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Optimal support reaction in elastic frame structures
Compufers & Sfrucrur~s, Vol. 14, No. Z-44. pp. 179485, Printed in Great Britain.
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0045-7~9/81/0301~7eo71M.oo10 Pergamon Press Ltd.
OPTIMAL SUPPORT REACTION IN ELASTIC FRAME STRUCTURES Z. MR& and T. LEKSZYCKI Institute of Fundamental Technological Research, Warsaw, Poland (Received 17 September 1980; received for publication
11November 1980)
Abstract-The conditions for optimal support reaction are derived allowing for translation or rotation of supports and requiring the minimization of global elastic compliance or any other functional of stress or displacements. Two illustrative examples are discussed in detail where besides the optimal solutions, also the sensitivity of design with respect to varying support position and direction is studied.
1.INTRODUCTION
The present paper supplements the previous works[l, 21 on optimal support reaction on structures under static loading. Whereas in [l] the optimal position of supports transferring force or couple reactions on beam structures was considered, in [2] the optimality conditions were derived in terms of a continuum theory assuming any stress or strain functional as an objective function. In this work, we shall further discuss these conditions and provide two illustrative examples. The problem of structure prestressing through initial displacement of supports turns out to belong to the class of problems of optimal supports reactions. One of the most natural objective functionals is the global compliance C measured by the elastic energy stored in the structure. The optimal support reaction corresponding to stationarity of C can easily be established; however, the conditions for global minimum of C are not likely to be satisfied in most cases and therefore the stationarity solutions should further be verified. When instead of the global compliance, some other stress functional is used as an objective function or a local stress intensity is to be minimized through optimal support location, the respective optimality conditions are derived using the concept of an adjoint structure. These conditions again correspond to a stationarity point of the objective function and need not assure a global minimum in all cases. In Section 2 we shall derive these stationarity conditions and in Section 3 two particular examples of optimal support reaction will be presented.
For a linear elastic material both W(Q) and U(q) are quadratic functions of their arguments and -
g=$+Q,_
Q=%&fq,
_
afl
&CM-’
-
(2)
where & and &f are the compliance and stiffness matrices. Assume that an additional support is introduced in order to minimize the stress functional G=
I
@(Q)dx -
(3)
which in general does not need to coincide with the global compliance C=
I
W(Q)dx -
(4)
measured by the complementary energy of the structure. Here x denotes the length measured along the central axis of a beam or a frame. It will be shown that by selecting a proper form of G, one can approximate the problem of minimization of local stress intensity by the global measure (3). To derive the conditions of optimal support reaction, let us, similarly as in [2,3], introduce an adjoint structure of the same form, support conditions but with no loading occuring in the primary structure, Fig. 1. On the other hand, the adjoint structure is assumed to be submitted to an initial strain field qi defined by the potential law a@
t.CONDITIONSFOROPTIMALSUPPORTRRACTlON
Consider a plane frame structure of specified configuration, cross sectional dimensions, loading conditions and some of supports. Let the positions and directions of some additional supports be at the designer choice together with magnitudes of support reactions. We shall use the term optimal reaction when the additional supports are selected in order to minimize the global static compliance, maximum stress concentrations or local deflections and strains. Let the generalized stresses and strains be denoted by Q _ and q_ and let the specific stress and strain energies be
179
$=@.
This initial strain field induces the residual stress state Q’(x) and the associated strain g’(x) = CO’, so that fr”=$+g’,
Q”=Q’ _ _
(6)
where q” and g” denote the resulting strain and displacement fields of the adjoint structure. Our analysis is valid for a hear elastic material and the matrices L and &I do not depend on stresses or strains Q and q. Let the position, direction and magnitude of support reaction be varied. Assume that from the position A the support is translated through the distance 8s along the
180
Z. MR~Zand T. LEKSZYCKI
Fig. 1. Primary and adjoint frames with unspecified support at B. beam axis, rotated through the angle Sd in the structure plane and its magnitude be varied from R to R’ = R t SR. Let the stress, strain and displacement fields in the primary structure before and after variation be Q, q, u and Q’=QtQ, q’=qtSq, g’=utSF and tlie displacement- and -strain fields of the adJoint structure before variation be &” and (I~. By the principle of virtual work there is
I-
p.@dx-RuRa=
Q.q”dx I _ -
Q’ *q” dx. I _ _
(8)
(9)
is assumed to be continuous within the adjoint structure. For small variations SR, 8s and S$, the higher order terms in (8) can be neglected and from (8) and (7) we obtain q” ’ SQ - dx = - SRuR“- Ru ;;.aSs - Ru,“&#J ( 10) I _ where it was assumed that cos S+ = 1 and sin Sd = S$. The reaction R is assumed to expend a negative work and therefore we put the negative sign before the term RuRa in (7) and subsequent formulae. Consider now the variation of the functional (3). In view of (5) and (6), it can be written
= 1(4.-4r).60dx=lq”.SQdx.
Q’.Sqdx=O. _ I _
@(Q)= W(Q), qi=gL1=fl,
g’=O, Ic”=lr
(12)
The last inteeral of (12) vanishes since 0’ is a self-
(14)
(15)
and the adjoint structure coincides with the primary structure. In this case, the optimal support conditions (14) are identical to those derived by Mrdz and Rozvany[l]. Let us note that when the compliance C is to be minimized, the optimality conditions (14) with g” = I( apply both for linear and non-linear elastic structures. The concept of an adjoint structure can also be applied when instead of the stress functional (3), the displacement functional H(u) =
I
‘I’(u) dx
(16)
is to be minimized. For the adjoint structure of the same form and support conditions, let the loading be defined by the relation (17)
(11)
In transforming (ll), we used the reciprocity relation valid for lineary elastic structures and q’.SQdx= I _
and since 6R, Ss and Se5are independent variations, the stationarity condition 6G = 0 implies that
Thus the optima1 support conditions (14) are directly expressed in terms of the displacement field of the adjoint structure: for R# 0, the two displacement components uR’ and u,” should vanish at the point A of the adjoint structure together with the derivative u&. These stationarity conditions (14) were first derived by Mroz in [2]. The case when the functional G(Q) coincides with the mean compliance C(Q) can be treated within the present formulation. In fact, when Q,(Q)= W(Q), then the initial strain qi defined by (5) equals-the strain q defined by the stress potential (2), thus
The external loading is denoted by p and uRa. us“ are the displacement components along the force R and in the perpendicular direction. The derivative
uks =
(13)
uRa = Ru& = Ru,” = 0.
p . u” dx -(R t SR)cos SC#J(U~” t u$,,Ss)
- (R t SR) sin S@,P =
SG = - uR”SR- Ru $,, 6s - Ru,“&$
(7)
and
I-
equilibrated stress field and Sq follows from a kinematically admissible displacemenf field &. Let us note that there is no force R” applied to the adjoint system at A. Comparing (11) and (lo), we obtain
and let corresponding state be u“, q”, Q”. Considering the variation of (16) we have - SH= =
f_
f
$+dx= Q”.Sgdx=
ff
p”.Sudx g”.SQdx.
(18)
Optimal support reaction
In transforming (18), we applied the reciprocity relation Q” * 6q = q0 . SQ valid for a linear elastic material satisfying 72). Combining now (18) with (IO), the optimality conditions (14) are again obtained. The derived conditions (14), are the necessary conditions for stationarity of the functionals G(Q) or H(F). In the examples discussed in the next section it will be shown that these conditions correspond also to a global minimum. 3. EXAMPLES
In this section we shall determine the optimal support conditions for a beam and a ring that correspond to a minimum of the global objective functional G,(Q) = [I,’ @“(Q) dx]“’
Consider a beam shown in Fig. 2(a), loaded by two concentrated forces P and kP. The position s of the support should be determined so that the functional
[r M2p
G,,(M) =
dx]“’
that is the functional G,(Q) tends to a supremum value of the integrand within the interval of integration (0,l). The proof of this property can be found, for instance in [6] or [7,8] and the functional (19) (or the so called “p-norm”) has already been applied in [4] and [5] in relation to maximal stress and maximal deflection minimization in beams and plates. In using (19), we therefore need not to separate problems of minimization of maximal local values of stress from those of minimization of global functional forms, but performing calculations for increasing p, the local minimax problems can be solved through minimization of (19). For instance when Q(Q) is the specific energy function, then for p +m the functional (19) tends to a maximal value of the stress energy, max @(Q) = max W(Q). On the other hand, for p = 1, G,(Q) wbuld represenf the global compliance C. Example 1. A cantilever beam with one unspecified support (Fig. 2)
(21)
be minimized. Since SG,=~~Mzp-‘GMdx=~~‘GMdx P
(22)
the initial curvature of the adjoint beam is given by the formula where M: = G,.
(19)
where O(Q) is a homogeneous function of generalized stress. One of characteristic properties of this functional is that
I
181
in elastic frame structures
Here M(x), K(X) and w(x) are bending moment, curvature and deflection fields of the beam Fig. 2, whereas K~(x), K’(X), K’(X) and w’(x) are the respective quantities for the adjoint beam. Let us note that M, = G,, occuring in (23) is not a constant parameter but tt depends on value of p and the distribution of the bending moment. However, eqn (23) indicates us a different norm, namely
K,(M) =
l4 M,, (F)‘”
dx
(24
with M0 as a constant parameter. The initial curvature field corresponding to (24) equals p = 2p (!t)“-
(25)
and hence the ratio of initial curvatures defined by (23) and (25) equals ’
.(I
$=g
M(;)=0203Pa
M
Ki
and Y? as
zp-I !?!A p
\
Fig. 2. (a) Optimal support position in a cantilever beam for mean compliance, (b) deflection and moment distribution.
(26)
182
Z. Ma6zand T. LEKSZYCKI
and is constant for given p. Therefore the stationarity of the functional (24) implies the stationarity of (21) and either K~(x) or Xi(x) can be used as the initial curvature fields for any finite p. On the other hand, when p + m, the value of MO should be properly resealed, so that MO+ max M for p + cc.In a numerical procedure carried out for increasing values of p, it can be set, for instance, that Mop = max Mb where ML denotes the bending moment field calculated at the preceding step corresponding to
the adjoint beam at x = s is obtained in the form w’(s) = XM:P+‘t /3,(M:P - M:P)s -&M :ps t M:P+‘(y, - y2)t y,M?'+' t M:“a(/3z-/31)
(31)
tihere Ml = Ml(O), M2= Mz(a), M3= MS(s). Since the adjoint beam is statically determinate (not supported at C), there is w’ = w”, w’ = 0. When the rigid support position is sought, the optiP’
Fig. 3. Optimalrigid support position for stress intensity design (p = IO),(a) moment and deflection fields for the beam, (b) deflection field for the adjoint beam.
Optimal support reaction in elastic frame structures
IM
183
M(ab0.167 R
Fig. 4. Optimal support action for stress intensity design: (a) moment and deflection fields for the beam,(b) deflection field for the adjoint beam.
where N and M denote the axial force and the bending moment, and y is the distance in the radial direction measured from the central line. The equilibrium equations take the form
(a)
rN’tM’=O,
-M”fiN=p.
(33)
Consider the global stress intensity measure to be minimized 2*r
G, =
(b)
[I 0
(a:P t &‘) ds
I/P 1
(34)
where U, and u2 are the stresses in the internal and external fibers. This measure can easily be expressed in terms of M and N since
u,&,!?+it+!fyr A
Ar
J’rty
where A denotes the cross sectional area. The elastic
Fig. S(a) Variation of maximal moment with y = s/a, (b) dependence of y on p.
Example 2. Circular ring under concentrated force Consider a circular ring loaded by a concentrated force at A and rigidly supported at B and C, symmetrically with respect to the axis AO. Let the position of supports and the direction of reactions by the angles j3 and 4 be unspecified in advance and are to be determined. Denoting by u and u the radial and circumferential displacements of the ring axis, we have N=y
[u+ru’+-&(u+r’u”) 1
M=-F(u+?u”),
J’=r
I
-&dy
(32)
Fig. 6. Circular ring with two unspecified supports at B and C.
184
Z. MR~Zand T. LEKSZYCKI
(34), that is
GM = [[VrM2Pds]“p
or G,,“=ln’($_)Zs
ds. (37)
The corresponding initial curvatures are then (a)
(38) Equations (38) are used to determine the initial displacement field U’(S) of the adjoint ring. The residual displacement field u’(s), u’(s) is found by requiring the total displacements uO(s), us(s) to satisfy the conditions tP = du”/ds = 0 for y = /3. The optimal angle and position of support are then found from the conditions
-1
Up = E Ju, / P,”
&!!C=~u” =() ds
(b)
.-. -90”
-60’
-30’
0”
J 300
60”
9
’
where uR“ = ua cos (/3- 4) t u’ sin (/_I- 4), u,” = u“ sin &I - 4)- v“ cos (/3 - 4). The detailed analysis is not presented here and only some results are shown in Figs. 7(a-c) for ring of rectangular cross section and the thickness/diameter ratio equal 0.01. Figure 7(a) shows the variation of the maximal bending moment with angle 4 of support for p = 90”. The optimal angle $J,,,,p 54” and represents a non-analytical point of intersection of moment variation at A and C. Figure 7(b) shows the variation of non-dimensional deflection at A with angle 4 for different values of /3. The lowest value of & corresponds to optimal mean compliance design based on the elastic energy (36). Finally, Fig. 7(c) shows the variation of the optimal angle $J against p for /3 = 90”. It is seen that this angle varies more significantly between its value for p = 1 and p = m. These results were obtained for the global stress intensity measures (37). When the global measure (34) is used and axial forces are included, the numerical results do not differ significantly from those shown in Fig. 7.
4. CONCLUDING REMARKS
p=T-----p=2
p=3
p=4
p:5
Fig. 7(a) Maximal moment variation depending on 4, (b) variation of deflection at A with 4, (c) optimal angle of force action Q vs exponent p.
The present paper illustrates the solution for a class of problems of optimal support for structures when the mean compliance or any global stress measure is to be minimized. It is interesting to note that not only support position but also the magnitude of reaction and structure deflection are determined at the supporting point from the optimality conditions. More complex structures such as plates or shells can be treated in a similar way by introducing the concept of an adjoint structure. Acknowledgemenf-The present paper was supported through
NSF grant No. INT 07-08722made available from the Maria Sk1odowska-Curie Fund established by contributions from the United States and Polish Governments.
energy is expressed as follows G=[=‘[&-&r+~~+&]ds.
(36) REFERENCES
An alternative simple measure of stress intensity can be proposed by neglecting the effect of axial force in
I. Z. Mrdz and G. 1. N. Rozvany, Optimal design of structures with variable support conditions. J. Opt. Theory Appl. 15, 85-101 (1975).
Optimal support reaction in elastic frame structures 2. Z. Mroz, On optimal force action and reaction on structures. Proc. IUTAM Symp. “Control of Structures” (Edited by H. Leipholz). University of Waterloo (1979). 3. Z. Mroz and A. Mironov, Optimal design for global mechanical constraints. Arch. Med. Appl. in press. 4. M. B. Fuchs and M. A. Brull, Norm optimization method in structural design. AZAA.Z.16, (1978). 5. N. V. Banitchuk, V. M. Kartvelitvili, A. A. Mironov, Opti-
185
mization problems with local criteria in the plate theory. Mech. TV.Tela, No. 1, 1978,(in Russian). 6. J. L. W. Jensen, Sur les fonctions convexes et les inegalites entre les valeurs moyennes. Acta Mathematics 30 (1906). 7. C. D. Johnson, Optimal control with Chebyshev minimax performance index. 1. Basic Engng,Proc. ASME, 1967. 8. R. P. Fedorenko, Approximate Solutions of Optimal Confrol Problems. Nauka, Moscow (1978)(in Russian).
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0045-7~9/81/0301~7eo71M.oo10 Pergamon Press Ltd.
OPTIMAL SUPPORT REACTION IN ELASTIC FRAME STRUCTURES Z. MR& and T. LEKSZYCKI Institute of Fundamental Technological Research, Warsaw, Poland (Received 17 September 1980; received for publication
11November 1980)
Abstract-The conditions for optimal support reaction are derived allowing for translation or rotation of supports and requiring the minimization of global elastic compliance or any other functional of stress or displacements. Two illustrative examples are discussed in detail where besides the optimal solutions, also the sensitivity of design with respect to varying support position and direction is studied.
1.INTRODUCTION
The present paper supplements the previous works[l, 21 on optimal support reaction on structures under static loading. Whereas in [l] the optimal position of supports transferring force or couple reactions on beam structures was considered, in [2] the optimality conditions were derived in terms of a continuum theory assuming any stress or strain functional as an objective function. In this work, we shall further discuss these conditions and provide two illustrative examples. The problem of structure prestressing through initial displacement of supports turns out to belong to the class of problems of optimal supports reactions. One of the most natural objective functionals is the global compliance C measured by the elastic energy stored in the structure. The optimal support reaction corresponding to stationarity of C can easily be established; however, the conditions for global minimum of C are not likely to be satisfied in most cases and therefore the stationarity solutions should further be verified. When instead of the global compliance, some other stress functional is used as an objective function or a local stress intensity is to be minimized through optimal support location, the respective optimality conditions are derived using the concept of an adjoint structure. These conditions again correspond to a stationarity point of the objective function and need not assure a global minimum in all cases. In Section 2 we shall derive these stationarity conditions and in Section 3 two particular examples of optimal support reaction will be presented.
For a linear elastic material both W(Q) and U(q) are quadratic functions of their arguments and -
g=$+Q,_
Q=%&fq,
_
afl
&CM-’
-
(2)
where & and &f are the compliance and stiffness matrices. Assume that an additional support is introduced in order to minimize the stress functional G=
I
@(Q)dx -
(3)
which in general does not need to coincide with the global compliance C=
I
W(Q)dx -
(4)
measured by the complementary energy of the structure. Here x denotes the length measured along the central axis of a beam or a frame. It will be shown that by selecting a proper form of G, one can approximate the problem of minimization of local stress intensity by the global measure (3). To derive the conditions of optimal support reaction, let us, similarly as in [2,3], introduce an adjoint structure of the same form, support conditions but with no loading occuring in the primary structure, Fig. 1. On the other hand, the adjoint structure is assumed to be submitted to an initial strain field qi defined by the potential law a@
t.CONDITIONSFOROPTIMALSUPPORTRRACTlON
Consider a plane frame structure of specified configuration, cross sectional dimensions, loading conditions and some of supports. Let the positions and directions of some additional supports be at the designer choice together with magnitudes of support reactions. We shall use the term optimal reaction when the additional supports are selected in order to minimize the global static compliance, maximum stress concentrations or local deflections and strains. Let the generalized stresses and strains be denoted by Q _ and q_ and let the specific stress and strain energies be
179
$=@.
This initial strain field induces the residual stress state Q’(x) and the associated strain g’(x) = CO’, so that fr”=$+g’,
Q”=Q’ _ _
(6)
where q” and g” denote the resulting strain and displacement fields of the adjoint structure. Our analysis is valid for a hear elastic material and the matrices L and &I do not depend on stresses or strains Q and q. Let the position, direction and magnitude of support reaction be varied. Assume that from the position A the support is translated through the distance 8s along the
180
Z. MR~Zand T. LEKSZYCKI
Fig. 1. Primary and adjoint frames with unspecified support at B. beam axis, rotated through the angle Sd in the structure plane and its magnitude be varied from R to R’ = R t SR. Let the stress, strain and displacement fields in the primary structure before and after variation be Q, q, u and Q’=QtQ, q’=qtSq, g’=utSF and tlie displacement- and -strain fields of the adJoint structure before variation be &” and (I~. By the principle of virtual work there is
I-
p.@dx-RuRa=
Q.q”dx I _ -
Q’ *q” dx. I _ _
(8)
(9)
is assumed to be continuous within the adjoint structure. For small variations SR, 8s and S$, the higher order terms in (8) can be neglected and from (8) and (7) we obtain q” ’ SQ - dx = - SRuR“- Ru ;;.aSs - Ru,“&#J ( 10) I _ where it was assumed that cos S+ = 1 and sin Sd = S$. The reaction R is assumed to expend a negative work and therefore we put the negative sign before the term RuRa in (7) and subsequent formulae. Consider now the variation of the functional (3). In view of (5) and (6), it can be written
= 1(4.-4r).60dx=lq”.SQdx.
Q’.Sqdx=O. _ I _
@(Q)= W(Q), qi=gL1=fl,
g’=O, Ic”=lr
(12)
The last inteeral of (12) vanishes since 0’ is a self-
(14)
(15)
and the adjoint structure coincides with the primary structure. In this case, the optimal support conditions (14) are identical to those derived by Mrdz and Rozvany[l]. Let us note that when the compliance C is to be minimized, the optimality conditions (14) with g” = I( apply both for linear and non-linear elastic structures. The concept of an adjoint structure can also be applied when instead of the stress functional (3), the displacement functional H(u) =
I
‘I’(u) dx
(16)
is to be minimized. For the adjoint structure of the same form and support conditions, let the loading be defined by the relation (17)
(11)
In transforming (ll), we used the reciprocity relation valid for lineary elastic structures and q’.SQdx= I _
and since 6R, Ss and Se5are independent variations, the stationarity condition 6G = 0 implies that
Thus the optima1 support conditions (14) are directly expressed in terms of the displacement field of the adjoint structure: for R# 0, the two displacement components uR’ and u,” should vanish at the point A of the adjoint structure together with the derivative u&. These stationarity conditions (14) were first derived by Mroz in [2]. The case when the functional G(Q) coincides with the mean compliance C(Q) can be treated within the present formulation. In fact, when Q,(Q)= W(Q), then the initial strain qi defined by (5) equals-the strain q defined by the stress potential (2), thus
The external loading is denoted by p and uRa. us“ are the displacement components along the force R and in the perpendicular direction. The derivative
uks =
(13)
uRa = Ru& = Ru,” = 0.
p . u” dx -(R t SR)cos SC#J(U~” t u$,,Ss)
- (R t SR) sin S@,P =
SG = - uR”SR- Ru $,, 6s - Ru,“&$
(7)
and
I-
equilibrated stress field and Sq follows from a kinematically admissible displacemenf field &. Let us note that there is no force R” applied to the adjoint system at A. Comparing (11) and (lo), we obtain
and let corresponding state be u“, q”, Q”. Considering the variation of (16) we have - SH= =
f_
f
$+dx= Q”.Sgdx=
ff
p”.Sudx g”.SQdx.
(18)
Optimal support reaction
In transforming (18), we applied the reciprocity relation Q” * 6q = q0 . SQ valid for a linear elastic material satisfying 72). Combining now (18) with (IO), the optimality conditions (14) are again obtained. The derived conditions (14), are the necessary conditions for stationarity of the functionals G(Q) or H(F). In the examples discussed in the next section it will be shown that these conditions correspond also to a global minimum. 3. EXAMPLES
In this section we shall determine the optimal support conditions for a beam and a ring that correspond to a minimum of the global objective functional G,(Q) = [I,’ @“(Q) dx]“’
Consider a beam shown in Fig. 2(a), loaded by two concentrated forces P and kP. The position s of the support should be determined so that the functional
[r M2p
G,,(M) =
dx]“’
that is the functional G,(Q) tends to a supremum value of the integrand within the interval of integration (0,l). The proof of this property can be found, for instance in [6] or [7,8] and the functional (19) (or the so called “p-norm”) has already been applied in [4] and [5] in relation to maximal stress and maximal deflection minimization in beams and plates. In using (19), we therefore need not to separate problems of minimization of maximal local values of stress from those of minimization of global functional forms, but performing calculations for increasing p, the local minimax problems can be solved through minimization of (19). For instance when Q(Q) is the specific energy function, then for p +m the functional (19) tends to a maximal value of the stress energy, max @(Q) = max W(Q). On the other hand, for p = 1, G,(Q) wbuld represenf the global compliance C. Example 1. A cantilever beam with one unspecified support (Fig. 2)
(21)
be minimized. Since SG,=~~Mzp-‘GMdx=~~‘GMdx P
(22)
the initial curvature of the adjoint beam is given by the formula where M: = G,.
(19)
where O(Q) is a homogeneous function of generalized stress. One of characteristic properties of this functional is that
I
181
in elastic frame structures
Here M(x), K(X) and w(x) are bending moment, curvature and deflection fields of the beam Fig. 2, whereas K~(x), K’(X), K’(X) and w’(x) are the respective quantities for the adjoint beam. Let us note that M, = G,, occuring in (23) is not a constant parameter but tt depends on value of p and the distribution of the bending moment. However, eqn (23) indicates us a different norm, namely
K,(M) =
l4 M,, (F)‘”
dx
(24
with M0 as a constant parameter. The initial curvature field corresponding to (24) equals p = 2p (!t)“-
(25)
and hence the ratio of initial curvatures defined by (23) and (25) equals ’
.(I
$=g
M(;)=0203Pa
M
Ki
and Y? as
zp-I !?!A p
\
Fig. 2. (a) Optimal support position in a cantilever beam for mean compliance, (b) deflection and moment distribution.
(26)
182
Z. Ma6zand T. LEKSZYCKI
and is constant for given p. Therefore the stationarity of the functional (24) implies the stationarity of (21) and either K~(x) or Xi(x) can be used as the initial curvature fields for any finite p. On the other hand, when p + m, the value of MO should be properly resealed, so that MO+ max M for p + cc.In a numerical procedure carried out for increasing values of p, it can be set, for instance, that Mop = max Mb where ML denotes the bending moment field calculated at the preceding step corresponding to
the adjoint beam at x = s is obtained in the form w’(s) = XM:P+‘t /3,(M:P - M:P)s -&M :ps t M:P+‘(y, - y2)t y,M?'+' t M:“a(/3z-/31)
(31)
tihere Ml = Ml(O), M2= Mz(a), M3= MS(s). Since the adjoint beam is statically determinate (not supported at C), there is w’ = w”, w’ = 0. When the rigid support position is sought, the optiP’
Fig. 3. Optimalrigid support position for stress intensity design (p = IO),(a) moment and deflection fields for the beam, (b) deflection field for the adjoint beam.
Optimal support reaction in elastic frame structures
IM
183
M(ab0.167 R
Fig. 4. Optimal support action for stress intensity design: (a) moment and deflection fields for the beam,(b) deflection field for the adjoint beam.
where N and M denote the axial force and the bending moment, and y is the distance in the radial direction measured from the central line. The equilibrium equations take the form
(a)
rN’tM’=O,
-M”fiN=p.
(33)
Consider the global stress intensity measure to be minimized 2*r
G, =
(b)
[I 0
(a:P t &‘) ds
I/P 1
(34)
where U, and u2 are the stresses in the internal and external fibers. This measure can easily be expressed in terms of M and N since
u,&,!?+it+!fyr A
Ar
J’rty
where A denotes the cross sectional area. The elastic
Fig. S(a) Variation of maximal moment with y = s/a, (b) dependence of y on p.
Example 2. Circular ring under concentrated force Consider a circular ring loaded by a concentrated force at A and rigidly supported at B and C, symmetrically with respect to the axis AO. Let the position of supports and the direction of reactions by the angles j3 and 4 be unspecified in advance and are to be determined. Denoting by u and u the radial and circumferential displacements of the ring axis, we have N=y
[u+ru’+-&(u+r’u”) 1
M=-F(u+?u”),
J’=r
I
-&dy
(32)
Fig. 6. Circular ring with two unspecified supports at B and C.
184
Z. MR~Zand T. LEKSZYCKI
(34), that is
GM = [[VrM2Pds]“p
or G,,“=ln’($_)Zs
ds. (37)
The corresponding initial curvatures are then (a)
(38) Equations (38) are used to determine the initial displacement field U’(S) of the adjoint ring. The residual displacement field u’(s), u’(s) is found by requiring the total displacements uO(s), us(s) to satisfy the conditions tP = du”/ds = 0 for y = /3. The optimal angle and position of support are then found from the conditions
-1
Up = E Ju, / P,”
&!!C=~u” =() ds
(b)
.-. -90”
-60’
-30’
0”
J 300
60”
9
’
where uR“ = ua cos (/3- 4) t u’ sin (/_I- 4), u,” = u“ sin &I - 4)- v“ cos (/3 - 4). The detailed analysis is not presented here and only some results are shown in Figs. 7(a-c) for ring of rectangular cross section and the thickness/diameter ratio equal 0.01. Figure 7(a) shows the variation of the maximal bending moment with angle 4 of support for p = 90”. The optimal angle $J,,,,p 54” and represents a non-analytical point of intersection of moment variation at A and C. Figure 7(b) shows the variation of non-dimensional deflection at A with angle 4 for different values of /3. The lowest value of & corresponds to optimal mean compliance design based on the elastic energy (36). Finally, Fig. 7(c) shows the variation of the optimal angle $J against p for /3 = 90”. It is seen that this angle varies more significantly between its value for p = 1 and p = m. These results were obtained for the global stress intensity measures (37). When the global measure (34) is used and axial forces are included, the numerical results do not differ significantly from those shown in Fig. 7.
4. CONCLUDING REMARKS
p=T-----p=2
p=3
p=4
p:5
Fig. 7(a) Maximal moment variation depending on 4, (b) variation of deflection at A with 4, (c) optimal angle of force action Q vs exponent p.
The present paper illustrates the solution for a class of problems of optimal support for structures when the mean compliance or any global stress measure is to be minimized. It is interesting to note that not only support position but also the magnitude of reaction and structure deflection are determined at the supporting point from the optimality conditions. More complex structures such as plates or shells can be treated in a similar way by introducing the concept of an adjoint structure. Acknowledgemenf-The present paper was supported through
NSF grant No. INT 07-08722made available from the Maria Sk1odowska-Curie Fund established by contributions from the United States and Polish Governments.
energy is expressed as follows G=[=‘[&-&r+~~+&]ds.
(36) REFERENCES
An alternative simple measure of stress intensity can be proposed by neglecting the effect of axial force in
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