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OPTIMIZATION OF STRUCTURAL FRAMES WITH ELASTICAND PLASTIC CONSTRAINTSBAHA AYDIN AKBORA a , ROSS B. COROTIS b & J. HUGH ELLIS Associate Professor ca Department of Civil Engineering , The Johns Hopkins University , Baltimore, Maryland,21218b Associate Dean for Academic Affairs and Hackerman Professor of Civil Engineering , TheJohns Hopkins University , Baltimore, Maryland, 21218c Department of Geography and Environmental Engineering , The Johns Hopkins University ,Baltimore, Maryland, 21218Published online: 04 Sep 2007.

To cite this article: BAHA AYDIN AKBORA , ROSS B. COROTIS & J. HUGH ELLIS Associate Professor (1993) OPTIMIZATIONOF STRUCTURAL FRAMES WITH ELASTIC AND PLASTIC CONSTRAINTS, Civil Engineering Systems, 10:2, 147-169, DOI:10.1080/02630259308970120

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OPTIMIZATION OF STRUJCTUJlRAIL IFWMIES WITH ELASTIC AND PLASTIC CONSTMINTS

BAHA AYDIN m O R A * , ROSS B. COROTIS~ and J. HUGH ELLIS'

*Department of Civil Engineering, The Johns Hopkins Universi@, Baltimore, Maryland 21218; now, Civil StafEngineer, Bechtel Co~orat ion, Gaithersburg, Maryland 20878 '~ssociate Dean for Academic Aflairs and Hackerman Professor of Civil Engineering,

The Johns Hopkins University, Baltimore, Maryland 2 12 18 '~ssociate Professor, Department of Geography and Environmental Engineering, The

Johns Hopkins University, Baltimore, Maryland, 21218

(Received, August 1992)

Structural design methods that employ optimization have traditionally followed one of two approaches: optimal elastic design which considers constraints on elastic stresses; and, optimal plastic design, involving ultimate load constraints. This study addresses elastic and plastic design, incorporated into an iterative scheme for checking both linear elastic stresses and plastic collapse load factors. Structural optimization of steel frames is achieved through tnatcrial reallocation, which is governcd by a gcneralized stress parameter in thosc cases where elastic design controls, and by a reduced cost vector from a linear program when the plastic collapse load factor controls. Regression analysis techniques among memberproperties are employed to reduce the number ofindependent design variables and to incorporate buckling and axial load effects for the elastic design. The linear programming optimization procedure is implemented in a computer system developed for thc minimum-weight design of steel frames. Our model is applied to one story-one bay, two story-two bay, and three story-five bay frames.

KEY WORDS: Structural Engineering, Optimization, Frames, Failurc.

Steel structures, while analyzed as a system, are designed principally on a member-by-mem- ber basis. Such a technique, as normally applied, leads to a practical structure that meets applicable code requirements. The designers, however, have no direct information on the implicit safety of their design with respect to ultimate collapse. While it is possible to conduct a plastic analysis subsequent to elastic design, such information would be more useful during the design process itself. This paper reports on a method that contains both elastically-based code conditions and system collapse load factor as constraints. An optimization approach is applied in this research to find a minimum weight structure, subject to constraints of the load and resistance factor design (LRFD) method'. At each stage of the optimization, material is reallocated within the structure to most efficiently meet the binding constraints of either

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148 H. A. AKBORA, R . B. COROTlS AND J . H . ELLIS

code requirement or plastic safety. The method is a practical design tool, producing an optimized frame from geometric and load input data. The structure meets code conditions, and provides the designer with a minimum load factor against collapse. all in an optimization context.

In the design method described in this paper. the process begins with the use of approximate solution techniques that reduce the indeterminate structure to statical detenni- nacy for generating an initial design. It is assumed that the best design is filly stressed, that is, each member is stressed to its full capacity (unless specified otherwise) under at least one loading condition. The design for each trial structure is then analyzed elastically with a linear-elastic program, and plastically by linear programming techniques. A different optimization scheme is employed, depending on whether the minimum weight design is governed by the elastic constraints or the plastic collapse load factor. Throughout this work the plastic moment-length o fa member will be used as a surrogate for its weight. The plastic moment-length is the product of the plastic moment capacity of the cross section and the length ofthe member.

The vast majority of existing steel structures have been designed by elas~ic design methods. The design profession is, however, moving rapidly in the directiori of load and resistance factor design, which includes many of the design concepts commonly associated with plastic design. In this study, both elastic and plastic constraints will be considered in the context of structural optimization.

APPROXIMATE ANALYSTS TECHNTQUES

In order to analyze a statically indeterminate structure, one needs to know the axial and flexural rigidities of its members. It is therefore necessary to first cany out an approximate analysis for member forces. Once tentative member sizes have been assigned, an elastic analysis may be performed. The approximate methods used herein are described very briefly, since they are well established. In this study, a separate approximate analysis is conducted for lateral and gravity loads, and the forces are superimposed.

For lateral loads, the analysis is based on the portal method, which assumes that each interior column carries twice as much story shear as each exterior column, and that there is a point of zero moment at the center of each column and girder. For gravity loads, each girder is assumed to have a point of inflection at one-tenth of the span length from each end joint. The moments at the ends of the girders are distributed between the columns in proportion to their stifbesses. At the initial stage of the design, it is assumed that all columns have the same moment of inertia.

CHARACTERIZATION OF MEMBER PROPERTIES

From the point of view of practical design of planar steel structures, the design variables - moment of inertia (I), cross-sectional area (A), plastic section modulus (Z) and radius of gyration (r) - are highly interdependent for a spectrum of cross sections. It is therefore

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STRUCTURAL OPTIMlZATION 149

P? 5 14000

F: ;t. 12000

et 'i: !OOOO k

8000 4

3 so00 Y

I, = 0.154A' + 33.265A - 0.743

$ 4000 Actual Regression Curve used in the analysis. Dote Indicate

$ 2000

0 0 40 80 120 160 200 240 280 320

Cross Sectional Area (inch8)

Figure I Regression Curve for Columns, Moment of Lnertia vs Area

convenient to vary only one decision variable during the optimization process and express the other three variables as a function of the fourth. This follows the lead of Grierson and camcronz, Grierson and ~ e e ~ and Ishikawa et al.4, who used regression analysis and concluded that it was as an effective tool to reduce the number of independent design variables. Their results were restricted to dimensionless relations. In this study a regression analysis is performed with cross-sectional area selected as the independent variable.

The range of sections for the regression analysis extends from W 14 x 30 to W 14 x 730 for columns and from W 18 x 35 to W 36 x 359 for girders (compact sections only). The results are presented in Figures 1 - 5, where all values refer to the strong axis. As can also be seen from the figures, the regression analyses agree with data points quite well. The regressions for columns yield closer fits than those for beams due to our considering only W 14 series beams. Figure 6 presents a regression curve for weak axis bending of columns, which is needed if the column is not braced in its weak axis. The equations are given on the figures, where all units are in terms of inches.

Relationships established up to this point between member properties still do not provide sufficient information to solve for member sizes of columns due to the effect of slenderness on column capacity. Considering a multi-story frame, it can be assumed for the initial design that the effective length parameter K is unity for sidesway-inhibited frames and K = 1.3 for sidesway-uninhibited frames with fixed support. More precise values are computed from actual section properties during subsequent iterations.

To determine the critical or buckling strength (F,,, which depends on the ratio KL/r) it is necessary to estimate the column radius ofgyration during the initial design phase. The value of r,, = 6.0 has been adopted for strong axis buckling. It can be seen from Figure 3 that this is a reasonable approximation for W14 column sections of widely varying cross-sectional areas. For weak axis buckling, regression curves with W 14 sections indicated three distinct regions, each of relatively constant values for radius of gyration. These regions were

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150 B. A. AKBORA, R. B. COROTIS AND J. H. ELLIS

I 1 I L I I I

- - w - d - - -

- - -

2, = 8.833 1om3A' + 5.864A - 6.231 - Actual Regression Curve used - in the analysis. Dots indicate -

data points. -

I I

Cross Sectional Area (inchx)

Figure 2 Regression Curve for Columns. Plastic Scaion Modulus vs Area

observed to correspond to flange width values. A value of r, = 2.0 was representative of scctions with areas below 15 in2 typical ofcolumns supportingup to four stories. For column sections with areas between 15 in2 and 40 in2, a value of r, = 3.0 was found to be representative. This will be assumed for initial design of columns supporting five to ten stories. Finally, larger columns are represented by r, = 4.0. It should be recalled that these are used for the initial design only, being replaced in subsequent cycles by the values for the actual sections.

Crass Sectional Area (inch')

Figure 3 Regression Curve for Columns, Radius of Gyration vs Area

I I T

2 10 9 - c

F:

3 2 6 - n 'a 5 -

'a 4 -

3 3 - 3 2 - d P: 1 -

0

I 1 I I I I 1 - - - - - - -

r, = -1.1 10-'A' t 1.307 10-'A f 5.737 - Actual Regression Curve used - in the analysis. Dots indicate -

data points. - I I I I I I I -

0 40 80 120 160 200 240 280 320

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STRUCTURAL OPTIMIZATION 151

12000 t I I I

v 7 f f000 - d

4 ; 10000 - 9000 - . I, = 3.845A1 f 27.836A - 288.271

8 8000 - Actual Regression Curve used f 7000 - in the analysis. Dots indicate

6000 - data points. 4

5000 - u 4000 - ' 3000 -

.oo* - C ,000 -

0 0 I 0 20 30 40 50

Cross Sectional A r e a (incht)

Figure 4 Regression Curve for Girders, Moment of Inertia vs Area

ELASTIC DESIGN

Cross Sectional Area (inchz)

Designers generally have used the allowable stress design method to proportion steel members. Today, much of the profcssion recognizes the advantages of load and resistance factor design (LRFD). The AISC specification presents load factors and load combinations that were selected for use with the recommended minimum loads given in the American

Figure 5 Regression Curve for Girders, Plastic Section Modulus vs Area

2400 2 2200

c. 2000

3 ieoo w 3 1600 u

- I I I I I 1 I I I I

- - - - 2, = 14.499A - 97.975 - - Actual Regression Curve used -

in the analysis. Dots indicate . s ;;;; 1

- data points.

too0 - - ..

% 800 - - 600 - -

2 roo - - 3 200 - 4

0 0 10 20 30 40 50 60 70 80 90 100 i f 0

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152 B. A. AKBORA. R . B. COROTIS AND J . H . ELLIS

0 40 80 120 160 200 240 280 320 Cross Sectional Area (inch*)

Figure 6 Regression Curvu for Columns @-y =IS) Moriitnt of Inertia vs Area

National Standards Institute (ANSI) specifications. These were developed from a detailed study6, the results of which will be used in this paper. The details of the approach that has been used for the elastic design can be found in Appendix I.

PLASTIC ANALYSIS

Modem design of structures involves principles of both elastic and plastic analyses. Plastic analysis can give useful information about the collapse load and the mode of collapse. Thc

- 25 ft. - Figure 7 Portal Frame

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STRUCTURAL OPTIMIZATION 153

main objective of plastic analysis is to determine the collapse load of a structure when the resisting capacities of its members are known. For plastic analysis in this study, the simplified rigid-plastic model of material behavior will be used, with no axial-moment interaction. The analysis procedures for plastic behavior are described in Appendix 11.

SOLUTION TECHNIQUES

In this study, elastic and plastic analyses are performed, and their results are compared. A single bay, single story frame is presented here as an example to illustrate the procedure. The planar steel framework has fixed-end supports and is subject to the two design load cases given by Eqs. (5) and (6) in Appendix I. Representative load values used in these expressions are D = 80 lb/ft2 (3800 ~ l m ' ) , L = 100 lb/ft2 (4800 ~ l m ~ ) and W = 2400 ib (10.7 KN). Dead load is computed based on a Cinch (15.2 cm) thick concrete slab, the value for the live load is obtained from the American National Standard (ANSI 1982) for office buildings, and wind load is computed using the provisions of the New York City Building Code as the reference, where the wind speed is 80 mph (129 km/h) (and pressure coefficient of 0.8 on the windward side and 0.3 on the leeward side). The geometry and configurntion of loads are presented in Figure 7. The spacing of the portals out of the plane is 15 A (4.6 m).

ELASTIC ANALYSIS SOLUTION

The linear behavior of the elastic analysis used in this study allows for scaling of all individual member sizes, keeping the relative rigidities constant. The appropriate scaling factor is obtained from the results ofthe elastic analysis program and the interaction formulas presented in Equations 10 and 1 1. In the first cycle of each resealing, an approximation is used due to the nonlinearity of the buckling equations.

For the simple portal frame chosen as the example, the values of the interaction formula (labelled as "Elastic Constraint Value") for each member are presented in Figures 8 and 9, corresponding to the load cases ofEq. (5) and Eq. (6), respectively. In this particular example it is assumed that both columns have the same member properties, which decreases the optimization space to two dimensions. Therefore, it is possible to analyze the structure for each column to girder moment of inertia ratio. Using the regression curves mentioned previously, the abscissa in Figures 8 and 9 has been expressed in terms of plastic moment capacity of the members. The total amount of material of the structure is kept at the value obtained from the approximate solution techniques used at the initial design step (1 0477.45 k-f?) (4329.63 K N - ~ ~ ) . For each moment ratio, the higher curve controls the elastic design. As can also be seen from Figures 8 and 9, for low plastic moment ratios, the elastic stress constraint value in columns (actually in the right column) has a higher value, which indicates that this is most critical in that range.

The graphs presented in Figures 8 and 9 are used to scale up or down member sizes in order to achieve a safe structure. The value obtained from the graph for a particular ratio is

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154 B. A. AKBORA. R . B. COROTlS AND J. H. ELLIS

I I I I I I I I I I I - L e f t C o l u m n - - * ..... *... R i g h t C o l u m n - - -.-.* G i r d e r d

- - - 7

-.-.dm.

*C.-.C-.- -.-. - -

- - - -

I I I 1 I I I I I 1 1

R a t i o o f Plastic M o m e n t s , Mp,/k$,

Figure 8 Elastic Analysis Results. Portal Frame with Loading presented in Equation 5.

used as the scaling factor for the structure to meet elastic design requirements with at least onc fully stressed member.

The plastic moment-length of thc new, scaled structure is computed for later comparison with thc plastic design solution. The plastic moment-length of the structure is defined as:

I I I 1 I I I I 1 I 1 - . - - , C e f t C o l u m n - - ., ..-..-.-.. Right C o l u m n - - .

-.-.a G i r d e r - - -

- - - -.. d -.. -. .... -..... - - - - .-.-.-

-.-.-.- - - - - - - - d

I I I I 1 1 1 t I I I

R a t i o o f P l a s t i c Moments. bC/MPp

Figure 9 Elastic Analysis Results, Portal Frame with Loading prescntcd in Equation 6 .

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STRUCTURAL OPTIMIZATION

n

in which Mpi = the plastic moment capacity of each member in k-ft,

L, = the length of each member inft, and

n = the number of members in the structure.

Plastic analysis solution

Using a method analogous to that for elastic analysis, it is possible to show different collapse mechanisms as a function of the ratio of the chosen plastic moment capacities. Figure 10 illustrates the change in minimum collapse load factor with respect to changes in moment capacity ratios. Due to the linearity of the mathematical program for the kinematic method, member moment capacities arc directly proportional to load. Therefore, the collapse load factor is directly related to the plastic moment capacity. The plastic failure envelope, presented in Figure 1 1, is derived from the minima ofthc curves shown in Figure 10, scaled to a collapse load factor of unity.

- Beam Uechani sm w i th . . . . . . . . . . Beam Mechanism w i th

Corner Hinges in Columns Corner Hinges in Beam

0 . 0 t 0 . 0 0 . 5 I .O 1 . S 2 . 0 2 .5 3 . 0

Ratio o f Plastic Moments. be/&

Figure 10 Collapse Load Analysis of the Portal Frame with Loading presented in Equation 5

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156 B. A. AKBORA. R. B. COROTIS AND 3. H . ELLIS

R a t i o o f P l a s t i c Momcnls. 4, / Lb,

Figure I 1 Plastic Analysis of the Portal Frame with Loading presented in Equation 5

The design ofthe structural frame will be achieved through elastic analysis and an additional plastic mechanism check. The solution for elastic design is compared to that for plastic design. Material reallocation to achieve a lighter structure (which still meets the safety requirements) is accomplished using either elastic or plastic optimization techniques, depending on which criterion requires more plastic moment-length. For instance, consider the case in which the elastic constraints can be met with a lighter (lower plastic moment- length) structure than is necessary to satisfy the specified collapse load factor. Then the total requisite weight of the structure is controlled by the collapse load factor constraint. Therefore. plastic optimization techniques are applied to reallocate the material among the members in the structure, bringing the total structural weight down.

OPTIMIZATION TECHNIQUES

Elastic method

The conventional approach to elastic design optimization is to proportion the members of the structure separately to satisfy a prescribed set of performance criteria, such as stress limits. Another approach to elastic optimization is the use of sensitivity analysis. Grierson and schmit7 have suggested this method of optimization by applying sensitivity analysis to

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STRUCTURAL OPTIMIZATION 157

approximate the performance constraints as linear functions of the design variables. In this study, an elastic analysis is performed on an initial design, and maximum elastic

constraint values (the left side of equations 10 and 1 1, here denoted as a) are computed for each member. These values indicate the material adjustment required for individual mem- bers. Rather than performing this adjustment individually on each member, the average of the maximum elastic constraint from each member is found:

in which

a,, = the average elastic constraint value,

a = the computed maximum elastic constraint value for the ith member, and

n = the number of members in the structure.

This average constraint value for the structure is used to modify member sizes by subtracting it from the actual constraint value of each member. This approach may be considered one of basic optimality criteria. Plastic moment capacity is chosen here as the independent variable to represent adjustment in member sizes. It is important to remember, however, that the plastic moment capacity is being used to select members that optimally satisfy the elastic design constraints. The change in member plastic moment capacity is given by:

AMP;= (a, - a,,)*Mp:

in which

AMPI = the change in the plastic moment capacity, and

Mi, = the current plastic moment capacity.

This optimization technique, as compared to the conventional approach, makes reduced modifications to the member sizes. Combined with the scaling mechanism presented earlier, this technique produces intermediate solutions during the iterative procedure that satisfy the elastic requirements.

Plastic method

The search technique used for the plastic mechanism check is completely different from that used for elastic design. This search is based on reduced cost information which is provided by the linear programming solver. The reduced cost indicates the amount by which a variable

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158 R. A. AKBORA, R. B. COROTIS AND J. H. ELLIS

(plastic moment-length) can be changed to improve the objective hnction without altering the basis (mode of collapse associated with the minimum collapse load factor). Thus, it directly provides the amount of material that can be removed from a hinge location without changing the failurc mechanism. This information is utilized to take out material in several members at once. A further constraint is imposed to keep all members prismatic. Because several members are altered in one iteration. a change in failure mechanism may occur with the new structure. Such an event is not a problem, since it is taken into account in subsequent iterations.

Among all critical locations along each member, the smallest reduced cost value is found for positive and negative rotation variables separately. Then the larger of these two values is used as the amount of material to be taken out from that member. Faster convergence to an optimal solution was noted when the larger of the negative or positive variables was used to reallocate material.

ITERATIVE PROCEDURE

The optimization technique to find the minimum weight structure undcr clastic and plastic constraints consists of an iterative procedure as described below:

1. Approximate solution techniques are applied to the indeterminate structure to obtain an initial design.

2. Based on an elastic structural analysis, the structure is scaled to satisfy clastic requirements.

3. Using the results of a plastic analysis, the structure is scaled to have a collapse load factor equal to unity.

4. The plastic moment-lengths derived with steps 2 and 3 are compared. If the value obtained in step 2 is greater, the optimization described in the earlier section entitled"E1astic Method" is performed. Otherwise, optimization as described in the section "Plastic Method" is performed.

5. The moditied structure is compared to the previous solution. Tf the current solution is lighter, it is stored and the process goes to step 2. If thc currcnt design is found identical to the previous solution, convergence has occurred and thc proccss stops. If the current design is heavier than the previous solution, an intermediate member size is computed by:

where Mp, is the new member size, Mp, is the current member size, Mq, is the previous member size and n is the number of successive trials in which a heavier structure has resulted. Mpp is not allowed to change unless a lighter structure is found. This search technique8 is based on the commonly used Fibonacci constant (i. e., 0.6 18.) If no structure is found after 3 successive iterations, the iterative process stops, presenting the last stored solution as the answer.

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STRUCTURAL OPTIMIZATION 159

Table 1 One Story, One Bay - Initial 'Trial' Structure

Section Marimum Graviiy Load Case Marimum Wind Load Case

APPLICATIONS

Three steel frames of ASTM A36 steel arc subjected to the loading conditions of equations 5 and 6, with the values of D, L and W used previously. The first example involves a single portal frame.

One story - one bay frame

The initial trial structure for the one story, one bay portal frame, presented in Figure 7, is obtained using the approximate solution technique. The results obtained are presented in Table 1.

Figure 12 illustrates the solution procedure for the maximum gravity load case (for this case Mpcl = M p F Z = MPc). Point 1.1 represents the initial structure given in Table 1. Reallocating and scaling according to the elastic criteria leads to point 1.2. Continued elastic optimization yields final points (1.3 and 1.4). An arbitrary starting point was also selected

1 1 I I I I 1 I I I I

- .......... E l a s t i c a l l y Safe Structure

- P l a s t i c a l l y Safe Structure Structures. obtained a t Intcrmadi a re - . Steps for Leas t-We ight Oasi gn

-

-

.... 1.1 .... ..... - ... .... ..-. ....

2.1 r

-

0 0 . 0 0 .5 I .O 1.5 2 . 0 2 . 5 3 . 0

Rat io o f P l a s t i c Moments, )$,/+

Figure 12 Search Procedure for D~ffcrent Starting Structures for Maximum Gravity Load Case

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160 B. A. AKBORA, K. B. CORO'TIS AND 3 . H . ELLIS

Table 2 One Story, One Bay - Minimum Weight Structure (wilh member restrictions)

Secrion Mr~rinirrnt Gruvip Load Case h.larinrimi Wind Load C~se

~ i f i , , I IS7.5 1 I;-fi

M P ~ Z 187.51 k-f? Mpp 155.02 k-A h L ,067 Failure hlcchanism Ucam

124.93 b-li 124.93 k-A 101.45 k-R 1.0794 Beam

for the maximum gravity load case ( M p c l = MpF~ = 250 k-fi (339 KN-m), Mpg = I00 k-ft (136 KN-m)). This is point 2.1 in Figure 12. Plastic optimization leads to point 2.2. Elastic criteria then govern and elastic optimization is used to reach point 2.3, which is identical to point 1.4.

The 'initial' structure (presented in Table 1) is optimized for both loading cases with the columns restricted to have the same member sizcs. The final design, given in Table 2, represents the minimum weight structure. Thc failure mechanism row indicates how the least weight structure will fail plastically when elastic constraints are ignored. As can be sccn from the result that the values of h are grcater than unity, elastic constraints control in both cases for the minimum weight structure.

It should also be noted that in case of structural optimization under multiplc loading conditions, the solution obtained for the maximum life-time gravity load satisfies both loading conditions. This is due to the grcatereffect of vertical loading on this structure. With an increase of the horizontal load, it was observed that separate solutions were needed for the two load cases.

Two ston, - two bav, frame

Vertical and portal approximate solution techniques are applied to obtain the trial structure for a two story, two bay frame, shown in Figure 13. using the same values for D, L, and W as previously. The results obtained are presented in Table 3 .

The optimization results obtained by restricting all columns and girders within each floor are also shown In Table 3, along with the results without any restnetions on member sizes. The decrease in the plastic moment capacity of the middle columns can be explained as a result of the balanced gravity loading. which reduces the effect of member end moments on the size of the column.

The same structure was subjected to both load cases with the wind load arbitrarily scaled up (after some experimentation, by a factor of 12.5). The resulting minimum weight structures were seen to have some member sizes controlled by the maximum wind load case, and some by the maximum gravity load case. A structure with the larger of each member size was analyzed under the two load cases separately. It was observed that the structure was overdesigned for both load cases, and member sizes could be reduced proportionately by 7%. Further material savings was possible, but only with a reallocation among members.

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Tab

le 3

M

embe

r Pro

pert

ies f

or th

e T

wo

Stor

y, T

wo

Bay

Str

uctu

re

Initi

al "

Tri

al" S

truc

ture

M

in.

Wei

ght S

truc

ture

with

Mem

ber

Res

tric

tions

M

in.

Wei

ght S

truc

ture

with

out M

embe

r R

estr

ictio

ns

Sect

ion

Max

. Gra

vity

Lou

d C

ase

Max

. W

ind

Loa

d C

ase

Max

. Gra

vity

Loa

d C

ase

Max

. W

ind

Loa

d C

ase

Max

. Gm

vity

Loa

d C

ase

Max

. W

ind

Loa

d C

ase

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162 B. A. AKRORA, R. B. COROTIS AND I. H. El-LIS

I2 ft.

t-- 25 R. - 25 ft. - Pigure 13 Two Bay, Two Story Portal Fnme

In order to illustrate the method on a larger structure, a three story, five bay frame was selected. From our experience with othcr structures, it was decided to use the maximum gravity load condition for design (equation 5). The geometry of thc strucrure was the same as that of Example 2 (Figure 13), with an additional story and three additional bays.

The program was allowed to determine the initial design from approximate methods with all section properties different. Although an elastic design check showed that this structure did not meet the design provisions in numerous cases, it was used as input to the iterations. The structure had a total amount of material of 1 1 1,620 k-j? (46,125 K N - ~ ~ ) . All subsequent optimizations were run with the restriction that the columns and girders within a single floor level would be identical.

Convergence to a final design was reached after four iterations of the program. In the first iteration, which resulted in total material of 141,918 k-ft2 (58,645 KN-m2), optimization by the "Elastic Method was called for by the program. This structure was heavier than the initial trial structure because the initial structure in this case did not satisfy the design check and as well, the revised structure was forced to have constant properties across bays. In the

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STRUCTURAL OPTIMIZATION

Table 4 Two Story, Five Bay - Minimum Weight Structure

Section Columns Girders - - - - - - - -

First Floor 312.07 k-A 2 11.89 k-A Second Floor 232.80 k-A 205.78 k-A Third Floor 139.32 k-A 229.1 5 k-A

second iteration, the "Plastic Method" was invoked by the program. The third and fourth iterations were both controlled by the elastic criteria. Final material in the structure was 130, 1 14 k:j? (53,767 KN-m2.)

SUMMARY

This paper has described modelling procedures involving the coordinated use of approxi- mate analysis, linear-elastic analysis, linear programming techniques for plastic design, and material reallocation methods. These procedures provide the designer with an effective tool to conduct the design of planar frames under the combined consideration of elastic design and the formation of plastic mechanisms.

The use of reduced cost information to reallocate material in the plastic analysis case provides an efficient tool to achieve optimality. Reduced costs appear to represent precisely the change to member sizes necessary to produce a plastically optimum structure.

It was observed that in the case of multiple load cases, it is not always possible to select one of the least weight solutions to satisfy all loading conditions simultaneously. The usual procedure of selecting the larger member size from each loading condition appears sub-op- timal (at least for these examples) and warrants further study.

Acknowledgments

Support for this research was provided by National Science Foundation under Grants MSM-88 144695 (RBC) and CES-87140 14 (JHE).

References

1. AISC. Load and resistance factor design spaification for structural steel buildings, American Institute of Steel Construction, Inc., Chicago, Illrnois, 1986.

2. Grierson, D.E. and Cameron, C.E. Computer-automated synthesis ofbuilding frameworks, Canadian Journal of Civil Engineering, 1984, 1 1 (4), 863-874.

3. Grierson, D.E. and Lee, W.H. Optimal synthesis of frameworks using standard sections, Journal of Computers and Smctures, 1984, 12(3), 335-370.

4. Ishikawa, N., M~hara, T, Katsuki, S. and Furukawa, K. Optimal design of skeletal structures under elastic and pla-.!ic design criteria, Proc. of JSCE Structural Engineering/Eanhquake Engineering. 1 984, Vol. 1, No. 2, October, 97- 104.

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1 64 R. A. AKBORA, R. B. COROTIS AND J . H . ELLIS

5. ANSI. Minimum desig~t lond~ for huildi~tgs and othersmrcrures. A58. 1-1982. American National Standards Institute, Inc., 1982, New York, NY.

6 . Ellingwood, B., Galambos. T.V., MacCregor. J.G. and Cornell, C.A. Develr~pmenl ofapmbahilih ba~edload critrrionfir Antprica~t nnrionnlrc1ndord.4TS. National B~~rcau o f Standi~rds Spccial Publication 5 7 7 , Junc. 1980.

7. Ciriersun, D.E. and Schurtt, L.A.. Jr. Synthes~s under service and ultimate performance constraints. .lmnrrrnrtl r<fComprc/rrs t r 1 1 ~ 1 Srnrcrlit-c,s, l 5(4), 19x2, 305-3 17.

8 Smith. A . A . . Hinton. E. and Lewis. R.W. Cit,if E ~ ~ g i ~ r ( ~ ~ r i ~ i g ~ ~ . s / ~ I ~ ~ s . ~ I I u ~ . Y ~ . s attdUesigrr, John Wiley & Sons, Inc., New York. 1984. 166-167.

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STRUCTURAL OPTIMIZATION

The abbreviations in these expressions are D for dead loads, L for live loads, W for wind loads, and U for ultimate load. The ultimate load for the lifetime maximum gravity load condition is given by:

and for the lifetime maximum wind load condition by:

There are as well, effective length interaction effects among all members in a frame. The most common procedure for obtaining effective length is to use alignment charts from the Structural Stability Research Council (SSRC) Guide (1976). The alignment chart method is also suggested in the AISC commentary (1986) as satisfying the 'rational method' require- ment. The computer coding ofthese alignment charts is achieved in a straightforwardmanner by solving the equations of the charts numerically for 'K'.

The equations are given as

for bruced frames and as

Ga GB ( x / K ) ~ - 36 - - n/K 6 (Go + GB) tan (n/K)

for unbraced frames, in which G is the usual joint column/girder stiffness factor. The values of G are computed at each stage with the member moments of inertia determined from the regression analysis.

The AISC Specification (1986) provides a formula for critical or buckling strength, F,,, in long columns with elastic buckling and another for short and intermediate columns with inelastic buckling. Both equations include estimated effects of residual stresses and initial out-of-straightness of the members. The design strength of a member is deter- mined as:

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166 9. A. AKBORA, R. 9. COROTIS AND J. H. ELLIS

with 4,. = 0.85. The inelastic formula is an empirical result that appears in the commentary of the ATSC

Manual ( 1986):

F,, = { exp [ -0.0424

for K L / r 5 4.7 1 &/F.

i n which E = the modulus ofelasticity of steel (29000 ksi) (2.0 x ~ O ~ K N I ~ ' ) and,

F, = specified minimum yield stress ofthe type of steel being used (36 ksi) ( 2 4 8 , 0 0 0 ~ ~ 1 m ~ ) .

For clastic or Euler buckling, the familiar Euler equation is multiplied by 0.877 to retlect the effcct of initial out-of-straightness.

for K L/r > 4,71 &/F" Axial forces in bending members also give rise to secondary, or P A, moments. Consistent

with most building Erame design, it will be assumed that such secondary effects are negligible.

By restricting beanis to compact sections, the full plastic moment, M, can be developed, as long as the compression flange is adequately braced. In many designs the floor diaphragm will provide this bracing. It will be assumed in this study that the actual unbraced length of a beam member does not exceed that which allows full development of the plastic moment. For long-span structures with relatively light members unbraced by the floor system, this condition might not be met, and the final design would have to be modified. The plastic moment may be determined as

with 4b = 0.9. In Section H1 of the AISC Specification, the following interaction equations are given

for symmetric shapes subjected simultaneously to bending and axial tension forces.

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STRUCTURAL OPTlMIZATlON

where P, and Mu are the required axial and flexural strengths, Pn and Mn are the nominal axial and flexural strengths and $t and $b = 0.9. The other terms in these equations have previously been defined. The same interaction equations are used for members subject to axial compression and bending, with $I replaced by +c = 0.85.

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B. A . AKBORA. R . B. COROTIS AND J . H . ELLIS

APPENDiX 11 - PLASTIC ANALYSIS PWOCElIDURlE

Existing methods for plastic analysis are based on either the kinematic or static theorems (Cohn et al. 1972). The combination of mechanisms technique indicates that all possible collapse mechanisms of a structure can be generated by linear combinations of rn indepen- dent mechanisms. This number. m, which is also the number of independent equilibrium equations. is given by:

in which NR = the degree of statical indeterminacy and

J = number of potential plastic hinges, or critical sections

According to the kinematic theorem ofplastic analysis, the load factor, ?,, and the associated collapse mode of the structure satisfy the following conditions:

in which 2.i =kinematic multiplier, MPj = plastic moments, B i j =hinge rotation rates, U, = total energy rate dissipated by plastic hinges, e, = external work rates of the unfactored loads and p = the total number of possible mechanisms.

The plastic analysis problem can be cast as a linearprogranz (Grierson and Gladwell 1971). The load factor h can then be computed by solving the following problem:

Find 0; and 9') (j = 1, ..., J) and t k (k = 1, ..., m) such that

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STRUCTURAL OPTIMIZATION 169

The problem has (2J+m) variables and J+l constraints. Using results from fractional programming (Charnes and Cooper 1962) permits the normalization of external work that yields the linear objective function (equation (14)) and associated constraint (equation (1 6)) from the original nonlinear objective (equation (13)).

The computer program based on the kinematic approach requires as input any set of elementary mechanisms (these may be selected from the hndamental equilibrium condi- tions if desired). A general procedure for identification of elementary mechanisms has been developed by Watwood (1979). A simpler procedure applicable only to regular frames has been developed for use here. The procedure has a built-in hinge identification mechanism, and assumes that two hinges can develop in columns (at the ends) and three hinges in girders (at the ends and center). A complete set of independent mechanisms is composed of beam mechanisms, sway mechanisms and joint mechanisms. These elementary mechanisms are stored for use later in formulating the linear program. The external work for each mechanism is determined by combining load data with the corresponding elementary mechanisms. Internal work is computed by using the rotations and the associated plastic moment capacities.

As a final step, a linear program input file is automatically prepared. This file specifies names for the constraints and variables, and defines all the constraints and bounds. It follows MPS format and thus permits the linear program to be solved with any of a large number of commonly available software packages.

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