Least weight structures for threshold frequencies in a seismic environment

LEAST WEIGHT STRUCTURES FOR THRESHOLD FREQUENCIES IN A SEISMIC ENVIRONMENT? R. D. MCCONNELL Structural Division, Veterans Administration Washington, DC 20420,U.S.A. (Received 30 November 1975) Abstract-Conventional component seismic design procedures fall into two categories: a table of component forces (which often result in grossly conservative or grossly underdesigned structures); or sophisticated computer analyses usingdetailedmodelswhichundergodynamictime-historyforcesimulations.Aerospaceengineers havedesignedmany component structures to a minimal-fundamental,or “threshold” frequency specification. Although it does not appear as generally applicable for ground-supported “base structures”, the design of substructures for earthquake induced forces can similarly adopt such specifications. For example, radio towers, water tanks, electrical gear, etc., which are supported by larger structures will be subjected to the filtered natural frequencies of the “base structure” regardless of the earthquake spectrum. By knowing approximate amplification factors for the principal contributing frequencies (and corresponding base-excitation levels) of its “base structure”, a substructure specification for the (1) dynamic amplitication factor, (2) arbitrary safety-factor and (3) threshold design frequency can be determined. This paper describes adjustment of a structure’s fundamental frequency to a prescribed minimum level by a modified Bayleigh method which simultaneously converges to the fundamental mode while being proportioned accordingly to a fully-stressed optimum design. INTRODUCTION Aerospace designers have designed many component structures to a minimal-fundamental, or “threshold”, frequency specification. In general, dynamic force levels

ture specification for the (1) dynamic amplification factor, (2) arbitrary safety-factor and (3) threshold design frequency can be determined. (This has assumed consideration, where necessary, for an assumed mass-spring system representing the substructure as part of the “base structure” model.) The arbitrary factor-of-safety would take into consideration the spread between the specified threshold design frequency and the fundamental (or major contributing) frequency of the “base structure”, in addition to the assumed possible accrued errors in the particular analyses. This paper describes a method for designing leastweight, or optimized, structures when a threshold frequency (with appropriate force factors) is known. The process is highly amenable to a job-by-job application due to its simplicity. Small mathematical models of a structure can be analyzed by a hand-calculator while standard static-load finite-element programs can be used for larger structures. Design (member-selection) methods, whether by hand or by special program algorithms, becomes the same common set of procedures as would be introduced in most optimum-design methods.

imposed on the component (at the fundamental mode) are determined by the principal inertial force excitations to which the structure under design will be subjected at frequencies known to be below the specified “threshold design frequency.” Undoubtedly such concepts may have been used in various earthquake-related analyses and designs. Conventional component seismic design procedures as specified by various codes, however, fall into two categories: a table of component forces (which often result in grossly conservative or grossly underdesigned structures); or sophisticated computer analyses using detailed models which undergo dynamic time-history force simulations. Such sophistication is well known to be expensive, time-consuming and somewhat arbitrary in eachquake analyses due to the approximations of the mathematical model and the forcing functions. In addition, large programs often have the hazards of inadequacies in the mathematical algorithms and errorconditioning problems with large models. Although it may not appear as generally applicable for ground-supported structures, the design of substructures for earthquake induced forces can similarly adopt such aerospace methods. By “substructures” here is meant any component structure for which the predominant frequency to which it may be subjected is known to a reasonable degree of accuracy. For example, radio towers, water-tanks, electrical gear, etc. which are supported by larger structures will be subjected to the filtered natural frequencies of the “base structure” regardless of the earthquake spectrum. (Technically, “larger” structure would imply considerably higher dynamic modal-mass.) By knowing approximate amplification factors for the principal contributing frequencies (and corresponding base-excitation levels) of its “base structure”, a substruc-

SFRUCTURAL. OPlT?&TATION METHODS

tpresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, D.C., 29-31 March 1976.

With the increasing availability and intricacy of finite-element matrix programs for structural analysis, there has been a growing interest in, and development of structural optimization procedures to augment those programs. They generally optimize by minimizing weight and/or cost or a similar objective function. Finite-element programs use matrix representations, or idealizations, of real structures for the purpose of behavioral analyses under various loading conditions and constraints. Most methods and applications involving optimization of dynamic properties and behavior of structures have, in general, incorporated the basic procedures of prior studies in optimization under static load conditions. The goal in structural optimization is to minimize a prescribed merit function. This merit function could be weight, cost, fabrication criteria, or a combination of these[l]. Active constraints on the minimization of the merit function (or “optimality criterion”) would be the analysis equations for the set of design variables, such as

157

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R. D. MCCONNELL

stress solutions for the design of the members. Sideconstraints would be limits on displacements, minimum member sizes, etc. These constraints must not be violated when optimizing the mathematical representation of the structure to a particular merit-function. In those cases where the geometry is to be optimized, volume or space constraints may be imposed. Structural optimization methods can be divided into two general categories: mathematical programming involving search techniques, and optimality procedures. The principally identified methods in the first category are feasible-usable direction[2], gradient projection[3], optimum vector [4,5], linear programming [6] and allocation[7], all being defined as explicit minimization processes. Implicit minimization methods in the same general mathematical search category are steepest descents [8], adaptive gradients [9], variable metric [ lo] and conjugate gradients[ 111.These identities are indicative of the redesign search routine algorithm. “Implicit” methods use penalty functions[l2] to transform a constrained minimization problem to an unconstrained one. Conceptually, these can also transform discrete values of the design variables into general unconstrained functions. The optimality procedures include fully-stressed design [I3-191, uniform strain-energy density [18,20,21], simultaneous buckling[22,23] and limit design[24-261. Most examples demonstrated in the mathematical search technique group have been small, simple models of structures, generally involving static loading conditions. These procedures are very time-consuming, may not monotonically converge and involve much programming effort[2,4,5]. Virtually all practical applications involving larger model representations of real structures have employed one of the optimality methods. Conceptually, the optimality methods require solution of first derivatives of the merit-function, or a substitute function, in which case the method is classified as a substitute optimality process. (For example, the use of fully-stressed convergence is substituted for minimum weight.) In the actual iterative procedures, derivatives are not calculated specifically, hence the redesign calculations are relatively few. In the mathematical search technique, however, second derivatives must be extracted or approximated at each iteration to determine sensitivity coefficients for redesign toward a minimum of the merit fuhction. The two substitute optimality methods which have been used with success are fully-stressed designed and uniform strain-energy density design. These have been used for linear-elastic design structures. With the exception of such techniques as penalty functions[l2], in the mathematical search procedures, parameters, in general, with the exception of specific side-constraints, must have equivalent linearized functional representation. Such linear representation of discrete values is generally approximate and cannot always be so expressed. In the realm of optimization of structures, the penalty function, or unconstrained optimization, techniques have not been very efficient. Such penalty functions may introduce near-singularities in the search process with consequent instabilities [ 11. Most literature on applications and comparative reviews conclude that the more efficient and successful methods have been those of the optimality, or substitute optimality category.

It has long been known that simple structures optimized under a one-load condition will converge to a determinate structure in the absence of side-constraints which maintain minimum sizes for all members in the original structure [16]. (One frequency constraint is considered the dynamic equivalent of a one-load condition.) That determinate structure will have member sizes determined only by the forces in the members. The allowable stress could be prescribed by strength, yield or buckling criteria, and may be different for each member. The two methods used today for practical size problems under dynamic constraints or loading conditions are, as in the case of static loads, the substitute optimality approaches of fully-stressed design and uniform strainenergy density design. These two .produce the same optimum structure when the structure is made of only one material. In the case of threshold frequency optimization by fully-stressed methods, the allowable stress is uniform for all members in the structure. This is to say that if strength criteria do not preclude the adjusted stress-level, fullystressed methods in the dynamic case will reduce the stresses uniformly (thereby increasing sizes) to provide the stiffness sought. This step is often called “proportioning”. Re-analysis is then necessary as the mode-shape, and consequently the effective force-distribution, changes. As noted earlier, side-constraints may be introduced to preserve strength in particular members. The primary use of structural optimization procedures in dynamics problems has been to optimally adjust the fundamental frequency of the structure while assuring all additional strength requirements. One reference, [ 161,is of particular note here in that it is a representative practical application of optimizing a real structure to a fundamental frequency threshold level by the fully-stressed substitute optimality approach. The authors, Young and Christiansen, point out that they do not find an absolute, or global, optimum, but they do produce a “good, efficient structure”. Their procedure was essentially: (1) produce an original design; (2) analyze for resonant frequencies in the reduced models, mode shapes and corresponding modal forces in the detailed model; and (3) redesign (or proportioning). These three steps, basic design, analysis and redesign, are the fundamental steps common to all iterative optimization methods. This paper describes a simple, stable method for adjustment of the entire frequency set to a minimum level. The structure adjusted by this method will be simultaneously optimized to a least weight criterion without requiring the use of eigenvalueleigenvector extraction routines, or search algorithms since using the “fully-stressed” approach. This procedure converges rapidly and requires only standard finite-element programs for static-load conditions. PROCEDURE

For those cases where the entire set of eigenvalues is shifted to provide a fundamental frequency above a prescribed minimum level, proportioning of the structure is possible. Methods for total adjustment with proportioning have been presented in the literature as described earlier. It is often possible, however, to obtain such results by a rapidly converging, modified Rayleigh method using finite-element programs written for static-load analyses. The economy so realized, as contrasted to the

159

Least weight structures for threshold frequencies

use of eigenvalue extraction routines, permits the use of larger, more detailed models in many instances. The element of instability of comparable methods in the literature is principally inherent in the algorithms used to transpose mode-shapes to relative forces in the structural elements [ 16, 191.Modal extraction and vector transposition are not explicitly required in the familiar Rayleigh energy method to obtain the fundamental frequency of a structure. Static equivalent loads which can be factored by any arbitrary constant, are imposed by ratio and direction in accordance with the assumed (approximate) fundamental mode-shape. Resulting forces are then extracted by the routine stiffness matrix procedures for static loads. The force set is then linearly proportioned to achieve a fully-stressed state in that member which experiences the numerically largest force using the prevailing size of that member. All members are then simultaneously proportioned to the forces and scaled by the ratio of hF/Ae where hp is the desired eigenvalue level and hb is the currently extracted approximation. The process is repeated using applied forces proportional to the latest mode-shape. The iteration is complete in the Rayleigh method when the As extracted approximates AFdesired. This modified Rayleigh method is simultaneously converging to the true-mode shape of the fundamental mode while being proportioned accordingly. In most cases, an approximate mode-shape is known. When this is not the case, it would be advisable to use a procedure such as the “inverse power method with shifts” to extract the fundamental frequency and its corresponding mode-shape for only the first step[27]. Following adjustments, the routine Rayleigh method, as modified above, would then be used. Convergence can be assured by checking the total structural weight upon each successive redesign step. If there is a divergence due to a poor estimated mode-shape, it is only necessary to carry the Rayleigh iteration to successive steps (without proportioning) until convergence is assured. The advantages of this method, in contrast to those in the literature, are (1) since only the fundamental frequency is of concern, the Rayleigh

method is efficient and economical; and (2) the element forces are determined without conventional eigenvalueleigenvector extraction. The following is a statement of the basic Rayleigh iterative steps:

A”$ Mr(4’Y r=* Where Fti are the inertia forces corresponding to an assumed (or prior calculated) mode-shape. A”& are the deflections due to Fti where 4” is the normalized vector[28]. i%f,are the masses. ExAMPLE A simple 3 x 3 matrix model is shown in Fig. 1. The schematic spring model corresponding to the structural model is identical to one used in Ref. [28]. The first iteration cycle is identical. The subsequent steps toward weight/frequency optimization can be compared with the elementary method, in that reference, whereby only the extraction of the original fundamental frequency was required. Table 1 illustrates the steps in a simultaneous Rayleigh extraction of the fundamental frequency of this structure while proportioning members to optimize its weight. It concurrently shifts the fundamental frequency to a prescribed threshold level of o = 30 rad/sec. The assumed mode-shape is obtained by a 1 g force on each mass.

,!1(=2 lb-&in. (772 lb) y= I lb-se&in (386 lb)

w

Fig. 1. 3 Degree-of-freedom example of optimal adjustment of fundamental frequency by Rayleigh method.

Table 1. Solution of optimal adjustment of fundamental frequency by Rayleigh method

2

:

2.02 1.00

2.02 2.00

4340 6000

0.002014 0.001207

2.00 1.00

4.04 2.00

4.00 2.00

3

3.22

3.22

2670

0.003621

3.00

9.66 I15.70

9.00 Ils.00

1.00 2.00 3.00

2.00 4.00 9.00

2.00 4.00 9.00

02

3

1 2 3

1.00 2.00 3.00

2.00 2.00 3.00

a2

rn

15.70 - 867 0.001207 x 15."" 6000 4285 2571

0'

29.45,

0.001167 0.002334 0.003501

19.00 I- 857 - 0.001167 x 15.00 ,

o-

u2/J 12

29.2

Springs Optimized too P 13501

- 30.

"I. 900/791 x 12000 - 136.53.

160

R. D. MCIZONNELL

The first cycle produces a frequency of 28.2rad/sec, and a mode-shape of 1, 2.02 and 3.22. The spring stiffnesses (used in the Fig. 1 schematic to illustrate the method) are then adjusted to 6000,434Oand 2670lb/in to result in a uniform stress. This is obtained in this case by assuming the effective spring lengths equal and spring stiffnesses proportional to area. If a spring, therefore, has twice the area, it has twice the weight (for the same lengths) and will take twice the force at the same stress. The mode-shape to which this model must converge can be recognized to be 1, 2, 3 for these conditions. That “computed mode-shape” is achieved in the second cycle following proportioning of the springs. That cycle produces a frequency w = 29.5 (increased due to the basis of the spring proportioning). The third cycle is exact, in this instance, as the assumed and computed mode-shapes are identical. The resulting frequency is 29.3. An adjustment by 022/o: of the stfinesses to produce 30 rad/sec results in the minimum total weight springs as shown. CONCLUSION

This paper has described a simplified means to obtain optimization concurrent with threshold-frequency adjustment. The use of a model from an earlier textbook (Fig. 1) was to permit comparison with this paper’s modified Rayleigh method. The three degree-of-freedom model was originally efficient and near-optimum (from the standpoint of fully-stressed criteria). A further efficiency in weight was obtained (Table 1). Larger models were similarly optimized with the efficiency in total “weight” savings dependent on the initial member sizes. REFERFACES

1. R. J. Melosh,A Plan for Computer-Assisted Optimization of Structures, NAS 5-11779,NASA, G.S.F.C., Greenbelt, Md. (1970). 2. G. Zoutendijk, Method of Feasible Directions. Elsevier, * Amsterdam (l%O). 3. J. B. Rosen, The gradient projection method for nonlinear programming. J. SIAM Appl. Math. IX(4), 514-532 (Dec. l%l). 4. R. A. Gellatly, Development of Procedures for Large-Scale Automated Minimum-Weight Structural Design. AFFDL-TR66-180, Air Force Flight Dynamics Laboratory, WrightPatterson AFB (Dec. 1966). 5. R. A. Gellatly and J. A. Bumo, Development of Procedures for Large-Scale

Automated Minimum- Weight Slrucfural Design,

Supplement 1, Programmer Documentation. AFFDL-TR-66180, Supplement 1, Au Force Flight Dynamics Laboratory, Wright-Patterson AFB (Dec. 1966). 6. G. G. Pope, Application of linear programming techniques in the design of optimum structures. AGARD Symposium on Structural Optimization, Istanbul (Oct. 1%9).

7. R. J. Melosh and R. Luik, Approximate Mulriple Configuration Analysis and Allocation for Least Weight Structural Design.

AFFDL-TR-67-59, Air Force Flight Dynamics Lab., WrightPatterson AFB (1%7). 8. A. L. Cauchy, Methode generale pour la resolution des systemes d’equations simultanee. Compt. Rend. XXV, 536538 (1947). 9. H. H. Rosenbrock, An automated method for finding the greatest or least value of a function. Computer J. III, 175-184 Il%O). IO. R. Fletcher and M. J. S. Powell, A rapidly convergent descent method for minimization. Computer J. VI, 163-168 (1963). 11. R. Fletcher and C. M. Reeves, Function .minimization b; conjugate gradients. Computer I. VII, 149-154 (1964). 12. A. V. Fiacco and G. P. McCormick, Computational algorithm for the sequential unconstrained minimization technique for nonlinear programming-II. Management Sci. X(4), 601-617. 13. J. C. Maxwell, Scientific Papers II. pp. 175-177.Cambridge University Press, Cambridge (1890), Reprinted by Dover Publications, New York (1952). 14. A. G. M. Michell, The limits of economy of material in frame-structures. Phil. Mag. (London), Ser. 6, VIII(47), 589-597 (Nov. 1904). 15. F. H. Cilley, The exact design of statically indeterminate frameworks: an exposition of its possibility, but futility. Trans. ASCE XLIU, 353-407 (June 1900). 16. J. W. Young and H. N. Christiansen, Synthesis of a space truss based on dynamic criteria. J. Struct. Div. ASCE XCII(ST6),425-442 (Dec. 1%6). 17. R. A. Gellatly, R. H. Gallagher and W. A. Luberacki, Development of a Procedure for Automated Synthesis of Minimum Weight Structures. FDL-TR-64-141, Wrigbt-

Patterson AFB, Ohio (1964). 18. V. B. -Venkayya, N. S. Khot and V. S. Reddy, Energy Distribution in an Optimum Structural Design. AFFDL-TR68-156,Wright-Patterson AFB, Ohio (Mar. 1%9). 19. C. P. Rubin, Minimum weight design of complex structures subject to a frequency constraint. A.I.A.A. J. 8(5) (1970). 20. W. Prager and J. E. Taylor, Problems of optimal structural design. Trans. J. Appl. Mech. ASME XXXV, Series E(3), (Mar. 1%8). 21. J. E. Taylor, Maximum strength elastic structural design. J. Engng Mech. Div. ASCE, EM3 653-663 (June 1969). 22. G. Gerard, Minimum Weight Analysis of Compressive Structures. New York University Press, New York (1960). 23. F. R. Shanley, Weight-Strength Analysis of Aircraft Structures. pp. l-260. McGraw-Hill, New York (1952). 24. R. von Mises, Mechanik der festen Korper im Plastishdeformablen Zustand. Nachr. Kgl. Ges. Wiss. pp. 582-592. Gottingen, Math-Physic Klasse (1912). 25. A. Nadai, Plasticity. A Mechanics of Plastic Stare of Matter. McGraw-Hi, New York (1931). 26. P. Hodge and W. Prager, A variational principle for plastic materials with strain hardening. J. Math. Phys. XXVII, l-10 (1948). 27. C. W. McCormick, (Editor), The NASTRAN User’s Manual. NASA SP-221, National Aeronautics and Space Administration, Washington, DC. (Oct. 1969). 28. J. M. Biggs, Introduction to Structural Dynamics. McGrawHill, New York (1964).