The modified fully utilized design (MFUD) method is undergoing development for use by engineers who favor traditional methods of designing structures over methods based on advanced calculus and nonlinear mathematical programming techniques. Thus far, the MFUD has been developed for trusses, with cross-sectional areas of truss members as design variables. Like nonlinear optimization methods, the MFUD method is iterative, but in comparison with those methods, the MFUD method involves less and simpler computation.

The MFUD method is derived from the fully stressed design (FSD) and fully utilized design (FUD) methods. The FSD method, based on a simple stress-ratio approach, is popular in civil, mechanical, and aerospace engineering. The FSD method is an elegant conceptual tool for arriving at stress-limited designs, but does not provide for displacement constraints, which are imposed with increasing frequency in designing modern structures.

An extension of the FSD method through provision for displacement constraints in addition to stress constraints yields the FUD method. The term "fully utilized design" signifies a design in which the number of active constraints equals or exceeds the number of design variables. One obtains the FUD of a structure by the following procedure:

1. Using the stress constraints only, generate the FSD.
2. Uniformly prorate the FSD to obtain the FUD, using a constant proration factor that satisfies the single most infeasible displacement constraint. For a truss structure, this entails multiplying the cross-sectional areas of all truss members by the same factor to strengthen all the members enough to limit the displacement, as required.

The FUD thus obtained is feasible but can be an overdesign; the weight of the FUD structure can be greater than that of an optimally designed structure.

The MFUD method accommodates both stress and displacement constraints simultaneously. The steps of this method applied to a simple truss structure are the following:

1. Identify the design variables to initiate iterations. Optionally, one can begin iterations from the FSD. For subsequent iterations, the stress constraints are expressed in terms of cross-sectional areas, given by

Ai = (Fi)max/ σi0,

where Ai is the area in question for the ith member, (Fi)max is the maximum force in ith member under all loading conditions, and σi0 is the maximum allowable stress in the ith member.

2. Identify the displacement constraints violated by the design obtained in step 1.
3. For each displacement constraint identified in step 2, update the design independently to satisfy the constraint. The update process comprises two subprocesses: (1) identification of a subset of design variables pertinent to that constraint and (2) computation, for each member, of a member-weighted parameter, which is a multiplicative parameter based partly on the sensitivity of the constraint-violating displacement to the cross-sectional area of the member. The member-weighted parameter supplants the constant proration parameter of the FUD method. The equations used in these subprocesses are derived from the integrated force method, which was described in "Constructing Finite Elements for the Integrated Force Method" (LEW-16421), NASA Tech Briefs, Vol. 21, No. 7 (July 1997), page 70.
4. Obtain the design update for the structure as the union of all of the designs updated for the constraints in step 3. If any member is affected by more than one of the constraint-updated designs in the union process, the cross-sectional area selected for that member should be the maximum one.
5. Repeat steps 1 through 4 until the design converges. The converged design will satisfy both stress and displacement constraints.

Despite its relative simplicity, and even though it does not incorporate an explicit minimum-weight condition, the MFUD method can yield solutions comparable to those obtained by nonlinear optimization techniques (for example, see figure). Even if one still prefers a full optimization, the MFUD method could be used to generate initial designs for subsequent optimization iterations, thereby alleviating some of the computational burden of optimization.

This work was done by Laszlo Berke and Dale Hopkins of Lewis Research Center, Surya Patnaik of Ohio Aerospace Institute, and Atef Gendy of the National Research Council.

Inquiries concerning rights for the commercial use of this invention should be addressed to

###### NASA Lewis Research CenterCommercial Technology OfficeAttn: Tech Brief Patent StatusMail Stop 7 321000 Brookpark RoadClevelandOhio 44135

Refer to LEW-16624.

##### NASA Tech Briefs Magazine

This article first appeared in the June, 1998 issue of NASA Tech Briefs Magazine.

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