Economical
value of the work-hardening.
Serviceability
Turning back to the design problem given by eq. (12), we see that in the case H = 0 it coincides with a standard plastic design problem, that is a design problem for plastic collapse. Calling @i the optimum value of the objective function in the case above, and @* that in the case H # 0, it is of course true that +i 2 cP*. In other words, the cost gi of the optimum design for plastic collapse is not less than the cost g* of the optimum design for work-hardening adaptation: g;+
g” .
(21)
Therefore, the work-hardening has an economical value since by taking it into consideration in the optimization process, one can save material. The amount of material saved, and hence the economical value of the work-hardening, depends on the work-hardening law. If the work-hardening is isotropic, its economical value is infinitely large because in this case [4] there is always adaptation capability, whatever the loading domain. The optimum design is the initial one, i.e. the design X* = o, and any load can be sustained without expense.
C. Polizzotto, C. Mazzarellaand T. Panzeca, Optimum design for work-hardeningadaptation
137
.
~,#0)[41,[7].Therefore,itis In the case of kinematic work-hardening, it is ti, =o(but seen that the terms depending on the self-weight and on the (fixed) load pf + p”, disappear from the dual formulation of the optimum design problem (19). As a result the optimum design is not influenced by the presence of any fixed load and can thus be sustained without expense. The results obtained above are drastically modified when limitations are imposed upon plastic deformations occurring in a real loading program. These limitations inevitably reduce the economical value of the work-hardening, and it would be interesting to explore this aspect of the problem. But it seems reasonable to expect a considerable saving of material even when serviceability requirements are imposed. Limitations on the changes of dimensions and size of the virgin yield domain, which are due to the work-hardening behavior, can easily be imposed by constraining the work-hardening displacement vector h = H1. Then a matrix condition of the form Hk
-Hl
(22)
d being a suitable limiting vector, should be added to the constraint set of the optimization problem of eq. (12). But, how must the vector d be selected in order to satisfy given serviceability require-
ments? To the authors’ knowledge this point is not clear and demands further investigation in the future.
6. Numerical applications
Two examples are presented for numerical applications: a plane truss and a plane frame, in the hypothesis of kinematical work-hardening. In this case we know [4], [7], [8], [ 141 that the work-hardening matrix Hh of an element is given by H” =
,*(Nh )‘Nh ,
(23)
where K* is a physical positive constant. Introducing the “initial” stress vector G, given by 6’ = [(ii’)‘, (ii*)‘, ... (P)‘]
)
ah = KhNhA* )
(24)
permits one to write the plastic potential vector + (5) in the form $=N’(a-if)-k. This form is computationally than that in k . Example
(25) convenient since the number of variables in the vector is is much less
1. Plane truss
Consider the plane truss subjected to the loading conditions described in tig. 3a. The loading domain R is the triangle of fig. 3b whose vertices O-l-2 are the three relevant loading conditions. The state of stress in a bar is described by a single stress component, the normal stress resultant
C. Polizzotto, C. Mazzarella and T. Panzeca, Optirnunl design for work-hardening adaptation
138
x
f, -
f,= Fr,
6
4
2
6f* f,. Fr,
B
10
-L-.-L a) Pig. 3. Truss layout
Nh ; the plasticity
subjected
conditions
to variable
-
D
7
loadings:
z1
b) a) geometrical
scheme
and loads; b) loading
domain
R
of the bar are
Given the material yield point uY and ignoring any stability requirement, the bar areas Ah are then proportional to the bar load capacities kh , which can be assumed to be design variables, i.e. xh = kh. Two types of optimum design are obtained, the Standard Limit Design (SLD) and the WorkHardening Adaptation Design (WHAD); each type is considered for the following cases: I. The lower bound of design variables is zero, i.e. xh > 0. II. The lower bound of design variables is different from zero, i.e. xh/F Z 1, where F is some force intensity. Table 1 contains optimum values of design variables and of (conventional) volume W = C;i=, Ihx”, and shows the percentage of material volume that can be saved if the work-hardening is taken into account. Table 2 contains the joint velocities of the optimum designs for work-hardening adaptation under the three relevant loading conditions O-l-2. We see that the resultant mechanism . . . . vanishes, as the adaptation conditions (17) demands in the present case of 'R = '(0) + ‘(1) + *(2) kinematical work-hardening [4], [ 71. 1. Optimum
Table Design case
Design
Optimum
,+l
Volume
bar areas Ahcry/F -
type 1
I. Xh> 0
bar areas and volume
2 ~_._~~.
3
SLD WHAD
3.000 3.000
2.829 1.415
2.000 1.500
SLD WHAD
3.000 3.000
2.829 1.000
2.000 1.500
4
1.000 1.000
5
6
7
5.000 3.000
1.415 0.708
3.000 1.500
5.293 3.000
1.000 1.000
2.708 1.500
8
9
10
1.415 0.708 1.000 1.000
1.415 1.122
W/FL
1.000 1.000
Material saving !C
18.66 11.83
36.6
21.24 15.14
28.8
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
139
Table 2. Collapse mechanisms for WHAD Relevant loading condition
Design case
I.xh do
Il.+
Joint horizontal velocities
Joint vertical velocities
A
B
C
D
A
B
C
D
0 1 2
0.500 -0.500
1.000 -1.000
-0.500 0.500
-0.500. -0.500 1.000
-0.708 1.208 -0.500
-1.415 2.708 -1.293
-0.708 1.208 -0.500
-0.915 2.708 -1.793
0 1 2
0.500 -0.500
1.000 -1.000
-0.500 0.500
-1.000 1.000
-0.708 0.500 0.208
-0.708 2.000 -1.293
-0.708 0.500 0.208
-0.708 2.000 -1.293
Fig. 4. Frame structure subjected to variable loadings: a) continuous model; b) discrete model; c) loading domain R.
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardrhR
140
Example
adaptatiorl
2. Plane frame
Consider the plane frame of fig. 4a. It is subjected to the action of vertical and horizontal loads which are uniformly distributed along the beams. The vertical load has equal intensity p, for the two beams, while the horizontal load has intensities 1.Sp, and I-)~,respectively, for the upper and lower beam. The loading conditions are described by pi = 12.00 r,F/L
,
pz
=
kO.88 r,FIL
>
o+<
1,
-71 G r* < 71
(where F and L are force and length factors, respectively), and hence by the loading domain R of fig. 4c, whose vertices O-l-2 are the relevant loading conditions. Any fixed load (like self-weight) can be ignored, since it will not affect the optimum design. In the spirit of the present discrete approach the lumped-deformability model of fig. 4b is adopted with 30 critical sections and rigid elements of equal length L/6. The loads are described as concentrated forces, i.e. the forces P, vertically applied to the ends of all the rigid elements of the two beams, and the forces 1.5P,, and P, horizontally applied to the centroids of the above rigid elements of, respectively, the upper and the lower beam. These forces are then P, = 1.00OF~,
,
P, =+0.148
Fr2.
In the hth critical section (h = 1, 2, .. . 30) the stress state and the “initial” stress state are
where Mh and Nh are the bending moment and the axial force, respectively, while the strain rate state is described by the vector (ih)’ = {i”
Ch} )
Fig. 5. Idealized
(virgin)
yield polygon
for standard
I-section
working
in flexure
about
the strong
axis.
C. Polizzotto, C. Mazzarellaand T. Panzeca, Optimum design for work-hardeningadaptation
141
where ih and ih are the rates of relative rotation and relative axial displacement, respectively. If standard steel I-sections of a given series are to be used, we remember that for this type of cross-section an idealized yield domain as in fig. 5 can be assumed [ 71, [ 141, [ 151, where M, , N,, are simple yield limits, i.e. the yield moment and the yield axial force, respectively. With the hypothesis NY = bM, , b being a suitable constant, there will be only one design variable for each critical section, i.e. the variable M,, but some constraints can be introduced for practical reasons so that the total number of independent design variables can be even less than the number of critical sections. 0924
23
l--l 1 \
\
\
1123
\
\
\
\
n
My- diagrams
Standard Limit Design (SLD) I -------
Work-Hardening Adaptation Design (WHAD)
Fig. 6. Optimum design of a plane frame in the design case III (free sections): My-diagrams.
C. Polizzotto, C. Mazzarella and T. Panan;eca,Optimum design for work-hardening adaptation
142
The &vector kh of plastic resistances relative to the hth section is expressed in the form kh = VhA4t, and the 6 X I matrix Vk is the same for all critical sections, i.e. [7] ~ [ 141 T/= (1 + (0.85)2b2}-‘/2
(I’?’ = { 1 bq bq 1 bq bq} , The 6
X
1 matrix Nh of the external unit normals is [ 71, [ 141
-I
Nh_lGt-l-E 07-i
t rl
0
-?-/ -q
I t= . ’
0.85bq
The equilibrium equations and plasticity conditions have been written in a dimensionless form. For this purpose a suitable numerical value has been given to the dimensionless product bL, i.e. bL = 38.7626 [14]. Simpli~cations that result frotn the symmetry of the frame geometry and relevant loading conditions have been taken into account in order to determine the minumum (conventional) volume W in the following three design cases: III. All 30 critical sections are independent. IV. Columns and beams must be prismatic, but each structural element is independent of the other (~ec~~oZog~c~Zco~lstraints). V. Columns and beams must be prismatic with the beams equal to each other (technological co~~truints), and the columns must be of nondecreasing cross-sections from the upper to the lower ends of each vertical axis (co?z~tr~ct~o~zreq~~re~lent co~str~~izt~). Again, the standard limit design (SLD) and work-hardening adaptation design (WHAD) have been obtained. Optitnum values of yield moments and volume in the three cases above, and material saving as well, are reported in table 3 (see also fig. 6). For the sake of simplicity, no mention is made of the resultant mechanism for WHAD which once more complies with the adaptation conditions (17), that is it equals zero [41, f7 1. The amount of material saving shown in tables 1 and 3 will quite probably decrease when the proper standard I-section - which is selected from some commercial series - is assigned to every structural element. This penalty is connected with the use of continuous variable methods in place of discrete variable methods. Table 3. Optimum yield moments and volume Design case ----.--.-III. Free sections
Design type __-.
Volume
Optimum yietd momentsM#X Column 1-4
Column 16-19
Beam 9-15
Beam 24-30
W/FL2
Material saving %
13.23
SLD (See fig. 6) WHAD
IV. Technological constraints V. Technological and construction constraints
8.74
SLD
0.584
0.793
1.039
WHAD
0.272
0.667
0.667
0.667
13.63
SLD
0.695
0.667
0.964
0.964
19.40
WHAD
0.667
0.661
0.667
0.667
16.00
0.750
33.9
18.99 28.2
17.5
C. Polizzotto,
C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
143
7. Conclusions We have studied here a method of designing optimum structures with the criterion of the workhardening adaptation. This method applies to general discrete structures which have a fixed node layout and are made up of structural elements belonging to a continuous spectrum. Every element has a piecewise linearized rigid-plastic behavior, and its yield surface is a polyhedron whose faces can only translate as a result of the work-hardening. This latter can be of any type that is symmetric with respect to the yielding modes. The self-weight of the structure, which is design-dependent, is taken into account. Since the cost function as well as the self-weight are supposed to be linearly dependent on the design variables, optimization problems of linear programming are encountered. There are two dual formulations, both based on the statical theorem of the work-hardening adaptation theory. The optimality conditions are discussed using also proper geometrical descriptions. The present method coincides with the classical limit design method in the case of rigid-perfectly plastic behavior. It permits one to obtain optimum designs which are of less cost than standard ones, i.e. designs for plastic collapse, as is shown also by the numerical example given in the paper. The apparent economical value of work-hardening would be much reduced if proper limitations were imposed upon plastic deformations occurring during any loading program within the limits, but it is expected to remain large enough. Limitations on plastic deformations should be introduced into the framework of the optimization problem because only in such a way can the optimum design prove to be also serviceable, i.e. able to escape dangers of excessive (plastic) deformations. In the present work insufficient attention has been given to this fondamental problem of serviceability of optimum designs. A specific paper will be devoted to it in the near future. The results obtained in the present paper are to be considered as a first step towards desirable future developments.
Acknowledgement The present paper is part of a research project sponsored by the Consiglio Nazionale delle Ricerche, CNR, Italy. The authors gratefully appreciate the financial support of CNR.
References [l] W. Prager, Bauschinger adaptation of rigid, work-hardening trusses, Mech. Res. Comm. 1 (1974) 253-256. [2] W. Prager, Optimal plastic design of trusses for Bauschinger adaptation, in: Omaggio a Carlo Ferrari (Libreria Editrice Universitaria Levrotto & Bella, Torino, 1974). [3] W. Prager, Adaptation Bauschinger d’un solide plastique B ecrouissage cinematique, C.R. Acad. Sci. Paris 280 Ser. B (1975) 585-587. [4] C. Polizzotto, Work-hardening adaptation of rigid-plastic structures, Meccanica 10 (1975) 280-288. [5] W.T. Koiter, General theorems for elastic-plastic solids, in: I.N. Sneddon and R. Hill (Eds.), Progress in Solid Mechanics 1 (North-Holland, Amsterdam, 1964) ch. 4. [6] J.A. Kbnig and G. Maier, Adaptation of rigid-workhardening discrete structures subjected to load and temperature cycles and second-order geometric effects, Comp. Meth. Appl. Mech. Eng. 3 (1976) 37-50.
144
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design jbr work-hardening adaptation
[ 71 C. Mazzarella and C. Polizzotto,
Optimization of rigid-workhardening structures (in Italian). Tech. Rep. SISTAR-75-OMS-2. Facolti di Architettura, Palermo (Apr. 1975) and Costruzioni Metalliche (in press). [8] G. Maier, A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes, Meccanica 5 (1970) 54-66. [9] C. Polizzotto, Optimum plastic design of structures under combined stresses, Int. J. Solids Structures 11 (1975) 539-553. [lo] C. Polizzotto, Optimum plastic design for multiple sets of loads, Meccanica 9 (1974) 206-213. [ 1 I] G.B. Dantzig, Linear programming and extensions (Princeton Univ. Press, Princeton, 1963). [ 121 O.L. Mangasarian, Nonlinear Programming (McGraw-Hill, New York, 1969). [ 131 D.C. Drucker and R.T. Shield, Design for minimum weight, Proc. Int. Congress Appl. Mech. 5 (1956) 212-222. [ 141 C.E. Massonnet and M.A. Save, Plastic Analysis and Design 1. Beams and Frames (Blaisdell, New York, 1965). [ 151 C. Polizzotto and C. Mazzarella, Limit design of frames with axial force-bending moment interaction, Tech. Report SISTAR-75-OMS-3 Facolta di Architettura. Palermo (Dec. 1975).
OPTIMUM DESIGN FOR WORK-HARDENING ADAPTATION * C. POLIZZOTTO, C. MAZZARELLA
and T. PANZECA
Facoltcidi Architettura, Universitridi Palermo, ViaMaqueda, 17.5. 90133 Palermo, Italy Manuscript received 23 December 1976 The finite element-linear programming approach and the work-hardening adaptation criterion are used to formulate a general theory of optimum design of rigid-work-hardening structures subjected to loads which vary statically within given limits. Self-weight, as well as some technological constraints, can be introduced into the framework of the optimization problem. The optimality conditions are discussed with the aid of geometrical descriptions as well, and a comparison is made with the standard limit design. Numerical applications are given for a plane truss and a plane frame with axial force-bending moment interaction.
Introduction The notion of Bauschinger adaptation is due to Prager [ 1 ] - [ 31. It is a special kind of shakedown in which the adaptation is produced only by (plastic) deformations occuring in a rigidkinematically hardening structure which is subjected to statically varying loadings. Some other contributions have followed Prager’s work. Polizzotto [41 has given a generalization of Prager’s results to a broader class of hardening rules, as well as an inadaptation theorem analogous to that of Koiter for elastic-plastic inadaptation [5 I. Second-order geometric effects and temperature cycles have been accounted for by KBnig and Maier [61, who have also given bounding criteria for plastic deformations and displacements. Finally, Polizzotto and Mazzarella [ 71 have adopted the work-hardening adaptation criterion for a formulation of the optimum design problem. The present paper considers the topic treated in [ 71, discussing a number of aspects more fully and introducing aspects that were disregarded before. The structure to be designed is a general structure conceived as a discrete model. This has a fixed node layout, and its finite elements have dimensions, sizes and physical properties to be determined in such a way that the resulting design satisfies the two following conditions: a) The cost is minimum. b) There is safety as regards quasistatic loads which vary with an unknown time sequence but remain in the interior of a given loading domain. To this goal some simplifying hypotheses are introduced. They are: 1) The typical finite element has an idealized piecewise linear rigid-plastic behavior. In any stage the yield surface is a polyhedron whose faces can only translate as a result of the work-hardening [41, [71, [81*
2) The work-hardening rule has symmetry with respect to the yielding modes [ 81. 3) The pEastic resistances, i.e. the distances from the origin of the faces of the virgin yield polyhedron, are linearly dependent on the design variables. * Friendly dedicated to Professor Ugo Fuxa.
C. Polizotto,
130
C. Maazzarellaand T. Panzera. Optinutm design fbr work-hardening adaptation
4) The cost is a linear function of the design variables. 5) A continuous spectrum of elements is available. 6) Geometrical effects on equilibrium equations are ignored. In the present context safet.~ means capability of the structure to adapt itself to a purely rigid state, generally after an initial period during which plastic dissipation occurs. A sufficient criterion for ruork-hardening adaptation, suitable for the present application, is that formulated in [4]. An optimized structure, designed to be safe in the sense explained above, can however suffer from dangers like fatigue failure or displacements which are too large and so become unserviceable. The amount of plastic dissipation energy involved in a real adaptation process, though limited, could in fact be excessive in comparison with some fatigue and service restrictions. Then an enlarged safety notion should be used as a basis for the optimization, but this will be the proposition of future work. The present paper only considers possible limitations on dimension and size changes of the virgin yield domain as a result of the work-hardening adaptation process.
1. Preliminary
propositions
The structure is acted upon by variable loads which, for the sake of simplicity, are considered to be applied directly on the nodes of the given layout. These loads, which are described by the vector f, are the sum of service, fixed, and self-weight loads, respectively p, pf and pw , i.e.
Let the vector u describe the (infinitesimal) displacements of the nodes, and let the vectors and u describe the (generalized) strains and stresses of the elements, respectively. After these definitions the compatibility and equilibrium equations are readily written as
E
E=CU,
(3)
C’a =f,
(3)
where C is the compatibility matrix, which depends only on the given node layout. The conditions of conformity, namely the conditions dictated by the plasticity laws, are written using a matrix description which is now becoming usual in structural analysis and synthesis (see e.g. [41, [6]-[8]). With re f erence to a single element of the structure, say the hth, these conditions are the following: ;h
,j/hih
,
+h
= (Nh)t&
_
Hh
kh
__ kh I? =
@GO,
ih>O,
(+h)tjh=o.
($‘)+=o
1)
2, . . . n .
(4)
’
Fig. 1 is an illustration of the meaning of eqs. (4). The superposed dot means derivative with respect to time t, N* is the matrix of unit external normals to the faces of the yield polyhedron, while Hh is the work-hardening matrix (by hypothesis symmetric and positive semidefinite).
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
Fig. 1. Sketch of yield surface in two dimensions: i: = N,i, 6, = @*= 0, @3,...@6< 0.
+ N,& + . .. + N,i,
= N,& + N,&;
131
i,, & > 0, i3 = ... i6 = 0;
Moreover k” is the vector of plastic resistances, and $* is the vector of plastic potentials. There is plastic yielding in the element, that is kh + o, only if(i) some components (at least one) of the plustic activation coefficient vector ih are nonzero, and (ii) the corresponding components of the vector +h, as well as those of the vector 9” = (Nh)‘ih - Hh ih, are zero. Relations analogous to eqs. (4) can be written for the overall structure by introducing supervectors and supermatrices which respectively collect all the pertaining vector and matrix quantities, for instance: N = [Nl, N2, ... N”]
H=
)
d = [(a’)’ , (u2)$ ... (a”)fl , Then the conformity i=Ni, +
pP,IP, ...H”J ,
k’= [(k’)‘,
(k2)‘,... (k”)‘] .
conditions for the overall structure are
+=N’a-HA-k i>o,
+ti=o,
&=o
*
(5)
It is worth noting that the two orthogonality conditions of eqs. (5) are equivalent to the analogous conditions of eqs. (4) written for all n elements because the relevant variables are all sign-constrained.
C. Polizzotto, C. Mazzarella and T. Panzera, Optimum design ftir work-hardenirzg adaptatiotl
132
2. Design variables A design is completely described by a number of nonnegative (geometrical or physical) parameters called design variables. These are collected in a design vector x > o. The design x = o is the weakest design, that is the design obtained using a material cover which is the lightest, taking into consideration given construction requirements. When there are no such requirements, the design x = o identifies with the uncovered layout. The cost g, the plastic resistances k and the self-weight p, are design-dependent variables. This dependence is generally nonlinear, but quite frequently it is supposed to be linear. Though this linear idealization causes of course some inaccuracy, solutions which are more easily obtained are however expected to give sufficient information about the nonlinear problem. As a result the following linear relations hold: g=gO
+dx,
(6)
k = k” + Vx , Pw =pw0
(7)
+wx,
03)
where the superscript ’ refers to the weakest (or initial) design. We observe: i) The cost gradient vector c has nonnegative components and depends on the geometric and physical properties of the elements as well as on the way the design variables are defined. ii) The matrix V, with nonnegative entries only, depends on the piecewise linearization of the element yield domains. It characterizes the way the element yield polyhedra expand when the design variables increase. Particular expansion modes are the free expansion mode (the faces of each polyhedron translate each other independently), and the uniform expamiorz mode (the yield domain remains geometrically similar to itself during the expansion). In any case the matrix V must be defined in such a way that the expanded polyhedra described by eq. (7) and the given matrix N could be considered as good piecewise linear approximations of the true yield domains of all the available elements. In the sequel the column vectors of V will be supposed to be unit vectors [71, [9], [ 101. iii) The self-weight matrix W depends on the given layout, as well as on some physical and geometrical properties of the elements [ 7 I , [ 9 I, [ 10 1.
3. The statical adaptation
criterion
In 141 two work-hardening adaptation criteria were given, one of statical type, the other of kinematical type. Both could be used as basic tools in the present optimization problem, but to achieve this goal the first seems to be more convenient than the second. The statical adaptation criterion named above is based on the notion of statically admissible yield surface. This is a subsequelzt yield surface (namely a surface obtained from the virgin one by changes which comply with the specific work-hardening law) such that for every loading condition there is an equilibrated stress distribution which is not in the exterior of the same yield surface.
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
133
Let the set of loading conditions be described by the homogeneous vector function
f(T)=P(T)+PF >
PF =pf +Pw
(9)
)
where r is a vector parameter belonging to an r-dimensional loading domain R. Now a statically admissible yield surface is defined taking a vector 1,> 0 (hardening intensity vector) such that the following conditions are met (see eqs. (3) and (5)): C%(t)-p(t)-pF=o,
N’s(r)-H)c-k
VTER.
(10)
The statical adaptation criterion says that is such a vector A exists, the structure possesses adaptation ability for all the loading conditions mp(r) +pF, whatever m is, but 0 < m < 1. It is easy to prove the following theorem on the statically admissible yield surfaces: THEOREM. Zf a yield surface is statically admissible with respect to a set of loading conditions, it is also statically admissible with respect to the loading conditions of the convex hull [ 111 of this set. Proof
Equilibrium and conformity (10) are by hypothesis and in particular in two certain points pi, f2 of it. Depending equations and on convexity of the yield domain, equilibrium be met also in the interior of the segment with end points rl {r=(l
-Q)ti
+(Yt2,
met in the set of loading conditions, on linearity of the equilibrium and conformity are easily shown to and f2, i.e. the segment
vol: o
to which the following loading set corresponds:
{f(T)=(l
-a)f(rl)+orf(t2),
Va:
o
Since every point of the convex hull belongs to at least one such segment, the theorem is proved. A general smooth loading domain is always supposed to be piecewise linearized. Then the convex hull of a (piecewise linear or linearized) loading domain R is an r-dimensional polyhedron of, say, p vertices. In the sequel we shall define the relevant loading conditions of a given loading domain R to be the smallest discrete set of loading conditions whose convex hull is also the convex hull of R. In other words, the relevant loading conditions are just the p vertices of the convex hull. In the frequent case of p- 1 alternative loading conditions, each consisting of loads which vary proportionally to a single nonnegative parameter, i;e. fi=ripi+pF,
O
1,
i= 1,2 ,... p-l,
there are p relevant loading conditions, that is the p- 1 points ri = 1, 7,. = 0, i # i (i, j = 1, 2, . . . p-l), as well as the origin r1 = T* = .. . = rP_ i = 0 . By virtue of the theorem above, to apply the statical adaptation criterion to a given loading domain R, it is sufficient to write the conditions (10) only for the p relevant loading conditions ofR.
134
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
4. The optimum design problem
Let the following loadings: fi=pi+pF,
ViEP,
(11)
withP= (1, 2, . .. p}, be the set of relevant loading conditions of a given (piecewise linearized) loading domain R. Taking into account all the hypotheses and positions of the previous paragraphs, eqs. (6)-Q) and (lo), (1 l), we write the following minimization problem: c*ai-
N’oi - Hk-
Min cP=c’x: i
wx-f;
I
X20,
=o, Vx -k”<
Vi E
o ,
120,
P
ViEP
,
(12)
I
wheref,? =pi+pr+pt, i E P, are the loadings which should be applied upon the weakest design, and the objective function Q differs from the cost g (6) by a constant term only. Following known classical methods [ 1 I 1, [ 121, the optimality conditions are now deduced, that is the necessary and sufficient conditions in order that the vectors x*, L* , tstY, Vi E P, and some other vectors zi,!, k;, Vi E P, satisfy the optimization problem (12). These conditions are written after some manipulations as follows (a being an arbitrary positive constant): C*a,! - Wx’-f:
=o,
ViEP
(equilibrium) ,
(13)
(conformity)
(14)
+fI = N’af I - Hn* - Vx* - k” , 6,: =Nif +T < 0 ) i;>o, Nif-Cti,?=o,
+Tif=O,
,
ViEP
ViEP
(compatibility)
,
(15)
JI* = V’i;; - W’&;; - (ye., &* = vx* (uniformity) +**x* = 0
$20,
x*20,
X* =Hi;
, h’ =H1*
xX0,
l.*>o,
if*‘&* =o
,
(16)
(adaptation) ,
(17)
(resultant mechanism) .
(18)
1
I
Eqs. (13)-( 18) describe how the optimum design - considered to be in the limit state where adaptation has occurred - behaves under the action of the p loading conditions. This behavior has the following characteristics: (a) Each of the p loading conditions produces in the structure a compatible strain field and an
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design jbr work-hardening adaptation
135
equilibrated
stress field which are conforming, that is they correspond to each other through the plasticity laws (13)-( 15). (b) The resultunt mechanism, that is the sum of the mechanisms respectively produced under the action of the p loading conditions, satisfies uniformity and adup tu tion requirements (16)-( 18). As it has been explained in [9], [ 101, eqs. (16) have been named “uniformity conditions” because they are substantially equivalent to the “uniform dissipation energy principle” of Drucker and Shield [ 131. Moreover, a physical analogy can be envisaged on the basis of the formal similarity between eq. (14) and eq. (16). By virtue of this analogy the optimum design x* can be viewed as the result of a growing process in which theplastic resistance vector Ak” = I/X* is linked to the resultant plastic strain rate intensity vector 5: through the uniformity conditions (16): just as plastic geformation is a result of the yielding process, in which the plastic strain rate vector Ef = NL; is linked to the stress vector Go!through the conformity conditions (14).
Fig. 2. Sketch of growing surfqce and hardening hypercone in two dimensions: Ak = Y,x,+ V,x, =V,x,;h=H,h,+H,h,=H,h,; x1 = 0,x, > 0; hI = 0, A2 > 0; rl,1 < 0, $2 = o;x1<0, x2 = 0.
The above analogy can also be described by using geometrical language [9], [ 101. In fact, in the i-space (see fig. 2) a (convex polyhedral) growing surface exists which has a,many faces as there are design variables and whose (external unit) normals are the column vectors of the matrix V. The distances from the origin of these faces in the case where the self-weight is ignored (i.e. if W = 0) are measured by the components of the vector (YC;in other words they are proportional to the components of the cost gradient vector c. In the more general case where the self-weight is present (i.e. if W # 0) these distances are measured by the vector CYC = 01c+ W'irg , and the modified distance aZj of the jth face is generally greater but even less than the simple distance aci. Then the presence of the self-weight produces translations of the faces of the growing surface, with consequent cost raising (or lowering) effects. Finally, we see that the vector Ak’ = VX* , which describes the plastic resistance increments of the optimum design, satisfies the following growing law: (i! it is displayed along the external generalized normal to_the growing surface at the point where AR touches this surfaces; (ii) it is zero in the case where A, does not touch the surface.
136
C. Polizzotto,
C. Mazzarella
and T. Panzeca,
Optimum
design for work-hardening
adaptation
The adaptation conditions (17) were discussed in 141. They give information about the yield domain of the optimum design in the limit state where work-hardening adaptation has been reached. In fact, they link the resultant mechanism &i, i;T given by eq. (18) with the ~z~orkhardening intensity vector I*, and therefore with the hardening displacement vector h* = H A*, whose components are the translations of the faces of the yield domain from the virgin state to the limit state above. These adaptation conditions can receive adequate description in the i-space, where a (polyhedral) adaptation hvpercone is defined. Its vertex is the origin, and the external normals of i!s faces are described by the column vectors of the hardening matrix H (see fig. 2). The vector Ai cannot be external to this cone: when it lies along a generatrix of the cone, the yector h* = HA* is perpendicular to the conical surface along the activated generatrix, but when LI; is internal to the cone, h” = o. The resultant mechanism, given by eq. (1 S), is a definition of the so-called Fotllkes mechanism
[91, [lOI. In concluding this paragraph, we observe that the primal problem given previously, can be reformulated in a dual form. Following known methods [ll], [12], wehave I
i Max q = (Y-’ c
(<,fio)‘zii - (k’)’
ii):
iEP
Of course, the optimum = c
((fo)‘;;
- Ctii = o ,
vi, - W’liR H&Go, i,
;
&X*
.
Nii
Vi E P
ii ) li,
iEP
\
(YCGO,
Vi E P
$20,
= c
values of the two objective
-
= c
1.
(19)
iii
iEP
functions
eq. (12).
coincide,
I
i.e.
- (P)’ i;).
iEP
5. Economical
value of the work-hardening.
Serviceability
Turning back to the design problem given by eq. (12), we see that in the case H = 0 it coincides with a standard plastic design problem, that is a design problem for plastic collapse. Calling @i the optimum value of the objective function in the case above, and @* that in the case H # 0, it is of course true that +i 2 cP*. In other words, the cost gi of the optimum design for plastic collapse is not less than the cost g* of the optimum design for work-hardening adaptation: g;+
g” .
(21)
Therefore, the work-hardening has an economical value since by taking it into consideration in the optimization process, one can save material. The amount of material saved, and hence the economical value of the work-hardening, depends on the work-hardening law. If the work-hardening is isotropic, its economical value is infinitely large because in this case [4] there is always adaptation capability, whatever the loading domain. The optimum design is the initial one, i.e. the design X* = o, and any load can be sustained without expense.
C. Polizzotto, C. Mazzarellaand T. Panzeca, Optimum design for work-hardeningadaptation
137
.
~,#0)[41,[7].Therefore,itis In the case of kinematic work-hardening, it is ti, =o(but seen that the terms depending on the self-weight and on the (fixed) load pf + p”, disappear from the dual formulation of the optimum design problem (19). As a result the optimum design is not influenced by the presence of any fixed load and can thus be sustained without expense. The results obtained above are drastically modified when limitations are imposed upon plastic deformations occurring in a real loading program. These limitations inevitably reduce the economical value of the work-hardening, and it would be interesting to explore this aspect of the problem. But it seems reasonable to expect a considerable saving of material even when serviceability requirements are imposed. Limitations on the changes of dimensions and size of the virgin yield domain, which are due to the work-hardening behavior, can easily be imposed by constraining the work-hardening displacement vector h = H1. Then a matrix condition of the form Hk
-Hl
(22)
d being a suitable limiting vector, should be added to the constraint set of the optimization problem of eq. (12). But, how must the vector d be selected in order to satisfy given serviceability require-
ments? To the authors’ knowledge this point is not clear and demands further investigation in the future.
6. Numerical applications
Two examples are presented for numerical applications: a plane truss and a plane frame, in the hypothesis of kinematical work-hardening. In this case we know [4], [7], [8], [ 141 that the work-hardening matrix Hh of an element is given by H” =
,*(Nh )‘Nh ,
(23)
where K* is a physical positive constant. Introducing the “initial” stress vector G, given by 6’ = [(ii’)‘, (ii*)‘, ... (P)‘]
)
ah = KhNhA* )
(24)
permits one to write the plastic potential vector + (5) in the form $=N’(a-if)-k. This form is computationally than that in k . Example
(25) convenient since the number of variables in the vector is is much less
1. Plane truss
Consider the plane truss subjected to the loading conditions described in tig. 3a. The loading domain R is the triangle of fig. 3b whose vertices O-l-2 are the three relevant loading conditions. The state of stress in a bar is described by a single stress component, the normal stress resultant
C. Polizzotto, C. Mazzarella and T. Panzeca, Optirnunl design for work-hardening adaptation
138
x
f, -
f,= Fr,
6
4
2
6f* f,. Fr,
B
10
-L-.-L a) Pig. 3. Truss layout
Nh ; the plasticity
subjected
conditions
to variable
-
D
7
loadings:
z1
b) a) geometrical
scheme
and loads; b) loading
domain
R
of the bar are
Given the material yield point uY and ignoring any stability requirement, the bar areas Ah are then proportional to the bar load capacities kh , which can be assumed to be design variables, i.e. xh = kh. Two types of optimum design are obtained, the Standard Limit Design (SLD) and the WorkHardening Adaptation Design (WHAD); each type is considered for the following cases: I. The lower bound of design variables is zero, i.e. xh > 0. II. The lower bound of design variables is different from zero, i.e. xh/F Z 1, where F is some force intensity. Table 1 contains optimum values of design variables and of (conventional) volume W = C;i=, Ihx”, and shows the percentage of material volume that can be saved if the work-hardening is taken into account. Table 2 contains the joint velocities of the optimum designs for work-hardening adaptation under the three relevant loading conditions O-l-2. We see that the resultant mechanism . . . . vanishes, as the adaptation conditions (17) demands in the present case of 'R = '(0) + ‘(1) + *(2) kinematical work-hardening [4], [ 71. 1. Optimum
Table Design case
Design
Optimum
,+l
Volume
bar areas Ahcry/F -
type 1
I. Xh> 0
bar areas and volume
2 ~_._~~.
3
SLD WHAD
3.000 3.000
2.829 1.415
2.000 1.500
SLD WHAD
3.000 3.000
2.829 1.000
2.000 1.500
4
1.000 1.000
5
6
7
5.000 3.000
1.415 0.708
3.000 1.500
5.293 3.000
1.000 1.000
2.708 1.500
8
9
10
1.415 0.708 1.000 1.000
1.415 1.122
W/FL
1.000 1.000
Material saving !C
18.66 11.83
36.6
21.24 15.14
28.8
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
139
Table 2. Collapse mechanisms for WHAD Relevant loading condition
Design case
I.xh do
Il.+
Joint horizontal velocities
Joint vertical velocities
A
B
C
D
A
B
C
D
0 1 2
0.500 -0.500
1.000 -1.000
-0.500 0.500
-0.500. -0.500 1.000
-0.708 1.208 -0.500
-1.415 2.708 -1.293
-0.708 1.208 -0.500
-0.915 2.708 -1.793
0 1 2
0.500 -0.500
1.000 -1.000
-0.500 0.500
-1.000 1.000
-0.708 0.500 0.208
-0.708 2.000 -1.293
-0.708 0.500 0.208
-0.708 2.000 -1.293
Fig. 4. Frame structure subjected to variable loadings: a) continuous model; b) discrete model; c) loading domain R.
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design for work-hardrhR
140
Example
adaptatiorl
2. Plane frame
Consider the plane frame of fig. 4a. It is subjected to the action of vertical and horizontal loads which are uniformly distributed along the beams. The vertical load has equal intensity p, for the two beams, while the horizontal load has intensities 1.Sp, and I-)~,respectively, for the upper and lower beam. The loading conditions are described by pi = 12.00 r,F/L
,
pz
=
kO.88 r,FIL
>
o+<
1,
-71 G r* < 71
(where F and L are force and length factors, respectively), and hence by the loading domain R of fig. 4c, whose vertices O-l-2 are the relevant loading conditions. Any fixed load (like self-weight) can be ignored, since it will not affect the optimum design. In the spirit of the present discrete approach the lumped-deformability model of fig. 4b is adopted with 30 critical sections and rigid elements of equal length L/6. The loads are described as concentrated forces, i.e. the forces P, vertically applied to the ends of all the rigid elements of the two beams, and the forces 1.5P,, and P, horizontally applied to the centroids of the above rigid elements of, respectively, the upper and the lower beam. These forces are then P, = 1.00OF~,
,
P, =+0.148
Fr2.
In the hth critical section (h = 1, 2, .. . 30) the stress state and the “initial” stress state are
where Mh and Nh are the bending moment and the axial force, respectively, while the strain rate state is described by the vector (ih)’ = {i”
Ch} )
Fig. 5. Idealized
(virgin)
yield polygon
for standard
I-section
working
in flexure
about
the strong
axis.
C. Polizzotto, C. Mazzarellaand T. Panzeca, Optimum design for work-hardeningadaptation
141
where ih and ih are the rates of relative rotation and relative axial displacement, respectively. If standard steel I-sections of a given series are to be used, we remember that for this type of cross-section an idealized yield domain as in fig. 5 can be assumed [ 71, [ 141, [ 151, where M, , N,, are simple yield limits, i.e. the yield moment and the yield axial force, respectively. With the hypothesis NY = bM, , b being a suitable constant, there will be only one design variable for each critical section, i.e. the variable M,, but some constraints can be introduced for practical reasons so that the total number of independent design variables can be even less than the number of critical sections. 0924
23
l--l 1 \
\
\
1123
\
\
\
\
n
My- diagrams
Standard Limit Design (SLD) I -------
Work-Hardening Adaptation Design (WHAD)
Fig. 6. Optimum design of a plane frame in the design case III (free sections): My-diagrams.
C. Polizzotto, C. Mazzarella and T. Panan;eca,Optimum design for work-hardening adaptation
142
The &vector kh of plastic resistances relative to the hth section is expressed in the form kh = VhA4t, and the 6 X I matrix Vk is the same for all critical sections, i.e. [7] ~ [ 141 T/= (1 + (0.85)2b2}-‘/2
(I’?’ = { 1 bq bq 1 bq bq} , The 6
X
1 matrix Nh of the external unit normals is [ 71, [ 141
-I
Nh_lGt-l-E 07-i
t rl
0
-?-/ -q
I t= . ’
0.85bq
The equilibrium equations and plasticity conditions have been written in a dimensionless form. For this purpose a suitable numerical value has been given to the dimensionless product bL, i.e. bL = 38.7626 [14]. Simpli~cations that result frotn the symmetry of the frame geometry and relevant loading conditions have been taken into account in order to determine the minumum (conventional) volume W in the following three design cases: III. All 30 critical sections are independent. IV. Columns and beams must be prismatic, but each structural element is independent of the other (~ec~~oZog~c~Zco~lstraints). V. Columns and beams must be prismatic with the beams equal to each other (technological co~~truints), and the columns must be of nondecreasing cross-sections from the upper to the lower ends of each vertical axis (co?z~tr~ct~o~zreq~~re~lent co~str~~izt~). Again, the standard limit design (SLD) and work-hardening adaptation design (WHAD) have been obtained. Optitnum values of yield moments and volume in the three cases above, and material saving as well, are reported in table 3 (see also fig. 6). For the sake of simplicity, no mention is made of the resultant mechanism for WHAD which once more complies with the adaptation conditions (17), that is it equals zero [41, f7 1. The amount of material saving shown in tables 1 and 3 will quite probably decrease when the proper standard I-section - which is selected from some commercial series - is assigned to every structural element. This penalty is connected with the use of continuous variable methods in place of discrete variable methods. Table 3. Optimum yield moments and volume Design case ----.--.-III. Free sections
Design type __-.
Volume
Optimum yietd momentsM#X Column 1-4
Column 16-19
Beam 9-15
Beam 24-30
W/FL2
Material saving %
13.23
SLD (See fig. 6) WHAD
IV. Technological constraints V. Technological and construction constraints
8.74
SLD
0.584
0.793
1.039
WHAD
0.272
0.667
0.667
0.667
13.63
SLD
0.695
0.667
0.964
0.964
19.40
WHAD
0.667
0.661
0.667
0.667
16.00
0.750
33.9
18.99 28.2
17.5
C. Polizzotto,
C. Mazzarella and T. Panzeca, Optimum design for work-hardening adaptation
143
7. Conclusions We have studied here a method of designing optimum structures with the criterion of the workhardening adaptation. This method applies to general discrete structures which have a fixed node layout and are made up of structural elements belonging to a continuous spectrum. Every element has a piecewise linearized rigid-plastic behavior, and its yield surface is a polyhedron whose faces can only translate as a result of the work-hardening. This latter can be of any type that is symmetric with respect to the yielding modes. The self-weight of the structure, which is design-dependent, is taken into account. Since the cost function as well as the self-weight are supposed to be linearly dependent on the design variables, optimization problems of linear programming are encountered. There are two dual formulations, both based on the statical theorem of the work-hardening adaptation theory. The optimality conditions are discussed using also proper geometrical descriptions. The present method coincides with the classical limit design method in the case of rigid-perfectly plastic behavior. It permits one to obtain optimum designs which are of less cost than standard ones, i.e. designs for plastic collapse, as is shown also by the numerical example given in the paper. The apparent economical value of work-hardening would be much reduced if proper limitations were imposed upon plastic deformations occurring during any loading program within the limits, but it is expected to remain large enough. Limitations on plastic deformations should be introduced into the framework of the optimization problem because only in such a way can the optimum design prove to be also serviceable, i.e. able to escape dangers of excessive (plastic) deformations. In the present work insufficient attention has been given to this fondamental problem of serviceability of optimum designs. A specific paper will be devoted to it in the near future. The results obtained in the present paper are to be considered as a first step towards desirable future developments.
Acknowledgement The present paper is part of a research project sponsored by the Consiglio Nazionale delle Ricerche, CNR, Italy. The authors gratefully appreciate the financial support of CNR.
References [l] W. Prager, Bauschinger adaptation of rigid, work-hardening trusses, Mech. Res. Comm. 1 (1974) 253-256. [2] W. Prager, Optimal plastic design of trusses for Bauschinger adaptation, in: Omaggio a Carlo Ferrari (Libreria Editrice Universitaria Levrotto & Bella, Torino, 1974). [3] W. Prager, Adaptation Bauschinger d’un solide plastique B ecrouissage cinematique, C.R. Acad. Sci. Paris 280 Ser. B (1975) 585-587. [4] C. Polizzotto, Work-hardening adaptation of rigid-plastic structures, Meccanica 10 (1975) 280-288. [5] W.T. Koiter, General theorems for elastic-plastic solids, in: I.N. Sneddon and R. Hill (Eds.), Progress in Solid Mechanics 1 (North-Holland, Amsterdam, 1964) ch. 4. [6] J.A. Kbnig and G. Maier, Adaptation of rigid-workhardening discrete structures subjected to load and temperature cycles and second-order geometric effects, Comp. Meth. Appl. Mech. Eng. 3 (1976) 37-50.
144
C. Polizzotto, C. Mazzarella and T. Panzeca, Optimum design jbr work-hardening adaptation
[ 71 C. Mazzarella and C. Polizzotto,
Optimization of rigid-workhardening structures (in Italian). Tech. Rep. SISTAR-75-OMS-2. Facolti di Architettura, Palermo (Apr. 1975) and Costruzioni Metalliche (in press). [8] G. Maier, A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes, Meccanica 5 (1970) 54-66. [9] C. Polizzotto, Optimum plastic design of structures under combined stresses, Int. J. Solids Structures 11 (1975) 539-553. [lo] C. Polizzotto, Optimum plastic design for multiple sets of loads, Meccanica 9 (1974) 206-213. [ 1 I] G.B. Dantzig, Linear programming and extensions (Princeton Univ. Press, Princeton, 1963). [ 121 O.L. Mangasarian, Nonlinear Programming (McGraw-Hill, New York, 1969). [ 131 D.C. Drucker and R.T. Shield, Design for minimum weight, Proc. Int. Congress Appl. Mech. 5 (1956) 212-222. [ 141 C.E. Massonnet and M.A. Save, Plastic Analysis and Design 1. Beams and Frames (Blaisdell, New York, 1965). [ 151 C. Polizzotto and C. Mazzarella, Limit design of frames with axial force-bending moment interaction, Tech. Report SISTAR-75-OMS-3 Facolta di Architettura. Palermo (Dec. 1975).