..:>~~

~ • ..2.cIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 362

A GENERAL FORMULATION FOR

THE OPTIMUM DESIGN OF

FRAMED STRUCTURES

by

W. J. McCutcheon

S. J. Fenves

A Technical Report of a Research Program

Spo nsored by

THE OFFICE OF NAVAL RESEARCH DEPARTMENT OF THE NAVY

Contract No. N 0014-67-A-0305-001 0 Project NA VY-A-0305-001 0

UNIVERSITY OF I LLI NOIS URBANA, I LLI NOIS

AUGUST, 1970

A GENERAL FORMULATION FOR

TEE OPTIMUM DESIGN OF

FRAr.:IED STRUCTURES

by

'(,V. J. r,;:cCutcheon

S. J. Fenves

A Technical Report of a Research Program

Sponsored by T}-<]!; OFFICE OF N.LI~VAL RESEARCH

DEPAR TMENT OF 'IlKS ~1A VY Contract No. N 0014-67-A-0305-0010

Project NAVY-A-0305-0010

UNIV-BRS ITY OF ILLIr'~OIS URBANA, ILLINOIS

AUGUST, 1970

AC KNOWLEDGENIENTS

This report was prepared as a doctoral dissertation

by I','Ir. William J. McCutcheon and was submi tted to the Gradua te

College of the University of Illinois at Urbana-Champaign in

partial fulfillment of the requireme~ts for the degree of

Doctor of Philosophy in Civil ~ngineering. The work was done

under the supervision of Dr. steven J. Fenves, Professor of

Civil Engineering.

iii

TABLE OF' CONTENTS

Page

ACKNOWLEDGEIvlENTS • . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . vii

Chapter

1 .L

2

INTRODUCTION . . . . . . . . . . . . . . . . 1

1.1 Purpose •••••••••••••••• 1

1.2 Scope ••••••••••••••••• 2

1.3 Assumptions and Limitations ••• • •• 4

1.4 Organization of Report • • • . . . · . . BAS Ie FOR MULli TION • • • • • • • • • • • • • •

2.1 Structural Variables and Matrices • • •

2.2 Yield Constraints • • • • • • • • • • • 2.2.1

2.2.2

Equilibrium constraints • Member force constraints

. . . . • • • •

5

7

7

9 10

11

2.3 Working Stress Constraints. • • • • •• 17 2.3.1 Equilibrium constraints. • • •• 18

2.3.2 Member force constraints • • •• 18 2.3.3 Compatibility constraints. • •• 21

2.4 Optimum Design as a Linear Programming

2.5

Problem • • • • • • • • • • • • • • •• 22 2.4.1 Equilibrium constraints. • • •• 23 2.4.2

2.4.3 2.4.4

2.4.5

Member force constraints • • • • Compatibility constraints.

Objective function

Basic formulation • • • • •

• • • •

• • •

• • •

• • •

Specification of Member Groups • • • • •

iv

24

25 27 27

30

v

Chapter Page

3 GENERALIZA'rION OF FORMULA.TIOr~ FOR ALTERNATIVE

4

5

6

LOADS • • • • • • • • • • • • • • • • • • • • 33

3.1 Independent Loads and Alternative Loading Combinations • • · • • · • • • • 33

3.2 Generalization of Formulation · • • • · 35 3.2.1 Redefinitions • • · • • • • • • • 35 3.2.2 Equilibrium constraints · • • 37 3.2.3 Member force constraints • • • · 38 3.2.4 Compatibility constraints • • · • 40

3.2.5 Generalized formulation · • • • • 42

IMPLEIvrENTATION • • • • • • • • • • • • • • • , 44

4. 1 POST - FORTRAN Program • • • • • • • •• 44

4.2 Linear Programming Solver • • • • • • •

ILLUSTRATIVE EXAMPLES • • • • • • • • • • · •

5.1 Plane Frame Design • • • • • • • • • · • 5.1.1 Effect of load factor and stress

ratio • . • • • • • • • • • • · •

5.1.2 Effect of member force inter-action . • • • • • • • • • • • •

5.1.3 Effect of alternative loadings

5.2 Space Frame Design • • • • • • • • • • •

5.3 Computer Program Efficiency • • · • • •

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY

6.1 8onclusions • • • • • • • • • • • • • •

6.2 Suggestions for Further Study • • • • •

47

49

49

52

54 55

56

60

63

63

64

FIGURES

APPE:'·rDIX

LIST OF REFERENCES

vi

Page

68

77

81

LIST OF FIGURES

Figure Page

1 Computer Program Flow Diagram · • • · • · • • 68

2 Two-Story Plane Frame . · · · • · • • • · • • 70

3 Volume vs. Load Factor · · • • • • • • · · · 71

4 Volume vs. Reciprocal of Stress Ratio • • • • 72

5 Independent Loads on Plane Frame · • · • • • 73

6 Staircase . • . • • . . · • • • · • • • · • • 74

7 Staircase Support Structure • • • • • • • • • 75

8 Staircase Dead Plus Live Load (kips) • • • • 76

vii

Chapter 1

INTRODUCTICK

The increased availability and use of digital com­

puters has caused rapid changes in practically all areas of

study. In structural engineering, two trends have become

evident. The first of these has manifested itself in the

increased use of general network and matrix formulations in

structural analysis. The second trend has been toward the

use of mathematical programming techniques in the solution

of problems in optimal design. While many types of optimi­

zation problems have been treated in recent literature, very

little has been done to present a unified formulation for

the optimum design of framed structures.

1.1 Purnose

The object of this study is to develop a unified

approach for the formulation and solution of problems in­

volving the optimum design of structural systems. It is not

the purpose of this study to develop any new methods of

structural analysis or new techniques of mathematical pro­

gramming, but rather to demonstrate that a single formulation

may be used to formulate and solve most problems of optimum

structural design.

1

2

1.2 Scone

Methods of plastic analysis and design have tradi­

tionally been treated separately from elastic methods, and

the same trend has been followed in studies of structural

optimization. While it is recognized that linear programming

techniques may be applied to optimum design when limit analy­

sis or the theory of plastic collapse mechanisms is used [7,

12, 23J,* optimization problems involving elastic behavior

have, in general, been formulated as non-linear programming

problems [2, 14, 26J. Moses introduced the concept of apply­

ing linear programming to elastic structures by replacing

the non-linear equations by their first-order Taylor series

term [22J.

While much has been written concerning structural

optimization, the majority of the literature has been directed

toward the solution of particular structural types. Very

little has been done to present a unified approach, based on

a general matrix formulation, to the optimum design of framed

structures.

Fenves and Branin [5J developed the network-topo­

logical formulation of elastic structural analysis and

Gonzales and Fenves [8J have presented a general formulation

* Numbers in brackets indicate works in the List of References.

:3

for bo~h the analysis and design of rigid-plastic structures.

The latter work considers only design variables and con­

straints which relate to ultimate capacity, and the equations

obtained are linear. Even when stress interaction is consi­

dered, reasonable assumptions can be made to linearize the

problem.

This study develops a general formulation for

optimum design which includes the entire range of design

variables and behavior constraints. Elastic behavior, in

the form of working stress limitations, and ultimate capacity

under multiple loading conditions are considered. In addi­

tion, provisions are made for designating groups of members

to be made identical.

The formulation is presented in the form of an

iterative set of linear prograwning problems. The design

variables relating to elastic and ultimate behavior are line­

arly related and the objective function to be minimized is

expressed as a linear function of the design variables. By

making assumptions about the member properties and sizes,

the constraints which define the problem are also expressed

in linear form.

It would be possible to develop a somewhat more

general formulation in the form of a non-linear programming

problem, but methods now available for solving such problems

4

[2, 30J are not readily applicable, since they involve

selecting a feasible solution, moving the solution until a

constraint is encountered, moving along that constraint

until another constraint is encountered, determining whether

or not the neVI constraint is active, etc., until the optimum

is rea.ched. Because of the large number of constraints

which are possible, such a procedure can be a very laborious

and time-consuming operation, even when carried out on a

large-scale digital computer. Therefore, this study is re­

stricted to the linear programming approach only.

1.3 Assumptions and Limitations

The assumptions and limitations used in this study

are:

(a) Structural geometry is known. It is the sizes

of the members, rather than their configuration, which is to

be optimized"

(b) The types of structures to be considered are

plane grids, plane frames, and space frames, since these

types can be represented as networks. It is possible to

extend the formulations which are developed to trusses also,

but portions of the formulations will then degenerate (e.g.,

member force interaction need not be considered).

(c) The material is linearly elastic - perfectly

5

plastic. When yielding occurs, there is no "spread length"

alon~ the member.

(d) All members are straight and prismatic with

cross sections having two orthogonal axes of symmetry.

(e) Buckling is not considered.

(f) Deflections are assumed to be small. There­

fore, secondary effects can be ignored.

(g) yield hinges and working stress limits can

occur only at member ends. Therefore:

(h) Loads are restricted to concentrated loads

(forces and couples) at the joints. If it is desired to

place a load elsewhere, a fictitious joint must be inserted

at that point.

1.4 Organization of Report

In Chapter 2, after a brief review of structural

vectors and matrices, a basic formulation for optimum design

is developed. The yield and working stress constraints are

presented separately and then combined to form a linear pro­

gramming problem. A method for specifying "member groups"

is also presented.

Chapter 3 generalizes the basic formulation to

include multiple loads and alternative loading combinations.

Chapter 4 discusses a computer program which was

6

written to implement the formulations developed in Chapters

2 and 3.

In Chapter 5, several sample problems are presented.

These are solved using the computer program described in

Chapter 4. The efficiency of the program in solving these

problems is discussed.

Finally, Chapter 6 presents a summary of conclu­

sions reached during the course of this study and several

suggestions for further investigation.

Chapter 2

BASIC FORMULATION

This chapter presents a basic formulation for the

optimum design of framed structures. First, some basic con­

cepts concerning structural variables and matrices are re­

viewed, and then the types of constraint equations and in­

equalities necessary to describe the problem are discussed.

Finally, the basic formulation is presented as an iterative

set of linear programming problems, and a method for defining

"member groups" is presented.

2.1 structural Variables and Matrices

Consider a structure having b members and n free

joints. A member is represented as an oriented branch going

from its positive or A-end to ·its negative or B-end. Each

member has its own member coordinate system, with the member

x-axis lying along, and oriented in the same direction as,

the member. Since the members can have completely different

orientations and coordinate systems from one another, it is

necessary to define a single global coordinate system for

the structure as a whole. In this study, only frames and

grids are considered. For such structures, the number of

force and distortion vector components (degrees of freedom)

7

8

in member and global coordinates are the same and is denoted

by f.

The following vectors are needed to define the

forces and distortions of the structure (the notation used

follows that of Fenves and Branin [5J as closely as possible):

P' is a vector of externally applied joint loads,

consisting of n subvectors (one for each free joint), each

of order f. It is expressed in global coordinates.

u' is a vector of joint displacements, also in

global coordinates and of the same order as pt.

R is a vector of member forces, expressed in mem-

ber coordinates at the B-ends of the members and containing

b subvectors of order f.

V is a vector of member distortions, also in mem-

ber coordinates and of the same order as R.

The structural matrices used in this study are the

following:

A is the branch-node incidence matrix, a typical

submatrix, Aij , of which is (Tl, -HIT{, 0) if member i is

(negatively, positively, not) incident on joint j. Hi is a

square translation matrix of order f which transfers the

member force from a negative end force to a positive end

force (i.e., RAi = HiRBi ), both in member coordinates. T. l

is a square rotation matrix of order f which transforms the

9

member force vector at the negative end of member i from

member to global coordinates.

k is the unassembled stiffness matrix, a diagonal

matrix of b submatrices k., where k. is a square matrix of 1. 1.

order I which defines the stiffness of member i (i.e., Ri =

k.V.) for working loads. 1. 1.

other vectors and matrices will be defined as they

are introduced.

The stiffness (or node) method is the "classical"

method for elastic analysis and presents greater assets for

computer implementation than does the flexibility (or mesh)

method. Gonzales and Fenves [8J have shown that the node

method is also more readily adaptable for plastic analysis

and design. For these reasons, the formulations developed

in the following sections are based on the node method.

Therefore, it is not necessary in this study to employ the

node-to-datum path and branch-circuit matrices or the redun-

dant member force vector, since these are needed only for

the mesh method.

2.2 Yield Constraints

The static or lower-bound theorem of plastic analy­

sis [9J states that the load-carrying capacity of a struc-

ture is the largest load which corresponds to a statically

10

admissible state of stress. A statically admissible state

of stress is defined as a state in which:

(a) the stresses are in internal equilibrium, as

well as in equilibrium with the external loads; and

(b) the yield limit is not exceeded anywhere in

the structure.

Thus, for plastic analysis, two types of constraints,

corresponding to (a) and (b) above, are requiredl equilibrium

constraints and member yield force constraints.

Gonzales and Fenves [8J have shovm that plastic

design problems can be formulated similarly to plastic analy-

sis problems and that the same types of constraints are re-

quired for design. Thus, equilibrium and member force con-

straints must be imposed for design.

2.2.1 Equilibrium constraints

In order for the structure to be in equilibrium

(both internal and external), the sum of the member forces

at each joint must be equal to the externally applied load(s)

at that joint. In matrix notation:

A tR = P' Y Y

(2-1)

where Ry is a vector of member yield forces at the B-ends of

the members, P' is a vector of externally applied joint yield y

loads, and A is as defined in Section 2.1. The superscript

11

"t" denotes matrix transposition.

2.2.2 l';'Iember force constraints

Recalling that yield hinges can occur only at

member ends, let R' denote a vector of yield forces at both y

ends of the members in the structure. Ry can be expressed

as:

R' =QR (2-2) Y Y

where Q is a diagonal matrix of submatrices Q .• A typical l.

subvector P , . \. . yl.

of R' is: y

R '. = _;t~~ {

R A~ yl. R B. Y l.

(2-3)

Since Rt\. = H. R3.' Eq. 2-2 can be rewritten as: nl l.: 1

R BO

Y l.

where I is an identity matrix of order f.

of the form:

(2-4)

Therefore, Q. is l.

(2-5)

The vector R' contains all the member yield force y

12

components. However, since not all these components are

necessarily used in the definition of yielding, it is neces-

sary to extract from H' the components actually needed. This y

can be accomplished by the premultiplication of Rt by an y

extractor matrix ~ , so that: y

rtf = ~ R' Y Y Y

(2-6)

where the vector Ry contains only those components which are

used to define yielding and ~y is a diagonal matrix of sub­

matrices ~ .• y~

If, for example, the structure of interest is a

plane frame, Ry contains six (2f) elements per members namely,

axial force, shear force, and bending moment at each end.

Assuming that axial force and bending moment are used in de­

fining the yield criterion, a typical submat~ix ~ . will be yl

of the form:

(2-7)

In general, the combinations of member forces

which define yielding are not described by linear relations.

However, it is possible to approximate the yield surface as

a series of linear segments [13J. To continue with the above

13

example of a planar frame in which axial force and bending

moment are the components of interest, the general form for

one segment of the linearized yield surface at one end of a

typical member i is:

(2-8)

where p and m are the axial force and bending moment at that

end of member i, Pult and IDult are the ultimate values of p

and m (i.e., Pul t is the value of p which will cause plastic

yielding to occur when only axial force is acting on the

member, etc.), and 91 and 92 are constants which define the

yield surface segment.

It is assumed in this study that Pult and mult can

be linearly related to a single member reference yield ca­

pacity, Py • Thus, Pul t and mult can be expressed as:

1 - (a) Pult = y Py 1

(2-9) 1 - (b) mult = y Py

2

The yield capacity of member i, p ., can be selected as any y~

section property (e.g., cross-sectional area, plastic modu-

lus, mult ' etc.) which can be linearly related to the sec­

tion's ultimate axial force and moment capacities, as in

14

Eqs. 2-9, by constants 1/Y1 and 1/Y2.

Substituting Eqs. 2-9 into 2-8 and rearranging

terms gives:

Eq. 2-10 can be cast in matrix form as:

-p . -yl

(2-10)

> 0 (2-11)

Thus, it can be seen from Eq. 2-11 that a vector R* of mem­y

ber yield forces, normalized with respect to the member

plastic capacities, p , can be defined as: y

R* = r R' y y y (2-12)

fy isa diagonal matrix of submatrices r . which contain the yl

constants y. Recalling that Eq. 2-11 represents only one end

of member i, fyi for the above example can be written as:

r . = yl.

in which Yl and Y2 are constants defined in Eqs. 2-9.

(2-13)

Finally, ~~ is defined as a vector of linear com­

binations of member yield forces, and (again referring to

15

Eq. 2-11) can be written as:

R* = e R* y Y Y

(2-14)

where 8 is a diagonal matrix of submatrices e .. Y yl For example, in the 1970 AlSO Specification, the

linearized yield surface for compression and bending is de-

fined by:

+ m < 1 (a) - mult -(2-15)

---.:E.- + .85 _m_< 1 (b) Pult - mu1t -

Assuming that the same relation holds for combined tension

and bending, a typical submatrix e . will be: Yl

o 1 I , o -1 I

1 .85 : 1 -.85:

-1 .85 I

o

-1 -.8~: -------~l--O---i--

o I 0 -1 : 1 .85 : 1 -.85 I -1 .85 : -1 -.85

(2-16)

Thus, if the yield surface is defined by s linear segments y

(s = 6 above), e ).. will contain 2s rows. y y y

Defining U . as a 2Sy vector of ones, the yield yJ.

16

force constraints for member i can be written as:

u .p . - ~*. > 0 yl yl Yl- (2-17)

and the entire set of member yield force constraints as:

(2-18)

in which Py is a'vector of member yield capacities and Uy is

a diagonal matrix of vectors U .• Since each vector U . is yl y~

of the order 2Sy ' the matrix Uy is therefore of the order

2bs x b. y

Substituting Eqs. 2-14, 2-12, 2-6, and 2-2 into

2-18, the yield force constraints can be written in terms of

the basic variables, namely the forces, R , used in the y .

equilibrium equations (Eq. 2-1) and the unknown plastic ref-

erence capacities, P J as: y

(2-19) .

In summary, the transformations on Ry are, from

right to left, as follows I

Q transforms R , a vector of member B-end yield y

forces, into a vector of forces at both ends of the members;

~y extracts the components which are used to define

yielding;

fy normalizes the force components with respect to

the member reference yield capacities, p ; and y

17

8 y combines the normalized forces as defined by

the linearized yield surface.

IntroducingITy to denote the matrix product 8y fy6yQ,

the yield force constraints can be written as:

Up -DR >0 Y y Y Y -

(2-20)

The constraints of Eq. 2-20 can be represented

graphically as:

(2-21)

where n . represents the matrix product 8 .f .~ .Q .• yl yl yl yl 1

2.3 Working Stress Constraints

In the "classical" elastic solution of a framed

structure, three types of equations must be satisfied. equi-

librium equations, member force-distortion equations, and

structural compatibility equations. Due to the design nature

of the problems considered in this study, it is necessary to

impose a fourth set of constraints, namely member force con­

straints, to insure that working stress limits are not ex-

ceeded anywhere in the structure.

It will be shown in the following sections that

18

the equilibrium and member force constraints for working

loads can be formulated in a manner analogous to those for

yield loads, and that the member force-distortion equations

and structural compatibility equations can be combined into

a single set of "compatibility" constraints.

2.).1 Equilibrium constraints

The equilibrium equations for working loads are

exactly analogous to those for yield loads, namely:

A tR = P' w w (2-22)

The subscripts "w" indicate working forces and loads.

2.).2 Member force constraints

The member force constraints for working loads are

also very similar in form to those for yield loads. R~ is a

vector of working forces at both ends of the members and can

be expressed as:

R' = QR w w (2-23)

where Q is exactly the same as defined by Eq. 2-5, since

working stress limits, like yield limits, are specified only

at member ends.

R~ is a vector of member force components which are

used to define working stress limits. It can be written ass

19

H' =L1 Rt w w w (2-24)

~w is an extractor matrix similar to ~y,and extracts those

components which enter into the definition of working stress

limits, vlhich mayor may not be the same components used to

define yielding.

Pw (similar to Py) is defined as a vector of member

working capacities which can be related linearly to the mem­

bers' maximum allowable force components by a matrix f (also w

s~milar to its yield counterpart, fy). Thus, R~J a vector of

member force components normalized with respect to Pw' can be

computed by premultiplying R~ by fw:

. R* = r R' w w w (2-25)

R*, a vector of combinations of member force coro­w

ponents which define working stress limits, is obtained by

premultiplying R* bye: w w

R* = e R* (2-26) w w w

8w is a diagonal matrix of submatrices ewi~ For example,

limiting working stress combinations can be defined by the

inequalities:

+~+ m <1 Pall - mall

(2-27)

20

where p and m are the axial force and bending moment at one

end of a typical member, and Pall and mall are the maximum

allowable values of p and m when they act separately. A

typical submatrix, e .J for such a case will be: Wl

1 1 I I

1 -1 I 0

-1 1 I I

-1 -1 I e . = -------}--r---r (2-28) Wl

t 1 -1

0 , I -1 1 I I -1 -1

Thus, the formulation of the member force con-

straints for working conditions is analogous to that for

yield conditions, and can be written as:

where Uw is a diagonal matrix of unit vectors U ., each of Wl

order 2s , and s is the number of segments defining working' 'N \~

stress limits (equal to 4 in Eqs. 2-27 and 2-28).

Substituting Eqs. 2-26, 2-25, 2-24, and 2-23 into

2-29. the member working force constraints can be written as:

(2-30)

Thus, the form of the working force constraints is

similar to that for the yield force constraints. However,

there is still a large degree of independence between the two

21

sets of force constraints in that:

(a) the number and type of force components used

to define working limits can be different from those defining

yield (~w vs. ~y);

(b) the relationships between the maximum allowable

member forces and the working capacities, Pw' can be entirely

different from those between the ultimate forces and the

yield capacities, Py (fw vs. fy);

(c) the number of linear segments defining working

stress limits and yield limits can be different from one

another (sw vs. Sy); and

(d) the linear combinations of forces which define

the working stress limits can be different from those which

define yield (8w vs. 8 y ).

Using I1w to denote the matrix product 8wfw~Qf the

working force constraints can be rewritten as:

(2-31)

2.3.3 Comnatibility constraints

The compatibility requirements for the structure,

namely that the distortions of each member must be equal to

the difference between the deflections of its end joints, can

be stated in matrix form as:

22

v = Au' (2-32)

where V is a vector of member distortions at working loads

and u' is a vector of joint displacements.

The member compatibility requirements are defined

by the force-distortion equations:

(2-33)

where k is the unassembled stiffness matrix defined in Sec-

tion 2.1.

Since the member distortions, V, are not of primary

interest and need not be solved for, the two sets of equa-

tions above can be combined by substituting Eq. 2-32 into

2-33 and rearranging terms, giving:

kAu' - R = 0 w (2-34)

Eq. 2-34 represents the compatibility constraints as they are

used in this study.

2.4 Optimum Design as a Linear Programming Problem

The relationships of Eqs. 2-1, 2-20, 2-22, 2-31,

and 2-34 represent the entire set of constraints which are

needed to define the problem. As these constraints are formu-

lated, however, it is assumed that the yield variables and

working stress variables are independent of one another. It

23

is therefore necessary to define the relationships between

P' and P' of the equilibrium constraints (Eqs. 2-1 and 2-22) y w

and between pyand Pw of the member force constraints (Eqs.

2-20 and 2-)1) before the yield and working stress constraints

can be combined into a single formulation. It is also neces-

sary to recognize that the stiffness matrix k is not actually

known, since for a design problem the sizes of the members

are not known.

Thus, it is necessary to modify the constraints be­

fore an objective function is defined and the basic formula­

tion is presented as a linear programming problem.

2.4.1 Eguilibrium constraints

The applied joint loads for yield and working con-

ditions, Py and P~J are not actually independent of one

another. The yield loads can be taken as being equal to the

working loads times a "load factor":

P' = tP' Y w (2-35)

where t is the load factor.

The equilibrium constraints can therefore be re-

written as:

AtR = tP' Y

A tR = P' w

(a)

(b) (2-36)

24

The subscripts By" and "w" can thus be dropped from P' since

it is the only independent vector of joint loads.

2.4.2 Member force constraints

As the member force constraints are formulated in

Eqs. 2-20 and 2-31, the member yield capacities and working

capacities, p- and p , are independent of one another. This, y -w -

of course, is not the case, and it is necessary to take into

account the relationship between the two sets of capacities.

It is possible to define p (no subscript) as a vec-

tor of member reference capacities. These reference capaci-

ties can be selected as any section property of the members

(e.g., area) which can be related to both their yield and

working stress capacities, Py and Pw' by linear relationships

of the form:

= r yp -= rw p

(a)

(b) (2-37)

where rand r are diagonal matrices (of order b) of con-y w

stants r . and r .• The selection of values for ry and rw yl Wl

can involve many considerations (such as the member shape

factors, assumed member proportions, etc.), as will be illus-

trated in the examples of Chapter 5.

Using the above relationships, the member force

constraints can be rewritten in the following form:

Uyryp - I\Ry > 0

uwr wP - I\;Rw > 0

2.4.3 Compatibility constraints

(a)

(b)

25

(2-38)

As the compatibility constraints are presented in

Eq. 2-34, it is assumed that the member stiffnesses, as ex-

pressed by the k matrix, are known. This is not the case,

however, since the stiffness of a member depends on its size.

which is unknown. It is possible, however, to represent the

stiffness matrix of any member as a matrix of constants times

that member's reference capacitys

k. = poke (2-39) ~ ~ ~

where Ki is a "scaled stiffness" matrix of member i. For

example, the stiffness matrix for a member of a plane frame

is:

EA ~ 0 0 p

0 EIz EIz (2-40) k. = 12- -6-

~ p3 22 . EIz EI

0 -6- 4_z p2 2

where E is the modulus of elasticity, A is the cross-sec­x

tional area, I z is the moment of inertia, and 2 is the length

26

of the member. The scaled stiffness matrix is therefore:

EAx 0 0

f E! Elz k. = 0 12 __ z -6- (2-41)

~ p3 12 EIz EI

0 -6- 4_z 22 2

inhere Ax is the scaled area (equal to Ax/Pi) and I z is the

scaled moment of inertia (equal to Iz/Pi). If, for example,

Pi is selected to represent the member's area, then Ax will

equal 1.0 and I z will equal the (assumed) ratio of the mem­

ber's moment of inertia to its area, i.e., the square of the

assumed value of the radius of gyration about the z-axis_

Substituting Eq. 2-39 directly into Eq. 2-34 would

result in non-linear compatibility equations, since the pro­

ducts of two sets of unknowns, p and u t, appear in the term

pE~u·. However, by replacing the unknowns Pi by a set of

trial, or assumed, member capacities, Pi' the compatibility

constraints can be rewritten in linear form as:

pKAu' - R = 0 w (2-42)

where K is a diagonal matrix of submatrices k. and p is a ~

diagonal matrix of submatrices which are identity matrices

(of order f) times the trial values Pi-

27

2~4.4 Objective function

-Any set of values for p, R , R , and u t which y w

satisfies the constraints of Eqs. 2-36, 2-38, and 2-42 rep-

resents a feasible design for the structure of interest.

Since the object of this study is to develop a formulation

for optimum design, it is necessary to define an objective

or cost function, which is used as the criterion for deter-

mining which of the feasible designs is optimal.

In this study, it is assumed that the cost per unit

length of each member is proportional to its reference capa­

city, Pi. Therefore, the cost of a member is equal to its

length times its reference capacit~ and the cost of the entire

structure is the sum of the member costs. In vector notation,

then, the objective function is:

minimize t­L P

where L is a vector of individual member lengths.

(2-43)

If, for example, the member areas are selected as

the reference capacities, Eq. 2-43 will yield a minimum

volume design.

2.4.5 Basic formulation

Using the objective function of Eq. 2-43 and the

constraints of Eqs. 2-36, 2-38, and 2-42, the problem of

optimum design can be cast in the following form:

minimize t­L P

subject to:

A tR Y

uyr yP - TIyRy

A. tR . w

U~wP -l\.Rw

= tP'

> 0 -= P'

> 0 -pkAu' - Rw = 0

P > 0

u' unrestricted in sign

28

(a)

(b)

(c)

(d) (2-44)

(e )

(f)

(g)

(h)

Constraints (b), (c), (d), (e), and (f) of Eq. 2-44

can be cast more graphically ass

f At , I = tp' , I I

-----~---~---~--- P I I I Uri -IT I I > 0 - ¥. -¥. - ~ - -~ -:- - - =t -:- - - - -

I I A I = p' (2-45) I I I

~----~---~---~---U I 'I\, I > 0 wrw I I - I -_____ L ____ ~ ___ ~---

I I -I I pKA 0 I I I =

Eqs. 2-44 represent the basic formulation for the

optimum design of framed structures, and is in the form of

a linear programming (L-P) problem, with the values of p, R , Rw' and u' forming the solution vector. y

An optimum solution is reached when the values of

the reference capacities, p, are the same as the trial values

29

-of p. Thus, the problem can be solved as an iterative set of

L-F problems, with the solution values, p, of one iteration

being used for the trial values, ~J of the next iteration.

If no reasonable values for p can be selected for the first

trial, it is possible to avoid this problem by neglecting

the constraints of Eq. 2-44(f) on the first iteration. Con-

vergence criteria (i.e., specifying how close the values of

p and p must be before the solution can be considered to have

converged) can be set to whatever is deemed necessary for the

problem being solved.

The initial selection of constants for the ry , rw'

r y , rw' and K matrices involve assumptions regarding the

member types and sizes. The iterative nature of the solution

process, however, allows revisions to be made in these matri­

ces if the initial assumptions prove to be invalid.

The solution vector contains the values for the

member reference capacities (p), member yield forces (R ), y

member working forces (Rw)' and elastic joint deflections

(u'). If desired, the ultimate and allowable member force

component capacities can be obtained by using the relation-

ships of Eqs. 2-9 and 2-37 to give:

1 - (a) Puki = -- r .p. Y k" yl. l. Y l.

(2-46) 1 - (b) Paki = --r wiPi Ywki

)0

where Puki and Paki are, respectively, the ultimate and

allowable capacities for the kth force component of member i,

Y kO and Y kO are the corresponding elements of the rand y 1 W 1 Y

r matrices, and r 0' r 0' and p. are as previously defined. 'IN yl Wl 1

2.5 Specification of Member Groups

When designing framed structures, it is usually

desirable to specify groups of members as being identical.

This type of "member group" designation can be incorporated

very simply into the basic formulation of Eqs. 2-44 by rede­

fining the p and L vectors and the Uy and Uw matrices as

follows I

(a) p is a vector of elements -Pk' where -Pk is the

reference capacity of all members in member group ki

(b) L is a vector of elements Lk , where Lk is the

sum of the lengths of all members in member group k;

(c) the unit vectors, Uyi ' Which comprise uy ' are

arranged so that every Uyi which corresponds to a member in

group k is placed in column k. Thus, there are as many col­

umns in Uy as there are member groups in the structure; and

(d) Uw is redefined similarly to Uyo

Obviously, the input values ryi' rwi' and Pi will

be the same for all members in the same member group.

As an example of the above redefinitions, consider

31

a structure of three members in which members 1 and 3 are to

be identical (i.e., comprise one member group). L will be

of the forml

(2-47)

-where the subscripts refer to individual members; p will be:

(2-48)

where the subscripts refer to member groups; and Uy and Uw

will appear as I

1 I I

1 I I

• 1

• I • I

I 1 I -...1--

: 1 : 1

U = I • (2-49) I • t • I I 1 _..J.._ I

1 I

1 : I • • I

• I I

1 I I

-Henceforth, L, p, Uy ' and Uw will be assumed to

conform to the redefinitions given in this section. The case

32

where all member capacities are independent of one another

is merely a special form of these redefinitions, with each

member comprising its own member group.

8hapter 3

GENERALIZATION OF FORMULATION FOR ALTERNATIVE LOADS

The basic formulation of Eqs. 2-44 represents a

linear programming formulation for the optimum design of a

framed structure acted upon by a single set of external joint

loads. Alternative loading combinations are not considered.

In this chapter, the basic formulation is general­

ized to consider alternative loads.

3.1 Independent Loads and Alternative Loading Combinations

The types of independent loads which can act on a

structure include a dead load; live load; wind load; and

forces due to earthquake, hurricane, or other natural phe-

nomenon.

It is grossly over-conservative in structural de­

sign to assume that the maximum values for all the indepen­

dent loadings act simultaneously. Therefore, it is neces­

sary to define the alternative loading combinations which

can reasonably be expected and which must be considered in

designing the structure.

For example, the 1970 AISC Specification requires

that the following loading combinations be considered in the

plastic design of a continuous frame subject to dead, live,

33

and wind load:

1.85 (dead + live)

1.40 (dead + live + wind)

(a)

(b)

34

(3-1)

The corresponding combinations for working stress design are:

1.00 (dead + live)

.75 (dead + live + wind)

(a)

(b) (3-2)

The alternative loading combinations defined by

Eqs. 3-1 and 3-2 can be expressed in matrix form. If A is y

a matrix which defines the yield combinations, containing

one column for each independent load and one row for each

alternative yield load combination, then for Eqs. 3-1:

A = [1.85

y 1.40

1.85

1.40 (3-3)

Similarly, the working load combinations of Eqs.

3-2 are defined by matrix

A = [1.00 w .75

A : w

1.00

.75 (3-4)

The number of columns in Ay and Aware the same

(equal to the number of independent loadings, which is de­

noted n i ), but the number of rows need not be the same.

35

Denoting the number of yield combinations and working stress

combinations as ny and nw' respectively, Ay contains ny rows

and AW contains nw rows.

3.2 Generalization of Formulation

In order to generalize the basic formulation to

consider alternative loading combinations, it is necessary

to redefine some of the structural vectors and matrices

which were introduced in Chapter 2 and to modify the equi-

librium. member force, and compatibility constraints.

3.2.1 Redefinitions

The vector of external joint loads, pt, considers

only one set of loads as defined in Chapter 2. To consider

multiple independent loads, p' for the generalized formula­

tion is defined as:

P' = (3-5)

where the subscripts ~, S, etc. indicate the independent

loads.

Since a separate load factor, t, can be associated

with each independent loading, t is redefined as a diagonal

matrix of submatrices which are identity matrices (of order

nf) times constants t , t~, etc. Cl .....

The member force vectors, Ry and Rw' are similar

in form to P' a

(3-6)

with the subvectors Rya' Rwa.' etc. corresponding to the mem­

ber yield and working forces for independent loadings a, etc.

It is possible to redefine the vector of joint dis-

placements, u t , similarly to pt, Ry ' and Rw' with each sub­

vector corresponding to an independent loading. However,

since it is the joint displacements of the alternative load­

ing combinations which are of practical interest to the

designer, u' is redefined as:

(" u' A I

u' B I u' = ) (3-7)

• ( • • I u' j ~)

37

where the subscripts A, B, etc. indicate the alternative

loads.

-The p vector of member reference capacities remains

as previously defined in Section 2.5.

3.2.2 Equilibrium constraints

The equilibrium constraints for a single indepen­

dent load case are presented in Eqs. 2-36. In the case of

multiple independent loadings, equations of the same type

are required for each loading:

t taP; (a) ARyo. =

t p' Cb) A Rwa. = a (3-8) t A Ryl3 = t~Ps (c)

t p' Cd) A RwS = S

etc.

Thus, the entire set of equilibrium constraints

can be written aSI

At /'

R ex I ta. P; _"'t_ At

R S ~~:~ ,.} _"'t~ = (3-9) )

• i • • !

t ~''''

and:

38

= (3-10) •

Using At to represent a diagonal matrix of n. sub-1

matrices At (one subrnatrix per independent load), the equi-

librium equations area

At R = tp' (a) - y (3-11) At R = p' (b) - w

where Ry ' Rw' p', and t are as defined in Section 3.2.1.

3.2.3 Member force constraints

Eqs. 2-38 give the member force constraints for a

single independent loading. For multiple independent loads

and alternative loading combinations, it is necessary to

consider combinations of the independent member forces (R , ya

Rwa' etc.), as defined by relationships similar to those of

Eqs. 3-1 and 3-2.

Consider, for example, a structure of two members

with the alternative loadings defined by Eqs. 3-1 and 3-2.

For simplicity, let the dead load and live load be combined

into a single independent loading so thats

r1 • 85 0 l Ay = ~ .40 1.4~ (3-12)

39

The member yield force constraints required for this struc-

ture are:

U r 13 - ~ 1 (1 • 8 5Rya 1 ) > 0 (a) yl yl 1 -

UylrylPl - ~1 (1.40Rya1 + 1.40RyS1 ) > 0 (b) - (3-13) Uy2ry2P2 - f\2 (1. 85Rytl2 ) > 0 (c) -U r -0 - f\2 (1.40nya2 + 1.4oR "2) > 0 (d) y2 y2~2 YP -

where the numerical subscripts refer to the members.

The constraints of Eqs. 3-13 can be expressed more ~

graphically in the form:

> 0

(3-14)

Using the symbol A to denote the matrix which nre­y

multiplies the Ry vector in Eq. 3-14, the member yield force

constraints for a general structure can be written in the

form:

Urp-AR >0 y y y y (3-15)

-where rand p are as previously defined, R is as defined y y

by Eq. 3-6, U is as defined in Section 2.5 except that y

there are n subvectors U . per member, and A is a bn x bn. y yl Y Y l

40

matrix of submatrices, each 2s x f, where a typical sub­y

matrix for member i, independent loading j, and loading

combination q, is:

A = "- II yi (a) Ygh Yqj

where g = n (i - 1) + q (b) y h = b (j - 1) + i (c )

(3-16) i = 1, · . . , b Cd)

j = 1, · . . , n. (e) l.

q = 1, • •• J n (f) y

All other submatrices are zero.

Redefining Uw similarly to Uy (i.e., nw subvectors

U . per member) and A similarly to A J the member working Wl. . W Y

force constraints can be written as:

(3-17)

3.2.4 Compatibility constraints

The compatibility constraints for a single loading

are given by Eq. 2-42. Since u' for multiple loadings is

defined as the joint deflections for the alternative loads

(Eq. 3-7), rather than for the independent loads, it is

necessary to combine the independent member forces of Rw

(Eq. 3-6) in accordance with the alternative loading combina-

tions defined by the ~ matrix. w

Using, for example, the alternative loads defined

by the AW matrix of Eq. 3-12, the compatibility constraints

required are:

pkAUA pkAuB

-- 0

( • 75 Rwa. + • 7 S RWd ) = 0

(a)

(b) (3-18)

where uA and uB are the joint deflections corresponding to

the alternative loads and Rwa and Rw~ are the working member

forces corresponding to the independent loads.

Eqs. 3-18 can be written in the form:

(3-19)

Thus, the compatibility constraints for alternative

loads can be written as!

PKA u' - A * R = 0 w w (3-20)

where pKA denotes a diagonal matrix of nw submatrices pkA and A: is a matrix whose typical submatrix is equal to the

corresponding coefficient of the Aw matrix times an identity

matrix (of order bf).

42

3.2.5 Generalized formulation

A generalized formulation which considers alterna-

tive loading combinations can be assembled from the con-

straints of Eq.s. 3-11, 3-15, 3-17, and 3-20 and the same

objective function (Eq. 2-43) used for the basic formula-

tions

minimize t­L P

subject to:

A tR - y

U r p y y - A R y y A tR - w

Uwrwp - A R w w

pkAu' A*R w w -p

= tp'

> 0 -= p'

> 0 -= 0

> 0 -R y' Rw' u' unrestricted

in sign

(a)

(b)

(c)

(d) (3-21)

(e)

( f)

(g)

(h)

Thus, the generalized formulation, like the basic

formulation, is an iterative set of linear programming prob­

lems. The final solution vector contains the values of the

member reference capacities, p, the yield and working member

forces, Ry and Rw' corresponding to the independent loads,

and the elastic joint deflections, u', corresponding to the

combined working loads.

In graphical form, the constraints of the general-

43

ized formulation are:

"' ::~~ ('

---P'

~~~ j (3-22)

I A t I I I -I 1

----~----~---_r---Uri -A I I _~_~JL __ ~_L __ %_L __ _

I I 11 I I I A I I 1-' -----,----r----.---

U r' I -A I W W I I W I ----~----~---~---, , -A* I piCA

I I Wi: I I I

= -p >

R ---~ =

Rw >

u' -=

In general, the t matrix in constraint (b) of Eq.

3-21 can be dropped from the generalized formulation since

the load factors are explicitly contained in the A and A y w

matrices.

It should be noted that the basic formulation (Eqs.

2-44) is merely a special case of the generalized formulation

with the number of independent loads and the number of alter­

native loading combinations for both yield and working loads

all equal to one (n i = ny = n = 1), and with A = A = 1.00. w y w

Chapter 4

IMPLEMENTATION

A computer program was written to implement the

generalized formulation of Eqs. 3-21. It consists of two

main parts: (1) a POST program with FORTRAN subroutines and

(2) a general-purpose linear programming solver. The pro­

gram was wri tten to run as a single job on the IBI'.'i/360

system. One run corresponds to one cycle of the iterative

process. Thus, all intermediate results are available and

changes can be made in the various matrices between itera­

tions. The output values of p can be compared to the input

-assumed values of p to determine whether the solution has

converged.

A flow diagram for the program is presented in

Fig. 1. The individual operations within the program are

discussed in the succeeding. paragraphs.

4.1 POST - FORTRAN Program

POST [20J is a computer language very similar to

FORTRAN, but includes implicit matrix operations and dynamic

storage allocation. It is therefore ideally suited for this

study in which large numbers of matrices of various sizes

44

45

must be generated and manipulated.

In the POST program, data which define the struc­

tural type (plane frame, plane grid, or space frame), size

(number of members, joints, and supports), geometry (joint

coordinates and~ember incidences), and member group desig­

nations are read first (Box 1 in Fig. 1). The program is

written so that a problem may be solved by considering- only

the yield constraints or only the working stress constraints,

as well as by considering the entire set of constraints.

Also, for the purpose of obtaining initial trial values for

p, the compatibility equations may be neglected even when

the other working stress constraints are considered. There­

fore, indicators which define the problem type (yield only,

working stress only, or both) and the status o~ the compati­

bili ty constraints (enforced 'or not enforced) are also read.

After certain values are initialized (e.g., f is set to 3 for

a plane frame or grid, to 6 for a space frame) in Box 2, the

rest of the data which define the assumed member properties,

yield and working stress limits, independent sets of joint

loads, and alternative loading combinations are read in Box 3.

Information which defines the number and types of constraints

of the L-P problem is then passed to a disk (Box 17), by

means of a FORTRAN subroutine (Box 4), in a format which can

be read by the L-P solver.

46

The objective function, the member IT matrices, and

the submatrices which comprise the A, k, etc. matrices are

computed by considering each member in turn (Boxes 5 through

10). Since much of the information can be passed to the

disk immediately, the amount of storage required is greatly

reduced. However, this sequence of computations causes the

order of the variables (columns) to be changed, since the

L-P matrix (the matrix in Eq. 3-22) must be input to the L-F

solver columnwise. The order in which the variables are

generated by the computer program is: first R, then u', and

finally p, where R combines Ry and Rw in the form:

Ry1J

~!! ~ R =

Ry2 (4-1)

Rw2 i --- ,

~ J where the numerical subscripts indicate the members. As

these computations are made, a FORTRAN subroutine (Box 11)

writes the columns corresponding to R onto the disk. Infor­

mation pertaining to the u t and p columns is retained in

primary storage.

In Boxes 12 through 14, the information needed to

47

define the u' columns is computed and written onto the disk,

if compatibility is enforced. Finally, two more FORTRAN

subroutines (Boxes 15 and 16) write information pertaining

to the p columns, including the coefficients of the objective

function, the load vectors, and the bounds (Eq. 3-21(h)) on

the disk.

4.2 Linear Programming Solver

The second part of the program consists of a Mathe­

matical Programming System/360 (MPS) routine which solves

the L-P problem defined by the generalized formulation. The

data is read (Box 18) from the disk file (Box 17) which was

created by the POST-FORTRAN part of the program. The problem

is solved in Box 19 and the solution printed i~ Box 20.

The !VIPS language [18, 19J, in addition to its large

capacity (up to 4095 constraints and unlimited number of

variables), allows many options in selecting solution stra­

tegies and specifying the type of documentation and output

desired. For example, it is possible to have all the input

data printed out in tabular form and/or to have the input

matrix represented in graphical form.

The MRS output gives the value of the objective

function, the row activities (i.e., an evaluation of the

terms to the left of the = and ~ signs of the constraints),

48

and the values of the variables (R, u', and p) which form

the solution vector.

Chapter 5

ILLUSTRATIVE EXAMPLES

This chapter describes a number of problems which

were solved using the formulations developed in Chapters 2

and 3 and the computer program discussed in Chapter 4.

The first structure considered is a two-story plane

frame. This structure is small enough that it can be solved

for many parameter variations without the computer time re­

quired becoming too great. In the second part of the chapter,

a larger structure, a space frame, is considered in order to

determine the applicability of the formulations and computer

program to structures of this type.

In all cases, the results presented consist only of

the value of the objective function and, for the space frame,

the values of the member capacities. The computer output,

however, contained complete information regarding the member

forces, joint deflections, etc.

5.1 Plane Frame Design

The first structure considered is the two-story

plane frame shown in Fig. 2. The beams are divided into two

members each in order to accommodate mid-span loads. The

beams constitute one member group (i.e., are to be made iden-

49

Chapter 5

ILLUSTRATIVE EXAMPLES

This chapter describes a number of problems which

were solved using the formulations developed in Chapters 2

and 3 and the computer program discussed in Chapter 4.

The first structure considered is a two-story plane

frame. This structure is small enough that it can be solved

for many parameter variations without the computer time re­

quired becoming too great. In the second part of the chapter,

a larger structure, a space frame, is considered in order to

determine the applicability of the formulations and computer

program to structures of this type.

In all cases, the results presented consist only of

the value of the objective function and, for the space frame,

the values of the member capacities. The computer output,

however, contained complete information regarding the member

forces, joint deflections, etc.

5.1 Plane Frame Design

The first structure considered is the two-story

plane £rame shown in Fig. 2. The beams are divided into two

members each in order to accommodate mid-span loads. The

beams constitute one member group (i.e., are to be made iden-

49

50

tical), the upper story columns a second, and the lower story

columns a third member group. Thus, the structure contains

six free joints, eight members, and three member groups.

As stated in Chapter 2, it is necessary to make

certain assurnntions about the members in deriving the fy' fw'

r , r , and k matrices. First, the member capacities are y w defined as:

-p = A x (a)

(b) (5-1)

(c)

-8hoosing p as A will result in a minimum volume x

design.

Selecting axial force and bending moment as the

member force components which define the yield and working

stress limits, the coefficients which are needed to define

the ryand fw matrices (see Eqs. 2-9 and 2-13) aret

- Z Z ~ mult a = = -X- = A Pult Pult ° A

Y x x (a)

~ = mult 1 (b)

- GaS S Pw mall = =

°aAx = A Pall Pall x

(5-2) (c)

-Pw 1 =

mall (d)

51

where 0 and 0 are the yield and allowable stresses, re-y a

spectively, and Z and S are the plastic and elastic section

moduli.

The coefficients needed to define the ry and rw

matrices are:

- Z ~ m ° = ult -L r = = y P A A x x

(a)

(5-3) -°as Pw mall

rw = = = - A A P x x

(b)

and the values needed to compute Ie are:

A A x 1 = = x -p

!z I z I z = = P A x

(a)

(5-4) (b)

It is assumed that the structure is to be made of

steel wide flange (WF) sections, with the beams 10 inches

deep and the columns (both upper and lower) 8 inches deep_

Selecting an "average" radius of gyration of 4.32

inches and a value of 0.9 for the ratio S/Z, the following

approximate relationships exist for the 10 WF membersz

(4.32)2 = 18.6 (a)

52

S Iz/Ax 18.6 A = ~ s.o = 3.72 (b)

d/2 x (5-5) z

"" .l!.E 4.13 (c ) = A '" .9 x

where d is the depth of the member.

Similarly, for the 8 WF columns, with an assumed

radius of gyration of 3.47 inches:

I z (3.47)2 = 12.0 (a) A ~

x

S 12.0 3.0 (b) (5-6) A ~ --zr.o = x

z l& - 3.33 (c) A ~ .9 -x

Thus, by substituting Eqs. 5-5 and 5-6 and the

values of cr and a into Eqs. 5-2, 5-3, and 5-4, all coef-y w

ficients of f , fw' r , r , and k are known. y y w

5.1.1 Effect of load factor and stress ratio

Initially, the plane frame is designed for the

single loading shown in Fig. 2, with only bending moment

defining the yield and working stress limits, i.e.:

m + --L < 1 (a) - m -ult

(5-7)

+..i< 1 (b) - mall -

53

where my and mw are the bending moments of the yield and

working member forces, respectively.

The ratio pw/Py' denoted r, for this problem is:

-pw

r = _ = n ~y

cr S a = OZ = .9 y

(5-8)

Thus, r is a "stress ratio", since its value depends solely

on the ratio 0 /0 • a y

In this phase of the problem, cr is held constant y

at 36 ksi and r is set to values of 0.45, 0.55, 0.65, and

0.75 by altering the value of Ga. In addition, the load

factor, t, is varied from 1.33 to 2.33 in increments of 0.33.

The results are presented in Figs. 3 and 4.

In Fig. 3, the objective function is plotted against

the load factor for the various values of r. It can be seen

that for low values of t, the curves are horizontal, indi-

cating that only the working stress constraints control the

solution. As the load factor increases, some of the yield

constraints become active, as indicated by the curved sec-

tions, until the inclined straight line is reached. This

line indicates that all the working stress constraints are

inactive and the solution depends only on the yield con-

straints, and the value of the objective function is propor-

tional to the load factor.

54

For a single-story frame, the transition from a

solution based solely on the working stress constraints to

a purely plastic solution was found to be very abrupt. On

the other hand, for a very large structure, the transition

curves can be expected to be much longer than in Fig. 3.

The results for the two-story frame are also pre-

sented in Fig. 4, with the volume plotted against the recip-

rocal of r for the various values of t. In this plot, the

horizontal portions of the curves represent solutions based

on the yield constraints and the inclined straight line indi­

cates that only the working stress constraints are active.

5.1.2 Effect of member force interaction

The solutions presented in the previous section

were obtained without considering member force interaction,

with bending moment only considered in the definitions of

the yield and working stress limits.

In order to demonstrate the effect of member force

interaction, the frame with t = 2.00 and r = 0.65 (corre­

sponding to 0a = 26 ksi) is redesigned considering the effect

of interaction between bending moment and axial force. The

yield surface defined by Eqs. 2-15 and the working stress

limits of Eq. 2-27 were considered in conjunction with one

another, as well as with the simple limits of Eqs. 5-7. The

volumes of the four structures thus designed are summarized

below:

Yield limits

Eq. 5-7(a)

Eq. 2-15

Working stress limits

Eq. 5-7 (b)

4578 in3

4641 in3

Eq. 2-27

4740 in3

4769 in3

55

As could be predicted, considering interaction in

either the yield or working stress constraints increased the

volume of the structure, and the largest volume was obtained

when interaction was considered in both.

5.1.3 Effect of alternative loadings

For the designs discussed in the previous two sec-

tions, the structure was considered to be acted upon by a

single set of loads, as shown in Fig. 2. In order to demon-

strate the effect of alternative loadings, the same plane

frame is designed for the independent loads shown in Fig. 5.

Thus, the vertical forces represent the dead plus live load,

while the .horizontal forces represent the wind load.

The alternative loading combinations to be consid­

ered are defined as follows:

(5-9)

Because the member group designations (Fig. 2) dictate that

the frame be symmetric, reversibility of the. wind load need

56

not be considered.

'rhe frame is first designed wi thout member force

interaction, using the yield and working stress limits de-

fined by Eqs. 5-7. Then, it is designed using the inter-

action relations of Eqs. 2-15 and 2-27. The results are:

without interaction, volume

with interaction, volume

= 3520 in3

= 3689 in3

For the purpose of comparing these results to those

from the previous section, it should be noted that the pre-

vious designs can also be obtained in terms of the independ-

ent loads of Fig. 5, by setting ~y = [2.00 2.00l and A = - w [1.00 1.00J. Thus, considering the less severe alternative

loads of Eq. 5-9 reduced the volume of the frame without

interaction from 4578 to 3520 in3 , and the volume of the

frame with interaction from 4769 to 3689 in3•

5.2 Space Frame Design

In order to determine the applicability of the

formulations and computer program to larger problems, the

staircase of Fig. 6 is considered. The structure which

supports the staircase (Fig. 7) lies along the centerline

and has fixed supports at the upper and lower landings, with

no intermediate supports of any kind. It is designed as a

space frame (f = 6) and made of 12-inch square structural

57

tUbing with 0y = 36 ksi and 0a = 22 ksi.

The design live load is taken as 100 psf and the

dead load as 20 psf. Since loads are restricted to concen-

trated loads at the joints, the dead load plus live load is

expressed as the set of 14 concentrated forces shown in

Fig. 8 and the structure is divided into 15 segments (mem­

bers). The load factor, t, is taken as 1.85.

The member force components which enter into the

definitions of the yield and working stress limits are the

torsional moment (m1 ) and the bending moments (m2 and m3).

The member capacities are defined as:

p = A x (a)

- (b) (5-10) py = m2ult = m3ult - (c) Pw = m2a11 = m)all

Selecting 4.66 inches as a typical radius of gyra-

tion and computing the torsional constant, Ix' the values

needed in the K matrix are:

A = 1 (a) x

Ix Ix

~ 35.9 (b) (5-11) = Ax

I lz I z (4.66)2 = 21.7 (c) = = A~ y x

The ultimate and allowable values of m1, m2 , and

m3 are determined by computing their theoretical values [21J

and reducing these by 10% (the same reduction required to

58

bring theoretical values of Ax' I z ' etc. into agreement with

the handbook [lJ values) to obtain:

m1u1t ~ 3.46 ° A (a) y x

m2u1t = m3u1 t ~ 4. 50 ° Y Ax (b) (5-12)

mlal1 ~ 3.46 ° A (c) a x

m2al1 = ffi3a11 ~ 3. 60 0a Ax (d)

Therefore, the coefficients needed for the ryl fw l

r y ' and rw matrices are:

P:i 4.50 a A = Y.. x = 1.30 m1u1t :3 .46 a A y x

(a)

P:y: = P:y: = 1.00 m2u1t m3u1t (b)

- 3.60 0a Ax Pw 1.04 = = m1a11 3.46 O"a Ax (c)

-Pw = = 1.00 m3a11 (d)

P 4.50 ° A r =:..x = Y.. x = 162.0 (e)

y p Ax

3.60 0a Ax

A x

= 79.2 (f)

(5-13)

The yield limits are defined by a fictitious unit

cube in ml - m2 - m3 space J i.e.:

59

+ m1

< 1 - m1ult - (a)

+ m2 < 1 - m2u1t - (b) (5-14)

+ m:2

< 1 - m3ult -(c)

'and the working stress limits are defined similarly (i.e.,

substitute "all" for "ult" in Eqs. 5-14).

Two design cases are considered: (1) all the mem­

bers are identical (i.e., comprise a single member group),

and (2) the horizontal members comprise one member group and

the inclined members a second.

The resulting designs are:

1 memo ~ou£ 2 memo grou"Os

horizontal members 15.62 in 2 16.74 . 2 area, In

inclined members 15.62 in 2 5.76 . 2 area, In

total volume 8597 in3 6597 in3

It should be noted that the area of 5.76 in2 com-

puted for the inclined members in the second case is consid-

erably smaller than the area of the smallest 12-inch square

tubing listed in the AISC handbook [lJ.

60

5.3 Computer Program Efficiency

In solving the problems described in the preceding

sections, there was a wide range of computer times required

to obtain solutions. For all problems, a one per cent con­

vergence criterion was used, i.e., the solutions were it­

erated until the output values, p, were within one per cent

of the input values, p. The plane frames designed considering only bending

moment in the definitions of the yield and working stress

limits (Section 5.1.1) required approximately 10 seconds each

of IBM 360/75 processor time (5 for POST-FORTRAN + 5 for MPS)

per iteration. On the average, five iterations were needed

for the solutions to converge.

Consideration of member force interaction (Section

5.1.2) increased the computer time only slightly to an aver­

age of approximately 12 seconds (5 + 7) per iteration. A

drop in the average number of iterations to four can be at­

tributed to a better selection of initial values for p.

The inclusion of alternative loadings (Section

5.1.3) caused a large increase in computer time. For the

structure with no member force interaction, iterations of 30

seconds (6 + 24) each were required. For the frame with in­

teraction, each iteration required 102 seconds (7 + 95) of

computer time. In both cases, five iterations were needed

61

for convergence.

The space frame designed with all members the same

required 95 seconds (11 + 84). Because all members are iden-

tical, no cycling was required. The frame with two member

groups required four iterations of approximately 106 seconds

(11 + 95) each for convergence.

It can be seen from the above that the time required

for the POST-FORTRAN part of the program increased very slowly

as the size of the problems increased, whereas the IvlPS time

increased very rapidly.

In addition to the structures presented in the pre­

ceding sections, it was attempted to obtain solutions to some

larger problems. Solutions were not obtained for one of two

reasonss (1) the structure was so highly constrained that

the MPS L-P solver could not reach a feasible solution, or

(2) the computer time required became so great that the cost

became prohibitive.

As an example of the former, it was attempted to

design the staircase structure of Section 5.2 considering the

yield limits defined by a unit tetrahedron, i.e.:

+ + + < 1 (5-15)

Even for the case with all members identical, a feasible

62

solution could not be obtained by the MPS program.

An example of (2) above occurred when it was at­

tempted to design the staircase for alternative loads, con­

sidering one reversible wind loading in addition to the dead

plus live load. Allowing the MPS program to run for twenty

minutes, no solution was obtained.

Chapter 6

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY

6.1 Conclusions

The formulation developed in this study represents

a unified approach to the optimum design of framed struc­

tures. The solution process takes the form of an iterative

set of linear programming problems.

The formulation presents the following advantagesz

(a) It is in a general form and can be applied to

framed structures of all types. While developed for frames

and grids, with minor modifications it can also be applied

to the design of trusses.

(b) Both ultimate capacity and worki~g stress

limitations are considered.

(c) The definitions of yield and working stress

limits include the effect of member force interaction.

(d) Multiple loads and alternative loading combi­

nations are considered.

(e) Groups of members may be specified as being

identical.

While it is necessary to make various assumptions

about the member sizes and shapes, this restriction does not

appear to be unreasonable. Due to the iterative nature of

64

the solution process, the initial assumptions may be modi-

fied as the solution progresses.

Satisfactory results were obtained for several sam-

pIe problems, as described in Chapter 5. Eowever, solutions

could not be obtained for some larger problems. Apparently,

general-purpose L-P programs, such as MPS, are not well

suited for solving problems of the type defined by the formu­

lation which has been developed.

6.2 Suggestions for Further Study

There are many ways in which the formulation and

implementation developed in this study can be extended or

improved %

(a) The objective function used in this study is

somewhat restrictive. It should be generalized to include

functions of all the variables (Ry ' Rw' andu', as well as

p). It is possible, for example, to express the cost of

connections as a function of the member forces [24J.

(b) It seems feasible to take into account second-

ary effects due to displacement. The iterative solution

process is ideally suited for this type of computation.

(c) Buckling constraints for individual members

can be considered either by redefining some of the matrices

or by adding new constraints.

(d) In this study, all loads are assumed to be

known. It is possible, however, to consider loads which

depend upon the member sizes (e.g., dead load due to the

weight of the members) in the following manner:

The loads of any independent loading, P', are

taken as:

P' = pI + P' 1 2 (6-1)

where Pi is known and P2 can be expressed as a function of

the member design capacities, in the form:

P2 = G p (6-2)

where G is a matrix of constants. The equilibrium equations

are:

(6-))

and can be rewritten by substituting Eqs. 6-1 and 6-2 into

Eq. 6-3 and rearranging terms to give:

(6-4)

(e) The formulation, though developed for design,

can be modified for use in analysis. For analysis, the sizes

of the members, as expressed by p, are known. The loads can

be expressed as a vector of constants, P', times an "analysis

66

factor," F. Thus, the equilibrium equations can be written

as:

and the objective function is to maximize F. Since D is

known, the compatibility constraints can be left in the form

of EQ. 2-34 and iteration is not necessary. For multiple

loads, the objective function can be defined as a linear

function of the several analysis factors F1 , F2 , ••• , Fni

•

(f) Imposing deflection constraints can cause spe­

cial computational difficulties [17J. While it would appear

that such constraints can be incorporated into the formulation

very simply by adding u'<u' to the constraints of Eqs. 2-44 - ~x

and 3-21, this approach was attempted for a very simple prob-

lem and it was found that the solution oscillated rather than

converged. This phenomenon can be attributed to the fact that

the stiffness matrix, as expressed by the product PK, is a

constant during any iteration and does not take into account

-changes in the member capacities, p.

(g) It has been noted previously that the general­

purpose L-P solver was poorly suited for this study. There­

fore, special algorithms, such as the decomposition method

[4J, should be investigated for possible use in solving the

L-P problem.

67

(h) As an alternative to, or in addition to, (g)

above, the possibility of employing a partially non-linear

formulation should be investigated. If the values pare

used in the compatibility constraints, rather than p, then

Eq. 2-42 will appear as:

pkAu t - R = 0 w (6-6)

where the first term is non-linear since the products of

variables p and u' appear.

Despite the difficulties which can be anticipated

in implementing such a formulation, it does present two dis-

tinct advantages over the purely linear formulation: (1) so-

lutions can be obtained in one step, without iteration, and

(2) this type of formulation appears to proviqe an answer to

the problem of including deflection constraints in the formu-

lation (see (f) above).

Start

Initialize ob j. func.,

f, etc.

3"- I - . \Read load';::7 \ member - I

4

properties I etc. I

Define types o~

constraints

KEY ----e. Program flow

- - - • Data flow

c=:) Terminal

c=J Compu ta tion

D Read data

c=) FORTRAN subroutine

C~ Loop start

o Loop end

<> Decision

D Printed output

o Disk file

\) Offpage connector

5 Do to a.

--______ ~w\ for \~ = 1 to b

6 __ .1 __ Compute

L., T., H., 1 1 1

(TH)i

7 -- In.crem~nt 1 obJ. func. i

by L. I

8

9

10

11

0-

1 I

yes

co~put~ k. I

1 I

Compute

IIyi ' I\ri

Figure 1: Computer Program Flow Diagram

68

Compute

15k

14 r--------, columns of L-P matrix

1 r--=----L.--. p columns of L-P

atrix (incl ob j. func.)

1~ _______ _

End POST-FORTRAN

-----"

Figure 1 (cont.)

C Start MPS

18 ....-_---' __ .....,

19 .-----&.----.

Solve L-P problem

20 r-----"------.

Print results

End

70

2Sk

10k > CD \V CD

10 WF '------r i I~ ~

0 ~ :;::

® ro co 2Sk

1 :

CD CD N

10k > C"-

10 WF @J

N

~ rx.. G) 3: ~ a> co <l)

rrtrr rr.'m~ ~ .. 2 @ 72" ----- - .. -~-.~

CD, 0, G) = member groups

Figure 2: Two-Story Plane Frame

71

7000

r = .45 • • • •

6000

II ......... C") 5000 ~

-M

---Q)

S ::s r-i 0 :>

4000 .75

3000--------~~------~--------~--------~---------

1.33 1.67 2.00 2.33

load factor (t)

Figure 3: Volume vs. Load Factor

(V"'\

~ .r-f

Q)

s ~

r-f 0 :::-

72

7000

6000

t = 2.33

5000

4000

3000~------~~----~----------~------------~---1

.75 1

:b3 1

.55

reciprocal of stress ratio (l/r)

Figure 4s Volume vs. Reciprocal of Stress Ratio

1 :43

! .-m-;

(a) Dead plus live load

10~ >-----------------~

10l): >

iT/Ti i7TTi

(b) Wind load

Figure 5: Independent Loads on Plane Frame

73

74

y

z

. staircase Figure 6·

75

y

y

z

Figure 7: Staircase Support structure

76

y

2.04

2.10 1.98 1.92 X 1,.98 2.10

2.16

1.92 /

2.04/ 1.98 ':J

2.10

z

Figure 8: staircase Dead Plus Live Load (kips)

A

A x A x Ai;

b

d

E

f

H. ~

I

Ix

I y ' I z

Ix' I y '

k

K

L

L. , f 1

m

m1

m2 , m)

n

I z

APPENDIX

NOTATION

branch-node incidence matrix

member cross-sectional area

scaled member area

diagonal matrix of n. submatrices At 1

number of members

depth of member

modulus of elasticity

number of degrees of freedom

translation matrix for member i

identity matrix

torsional constant

moments of inertia about member y- and z-axes

scaled values of Ix' I y ' and I z unassembled stiffness matrix

scaled stiffness matrix

vector of member lengths

member length

bending moment

torsional moment

bending moments

number of free joints

77

P'

p' p' y' w

p

-p

= P

piCA

R

Ry ' Rw

R' R' y' w

R' R' y' w

R*, R* Y w -* r\ Ry ' n: r

r y , rw

S

T. 1.

t

u'

78

number of independent loads

number of yield and working alternative loading combinations

vector of joint loads

vectors of yield and working joint loads

axial force

vector of member reference capacities

member yield and working stress capacities

matrix containing trial member capacities

diagonal matrix of nw submatrices pkA vector of member forces

vectors of yield and working member forces

vectors of forces at both ends of the members

vectors of force components used to define yield and working stress limits

vectors of normalized member forces

vectors of combined member forces

stress ratio pwipy

diagonal matrices of ratios PyiP and pwip elastic section modulus

number of linear segments defining yield and working stress limits

rotation matrix for member i

load factors

matrices used in member force constraints

vector of elastic joint displacements

v

z

fy' fw

Yl' Y2

~y' !J. w 8, y 8 w

9 1 , 92

A y ' Aw

A y' AW

A* w

IIy ' l\;

cry' O"a

Q

Subscri:Qts

A, B

A, B, ... a, all

i

u, ult

w

y

vector of elastic member distortions

plastic section modulus

normalization matrices

coefficients of the fy matrix

extractor matrices

79

matrices defining linear combinations of normalized forces

constants defining yield surface segment

matrices used in member force constraints for alternative loads

matrices which define alternative loading com­binations for yield and working loads

matrix used in compatibility constraints for alternative loads

matrix products 8yr~yQ and ewfw~Q

yield and allowable stresses

matrix which transforms vector of member B­end forces into vector of forces at both ends of the members

positive and negative member ends

combined working loads

allowable

typical member i

ultimate

working stress

yield

80

a, S, •.. independent loads

Superscript

t matrix transposition

1 •

2.

4.

5.

6.

8.

10.

11.

LIST OF REFERENCES

American Institute of Steel Construction, Manual of Steel Construction, AISC, 1963.

Brown, D. M. and A. H.-S. Ang, "Structural Optimization by Nonlinear Programming," Journal of the Structural Division, ASCE, Vol. 92, No. ST6, December 1966.

Dantzig, G. B., Linear Programming and Extensions, Princeton University Press, 1963.

Dantzig, G. B. and P. Wolfe, etA Decomposition Principle for Linear Programs," Operations Research, Vol. 8, No.1, January-February 1960.

Fenves, S. J. and F. H. Branin, "Network-Topological Formulation of Structural Analysis," Journal of the structural Division, ASCE, Vol. 89, No. ST4, August 1963.

FORTRAN IV Language (C28-6515-6), IBM Systems Library, IBM, 1966.

Foulkes, J., "Minimum Weight Design and the Theory of Plastic Collapse," Quarterly of Applied Mathematics, Vol. 10, No.4, January 1953. .

Gonzales C., A. and S. J. Fenves, "A Network-Topological Formulation of the Analysis and Design of Rigid-Plastic Framed Structures," Civil Engineering Studies, struc­tural Research Series No. 339, University of Illinois, September 1968.

Greenberg, H.J. and W. Prager, "Limit Design of Beams and Frames," Proceedings of the American Society of Civil Engineers, Vol. 77, Separate No. 59, February 1951.

Hadley, G., Linear Programming, Addison-Wesley Publishing Company, 1962.

Hall, A. S. and R. W. Woodhead, Frame Analysis, John Wiley and Sons, 1961.

81

82

12. Heyman, J., "Plastic Design of Beams and Frames for Minimum Material Consumption," Quarterly of Applied Mathematics, Vol. 8, No.4, January 1951.

13. Hodge, P. G., Plastic Analysis of structures, McGraw­Hill Book Company, 1959.

14. Klein, B., "Direct Use of Extremal Principles in Solving Certain Optimizing Problems Involving Inequalities," Operations Research, Vol. 3, No.2, May 1955.

15. Livesley, R. Ke, "The Automatic Design of Structural Frames, tI Quarterly Journal of IVlechanics and Applied Mathematics, Vol. 9, Part 3, September 1956.

16. Livesley, R. K., Matrix Methods of structural Analysis, Pergamon Press, 1964.

17. Livesley, R. K., "Automatic Design of Guyed Masts Subject to Deflection Gonstraints,1t International Journal for Numerical Methods in Engineering, Vol.·2, No.1, January­NIarch, 1970.

18.

19.

20. Melin, J. 1,,/., "POST: Problem-Oriented Subroutine Trans­lator," Journal of the Structural Division, ASCE, Vol. 92, No. ST6, December 1966.

21. Morris, G. A. and S. J. Fenves, "A General Procedure for the Analysis of Elastic and Plastic Frameworks," Civil Engineering Studies, Structural Research Series No. 325, University of Illinois, August 1967.

22. Moses, F., "Optimum Structural Design Using Linear Pro­gramming," Journal of the Structural Division, ASCE, Vol. 90, No. ST6, December 1964.

23. Prager, W., "Minimum-Weight Design of a Portal Frame," Journal of the Engineering Mechanics Division, ASCE, Vol. 82, No. EM4, October 1956.

24.

25.

26.

28.

30.

8)

Ridha, R. A. and R. N. Wright, "Minimum Cost Design of Frames," Journal of the Structural Division, ASCE Vol. 93, No. ST4, August 1967.

Schmit, L. A., "Structural Synthesis 1959 - 1969, a Decade of Progress," Survey paper presented at the Japan-U. S. Seminar on Matrix Methods of Structural Analysis and Design, Tokyo, August 1969.

Schmit, L. A., T. P. Kicher, and W. M. Morrow, "Struc­tural Synthesis Capability for Integrally Stiffened Waffle Plates," AL~ Journal, Vol. 1, No. 12, December 1963.

Shames, I. H., I'.:Iechanics of Deformable Solids, Prentice­Hall, 1964.

Sheu, C. Y. and W. Prager, "Recent Developments in Opti­mal Structural Design," Applied Mechanics Reviews, Vol. 21, No. 10, October 1968.

Smith, J. O. and O. M. 'Sidebottom, Inelastic Behavior of Load-Carryinp: Members, John Wiley & Sons, 1965.

Wright, R. N., COSD (Constrained Optimum Stee"Dest De­scent), Department of civil Engineering, university of Illinois, December 1967.

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U. S. Naval Ordnance Station Attn: Mr. Garet Bornstein Research & Development Division Indian Head, Maryland 20640

Chief of Naval Operation Department of the Navy Washington, D. C. 20350 Attn: Code Op-03EG

Op-07T

Spec i a 1 Proj ects Off ice (CNM- PM- 1) (MUN) Depa rtment of t he ,Navy Washington, D. C. 20360 Attn: NSP-OOl Dr. J. P. Craven

Deep Submergence Sys. Project (CNM- PM-11) 6900 Wisconsin Ave. Chevy Chase, Md. 20015 Attn: PM-1120 S. Hersh

U. S. Naval Appl ied Science Lab. Code 9832 Technical Library B u i 1 din g 291, Na va 1 Ba s e Brooklyn, New York 11251

Director, Aeronautical Materials Lab. Naval Air Engineering Center Nava 1 Ba se Ph i 1 ad e 1 phi a, Pe n n s y 1 van i a 1 91 1 2

Navy (cont'd)

Naval Air Systems Command Dept. of the Navy Wash ington, D. C. 20360 Attn: NAIR 03 Res. & Technology

320 Aero. & Structures 5320 Structures

604 Tech. Library

Naval Facil ities Engineering Command

Dept. of the Navy Washington, D.C. 20360 Attn: NFAC 03 Res. & Development

04 Engineering & Design 04128 Tech. Library

Naval Ship Systems Command Dept. of the Navy Washington, D.C. 20360 Attn: NSHIP 031 Ch. Scientists for R&D

0342 Ship Mats. & Structs. 037 Ship Silencing Div. 052 Shock & Blast Coord.

2052 Tech. Library

Naval Ship Engineering Center Main Navy Building Washington, D.C. 20360 Attn: NSEC 6100 Ship Sys. Engr. & Des. Dept.

61020 Computerated Ship Des. 6105 Ship Protection 6110 Ship Concept Design 6 1 20 H u 1 1 D i v. - J. Na c h t she i m 61200 Hull Div. - J. Vasta 6132 Hull Structs. - (4)

Naval Ordnance Systems Command Dept. of the Navy Washington, D.C. 20360 Attn: NORD 03 Res. & Technology

035 Weapons Dynamics 9132 Tech. Library

Air Force

Commande r WADD Wright-Patterson Air Force Base Dayton, Ohio 45433 Attn: Code WWRMDD

AFFDL (FDDS)

-3-

Wright-Patterson AFB (cont'd) Attn: Structures Division

AFLC (MCEEA) Code WWRC AFML (MAAM)

Commander Chief, Appl ied Mechanics Group U. S. Air Force Inst. of Tech. Wright-Patterson Air Force Base Dayton, Ohio 45433

Chief, Civil Engineering Branch WLRC, Research Divis ion Air Force Weapons Laboratory Kirtland AFB, New Mexico 87117

Air Force Office of Scientific Res. 1400 Wilson Blvd. Arlington, Virginia 22209 Attn: Mechs. Div.

Structures Research Division National Aeronautics & Space Admin. Langley Research Center Langley Station Hampton, Virginia 23365 Attn: Mr. R. R. He1denfels, Chief

National Aeronautic & Space Admin. Associate Administrator for Advanced

Research & Technology Washington, D.C. 20546

Scientific & Tech. Info. Facil ity NASA Rep res e nt a t i ve (S-A KID L) P. O. Box 5700 Bethesda, Maryland 20014

National Aeronautic & Space Admin. Code RV-2 Washington, D.C. 20546

Other Government Activities

Commandant Chief, Testing & Development Div. U. S. Coast Guard 1300 E Street, N. W. Washington, D. C. 20226

Director Marine Corps Landing Force Dev. Cen. Marine Corps Schools Quantico, Virginia 22134

Director National Bureau of Standards Washington, D. C. 20234 Attn: Mr. B. L. Wilson, EM 219

National Science Foundation Engineering Division Washington, D. C. 20550

Science & Tech. Division Library of Congress Washington, D. C. 20540

Director STBS Defense Atomic Support Agency Washington, D. C. 20350

Commander Field Command Defense Atomic Support Agency Sandia Base Albuquerque, New Mexico 87115

Chief, Defense Atomic Support Agency Blast & Shock Division The Pentagon Washington, D. C. 20301

Director, Defense Research & Engr. Technical Library Room 3C-128 The Pentagon Washington, D. C. 20301

Chief, Airframe & Equipment Branch FS-120 Office of Fl ight Standards Federal Aviation Agency Washington, D. C. 20553

Chief, Division of Ship Design Maritime Administration Washington, D. C. 20235

4

Deputy Chief, Office of Ship Constr. Maritime Administration Washington, D. C. 20235 Attn: Mr. U. L. Russo

Mr. Milton Shaw, Director Div. of Reactor Devel. & Technology Atomic Energy Commission Germantown, Md. 20767

Ship Hull Research Committee National Research Council National Academy of Sciences 2101 Constitution Avenue Washington, D. C. 20418 Attn: Mr. A. R. Lytle

PART 2 - CONTRACTORS AND OTHER TECHNICAL COLLABORATORS

Universities

Professor J. R. Rice Division of Engineering Brown University Providence, Rhode Island 02912

Dr. J. Tinsley Oden Dept. of Engr. Mechs. University of Alabama Hunstvi11e, Alabama

Professor M. E. Gurtin Dept. of Mathematics Carnegie Institute of Technology Pittsburgh, Pennsylvania 15213

Professor R. S. Riv1 in Center for the App1 ication of Mathematics Lehigh University Bethlehem, Pennsylvania 18015

Professor Jul ius Mik10witz Division of Engr. & Appl ied Sciences Cal ifornia Institute of Technology Pasadena, California 91109

Professor George Sih Department of Mechanics Lehigh University Bethlehem, Pennsylvania 18015

Dr. Harold Liebowitz, Dean School of Engr. & App1 ied Sciences George Washington University 725 23rd Street Washington, D. C. 20006

Universities (cont1d)

Professor El i Sternberg Div. of Engr. & Appl ied Sciences Cal ifornia Institute of Technulogy Pasadena~ Cal ifornia 91109

Professor Paul M. Naghdi Div. of Appl ied Mechanics Etch ever ry Ha 1 1 University of Cal ifornia Berkeley~ Cal ifornia 94720

Professor Wm. Prager Reve 1 1 e Co 11 ege University of Cal ifornia P. O. Box 109 La Jolla~ Cal ifornia 92037

Professor J. Baltrukonis Mechanics Division The Catholic Univ. of America Washington, D. C. 20017

Professor A. J. Durelli Mechanics Division The Catho1 ic Univ. of America Washington, D. C. 20017

Professor H. H. Bleich Department of Civil Engineering Columbia University Amsterdan & 120th Street New York, New York 10027

Professor R. D. Mindl in Department of Civil Engineering Columbia University S. W. Mudd Building New York, New York 10027

Professor F. L. DiMaggio Department of Civil Engineering Columbia University 616 Mudd Building New York, New York 10027

Professor A. M. Freudenthal Department of Civil Engr. &

Engr. Mechs. Columbia University New York, New York 10027

5

Professor B. A. Boley Dept. of Theor, & Appl. Mechs. Cornell University Ithaca, New York 14850

Professor P. G. Hodge Department of Mechanics III inois Institute of Technology Chicago, III inois 60616

Dr. D. C. D rue ke r Dean of Engineering University of Illinois Urbana, III inois 61803

Professor N. M. Newmark Dept. of Civil Engineering University of Illinois Urbana, III inoi s 61803

Professor A. R. Robinson Dept. of Civil Engineering University of III inois Urbana, III inois 61803

Professor S. Taira Department of Engineering Kyoto University Kyoto, Japan

Professor James Mar Massachusetts Inst. of Tech. Rm. 33-318 Dept. of Aerospace & Astro. 77 Massachusetts Avenue Cambridge, Massachusetts 02139

Professor E. Reissner Dept. of Mathematics Massachusetts Inst. of Tech. Cambridge, Massachusetts 02139

Professor William A. Nash Dept. of Mechs. & Aerospace Engr. University of Massachusetts Amherst, Massachusetts 01002

Library (Code 0384) U. S. Naval Postgraduate School Monterey, Cal ifornia 93940

Professor Arnold Allentuch Department of Mechanical Engineering Newark College of Engineering 323 High Street Newark, New Jersey 07102

Universities (cont'd)

Professor E. L. Reiss Courant Inst. of Math. Sciences New York University 4 Washington Place New York, New York 10003

Professor Bernard W. Shaffer School of Engrg. & Science New York University University Heights New York, New York 10453

Dr. Francis Cozzarel1 i Div. of Interdiscipl inary

Studies and Research School of Engineering State University of New York Buffalo, New York 14214

Professor R. A. Douglas Dept. of Engr. Mechs. North Carol ina St. Univ. Raleigh, North Carol ina 27606

Dr. George Herrmann The Technological Institute Northwestern University Evanston, Illinois 60201

Professor J. D. Achenbach Technological Institute Northwestern University Evanston, Illinois 60201

Director, Ordnance Research Lab. Pennsylvania State University P. O. Box 30 State College, Pennsylvania 16801

Professor Eugene J. Skudrzyk Department of Physics Ordnance Research Lab. Pennsylvania State University P.O. Box 30 State College, Pennsylvania 16801

Dean Oscar Baguio Assoc. of Struc. Engr. of

the Ph j 1 i pp i nes University of Phil ippines Manila, Philippines

6

Professor J. Kempner Dept. of Aero. Engr. & Appl ied Mech. Polytechnic Institute of Brooklyn 333 Jay Street Brooklyn, New York 11201

Professor J. Klosner Polytechnic Institute of Brooklyn 333 Jay Street Brooklyn, New York 11201

Professor A. C. Eringen Dept. of Aerospace & Mech. Sciences Princeton University Princeton, New Jersey 08540

Dr. S. L. Koh School of Aero., Astro. & Engr. Science Purdue University Lafayette, Indiana 47907

Professor R. A. Schapery Purdue University Lafayette, Indiana 47907

Professor E. H. Lee Div. of Engr. Mechanics Stanford University Stanford, Cal ifornia 94305

Dr. Nicholas J. Hoff Dept. of Aero. & Astro. Stanford University Stanford, Cal ifornia 94305

Professor Max An1 iker Dept. of Aero. & Astro. Stanford University Stanford, California 94305

Professor J. N. Goodier Div. of Engr. Mechanics Stanford University Stanford, Cal ifornia 94305

Professor H. W. Liu Dept. of Chemical Engr. & Metal. Syracuse University Syracuse, New York 13210

Professor Markus Reiner Technion R&D Foundation Haifa, Israel

Universities (cont'd)

Professor Tsuyoshi Hayashi Department of Aeronautics Faculty of Engineering University of Tokyo BUNKYO- KU Tokyo, Japan

Professor J. E. Fitzgerald, Ch. Dept. of Civil Engineering Univers ity of Utah Sa 1 t La ke Cit y, Ut a h 84 11 2

Professor R. J. H. Bollard Chairman, Aeronautical Engr. Dept. 207 Guggenheim Hall University of Washington Seattle, Washington 98105

Professor Albert S. Kobayashi Dept. of Mechanical Engr. University of Washington Seattle, Washington 98105

Off ice r- i n- Charge Post Graduate School for Naval Off. Webb Institute of Naval Arch. Crescent Beach Road, Glen Cove Long Is 1 and, New York 11542

Librarian Webb Institute of Naval Arch. Crescent Beach Road, Glen Cove Long Island, New York 11542

Sol id Rocket Struc. Integrity Cen. Dept. of Mechanical Engr. Professor F. Wagner University of Utah Salt Lake City, Utah 84112

Dr. Daniel Frederick Dept. of Engr. Mechs. Virginia Polytechnic Inst. B1acksburgh, Virginia

Industry and Research Institutes

Dr. James H. Wiegand Senior Dept. 4720, Bldg. 0525 Ball istics & Mech. Properties Lab. Aerojet-General Corporation P. O. Box 1947 Sacramento, Cal ifornia 95809

-7-

Mr. Ca r 1 E. Ha r t b owe r Dept. 4620, Bldg. 2019 A2 Aeroject-General Corporation P. O. Box 1947 Sacramento, Cal ifornia 95809

Mr. J. S. Wise Aerospace Corporation P. O. Box 1300 San Bernardino, Cal ifornia 92402

Dr. Vi to Sa 1 e r no Appl ied Technology Assoc., Inc. 29 Church Street Ramsey, New Jersey 07446

Library Services Department Report Section, Bldg. 14-14 Argonne National Laboratory 9700 S. Cass Avenue Argonne, III inois 60440

Dr. M. C. Junger Cambridge Acoustical Associates 129 Mount Auburn Street Cambridge, Massachusetts 02138

Dr. F. R. Sc hwa r z 1 Central Laboratory T.N.O. Schoenmakerstraat 97 Delft, The Netherlands

Research and Development Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340

Supervisor of Shipbuilding, USN, and Naval Insp. of Ordnance

Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340

Dr. L. H. C he n Basic Engineering Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340

Dr. Wendt Valley Forge Space Technology Cen. General Electric Co. Valley Forge, Pennsylvania 10481

Dr. Joshua E. Greenspon J. G. Engr. Research Associates 383 1 Me n 1 0 D r i ve Baltimore, Maryland 21215

Industry & Research Inst. (cont'd)

Dr. Wa 1 t. D. Pi 1 ke y lIT Research Institute lOWe s t 35 S t r e e t Ch i cago, III i no is 60616

Library Newport News Shipbuilding & Dry Dock Company

Newport News, Virginia 23607

Mr. J. I. Gonzalez Engr. Mechs. Lab. Ma r tin Ma r jet t a MP - 233 P. O. Box 5837 Orlando, Florida 32805

Dr. E. A. Ale xa nde r Research Dept. Rocketdyne D.W., NAA 6633 Canoga Avenue Canoga Park, Cal ifornia 91304

Mr. Cezar P. Nuguid Deputy Commissioner Phil ippine Atomic Energy Commission Ma nil a, Phi 1 i p pin e s

Dr. M. L . Me r r itt Divis ion 5412 Sandia Corporation Sand i a Base Albuquerque, New Mexico 87115

Director Ship Research Institute Ministry of Transportation 700, SHINKAWA M i taka Tokyo, Japa n

Dr. H. N. Abramson Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206

Dr. R. C. DeHart Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206

Dr. M. L. Baron Paul Weidlinger, Consulting Engr. 777 Third Ave - 22nd Floor New York, New York 10017

Mr. Roger Weiss High Temp. Structs. & Materials Appl ied Phys ics Lab. 8621 Georgia Ave. Silver Spring, Md.

Mr. Will iam Caywood Code BBE App 1 i ed Phys i cs Lab. 8621 Georgia Ave. Silver Spring, Md.

Mr. M. J. Berg Engineering Mechs. Lab. Bldg. R-1, Rm. l104A TRW Sys tems 1 Space Pa rk Redondo Beach, Cal ifornia 90278

-8-

Unclassified Security etas sification

DOCUMENT CONTROL DATA· R&D

L .5,'curiry cJassi fication of title, body of abs/wct and indexinl5 annolilrion must be entered when tile overall report is cJ,u;sified) '. -~:;';";";";"";;;":;";:';"';:":';';;';~~~~~~~~~ _____ --~-~--t , 1 ORIGINATING ACTIVI TY (Corporate author) 28. REPORT SECURITY CLASSIFICATION

University of Illinois at Urbana-Champaign Department of Civil Engineering

Unclassified 2b. GROUP

3. REPORT TITLE

A GEf\TERAL FORMULATION FOR THE OPTIIVIUM DESIGN OF FRAMED STRUCTURES

4. DESCRIPTIVE NOTES (Type of report and.inclusive dates)

Progress: April 1969 - July 1970 5· AU THOR(S) (First name, middle initial, last name)

William J. McCutcheon Steven J. Fenves

6. REPORT OATE

August 1970 8a. CONTRACT OR GRANT NO.

N 0014-67-A-0305-0010 b. PROJECT NO.

Navy A-0305~0010

78. TOTAL NO. OF PAGES

83 9a. ORIGINATOR'S REPORT NUMBER(S)

Structural Research Series No. SRS 362

c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assi~ned this report)

d.

10. DISTRIBUTION STATEMENT

Qualified requesters may obtain copies of this report from DDC.

11. SUPPLEMENTARY NOTES

13. ABSTRACT

12. SPONSO RING MI LI TARY AC TI VI TY

Office of Naval Research Structural Mechanics Branch Department of the Navy

A unified approach to the optimum design of framed structures based on a general matrix formulation is presented. Both elastic' behavior, in the form'of working stress limitations, and ultimate capacity are considered.

The formulation is presented in the form of an iterative set of linear programming problems. The design .variables relating to elastic and ultimate behavior are linearly related and the objective function is expressed as a linear function of the design variables. By making assumptions about the member properties and sizes, the constraints are also expressed in linear form.

The formulation is initially developed for a structure acted upon by a single set of external loads, and is subsequently generalized to include multiple loads and alternative loading combinations. Yield and working stress limits, which are defined to include the effect of stress interaction, are specified only at member ends. Therefore, loads are restricted to concentrated forces and couples at the joints.

A computer program was written to implement the formulation and was used to solve several sample problems. However, solutions could not be obtained for some large, highly constrained problems. It is expected that the implementation can be improved by using special linear programming algorithms or by employing a partially non-linear formulation.

DO lFN°o~~\51473 (PAGE 1) -Unclassified

SIN 0101-807-6811 Security Classification A-31~08

Unclassified Security Classification

14 LIN K A LIN K B LIN K C KEY WORDS

ROLE WT ROLE WT ROLE WT

Optimum design

Structural design

Matrix methods

Computer programming

Linear programming applications

Framed structures

;

,

Unclassified SIN 0101-807-61321 Security Classification A-31409