modelling considerations in optimum design of reinforced structures

Modelling considerations in optimum design of reinforced structures B. Prasad

Engineering and Research Staff, Ford Motor Company, Dearborn, MI 48121, USA (Received March 1982; revised November 1982)

The paper provides some potentially useful modelling schemes for reducing computational cost in designing reinforced structures. The schemes are based on the concepts of element and cross-section idealizations while maintaining the characteristics of the load path. An example of a double layer reinforced panel is given and it is shown how these schemes can lead to increased flexibility and reduced cost in obtaining minimum weight designs.

Key words: mathematical model, optimum design, reinforced structures

In 1960 it was suggested’ that finite element structural analysis methods and nonlinear programming techniques could be coupled together to generate automated structural design optimization capabilities. Since then we have seen the emergence of several powerful and general design program? capable of handling a large number of design variables. The progress in this field has been enormous as evidenced by the large number of papers published (nearly 200 references in reference 5, for example). How- ever, it is still not possible to carry out an efficient detailed configuration design of the structure as is often required by the automotive and aerospace industries. For example the minimum weight design of a multicomponent structure using currently available computer programs requires far more computer time, storage and man effort than is economically viable. Consequently in recent years, a re- appraisal of the objectives of structural optimization has occurred leading to a change in attitude toward a process more akin to exploiting the inherent structural characteristics. Many researchers believe that optimum design of large multicomponent structures requires an innovative use of approximation concepts and model simplification procedures.6’7

This paper therefore advances some additional cost saving design concepts. The procedure is built in cross- section and element idealizations schemes and ranking of the design variables by importance. The procedures for staging of the design process are also discussed.

Practical design

A practical design of a component herein means obtaining a fairly detailed design while using a minimum amount of computer time, man hours and testing resources. Usually

the result of increased design detail is an increase in the amount of computer time and effort. Significant detail in the initial phase of the design may not be necessary for two reasons. First, there exist both structural and nonstructural requirements (such as that given in reference 8) but usually it is not possible to meet them all simultaneously. After the design is initially!completed for some, it is checked for the others. The design therefore keeps chang- ing. After undergoing one or more perturbations in search of the best compromise, the resulting design often takes on a configuration which is significantly different from the initial design. Second, structures are usually designed to be somewhat on the conservative side.

Simplified design approaches offer a potentially superior and faster way of introducing new design concepts, creat- ing or checking new ideas, or obtaining the final design itself. The following factors which have been found useful in obtaining cost effective practical designs are discussed in this paper:

1. 2. 3. 4. 5.

Model simplification for design Cross-sectional idealization Element-type idealization Ranking of design variables Configuration mapping

Staging of the design process is considered as an integral part of a minimum weight design. This is shown in Figure 1. The ‘design program’ forms an important part of this process as shown in stage 2. The program is assumed to be efficient and is based on the innovative use of the approximation concepts for reducing the total number of iterations. There are several papers which deal with this aspect in greater detail. 2,3,6 No discussions of these are therefore given here. The PARS (Programs for Analysis and Resizing

146 Appl. Math. Modelling, 1984, Vol. 8, June 0307-904X/84/03146-11/$03.00 0 1984 Butterworth &Co. (Publishers) Ltd

Optimum design of reinforced structures: 9. Prasad

of Structures) system,’ which is based on the aforementioned concepts, has been used here during stage 2.

Computer program

PARS is a companion code to the SPAR” finite element program and like SPAR, it is a modular set of pro- cessors communicating through a data base. PARS is capable of performing integrated interdisciplinary analyses (such as stress and flutter analyses, computation of sensitivity vectors, constraint formations and deletions, etc.). It also provides an efficient code for sizing large or small scale finite element models in the presence of strength, thermal and aeroelastic constraints with minimum and maximum bounds on structural dimensions. Detailed discussion of this program can be found in a recent paper.’

The optimization algorithm is based on a penalty approach called the Variable Penalty Method” which combines the SUMT (Sequence of Unconstrained Minimiza- tion Technique) procedure and Newton’s method in such a way that the second derivatives of the constraints required for Newton’s method are supplied approximately in terms of only the first derivatives of the constraints. This retains the features of the second order minimization techniques at a cost no larger than that required for the first order methods. The approach provides schemes for minimizing the errors in the approximation of the second derivatives matrix while providing the best penalty for the constraints during each unconstrained minimization sequence. A detailed discussion of this approach can be found in reference 11.

Staging of the design process

There are three basic stages which can be identified in an overall design process. The first consists of examining the large structure and identifying within it a number of component small subsystems. Each must be represented in a simple idealized form in which the structural behaviour is governed by a minimum number of parameters yet detailed enough so as not to lose any significant physical characteristics of the component so idealized. The second stage in the overall design process is the subsystem design. Best values of the design parameters governing the idealized sections have to be found to conform with the design requirements and to have minimum weight. The third stage in the design process is to design each subsystem in detail, i.e. to choose the necessary detailed configuration of the sections and shapes so that their equivalent cross-sectional properties match with the idealized section values found in

the stage 2 process. Here the cross-sectional properties are also chosen to satisfy some of the remaining nonstructural requirements such as manufacturing. packaging or merely styling.

There should always be a fourth stage in the design process, that of carrying out a final analysis on the complete structure with the individual detailed design of the subsystem in place to check that the entire structure does not violate the desired conditions s,et forth earlier. A schematic diagram depicting various stages leading to the final design is shown in Figure 1. The detailed design of the subsystem is deferred until stage 3. It is then more cost effective to do a simplified design of the subsystem first and then based on this, obtained a detailed design later.

STAGE 1

; SIMPLIFIED a REDUCED

+ AUTOMATED

DESIGN PROGRAM

STAGE II

OPTIMUM DESIGN

DETAILED DESIGN

L-*c Figure 7 Stages in a practical design process

STAGE XI

STAGE Zp

Model simplification fbr design

Amongst various other things cost has always been a critical factor in optimizing a large structure. Frequently, the time involved in each analysis is large and it becomes impractical to repeat the analysis several times for a minimum weight design. The factors which affect the cost of analysis in a finite element model are (a), number of nodes; (b), number of elements; (c), band width; and (d), equation solver (Figure 2). In a given finite element program we generally do not have much control of(c) and (d). (a) and (b), are generally the factors which affect the analysis costs. They are also the ones that affect the design costs since the stress constraints depend on the number of elements and the deflection constraints on the nodes of the finite element. Model simplification can reduce both the analysis and design costs. From the analysis point of view, simplification means obtaining a finite element model with minimum number of nodes and elements but yet detailed enough to be able to predict the correct behavioural response. From the design point of view it means linking the finite element. The process is often called design variable linking.’ Various linking approaches are available; all share the basic strategy that one or, at most, a few independent design variables Vj control the structural parameters associated with all finite elements in that linking group (see for example reference 2). A struc- tual parameter, in this context can be either a size (thickness or area) of the finite elements or any physical dimension associated with the structural geometry. The parameters are

Appl. Math. Modelling, 1984, Vol. 8, June 147

Optimum design of reinforced structures: B. Prasad

0 TYPES OF ELEMENTS , DEGREES OF FREEDOM , NUMBER OF ELEMENTS , MOOELLING ASSUMPTIONS

, APPROXIMATE LINE SEARCH

, CONSTRAINT APPROXIMATION

, EQUATION SOLVER

, K-DECOMPOSITION

0 THROW AWAY CONCEPTS

0 S:CEoTN;PRDER , DESIGN VARIABLES

(NDV)

0 CRITICAL CONSTRAINTS (NCON

, LOAD CASES (LCASEI

, K-MATRIX PARTITIONING , IMPLICIT METHOD

figure 2 Parameters affecting efficiency of a design process

linked by specifying their dependence on ‘master’ design vaiiables Vi, i = 1, . . . , n. The relationship is often established as:

di = f FijVj

j=l (1)

or

@I= [F](v)

where D is a vector of resized quantities and F is a matrix of coefficients.

Design variable linking makes it possible to reduce the number of independent design variables while at the same time imposing constraints that can make the final designs look more realistic. Linking also facilitates the introduction of constraints, reflecting symmetry considerations, designer insight based on prior experience, as well as fabrication and cost consideration associated with the number of parts to be assembled.

Cross-section idealization In automotive structures, there exist numerous forms of

open and closed stiffeners made out of thin sheet stock which can be modelled as beam elements. In most finite element programs there exist, however, a limited number of cross-sectional types (most common are I, tee, angle, waffle, channel, box, etc.). Thus one cannot be very precise in modelling stiffeners with veam elements. Also the resulting finite element model with stiffeners as beams contains a significant number of design parameters per cross-section. Use of these as design parameters often requires high cost. It does not provide much gain in the overall objective of the design, that is to minimize the weight. The cost can, however, be reduced significantly by employing a so-called ‘carry through concept’. In Figure 3 an I, T and a hat section are shown. When this type of beam cross-section is used as a stiffener, it is known that normal stress is taken by the flanges of the beam and the shear stress is borne by the web. One can, therefore, model a stiffener (such as an I-section) as two parallel sets of rod elements, offset by a series of shear panel elements. There are two distinct

advantages in this concept. First, by partitioning a stiffener beam into basic element types, one avoids the complexity on modelling. The loss in accuracy is significantly small; the normal stress is borne by the series of rod elements con- stituting the flanges of the idealized sections. The shear stress is taken by the shear panel members representing the web. Part of the axial load carrying capacity of the web, is, generally lost by this idealization, causing the resulting design to be somewhat on the conservative side. This may add to the reliability of the structure. Secondly, it effectively reduces the number of design parameters to a smaller set of primary design variables. In the example shown in Figure 3, there are only four design variables per cross- section (i.e. the areas at the top and bottom rod elements, height of the web and the thickness of the shear panel elements).

Element idealization in resizing

The term ‘element idealization’ has been used here to indicate a process whereby complicated large substructures can be idealized for design purposes as an assemblage of simple finite elements. The theoretical justification for its use is given in appendix 1. It should be noted that it is possible to have more than one such assemblage with different element types possessing about the same merit (e.g. the weight). (As shown in the section on ‘Basis for simplification’, it is essential that the characteristics of the load path are not changed during such element idealization.)

An example of this is shown in Figure 4. Here the inner panel is modelled using rod and shear panel elements. If one decides to use beam finite elements, it would simply add to the cost of the optimization. The final optimal weight of these ‘two models’ (see Figure 4) are expected to be close to each other. For the purpose of discussion let us distinguish between the finite elements whose stiffness matrices are linear (such as bar, shear panel or membrane plate elements) functions of the design variables as opposed to those whose stiffness matrices are nonlinear functions of the design variables (e.g. beam and plate bending elements). The derivative computations for constaints are quite easy with linear size-stiffness elements and require reasonably small computational cost compared to those having nonlinear size-stiffness relations. Linear size-stiffness finite elements are therefore generally preferred in structural

(a) Original offset lb) Simplified equivalent beam sections section

Figure 3 Cross-sectional idealization for resizing

148 Appt. Math. Modelling, 1984, Vol. 8, June

B:

c:

MODEL I

---_----- ‘\ /’ f

l!7I23l ‘---\ 0’ 1 ,A

/’ ‘\A ------ \ -----_I

I ‘\ m I ‘\\ ‘\

\ I

MODEL II

(a 1 Inner panel is simplified (b) Inner panel is simplified as beam elements as rod and shear panel

elements

Figure 4 Skeleton (simulation) diagrams for double-layered

panels (three patterns shown)

models for resizing. Amongst the nonlinear size-stiffness elements, the bending plate element is recognized to be essential for design. Unlike the beam elements, it is difficult to replace plates with elements of other types.

Plate elements. In recent years, the use of plate elements (bending) has increased sharply. One of the obvious reasons is that it is indispensible for modelling a variety of structures. It is also one of the simple elements from the class of bending which take point as well as pressure loading. Its application in component designs is widespread and since in the automotive industry most of the com- ponents are made out of sheet stock, one cannot ignore plate elements while designing car structures.

Beam elements. The use of beam elements for design has not been very popular mainly due to the following two reasons. First, there exist numerous cross-sectional configurations for beam elements each requiring a significant number of design parameters such as thickness, width, height of the flange, web, etc. In most applications,4 in order to reduce the number of design variables the following forms of relations are often used:

Ii = ajAPi j= 1,2

Zi = bjAqi j= 1,2

where A is the area of cross-section, I,, I2 are the moment of inertia and Zj the section modulus in principal directions. aj, bj, pi and qj are the cross-sectional constants. AS one might anticipate, it is difficult to express all possible configurations in the power form of equation (2), even if a structure is so designed, the resulting values for A are left with limited meaning for practical use. Secondly, the optimization procedure, which requires the derivatives of the constraints with respedt to design variables, is difficult with beam elements since the stiffness matrices are highly nonlinear functions of the design variables. Finite difference derivative computations on the other hand can be very costly.13


Modelling structure through simple elements is not enough, one must choose the correct element types so that the resulting representation possesses some practical value. For an example, in reference 2 a wing is represented by an assemblage of truss elements carrying axial tension or com- pressions. Though it provides a very good academic problem for checking the algorithms, etc., it has limited influence on the actual design construction. For instance it would be very difficult to derive the dimensions for the actual members (e.g. plates, spars, shear panels, struts) from the cross-sectional area of the axial load-carrying members after the wing is designed for minimum weight.

Ranking of design variables

The process of cross-section and element idealizations for redesign have been discussed earlier. One of the benefits that was not so apparent is that it implicitly reduces the size of the design problem by ranking the design variables by importance, i.e. primary and secondary design variables. Primary design variables are those which govern the generic parameters of a finite element model. By virtue of the definition, the distinctions are such that the secondary variables are directly linked to the primary. Once the primary variables conforming to the minimum weight design are determined, evaluations of the secondary variables remain a matter of routine computations. This is discussed in the next section.

The advantages of distinguishing among primary and secondary design variables are manifold. First, primary design variables are small in number and therefore can be obtained through a relatively inexpensive computation. Second, this approach has the added flexibility of choosing the desired cross-sectional configuration, even after the design is completed. A typical example of this case can be envisaged in the design of double-layered panel structure where one chooses to use a different stiffener cross-section for the inner panel instead of the one already designed for. With the use of the aforementioned distinction it would no longer be essential to re-run the optimization program (unless significant changes in the layout or locations of the stiffeners are desired), but to use the existing information in obtaining a desired configuration, this is discussed in the next section. Third, it can also serve as a convenient means for understanding the behaviour of the structure which is usually incomplete in the initial design stage or to carry out some initial trade-off studies. With the carry through concepts introduced, the unit design cost is fairly small and therefore it becomes inexpensive to carry out these studies repeatedly.

Configuration mapping*

One of the important aspects of design is to determine the layout of the desired stiffener cross-section (i.e. ffl, tfz, Wfl, Wf2, etc., referred to here as secondary variables; see Figure 6) after the optimum design is completed in stage 2. To illustrate this concept let us assume that one is primarily interested in back transformation, i.e. mapping of the primary design variables (A,, AZ, t and h) as obtained through the stage 2 process into one of the cross-sectional configurations shown in Figure 6. A typical set of parameters (secondary variables) associated with the mapping

*The term ‘configuration mapping’ is used here to indicate the process of determining the magnitudes of secondary design variables once the primary design variables are found.

Appt. Math. Modelling, 1984, Vol. 8, June 149


2 22 35 48 61 74 07 100 II3 126 146 159 172 I98 211 224 237 250

Figure 5 Finite element model of a typical reinforced panel (18 in X 54 in)

are given in Figure 6. The relationships between primary and secondary variables can be established typically as:

A 1 = tf, Wf, + CYr;, (3)

A, = tf2 Wf2 + fit’, (4)

h=(l--)h, (5)

t=t w (6)

There are more unknowns than the number of equatio;is. Some of the variables are, therefore, evaluated empirically or are based on other conditions. The value for tw is detcr- mined from equation (6), first. The values of Wf, and US2 are usually governed by nonstructural requirements such as packaging, manufacturing, welding, etc., and therefore they are selected next. 01, p and y are small constants which account for the loss of axial load carrying capacity of the web. The designers usually have a good estimate of their values based on prior experience. The values, unfortuna- tely, depend upon the configuration types (such as I, T or V section) and are essentially empirical in nature. Having estimated the values of CY,~ and y for a particular cross- section, tfi and ff2 can be determined from equations (3) and (4), respectively; h, can be determined from equation (5). This essentially completes stage 3 of the design process as shown in Figure 1.

The fourth stage of the design is to carry out the complete analysis of the structure with the detailed configuration found in stage 3 to check the status of the design requirements and also to determine some additional effects (such as fatigue life, etc.) requiring the use of such a detailed model. The staging of the design processes thus fits very well with the types of performance criterion, one is normally interested in.

Basis for simplification Before we give any justification for using the simplified approach it is worth briefly reviewing some of the known characteristics of a general structure pertaining to an optimal design.

(a) Mulfiple minima. The existence of multiple minima in structural optimization is beyond dispute. Some of these minima may represent the same merit value (such as weight) but the designs can be distinctly different. In such a situa- tion an optimization procedure can only lead to and at best guarantee convergence to a relative minimum. But which of

150 Appl. Math. Modelling, 1984, Vol. 8, June

these optima is the desired optimum is difficult to predict beyond prior knowledge of all the possible designs.

(b) Alterrlate load path. A load path is characterized by the behaviour of the response functions or constraints (for example deflection, frequency, mode shape, etc.) relative to a given load input (such as applied force, moment, pressure, motion, etc.). A determinate structure represents a convex design problem and it has a unique load path. An indeterminate structure, on the other hand, can be viewed as a com- posite of a number of determinate combinations. Each determinate combination can represent a distinct load path, corresponding to each load path there will be a relative minimum. For an indeterminate structure it is thus possible to have as many relative minima as the possible determinate or equivalent combinations.

(c) Uniqueness. It is also known that for a single loading case a determinate structure gives a lowest weight design, i.e. it has a global minimum.

Design analogy

The facts mentioned earlier seem to have very little influence in the present forms for the practical design of

WNEL *

7 t 2

(b) SIMPLIFIED SECTION (01 ORIGINAL OFFSET AS DESIGNED OPTIMALLY BEAM SECTIONS

Figure 6 Mapping of idealized section to an original form


Chang and Barone l2 have also made a preliminary design study of the effect of alternate reinforcing patterns on panel weight. However, the simplifying assumption had resulted in the entire panel model having only three design variables including the thickness of the outer panel. These limitations affect the least weight of the structural pattern which can be possibly achieved. The present approach allows each stiffener to vary independently. The number of secondary design variables are still manageable since a significant number of secondary design variables are effectively merged into primary. A maximum of four design variables are allowed per stiffener cross-section but the user can still reduce them by using design variable linking. The cost of running this model is small because it cosists of an assemblage of basic elements. One can, therefore, use a large number of design variables to arrive at the most reasonable and practical design.

structures. It is noted that they convey an important meaning if viewed somewhat differently. Combining (a), (b) and (c), the characteristics can also be stated as follows:

‘For a well defined load path any optimized assemblage of indeterminate structure will have the same weight.’ This represents the backbone for the simplification advanced earlier. As long as by making such simplifications, the flow of load path is not changed, we do not expect to get very significantly different minimum weight designs. The proof of this analogy is given in appendix 2. In a simple structure it is easily possible to meet this requirement. This is shown in Figure 4. Two different models of a rectangular panel (18 in x 56 in) are obtained. In one case it consists of plate and beam elements and in the other case it consists of an assemblage of plate, rod and shear panel elements. Thus in case (b), the inner panels which allow the loads to be ditributed along the path of the stiffeners, have been replaced by an assemblage of rod and shear panel elements. With that in place, the load path is not changed and the resulting weight of the structure is expected to be the same. Replacing the plate for the outer panel by an assemblage of nonbending elements in the above example would not be the same thing since the plate element provides a continuous assemblage of an infinite number of beam elements and thus it has provision for allowing the load to vary in a random manner. With the use of a few discrete beam elements instead of plate elements, the load path is restricted and therefore the final designs are not expected to be the same. It is clear that valid assemblages of a structure are only those which maintain a unique load path.

Design examples: reinforced panels

In order to illustrate the benefits of structural simplification for design, a double layer stiffened panel assembly is chosen as an example. A typical panel assembly together with a corresponding finite element is shown in Figure 5. The size of the baseline panel is (18 in x 54 in) with an aspect ratio of 1: 3. The outer panel is modelled using plate (bending) finite elements and the inner panel using an assemblage of bar and shear panel elements. The stiffened cross-section of the inner panels is defined in terms of three independent design variables; namely the area of the top bar ai, area of the bottom bar Q: and the thickness of the interconnecting shear panel element, ti. Thus there were three primary design variables per reinforcing member of the inner panel. A single design variable is used through linking to control the thickness of the entire outer panel. The total number of design variables is clearly a function of the reinforcing members used for the inner panel. Three distinct patterns of the inner panel layout, as shown in Figure 4 are chosen for this design evaluation. Except for the outer panel thickness, the cross-sectional dimensions of all the stiffeners were allowed to vary independently. There were 16 design variables controlling the sections for reinforcing pattern A, 22 for pattern B and 25 for pattern C. In each of the patterns A, B and C design variable linking is imposed to account for some nonstructural requirement. In the present case this has been used to make the final design symmetric. This has resulted in reducing the effective number of primary design variables from16to13inA,from22to16inBandfrom25to13 in pattern C. Nonetheless, this provides for some additional cost saving without endangering any significant weight penalty.

Constraints and boundary conditions

The combined weight of the outer panel and the stiffeners is chosen as the objective function in the optimization process. Since the main purpose of this study was to show the importance of previously outlined cost saving concepts, the overall bending and torsional stiffness were only included as major critical constraints. They correspond to the two independent test conditions which are often per- formed on most double-layered panel structures (such as deck-lid) when used in automotive applications. SPAR,” a general purpose finite element program, is used for obtaining overall bending and torsional stiffnesses. In the bending case the reinforced structure is supported at the four corner points and the load is applied in the centre of the top plate (Figure 7). The bending stiffness is computed using:

(7)

where P is the applied load and 6 is the deflection measured under the load. Since the structure is linear, the bending stiffness will be independent of the load magnitude. Any reasonable value for P can thus be used. To obtain designs with different stiffness requirements, the load P is kept constant and the deflection 6 is varied in such a way that the resulting stiffness corresponds to that specified for each new design.

In the case of torsional requirements, the three corner points were only supported, the fourth corner point was set free. The torsional stiffness is defined as:

T Pb=

Kr=2Y-

where T is a torque produced by the applied force P at the fourth corner point and 0 the corresponding twist angle. This can also be expressed in terms of P, deflection under the load 6, and the smaller length of the two sides, b. To provide at least the same torsional stiffness to the reinforced structure as that initially prescribed for bending (K, > Kb), the allowable limits of F in equation (8) was accordingly determined in each case and were included as deflection constaints in the optimization procedure. The results of analysis for two boundary conditions were thus considered simultaneously during each iterative cycle of PARS.a

Appt. Math. Modelling, 1984, Vol. 8, June 151


Initial conditions

Three different finite elements models14 were developed for each of the reinforcing patterns. In pattern A, the reinforced structure were simulated using 190 grid points, 108 plate elements, 60 shear panels and 177 bar finite elements. In pattern B there were 268 grid points, 162 plate elements, 81 shear panels and 239 bar elements. In structural pattern C, these were 186, 108, 54, and 128, respectively.

The initial size of the outer panel was taken as 0.20 in and that of the shear panel and bar elements which represent the reinforcing members were 0.01 in and 0.001 in’, respectively. There were 15 independent design runs made (3 reinforcing patterns x 5 prescribed stiffness points). Each run had two constraint conditions and one load case.

The material properties of the double-layered panel used were:

Young’s modulus E = 29 x 1 O6 psi (207 GPa)

Poisson’s ratio v= 0.3

Weight density p = 0.283 lb/in3 (7.83 g/cm3)

In most panel structures the outer skin is exposed and the minimum allowable thickness is guided by the requirements such as good surface finish, corrosion resistance and local oil canning effects. The manufacturing process also limits the value of the panel thickness that could be used. These requirements were imposed on the design by limiting the permissible size of the design parameters so as to not exceed the minimum gauge standards. As an illustration of the concept, the paper describes an example where minimum gauge standards and overall bending and torsional stiffness requirements are chosen as key constraints. The baseline bending stiffness is taken as 250 lb/in and the baseline minimum allowable thickness of the outer panel is set at 0.020 in. The section depth of the reinforcing cross-sections is kept equal to the maximum allowable by space requirements. The problem thus posed is to minimize the panel weight without violating these conditions.

Reinforcing pattern design

In order to determine the effect of baseline stiffness on the optimal weight of the panel, a series of minimum weight design problems was solved with different stiffness constraints imposed at the centre of the panel. As noted earlier three reinforcing patterns A, B and C were chosen for this study. The procedure for finding the series of minimum weight was thus repeated for each pattern in turn. In each case the inner panel was represented by the series of rod and shear panel elements and the outer panel by bending plate elements (see for example Figure 5). The stiffness constraints were allowed to vary in an extended range to obtain the behaviour of the structure in an ex- tremely large number of situations.

The effect of the baseline stiffness on the optimal weight of the structure is shown in Figure 8. Three curves are plotted corresponding to three reinforcing patterns chosen for this study. The results indicate that the ‘best’ minimum weight design is attained with pattern A, up to a certain stiffness value. But if the stiffness value is increased to more than 9501b/in, pattern A no longer remains the best pattern. In this case the best design is dictated by pattern B. It is interesting to note that for lower values of stiffness (P/6 < 500 lb/in) the variation of the optimal weight is practically

linear. The nonlinear behaviour is mainly due to the increase in bending plate thicknesses above the minimum gauge value of 0.020 in. Tables I, 2 and 3 give the values for the design parameter at the optimum corresponding to three different patterns A, B and C. Up to a definite stiffness value (varies with the pattern) the minimum gauge constraints override the thickness of the outer panel. After that, thickness is controlled by the overall stiffness constraints used. In this case, pattern B requires the smallest thickness of the outer panel and pattern C requires the largest.

In order to obtain these plots a significant number of runs (5-7 runs for each pattern) were required on PARS; each run requiring an average of 8-l 5 iterations for convergence. Analytical derivatives of the constraints (both for bending plate and linear size-stiffness elements) were employed to reduce the costs of iterations.’ Hybrid constraint approximations13 were used during the uni- dimensional search.

Conclusions

A simple example of a reinforced panel demonstrates that the use of model simplification techniques, element idealizations and ranking of design variables can result in consider- able computational time saving and increased flexibility. The cost saving is achieved because we deal with simple elements and those analytical derivate computations,

( i 1 BENDING STIFFNESS

(i i 1 TORSIONAL STIFFNESS

Figure 7 Constraint conditions used for reinforced panels



Table 2 Final design data (pattern B)

Design Type of Overall Stiffness (lb./inch) Variable Variable Number (Unit) 250 333.33 500 1000 Remark

241 I .

Aspect Ratio I: 3 I- = 16

w 3 14

: = I2 F

% IO

8

t

-0 250 500 750 1000 1250

OVERALL STIFFNESS ( lb. / inch 1 Figure 8 Effect of stiffness on optimal weight of reinforced

panels

required for resizing are easy and economical. The idealizations retain all the essential characteristics of the base structure while keeping the design and analysis costs low. The design provides the information about the effective section properties (referred to here as primary design variables) of the reinforcing members, and decisions regarding the

Tab/e 1 Final design data (pattern A)

Design Typeof Overall Stiffness (lb./inch) Variable Variable Number (unit) 250 333.33 500. 1000. Remark

Area(in') . . . . . . . . . . indicates II . . . . . . minimum ,I ,1 o.Oio 11 ,1 o.Oio II . . ,1 0.048

Thickness . . (inch) II . . II . .

. . . . 0:ozs gage value

. . . . . .

. . . . . .

. . . . 0:iss

0.;)69 oIi24 0.278 . . . . . .

. . . . . .

. . . . . . <I

o.o;o 0.036 0.018 0.044

II 0.020 0.020 0.0210

Total Weight

Legend:

Ibs. 7.582 8.354 10.013 20.652

4 5 6 7 8 9 IO II

I2 13 14 I5 I6

Area(in') . . Are&"') . . I, 0.008 I, 0.040 I! 0.010 I, 0.072 II 0.006 0 0.025 II . . I, 0.007

Thickness 0.006 (in) ,, II o.bbs ,I 0.012 II . . !, 0.020

0.007 . . . . . . mdicates 0.007

O.&S 0. iS5 minimum gage

0.024 0.076 0.161 0.248 0.008 0.082 0.135 0.077 0.157 0.198 O.&S 0.006 0.092 0.201 0.112

. . . . . .

O.&l7 :: ::

0.014 0.011 0.056 0.011 0.023 0.040 0.028 0.089 0.048

0.007 .. 0.020 o.oio 0.0202

Total Weight

Ibs. 7.892 8.897 10.925 19.993

Pattern B : II II

Table 3 Final design data (pattern C)

Design Type of Overall Stiffness (Ib./mch) Varrable Variable Number (Unit) 250 333.33 500 IO00 Remark

Thickness (inch) I,

0.0175 0.012 . . 0.0175 0.011 0.0176 0.056 0.0;2 0.050 0.096 0.131 0.014 0.039 0.0&3 0.043 0.087 O.lZl 0.024 0.034 0.043 0.039 0.088 0.129 0.022 0.015 0.016

0.014 . . indicates 0.013 mmimum 0.189 .w.= 0.316 0.132 0.282 0.092 0.170 0.019

0.019 0.015 0.012 0.038 0.019 0.015 0.011 0.032 0.034 0.031 0.037 0.112 0.020 0.020 0.020 0.026

Total Weight

Ibs. 8.614 9.856 12.134 23.517

5

--- ---~----6----1

2 I el 21 7 I

Patter" c: I

9 r 12$ I g

I3

4 IO

details of the member-configurations are left with the designer. The designer thus has the added flexibility of choosing the proper configurations to match the computed properties predicted by the optimizer. The design does not have to be re-optimized but the primary variables are simply used to determine the values of the so- called secondary variables in some effective way. Thus, a number of sectional configurations can be stipulated for any given reinforcing pattern, such as A, B or C. The particular one that satisfies some additional nonstructural requirements such as space, size or style, could be chosen as the final one. The design charts, similar to that shown in Figure 8, provide means of designing structures to its near



optimal. These curves can be generated for a wide range of aspect ratios (length to width ratio) and for different constraints like overall bending, torsion or both combined. These charts can also be generated for a wide range of materials such as aluminium, SMC, XMC, etc. The simplifications outlined within the body of this paper provide means of generating these curves and tables rapidly and economically. More importantly it makes the vast majority of the cross-sectional configurations available for a reinforcing pattern design.

Acknowledgement

The author thanks Dr P. Beardmore for careful reading of the manuscript and providing help for its timely com- pletion. Thanks are also due to Drs C. T. Chon, H. T. Kulkarni and S. Tang for their suggestions.

References

1

2

8

9

Schmit, L. A. ‘Structural design by systematic synthesis’, Proc. 2nd Con/: Electronic Cornput., ASCE, New York, 1960, pp. 105-122 Schmit, L. A. and Miura, H. ‘An advanced structural analysis/ synthesis capability - ACCESS 2’, Int. J. Numer. Meth. Engg. 1978.12.353 Haftka, R. T. and Prasad, B. ‘Programs for analysis and resizing of complex structures’, Comput. Struct. 1979, 10, 323 Isreb, M. ‘DESAPI: A structural sysnthesis with stress and local instability constraints’, Comput. Struct. 1978, 8, 243 Venkayya, V. B. ‘Structural optimization: a review and some recommendations’,l)zt. J. Numer. Meth. Engg. 1978, 13, 203 Schmit, L. A. and Miura, 11. ‘Approximation concepts for efficient structural synthesis’, NASA CR 2552, 1976 Rosen, A. and Schmit, L. A. ‘Design oriented analysis of imperfect truss structures ~ part II - approximate analysis’, Int. J. Numer. Meth. Engg. 1980, 15,483 Chang, D. C. and Justusson, J. W. ‘Structural requirements in material substitution for car weight reduction’, SAT: publication 760023,1976 Prasad, B. and Ilaftka, R. T. ‘Organization of PARS - a structural resizing system’, Adv. in Bngng Software 1982,4

Cl),9 10

11

12

13

Whetstone, W. D. ‘EISI-EAL: Engineering analysis language’, Proc. 2nd Speciality Conf: on Comput. Civ. Engng., ASCE, Baltimore, Maryland, 9-13 June, 1980 Prasad, B. ‘An improved variable penalty algorithm for automated structural design’, Comput. Meth. in Appl. Mech. Engng 1982,30,245 Chang, D. C. and Barone, M. R. ‘Structural optimization in panel design’, SAE Publication 770610, 1978 Prasad, B. and Haftka, R. T. ‘Optimal structural design with plate finite elements’, J. Struct. Div., AS@, 1979, 105 (ST1 l), 2367

14

15

16

17

Prasad, B. ‘Some considerations in efficient design of light- weight structures’, 4th Int. Conf: Vehicle Structural Mech., Detroit. Michigan, 18-20 November! 1981 Zangwill, W. I. ‘nonlinear programmmg: a unified approach , Prentice-Hall, Englewood Cliffs, New Jersey, 1969 Luenberger, D. C. ‘Introduction to linear and nonlinear programming’, Addison-Wesley, Reading, Massachusetts, 1973 Svanberg, K. ‘On local and global minima in structural optimization’, Proc. Int. Symp. on Optimum Struct. Design, 19-22 October, 1981, Tucson, Arizona, pp. 6.3-6.9


Appendix 1

Basic definitions

Here, we outline some basic definitions and results concern- ing nonlinear optimization in general and convex problems in particular. These will be utilized in appendix 2. Some of the results collected have been known for several decades and therefore we defer the proofs. A more general descrip- tion and proof in some cases may be found in references 15-I 7.

Structural problem Q

A structural problem (say Q) for a typical minimum weight design may be posed as:

Q: Min. M(v); v E Rn (Al .l)

subject to:

gj(V)>O; j= 1, . . ..m (Al .2)

and

@” < vi f pimax; i= 1, . . ..n (Al .3)

where M is the mass and gj represents a behviour constraint (such as deflection, stress, etc.). vTin and vLpax are bounds on the design variables.

Definition A. 1. A point v* of the design space D is said said to be a global minimum point of Q if:

M(v*) <M(v); for Vv E L? (Al .4)

where D and GI are defined as:

D={vER~~v~~*<v~<v~“~~ i= 1, . . ..n>

L?={(vEDlg;(v)>O j=l ,...,mI

Definition A.2. A point v* of the design space D is said to be a local minimum point of Q if:

M(v*) S M(v) for Vv E L2

such that [(v - v*ll < 6; F > 0. If v* is a global minimum point then v* is obviously a local minimum point, but the converse is in general not true.

Definition A.3. A design space D is said to be convex if:

{~‘r+(l-~)vzIED;

O<p< 1 foreveryvI,v2ED (A1.5)

Definition A.4. A real valued function g, defined on a design space D is said to be convex over D if:

g{LW + (1 -V) PZI G N(V1) + (1 - cl)g(vz);

0 </J< 1 (A1.6)

for every vr, v2 ED. For continuously differentiable functions, there is an

alternative characterization of convexity (more relevant to structures).

Definition A.5. A continuously differentiable function g, defined on a convex design space D, is said to be convex over D, if and only if the Hessian matrix of g is semi- positive definite throughout D, i.e.:


a*g XT[H]X = cc--- XiXl 2 0

i I aVi aV,

for all v ED and for every x E Rn.

(Al .7) With indeterminate systems, g, in general, cannot be

expressed into the form of equation (A2.4). This gives rise to the following very important theorem.

Definition A.6. For structural problem Q: let us assume that there exists a response function or a constraint which can be expressed into a matrix form:

gi = c [YTLAIPl (Al .S)

where [A] is a square matrix, p and qi are vectors and c is a constant. The load path, associated with a constraint gi in such a case, is said to be unique if and only if the vector qj can be expressed as:

4j = h;P (Al .9)

where Xi is a constant.

Theorem I. For those structural design problems in which the constraints gi can be expressed as:

gi = q,T[k]-‘p or 41T[A]-‘p (A2.5)

then the constraints will be convex over D if and only if the pre- and post-multiplying vectors are related by:

4j = $P (A2.6)

where K and A are symmetric matrices. To prove this theorem we utilize the results of the defi-

nition A2.5 and show that the resulting Hessian matrix of g is positive semi-definite for all v E s2. If we differentiate the constraintsgi in equation (A2.5) once, we can write (dropping subscript j):

Definition A. 7. If a response function is convex over a specified design space D or its subspace then the load path associated with that function will also be unique over that space or subspace.

Definition A.8. A structural design problem Q is said to be a convex (optimization) problem, if the feasible set s2 is convex over the design space D and the mass M(v) is convex over a.

dg _ = +k-’ [ I ; k-‘p

dVi I

where we use the formula:

ak-’

(A2.7)

av-=-k-l ak k-’ I 1 hi

(A2.8) I

For some finite elements, which exhibit a linear relationship between the stiffness matrix and the resized dimensions (such as truss, shear pannel and membrane plate elements), we can express equation (A2.7) as:

Appendix 2

Proof for design analogy

According to the results of appendix 1, the sufficient condi- tion for structural design problem Q to be convex is to show that the mass M(U) and the constraint functions gj; j=l > ..., m are convex over the design space D. Since mass is linear functions of design variables, using the definition A.4 it is not difficult to show that M(v) is convex. The constraint function, in general, could be both convex and nonconvex. Nonconvexity will not hold, however, for some structures (such as determinate truss, etc.) which give rise to behaviour constraints that can readily be expressed in a special form qiT[A]p.

With the displacement constraint case, for example, we have the basic finite element equation:

ku =p (A2.1)

as - = - qTk-‘kik-‘p avi

(A2.9)

where ki is a part of the matrix k associated with Vi such that:

k=k,+ t kiVi i=l

(A2.10)

Differentiating equation (A2.9) once again:

3% as &Vi

= qTk-’ ak k-‘kik-lp + qT~-l/#-l~k-lp au1 I

=qTk-‘[kik-‘kit kik-‘kl]k-‘p

To prove that:

(A2.11)

or

u = [kj-‘p for all x.

We form using equation (A2.11):

If the displacement is measured in the direction of the load, we can formulate g for displacements in determinate systems as:

g = e’[k]-‘p

where (e} is a vector given by:

{e> = M

(A2.2) = C CqTk-‘[Xlklk-lXiki + Xikik-‘X,kl] k-‘p I i

(A2.12)

Denoting S = CXiki, which is symmetric, equation (A2.13) (A2.3) results:

or

g = xPT[kl-‘p (A2.4)

a*g - xixl = 2qTk-‘Sk-‘Sk-‘p aVi av,

(A2.13)



with 4 = @, we then obtain:

2h[pWW’Sk-‘p] = 2X(Wp)%‘(Wp)

(A2.14)

Since right-hand side of equation (A2.14) is a quadratic form and k-’ is positive definite, we conclude that:

Thus according to definition (A.7), if the problem Q consists of a single load case, the load path corresponding to this type of constraint function will also be unique. This implies that if one creates two models of the same structure in such a way that the corresponding load paths of each model are essentially the same, both the problems, according to the above theorem, will be convex and therefore it

a28 would result into the same minimum weights.

aVi aV)

xix,~O forVxED (A2.15)


modelling considerations in optimum design of reinforced structures

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