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Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints
Computers and Structures 75 (2000) 167±185
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Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints O. Sergeyev 1, Z. MroÂz* Institute of Fundamental Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Poland Received 29 April 1998; accepted 3 March 1999
Abstract The present paper deals with the problem of determining the optimal joint positions and cross-sectional parameters of linearly elastic space frames with imposed stress and free frequency constraints. The frame is assumed to be acted on by dierent load systems, including temperature and self-weight loads. The stress state analysis includes tension, bending, shear, and torsion of beam elements. By a sequence of quadratic programming problems, the optimal design is attained. The sensitivity analysis of distinct as well as multiple frequencies is performed through analytic dierentiation with respect to design parameters. Illustrative examples of optimal design of simple and medium complexity frames are presented, and the particular case of bimodal optimal solution is considered in detail. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction In this paper, we shall discuss the problem of sensitivity analysis and optimal design of frame structures for which both cross-sectional and con®guration design parameters are to be determined from the solution. We shall impose the stress and free frequency constraints on the optimal design. The ®rst constraint provides proper stress levels under speci®ed loads, the other constraint is aimed to assume proper structure response under dynamic excitation for which the resonant conditions are avoided and proper frequency spectrum is obtained. The optimal design problems with free frequency constraints were treated in numerous papers (cf. [5,9,10] and references cited therein).
* Corresponding author. 1 Visiting Research Fellow from the Department of Theoretical Mechanics, Technical University of Nizhni Novgorod, Russia.
Similarly, the sensitivity analysis for single and multiple eigenfrequencies was considered by numerous authors (cf. Wittrick [22], Masur and Mroz [11,12], Haug and Rousselet [4], Pedersen [16, 17], Mills-Curran [14], Mc Gee and Phan [13], Olho et al. [15], Krog and Olho [8]). The aim of this paper is to extend the previous analyses and consider variation of both cross-sectional and con®gurational parameters. It turns out that structures are much more sensitive to con®guration changes, so search for optimal structure joint positions provides much more ecient designs. The use of stress and frequency constraints assures practical designs for which both static and dynamic responses are controlled. One of the characteristic features of free frequency constrained designs is occurrence of multiple or nearly equal eigenvalues. Such coincidence of free frequencies is associated with structural symmetry or is induced by the evolution of eigenvalue spectrum due to redesign process toward an optimum with constraint set on the fundamental frequency. It is well known that multiple
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 8 8 - 7
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frequencies are not dierentiable in the common sense (that is are not Frechet dierentiable) and only directional sensitivity derivatives can be calculated. This fact creates some diculties in ®nding sensitivities of multiple frequencies with respect to design parameters and in applying the eective gradient optimization techniques. The present paper is devoted to development of an ecient method of sensitivity analysis and optimization of space frame structures with single and multiple eigenfrequencies. The sequential quadratic programming scheme is used with direct application of sensitivity derivatives. It was found that the applied method is reliable and accurate. The speci®c examples are concerned with a two-beam space frame, frame dome with 52 beam elements, and ®nally with a frame of mobile crane.
2. Sensitivity analysis for single and multiple eigenfrequencies Consider a linear elastic discretized frame structure for which the static equilibrium equation takes the form S fD g fA1 g fA2 g fA3 g 1 where [S ] is the global stiness matrix, {D } is the vector of joint displacements, and fA1 g is the vector of external joint loads, fA2 g is the vector of distributed equivalent loads, fA3 g is the vector of temperature equivalent loads. The state of free vibrations is governed by the symmetric eigenvalue problem ÿ S ÿ lj M Fj 0, j 1, . . . ,n 2
matrices) we have S
X C4e T Se C4e , e
M
X C4e T Me C4e
5
e
The transformation matrix C4e is a block-diagonal matrix composed of four orthogonal transformation matrices Ce ,Ce ÿ1 Ce T relating the beam element to the global Cartesian reference frame. Considering two basic types of constraints, namely frequency and stress constraints, further, selecting the set of design parameters contained in the design vector fX g, the optimization problem can be stated as follows. It is required to determine from a given range fX gmin ,fX gmax the joint positions (shape parameters) as well as cross-sectional dimensions (size parameters) fX g for which the frame mass attains its minimum subject to stress, frequency and side constraints, so that M fX g min M fX g fX g2F
6
where the feasible domain is speci®ed by the inequalities F fX g:seff fX gRsaeff , oj fX groaj ,
j 1,2, . . . ,k,
fX gmin RfX gRfX gmax
7
The eigenvectors fFj g are all [S ]-orthogonal and [M ]normalized, so that fFi gT S Fj lj dij ,
Situations where several frequencies coalesce and become a multiple frequency may naturally occur. They constitute one of the main diculties for gradient optimization methods as multiple frequencies are only directionally dierentiable. In order to generate correct optimization results, we have to avoid non-dierentiability of multiple frequencies in the active set of constraints. In the following sections, we summarize results of design sensitivity analysis of distinct and multiple frequencies, carry through an analytical sensitivity analysis of extremum eective stresses and next the gradient optimization method will be used to solve the optimal design problems with stress and frequency constraints.
fFi gT M Fj dij ,
2.1. Design sensitivity analysis of distinct frequencies
where [M ] is the global mass matrix, lj o2j is the jth squared angular vibration frequency, fFj g is the associated eigenmode and n denotes the number of degrees of freedom. The explicit forms of [S ] and [M ] are provided in the Appendix A. The eigenvalues are all real and can be ordered in the following manner: 0 < l1 Rl2 R Rlj R Rln
i,j 1, . . . ,n
3
4
The system stiness and mass matrices are accumulated from beam element stiness Se and mass matrices Me , i.e., symbolically (omitting Boolean
Let X be one of scalar design parameters. Assuming that lj are simple, the frequency sensitivities are obtained from Eq. (2) by direct dierentiation
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
@ S @ lj @ M ÿ M Fj ÿ lj Fj ÿ F j S @X @X @X @ Fj ÿ lj M @X
the analysis from Refs. [14,21,22], we introduce linear ~ j g associated with the combinations of eigenvectors fF multiple frequency l~
0
8
By premultiplying Eq. (8) by fFj g and making use of Eq. (2), the following expression is obtained for the frequency sensitivity in the case of simple frequencies lj @ lj T @ S @ M ÿ lj Fj 9 Fj @X @X @X The derivatives of matrices [S ] and [M ] are calculated analytically at the element level (cf. [20] and see Appendix A), thus X @ C4e T @ S @ C4e Se C4e C4e T Se @X @X @X e T @ Se C4e , C4e @X X @ C4e T
@ M @X
e
@X
T
10
If all design variables Xj are changed simultaneously then we can ®nd the linear increment of the distinct frequency lj in the form Dlj
X @ lj i
@ Xi
DXi rT lj fDX g
j 1,2, . . . ,N
13
k1
where bjk are unknown coecients to be speci®ed. Consider a variation of the design vector fX g in the form fX g efeg, where the unit vector feg represents a direction in the design space along which the design parameters are changed and denotes the magnitude of perturbation in this direction. For the increment of a single parameter X, the matrices [S ] and [M ] undergo variation and we have
@ S S X eex S eex , @X
@ M M X eex M eex @X
14
lj X eex l~ eex
@ l~ , @X
n o ~j n o @ F ~ j eex Fj X eex F , @X
15
j 1,2, . . . ,N Substituting Eqs. (15) and (14) into Eq. (2), we obtain within linear approximation
Let us consider the case of N-multiple frequency j 1,2, . . . ,N
o X N ~j F bjk fFk g,
11
2.2. Design sensitivity analysis for multiple frequencies
l~ lj ,
n
Along the direction of design variation, the following linear approximations of frequencies and eigenvectors are obtained
Me C4e
@ C4e @ Me C4e C4e Me C4e T @X @X
169
12
where l~ is the common value of repeated eigenfrequencies. The computation of sensitivities of this frequency is not straightforward. This is due to the fact that the eigenvectors fFj g,j 1,2, . . . ,N of the repeated eigenfrequencies are not unique. Any linear combination of eigenvectors will satisfy the eigenvalue problem (2). It should be noted that multiple frequencies are not differentiable in the common sense (i.e., the Frechet derivative does not exist), cf. [3,11]. Thus, to ®nd the sensitivities of multiple frequencies, we have to use directional derivatives in the design space. Following
n o n o ~j ÿ @ F @S @M ~ j ex S ÿ l~ M ex ÿ l~ F @X @X @X n o @ l~ ~j 0 ex M F ÿ @X
16
Premultiplying this equation by fFj gT one obtains ~ T @ S ~ @ M ~ s ex ÿ @ l ex Fj T M ÿl F Fj @X @X @X ~ s 0, j,s 1,2, . . . ,N F
17
In view of representation (13), we obtain from Eq. (17) the system of linear algebraic equations for unknown coecients bsk
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170
" N X T @ S ~ @ M fFk gex ÿl bsk Fj @X @X k1 @ l~ ex djk ÿ @X
#
s T N Ae 18
j,k 1,2, . . . ,N
Introducing the vector ffjk g in the form T T @ S ~ @ M i fFk g fjk Fj i ÿ lh h @ fX g @ fX g
s T N qe
20
None of the sensitivities mj can be set in a one-toone correspondence with any particular eigenvector fFj g: In many cases, it is expedient to eliminate the unit vector feg from Eq. (21) and generate the explicit formula for determining the increments Dlj , j ~ As there is 1, . . . ,N of the N multiple frequency l: efeg fDX g and em Dl, and we obtain h i T det fjk fDX g ÿ Dldjk 0, j,k 1, . . . ,N 22
) y1 x 1 , ÿ Iz1 Le
ÿLe x 1 1 ÿy1 L2e x 21 x 1 ÿ ÿ , 2 ae Le 2Iz1 L2e Le
! !) 1 ÿz1 L2e x 21 x 1 1 ÿ , 2Iy1 L2e Le 6 6
sYe
YE ÿ YB ÿaE x 1 YB ÿ Y0 Le
24
3. Sensitivity of the maximal eective stress Every beam is acted upon by the joint load vector fAe g uniform distributed load fqe g and temperature in a beam span. Based on the superposition principle, the axial normal stress can be written in the form, cf. [6] T ÿ s N sAe x 1 ,y1 ,z1 ,fX g fAe g
! 25
where x 1 0Rx 1 RLe is the local longitudinal coordinate and y1 ,z1 are the lateral coordinates, Le is the length of a beam, ae is the cross-sectional area, Iy1 ,Iz1 are the cross-sectional moments of inertia, a is the thermal expansion coecient, E denotes the Young's modulus, Y0 is the reference temperature and the linear temperature distribution is assumed between by the joint temperatures YB and YE : Note that warping stresses are neglected. The shear stresses are determined by
0 t
Mx 1 Qy B 1B S ÿ 2Ot tIz1 @ z1 0
Qz 1 B BS ÿ tIy1 @ y1
1 1 sz1 ds C C t ds A t 1
1 sy1 ds C C t ds A t
26
where s is the arc length of a middle section line, S y1 ,S z1 are static moments of the section part between the origin and the current point, O is the area bounded by the contour middle line and t is the current thickness of the section wall. The main assumptions behind Eq. (26) are closed thin walled cross-sections, and for details we refer to Ref. [7]. The shear stress can be presented in the matrix notation T ÿ t N tAe x 1 ,y1 ,z1 ,fX g fAe g
n ÿ oT N sqe x 1 ,y1 ,z1 ,fX g qe fX g
with the vector de®nitions
(
y1 x1 1ÿ , Iz 1 Le
and with temperature stress
where h i denotes a matrix package, the fundamental equation for determining the directional derivatives m of the multiple frequency takes the form h i T det fjk feg ÿ mdjk 0, j,k 1, . . . ,N 21
sYe x 1 ,fX g
ÿ1 ÿz1 x1 ,0,0,0, 1ÿ , ae Iy1 Le z1 x 1 , Iy1 Le
0,0,0,0,
0
A non-trivial solution to these equations exists only if the determinant of the system is equal to zero, thus " # T @ S ~ @ M @ l~ fFk gex ÿ det Fj ex djk ÿl @X @X @X 19 0,
(
n ÿ oT N tqe x 1 ,y1 ,z1 ,fX g fqe g 23
27
Assuming that shear stresses are generated only from shear forces and the torsional moment we get
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
t T N Ae 0, Fy1 y1 ,z1 , Fz1 y1 ,z1 , Fx 1 y1 ,z1 , 0, 0, 0, 0, 0, 0, 0, 0 , t T x1 1 x1 1 N qe 0, Le Fy1 , Le Fz1 ÿ ÿ 2 2 Le Le
28
where the functions Fy1 ,Fz1 of shear forces and Fx 1 of torsional moment are dependent on the cross-sectional shape. The eective stress seff at a given point of a beam and corresponding to a given load case is de®ned by p ÿ seff x 1 ,y1 ,z1 ,load case,fX g s2 Gt2 29 where G 3 for the von Mises hypothesis and G 4 for the Tresca hypothesis. Its maximum value equals ÿ ÿ seff x 1 ,y1 ,z1 ,fX g max seff x 1 ,y1 ,z1 ,fX g Volume
30
From Eq. (29) by the implicit dierentiation we get @ seff 1 @s @t Gt 31 s seff @X @X @X We need the sensitivities at a beam point where seff reaches its maximum value. Now taking the derivative of Eq. (30) and assuming that coordinates of the extremum point depend on the design parameter X, we get ÿ d max seff x 1 ,y1 ,z1 ,fX g volume
dX
@ seff @ x 1
@ seff @ y1
@ seff @ z1
@x1 @X @ y1 @ X @ z1 @ X @ seff @ X x ,y ,z ,fX g 1
1
that T @ N sAe T @ fAe g @s fAe g N sAe @X @X @X s T @ N qe s T @ fqe g @ sYe fqe g N qe @X @X @X
32
1
Let us demonstrate how to obtain particular derivatives occurring in Eq. (32). If the point x 1 ,y1 ,z1 lies inside beam segment then from stationarity of seff it follows that ®rst three terms of Eq. (32) are equal to zero. In the case of x 1 ,y1 ,z1 being at a beam boundary we have to study the boundary behaviour with respect to the design parameter X. The derivatives @ seff @ seff @ seff @ x 1 , @ y1 , @ z1 in Eq. (32) follow from the de®nitions (24) of fN sAe g,fN sqe g and from Eqs. (28) and (25) specifying fN tAe g,fN tqe g,sYe : The next step is to calculate the last term @@sXeff of the desired derivative (32). From the de®nitions (Eqs. (23) and (27)) of s and t it follows
33
T @ N tAe T @ fAe g @t fAe g N tAe @X @X @X t T @ N qe t T @ fqe g 34 fqe g N qe @X @X @ Ns @ Nt @ Ns @ Nt The terms f@ XAe g , f@ Xqe g , f@ XAe g , f@ Xqe g and @@sXYe can be obtained from Eqs. (24), (28) and (25). @ q The term @fXg non-vanishing only in the case of iner@ fqe g q tia loads is @ X faee g @@ aXe : The term @ @fAXe g follows from the relation fAe g Se C4e fDe g
4. Sensitivity analysis for eective stresses
171
35
where Se is the beam element stiness matrix speci®ed in a local coordinate system, fDe g is the joint displacement vector of the beam element. Now taking the derivative of Eq. (35), we obtain @ fAe g @ Se @ C4e C4e fDe g Se fDe g Se @X @X @X C4e
@ fDe g @X
36
By the direct dierentiation we obtain from the equilibrium equation (1) @ fD g @ fA2 g @ fA3 g @ S S f g D 37 ÿ @X @X @X @X 2g and thus with known sensitivities for loads @ fA @ X and @ fA3 g @ S as well as sensitivities for the stiness matrix @X @X , we can by a simple forward and backward substitutions determine @@fDX g : Performing calculations with derived formulas of this section, one can obtain the derivative of extremum eective stress (32) for every beam.
5. Optimization procedure with single and multiple frequency constraints The single frequency sensitivity is speci®ed by Eq. (9); however, for the state of multiple frequency, the sensitivities can be determined by solving the non-linear equation (22). In order to simplify the analysis, the design evolution is assumed to follow the path along which the o-diagonal terms of Eq. (22) vanish, so
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172
that T fjk fDX g 0,
j 6 k,j,k 1,2, . . . ,N
38
Young's modulus of elasticity, E = 200 GPa Shear modulus of elasticity, G = 80 GPa Allowable eective stress, saeff 147 MPa Mass density, r 7799 kg/m3
Then, each of increments Dlj in Eq. (22) is directly related to a particular eigenvector fFj g, and is explicitly expressed, namely
Temperature stresses are neglected. PC-486 computer was used and CPU-time is given in seconds.
T Dlj fjj fDX g
6.1. Example 1: two-beam frame
39
Now there is no need to apply the directional derivatives for multiple frequencies, as there simply is rlj ffjj g: As optimal design (Eqs. (6) and (7)) is a highly nonlinear problem, we change the problem by a sequence of optimal redesigns Xp1 Xp DXp 40 where fXp g is a given design after p redesigns. Omitting the index p, we can now formulate the optimization problem as follows 1 T min DM rMT fDX g fDX g H fDX g 2
41
where [H ] is the Hessian matrix and it is determined in a way discussed in detail by Reklaitis et al. [19]. The constraint inequalities are approximated by linear terms seff rT seff fDX gRsaeff , lj rT lj fDX grlaj T fjk fDX g 0,
j 6 k,j,k 1,2, . . . ,N
42
The side constraints (7) fDX gmin RfDX gRfDX gmax
43
are speci®ed by the move limits in the way discussed by Pedersen [17]. The advantage of this formulation is that it gives a uni®ed approach to handle problems with both simple and multiple frequencies, and that ordinary design sensitivity analysis and gradient optimization methods can be applied to solve the optimization problem.
The ®rst example is aimed at illustrating the optimal design procedure in the case of occurrence of multiple eigenfrequencies. The phenomenon of veering or intersection of frequency lines is then observed with the associated mode exchange for varying design parameter. Consider a two-beam structure built-in at two ends 1 and 3 and rigidly connected at the node 2 to which a non-structural mass M with moments of inertia IX IY 0:147 kg m2 and moments Mx My 784:8 Nm are added, Fig. 1. The beam crosssections are tubular with the interior diameter d = 0.04 m and wall thicknesses t1 and t2. The distributed mass eect of beams is included into the analysis. The structure has two degrees of freedom with the generalized coordinates assumed as rotation angles of the lumped mass with respect to the reference axes X and Y. The free vibration frequencies and modes are speci®ed from the eigenvalue problem s11 ÿ lm11 0 S ÿ lM fvg s22 ÿ lm22 0 44 0 v1 0 v2 where 4EIy2 GIx 1 s11 , L1 L2
s22
4EIy1 GIx 2 , L2 L1
6. Examples We shall now present four numerical examples starting from a two beam structure and ending with a medium complexity space structure. Common data for all examples are:
Fig. 1. Two-beam space frame.
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
m11
Ix 1 Ix 2
ra2 L32 IX , 105
m22
ra1 L31 IY , 105
pÿ 3 d t1 3d 2 t21 4dt31 2t41 , 4 pÿ d 3 t2 3d 2 t22 4dt32 2t42 , 4
ÿ a1 p dt1 t21 , Iy1
ÿ a2 p dt2 t22 ,
Ix 1 Ix ,Iy 2 : 2 2 2
45
Here Ix 1 ,Ix 2 ,Iy1 and Iy2 are cross-sectional moments of inertia, a1, a2 are the cross-sectional areas of tubular cross-sections. The characteristic equation is generated from the condition detS ÿ lM 0
46
providing the values of l1 o21 and l2 o22 : The lines of constant frequencies and eigenmodes are plotted in the design plane t1, t2 in Fig. 2. It is evident that for t1 t2 the bimodal states o1 o2 occur. Requiring symmetric response with respect to the bimodal line Bml, the mode exchange must be assumed when passing from the domain t1 > t2 to the domain t2 > t1 : The respective portions of frequency lines o1 const, and o2 const intersect at points distant from the bimodal line. However, there is no intersection of the lines and o1 const and o2 const on the bimodal line since these lines approach, touch each other, and
173
separate. This situation corresponds to the curve veering eect, investigated in numerous papers, see e.g. Happawana et al. [2]. Physically, the veering eect is associated with the mode exchange when passing through the bimodal state. The frequency lines are then exchanged and there is no intersection of o1 and o2 lines. However, if the ®rst and second frequencies were associated with respective modes throughout the whole range of design parameter variation, then on the bimodal line we would have the frequency line intersection, as it is illustrated in Fig. 3. Such interpretation of frequency lines would obviously destroy the symmetry of response with respect to the bimodal line as the eigenmodes for t1 a,t2 b and t1 b,t2 a would be dierent. Fig. 4 presents the three-dimensional visualization of frequency variation for the case of veering and intersection of frequency surfaces. The solution of Eq. (44) provides the eigenvectors 1 T 2 T T T fv 1 and fv 2 which are 1 ,v2 g f0,1g 1 ,v2 g f1,0g exchanged on the Bml when passing from the domain t1 > t2 to the domain t2 > t1 : The present example illustrates the existence of the bimodal line in the design plane and the curve veering eect occurring on this line. Consider now the optimal frame design aimed at minimizing its total mass with the constraint set on eective stresses and frequencies, namely seff t1 ,t2 ÿ 1R0, saeff
1:0 ÿ
oj t1 ,t2 R0, oa
j 1,2
Fig. 2. Contour plots of free frequencies o1 and o2 with veering points on Bml: (a) o1 = const, (b) o2 = const.
47
174
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
Fig. 3. Contour plots of free frequencies o1 and o2 with intersection points on Bml: (a) o1 =const, (b) o2 =const.
where oa 118:96 Hz, saeff 147 MPa. The constraints imposed on design variables are: 1Rt1 =t0 R5,1Rt2 =t0 R5 where t0 0:001 m. The optimal design of two-beam frame lies on the boundary of feasible domain at the point 0 t1 3:27 mm, t2 3:27 mm, Mopt 6:94 kg), Fig. 5. This point is speci®ed by the intersection of the bimodal line Bml with the boundary of feasible domain. The optimal solution obtained numerically is clearly illustrated in Fig. 5. In fact, the locus of bimodal frequencies constitutes a set of optimal designs for varying frequency constraint levels.
6.2. Example 2: two-beam frame with three degrees of freedom Consider now the optimal design of a similar twobeam structure shown in Fig. 1, where L = L1 = L2 = 3 m., and with two design variables: thickness of circular cross-sections t t1 t2 and vertical coordinate Z2 of node 2. The frame is also loaded by a concentrated force P = 2901.1 N at the node 2 acting along the negative Z-direction, and mass M = 147.15 kg. The structure has three degrees of freedom with the generalized coordinates corresponding to trans-
Fig. 4. Variation of the ®rst two frequencies o1 , o2 : (a) veering, (b) intersection.
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175
Fig. 5. Illustration of constraints and optimal design in the plane t1, t2.
lation of the joint 2 along the axis Z and rotation of the nonstructural mass M around the axes X and Y. Two independent cases will be carefully studied for this example in order to clarify . eect of self-weight on optimal design occurring for bimodal state with single frequency constraint . eect of frequency constraints set on three eigenfrequencies. The structure is symmetric and therefore multiple frequencies are expected. Three free vibration frequencies and modes are speci®ed from the eigenvalue problem
s11 ÿ lm11 s12 ÿ lm12 4 s12 ÿ lm12 s22 ÿ lm22 s13 ÿ lm13 0 8 9 8 9 < v1 = < 0 = v2 0 : ; : ; v3 0 where
s12 Z2
s22 s33
3 s13 ÿ lm13 5 0 s33 ÿ lm33
6EIy1 , L2
s13 Z1
6EIy1 , L2
ÿ 1 2 g1 GIx 1 4EIy1 1 g21 , L ÿ 1 2 m2 GIx 1 4EIy1 1 m22 , L
m11
S ÿ lM fvg 2
s11
ÿ 2 12Iy1 ÿ 2 E 2 2 Z1 Z2 , a1 Z1 Z2 L2 L
ÿ ra1 L ÿ ÿ 2 140 Z1 Z22 156 Z21 Z22 M, 420
m12 48 m22
22ra1 L2 Z2 , 420
ra1 L3 ÿ 2 g 1 , 105 1
m13
22ra1 L2 Z1 , 420
m33
ra1 L3 ÿ 2 m2 1 , 105
Z2 Z1 m2 ÿZ2 ÿg1 q , 2 3 Z 22
176
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
Fig. 6. Surface plot of three frequencies.
3 g1 m2 Z1 Z1 Z2 q : 32 Z 22 The frequency surfaces o1 o1 z,t,o2 o2 z,t, and o3 o3 z,t are plotted in the design space, Fig. 6. It
is seen in Fig. 7 that the initially distinct frequencies o1 and o2 coalesce when coordinate Z2 increases. The eective stress constraints and two dierent sets of frequency constraints are imposed on the optimal design. The ®rst problem is concerned with the constraint set on the fundamental frequency o1 r4:18 Hz. The optimal design without self-weight corresponds to mass Mopt 12:37 kg and the frequencies at the optimum are: o1 4:18 Hz, o2 29:79 Hz, o3 30:01 Hz. This is interesting and principally important to analyse how the optimal design is changed, when the selfweight is taken into account for the same formulation of optimization problem. The optimal design with selfweight corresponds to mass Mopt 12:39 kg and the frequencies at the optimum are: o1 6:06 Hz, o2 29:68 Hz, o3 29:9 Hz. With respect to the initial design the frame mass decreased by 7.97 kg without selfweight and by 7.95 kg with self-weight. The analytical solutions are illustrated in Fig. 8 and Table 1 provides the initial and optimal design variables. The optimal solutions were attained in 14 iterations. The next optimal design with self-weight and force P = 33,557.85 N was obtained by setting three frequency constraints besides the eective stress constraints, namely o1 r18 Hz, o2 r18 Hz and o3 r34 Hz. The optimal mass is now Mopt 12:78 kg and the three frequencies at the optimum are: o1 18 Hz, o2 18:017 Hz, o3 37:76 Hz, see Fig. 9. First two frequencies are equal, provided a criterion for identical frequencies allows for relative dierence between the frequencies
Fig. 7. Variation of o1 ,o2 and o3 in function of node position Z2 for t = 2 mm and the respective three modes when coordinate Z2 = 2000 mm.
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177
Table 1 Values for initial design, side constraints and for optimal design without and with self-weight. The concentrated force P is equal to 2901.1 N and the mass M at node 2 is equal to 147.15 kg Design variables
t Z2 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
Minimum
2 100
Initial values
3.2 120 20.339
Maximum
4 400
Optimal values (mm) Without self-weight
With self-weight
2 159.786 12.366 5 14 11
2 234.980 12.386 4 14 11
DoR10ÿ3 : Table 2 presents the optimal set of design variables and side constraints. The analytical solution shown in Fig. 9 is in good agreement with numerical solution, Table 2. The ®rst two examples were treated in order to verify the numerical procedure and study the evolution of bimodal states in the design space. Let us note that in the ®rst example the bimodal designs evolved along the line in the design plane, but in the second example the bimodal states occurred within the common portion of two frequency surfaces. The constraints set on eigenfrequencies may aect signi®cantly the optimal con®guration. On the other hand, the eect of self-weight is not signi®cant especially with respect to higher frequencies. 6.3. Example 3: ®fty-two-beam frame dome The frame dome consists of 52 beam elements. The
Fig. 8. Analytical solutions for two-beam space frame: (a) without self-weight and (b) with self-weight.
Fig. 9. Analytical solution for two-beam space frame. Optimum is attained on the bimodal surface o1 o2 :
178
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Table 2 Values for initial design, side constraints and for optimal design with account for self-weight Design variables
Minimum
Initial values
Maximum
Optimal values (mm)
t Z2 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
1.5 100
2.5 2400 20.003
4 2500
1.665 2266.281 12.779 7 15 13
Fig. 10. Frame dome with 52 beam elements. Joint numbers and initial joint positions are shown together with linkings of beams as indicated by a±h.
elements are linked into eight types of tubular crosssections, Fig. 10. We also link node coordinates 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 into 5 groups speci®ed in terms of design variables Z1, X2, Z2, X6, Z6, Table 4. The inside diameter of tubular cross-sections is kept constant at 86 mm. The problem was treated in two versions, ®rstly a single load case without and with self-weight and next a multiple load case incorporating
four independent loading conditions without and with self-weight are considered with constraints on eective stresses seff R147 MPa for all load cases and constraint on fundamental frequency (Table 3). Figs. 11±14 illustrate optimal layouts of the dome, for which the numerical values are given in Table 4. It turns out that the optimal solutions are characterized by a distinct fundamental frequency, Fig. 15. If we use
Table 3 Loading speci®cations for the two solved problems Problem No.
Load condition
Nodes
Load direction Z Load P (N)
1
CASE CASE CASE CASE
1 1 1,2,3,4,5,6,7,8,9,10,11,12,13 1,4,5
ÿ632745 ÿ300000 ÿ30000 ÿ150000 ÿ100000 ÿ150000 ÿ70000
2
1 1 2 3
CASE 4
1,2,3,4
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
Fig. 11. Optimal layout of the dome with a single load P = 632745 N at the top and without self-weight Mopt 1981:595 kg).
Fig. 12. Optimal layout of the dome with a single load P = 632745 N at the top and self-weight. Mopt 2069:145 kg.
179
Fig. 13. Optimal layout of the dome with four independent load cases and without self-weight Mopt 1782:807 kg).
Fig. 14. Optimal layout of the dome with four independent load cases and self-weight Mopt 1880:445 kg).
Minimum
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 7000 18000 6000 23000 3000
DV
ta tb tc td te tf tg th Z1 X2 Z2 X6 Z6 Optimal mass (kg) Number of iterations Number of active constraints for optimum Calculation time (s) 10 10 10 10 10 10 10 10 11000 22000 10000 27500 7000
Maximum
5.809 1.5 4.046 2.149 1.5 3.907 1.5 1.5 10428.27 18000 7597.405 23000 4015.105 2069.145 19 46 50
60
Single load condition with self-weight
5.444 1.5 4.136 2.063 1.5 3.565 1.5 1.5 10052.8 18000 7078.23 23000 3806.667 1981.595 21 44
Single load condition without self-weight
Optimal values (mm) Allowable fundamental frequency oa 5:9 Hz
100
2.862 1.5 1.911 2.380 1.893 2.961 1.5 1.5 11000 18459.65 7761.153 23000 3272.762 1782.807 23 40
Four independent load cases without self-weight
Table 4 Values for side constraints and optimal design for problem 1 (single load condition) as well as for problem 2 (four independent load cases)
150
3.480 1.5 2.064 2.617 1.893 3.211 1.5 1.5 10544.55 18282.51 8128.216 23000 3579.28 1880.445 33 38
Four independent load cases with self-weight
180 O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
181
Fig. 15. Iteration history, optimization of the dome with single load including self-weight. The optimum design is shown in Fig. 12.
as a criterion for identical frequencies that the relative dierence between the frequencies must be R10ÿ4 , then the second and third frequencies should be considered as identical. The number of redesign iterations from the initial to optimal design was equal 15, 19, 23 and 33, respectively. The number of redesigns depends on the initial design and the strategy for choosing move limits. For the present examples we applied simple strategies of reducing move limits which are described in Pedersen [17]. 6.4. Example 4: frame of mobile crane Fig. 16. Mobile crane frame. Initial joint positions are shown together with linkings of beams as indicated by a±f.
The frame of a mobile crane, also treated by Apostol et al. [1], contains 26 beam elements, 18 nodes and
Table 5 Values for side constraints and optimal design of the mobile crane frame for single load condition with self-weight DV
Minimum
Initial values
Maximum
Optimal values (mm) Allowable fundamental frequencyoa 3:3 Hz
ta tb tc td te tf X3 Y3 Z3 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
0.8 0.8 0.8 0.8 0.8 0.8 100 300 4500 1820.585
4 3 2 2 2 4 600 600 5000
10 10 10 10 10 10 4700 2200 5350
4.573 3.716 0.8 0.930 0.931 0.8 100 300 5056.695 1603.918 28 13 18
182
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
Fig. 18. Iteration history, optimization of the mobile crane frame. The optimal design is shown in Fig. 17.
72 degrees of freedom and is subjected to one load condition including self-weight. The loads are all acting in the negative Z-direction at joints 3, 4, 9, 10, 15, 16 and the actual value at these joints is 123607 N. Beams of tubular cross sections with constant interior diameter 86 mm are linked into six groups as shown in Fig. 16. The coordinates of nodes 3, 4, 9, 10, 15, 16 are linked into three groups speci®ed in terms of design variables X3, Y3, Z3, Table 5. Therefore, beams occurring in the roof of the frame take part in size and shape optimization. Optimal solution with the constraint set on lowest frequency o1 r3:3 Hz and constraints on eective stresses seff R147 MPa is shown in Fig. 17 with the actual values in Table 5. Fig. 18 also
Fig. 17. Optimal shape of mobile crane frame.
shows our monitoring of ®rst lowest frequencies o1 ,o2 ,o3 , which turn out to remain distinct frequencies. 7. Conclusions A general optimization formulation is presented for three-dimensional frames for stress and multiple frequency constraints. The optimization can be performed using ordinary methods of design sensitivity analysis and eective gradient optimization methods. The eciency and accuracy of the optimization method has been demonstrated by a thorough numerical study of space frames. The present work extends the previous treatments of optimal design of space frames as both size and con®guration design parameters are treated in the uniform way using the analytically derived sensitivity derivatives. It turns out that the structure response is much more sensitive with respect to joint positions variation, and more eective designs can be generated by optimizing both shape and size parameters. The phenomenon of veering at the bimodal state is one of the interesting observations following from this paper. When curve veering occurs, there is an exchange of eigenmodes at the bimodal point and the frequency lines do not intersect. On the other hand, when intersection of frequency lines occurs, the eigenmodes are preserved along the same lines.
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
2
0 0 6 6 60 0 S3 6 6 6 6Ee Iy1e 4 0 ÿ L2e
Apppendix A
A1. Element matrices for beam vibrations The system stiness matrix [S ] and mass matrix [M ] are accumulated from beam element stiness and mass matrices referred to the system coordinate axes, i.e. symbolically (omitting Boolean matrices) X T S c4e Se c4e , e
M
X T c4e Me c4e
A1
e
c4e is a block-diagonal matrix generated by four orthogonal transformation matrices ce : The matrix ce is the global coordinate system±local coordinate system transformation matrix. Here are the beam stiness and mass matrices and speci®ed in a local coordinate system as shown below 2
S1 6 S3 T 6 Se 4 ÿS1 S3 T
S3 S2 ÿS3 S4
ÿS1 ÿS3 T S1 ÿS3 T
3 S3 S4 7 7 ÿS3 5 S2
where 3
2
Ee ae 6 Le 6 6 6 0 S1 6 6 6 6 4 0
0
0
12Ee Iz1e L3e
0
0
12Ee Iy1e L3e
2
Ge Ix 1c 6 L 6 e 6 6 S2 6 60 6 6 4 0
7 7 7 7 7, 7 7 7 5
0
0
4Ee Iy1c Le
0
0
4Ee Iz1e Le
3 7 7 7 7 7, 7 7 7 5
0 6Ee Iz1e L2e 0
183
3 7 7 7 7, 7 7 5 3
2
Ge Ix 1e 6ÿ L 6 e 6 6 6 S4 6 0 6 6 4 0
0
0
2Ee Iy1e Le
0
0
2Ee Iz1e Le
7 7 7 7 7, 7 7 7 5
where Ee ,Ge ,Ix 1e ,Iy1e ,Iz1e and ae ,Le are Young, shear moduli, moments of inertia, and cross-sectional area, length of a beam, respectively. The mass matrix is 2 3 M1 M3 M4 M5 T 7 r ae Le 6 6 M3 M2 T M5 M6 7 A3 Me e 420 4 M4 M5 M1 ÿM3 5 M5 T M6 M3 M2 where 2
3 140 0 0 M1 4 0 156 0 5, 0 0 156 3 2 140 0 0 4L2e 0 5, M2 4 0 0 0 4L2e 2
A2
3 0 0 0 M3 4 0 0 22Le 5, 0 ÿ22Le 0 2 3 70 0 0 M4 4 0 54 0 5, 0 0 54 2
3 0 0 0 M5 4 0 0 ÿ13Le 5, 0 13Le 0 2 3 70 0 0 2 5, ÿ3Le 0 M6 4 0 0 0 ÿ3L2e where re is density of a beam. These results may be con®rmed in the book by Przemieniecki [18]. Let us specify the rules of prescribing the local coordinate system x 1e y1e z1e : 1. The global coordinate system XYZ is right-hand. 2. The local coordinate system x 1e y1e z1e is right-hand. 3. Axis x 1e is the beam axis, axes y1e ,z1e are the princi-
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
184
@ ge wZ g ÿ i e e, @ Zi Le
The transformation matrix [ce] 2 cosX,x 1e cosY,x 1e ce 4 cosX,y1e cosY,y1e cosX,z1e cosY,z1e 2 3 ge me Ze 4 g e m e Z e 5 g m Z e
ÿ ÿ @ m e ÿwi me ge Ze 2 ÿ Z2e cosa ÿ me sina = Le b3e , @ Xi
is given by 3
cosZ,x 1e cosZ,y1e 5 cosZ,z1e A4
where ÿ g e ge Ze cosa ÿ me sina =be ,
Z e ÿbe cosa,
ÿ @ Z e ÿwi ge Z2e cosa= Le be , @ Xi ÿ ÿ @ g e ÿwi ge me Ze 2 ÿ Z2e cosa ge me sina = Le b3e , @ Yi
ÿ @ Z e ÿwi me Z2e cosa= Le be , @ Yi
ÿ
g e ÿ me cosa ÿ ge Ze sina =be , ÿ m e ge cosa ÿ me Ze sina =be ,
@ g e wi ge be cosa=Le , @ Zi
Z e be sina,
@ m e wi me be cosa=Le , @ Zi
q g2e m2e :
@ Z e wi Ze be cosa=Le , @ Zi
A2. Sensitivity analysis for the system stiness and mass matrices For frames with a particular way of prescribing local coordinate systems of the beams, we managed to obtain analytical expressions of the derivatives h @@fXS g i and h @@ fM X g i, where h i denotes matrix package. Derivatives of transfer matrix ce to node coordinates are de®ned as follows ÿ wi 1 ÿ g2e @ ge @ me wm g @ Ze wZ g , ÿ i e e, ÿ i e e, Le @ Xi @ Xi Le @ Xi Le @ ge wm g ÿ i e e, @ Yi Le
ÿ ÿ @ g e ÿwi Ze g4e g2e m2e ÿ m2e cosa @ Xi ÿ ge me sina = Le b3e ,
ÿ ÿ @ m e ÿwi Ze m4e m2e g2e ÿ g2e cosa @ Yi ge me sina = Le b3e ,
ÿ m e me Ze cosa ge sina =be ,
be
@ me wZ m ÿ i e e, @ Zi Le
ÿ w 1 ÿ Z2e @ Ze i , Le @ Zi
pal cross-section axes. 4. The auxiliary z axis is normal to the axes x 1e and Z and forms a right-hand triplet PrXY x 1e zZ: 5. The z1e axis forms with an auxiliary axis z an angle a counted o from z to z1e around the x 1e axis by the principle of the right-hand screw. 6. If the beam x 1e axis is parallel to the Z axis then z axis is parallel to the global X axis.
ÿ wi 1 ÿ m2e @ me , Le @ Yi
@ Ze wZ m ÿ i e e, @ Yi Le
ÿ ÿ @ g e ÿwi ÿ Ze g4e g2e m2e ÿ m2e sina @ Xi ÿ ge me cosa = Le b3e , ÿ ÿ @ m e ÿwi me ÿ ge Ze 2 ÿ Z2e sina ÿ me cosa = Le b3e , @ Xi ÿ @ Z e wi ge Z2e sina= Le be , @ Xi ÿ ÿ @ g e ÿwi ge ÿ me Ze 2 ÿ Z2e sina ge cosa = Le b3e , @ Yi
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
ÿ ÿ @ m e ÿwi ÿ Ze m4e m2e g2e ÿ g2e sina @ Yi ge mcosa = Le b3e , ÿ @ Z e wi me Z2e sina= Le be , @ Yi @ g e ÿwi ge be sina=Le , @ Zi @ m e ÿwi me be sina=Le , @ Zi @ Z e ÿwi Ze be sina=Le @ Zi where ÿ1, node is at the beginning of a beam, wi 1, node is at the end of a beam If the beam axis x 1e is parallel to the global Z axis @ c then the derivative of transformation matrix h @ f Xe g i @ ce does not exist. An additional speci®cation h @ fX g i in this case can be performed by assuming that no discontinuity of the cross-section orientation in space can be observed for the case of small coordinate variation. This assumption leads to a jump-like variation of angle a depending on which coordinate of the derivative is considered. The nondierentiability is associated with an additional de®nition of the local coordinate system in a special case (cf. point 6 of rules for local coordinate systems).
References [1] Apostol V, Santos JLT, Goia I. Design sensitivity analysis and optimization of truss/beam structures with arbitrary cross-sections. In: Olho N, Rozvany GIN, editors. Proc. WCSMO-1. Structural and multidisciplinary optimization. Oxford: Pergamon Press, 1995. [2] Happawana GS, Bajaj A, Nwokash ODI. A singular perturbation analysis of eigenvalue veering and modal sensitivity in perturbed linear periodic systems. Journal of Sound and Vibration 1993;160(2):225±42. [3] Haug EJ, Choi KK, Komkov V. Design sensitivity analysis of structural systems. New York: Academic Press, 1986. [4] Haug EJ, Rousselet B. Design sensitivity analysis in structural mechanics. Part II: eigenvalue variations. J Struct Mech 1980;8:161±86.
185
[5] Khot KS. Optimization of structures with multiple frequency constraints. Computers and Structures 1985;20:869±76. [6] Kiselev VG. Sensitivity analysis of extreme strength criteria. Applied problems of strength and plasticity. Analysis and Optimization 1997;91±100 (in Russian). [7] Kollbrunner CF, Hajdin N. Dunnwandige StaÈabe, Band 1. Berlin: Springer, 1972. [8] Krog LA, Olho N. Topology optimization of plate and shell structures with multiple eigenfrequencies. In: Olho N, Rozvany GIN, editors. Proc. WCSMO-1. Structural and multidisciplinary optimization. Oxford: Pergamon Press, 1995. p. 675±82. [9] Lee IW, Jung GH. An ecient algebraic method for the computation of natural frequency and mode shape sensitivities. Part I: distinct natural frequencies. Computers and Structures 1997;62:429±35. [10] Lee IW, Jung GH. An ecient algebraic method for the computation of natural frequency and mode shape sensitivities. Part II: multiple natural frequencies. Computers and Structures 1997;62:437±43. [11] Masur EF, MroÂz Z. Non-stationarity optimality conditions in structural design. Int J Solids Structures 1979;15:503±12. [12] Masur EF, MroÂz Z. Singular solutions in structural optimization problems. In: Nemat-Nasser S, editor. Variational methods in the mechanics of solids. New York: Pergamon Press, 1980. p. 337±43. [13] Mc Gee OG, Phan KF. Adaptable optimality criterion techniques for large-scale space frames with multiple frequency constraints. Computers and Structures 1992;42:197±210. [14] Mills-Curran WC. Calculation of eigenvector derivatives for structures with repeated eigenvalues. AIAA J 1988;26(7):867±71. [15] Olho N, Krog LA, Lund E. Optimization of multimodal structural eigenvalues. In: Olho N, Rozvany GIN, editors. Proc. WCSMO-1. Structural and multidisciplinary optimization. Oxford: Pergamon Press, 1995. [16] Pedersen P. Design with several eigenvalue constraints by ®nite elements and linear programming. J Struct Mech 1995;10(3):243±71. [17] Pedersen P. Optimal joint positions for space trusses. J Structural Div 1973;ASCE-99:2459±76. [18] Przemieniecki JS. Theory of matrix structural analysis. New York: McGraw-Hill, 1968. [19] Reklaitis GV, Ravindran A, Ragsdell KM. Engineering optimization methods and applications. New York: Wiley-Interscience, 1983. [20] Sergeyev O, Pedersen P. On design of joint positions for minimum mass 3D frames. Structural Optimization 1996;11:95±101. [21] Seyranian AP, Lund E, Olho N. Multiple eigenvalues in structural optimization problems. Structural Optimization 1994;8:207±27. [22] Wittrick WH. On eigenvalues of matrices dependent on a parameter. Numer Mathematik 1962;6:377±87.
www.elsevier.com/locate/compstruc
Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints O. Sergeyev 1, Z. MroÂz* Institute of Fundamental Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Poland Received 29 April 1998; accepted 3 March 1999
Abstract The present paper deals with the problem of determining the optimal joint positions and cross-sectional parameters of linearly elastic space frames with imposed stress and free frequency constraints. The frame is assumed to be acted on by dierent load systems, including temperature and self-weight loads. The stress state analysis includes tension, bending, shear, and torsion of beam elements. By a sequence of quadratic programming problems, the optimal design is attained. The sensitivity analysis of distinct as well as multiple frequencies is performed through analytic dierentiation with respect to design parameters. Illustrative examples of optimal design of simple and medium complexity frames are presented, and the particular case of bimodal optimal solution is considered in detail. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction In this paper, we shall discuss the problem of sensitivity analysis and optimal design of frame structures for which both cross-sectional and con®guration design parameters are to be determined from the solution. We shall impose the stress and free frequency constraints on the optimal design. The ®rst constraint provides proper stress levels under speci®ed loads, the other constraint is aimed to assume proper structure response under dynamic excitation for which the resonant conditions are avoided and proper frequency spectrum is obtained. The optimal design problems with free frequency constraints were treated in numerous papers (cf. [5,9,10] and references cited therein).
* Corresponding author. 1 Visiting Research Fellow from the Department of Theoretical Mechanics, Technical University of Nizhni Novgorod, Russia.
Similarly, the sensitivity analysis for single and multiple eigenfrequencies was considered by numerous authors (cf. Wittrick [22], Masur and Mroz [11,12], Haug and Rousselet [4], Pedersen [16, 17], Mills-Curran [14], Mc Gee and Phan [13], Olho et al. [15], Krog and Olho [8]). The aim of this paper is to extend the previous analyses and consider variation of both cross-sectional and con®gurational parameters. It turns out that structures are much more sensitive to con®guration changes, so search for optimal structure joint positions provides much more ecient designs. The use of stress and frequency constraints assures practical designs for which both static and dynamic responses are controlled. One of the characteristic features of free frequency constrained designs is occurrence of multiple or nearly equal eigenvalues. Such coincidence of free frequencies is associated with structural symmetry or is induced by the evolution of eigenvalue spectrum due to redesign process toward an optimum with constraint set on the fundamental frequency. It is well known that multiple
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 8 8 - 7
168
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
frequencies are not dierentiable in the common sense (that is are not Frechet dierentiable) and only directional sensitivity derivatives can be calculated. This fact creates some diculties in ®nding sensitivities of multiple frequencies with respect to design parameters and in applying the eective gradient optimization techniques. The present paper is devoted to development of an ecient method of sensitivity analysis and optimization of space frame structures with single and multiple eigenfrequencies. The sequential quadratic programming scheme is used with direct application of sensitivity derivatives. It was found that the applied method is reliable and accurate. The speci®c examples are concerned with a two-beam space frame, frame dome with 52 beam elements, and ®nally with a frame of mobile crane.
2. Sensitivity analysis for single and multiple eigenfrequencies Consider a linear elastic discretized frame structure for which the static equilibrium equation takes the form S fD g fA1 g fA2 g fA3 g 1 where [S ] is the global stiness matrix, {D } is the vector of joint displacements, and fA1 g is the vector of external joint loads, fA2 g is the vector of distributed equivalent loads, fA3 g is the vector of temperature equivalent loads. The state of free vibrations is governed by the symmetric eigenvalue problem ÿ S ÿ lj M Fj 0, j 1, . . . ,n 2
matrices) we have S
X C4e T Se C4e , e
M
X C4e T Me C4e
5
e
The transformation matrix C4e is a block-diagonal matrix composed of four orthogonal transformation matrices Ce ,Ce ÿ1 Ce T relating the beam element to the global Cartesian reference frame. Considering two basic types of constraints, namely frequency and stress constraints, further, selecting the set of design parameters contained in the design vector fX g, the optimization problem can be stated as follows. It is required to determine from a given range fX gmin ,fX gmax the joint positions (shape parameters) as well as cross-sectional dimensions (size parameters) fX g for which the frame mass attains its minimum subject to stress, frequency and side constraints, so that M fX g min M fX g fX g2F
6
where the feasible domain is speci®ed by the inequalities F fX g:seff fX gRsaeff , oj fX groaj ,
j 1,2, . . . ,k,
fX gmin RfX gRfX gmax
7
The eigenvectors fFj g are all [S ]-orthogonal and [M ]normalized, so that fFi gT S Fj lj dij ,
Situations where several frequencies coalesce and become a multiple frequency may naturally occur. They constitute one of the main diculties for gradient optimization methods as multiple frequencies are only directionally dierentiable. In order to generate correct optimization results, we have to avoid non-dierentiability of multiple frequencies in the active set of constraints. In the following sections, we summarize results of design sensitivity analysis of distinct and multiple frequencies, carry through an analytical sensitivity analysis of extremum eective stresses and next the gradient optimization method will be used to solve the optimal design problems with stress and frequency constraints.
fFi gT M Fj dij ,
2.1. Design sensitivity analysis of distinct frequencies
where [M ] is the global mass matrix, lj o2j is the jth squared angular vibration frequency, fFj g is the associated eigenmode and n denotes the number of degrees of freedom. The explicit forms of [S ] and [M ] are provided in the Appendix A. The eigenvalues are all real and can be ordered in the following manner: 0 < l1 Rl2 R Rlj R Rln
i,j 1, . . . ,n
3
4
The system stiness and mass matrices are accumulated from beam element stiness Se and mass matrices Me , i.e., symbolically (omitting Boolean
Let X be one of scalar design parameters. Assuming that lj are simple, the frequency sensitivities are obtained from Eq. (2) by direct dierentiation
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
@ S @ lj @ M ÿ M Fj ÿ lj Fj ÿ F j S @X @X @X @ Fj ÿ lj M @X
the analysis from Refs. [14,21,22], we introduce linear ~ j g associated with the combinations of eigenvectors fF multiple frequency l~
0
8
By premultiplying Eq. (8) by fFj g and making use of Eq. (2), the following expression is obtained for the frequency sensitivity in the case of simple frequencies lj @ lj T @ S @ M ÿ lj Fj 9 Fj @X @X @X The derivatives of matrices [S ] and [M ] are calculated analytically at the element level (cf. [20] and see Appendix A), thus X @ C4e T @ S @ C4e Se C4e C4e T Se @X @X @X e T @ Se C4e , C4e @X X @ C4e T
@ M @X
e
@X
T
10
If all design variables Xj are changed simultaneously then we can ®nd the linear increment of the distinct frequency lj in the form Dlj
X @ lj i
@ Xi
DXi rT lj fDX g
j 1,2, . . . ,N
13
k1
where bjk are unknown coecients to be speci®ed. Consider a variation of the design vector fX g in the form fX g efeg, where the unit vector feg represents a direction in the design space along which the design parameters are changed and denotes the magnitude of perturbation in this direction. For the increment of a single parameter X, the matrices [S ] and [M ] undergo variation and we have
@ S S X eex S eex , @X
@ M M X eex M eex @X
14
lj X eex l~ eex
@ l~ , @X
n o ~j n o @ F ~ j eex Fj X eex F , @X
15
j 1,2, . . . ,N Substituting Eqs. (15) and (14) into Eq. (2), we obtain within linear approximation
Let us consider the case of N-multiple frequency j 1,2, . . . ,N
o X N ~j F bjk fFk g,
11
2.2. Design sensitivity analysis for multiple frequencies
l~ lj ,
n
Along the direction of design variation, the following linear approximations of frequencies and eigenvectors are obtained
Me C4e
@ C4e @ Me C4e C4e Me C4e T @X @X
169
12
where l~ is the common value of repeated eigenfrequencies. The computation of sensitivities of this frequency is not straightforward. This is due to the fact that the eigenvectors fFj g,j 1,2, . . . ,N of the repeated eigenfrequencies are not unique. Any linear combination of eigenvectors will satisfy the eigenvalue problem (2). It should be noted that multiple frequencies are not differentiable in the common sense (i.e., the Frechet derivative does not exist), cf. [3,11]. Thus, to ®nd the sensitivities of multiple frequencies, we have to use directional derivatives in the design space. Following
n o n o ~j ÿ @ F @S @M ~ j ex S ÿ l~ M ex ÿ l~ F @X @X @X n o @ l~ ~j 0 ex M F ÿ @X
16
Premultiplying this equation by fFj gT one obtains ~ T @ S ~ @ M ~ s ex ÿ @ l ex Fj T M ÿl F Fj @X @X @X ~ s 0, j,s 1,2, . . . ,N F
17
In view of representation (13), we obtain from Eq. (17) the system of linear algebraic equations for unknown coecients bsk
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
170
" N X T @ S ~ @ M fFk gex ÿl bsk Fj @X @X k1 @ l~ ex djk ÿ @X
#
s T N Ae 18
j,k 1,2, . . . ,N
Introducing the vector ffjk g in the form T T @ S ~ @ M i fFk g fjk Fj i ÿ lh h @ fX g @ fX g
s T N qe
20
None of the sensitivities mj can be set in a one-toone correspondence with any particular eigenvector fFj g: In many cases, it is expedient to eliminate the unit vector feg from Eq. (21) and generate the explicit formula for determining the increments Dlj , j ~ As there is 1, . . . ,N of the N multiple frequency l: efeg fDX g and em Dl, and we obtain h i T det fjk fDX g ÿ Dldjk 0, j,k 1, . . . ,N 22
) y1 x 1 , ÿ Iz1 Le
ÿLe x 1 1 ÿy1 L2e x 21 x 1 ÿ ÿ , 2 ae Le 2Iz1 L2e Le
! !) 1 ÿz1 L2e x 21 x 1 1 ÿ , 2Iy1 L2e Le 6 6
sYe
YE ÿ YB ÿaE x 1 YB ÿ Y0 Le
24
3. Sensitivity of the maximal eective stress Every beam is acted upon by the joint load vector fAe g uniform distributed load fqe g and temperature in a beam span. Based on the superposition principle, the axial normal stress can be written in the form, cf. [6] T ÿ s N sAe x 1 ,y1 ,z1 ,fX g fAe g
! 25
where x 1 0Rx 1 RLe is the local longitudinal coordinate and y1 ,z1 are the lateral coordinates, Le is the length of a beam, ae is the cross-sectional area, Iy1 ,Iz1 are the cross-sectional moments of inertia, a is the thermal expansion coecient, E denotes the Young's modulus, Y0 is the reference temperature and the linear temperature distribution is assumed between by the joint temperatures YB and YE : Note that warping stresses are neglected. The shear stresses are determined by
0 t
Mx 1 Qy B 1B S ÿ 2Ot tIz1 @ z1 0
Qz 1 B BS ÿ tIy1 @ y1
1 1 sz1 ds C C t ds A t 1
1 sy1 ds C C t ds A t
26
where s is the arc length of a middle section line, S y1 ,S z1 are static moments of the section part between the origin and the current point, O is the area bounded by the contour middle line and t is the current thickness of the section wall. The main assumptions behind Eq. (26) are closed thin walled cross-sections, and for details we refer to Ref. [7]. The shear stress can be presented in the matrix notation T ÿ t N tAe x 1 ,y1 ,z1 ,fX g fAe g
n ÿ oT N sqe x 1 ,y1 ,z1 ,fX g qe fX g
with the vector de®nitions
(
y1 x1 1ÿ , Iz 1 Le
and with temperature stress
where h i denotes a matrix package, the fundamental equation for determining the directional derivatives m of the multiple frequency takes the form h i T det fjk feg ÿ mdjk 0, j,k 1, . . . ,N 21
sYe x 1 ,fX g
ÿ1 ÿz1 x1 ,0,0,0, 1ÿ , ae Iy1 Le z1 x 1 , Iy1 Le
0,0,0,0,
0
A non-trivial solution to these equations exists only if the determinant of the system is equal to zero, thus " # T @ S ~ @ M @ l~ fFk gex ÿ det Fj ex djk ÿl @X @X @X 19 0,
(
n ÿ oT N tqe x 1 ,y1 ,z1 ,fX g fqe g 23
27
Assuming that shear stresses are generated only from shear forces and the torsional moment we get
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
t T N Ae 0, Fy1 y1 ,z1 , Fz1 y1 ,z1 , Fx 1 y1 ,z1 , 0, 0, 0, 0, 0, 0, 0, 0 , t T x1 1 x1 1 N qe 0, Le Fy1 , Le Fz1 ÿ ÿ 2 2 Le Le
28
where the functions Fy1 ,Fz1 of shear forces and Fx 1 of torsional moment are dependent on the cross-sectional shape. The eective stress seff at a given point of a beam and corresponding to a given load case is de®ned by p ÿ seff x 1 ,y1 ,z1 ,load case,fX g s2 Gt2 29 where G 3 for the von Mises hypothesis and G 4 for the Tresca hypothesis. Its maximum value equals ÿ ÿ seff x 1 ,y1 ,z1 ,fX g max seff x 1 ,y1 ,z1 ,fX g Volume
30
From Eq. (29) by the implicit dierentiation we get @ seff 1 @s @t Gt 31 s seff @X @X @X We need the sensitivities at a beam point where seff reaches its maximum value. Now taking the derivative of Eq. (30) and assuming that coordinates of the extremum point depend on the design parameter X, we get ÿ d max seff x 1 ,y1 ,z1 ,fX g volume
dX
@ seff @ x 1
@ seff @ y1
@ seff @ z1
@x1 @X @ y1 @ X @ z1 @ X @ seff @ X x ,y ,z ,fX g 1
1
that T @ N sAe T @ fAe g @s fAe g N sAe @X @X @X s T @ N qe s T @ fqe g @ sYe fqe g N qe @X @X @X
32
1
Let us demonstrate how to obtain particular derivatives occurring in Eq. (32). If the point x 1 ,y1 ,z1 lies inside beam segment then from stationarity of seff it follows that ®rst three terms of Eq. (32) are equal to zero. In the case of x 1 ,y1 ,z1 being at a beam boundary we have to study the boundary behaviour with respect to the design parameter X. The derivatives @ seff @ seff @ seff @ x 1 , @ y1 , @ z1 in Eq. (32) follow from the de®nitions (24) of fN sAe g,fN sqe g and from Eqs. (28) and (25) specifying fN tAe g,fN tqe g,sYe : The next step is to calculate the last term @@sXeff of the desired derivative (32). From the de®nitions (Eqs. (23) and (27)) of s and t it follows
33
T @ N tAe T @ fAe g @t fAe g N tAe @X @X @X t T @ N qe t T @ fqe g 34 fqe g N qe @X @X @ Ns @ Nt @ Ns @ Nt The terms f@ XAe g , f@ Xqe g , f@ XAe g , f@ Xqe g and @@sXYe can be obtained from Eqs. (24), (28) and (25). @ q The term @fXg non-vanishing only in the case of iner@ fqe g q tia loads is @ X faee g @@ aXe : The term @ @fAXe g follows from the relation fAe g Se C4e fDe g
4. Sensitivity analysis for eective stresses
171
35
where Se is the beam element stiness matrix speci®ed in a local coordinate system, fDe g is the joint displacement vector of the beam element. Now taking the derivative of Eq. (35), we obtain @ fAe g @ Se @ C4e C4e fDe g Se fDe g Se @X @X @X C4e
@ fDe g @X
36
By the direct dierentiation we obtain from the equilibrium equation (1) @ fD g @ fA2 g @ fA3 g @ S S f g D 37 ÿ @X @X @X @X 2g and thus with known sensitivities for loads @ fA @ X and @ fA3 g @ S as well as sensitivities for the stiness matrix @X @X , we can by a simple forward and backward substitutions determine @@fDX g : Performing calculations with derived formulas of this section, one can obtain the derivative of extremum eective stress (32) for every beam.
5. Optimization procedure with single and multiple frequency constraints The single frequency sensitivity is speci®ed by Eq. (9); however, for the state of multiple frequency, the sensitivities can be determined by solving the non-linear equation (22). In order to simplify the analysis, the design evolution is assumed to follow the path along which the o-diagonal terms of Eq. (22) vanish, so
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
172
that T fjk fDX g 0,
j 6 k,j,k 1,2, . . . ,N
38
Young's modulus of elasticity, E = 200 GPa Shear modulus of elasticity, G = 80 GPa Allowable eective stress, saeff 147 MPa Mass density, r 7799 kg/m3
Then, each of increments Dlj in Eq. (22) is directly related to a particular eigenvector fFj g, and is explicitly expressed, namely
Temperature stresses are neglected. PC-486 computer was used and CPU-time is given in seconds.
T Dlj fjj fDX g
6.1. Example 1: two-beam frame
39
Now there is no need to apply the directional derivatives for multiple frequencies, as there simply is rlj ffjj g: As optimal design (Eqs. (6) and (7)) is a highly nonlinear problem, we change the problem by a sequence of optimal redesigns Xp1 Xp DXp 40 where fXp g is a given design after p redesigns. Omitting the index p, we can now formulate the optimization problem as follows 1 T min DM rMT fDX g fDX g H fDX g 2
41
where [H ] is the Hessian matrix and it is determined in a way discussed in detail by Reklaitis et al. [19]. The constraint inequalities are approximated by linear terms seff rT seff fDX gRsaeff , lj rT lj fDX grlaj T fjk fDX g 0,
j 6 k,j,k 1,2, . . . ,N
42
The side constraints (7) fDX gmin RfDX gRfDX gmax
43
are speci®ed by the move limits in the way discussed by Pedersen [17]. The advantage of this formulation is that it gives a uni®ed approach to handle problems with both simple and multiple frequencies, and that ordinary design sensitivity analysis and gradient optimization methods can be applied to solve the optimization problem.
The ®rst example is aimed at illustrating the optimal design procedure in the case of occurrence of multiple eigenfrequencies. The phenomenon of veering or intersection of frequency lines is then observed with the associated mode exchange for varying design parameter. Consider a two-beam structure built-in at two ends 1 and 3 and rigidly connected at the node 2 to which a non-structural mass M with moments of inertia IX IY 0:147 kg m2 and moments Mx My 784:8 Nm are added, Fig. 1. The beam crosssections are tubular with the interior diameter d = 0.04 m and wall thicknesses t1 and t2. The distributed mass eect of beams is included into the analysis. The structure has two degrees of freedom with the generalized coordinates assumed as rotation angles of the lumped mass with respect to the reference axes X and Y. The free vibration frequencies and modes are speci®ed from the eigenvalue problem s11 ÿ lm11 0 S ÿ lM fvg s22 ÿ lm22 0 44 0 v1 0 v2 where 4EIy2 GIx 1 s11 , L1 L2
s22
4EIy1 GIx 2 , L2 L1
6. Examples We shall now present four numerical examples starting from a two beam structure and ending with a medium complexity space structure. Common data for all examples are:
Fig. 1. Two-beam space frame.
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m11
Ix 1 Ix 2
ra2 L32 IX , 105
m22
ra1 L31 IY , 105
pÿ 3 d t1 3d 2 t21 4dt31 2t41 , 4 pÿ d 3 t2 3d 2 t22 4dt32 2t42 , 4
ÿ a1 p dt1 t21 , Iy1
ÿ a2 p dt2 t22 ,
Ix 1 Ix ,Iy 2 : 2 2 2
45
Here Ix 1 ,Ix 2 ,Iy1 and Iy2 are cross-sectional moments of inertia, a1, a2 are the cross-sectional areas of tubular cross-sections. The characteristic equation is generated from the condition detS ÿ lM 0
46
providing the values of l1 o21 and l2 o22 : The lines of constant frequencies and eigenmodes are plotted in the design plane t1, t2 in Fig. 2. It is evident that for t1 t2 the bimodal states o1 o2 occur. Requiring symmetric response with respect to the bimodal line Bml, the mode exchange must be assumed when passing from the domain t1 > t2 to the domain t2 > t1 : The respective portions of frequency lines o1 const, and o2 const intersect at points distant from the bimodal line. However, there is no intersection of the lines and o1 const and o2 const on the bimodal line since these lines approach, touch each other, and
173
separate. This situation corresponds to the curve veering eect, investigated in numerous papers, see e.g. Happawana et al. [2]. Physically, the veering eect is associated with the mode exchange when passing through the bimodal state. The frequency lines are then exchanged and there is no intersection of o1 and o2 lines. However, if the ®rst and second frequencies were associated with respective modes throughout the whole range of design parameter variation, then on the bimodal line we would have the frequency line intersection, as it is illustrated in Fig. 3. Such interpretation of frequency lines would obviously destroy the symmetry of response with respect to the bimodal line as the eigenmodes for t1 a,t2 b and t1 b,t2 a would be dierent. Fig. 4 presents the three-dimensional visualization of frequency variation for the case of veering and intersection of frequency surfaces. The solution of Eq. (44) provides the eigenvectors 1 T 2 T T T fv 1 and fv 2 which are 1 ,v2 g f0,1g 1 ,v2 g f1,0g exchanged on the Bml when passing from the domain t1 > t2 to the domain t2 > t1 : The present example illustrates the existence of the bimodal line in the design plane and the curve veering eect occurring on this line. Consider now the optimal frame design aimed at minimizing its total mass with the constraint set on eective stresses and frequencies, namely seff t1 ,t2 ÿ 1R0, saeff
1:0 ÿ
oj t1 ,t2 R0, oa
j 1,2
Fig. 2. Contour plots of free frequencies o1 and o2 with veering points on Bml: (a) o1 = const, (b) o2 = const.
47
174
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
Fig. 3. Contour plots of free frequencies o1 and o2 with intersection points on Bml: (a) o1 =const, (b) o2 =const.
where oa 118:96 Hz, saeff 147 MPa. The constraints imposed on design variables are: 1Rt1 =t0 R5,1Rt2 =t0 R5 where t0 0:001 m. The optimal design of two-beam frame lies on the boundary of feasible domain at the point 0 t1 3:27 mm, t2 3:27 mm, Mopt 6:94 kg), Fig. 5. This point is speci®ed by the intersection of the bimodal line Bml with the boundary of feasible domain. The optimal solution obtained numerically is clearly illustrated in Fig. 5. In fact, the locus of bimodal frequencies constitutes a set of optimal designs for varying frequency constraint levels.
6.2. Example 2: two-beam frame with three degrees of freedom Consider now the optimal design of a similar twobeam structure shown in Fig. 1, where L = L1 = L2 = 3 m., and with two design variables: thickness of circular cross-sections t t1 t2 and vertical coordinate Z2 of node 2. The frame is also loaded by a concentrated force P = 2901.1 N at the node 2 acting along the negative Z-direction, and mass M = 147.15 kg. The structure has three degrees of freedom with the generalized coordinates corresponding to trans-
Fig. 4. Variation of the ®rst two frequencies o1 , o2 : (a) veering, (b) intersection.
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175
Fig. 5. Illustration of constraints and optimal design in the plane t1, t2.
lation of the joint 2 along the axis Z and rotation of the nonstructural mass M around the axes X and Y. Two independent cases will be carefully studied for this example in order to clarify . eect of self-weight on optimal design occurring for bimodal state with single frequency constraint . eect of frequency constraints set on three eigenfrequencies. The structure is symmetric and therefore multiple frequencies are expected. Three free vibration frequencies and modes are speci®ed from the eigenvalue problem
s11 ÿ lm11 s12 ÿ lm12 4 s12 ÿ lm12 s22 ÿ lm22 s13 ÿ lm13 0 8 9 8 9 < v1 = < 0 = v2 0 : ; : ; v3 0 where
s12 Z2
s22 s33
3 s13 ÿ lm13 5 0 s33 ÿ lm33
6EIy1 , L2
s13 Z1
6EIy1 , L2
ÿ 1 2 g1 GIx 1 4EIy1 1 g21 , L ÿ 1 2 m2 GIx 1 4EIy1 1 m22 , L
m11
S ÿ lM fvg 2
s11
ÿ 2 12Iy1 ÿ 2 E 2 2 Z1 Z2 , a1 Z1 Z2 L2 L
ÿ ra1 L ÿ ÿ 2 140 Z1 Z22 156 Z21 Z22 M, 420
m12 48 m22
22ra1 L2 Z2 , 420
ra1 L3 ÿ 2 g 1 , 105 1
m13
22ra1 L2 Z1 , 420
m33
ra1 L3 ÿ 2 m2 1 , 105
Z2 Z1 m2 ÿZ2 ÿg1 q , 2 3 Z 22
176
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Fig. 6. Surface plot of three frequencies.
3 g1 m2 Z1 Z1 Z2 q : 32 Z 22 The frequency surfaces o1 o1 z,t,o2 o2 z,t, and o3 o3 z,t are plotted in the design space, Fig. 6. It
is seen in Fig. 7 that the initially distinct frequencies o1 and o2 coalesce when coordinate Z2 increases. The eective stress constraints and two dierent sets of frequency constraints are imposed on the optimal design. The ®rst problem is concerned with the constraint set on the fundamental frequency o1 r4:18 Hz. The optimal design without self-weight corresponds to mass Mopt 12:37 kg and the frequencies at the optimum are: o1 4:18 Hz, o2 29:79 Hz, o3 30:01 Hz. This is interesting and principally important to analyse how the optimal design is changed, when the selfweight is taken into account for the same formulation of optimization problem. The optimal design with selfweight corresponds to mass Mopt 12:39 kg and the frequencies at the optimum are: o1 6:06 Hz, o2 29:68 Hz, o3 29:9 Hz. With respect to the initial design the frame mass decreased by 7.97 kg without selfweight and by 7.95 kg with self-weight. The analytical solutions are illustrated in Fig. 8 and Table 1 provides the initial and optimal design variables. The optimal solutions were attained in 14 iterations. The next optimal design with self-weight and force P = 33,557.85 N was obtained by setting three frequency constraints besides the eective stress constraints, namely o1 r18 Hz, o2 r18 Hz and o3 r34 Hz. The optimal mass is now Mopt 12:78 kg and the three frequencies at the optimum are: o1 18 Hz, o2 18:017 Hz, o3 37:76 Hz, see Fig. 9. First two frequencies are equal, provided a criterion for identical frequencies allows for relative dierence between the frequencies
Fig. 7. Variation of o1 ,o2 and o3 in function of node position Z2 for t = 2 mm and the respective three modes when coordinate Z2 = 2000 mm.
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177
Table 1 Values for initial design, side constraints and for optimal design without and with self-weight. The concentrated force P is equal to 2901.1 N and the mass M at node 2 is equal to 147.15 kg Design variables
t Z2 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
Minimum
2 100
Initial values
3.2 120 20.339
Maximum
4 400
Optimal values (mm) Without self-weight
With self-weight
2 159.786 12.366 5 14 11
2 234.980 12.386 4 14 11
DoR10ÿ3 : Table 2 presents the optimal set of design variables and side constraints. The analytical solution shown in Fig. 9 is in good agreement with numerical solution, Table 2. The ®rst two examples were treated in order to verify the numerical procedure and study the evolution of bimodal states in the design space. Let us note that in the ®rst example the bimodal designs evolved along the line in the design plane, but in the second example the bimodal states occurred within the common portion of two frequency surfaces. The constraints set on eigenfrequencies may aect signi®cantly the optimal con®guration. On the other hand, the eect of self-weight is not signi®cant especially with respect to higher frequencies. 6.3. Example 3: ®fty-two-beam frame dome The frame dome consists of 52 beam elements. The
Fig. 8. Analytical solutions for two-beam space frame: (a) without self-weight and (b) with self-weight.
Fig. 9. Analytical solution for two-beam space frame. Optimum is attained on the bimodal surface o1 o2 :
178
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Table 2 Values for initial design, side constraints and for optimal design with account for self-weight Design variables
Minimum
Initial values
Maximum
Optimal values (mm)
t Z2 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
1.5 100
2.5 2400 20.003
4 2500
1.665 2266.281 12.779 7 15 13
Fig. 10. Frame dome with 52 beam elements. Joint numbers and initial joint positions are shown together with linkings of beams as indicated by a±h.
elements are linked into eight types of tubular crosssections, Fig. 10. We also link node coordinates 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 into 5 groups speci®ed in terms of design variables Z1, X2, Z2, X6, Z6, Table 4. The inside diameter of tubular cross-sections is kept constant at 86 mm. The problem was treated in two versions, ®rstly a single load case without and with self-weight and next a multiple load case incorporating
four independent loading conditions without and with self-weight are considered with constraints on eective stresses seff R147 MPa for all load cases and constraint on fundamental frequency (Table 3). Figs. 11±14 illustrate optimal layouts of the dome, for which the numerical values are given in Table 4. It turns out that the optimal solutions are characterized by a distinct fundamental frequency, Fig. 15. If we use
Table 3 Loading speci®cations for the two solved problems Problem No.
Load condition
Nodes
Load direction Z Load P (N)
1
CASE CASE CASE CASE
1 1 1,2,3,4,5,6,7,8,9,10,11,12,13 1,4,5
ÿ632745 ÿ300000 ÿ30000 ÿ150000 ÿ100000 ÿ150000 ÿ70000
2
1 1 2 3
CASE 4
1,2,3,4
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Fig. 11. Optimal layout of the dome with a single load P = 632745 N at the top and without self-weight Mopt 1981:595 kg).
Fig. 12. Optimal layout of the dome with a single load P = 632745 N at the top and self-weight. Mopt 2069:145 kg.
179
Fig. 13. Optimal layout of the dome with four independent load cases and without self-weight Mopt 1782:807 kg).
Fig. 14. Optimal layout of the dome with four independent load cases and self-weight Mopt 1880:445 kg).
Minimum
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 7000 18000 6000 23000 3000
DV
ta tb tc td te tf tg th Z1 X2 Z2 X6 Z6 Optimal mass (kg) Number of iterations Number of active constraints for optimum Calculation time (s) 10 10 10 10 10 10 10 10 11000 22000 10000 27500 7000
Maximum
5.809 1.5 4.046 2.149 1.5 3.907 1.5 1.5 10428.27 18000 7597.405 23000 4015.105 2069.145 19 46 50
60
Single load condition with self-weight
5.444 1.5 4.136 2.063 1.5 3.565 1.5 1.5 10052.8 18000 7078.23 23000 3806.667 1981.595 21 44
Single load condition without self-weight
Optimal values (mm) Allowable fundamental frequency oa 5:9 Hz
100
2.862 1.5 1.911 2.380 1.893 2.961 1.5 1.5 11000 18459.65 7761.153 23000 3272.762 1782.807 23 40
Four independent load cases without self-weight
Table 4 Values for side constraints and optimal design for problem 1 (single load condition) as well as for problem 2 (four independent load cases)
150
3.480 1.5 2.064 2.617 1.893 3.211 1.5 1.5 10544.55 18282.51 8128.216 23000 3579.28 1880.445 33 38
Four independent load cases with self-weight
180 O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
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181
Fig. 15. Iteration history, optimization of the dome with single load including self-weight. The optimum design is shown in Fig. 12.
as a criterion for identical frequencies that the relative dierence between the frequencies must be R10ÿ4 , then the second and third frequencies should be considered as identical. The number of redesign iterations from the initial to optimal design was equal 15, 19, 23 and 33, respectively. The number of redesigns depends on the initial design and the strategy for choosing move limits. For the present examples we applied simple strategies of reducing move limits which are described in Pedersen [17]. 6.4. Example 4: frame of mobile crane Fig. 16. Mobile crane frame. Initial joint positions are shown together with linkings of beams as indicated by a±f.
The frame of a mobile crane, also treated by Apostol et al. [1], contains 26 beam elements, 18 nodes and
Table 5 Values for side constraints and optimal design of the mobile crane frame for single load condition with self-weight DV
Minimum
Initial values
Maximum
Optimal values (mm) Allowable fundamental frequencyoa 3:3 Hz
ta tb tc td te tf X3 Y3 Z3 Mass (kg) Number of active constraints for optimum Number of iterations Calculation time (s)
0.8 0.8 0.8 0.8 0.8 0.8 100 300 4500 1820.585
4 3 2 2 2 4 600 600 5000
10 10 10 10 10 10 4700 2200 5350
4.573 3.716 0.8 0.930 0.931 0.8 100 300 5056.695 1603.918 28 13 18
182
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Fig. 18. Iteration history, optimization of the mobile crane frame. The optimal design is shown in Fig. 17.
72 degrees of freedom and is subjected to one load condition including self-weight. The loads are all acting in the negative Z-direction at joints 3, 4, 9, 10, 15, 16 and the actual value at these joints is 123607 N. Beams of tubular cross sections with constant interior diameter 86 mm are linked into six groups as shown in Fig. 16. The coordinates of nodes 3, 4, 9, 10, 15, 16 are linked into three groups speci®ed in terms of design variables X3, Y3, Z3, Table 5. Therefore, beams occurring in the roof of the frame take part in size and shape optimization. Optimal solution with the constraint set on lowest frequency o1 r3:3 Hz and constraints on eective stresses seff R147 MPa is shown in Fig. 17 with the actual values in Table 5. Fig. 18 also
Fig. 17. Optimal shape of mobile crane frame.
shows our monitoring of ®rst lowest frequencies o1 ,o2 ,o3 , which turn out to remain distinct frequencies. 7. Conclusions A general optimization formulation is presented for three-dimensional frames for stress and multiple frequency constraints. The optimization can be performed using ordinary methods of design sensitivity analysis and eective gradient optimization methods. The eciency and accuracy of the optimization method has been demonstrated by a thorough numerical study of space frames. The present work extends the previous treatments of optimal design of space frames as both size and con®guration design parameters are treated in the uniform way using the analytically derived sensitivity derivatives. It turns out that the structure response is much more sensitive with respect to joint positions variation, and more eective designs can be generated by optimizing both shape and size parameters. The phenomenon of veering at the bimodal state is one of the interesting observations following from this paper. When curve veering occurs, there is an exchange of eigenmodes at the bimodal point and the frequency lines do not intersect. On the other hand, when intersection of frequency lines occurs, the eigenmodes are preserved along the same lines.
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
2
0 0 6 6 60 0 S3 6 6 6 6Ee Iy1e 4 0 ÿ L2e
Apppendix A
A1. Element matrices for beam vibrations The system stiness matrix [S ] and mass matrix [M ] are accumulated from beam element stiness and mass matrices referred to the system coordinate axes, i.e. symbolically (omitting Boolean matrices) X T S c4e Se c4e , e
M
X T c4e Me c4e
A1
e
c4e is a block-diagonal matrix generated by four orthogonal transformation matrices ce : The matrix ce is the global coordinate system±local coordinate system transformation matrix. Here are the beam stiness and mass matrices and speci®ed in a local coordinate system as shown below 2
S1 6 S3 T 6 Se 4 ÿS1 S3 T
S3 S2 ÿS3 S4
ÿS1 ÿS3 T S1 ÿS3 T
3 S3 S4 7 7 ÿS3 5 S2
where 3
2
Ee ae 6 Le 6 6 6 0 S1 6 6 6 6 4 0
0
0
12Ee Iz1e L3e
0
0
12Ee Iy1e L3e
2
Ge Ix 1c 6 L 6 e 6 6 S2 6 60 6 6 4 0
7 7 7 7 7, 7 7 7 5
0
0
4Ee Iy1c Le
0
0
4Ee Iz1e Le
3 7 7 7 7 7, 7 7 7 5
0 6Ee Iz1e L2e 0
183
3 7 7 7 7, 7 7 5 3
2
Ge Ix 1e 6ÿ L 6 e 6 6 6 S4 6 0 6 6 4 0
0
0
2Ee Iy1e Le
0
0
2Ee Iz1e Le
7 7 7 7 7, 7 7 7 5
where Ee ,Ge ,Ix 1e ,Iy1e ,Iz1e and ae ,Le are Young, shear moduli, moments of inertia, and cross-sectional area, length of a beam, respectively. The mass matrix is 2 3 M1 M3 M4 M5 T 7 r ae Le 6 6 M3 M2 T M5 M6 7 A3 Me e 420 4 M4 M5 M1 ÿM3 5 M5 T M6 M3 M2 where 2
3 140 0 0 M1 4 0 156 0 5, 0 0 156 3 2 140 0 0 4L2e 0 5, M2 4 0 0 0 4L2e 2
A2
3 0 0 0 M3 4 0 0 22Le 5, 0 ÿ22Le 0 2 3 70 0 0 M4 4 0 54 0 5, 0 0 54 2
3 0 0 0 M5 4 0 0 ÿ13Le 5, 0 13Le 0 2 3 70 0 0 2 5, ÿ3Le 0 M6 4 0 0 0 ÿ3L2e where re is density of a beam. These results may be con®rmed in the book by Przemieniecki [18]. Let us specify the rules of prescribing the local coordinate system x 1e y1e z1e : 1. The global coordinate system XYZ is right-hand. 2. The local coordinate system x 1e y1e z1e is right-hand. 3. Axis x 1e is the beam axis, axes y1e ,z1e are the princi-
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
184
@ ge wZ g ÿ i e e, @ Zi Le
The transformation matrix [ce] 2 cosX,x 1e cosY,x 1e ce 4 cosX,y1e cosY,y1e cosX,z1e cosY,z1e 2 3 ge me Ze 4 g e m e Z e 5 g m Z e
ÿ ÿ @ m e ÿwi me ge Ze 2 ÿ Z2e cosa ÿ me sina = Le b3e , @ Xi
is given by 3
cosZ,x 1e cosZ,y1e 5 cosZ,z1e A4
where ÿ g e ge Ze cosa ÿ me sina =be ,
Z e ÿbe cosa,
ÿ @ Z e ÿwi ge Z2e cosa= Le be , @ Xi ÿ ÿ @ g e ÿwi ge me Ze 2 ÿ Z2e cosa ge me sina = Le b3e , @ Yi
ÿ @ Z e ÿwi me Z2e cosa= Le be , @ Yi
ÿ
g e ÿ me cosa ÿ ge Ze sina =be , ÿ m e ge cosa ÿ me Ze sina =be ,
@ g e wi ge be cosa=Le , @ Zi
Z e be sina,
@ m e wi me be cosa=Le , @ Zi
q g2e m2e :
@ Z e wi Ze be cosa=Le , @ Zi
A2. Sensitivity analysis for the system stiness and mass matrices For frames with a particular way of prescribing local coordinate systems of the beams, we managed to obtain analytical expressions of the derivatives h @@fXS g i and h @@ fM X g i, where h i denotes matrix package. Derivatives of transfer matrix ce to node coordinates are de®ned as follows ÿ wi 1 ÿ g2e @ ge @ me wm g @ Ze wZ g , ÿ i e e, ÿ i e e, Le @ Xi @ Xi Le @ Xi Le @ ge wm g ÿ i e e, @ Yi Le
ÿ ÿ @ g e ÿwi Ze g4e g2e m2e ÿ m2e cosa @ Xi ÿ ge me sina = Le b3e ,
ÿ ÿ @ m e ÿwi Ze m4e m2e g2e ÿ g2e cosa @ Yi ge me sina = Le b3e ,
ÿ m e me Ze cosa ge sina =be ,
be
@ me wZ m ÿ i e e, @ Zi Le
ÿ w 1 ÿ Z2e @ Ze i , Le @ Zi
pal cross-section axes. 4. The auxiliary z axis is normal to the axes x 1e and Z and forms a right-hand triplet PrXY x 1e zZ: 5. The z1e axis forms with an auxiliary axis z an angle a counted o from z to z1e around the x 1e axis by the principle of the right-hand screw. 6. If the beam x 1e axis is parallel to the Z axis then z axis is parallel to the global X axis.
ÿ wi 1 ÿ m2e @ me , Le @ Yi
@ Ze wZ m ÿ i e e, @ Yi Le
ÿ ÿ @ g e ÿwi ÿ Ze g4e g2e m2e ÿ m2e sina @ Xi ÿ ge me cosa = Le b3e , ÿ ÿ @ m e ÿwi me ÿ ge Ze 2 ÿ Z2e sina ÿ me cosa = Le b3e , @ Xi ÿ @ Z e wi ge Z2e sina= Le be , @ Xi ÿ ÿ @ g e ÿwi ge ÿ me Ze 2 ÿ Z2e sina ge cosa = Le b3e , @ Yi
O. Sergeyev, Z. MroÂz / Computers and Structures 75 (2000) 167±185
ÿ ÿ @ m e ÿwi ÿ Ze m4e m2e g2e ÿ g2e sina @ Yi ge mcosa = Le b3e , ÿ @ Z e wi me Z2e sina= Le be , @ Yi @ g e ÿwi ge be sina=Le , @ Zi @ m e ÿwi me be sina=Le , @ Zi @ Z e ÿwi Ze be sina=Le @ Zi where ÿ1, node is at the beginning of a beam, wi 1, node is at the end of a beam If the beam axis x 1e is parallel to the global Z axis @ c then the derivative of transformation matrix h @ f Xe g i @ ce does not exist. An additional speci®cation h @ fX g i in this case can be performed by assuming that no discontinuity of the cross-section orientation in space can be observed for the case of small coordinate variation. This assumption leads to a jump-like variation of angle a depending on which coordinate of the derivative is considered. The nondierentiability is associated with an additional de®nition of the local coordinate system in a special case (cf. point 6 of rules for local coordinate systems).
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