Niels L. Pedersen* and Anders K. Nielsen Department of Mechanical Engineering, Solid Mechanics
Technical University of Denmark
Nils Koppels Allé, Building 404, DK–2800 Kgs. Lyngby, Denmark email: nlp@mek.dtu.dk
ABSTRACT
Summary We consider general 3D truss structures. The design variables are the cross–sections of the truss bars together with the joint coordinates and these design variables are considered to be continuos variables.
Using the design variables we will simultaneously do size optimization (areas) and shape optimization (joint positions). The structures are subjected to multiple load cases and the objective of the optimizations is minimum mass with constraints on eigenfrequencies (possible multiple), displacements and stresses.
1. Introduction
Optimization of trusses using numerical methods has been an active area of research since it was initiated by the work of [2] and of [3]. In the paper [5] it was shown that we have the possibility of simultaneously using the area of the cross section of the bar and the joint coordinates as the design variables. A large number of papers have been published in the area of truss optimization, for a re-view see [1] and references therein. There is however a very limited amount of papers dealing with the introduction of constraints on eigenfrequencies in connection with the optimization of truss structures, but from a practical point of view it seems reasonable to include this. The introduction of constraints on eigenfrequencies also have the possibility of removing the degenerated structures you might find in optimization. The main contribution of the present paper is the inclusion of constraints on eigenfrequencies, in a general setting of truss optimization. In the following sections we give an outline, for a more thorough explanation see [7].
2. Stress Constraints
If a bar is in tension we have an simple constraint specifying that the stress ( 0 ) must be smaller than a stress related to the yield stress ( y ).
y (1)
where is a safety factor. In compression we have a similar limit on the stress but we also want to avoid buckling in the slender members. To be able to make a more clear comparison between the optimized structures shown in the example and the actually used designs we will apply a stress constraint that is given by the Danish standards DS409,DS410 and DS412. In figure 1 we show the allowable compressive stress in bars for different profile types together with the allowable stress for the ideal truss.
d2
dh {}T(d[S]
dh 2d[M]
dh ){} (2)
Nm2
1·108 2·108 3·108
00 50 100 150 200
4·108
250 300
Open or close thick–wall welded profiles Cold formed I–profiles Annealed rolled I–profiles Ideal truss
Figure 1. Allowable compressive stress in bars as a function of slenderness ratio, , using DS409,410,412. (Steel S355, y 3.55 108Nm2 E 2.1 1011Nm2).
3 Sensitivities
When we are using SLP (Sequential Linear Programming). as the optimization method we need the sensitivities of the objective as well as of the constraints with respect to the design parameters.
Most of the sensitivities can be found straight forward, see e.g. [4]. In the following we will only describe the sensitivities of eigen frequencies. If we have a distinct eigenpair, (2, {}) , then the sensitivity with respect to a design parameter, h , is given by
where [M] is the mass matrix and [S] is the stiffness matrix, and we have assumed the normaliza-tion of the eigenvector . If we have a multiple eigen frequency we can not use (2). The derivanormaliza-tion below is primarily taken from [6]. We will assume that we have a double eigenfrequency with two corresponding eigenvectors, (2, {}1, {}2) . The derivations for higher order of multiplicity can be done in a similar way. We will again assume that the eigenvectors are normalized and that the two eigenvectors are orthogonal.
{}T1[M]{}1 1 {}T2[M]{}2 1 {}T1[M]{}2 0 (3) The problem is that any linear combination of the two eigenvectors will also be an eigenvector with the same eigenfrequency.
{_} c1{}1 c2{}2 (4)
c21 c22 1 {_}T[M]{_} 1 (5) The sensitivity is therefore not only related to the change in the design space, given by the change in design parameter h , but also by the choice of the eigenvector. Only for two specific eigenvectors will the sensitivities have physical meaning. By inserting (4) in the sensitivity (2) we get
d2
dh c21g11 c22g22 2c1c2g12 (6) gnm {}Tn(d[S]
dh 2d[M]
dh ){}m (7)
To find the extreme values of (6) we differentiate with respect to the two constants, c1, c2 , and set this equal to zero.
gg1112 gg1222cc1200 (8)We now solve the eigenvalue problem (8) and find the eigenpairs.
(ga, {c}a) , (gb, {c}b) (9)
The physical sensitivities of the double eigenfrequency is given directly by ga and gb , and the corresponding eigenvectors are given by (4) with the constants taken from the vectors {c}a and
{c}b .
4 Example
In this example we will compare an optimized result to a real life application. In the spring of 2001 the national football stadium of Denmark was fitted with a roof. The size of the roof is 140 m 95 m . The actual end beam used in the roof is shown in figure 2 and has a mass of 65 metric ton. The roof is supported at four points indicated by balls in figure 2.
3
1 2
Figure 2. Original end beam of the folding roof, mass 65 metric ton.
To be able to compare the designs from optimization with the design in figure 2 we will constrain the optimization so that it will fulfill the same safety as specified by Danish standard (DS409–412).
According to the standards we have six loads:
Snow load, {Fs} (900 Nm2) self–weight, {Fg}
Lift due to wind, {Fwl} (wind speed 24 ms) Downward force due to wind, {Fwd}
Horizontal wind load, {Fw} Horizontal force, {Fc} (3500 Nm)
From these loads we get three different load cases:
Load case 1 : {Fg} {Fc} 1.5{Fs} 0.5{Fwd} 0.5{Fw} Load case 2 : {Fg} {Fc} 0.5{Fs} 0.5{Fwd} 1.5{Fw} Load case 3 : {Fg} {Fc} 1.5{Fwl} 0.5{Fw}
The constraints on the eigenfrequency is that the first eigenfrequency should be greater than 1 Hz . At the same time we will apply the displacement constraint that the maximum deflection is 1/200
of the total span of the beam. Figure 3 shows the optimized structure, it should be noted that nodes which are loaded or constrained are not allowed to change position. From figure 3 we see that the design has changed compared to figure 2. The original beam had a maximum height of 6.75 m this is in the optimized design reduced to 5.85 m . At the same time the width of the lower flange is reduced from 3.40 m to 1.66 m . The objective (mass) of the optimization was reduced to 41.2 metric ton. For this example the constraints on displacement and eigenfrequency is not active so the design is fully controlled by the stress constraints.
3
1 2
Figure 3. Optimized design of end beam, mass 41.2 metric ton.
Acknowledgments. The work of Niels L. Pedersen was supported by the Danish Technical Re-search Council through the THOR–programme (Technology for Highly Oriented ReRe-search): “Sys-tematic design of MEMS”. This support is highly appreciated.
REFERENCES
[1] Bendsøe, M. P., Ben–Tal, A. and Zowe, J. (1994) “Optimization methods for truss geometry and topology design”. Structural Optimization 7, 141–159
[2] Dorn, W., Gomery, R. and Greenberg, M. (1964) “Automatic design of optimal structures”. J.
de Mecanique 3, 25–52
[3] Fleron, P. (1964) “The minimum weight of trusses”. Bygningsstatiske Meddelser 35, 81–96 [4] Nielsen, A. K. (2001) “Tredimensionale gitterkonstruktioner: optimering for minimum mass
med restriktioner på dynamiske og statiske egenskaber”. Master thesis, Department of Mechan-ical Engineering, Solid Mechanics, TechnMechan-ical University of Denmark, (In Danish)
[5] Pedersen, P. (1970) “On the minimum mass layout of trusses”. AGARD–CP–36–70
[6] Pedersen, P. (1994) “Non–Conservative Dynamic Systems: Analysis, Sensitivity Analysis, Optimization”. Lecture notes from IUTAM fifth summer school on mechanics, Concurrent En-gineering Tools for Dynamic Analysis and Optimization, Aalborg University.
[7] Pedersen, N.L. and Nielsen, A.K. (2001) “Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses and buckling”. (submitted)
On sensitivity computations of geometrically and materially non-linear structural response
Sami Pajunen
Department of Mechanical Engineering Tampere University of Technology, Tampere, Finland
e−mail: sami@mohr.me.tut.fi
ABSTRACT
Summary In this study we consider the design sensitivity analysis of structures exhibiting large deflections. Based on governing FEM equilibrium equations and direct differentiation method, we derive both total and incremental sensitivity formulas for elastic and elasto-plastic responses, respectively.
Introduction
The design sensitivity analysis is an essential part of a general optimisation problem. A well-posed, standard-form optimisation problem can be solved using a suitable mathematical minimisation algorithm, whereas the data regarding the physical character of the problem is brought into the computation via requested function values and their derivatives with respect to design variables. In a general structural optimisation problem, the finite element method serves as a standard tool for evaluating the objective and constraint function values and their design derivatives.
In this study we concentrate on design sensitivity analysis of quasi-static structural systems that are discretised using the finite element method. We consider the displacement sensitivity both for large deflection problems as well as rate-independent elasto-plasticity. However, we restrict the discussion on problems that exhibit monotonically increasing unique load-deflection paths, i.e. we handle only structures that are not prone to buckling. As concluded in many research papers, see e.g. [1,2], the direct differentiation method (DDM) is superior to the adjoint system method, especially in elasto-plastic problems. Indeed, throughout this study also we use the DDM.
Displacement sensitivity
During the minimisation process, the design derivative dq/dx of nodal displacements q with respect to design variables x at the final load level λk+1pref, say, is needed. Here pref is the reference load vector, λ is the external load factor and the subscript denotes increment number.
In the so called total sensitivity approach, the governing equilibrium equation
r q( )−λpref =0 (1)
is differentiated with respect to design variable vector x leading, after standard manipulation, to
K q
x r
T x d d = − ∂
∂ . (2)
In above r is the internal nodal force vector and KT is the tangent stiffness matrix.
After regular path continuation up to desired load level λk+1pref, the design sensitivity can be obtained from (2) just by formulating the right-hand side of (2) for the used element type and adopting an already manipulated stiffness matrix KT. Thus, whenever the total sensitivity approach gives appropriate results, it should be adopted because of its low computational costs.
Because the total sensitivity approach is based on differentiation of the equilibrium equation at fixed but arbitrary state, the obtained sensitivity equation (2) can not give accurate results if the nodal displacements are somehow history-dependent. For example, in elasto-plastic problems (2) must not be used.
Opposite to the total sensitivity scheme, path-dependent problems can be handled using an accumulated sensitivity approach that is incremental in nature. To begin with, we first write the equilibrium equation (1) in an incremental form as
∆rk +1=λk+1pref−rk, (3)
in which the right-hand side terms contain fixed values of external and internal force vectors at equilibrium states k and k+1 and the left-hand side term is due to the iteration process from equilibrium state k to k+1. Differentiation of (3) with respect to design variable vector x leads us, after some computations, to
K q
Using (4) it is straightforward to compute the required design derivative at equilibrium state k+1 as
In (4) Kg is the geometric stiffness matrix , B is the kinematic matrix and σ is the vector of stresses. Compared to total sensitivity formula (2), one additional stress sensitivity term dσ/dx is required in the incremental approach (4). Indeed, this single term contains all the information concerning the adopted constitutive model. Computation of this term during the elasto-plastic analysis can be briefly described as follows.
Stress sensitivity Up to equilibrium state k the stress is updated as
σk =σk 1− +∆σk, (6)
in which the increment ∆σσk is accumulated during the increment from equilibrium state k to k+1. Differentiation of (6) with respect to design variable vector x gives
d
in which the latter derivative is dependent on the used elasto-plastic stress updating algorithm.
In the case of explicit time integration algorithm, the latter derivative can be written, after some manipulations, as