OPTIMAL DESIGN OF PLANAR FRAMES BASED

ON STABILITY CRITERION

By S. Pezeshk,' Associate Member, ASCE, and K. D. Hjelmstad,2

Member, ASCE

ABSTRACT: This paper suggests an optimization-based design methodology for improving the strength and overall stability of framed structures, the capacities of which are governed by inelastic limit-load behavior. The optimization objective function, comprising the dominant linearized buckling eigenvalue of the structure weighted by a frequency-dependent penalty function, is motivated by a simple model of nonlinear frame behavior. Designs are constrained to have constant weight. The method requires only the linearized buckling eigenvalues and eigenvectors of the structure, avoiding computationally intensive nonlinear structural analyses in the design cycle. An iterative optimality-criteria method is used to solve the op­timization problem. Several examples are given to examine the performance of the procedure, both in terms of robustness of the numerical algorithm and the quality of the designs it produces. By way of example, it is shown that by improving the overall stability characteristics of a structure under static loading, the dynamic per­formance of the structure is often improved.

INTRODUCTION

This paper is about stability, optimization, and the limit design of framed structures. One of the earliest efforts at formal structural optimization was by Lagrange and, coincidentally, concerned the stability of an elastic col­umn. Over two centuries later, the purview of structural optimization has widened considerably, but optimization to enhance elastic structural stability continues to be an active area of research. Unfortunately, many of the results in elastic stability are only of academic interest for building frames because the capacities of these structures are generally limited by inelastic instability.

In the 1950s advances in the theory of plastic structures led to the de­velopment of limit-design methods for minimum-weight structures. Some of the fundamental contributions in this area are attributed to Foulkes, Heyman, and others (Neal 1981). The idealization of structural behavior as rigid plas­tic allows the problem of finding the minimum-weight structure to be for­mulated as a linear programming problem that can then be subjected to ef­ficient and systematic solution methods. These methods are amenable to large-scale computation, but they have been slow to make an impact on practical design. Classical limit-design methods are based on a geometrically linear description of the structure that obviates accounting for stability in the design process. As a consequence, slender structures can be grossly underdesigned.

Modern framed structures are more slender than their forerunners, but they still tend to suffer inelastic deformation prior to achieving a limit load. While limit design has proven to be a pleasingly rational approach to proportioning

'Asst. Prof., Memphis State Univ., Dept. of Civ. Engrg., Memphis, TN 38152. 2Assoc. Prof., Univ. of Illinois at Urbana-Champaign, 205 North Mathews, Ur-

bana, IL 61801. Note. Discussion open until August 1, 1991. Separate discussions should be sub­

mitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 16, 1990. This paper is part of the Journal of Structural Engineering, Vol. 117, No. 3, March, 1991. ©ASCE, ISSN 0733-9445/91/0003-0896/$1.00 + $.15 per page. Paper No. 25630.

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structures, stability has become more important. There is a clear need for design procedures that account for both of these behavioral aspects. The estimation of the limit load of a complex inelastic structure is well within current computational technology. Direct maximization of the limit load of a structure is possible, but remains a prohibitively onerous task. In addition, the structural configuration with the maximal limit load might be fragile in the sense that the postlimit loss of carrying capacity may be abrupt.

In this paper we present an algorithm for improving the limit and postlimit response of nonlinear framed structures. The algorithm avoids nonlinear analysis in the optimization cycle and seeks to maximize the linearized buck­ling eigenvalues of the structure while holding the weight constant. We dem­onstrate how accomplishing this objective improves the inelastic static, and often dynamic, response of the structure. The approach is indirect, but ap­parently effective and efficient.

The principal contributions of the present work are: (1) The recognition of the connection between the linearized buckling problem and the nonlinear performance of framed structures; and (2) the exploitation of this connection in the development of a systematic approach to enhancing structural stability characteristics. Weighting of the objectives with a frequency-dependent function to account for the vibrational characteristics of the structure also appears to be novel. In what follows we present a brief motivation for the proposed approach. We then document a method of computation (based on existing optimality criteria techniques) and finally we illustrate the method through two design examples.

MOTIVATION

A plausible design-objective function for improving the limit and postlimit behavior of structures can be motivated by simple observations of the non­linear behavior of planar frames. Hjelmstad and Pezeshk (1988) presented an approximate model that demonstrated the effect of geometric nonlinear-ities on the performance of framed structures through a relation that gives the fully nonlinear response of a structure in terms of its geometrically linear response. We take this model as our point of departure. A brief sketch of the approximate model is recounted in the following to justify our subsequent choice of an objective function for optimization.

Consider a structure subjected to a combination of proportional (XRo) and nonproportional loads (R,), where X is the proportionality factor of the pro­portional loads. Further, let us introduce an associated buckling eigenvalue problem

K<|> = u.G<|> (1)

where K = the linear structure stiffness matrix; {jx,<}»} = the fundamental eigenpair; and G = the geometric stiffness matrix defined in terms of the two loading cases as

G = Go + - G, (2)

where G0 = the linearized geometric stiffness matrix for the proportional loads with A. = 1; and G, = the linearized geometric stiffness matrix for the

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nonproportional loads. Assume that the eigenvector is normalized, such that <|>'Gc|) = 1, and let a = u'G<|> be a parameter that measures the magnitude of the displacement vector u. If the fundamental mode dominates the non­linear response, the nonlinear load factor can be expressed in terms of the geometrically linear load factor, \L, as

M°0 = (3) a0(Ji. + ory0

where a0 = udG<|> is the value of a for the displacements under proportional loads only with \ = 1; -y0 = <J>'G0<|>; and 71 = <J>'Gi<|>. From Eq. 2 and the previously mentioned definitions, we note that 70 + "yi/V = 1- Consequently Eq. 3 takes the simplified form

H&) = ¥ " > - * * - .0)

where a = a /a 0 is a normalized displacement measure. From its definition one can observe that 70 is a number in the range 0 -1 . The case 70 —* 0 indicates greater relative importance of the dead loading to the eigenvalue problem, while the case 70 —> 1 indicates greater relative importance of the proportional loading.

With a few modest assumptions, the limit load can be approximated by a Rankine-type estimate as

* . — ^ (5 )

^>^

where \p = the geometrically linear plastic capacity of the structure. In the postlimit regime, the geometrically linear capacity )<L(a) is generally

constant or nearly constant. Eq. 4 suggests that the slope of the postlimit response curve in the neighborhood of the limit load is —(1 — 70 + 7oW |x). Thus, the larger is (A, the smaller will be the postlimit loss of carrying capacity. It is also evident from Eq. 5 that the larger is |A, the larger will be the limit capacity of the structure. In each case, it is apparent from this simple model that maximizing the buckling eigenvalue will lead to more robust structure, as far as overall stability is concerned.

FORMULATION

We propose maximizing the dominant buckling eigenvalue as a means of improving the stability of the structure. As is shown by Hjelmstad and Pe-zeshk (1988), the buckling mode that dominates the response of the structure is not known a priori. For planar moment resisting frames it is likely that the fundamental mode dominates the response, but for braced and three-dimensional frames, the first mode might not dominate. One can easily dem­onstrate that the argument of the previous section holds for any eigenpair {|x,<|>}. In the following developments we assume that the buckling eigen­value that appears in the formulation is the one corresponding to the mode

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that dominates the inelastic buckling response. Even though the dominant mode is generally not known in advance, an a posteriori analysis can de­termine the dominant mode and the procedure of optimization can be re­peated if necessary. The alternative of maximizing a weighted sum of ei­genvalues is treated in the companion paper (Hjelmstad and Pezeshk 1990).

To account for dynamic effects, we penalize the buckling eigenvalue ob­jective with a function that reflects the dynamic characteristics of the struc­ture. The dynamic weighting function will be expressed as squares of the natural frequencies of vibration, w(x) = [a)1(x),(o2(x) <»N(X)]. Further­more, we constrain the problem by looking at designs having a given weight, W0.

Assume that the members of the structure are grouped in such a way that all members of the same group have the same design variable. Let x = (xux2,.. -,xM) be the vector of design variables, where M is the number of groups. For the sake of the present discussion we will assume that there is a single design variable for each member, either the area or the moment of inertia. We wish to maximize the following objective, expressed in terms of the buckling and vibration eigenvalues:

max (A(x)p[w(x)] (6)

where u, = the dominant linearized buckling eigenvalue, and p(ta) = the dynamic penalty function. The optimum is subject to the constraints M

2 A,(JCi)w, = W0, x, < xt < xi (7) ;=i

where x —» A,(x) = a function that gives the area of member i from the associated design variable x. The rth design variable, JC, has a minimum per­missible value X), and a maximum permissible value of xt. The specific mass, w„ of group i is defined as the sum over all elements in group i of the product of length, L„, and mass density, p„:

wt = 2 p*L» (8>

A Lagrangian functional can be constructed from the objective and equal­ity constraint as

/(*,© = (x(x)p[w(x)] - i M

2 AM*, - W0 (9)

where £ = the Lagrange multiplier. The constraints on the extreme values of the design variables will be treated with an active-set strategy; to wit, whenever a design variable violates an extreme size constraint, that design variable is set to its limiting value and removed from the active set. A more detailed discussion of the active-set strategy is given later.

Taking the derivative of the Lagrangian with respect to the rth design vari­able, xi, and setting the corresponding equation to zero results in the first order necessary conditions for an optimum:

— = — p(w) + |J, 2J ~ 1 &4.r(*f)w, = 0 (10) dXj dXi k=i oco t dXj

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1 p(m)

(a)

FIG. 1. Possible Frequency Weighting Functions: (a) Sinusoidal Response Spectrum; and (b) Mexico's Federal District Code

Rearranging Eq. 10 gives the following optimality criteria:

C,(x)

&,(x)

where

1, i = 1, .,M

du, v-i dp dwk

G,(x) - fp(o>) + u. £ - ^ —

is the fth component of the gradient of the objective function and

q,(x) = A,'(JC,)W,

(11)

(12)

(13)

is the rth component of the gradient of the equality constraint. The weighting function /?(«) introduces information about the vibrational

characteristics of the structure. These functions can be used to avoid un­desirable dynamic effects such as resonance by pushing the structure away from designs with those dynamic properties. The choice of weighting func­tion depends on the specific application and can depend explicitly on the entire frequency spectrum, only a selected part of it, or may be completely independent of it. One possible choice would be the frequency response function H(w), which gives the ratio of input to output in the frequency domain. Examples of some possible weighting functions are given in Fig. 1. If the inverse of the frequency weighting function has a positive slope, then the design is pushed toward having shorter period, whereas when the slope of the weighting function is negative, then the design is pushed toward having a longer period. Weighting functions with steeper slopes result in a greater encouragement to change the frequency of the design. If the weight­ing function is flat, then there is no encouragement for design to change its frequency content.

Sensitivity Analysis To evaluate the optimality criterion given in Eq. 11, one needs to deter­

mine the sensitivities of the buckling loads and vibration frequencies of the structure with respect to the design variables. To determine the sensitivity of the buckling eigenvalues with respect to the design variable JC, consider the following eigenvalue problem:

Kfy = tyG+j (14)

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where fy and JUL = the yth eigenvector and eigenvalue respectively; K = the elastic stiffness matrix; and G = the geometric stiffness matrix. Differen­tiating Eq. 14 with respect to design variable xt and premultiplying by <|>j yields the expression for the eigenvalue sensitivity. The gradient of the buck­ling eigenvalue is given by the expression

3 ^ = <j>j[K,, + MyG,,)^

dxt tyGfyj

where ( )„• = d( )/dxt. The sensitivities of the frequencies with respect to the design variables x can be determined by applying the same procedure to the frequency eigenvalue problem K.tyj = co/Mi . Here *!»,• and to, represent the jth mode shape and (square of) frequency, respectively, and M repre­sents the mass matrix of the structure. The resulting sensitivity is

da>j i|»j(K„- + WjM„>|ij — = ( lo ) dXi ^jMljl;

A method for computing the derivative of the geometric stiffness matrix G„ is given by Pezeshk and Hjelmstad (1989). However, this term is nearly zero for gravity-loaded indeterminate framed structures and is identically zero for a statically determinate structure since the internal force distribution does not depend upon the relative rigidities of the members. Consequently, the term involving the gradient of the geometric stiffness is generally ignored in practical computations. Along these same lines, the mass distribution of a structure is often assumed to be independent of the design variables for framed structures since most of the mass is associated with nonstructural elements. Thus, the term involving the gradient of the mass matrix is also ignored in practical computations.

The linear elastic structural stiffness, K(x), exhibits the following explicit form in terms of cross-sectional properties

K(X) = 2 Mxd E K» + «*i) 2 K™ (17)

where K^ = the stiffness kernel of element m, linear in the cross-sectional area (i.e., the axial and shear components); and K'„ = the stiffness kernel of element m, linear in the moment of inertia (i.e., the bending part). The function x —> I,{x) gives the moment of inertia of the rth element cross section in terms of the design variable. The gradient of the structural stiffness ma­trix, with respect to design variable xh is then easily computed as:

Y = A'(Xi) 2 K£ + /'(*,) 2 K™ <18>

where x —> A,'(x) gives the rate of change of the area of element;' with respect to x; and x —> I',(x) gives the rate of change of moment of inertia of element / with respect to x.

Choice of Design Variables We confine our attention to selection of members from I-type steel sec­

tions. Ideally, these members would be selected from the collection of avail-

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able wide-flange rolled steel shapes, a set that is generally discrete. Efforts have been made to formulate algorithms to optimize on the discrete set (e.g., Liebman et al. 1981). It has also been common to attempt to cover the discrete set with a set of continuous functions (e.g., Brown and Ang 1966; Walker 1977) because optimization on a continuous domain is far more ef­ficient than optimization on a discrete domain. One can also increase the number of design variables used to describe a cross section (e.g., take height, depth, web thickness, and flange thickness as independent design variables) and evolve a design which could be fabricated from plates (Pezeshk and Hjelmstad 1989).

The best method of representing the design space is an issue far from being settled, but it is also tangential to the present discussion. For sim­plicity, we adopt a continuous functional dependence between cross-sec­tional area and moment of inertia and use one design variable for each group. The relationship adopted here is a slight modification of the empirical re­lations proposed by Walker (1977) for economy wide-flange steel sections:

5.87 + 0.0224/ columns: D = 0.00042/ + — (19a)

1 + (0.425 + 0.00162/)1/7

A = — (19b)

0.1521D208

girders: D = 2.660/° 287 (19c)

A = 0.61124/04719 (19rf) where D = the section depth in inches (1 in. = 25.4 mm) and A = the area in square inches. In the computations presented next the moment of inertia was taken as the single design variable for each group.

SOLUTION PROCEDURE

The optimum structure must satisfy the optimality criteria and the weight constraint. Since these equations are nonlinear, they must be solved by an iterative scheme. The iterative algorithm suggested here consists of using a set of recurrence relationships based on the optimality criteria. Repeated use of the recurrence relation in conjunction with a scaling procedure will move the initial design toward a configuration that satisfies the optimality criteria and the constraints. The design cycle consists of three parts: (1) Estimation of the Lagrange multiplier of the equality constraint; (2) updating the design vector in accord with the optimality conditions; and (3) scaling the design to maintain feasibility. The coefficients Q, and q, can be evaluated at any configuration with the information obtained from solving the linearized buckling and vibration eigenvalue problems and the associated sensitivities with re­spect to the design variables. The Lagrange multiplier is determined from the condition that the design must satisfy the linearized weight constraint. With £ and Q, known, the design variables can be updated with a recurrence formula. Finally, the design is scaled linearly to exactly satisfy the weight constraint. A brief overview of these steps is presented in the following sec­tions.

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Recurrence Relations Various forms of recurrence relations have been developed and used to

update the configuration in an optimization problem. Berke (1970) used a recurrence relation in a virtual strain energy formulation for minimizing weight with prescribed displacement constraints. The same recurrence relation was effectively used by Gellaty and Berke (1971) for design problems with stress and displacement constraints. Later, Venkayya et al. (1973) and Khot et al. (1973, 1976) derived different forms of the recurrence relations for displace­ment constraints, stress constraints, and dynamic stiffness requirements.

The recursive approach has the advantage that it eliminates the need for the Hessian of the Lagrangian functional required in some nonlinear pro­gramming algorithms. The Hessian is often expensive to compute or may not be known at all. One form of recursion is the exponential relation

GXxK)l' /r

i=\,...,M (20) *r+ 1 = *: .6W)J where K = the iteration counter; and r = a parameter that determines the magnitude of the adjustment in the design variable. The central idea of the recurrence relationship is that the ratio of Q, to £#, is unity at the optimum, and a deviation from unity indicates a need for adjustment in the design variables. If the ratio is less than one, the associated design variable is dom­inating and needs to be reduced. If the ratio is less than one, the associated design variable is dominating and needs to be reduced. If the ratio is greater than one, the associated design variable needs to be increased. The recur­rence relation treats the design variables as uncoupled when, in fact, a change in design variable i will change the value of Qj. The parameter r > 1 keeps the adjustment factors closer to unity to stabilize the iteration. At the opti­mum, the design variables will be unchanged by Eq. 20. A parameter study on the magnitude of the step length and its effect on the convergence of the algorithm can be found in the work of No and Aguinagalde (1987).

Near the optimum, the term in brackets in Eq. 20 will be near unity. Linearizing the exponential recursion about g,/(i<7, = 1 gives

xr-xf\l+-\^--l]}, i=l,...,M (21) I r Lr?,(xK) J J

This equation is referred to as the linear recurrence relation for the design variables and will be used to help estimate the Lagrange multiplier.

Estimation of Lagrange Multiplier The Lagrange multiplier £ must be determined in order to use the recur­

rence relation Eq. 21 to update the configuration. An equation to determine the Lagrange multiplier can be obtained by linearizing the constraint about the current iterate. We express the weight constraint in the form

M

c « = 2 A'(x1)wl - W0 = 0 (22)

Linearizing C(x) about the current iterate xK one obtains: M

L(C)X^ = C(xK) + ^ C*i - <)A,'(x,)w, (23)

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At the current iterate, C(xK) 5 0, since the design variables are updated by the recurrence relationship after the Lagrange multiplier is estimated. Thus, the linearized constraint becomes:

M M

2 Ai(xf)wi - Wo + 2 (*i - *T)Ai'W)w, = 0 (24) 1=1 i=\

We estimate the Lagrange multiplier by satisfying the linearized constraint at the new iterate xK+1. Substituting x = xK+1 into Eq. 24, estimating xK+1

from Eq. 21, and solving for the Lagrange multiplier we get M

J 42,<xKK(*,K) -

€ = 15 TH (25)

Since the constant weight constraint is an equality, £ can be either positive or negative.

Active-Set Constraint Strategy After each iteration a set of new design variables is obtained. If a design

variable lies within its permissible range, it is placed in the active set, oth­erwise it is placed in the passive set so that a proper scaling can be performed before the next iteration. At the start of each iteration, formerly passive vari­ables can either remain in the passive set or be reactivated. In general, it is not known a prior if a variable will be active at the optimum. Allwood and Chung (1984) have suggested that if a design variable is moved to the pas­sive set in two consecutive iterations, it will probably be passive at the op­timum. In principle, the method suggest by Allwood and Chung was fol­lowed in the computations reported here. However, it was found that in the early stages of optimization, the iterations can be erratic and it is profitable to return all variables to the active set at each iteration until the algorithm settles down.

Scaling Procedure Since the volume constraint is not exactly enforced at each iteration, it is

necessary to scale the design variables to keep the design feasible. In order to satisfy the weight constraint after each iteration, each active design vari­able is scaled such that

2 A,(S*,)w, + ^ Afadw, = W0 (26)

where /„ = the set of indexes in the active set; and Ip = the set of indexes in the passive set. Eq. 26 is a nonlinear scalar equation in the variable £ if A(x) is a nonlinear function. Scaling can be accomplished using, for ex­ample, a Newton iteration. If A(x) is linear in x, as it would be if the element areas are used as the design variables, iteration is necessary only if a variable becomes passive as a consequence of scaling.

The remainder of the paper is devoted to application of the foregoing method to two framed structures. The examples will demonstrate the performance

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of the design procedure. Presentation of the examples includes a discussion of the initial design method, a description of the optimization process, and subsequent static and dynamic nonlinear analyses of the initial and the op­timized designs.

THREE-STORY FRAME

The first example is a three-story frame with topology as given in Fig. 2. This frame was designed to meet the Uniform Building Code (UBC) (1979) specification. The loads on the structure included: (1) Dead load—80 psf (3.83 kPa); (2) live load—40 psf (1.91 kPa) for a typical floor, 20 psf (0.96 kPa) for the roof; and (3) exterior walls—50 lb/ft (0.73 kN/m). Lateral loads and their distribution were computed for zone 4 following UBC rec­ommendations.

A preliminary design was performed using full dead and live load to de­termine maximum moments in girder sections. Following UBC recommen­dations the base shear was determined as V = ZIKCSW = (1)(1)(0.67)(0.78) (139.2) = 72.74k (323 kN), in which Z = 1 (high seismicity zone), / = 1 (importance factor), and K = 0.67 (moment resisting frame). The weight of the structure was 139.2k (619 kN) and the base shear coefficient was es­tablished from a response spectrum for sinusoidal response and had the value CS = 0.78. The lateral forces were distributed according to the UBC and are shown in Fig. 2.

For the combination of dead plus live plus earthquake loads (D + L + Q), the allowable increase in working stress of 33% was followed (UBC 1979). Analysis of the preliminary design were done for load combinations {D + L) and 0.75 (D + L + Q).

The American Institute of Steel Construction (AISC) (1980) specification was used for the design requirements on steel sections. Yield stress was taken to be 36 ksi (248 MPa) for all sections. The sections were assumed fully braced against lateral buckling giving an allowable bending stress of Fb = 24 ksi (165 MPa) for the girders. The columns were assumed to be braced in the out-of-plane direction. After analyzing the structure and check­ing design requirements, W24 x 55 sections were chosen for girders and W12 X 96 for columns for the initial design.

The UBC design was used as the initial design for the optimization. The initial design was optimized by maximizing the first buckling eigenvalue of

216 in

120 in

120 in

144 in

t 0.985 k-sec2lft

° 1.124 k-sec2lft

° 1.152 k-sec2/ft

- r- Vn = 25.4mm

33.8 k

- > • 25.0 k

14.0 k

\k = 4.448/fcAT

FIG. 2. Three-Story Frame Topology, Mass Distribution, and Code Lateral Force Distribution

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TABLE 1. Properties of Optimized Designs without Frequency Penalty

Member

(D 1st story columns 2nd story columns 3rd story columns 1st story girders 2nd story girders 3rd story girders

Stiffness Properties

A (in.2) (2)

38.2 21.6 12.3 23.9 22.9 13.4

/ (in.4) (3)

1,500 807 385

2,360 2,163

698

Yield Properties

AUk) (4)

1,376 777 443 859 825 484

Vo(k) (5)

795 449 256 496 476 279

M0 (in.-k) (6)

7,484 4,122 2,143 6,875 6,462 2,884

Note: 1 in. = 25.4 mm; 1 k = 4.448 kN.

the structure under combined lateral and dead loads with no frequency pen­alty. Six design variables were used: one for each column and girder in each of the three stories. The optimization converged in 35 iterations resulting in an optimized design with the properties given in Table 1, which lists the cross-sectional area (A), the bending moment of inertia (/), the axial capacity (N0), the shear capacity (V0), and the flexural capacity of (M0). The fun­damental buckling eigenvalue of the design increased from 98.7 to 141.5 and the vibrational frequency decreased from 0.538 seconds to 0.444 sec­onds for the optimized design. The optimized design will be referred to as opt in future comparisons.

Response to Static Loads To investigate the performance of the optimized design, both the initial

design and the optimized designs were analyzed under statically applied lat­eral and dead loadings. Nonlinear analyses were performed to evaluate the overall stability of the initial and the optimized designs. [Note: The approach to analyzing the planar frames considered in this paper is that proposed by Simo et al. (1984). The finite element discretization of the frames analyzed throughout this paper consists of two elements between each structural joint. Quadratic interpolation was employed for all the elements.] The results of the analyses are given in Fig. 3. One can observe that the load-carrying capacity of the optimized structure increased by 44% without any increase in the rate of postlimit load degradation. The analysis indicates that the op­timized design is better than the initial design as far as overall strength and stability are concerned.

Response to Dynamic Loads To examine the performance of the structures under dynamically applied

loads, both the initial and the optimized designs were analyzed and compared under sinusoidal base acceleration and earthquake base acceleration. [Note: A Newmark constant-average-acceleration method ((J = 0.25) was used for dynamic analysis. Rayleigh damping was assumed, with damping ratios specified for the first two modes.]

The performance of both the initial and the optimized designs under si­nusoidal base acceleration is given in Fig. 4. To make the comparison more fair, the sinusoidal base acceleration was applied in the two cases with the

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1 h

\s if 1

1 , ! .., ! ,

___ s- opt Design •

optf Design ~ ~ :

Initial Design

\in = 25.4mm

I , i , i

10 20 30 TOP DISPLACEMENT (inches)

40

FIG. 3. Static Analyses of Initial and Optimized Designs with and without Fre­quency Penalty

same frequency but with an amplitude that gave the initial and optimized designs the same elastic dynamic magnification factor. A damping ratio of 3% critical was used for the first and the second modes. The base shears obtained from time-history analysis and plotted against the lateral top dis­placement of the frames in Fig. 4. One can see that the initial design tends to drift cyclically whereas the optimized design does not, indicating that the latter is more robust.

Optimization with Frequency Weighting Function To demonstrate how the dynamic weighting function influences the de­

sign, the three-story frame was optimized again with a frequency weighting function. Supposing that the structure was to be designed to resist a sinu­soidal loading of frequency w, we took the frequency weighting function to

-100

-200

I

(a)

i i

i i

/

i

—1

i

1

-

i

- 20 0 20 40 TOP DISPLACEMENT (Inches)

(b)

li'n = 25.4mm

U = 4.448WV

- 20 0 20 40 TOP DISPLACEMENT (Inches)

60

FIG. 4. Dynamic Response of Three-Story Frame under Sinusoidal Base Accel­eration: (a) Initial Design, Amplitude of Sinusoidal Base Acceleration; A„,al = 500; and (b) Optimized Design, Amplitude of Sinusoidal Base Acceleration /!„,„ = 228

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TABLE 2. Properties of Optimized Design with Frequency Penalty

Member 0)

1st story columns 2nd story columns 3rd story columns 1st story girders 2nd story girders 3rd story girders

Stiffness Properties

A (in.2) (2)

39.4 31.9 12.4 23.6 15.8 7.7

/ (in.4) (3)

1,550 1,230

389 2,311

992 217

Yield Properties

ATo(k) (4)

1,282 715 465

1,049 863 427

Vo(k) (5)

740 413 268 606 498 246

M0 (in.-k) (6)

6,947 3,774 2,099 9,295 6,918 2,390

Note: 1 in. = 25.4 mm; 1 k = 4.448 kN.

be the inverse of the sinusoidal response spectrum

p(co) = [(1 - (32)2 + (2(3g)2]1/2 (27)

where (3 = w/w. The properties of the resulting optimized design (optf) are given in Table 2. The first buckling eigenvalue of optf is 156 with the fun­damental period of 0.48 seconds. Fig. 5 shows where the fundamental pe­riods of the various designs are relative to each other. One can observe that the period of optf was longer than opt because the designs are on the down­hill portion of the weighting function. The structure optf is a compromise between the initial design and the optimized design opt.

The structure optf was analyzed under static loading with the results as shown in Fig. 3. The performance of opt is similar to optf, and both have higher load-carrying capacities than the initial design. The postlimit slopes are almost the same as the initial design.

The initial design and the two optimized designs were analyzed under the

en a h-

<

z < O "S. < z

Optimized Design Without Frequency Penalty

Optimized Design With Frequency

Penalty

1 2 PERIOD RATIO

FIG. 5. Sinusoidal Response Spectrum

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h-

/ I I I

? Ld C)

s n in Q

n o 1—

w

¥ o 1 -

s—-

1?

H

n

- 6

- 1 2

Initial Design opt Design optf Design

lm = 25.4mm

4 5 6 TIME (sec)

10

FIG. 6. Three-Story Frame under Pacoima Dam 1971 Earthquake Base Acceler­ation

1971 Pacoima Dam earthquake record. The response history of both the ini­tial and the optimized designs are given in Fig. 6. Again, one can see that the initial design drifts a great deal. The optimized design has controlled drift even though the base shears are higher. Although no failure is predicted for the initial design under Pacoima Dam earthquake, the optimized design is more desirable because of its ability to control drift. Observe that opt behaves like optf and both control drift better than the initial design.

EIGHT-STORY FRAME

The eight-story frame shown in Fig. 7 is a modified version of a design given by Korn and Galambos (1968). The properties of the structure are given in Table 3. This design was checked with UBC lateral load provisions (1979) and the AISC specification (1978). All the requirements were sat­isfied. The loading on the structure consisted of dead loads of 0.25 k/in (44 kN/m) for roof level and 0.30 k/in (52.5 kN/m) for typical floor levels. The lateral force distribution on the structure was obtained following the UBC lateral load provisions. The calculated lateral force distribution along

120 In

o to

o CM

CO

FIG. 7. Eight-Story Frame Topology

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TABLE 3. Properties of Eight-Story Frame

Story

(D 1 2 3 4 5 6 7 8

Column

Initial (2)

W14 x 99 W14 x 99 W14 x 90 W14 X 90 W12 X 79 W12 X 79 W10 X 49 W10 x 49

Optimized (3)

W14 x 99 W14 x 99 W14 x 74 W14 x 74 W14 x 48 W14 x 48 W12 x 35 W12 x 35

Girder

Initial (4)

W14 x 38 W14 x 38 W14 x 38 W14 x 38 W14 x 30 W14 x 30 W12 x 26 W12 x 26

Optimized (5)

W24 x 76 W24 x 76 W24 x 76 W24 x 76 W21 x 57 W21 x 57 W12 x 35 W12 x 35

Mass (k-sec2/in.)

(6)

0.056 0.056 0.056 0.056 0.056 0.056 0.056 0.056

F, (k) (7)

0.315 0.629 0.943 1.258 1.572 1.888 2.200 2.984

Note: 1 in. = 25.4 mm; 1 k = 4.448 kN.

with the story mass of the structure is given in Table 3. The eight-story frame was optimized by maximizing the first buckling ei­

genvalue of the structure under dead loading only, keeping the weight of the structure constant. There were 16 design variables: eight representing the moments of inertia of the columns and eight representing the moments of inertia of the girders. The properties of the optimized design area also given in Table 3.

Response to Static Loads The result of the static analyses are given in Fig. 8. Observe that the load-

carrying capacity of the optimized structure increased from a load factor of 5.0 for the initial design to 9.8 for the optimized design (an increase of about 100%) without any increase in the rate of postlimit load degradation. The

12

9 QL O I— O

£ 6 Q <

0 20 40 60 80 100 120 TOP DISPLACEMENT (inches)

FIG. 8. Static Response of Eight-Story Frame

910

1 ' I ' I r

Optimized Design

—— Initial Design

\in = 25.4mm

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150

100

05 5 0

x " i 0 u)

3 - 5 0

-100

-150

I

(a)

i

^ P

V t i ft U

i

ft

1

-

-

-

1

j ^ I I I

nj\] ft)

(in li/j = 25.4mm ~ HI Xh = 4.448/UV

1 1 1 1 1

-50 0 50 100 150 TOP DISPLACEMENT (Inches)

-50 0 50 100 150 TOP DISPLACEMENT (Inches)

FIG. 9. Dynamic Response of Eight-Story Frame under Sinusoidal Base Accel­eration: (a) Initial Design, Amplitude of Sinusoidal Base Acceleration, A:tm = 500; and (b) Optimized Design, Amplitude of Sinusoidal Base Acceleration A,,,., = 500

conclusion from static analysis is that the optimized design is better than the initial design as far as overall strength and stability is concerned.

Response to Dynamic Loads Each design was analyzed under a sinusoidal base acceleration. To make

the comparison of the initial and optimized designs fair, the amplitudes of the base accelerations were chosen to be the same at their respective resonant frequencies. Thus, the dynamic magnification factors for both cases were the same. A damping ratio of 5% critical was used for the first and the second modes. Fig. 9 presents the base shear versus the lateral top displace­ment of the structure under the sinusoidal base acceleration. Observe that the initial design has a tendency to drift whereas the optimized design has stable drift.

Both the initial and the optimized designs were analyzed under the 1971 Pacoima Dam earthquake. The response history and base shear history of both initial and the optimized designs are given in Fig. 10. Observe the severe drift of the initial design versus the controlled drift of the optimized design. The base shears are higher for the optimized design as expected due to the fact that the optimized design is stiffer and stronger.

t -z Ld 2 Ld O

5 0_

(/) Q

CL O t -

^ (0

I o

,g t >

60

30

0

- 3 0

- 6 0

1 l ' 1 ' 1 ' l ' l ' l '

f\ \ /^s^J

" —"x/ \ y\/\y\^, 1 , 1 . 1 , 1 , 1 , 1 ,

1 ' 1 >v L*^J

•

V ^ \ / r \ / '

Hit = 25.4mm , i . i . i ,

4 5 6 TIME (sec)

10

FIG. 10. Dynamic Response of Eight-Story Frame under Pacoima Dam 1971 Earthquake Base Acceleration

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CONCLUSIONS

Motivated by a simple model of the limit and postlimit response of planar frames, a design objective has been proposed to improve the overall strength and stability of inelastic framed structures. The procedure seeks to maximize the product of the dominant buckling eigenvalue and a frequency-dependent penalty function, subject to a constant weight constraint. The algorithm used for optimization was based on a classical optimality criteria approach, using an active-set strategy for extreme limit constraints on the design variables. The method is suitable for large-scale computation, requiring only a linear­ized buckling and vibration analysis of the structure at each iteration. It would appear, based on limited computational experience, that the algorithm is rel­atively insensitive to the initial design.

Example computations have verified that maximizing the buckling eigen­value increases the static capacity of a structure without degrading the post-limit response, thereby increasing overall toughness. Slender buildings are particularly well suited to the proposed optimization method since stability effects are more important in those cases. The algorithm generally yields a stiffer structure, and thus the optimized structures also tend to improve drift control under dynamic loads. The frequency weighting function can help control the vibration spectrum, but the benefits that accrue from the fre­quency penalty are application dependent.

ACKNOWLEDGMENTS

The writers thank Narbey Khachaturian for illuminating discussions re­lated to the topic discussed herein. The support of the National Science Foundation, grant number CES-8658019, and the American Institute of Steel Construction is gratefully acknowledged. The opinions expressed in this pa­per are those of the writers and do not necessarily reflect the views of the sponsors.

APPENDIX I. REFERENCES

Allwood, R. J., and Chung, Y. S. (1984). "Minimum-weight design of trusses by an optimality criteria method." Int. J. Numer. Methods Engrg., 20(4), 697-713.

Berke, L. (1970). "An efficient approach to the minimum weight design of deflection limited structures." Report No. AFFDL-TM-70-4-FDTR, U.S. Air Force Flight Dynamics Lab., Wright-Patterson Air Force Base, Ohio.

Brown, D. M., and Ang, A. H.-S. (1966). "Structural optimization by nonlinear programming." J. Struct. Div., ASCE, 92(6), 319-340.

Gellaty, R. A., and Berke, L. (1971). "Optimal structural design." Report No. AFFDL-TR-70-165, U.S. Air Force Flight Dynamics Lab., Wright-Patterson Air Force Base, Ohio.

Hjelmstad, K. D., and Pezeshk, S. (1988). "Approximate analysis of post-limit re­sponse of frames." J. Struct. Engrg., ASCE, 114(2), 314-331.

Hjelmstad, K. D., and Pezeshk, S. (1991). "Optimal design of frames to resist buck­ling under multiple load cases." J. Struct. Engrg., ASCE, 117(3), 914-935.

Khot, N. S. (1981). "Optimality criteria methods in structural optimization." Report No. AFWAL-TR-81-3124, U.S. Air Force Flight Dynamics Lab., Wright-Patterson Air Force Base, Ohio.

Khot, N. S., Venkayya, V. B., and Berke, L. (1973). "Optimization of structures for strength and stability requirements." Report No. AFFDL-TR-73-98, U.S. Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio.

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Khot, N. S., Venkayya, V. B., and Berke, L. (1976). "Optimum structural design with stability constraints." Int. J. Numer. Methods Engrg., 10(5), 1097-1114.

Korn, A., and Galambos, T. V. (1968). "Behavior of elastic-plastic frames." J. Struct. Div., ASCE, 94(5), 1119-1142.

Liebman, J. S., Khachaturian, N., and Chanaratna, V. (1981). "Discrete structural optimization." J. Struct. Div., ASCE, 107(11), 2177-2197.

Manual of Steel Construction. (1980). 8th Ed., Amer. Inst, of Steel Constr. (AISC), Chicago, 111.

No, M., and Aguinagalde, J. M. (1987). "Finite element method and optimality criterion based structural optimization." Comput. Struct., 27(2), 287-295.

Neal, B. G. (1977). The plastic methods of structural analysis. Chapman and Hall, London, England.

Pezeshk, S., and Hjelmstad, K. D. (1989). "Optimal design of nonlinear framed structures under multiple loading conditions based on a stability criterion." Report No. SRS-547, Univ. of Illinois, Urbana, 111.

Simo, J. C , Hjelmstad, K. D., and Taylor, R. L. (1984). "Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear." Com­put. Methods Appl. Mech. Engrg., 42(3), 301-330.

Uniform Building Code. (1985). Int. Conf. of Building Officials, Whittier, Calif. Venkayya, V. B., Khot, N. S., and Berke, L. (1973). "Application of optimality

criteria approaches to automated design of large practical structures." 2nd Symp. on Struct. Optimization (AGARD Conf. Proc. No. 123), Milan, Italy, April 1973.

Walker, N. D., Jr. (1977). "Automated design of earthquake resistant multistory steel building frames." Report No. UBC/SESM-77/12, Univ. of Calif., Berkeley, Calif.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A C(x)

D G.Go.G]

/ K

/(x,9 M

p(co) Q q

w0 w X

a

P 7o.7i

K,Xp,\cr

M-€ P

<!> to

= = = = = = = = = = = = = = = = = = = = = = =

area; constraint; beam depth; geometric stiffness matrices; moment of inertia; stiffness matrix; Lagrangian of objective function; mass matrix; frequency-dependent weighting function; gradient of objective function; gradient of constraint; structure weight; specific weight; design variables; participation factor; forced frequency ratio; dimensionless factors; load factor, plastic load factor, critical load factor; buckling eigenvalue; Lagrange multiplier, damping ratio; density; buckling eigenvector; and frequencies.

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