I I

:;'1~ i::S NO. 339

I

I

I I

By

A. GONZALEZ CARO

S. J. FENVES

A T echn i cal Report

of a Research Program

Spon so red by

THE OFFICE OF NAVAL RESEARCH

DEPARTMENT OF THE NAVY

Contract No. N00014-67-A-0305-00W

Proj ect NA VY -A-0305-00 10

UNIVERSITY OF ILLINOIS

URBANA, ILLINOIS SEPTEMBER, 1968

A NEThlO RK- TO PO LO GICAL FO RMULATIO N

OF THE ANALYSIS AND DESIGN

OF RIGID-PLASTIC FRAMED STRUCTURES

by

A. Gonzaiez Caro

S. J. Fenves

A Technical Report of a Research Program

Sponso red by TH E 0 FFIC E 0 F NAVAL RES EARCH

DEPARTMENT OF THE NAVY Contract No. N00014-67-A-0305-0010

Project NAVY-A-0305-0010

UNIVERSITY 0 F ILLINO IS URBANA, I LLINO IS SEPTEMB~R, 1968

ACKNOWLEDGEMENT

This report was prepared as a doctoral dissertation by

Mr. A. Gonzalez Caro and was submitted to the Graduate College of the

University of Illinois in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Civi I Engineering. The work

was done under the supervision of Dr. Steven J. Fenves, Professor of

Civil Engineering.

The authors wish to thank Dr. John W. Mel in, Associate

Professor of Civil Engineering, for his invaluable advice and assistance.

TABLE OF CONTENTS

A C KNO VI LED GEM EN T S .... " • e .... e e .... " G G 0 e " a • '" .... 0 G $ .... Q e <I) .. • • • • • • <> • • • • • • • • • •• iii LIST OF FIGURES. 0 0"" e 0 G. Q" e e •• Q .4) 0 0 0 •• " .. 0 G 11.0 ct e -90 fI I" 0 •• e Q. D. e e It ••• e vi LIS T 0 F TABLES 0 9 .e G • e I/) G 6 0 '3 e 0 0 ,. G -0 !Ii) 0 0, " 0 til 0 eGO 0 0 0, 0 C .0 .. -0 0 0 (I e 0 SI 4) • 0 e $ 0 • ., 0 Q C vii i NOTATION G ..... e <I) II GO". G ...... Q <I)" 10 <I)"" e ...... "" II .. 8 II II .. "" .... II e e e II e .... !!l 0 e""" .. " .. " II" ix

Chapter Page

2

3

INTRODUCTION.

LI Object of Study · . " 1.2 Background. .. " . .. " . 0 · . · 1.3 Assumptions and Lim i tat ions . · .. · 1.4 Organization of Report. . . · . · . e

PLASTIC ANALYSIS AND LINEAR PROGRAMMING CONCEPTS

" . . " . . . . . ..

1 2 l+

5

7

2.1 Plastic Analysis Theorems. " ••• " • • • 7 2e2 Mathematical Programming Pre} iminaries • 0 § • 8

2.2.1 Linear Programming and Dual ity Concepts 9

2.3 Use of Mathematical Programming in Plastic Analysis 11

2.3.1 'Example 1: System of Coll inear Springs. i 1 2.3.2 Example 2: Planar Frame".. ". • .. 17

2.4 Use of Collapse Mechanisms

2.4. 1 2 .. 4.2

Examp' e I: Example 2:

System of Collinear Springs. " .. Planar Frame "

2.5 Minimum Weight Design ...

205. 1 2 .. 5.2 2 .. 5.3

Example 1: System of Call inear Springs Example 2: Planar Frame. 0 ... e • e •

Use of Co11apse Mechanisms for Minimum Weight Design. Q ......... .

THE NETWORK FORMULATION .. 0 •

20

20 21

23

23 25

27

31

3. J' Summary of Linear Graph Theory... • .. .. .. 31 3.2 Graph Theory in Structural Analysis. .. .. • • • 33 3.3 Network-Topological Formulation of the Static

Theorem e· ...... : ... 0 • 0 ••• 0 •• eo .. o. 37 3.4 Role of a Typical Member in the Formulation 41 3.5 Example J: System of ColI inear Springs. eo. 43 3.6 Example 2: Planar Frame ....... 0 • • • .. 45 3.7 Network Formulation of the Kinematic Theorem. 47 3.8 Network Formulation of the Minimum Weight Design

P rob 1 em • • .. • GOO •• OO.$DGeee.& 51

iv

Chapter

5

GENERALIZATIO N 0 F FO RMULATIO N • .

4. 1 4.2

4,,3

4.4

The Yield Surface. " • 0

General ization of the Network-Topological Fa rmu J at ion .• • .. .. .. " • • • e _ • • 6

Role of a Typical Member in the General ized Formulation •• 0 •• ., .... " .... " ••

General ization of the Network Formulation for

. .. ..

Minimum Weight Design .............. " 0 •••

IMPLEM ENTATION ".'.' ... c • " ........ ..

v

Page

54

54

55

58

63

67

5.1 Introduction ................... "". 67 5.2 Use af General Purpose Linear' Programming Routines 68 5.3 Use of Special Algorithms. • .. • • • • • • • .. •• 70

5.3.1 Analysis of the General ized Formulations from the Point of View of the Decomposition

6

7

Principle ......... " ........ " ...

5.4 Implementation of Computer Program. ea.

5.5 Nonl inear Programming in Plastic Analysis ••

ILLUSTRATIVE EXAMPLES ..

Comparison of Mesh and Node Solutions Effect of Stress Interaction" •.•• General Case of Interaction: A Space Frame. Minimum Weight Design •• " •

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY.

7.1 Conclusions ••••• 7.2 Suggestions for Further Study ••

o •• Olto •• ·o.".

. . .

FIGURES " II

TABLES APPENDIX A NETWORK FORMULATION OF PLASTIC ANALYSIS BY USING

THE COLLAPSE MECHANISMS OF THE STRUCTURE. LIST OF REFERENCES. • • •• • •••• VITA. • it & D • • 0' C • a " 0 e. G e 4) • • .. .. • e • •

72

75 79

81

81 84 89 90

94

94 96

99 130

135 145 148

Figure

20 1

2.2

2.3

2.4

3 it 1

3.2

4.2

50 1

6. 1

LIST OF FIGURES

System of Call inear Springs.

Planar Frame .• c • " .. . Collapse Mechanisms for Collinear Springs.

Collapse Mechanisms for Planar Frame

Typical Structural Member ....... .

Linear Graph and Primary Strwcture for System of Collinear Springs. 0 ••••••••••••

Linear Graph and Primary Structure for Planar Frame

Roth's Diagram for Mesh Formulation. 0

Roth1s Diagram for Node Formulation.

Combined Roth's Diagram.

Yield Surface.

Yield Surfaces for Rectangular Section.

Multistory Frame and Systems of Releases.

Multistory Frame

Page

99

99

100

101

102

103

104

105

106

)07

J08

109

110

111

6.2 Collapse Mechanism for Multistory Frame (No interaction) 112

6.3 Normal ized Moment Diagram at Collapse For Multistory Frame (No Interaction). • • • •. • .. 0 ....... 113

6.4 Collapse Mechanism for Multistory Frame (Interaction Included) .. 0 • .. • • • • • .... .. • ....... " 114

6.5 Normal ized Moment Diagram at Collapse for Multistory Frame (Interaction Included). • • • • • .. 0 •• 115

6.6 Plane Frame Under Heavy Axial Loading 116

6.7 Normal ized Force Components at Collapse .for Planar. Frame (Linear Case) • . • . . . . . . . . . . . . . 117

vi

Figure

6.8 Normal ized Force Components at Collapse for Planar Frame (Nonlinear Case). Q 0 G ••••••

6,.9 Gable Frame and Mode of Collapse.

6.10 P1anar Grid

6. 11 Normal ized Force Components at Collapse for Planar Grid

6. J 2 Ci rcular Hel ix ..

6.13 Normal ized Values for the Stress Components and Stress

6. 14

6. 15

68 J 6

6. 17

6. 18

6 .. 19

Vector for Unfolded Hel ix

Moment Diagram for Minimum Weight Design (Case 1)

Collapse Mechanism for Minimum Weight Design (Case 1) •

Moment Diagram for Hinimum Weight Design (Case 2) •••

Collapse Mechanism for Minimum Weight Design (Case 2) 0

Moment Diagram for Minimum Weight Design (Case 3) •••

Collapse Mechanism for Minimum Weight Design (Case 3) •

vi i

Page

118

119

120

12 I

122

123

124

125

126

127

128

129

LIST OF TA.BLES

Number Page

Expanded Linear Programming Formu 1 at ion ... Mesh Me thod 130

2 Expanded Linear Programming Formulation ... Mesh Method (Reordered Constraints) . . . . II . . . . " . . . . . . 131

3 Expanded Li near Programming Formulation ... Mesh Method (Membe rs Renumbe red) • . . . a . " . . . . " . . 0 . . . 132

4 Expanded Linear Programming Formulation - Mesh Method (Alternate System of Releases). • • • • • • • • • • •• 133

5 Expanded Linear Programming Formulation - Node Method c 134

vi i i

A

B

b

c

C

E

f

H

HI

k

L

Mp

NOTATION

; branch-node incidence matrix

.= node-to-datum path matrix

matr i x defined by Eq. (3. 3)

:::; ma tr i x defined by Eq. (3. J 5)

:::; matr i x def i ned by Eq. (3. 17)

; number of members in a structure

:::; branch-circuit matrix

:::; matrix defined by Eq. (3.16)

:::; matrix extracting from RI those components defining yielding

:::; matrix defining] inear combinations of the 1 inear components of the yield surface

:::; matrix defining 1 inear combinations of the nonl inear components of the yield surface

:::; matrix defining 1 inear combinations of the components defining J inearized yield conditions

:::; matrix defining basic and joint mechanisms

:::; submatrix of basic mechanisms

:::; submatrix of joint mechanisms

:::; reference load

:::; number of force and dist~rtion vector components

:::; translation matrix (member coordinates)

~ translation matrix (global coordinates)

= vector relating the weight of the me~ber to the reference fully plastic values

:::; vector of member lengths

:::; rotation matrix

:::; vector of plastic capacities (simpJe plastic case)

ix

m

m ,m ,m x y z

N

n

n'

n

pI

p. I

p'

Q

R

R'

* R

* ~>

s

u

w

number of redundant members in a structure

~ fully plastic values of the torsion and bending moments

~ torsion and bending moments

= diagonal matrix containing the inverses of the fully plastic values

= number of joints in a structure

~ number of free joints in a structure

= matrix converting R into RI

vector of joint loads

= vector defining the joint loads with respect to the reference load

= reference fully plastic value for member

= vector of redundant forces

= fully plastic values of the axial and shear forces

~ axial and shear forces

~ vector of fully plastic reference values

= vector of negative end member forces

= vector of forces at both member ends

= vector of components defining yielding at both member ends

= vector of normal {zed forces at both member ends ~

= vector containing the squares of the elements of Rn

= number of segments in the yield surface

= vector of member distortions

= vector of joint displacements

= vector of hinge distortions

superscript t denotes transposition

x

CHAPTE~ 1

I NTRO DUCTIO N

1.1 Object of Study

[he object of this study is to present a unified network

approach to the fonnulation and solution of the analysis and minimum

weight design of rigid-plastic structural systems. These problems have

been widely treated in the 1 iterature, but, to the author1s knowledge,

no single unified formulation has been presented.

The methods to be considered in this study are those general Jy

classified as optimization techniques. The strict relationship between

the theorems of plastic analysis and the dual ity concepts of 1 inear

programming are introduced and carried out throughout the study.

This study does not intend to introduce new concepts of

plastic analysis or design, or to develop new mathematical programming

techniques. Rather, its aim is to show that the vast majority of the

problems of plastic analysis and design, as presented in the 1 iterature,

are only special cases of the formulations developed here. These

formulations are especially appropriate for digital computer

implementation. Furthermore, the network formulation makes it possible

to study the convenience of applying speciaJ ized optimization techniques

to the problem.

In order to develop the formulations an,approach similar to

that used for the network ana1ysis of elastic or elastic-plastic structures

is introduced. However, whereas in the above cases the problem is

established as a set of simultaneous equations, the present problem is

establ ished as one of mathematical programming.

2

The structures under consideration are those which can be

represented as networks, i.e., plane frames, plane grids and space

frames.. The structures are assumed to be subject to proportional loading

and to be formed of a rigid-plastic material. Thus the structure

col lapses when a sufficient number of yield hinges is created to trans-

form the structure into a mechanism.

The conventional methods of plastic analysis and design

usually consider only one of the stress components acting on a section

as decisive for the definition of yielding. However, each one of the

stress components may produce yielding when acting alone. Recent

studies, based on the theory of combined stresses, have developed the

concept of the yield surface for the definition of yielding. The

concept of a yield surface is the basis for the general ized network

formulation presented in this study.

1.2 Background

In this section only a few of the large number of studies

related to optimization techniques for plastic analysis and design and

to the app) ication of network topology in structural analysis are

described. The works included are those which are considered to be most

relevant for the present study.

a) Survey of plastic analysis and design. In 1949 Greenberg

and prager(13) stated and proved the static and kinematic theorems of

plastic ana1ysis. Charnes and Greenberg in 1951 (6) proved that these

two theorems (for the case of trusses) could be represented as dual

1 inear programming problems.

By assuming the weight of the structural members to be 1 inear

functions of their plastic moment capacities, Heyman in 1951 (17)

establ ished the minimum weight design of frames in a 1 inear programming

form. Foulkes in 1953(12) approached the same problem from the point of

view of I inear programming by using the possible collapse mechanisms of

a structure and Prager in J956(3 0) studied the problem in greater detail.

Dorn and Greenberg in 1955(9) studied the relationship of the

plastic analysis theorems and the dual ity concepts of 1 inear programming

as appl ied to trusses and Prager in 1962(3 1) as appl ied to frames.

Prager in 1965(32) showed the analogy ~etween the problem of plastic

analysis and that of network flows, the latter being common problem in

ope ra t ions resea r'ch'. (5)

More recent studies have dealt with the practical design of

steel frames as reported bv Bigelow and Gaylord in 1967(2), who included

within the model buckl ing constraints for the structural members, and by

Ridha and Wright in 1967(3~), who treated the problem of the minimum

cost of frames by using nonl inear programming.

The present trend in the development of plastic analysis and

design methods is mainly oriented toward the study of minimum weight

design for the case of alternative loads (33), and the theoretical and

experimental studies of the prob1em of overall frame instabil ity (as

given by the P-6 effect) which has proven to be an all-important

problem for the case of multistory steel frames. (20)

b) Network-Topology in structural analysis. Since 1955 the

digital computer has become an essential tool fo~ the structural

engineer and has changed the entire approach to structural analysis.

Several matrix formulations and programming techniques have been

presented which showed the ease of automation of one or more aspects of

the analysis problem.

Fenves and Branin in 1963(10) showed that the problem of

elastic analysis of structures is just a particular case of the more

general network theory of 1 inear systems and developed the network-

topological formulation of structural analysis. This formulation is

particularly well suited for programming on a digital computer because

the formulation is completely general and can take advantage of the

programming techni~es developed for general networks. The previous

development settled the foundations for the development of STRESS(11)

and other problem-oriented languages WhiCh have enjoyed wide success in

the engineering profession.

In recent studies Bruinette and Fenves in 1966(4) and Morris

and Fenves in 1967(27) developed network formulations for the elastic-

plastic analysis of structures. The procedures uti1 ized consist of

tracing the load=deformation behavior of a structure as it is subjected

to cumulative sets of loads. By incrementing the load factor, the

locations of the successive yield hinges are determined until the

condition of collapse of the structure is attained. The representation

of the plastic characteristics of the structural eiements is based on

the theory of combined stresses which is described, for instance, in

H d (19)

o gee

1.3 Assumptions and Limitations

The assumptions and limitations uti! ized in this study can be

classified as: those relating to the material,. those relating to the

structurai members and those reiating to the structure as a whole.

(i) Material

a) The Tresca or von Mises yield conditions apply.

b) The stress-strain diagram is rigid-plastic.

4

c) The rna te ria 1 iss tab 1 e.

(i i) Structural Members

a) All members are prismatic and straight with two axes

of symmetry.

b) Buck1 ing of members is not considered.

c) Plastic flow can only take pJace at discrete points

in the members ..

(iii) Structure as a whole

a) The loading is restricted to concentrated loads.

b) A single set of proportional loads is considered.

c) Overall frame instabil ity is not considered.

1.4 Organization of Report

In Chapter 2 the plastic analysis theorems and the dual ity

concepts of 1 inear programming are initially introduced. Two examples

of plastic analysis arealscussed in detail in order to show the rela­

tionship between plastic analysis and 1 inear programming_ These two

examples are also used to illustrate the method of collapse mechanisms

and to present the minimum weight design problem.

Chapter 3 starts with an introduction to graph theory and

:network-topology. Then a definition of the matrices involved in plastic

analysis is given and the network formulations are establ ished for the

s impJe plast ic case, i.e., wi thout stress interaction. The network

formulation is extended for the case of minimum weight design.

Chapter 4 starts with a brief discussion of the yield surface

and then proceeds to general ize the network formulation of the previous

chapter so as to include the effect of stress interaction. The

5

general ized formulation of the minimum weight design probJem is also

carried out.

6

In Chapter 5 the analysis problem is first reduced to a 1 inear

programming problem. The relative size of the problems obtained through

the appl ication of the mesh and node formulations is then compared, and

a study of the constraints is carried out in order to discuss the

appl ication of special ized procedures, e.g., ·the decomposition principle,

to the solution of the problem. A description of the imp1ementation of

the computer program is given. Finally, the appl ication of nonl inear

programming in plastic analysis is considered.

Chapter 6 presents several examples which emphasize the main

points developed in this study, i.e., analysis and design problems using

both the mesh and node formulations, and including the effect of stress

interaction for different types of structures.

Chapter 7 presents a summary of the conclusions reached in the

study and some suggestions for further investigation.

CHAPTER 2

PLASTIC ANALYSIS AND LINEAR PROGRAMMING CONCEPTS

In this chapter the plastic analysis theorems and some of the

basic concepts of 1 inear programming are initially stated# Two examples

which illustrate the relatibnship between plastic analysis and 1 inear

programming are presented and discussed in detail. Finally, the use of

collapse mechanisms and the minimum weight design problem are considered.

2.1 Plastic Analysis Theorems

In this section, the basic theorems of plastic analysis are

summarized. These theorems were initially stated by Greenberg and

Prager. (13)

The load-carrying capacity of a rigid-plastic structure can

be calculated by using either one of the two fundamental theorems of

pla~tic analysis. Before stating these theorems it is necessary to make

the following definitions.

Definition I. A statically admissible state of stress is

defined as follows:

a) the stresses are in internal equil ibrium.

b) the stresses are in equi 1 ibrium with the external loads"

c) the yield 1 imit is not exceeded anywhere in the structure.

Definition II. A kinematically admissible (or compatible)

system of velocities is defined as follows:

a) the system does not violate the kinematic constraints

(support conditions) of the structure.

b) the total external power is positive.

The two fundamen ta 1 thea rems rna y nov! be s ta ted as fo 11 ows :

7

First Theorem (static theorem):: The load-carrying capacity nf

a structure is the largest of all loads corresponding to statical1y

admiss ible states of stress.

Second Theorem (kinematic theorem): The load-carrying

capaci ty of a structure is the smallest of all loads corresponding to

kinematically admissible systems of velocities.

Corollary. If there exists simultaneously a statical 1y

admiss ible state of stress and a kinematical 1y admissible system of

veloci ties, then the load appl led on the structure is equal to the load-

carrying capacity, as this load cannot be greater than the load-carrying

capaci ty according to the static theorem and cannot be smaller than it

according tn the kinematic theorem.

2.2 0athematical Programming Prel iminaries

Mathematical programming deals with the problem of maximizing

(or minimizing) a function of one or several variables which must

satisfy a series of functions cal led constraints. The function to be

maximized is called the objective function. The typical mathematical

programming problem can be formulated as follows:

maximize f(x 1 , x~, ••. , X ) I... n

subject to g i (x I ' x 2 ' e e G , X ) > 0 for == 1 , -0 0 • , u (2 • 1 ) n

g i (xl' x2 ' • G • , X ) n < 0 for == u+l, • .0 • , v

g i (xl' x2 ' .. II! tI , X ) == 0 for = v+l, ... , m n

The simplest mathematical programming problem is obtained when

the objective function and the constraints are 1 inear functions of the

variables. The probiem is then called a 1 inear programming problem.

All other types of mathematical programming problems are generally

9

classified as nonI inear. While the methods available for solving linear

programming problems follow a fairly unified pattern which has been

successfully automated, this is not the case for non1 inear programming

problems. Many methods for solving non1 inear programming problems have

been developed, but each one of them is in general appl icabJe only to

certain types of problems presenting defined characteristics. (15)

In this study, primary attention is given to the appl ication

of 1 inear programming to rigid-plastic analysis and design. The

appJ ication of non1 inear programming is briefly discussed in Section

5.5.

2.2.1 Linear Programming and Dual lty Concepts

In this section the 1 inear programming problem is establ ished

and the dual ity concepts which are essential for the present study are

stated.

The typical 1 inear programming problem is as fol lows:

+ e •• + a x < b mn n - In

Xl' ••• , X > 0 n -

(a)

(b) (2.2)

The values of the coefficients c 1, ••• , cn,a 11 , ••• , amn,b 1,

•• 0, b are known, and the problem consists of finding a set of non-n

negative values for the variabies Xl' .e., xn which maximize the

objective function (a) and at the same time fulfill the constraints (b).

Any set of variables which satisfy the constraints is cal led a feasible

solution. The feasible solution which maximizes the objective function

is called ,the optimal solution.

10

The p ro b 1 em rep res e n ted byE q • (2. 2 ) can be cas tin rna t r j x

form as fo Ilows:

maximize ex

subj ec t to AX ~ B (2.3)

In which e

X > ° :::;: [c

1c

2 ... c ]

n

a i i

A :::;:

a ml

cl n

X a mn

xl

x n

and B :::;:

b m

Problem (2.3) is cal Jed the primal 1 inear programming

problem. Corresponding to this formulation there exists another

related 1 inear programming problem, called the dual, the formulation

of which is as follows:

.. , Btu minimize \"t

1,4,2:0

In wmich C, A and B are the matrices defined above for the primal

problem, and Wt

::; [wi' e • ., Wm

] is a vector of dual variables ..

It is to be noted that problems (203) and (2.4) are mutually

dua' !I i. e., e i the r one of them can be cons i de red as the pr ima 1 and the

other as the dual.

The following fundamental properties of dual 1 inear program-

. bi f' J • • h' d (.14) mlng pro ems are 0 speCIe Interest In t IS stu y:

(i) If X is a feasible solution to the pr imal and W is a

feasible so 1 uti on to the dua 1 , t then ex $. B W.

(i i) If X is a feasible solution to the pr ima 1 and 1,4 is a

feas i b 1 e solution to the dual such ... t-

that ex = B 1,4, then X is an op t i rna 1

solution to the primal and W is an optimal solution to the dual.

(i i i) If one of the two problems has an optimal solution,

then the other also has an optimal solution and the maximum of CX is

equal to the minimum of stW.

11

(iv) If the ith constraint in the primal is an equal ity then

the ith dual variable is unrestricted in sign and if the jth variable

in the primal is unrestricted in sign the jth constraint in the dual

wi 11 be a strict equal ity.

(v) If the ith constraint in the primal is fulfilled as

strict inequal ity in an optimal solution then the ith dual variable is

zero in an optimal solution to the dual. If the jth variable appears in

an optimal sO'lution to the pri:nal then the jth dual constraint holds as

a strict equality in the optimal solution for the dual (principle of

complementary slackness).

well as some other properties of dual problems have been widely

described in the 1 i terature. (7~ 14)

2.3 Use of Mathematical Programming in Plastic Analysis

In the following sections two examples of the calculation of

the load-carrying capacity of a structure will be presented by applying

the theorems of plastic analysis through a mathematical programming

formulation (linear for the present cases). The purpose of these

examples is to establ ish a complete relationshp between the concepts of

plastic analysis and 1 inear programminge

2.3.1 Example 1: System of Coll inear Springs

As a first illustration of the relationship between 1 inear

12

programming and the plastic analysis theorems, a system of colJ inear

springs is considered. This is an ideal ized system in which all the

members (rigid-plastic springs or bars) are colI inear. Thus only axial

forces are involved and no shear forces or couples have to be con=

side red.

The system shown in Fig. 2.1 was discussed by prager(3 2) as

an illustration of the relationship between plastic analysis and network

flows.. Jo i n t is the support of the· structure. The bars and the

corresponding plastic capacities (yieJd points for the present problem)

are as follows:

Bar

1-2

2-3

1"'3

Plastic Capacity

P12

'P23

P13

In this example, the static theorem will first be used to formulate the

analysis as a 1 inear programming probiem of maximizing the external

load F under a set of constraints specifying admissible states of stress.

If the bar forces are denoted by Px12' Px13' and Px23 ' then

the 1 inear programming formulation becomes:

maximize F 1- \ V:J/

subject to F - Px13 .. Px23 = 0 (b)

Px23 ... Px12 = 0 (c)

Px12 $: P12 Cd) (2.5)

-Px12 ~ P12 (e)

Px13~P13 (f)

-Px13 s: Pl3 (9)

(h)

( i )

1 3

Px23 ~ P23

·Px23 :s;; P23 (2.5) F 2: 0 U)

Px12' Px13' Px23 unrestricted (k) in sign

In the above formulation:

(i) (a) represents the appl ied load to be maximized.

(i i) (b) and (c) represent the equi 1 ibrium equations at

joints 3 and 2, respectivelyo

(iii) (d), (e), (f), (g), (h) and (i) specify that the bar

forces must not exceed the plastic capacities.

(iv) U) and (k) specify the signs of the unknown variables.

The solution yields the value of the 10ad-carryin9 capacity,

F, as well as the corresponding values of the bar forces. Through this

formulation the load carrying capacity is obtained as the least upper

bound of all loads corresponding to admissible states of stress.

The kinematic theorem can also be used to formulate the

analysis as a 1 inear programming problem of minimizing the external load

which acts on kinematically admissible systems of velocities. By giving

a unit velocity to the 10aded joint Ooint 3), a set of consttaints for

compatible velocities can be establ ished. This is done by requiring

that the velocities of the two chains 1-2-3 and 1-3i which transmit the

load to the support, be at least equal to the velocity of the loaded

joint. It is also neces~ary to establ ish a compatibil ity equation

between the ve 1 oc it i es of the two cha i n's -by spec i fy i n9 the' con t i nu j ty of

velocities in the mesh 1-3, 3-2, 2-1. Thus, the minimization of the

applied load is equivalent to the minimization of the external power,

14

since the velocity of the loaded joint has been set equal to one. By

equating external and internal power, it is also equivalent to the

minimization of the power dissipated in the barso

If w12 ' w23 ,. and w13 denote the velocities of extension at

yielding of the bars, then the 1 inear programming formulation of the

kinematic theorem becomes:

minimize Pl2w12 + P1..,w 1.., + P23w23 (a) j )

subject to w 12 + w23 ~ 1 (b) (2.6)

w13 = w12

-w23 0 (c)

\;.,1 12' w13 ' w2J ~ 0 (d)

In the above formulation, (b) and (c) define a kinematically admissible

system of velocities.

The solution provides an immediate answer for the load-carrying

capacity of the structure, F, i.e.:

F :;;; (2.7)

in which wJ2 ' w13 and w23 are the values of the variables given by the

optimai solution to Eq. (2.6). It can be seen that bv using this

formulation the load-carrying capacity is obtained as the greatest lower

bound of all loads corresponding to kinematically admissible systems of

velocities.

From the above two formulations, it can be seen that by using

the static theorem the load-carrying capacity is obtained as the least

upper bound of all loads corresponding to admissible states or stress,

and that by! using the kinematic theorem this load is obtained as the

3 reates t 1 O\'/e r bound of all ] oads co r respond i ng to k i nemat i ca 11 y

admissible systems of velocities. Furthermore, in the first formulation

there exists a set of equil ibrium constraints while in the second

15

formulation the constraints represent compatibil ity conditions. It

therefore appears obvious that the two formulations are dual in the sense

of mathematical programming. That this is indeed the case is confirmed

by writing the dual 1 inear programming problem of the formulation corres-

ponding to the first theorem, i.e., Eq. (2.5). The dual problem is:

subject to w3 > 1

"'w., + w!., - W', '., = 0

) I) I)

... w + w2 + w~3 - Wi' == 0 3 23

(2.8)

-w + Wi co wI! == 0 2 J2 12 I If I II I' If 0 w12 , w12 , w13 ' w13 ' w23' w23 ~

w2, w3 unrestricted in sign.

The new variables w2

and w3 will be interpreted below. Also, two

variables appear per bar, representing the two possible non-negative

velocities of yielding. This is due to the fact that the dual

formulation is completely general, and contemplates the possibll ity of

bars yielding in tension as well as in compression. Since for the

loading shown the bars cannot yield in compression, it is possible to

wr i te:

w" == Wi f == w" == 0 12 13 23

(2.9)

and Eq. (2.8) become:

subject to w3 2: 1

"'w 3 + wI3 == 0

-w + w2 + w23 = 0 3 (2. 10)

-w2 + Wi == 0

12

16

wz, w3 unrestricted in sign (2. 10)

The last two constraints define w2

and w3 as the velocities of joints 2

and 3, respectively. Substituting these values into the first two

constraints reduces the formulation to:

minimize PIZw 12 + P13w13 + P23w23

subject to wi2 + w~3 > 1

-Wi "'w I + w: 3 = 0 12 23 l

w1 2 ' wi3' w23 ~ 0

(2. 11)

The formulation of Eq. (2.11) is identical to the one given by Eq. (2.6),

with the obv ious change in nam i ng the va r i ab 1 es 0 Thus, it has been shown

that the appl ication of the first and second theorems of limit analysis

yield problems which are dual in the sense of mathematical programming.

This fact was first shown by Charnes and Greenberg. (6)

It can also be seen that for any reasonable values of the

plastic capacities of the bars, the solution to the problem formulated in

Eq. (2.5) wii; suppiy a value of F that is larger than zero. By using the

principle of complementary slackness of J Inear programming, it can be

concluded that the first constraint of the duaJ formulation is fulfilled

as a strict equal fty and therefore the velocities of extension at yielding

of bars 1-2 and 2-3, and thus the velocity of bar 1-3, will be equal to

the velocity of the loaded jointo This fact could have been easi1y pre-

dicted purely by compatibil ity considerations.

It should be noted that whereas the static' theorem was for-

mulated in terms of joint (nodal) equilibrium equations, the kinematic

theorem was estabJ ished in terms of mesh continuity equations. The

relationship between these two approaches will be discussed in Chapter 3,

after a network notation is introduced.

17

233.2 Example 2: Planar Frame

As an additional j 11ustration of the mathematical dual ity of

the two formulations, a planar frame subject to proportional loading will

be discussed in this section. The load-carrying capacity will be first

obtained by applying the static theorem. Thereafter the dual of the

initial formulation will be written and it will be shown that the

formulation so obtained is a direct statement of the kinematic theorem.

Consider the frame represent~d in Fig. 2.2(a), subject to the

indicated loading in which the parameter a is a constant that defines the

ratio between the horizontal and vertical external loads. For simpl icity,

it is assumed that the bending moment is the only stress component which

enters into the definition of yielding at a given cross-section. For

this particular problem there are five potential hinge locations as

indicated by the numbers 1 to 5 in figo 2.2(a).

In order to avoid the probJem of determining which one of the

members meeting at a joint will contain the plastic hinge, it is assumed

that the plastic moment capacity of the beam as well as that of the upper

half of the columns is m2

, and that of the lower half of the columns is

mI" Since the structure is indeterminate to the third degree~ a complete

release at the member end located just below hinge location 5 is intro-

duced (see Fig. 2.2 (b )).. I f the two redundan t fo rces and the redundan t

bending moment are denoted by pI, pI and ml as in the figure, then the x y z

1 inear programming formulation corresponding to th~ static theorem is as

follows:

18

maximize F

subJ"ect to (ah + £/2)F - £ps - m' < m y z - 1

1/2F - hp· - £p! - ml < m x y z ..... 2

_hpl - il2p~ - ml < m x y z ..... 2

-hp~ - m~ ~ m2 (2. 12) -m~ :;;: mj

-(ah+tI2)F + lp~ + m~ S m,

-112F + hp' + £pl + ml ~ m2 x y ,z

hp' + £/2pl + m' < m2 x y z -

hp' + rn l ~ m2 x z

m~ ~ m)

F 2: 0

pi, pi, m' unrestricted in sign -x-- Y- -z

The problem calls for the maximization of the reference load FQ

The two groups of five constraints in Eq. (2012) specify that the absolute

value of the moment at the five potential hinge locations does not exceed

the plastic capacity. According to the static theorem any load which

satisfies the constraints corresponds to an admissible state of stress and

the largest of all these loads is the load-carrying capacity of the

structure. The solution gives this load as well as the values of the

redundan ts ..

In order to be able to study the problem from the point of view

of the kinematic theorem the dual of the probiem formulated in Eq. (2. i2)

is discussed in the following. The new problem is as follows:

19

minimize m1 (wI + Ws + w6 + wIO) + m2 (w2 + w3 + w4 + w7 + w8 + w9)

subject to

-Wl-W2-W3-W4-w5 + w6 + w7 + wa + w9 + wlO

w1' ••• , WID 2 0

= 0

(2. 13)

The problem calls for the minimization of the internal power dissipated at

the hinges subjected to two types of constraints, as follows:

(i) the first constraint is a compatibil ity condition corres'"

ponding to a unit velocity given to the general ized external force. Since,

_i_~_<]enera]_, F wi 1 1 be gfeater ~han z~ro in t~e_o?~_imal solution to the

primal problem, this constraint will be fulfilled as a strict equal ity

according to the principle of complementary slackness. This constraint

g ua ran tees tha t the ~ .. JO rk of the ex te rna J fo rees is pas it i ve ~

(ii) the last three constraints are statements of the fact that

the resultant velocities at the re1eases should vanish. That is to say,

the final vertical, horizontal and angular velocities at the point where

the releases were introduced must be equal to zero.

It is interesting to note that in the optimal solution to Eq.

(2.13) only one of the pairs of velocities (wJ-w6), (W2-w

7), (W

3""W8),

(w4-w9

) and (wS-w lO ) can appear in the bas is since ,the corresponding

vectors are linearly dependent, i.e., if WI is in the optimal bas is it

precludes the existence of w6 in the same basis. This is physically

obvious since Wi and wi+S (i=1, ••• , 5) correspond to the rotation at hinge

location i and the appearance of either Wj or wi+5 in the solution only

indicates the direction of the rotation. Thus, if w. appears in the I

20

solution then compression is produced on the inside face of the hinge and

if wi+

5 appears in the solution then tension is produced on the same side.

The solution yields the minimum of the dissipated power

corresponding to kinematically admissible velocities, since the velocity

associated with the general ized load has been assumed to be equal to one.

The solution also gives the values of the load-carrying capacity

(maximum value of the reference load F) according to the equal ity of

internal and external power. Hence the prob1em calls for the minimization

of the general ized reference load corresponding to kinematically

admissible systems of velocities and is therefore a restatement of the

kinematic theorem.

2.4 Use of Collapse Mechanisms

A method often used for the ca1culation of the load-carrying

capacity of a structure consists of setting up the set of possible

collapse mechanisms and estabJ ishing for each mechanism an equation of

conservation of power. (19) Each collapse mechanism suppl ies a value

for the load-carrying capacity and, according to the kinematic theorem,

.. 1-._.1 ___ -' _ ......... .,.\1: ..... """ ~ ........ "",.: .... t.~ , ... ___ ....... 1 +_ .... "' _ _ ....... _11 __ ~ _.&. .... L.._ ... _1 .. ___ _ ~1It: Ivau \..Clllylll::l \"'ClI-'CI\..II..Y I;:) I;\.jUClI I..V 1.111; ;:)IIIClIII:;:;:)1. VI 1.111; value;:);:,v

obtained.

In order to illustrate this method, the two examples used in

the previous section will be considered again.

2.4. 1 Examp 1 e I: Sys tem of Co J 1 i near Spr i ngs

The system represented in Fig. 2.1 has three bars and one degree

of indeterminacy. Therefore, there should exist two independent equi 1 ibrium

equations between the bar forces or, equivalently, two basic collapse

21

mechanisms. These mechanisms are represented in Fig. 2.3.

For mechanism 1, it is possible to write the following power

equation (see Fig- 2.3 (a)):

Flel

~ P23 el + P13 e' (2. 14)

in which F) represents the load-carrying capacity as given by this

mechanism, e' the veJocity of collapse, and P23 and PI3 are the plastic

capacities as defined in Section 2.3. L Equation (2.14) suppl ies the

following value for F1 :

F] ::: P~3 + P13 (2.15)

Sif~lilarly, the power equation for mechanism 2 is given by (see Fig. 2.3(b)):

F2e'\ '0" P12eli -1- r13e" (2.1()

In which the quantities are defined similarly as in the case of mechanism

1. Equation (2. Ie) reduces to

(2. J 7)

In Eqs. (2.1L~) and (2. ]6), e' and e'l can be cons idered as

displacements or as velocities by' imagining the collapse of the structure

taking place during a unit of time.

The actual load-carrying capacity F of the structure is given

by

F == min I.... 1 Q \ \L.. I UJ

The solution yields the value of the load-carrying capacity,

F, and also gives the controll ing mechanism.

2.4.2 Example 2: Planar Frame

The two basic mechanisms and the one possible combined'

mechanism for the frame represented in Fig. 2.2 are shown in Fig. 2.4.

For mechanism 1 (Fig. 2.4(a):) the power equation is:

(2. 19)

and the load suppl ied by this mechanism is

Mechanism 2 (Fig. 2.4(b)) yields the following power equation

and a va1ue of the load

F 2 = 8m2/.e

Mechanism 3 (Fig. 2.4(c)) yields the following·power equation

(ah+212)F3

= 2m] + 4m2

and a value of the load

(2m j +4m2

)

F3 = (ah+112)

22

(2,,20)

(2" 2 i)

(2.22 )

(2.23 )

(2.24 )

According to the kinematic theorem the' load-carrying capacity F is given

by

(2.25)

By considering the two examples presented above, it could be

argued that the use of collapse mechanisms is a much faster and sJmpJer

method for the calculation of the load-carrying capacity of a structure

than the appJ ication of 1 inear programming. This is only true for very

simple structures. As will be seen later the enumeration of the basic and

combined mechanisms is for complex structures a very difficult task

presenting serious drawbacks for computer implementation. On the other

hand, the 1 inear programming formulation can be readi Iy systematized.

23

2~5 Minimum Weight Design

While the classical methods of analysis and design of elastic

structures follow different patterns, in the case of rigid-plastic

structures the methods considered in the previous sections for analysis

can also be appl ied for design with only a s1 ight change in the

formulation. Since an objective function has to be defined in order to be

able to handle the problem as a mathematical programming problem, it is

convenient to choose the weight of the structure as the design objective

function to be minimized. Furthermore, the weight of a structural member

per unit of length is a function of the plastic capacity of the member.

As a first approximation, this function can be assumed to be 1 inear. (22)

The two problems of analysis and design of rigid-plastic structures follow

to a certain extent a reverse pattern. While in the case of analysis

the p]astic capacities of the structural members are assumed as known and

it is required to calculate the load-carrying capacity, in the case of

design the system of loads is completely specified and it is required to

assign plastic capacities to the members in such a way that the structure

will safely carry the loads and at the same time furnish a minimum weight.

The two structures considered in the preceding sections wiii

again be considered, but in this case the externa1 'loads are considered

to be known and the structure is to be designed fOi minimum weight.

2.5.1 Example 1,: System of CoIl inear Springs

In the structure represented in Fig. 2. 1~ the value of the

app1 ied load F is known and it is required to determine the plastic

capacities of the bars, i.e., the values P12' P23 and P13' which give a

minimum weight for the structure when the weight of each bar per unit of

length is assumed to be proportional to its plastic capacity.

24

By applying the static theorem and establ ishing the con-

straints in a manner similar to that done in Section 2.3.1, the minimum

weight design problem can be formulated as follows:

m j n i m i z e .Pt 1 2 P 1 2 + i J3 P13 + i 23 P23

subject to Px13 + Px23 :::;

-Px23 + Px12 :::;

Pl2 - P)C12 2:0

P12 + PxJ2 ~ 0

P13 "" Px13 ~O

PI3 + Px13 ~ 0

Pi3 -Px23 2: 0

P23 + Px23 ~ 0

F

0

Px12' Px13' Px23 unrestricted in sign

(2.26)

In this formulation £;2' e23 , £13 are the known lengths of the bars, F

the known external load, P12' P23' PJ3 the unknown plastic capacities

of the bars and Px 12' Px13' Px23 the bar forces. The solution yields the

plastic capacities of the bars which furnish a minimum weight for the

structure.

The dual prob 1 em of Eq .. (2.26 ) is:

maximize FW3

subj ect to w3 - w .. Wi + w" :::; 0 2 23 23

w2 ... Wi + wf

' :::; 0

12 12

w3 CD Wi + wll'3 :::; 0 13

w12 + w','2 ~ £12 (2.27)

wI3 + wY3 S £T3

w23 + w23 ~ £23

w2' w, unrestricted in sign J

I 11 I W'l I W'l 0 w12 ' w12 ' wI3 ' 13' w23 ' 23 >

25

(2 .27)

In the same manner as was done in Section 2.3.1, it is. possible to set

w'l:2' w'113' w23 ell 11 equa 1 to zero and to subs t i tute Wz and w3 as given by

the second and third equal ity constraints into the first equal ity

constraints. The formulation reduces to:

•• F I + F J maXimize w12 w23

subject to w12 ~ £12

w23 ~ £23

w13 ~ £13 (2.28)

This formulation calls for the determination of a set of compatible

velocities subject to the constraints w •. < l .. and which maximize the IJ - IJ

externa 1 power. For this case the soiution is trivial, i.e., w12 = IJ .(.112'

W13 ~ £13' w23 = £23' and the problem does not deserve further dis-

cuss ion.

2.5.2 Example 2: Planar Frame

In the frame represented in Fig. 2.2 the appl ied external loads

are assumed to be specified and it is required to assign plastic moment

capacities to the members in such a manner that the weight of the

structure is minimum.

It is to be recalled from Section 2.3.2 that h is the member

length ass9clated with m1 and h+t the member length associated with m2•

The unknown plastic moment capacities m1 and m2 are to be determined in

26

such a way that the weight of the structure is minimum. The static

theorem can be appl ied to the solution of the minimum weight design

problem as follows:

minimize m1

(h/2+h/2) + m2 (h/2+h/2+£)

subj ec t to m1

+ .epl + m' > (ah+~/2)F y z -

m2 + hp' + x .epl + ml > 112F Y z -

m2 + hpJ + 1J20' + m'

> ° x ._- -, yz

m2 + hp' + m' x z ,2:0

m1 + m' >0 z -

mJ

... .ep' - m' > -(ah+tI2)F (2:29) y z -

m2 ... hpj - £p' ... ml > -£/2F x y z -

m2 - hpj ... £/2p' - m' > ° x y z -

m2 ... hp' ... mB ,2:0 x z

rn] ... m' > ° z -

m) , m2 .? 0

I I rn l unrestricted in sign px' Py' z

The constraints are the same as in Eq. (2.12), i.e., they specify that

the plastic capacity must not be exceeded at any potential hinge

iocation. The solution to the problem yields the plastic capacities

which minimize the weight of the structure.

The dual of the problem given by Eq. (2.29) is as follows:

maximize (ah+£/2)FwJ

+ £12Fw2 - (ah+£/2)Fw6 - £/2Fw7

subject to WI + Ws + w6 + w10 ~ h

W#\ + W., + w,. + w'"7 + wQ + wn S h+£ I:. ;) "T I v :; .

hW2 + hW3 + hW4 ... hw7

-hw8 - hw9 = 0

£w 1 + tW2 + tl2w3 - £w6 - lW7 -£/2w8 = 0

(2.30)

27

W1 + w2 + w3 + WLt + W5 ... W6 - W7 ... W8 ... W9 - WJ 0 = 0

(2 e 30)

The problem calls for the maximization of the external power corres-

ponding to kinematically admissible systems of velocities. The values

of WI' .e., WID have the same meaning as in the case of analysis.

2.5.3 yse of Collapse Mechanisms for Minimum Weight Design

As a final demonstration of the relatIonship existing between

the concepts already discussed, the problem of minimum weight design

will be treated by considering the possible collapse mechanisms for the

two structures studied above.

For the colI inear system of Fig. 2.1 consider the collapse

mechanisms of Fig. 2.3. The static theorem can be appl ied to minimize

the weight of the structure under a set of constraints specifying

admissible states of stress. The problem becomes

minimize £lZ P12 + £Z3 P23 + £13 P13

subject to P23 + P13 ~ F

P12 + P13 ~ F

PIZ' P13' P23 > 0

Note that the first two constraints correspond to the two possible

(2. 3 I )

collapse mechanisms. The solution to this problem yields the plastic

capacities of the bars, i.e., PIZ ' P13 and PZ3 which minimize the weight

of the structure.

If, in Eq. (2.26) the first two equality constraints are sub-

stituted into the inequal ity constraints and the constraints defining

yield restrictions for the case of compression in the bars are neglected,

28

then the system given by Eq. (2026) reduces to that given by Eqa (2.31).

Thus the equivalence of the two formulations is establ ished. Now if the

two primal problems are equivalent then it is obvious that the two dual

problems will also be equivalent. Accordingly, the dual of Eq. (2.31)

will not be discussed.

For th~ frame represented in Fig. 2.2 the mechanism of Fig.

2.4 furnish the following minimum weight design problem;

minimize hm} + (n+t)m2

subject to 2m1

+ 2m2 ~ ahF

4m2 ~ £/2F

2m 1 + 4m2 2:: (ah+.212) F

(2.32)

The solution yields the values of the plastic moment capacities m1

and

m2

which minimize the weight of ·the structure.

In order to compare the 1 inear programming formulation

establ ished by direct appl ication of the plastic analysis theorems

with the present formulation as given by Eq. (2.32), it is more can'"

venient to compare the dual formulations. It is obvious that if the

dual formulations are equivalent then the primal formulations wi 11 also

be equivalent. The dual of Eq. (2.32) is:

maximize ahFw l + £/2Fw2 + (ah+£/2) Fw)

subj ec t to 2w' + 2w' ~ h 1 3 (2.33) 2w'

1 + 4w' + 4w' 2 3 ~ h+.£

I I Wi :> 0 WI' w2' 3 -

It will now be shown that Eq. (2.33) are equivalent to Eq. (2.30).

Note first that inEq. (2.30) no Rprticular modes of collapse were

29

assumed and hence the formulation is completely general. That is to say,

after examining the collapse mechanisms of Fig. 2.4, it can be con-

eluded that the velocities w6 ' wJ

' Wg and Ws of Eq. (2.30) have to vanish

in the solution \'Jhich corresponds to this loading. This is clear since

for none of the modes of collapse do the follovvin~J conditions exist:

(i) Tension on the inner face at hinne location 1 (w6 0);

(ii) Compression on the inner face at hinge location 3

(W3

~ 0);

(iii) Tension on the inner face at hinge location 4 (wg 0);

and

(iv) Compression on the inner face at hinge location S

(WS

= 0).

After these remarks, Eq. (2.30) reduce to: g

maximize (ah+lI2)Fw 1 + .g12Fw2

- 2 FW7

subject to wI + w10 ~ h

w2 + w4 + w7 + Ws ~ h+£

~w2 + hW4 - hW7 - hWa = 0

£w 1 + £w2 - £w7 - f wa = 0

w1 + w2 + w4 - w7 - wa - wIO

w1' w2' w4, w7' wS' w10 ~ 0

= 0

(2.34 )

This formulation can be further simpJ ified by using the three equal ity

constraints to obtain w4' wa and w10 in terms of wI' w2 and w70 A

simple substitution yields:

WLt = 2w 1 + W2 .. W

7

wa = 2w1

+ 2w2

.. 2w 7

wIO = w1

(2035 )

Using Eq« (2.35) the problem represented by Eq. (2.34) reduces to

By setting

max im i ze (ah+£/2) Fw 1 + .e12Fw2

.- .e12Fw7

subject to 2w 1 ~ h

4wl + 4w2 - 2w7 ~ h+£

w, J w2 ' w7 ~ 0

W1 = Wi

1 + W' 3

w2 = w'

2

w7 ::: Wi

1

30

(2.36 )

(2 .37)

Equation (2 .. 36) reduce to Eq. (2.33). Thus the equivalence between the

formulation corresponding to the appl ication of the plastic analysis

theorems and that of the collapse mechanisms has also been shown.

CHAPTER 3

THE NETWORK FORMULATION

In this chapter the concepts of network topology, which arise

from the mathematical theory of graphs, will be util ized for the

formulation of the analysis and design of rigid-plastic structures.

The formulations to be presented are based on the plastic analysis

theorems and mathematical programming concepts developed in the previous

chap ter.

3.1 Summary of Linear Graph Theory

The theory of graphs, which is a branch of topology, has

recently found wide appl ication in many areas of science and technology.

The concepts of graph theory which are relevant to this study(S) are

summarized in this section.

A 1 inear graph is a non-empty set, of po iots cal Jed nodes, .............. -..

a set, E, of Jines called branches and a mapping,., of E into V&V.

For each branch of E the associated two nodes of V are said to be incident

with the branch and the branch is said to be incident with each one of

the nodes. A sequence of connected branches and nodes of a graph is

called a circuit or a mesh. A tree is a graph without circuits and

therefore there exists one and only one path between any two nodes of a

tree. For any graph it is always possible to remove a certain number of

branches (possibly zero) from it in such a manner. that the remaining

graph is a tree. The removed branches are called 1 inks. Each J ink

determine~ a circuit in the graph.

3 ,

32

The basic structure of a graph is defined by the manner in

which the branches and nodes are connected and is characterized by the

trees and circuits existing in the graph. These properties are con­

veniently described by a series of matrices called topological matrices.

In some cases, as in structural analysis, it is convenient

to ass ign an orientation to the branches of a graph. This can be

accomp) ished by defining one of the nodes incident with a branch as

the positive node and the other as the negative node. The orientation

of a branch is assumed to be from the positive to the negative node.

Furthermore, one of the nodes of the graph is usually des ignated as

the da tum node.

Consider an oriented graph having n nodes, b branches and m

J inks. These three quantitites are related by the Euler-Poincare

re 1 at ion:

b :::: m+n-l (3. 1 )

The topological matrices are defined as follows:

a) Branch-Node Incidence Matrix A. This is a b by (n-1)

matrix defining the connectiveness of the branches and non-datum nodes.

The typical element aU is (+1, ... I! 0) depending on whether branch i

is (positively, negatively, not) incident with node j.

b) Node-to-Datum-Path Matrix Br" It is usually convenient to

select a tree from a given graph. The matrix 8T

is a square matrix· of

order n-1 and the typical element bik

is (+1, -1, 0) depending on

whether branch i Is (positively, negatively, not) included on the unique

path going from node k to the datum node.

c) Branch-Circuit Matrix C. This is a b by m matrix Iden-

tifying the circuits in the graph. The orientation of each circuit is

33

defined by the orientation of the associated J ink. The typical element

c. is (+1, -1, 0) depending on whether branch i is (positively, i r

negatively, not) included in circuit r.

A network is a 1 inear graph in which algebraic variables

describing the behavior of the elements of the graph are associated

with the branches and nodes of the graph.

3.2 Graph Theory in Structural Anaiysis

Any skeletal structure can be represented by a 1 inear graph

by associating a branch of the 9r~ with each member of the structure

and a node of the graph with each joint of the structure. However, with

the exception of coIl inear structures (systems in which the forces and

displacements can only occur along a unique direction and which are, in

general, of no practical interest)" the elements of the topological

matrices have to be redefined in order to be useful for the represen-

tat ion of the static and kinematic relationships which arise in

structural analysis.

In general, a structural member is subjected to end actions as

well as external loads located along the member. A typical member i of

at structure is represented in Fig. 3 .. 1 (a).. The member ends are denoted

byA and B and the orientation is ~ssuj'f'ted to 1...- c ............. ! ..... =_+- A ~ ... fAt ..... ~ ue I 'VIII JV III\, ,., LV JVIIII. "".

The vectors RAi and Ra i represent the actions exerted by the

res t of the structure on the member ends and the vectors PJ and PI(

represent concentrated loads at points J and K. Each one of these

vectors consists, in general, of six components which can be described

by three forces along three orthogonal axes and three moments about

these axes •. The action RI that one side of the member exerts on the

other s ide (for example, the segment 18 on the segment AI in Fig. 3.1 (b)

34

can also be represented by a vector of the type described above.

For a typical section at point I in Fig. 3.1 (b), Rr can be

wr i tten as

Px Py

Ry Pz (3.2) ==

l:;J .&

where p , p , p , m , m ,m represent the action components in the given x y z x y z

order. These forces are defined in a member coordinate system, as shown

in Fig. 3.1 (a), in which the x axis coincides with the centroidal axis

of the member and the y and z axes coincide with the principal axes of

inertia of the cross-section.

Since a given structural member may follow any orientation in

space, it is necessary to define a global system of coordinates so that

the static and kinematic relationships for the entire structure can be

defined with respect to a unique reference system. Thus, the forces and

distortions of a member defined in member coordinates have to be trans-

formed into global coordinates when the equil ibrium or compatibil ity of

the structure is considered. These transformations may be carried out

through rotation and translation matrices defining the relationships

between member and global coordinates at different points of the

structure. The specific nature of these matrices ha.ve been presented

1 L.._ (16,21 ) e sewlilCre.

Consider a structure having b members, n l free joints and m

redundant members. For the structures considered in this study, the

35

number of force and distortion vector components in local and global

coordinates is the same, and is denoted by f.

For the purpose of representing structural analysis as at

network problem it is necessary to define the following quantities:

a) Let the vector of externally applied loads, expressed in

global coordinates, be denoted by pl. pi is a vector containing n l

subvectors of order f, each subvector corresponding to a free joint.

b) let the vector of unknown .redundant forces at the releases,

taken at the negative ends of the redundant members and expressed in

global coordinates, be denoted by p'. pi is a vector containing m sub-

vectors of order f.

c) let the actions at the negative ends of all member$,

expressed in member coordinates be denoted by R. R is a vector con-

taining b subvectors of order f.

d) let the elements of the matrix ST be redefined as follows:

the typical submatrix 'Sr of ST is (-A~H~k' A~H!k' 0) depending on i k I I I I

whether the negative end of member i is (positively, negatively, not)

included in the unique path from node k to the datum node. The matrix

Hik is a translation matrix representing the effect of the load vector

Pk at joint,~ expressed in global coordinates, on the action vector

at the negative end of member i, expressed in global coordinates. Hik

is a square~trix of order f. The matrix A. transforms the force I

vector at the negative end of member i from member to globai coordinates.

Thus A. is a rotation m~trix of order f. I

e) let the matrix B be defined as

B == [~T.l oj

36

in which 0 is a null matrix having as many rows as there are redundant

forces in the structuree

f) let the elements of the matrix C be redefined as foJ1ows:

the typical sub-matrix C. of C is (A:H! , -A:H! , 0) depending on Dr I I S I IS

whether member i is (positively, negatively, not) included in the

circuit defined by the rth 1 ink. In the definition of H! ,sis the IS

joint incident with the negative end of 1 ink r.

g) let the elements of the matrix A be redefined as follows:

the typical submatrix A .. of A is (A~, -H~A:J 0) depending on whether IJ I I I

member i is (negatively, positively, not) incident with joint j. The

matrix Hi is the matrix transferring the force RB.from the negative to D,

the pos itive end of member i, both in member coordinates. H. is a I

matrix of order f."

Based upon the previous definitions, it is seen that the

equil ibrium requirements for the structure can be written as:

(3.4)

Equation (3.4) states that the total member forces are equal

to the sum of the forces in the statically determinate primary

structure plus the forces due to the fedundants at the releases.

Al ternat ive1 y, the equ i1 ibrium requ i rements for the structure

can be written as

(305)

Equation (3.5) states that the sum of the forces at the joints of the

structure is equal to zero.

37

3.3 Network-Topological Formulation of the Static Theorem

It is assumed in this study that yield hinges can only form

at member ends. Thus it is necessary to restrict the loads to con-

centrated loads and to insert fictitious joints at those points where

concentrated loads are acting along a member. Let RI denote the

vector of forces at each member end in the structure. R' can be

written as

R I ,= fiR ( 3 • 6 )

in which n is a diagonal matrix of submatrices n .. The typical element I

R! of R 'is: I

B ys tat i c s, R A i is 9 i ve n by:

RAi

R~ can therefore be written as: I

R~ = I

Hence the typical submatrix n. I

n. I

= H.RS· I I

[~.] RBi

of n is given

=G]

(3. 7)

(3.8)

(3.9)

by:

0.10)

In the examples presented in Chapter 2 it was'assumed that the

definition of yieldin~ at a given section of a structural member depends

only on one of the components of the stress vector acting on the

section, i.e., axial force for call inear structures or bending moment

for framed structures. On the other hand, the vector R' includes all

stress components. Thus, it is required to select the components

defining yielding out of the vector R'. This selection can be

38

represented as the premultipl ication of the vector R' by an extractor

matrix~. Then a vector R', which includes only the components of

interest, is defined as:

O.lJ)

The vector of plastic capacities (i.e., yield points for coIl inear

structures or fully-plastic moment for rigid frames) is denoted by M • p

M is a vector of the same order as RI. P

The definitions in this and the pr~ceding section provide a11

the necessary preliminaries for the network-topological formulation of

rigid-plastic analysis and design.

It is further assumed in this study that only proport ionaJ

loads act on the structure, i.e., that each element of pi is proportional

to the reference load F. Thus, the vector pi can be written as a vector

of constants P' times the scalar reference load F.

It was seen in Chapter 2 that the calculation of the 1oad-

carrying capacity of a rigid-plastic structure can be formulated as a

1 inear programming probJem by maximizing the reference load F subjected

to a series of constraints specifying the equil ibrium requirements and

the yield restrictions for the structure. As shown in Eq. (3.4) and

0.5), the equil ibrium requirements may be stated in two alternate forms.

Us j ng Eq. 0.4), the J i nea r p rog ramm i ng fo rmu 1 at i on becomes

maximize F

subject to R = BP'+Cp' I \ \a)

-M < R' <M (b) p - - P

0.12)

F 2: 0

pi unrestricted in sign

In this fo rmu 1 at ion the equi 1 ibrium requirements are given by

39

constraints (a) and the yield restrictions by constraints (b). Using

E q s • (3 • 6 ) and ( 3. 1 1) co n s t r a i n t s (b) 0 f E q • (3. 1 2 ) can be r ew r itt en

as

-M < dlR < M p -. - p

(3. i 3)

Replacing the equal ity constraints (a) of Eq. CL J2) into Eq. (3.13),

the formulation given by Eq. (3 .. 12) can be reduced to

maximize F

subj ec t to -Mp ~ til (BP I+cp I) ~ Mp

F ~ 0

pi unrestricted in sign

Using the following definitions:

B = t:re

C = die

the formulation given by Eq. (3.14) can be rewritten as

maximize F

subject to 8I F+Cpl ~ Mp

-S'F-Cp' ~ Mp

F ~ 0

p' unrestricted in sign

(3.14)

(3.15)

(3.16)

(3. 17)

(3. 18)

The formulation given by Eq. (3.18»)1 which uses the circuits

definE!d by the redundants, will be called the circuit or~'

formulation of plastic analysis. The solution 'to this problem consists

of:

~) the vaiue of the reference load, F, 'correspondIng to the

load-carrying capacity of the structure;

b) the value of the redundant! at the releases, pi;

c) the location of the yield hinges at collapse.

The interrelations between the different quantities of Eq.

(3.18) are represented in the transformation diagram of Fig. 3.4(a)

which is patterned after Roth·s diagram(3) developed for general

networks. The matrix appended to each arrow may be regarded as a

40

transformation which converts the object of the tail of the arrow into

the object at the head of the arrow.

An alternate formulation can be obtained if the equil ibrium

requirements specified by Eq. (3.5) are used. The formulation can be

wr} tten as

maximize F

subject to AtR_pa = 0

-Mp ~ RI ~ Mp

F ~·O

R unrestricted in sign

(a)

(b) (3. 19)

Replacing pI by pUF and using Eq. (3.6) and (3.11), the formulation

given by Eq. (3.19) can be rewritten as

maximize F t ...

sub J ec t to A R aD P t F :B 0 (a)

(b)

(3.20) F ~ 0

R unrestricted in sign

The formulation given by Eq. (3.20), which Is based on the

equ i 1 ibri urn requ i rements at the Jo ints (nodes) of the structure, wi 11

be called the nodal formulation of plastic ~nalysis.

41

The solution to this problem yields:

a) the value of the reference load F corresponding to the load

carrying capacity of the structure;

b) the values of the forces at the negative member ends R;

c) the location of the yield hinges corresponding to the

col lapse of the structure.

The re 1 at ions exp res sed by Eq. (3.20) a re rep resen ted in the

diagram of Fig. 3.5(a).

3.4 ·Role of aTypical Member in the Formulation

In order to illustrate the formulations presented in the

previous section, it is worthwhile to summarize the contribution of a

typical member to the matrices and equations that constitute the

formulations. It should be point out that the formulations developed

above consider the most general case, i.e., a space frame. On the

other hand, for simp) icity, a typical member of a plane frame will be

discussed in this section. A section of the member is acted upon by

three components: axial force p , shear force p and bending moment m • x y z

The value of the fully p1astic moment is m3"

The contribution of member i to the vectors R, RI and RI as

well as to the matrices nand 6 will be discussed next. The vector R

is of the form

R == {3.21}

In which RBi is the vector of forces at the negative end of member i and

is given by

The vector R i is of the form

,,; r:"l . "

R' :::: = Rs· H ~ ~_ •

L- L· :~J in which

0 0

Hi :::: 0 1 0

0 It 1

and l is the 1 eng th 0 f the member ..

Thus

0

n = n o L @

I L -J

in which

II l n. ::!II L HIJ I

Now the subvecto r R ~ is of the fo rm I

R! :::: :::: I

m • %1

Pxi

Pyi lp .-kn i yl %

42

(3 .. 23)

(3.24 )

(3.25)

(3.26 )

(3.27)

43

Since the elements of Ri which define yielding have to be extraeted, ~i

is of the form

and R~ becomes I

R ~ =.6. R! = I I I

o 0 0

o 0 0 0

r-m • l ZI

L lpyj+mzi J

The yield restrictions for the member are

"'m3 < m • < m3 ..... 2:1 ~

-m3 ~ £Pyi+.mzi ~ M3

-0.28)

0.29)

(3. 30)

For the mesh formulation (Eq. (3.18» p . and m i of Eq. (3.30) yl Z

are functions of the reference load F as well as the redundants pi. On

the other hand the restrictions of Eq_a (3.30) appear _explicitLy in the

node formulation (Eq. (3.20·». However this formulation requires the

inclusion of the equilibrium constraints given by Eq. (3.20(a».

3.5 Example 1: System of Collinear Springs

In this section it is shown how the node formulation given by

Eq. (3.20) is appl ied to the example presented in Section 2.3.1. In

the next section the mesh formulation given by Eq. (3 .. 18) is appJ ied to

the case of the planar frame studied in Section 2.3.2.

The 1 inear graph corresponding to the system is represented in

Fig. 3. 2 (a) • The rna t r i x A i s 9 i ve n by

A ==

The vector pI is given by .

1-2

2 ... 3

1-3

2

1

... 1

o

3 o

(3.31 )

44

For this particular problem R can be used instead of R' since the forces

at both ends of each bar are obviously equal. Furthermore, since only

axial forces have been included in R, no extraction is necessary, and R

can replace R'. R can be written as

R ::

in which Px12' Px23 and Px13 are the unknown bar forces. The vector Mp

of known plastic capacities is given by

(3.34 )

The formulation given by Eq. (3.20) becomes

maximize F

subj ect to r. ... ) ... 1

[~J [~J l~ ~J Px12 F =

Px23

Px13

Px12 P12

Px23 ~ P23 (3.35)

Px13 Pl3

[:x~~ l <" r:~~l l:~:~~ J

......

L~~;J F ~ 0

Px12 11 Px23' Px13 unrestricted in sign

This formulation Is identical to the fa rmu 1 at ion given by Eq. (2.5) •

45

306 Exampie 2: Pianar Frame

As an illustration of the mesh formulation the planar frame

studied in Section 2.3.2 is again considered.

The linear graph and the primary structure (a tree) associated

with this frame are presented In Fig. 3.3(a) and 3.3(b) respectively.

The vec to rs P' and pi are given by

2 r~ l 0 pi 0

X

pi = 3 1 F pI == p',' o. 36)

0 y

a mt 4 0

z

0

The matrix B is given by

2 '" J. ;) "1'

IA~:12 I\~Hi3 AfH 14 """"'I

2 I\~H23 J\~H24 B lIIIII (3. 37j

3 l 0 0 A~:34J r-

4 0 0

in which

0 0

=r-~ 0 ~ l AtH' = [-~ :] A~H12 lIIIII ... 1 0 0 A~Hi3 0 1 14

0 0 1 o 1,/2 1 , 0 t - ..J (3. 38)

0 0 0 0 0 0

A~H23 :I 0 1 0 A~H24 = 0 1 0 A~H34 = 0 0

0 0 I o /'/2 1 0 0

The "'!!!IIfo ... ;.., r . gIven by tUg """ I" ... IS

46

A~H15 2 A~H25

c == 3 J\~H35 4 A!H45

in which

rO ol I, 0 ol h~H15 I-I 0 ~J

tHI I 0 1 ~J = 1\2 25 =

Lh l L h £/2

(3.40)

0 0 0 .... 1 0

tHI = 0 1 0 I\~H45 = 0 0 1\3 35

h 0 1 0 0 1

n can be wr i tten as

I

H] 0

I

n = H2

'II' J.

(3.41 )

0 H3

I

H4

in which

0 0 0 0

Hl ::11 H4 :II 0 1 0 H :;: H = 0 i 0

2 3 0 h 1 o 112 1

Since only the values of the bending moments at the po~ential hinge

locations should enter into R', the vector R' = nR must be premultipl ied

by tJ., given by

47

001 001 o

001 001

001

o 001 001

001

The plastic moment capacities represented by Mp are as follows

m1 m

2 --_ .. 2

m2

Mp = -~~- (3.44 ) m

2 m

2 3

----4

m2 m1

The linear programming formulation, as given in the mesh formulation of

Eq. (3,,18), becomes

maximize F

sub j e c t to R I ~ Mp

-R ' ~ Mp

F 2! 0

pI unrestricted

(3.45 )

After expanding the matrix inequalities and el iminating redundant con-

straints Eq. (3 .. 45) reduces to Eq. (2.12).

3.7 Network Formulation of the Kinematic Theorem

According to the kinematic theorem the load-carrying capacity

of a rigid-plastic structure can be calculated by minimizing the

diss ipated internal power corresponding to a kinematically admiss ible

48

system of velocities. This system can be obtained by giving a unit

velocity to the general ized reference load (thus guaranteeing that the

external power is positive), provided that the velocity constraints of

the structure are preserved.

The linear programming problem is obtained directly by

writing the dual of the mesh formulation of the static theorem, i.e.,

Eq. (3. 18). The prob J em becomes

t t min im ize Mp,w 1 + MP.w2

- t ... t subject to 8' w,-B' w2 ~

-t -t C w1-C w

2 ::: 0

wI' w2 ~ 0

or, after exp~nding the matrices,

t t minimize MPWI + MPW2

subJ"ect t~ ~,tBtnt~tw _~,tBtnt6tw ~ "1 " "2

Ct nt6 tw -c t nt 6 tw = 0 1 2

In the above formulation:

(a)

(b)

(c)

(d)

(3.46 )

(3.47)

(i) wI and w2 are vectors of the same order as R', repre­

senting the distortions (or rates of distortion) at the potential hinge

locations. To every element of w1 there corresponds an element of w2 "

Since according to constraints (d) WI and w2 are non-negative then the

elements of Wi or w2 which enter into the solution indicate the sign of

the distortion at the yield hinges.

(ii) (a) represents the dissipated internal power which

should be minimized.

(iii) Constraints (b) and (c) specify the kinematically

admissible system of velocities. Constraint (b) defines the unit

49

velocity given to the general ized reference load. Since in any optimal

solution this constraint is fulfilled as strict equal ity it is a con-

tinuityequation" Constraints (c) specify continuity at the points

where the releases were introduced. There are as many constraints in

(c) as there are degrees of indeterminacy in the structure.

The solution to the problem yields the minimum value of the

dissipated internal power~ According to the principle of conservation

of energy, this is also the value of the external power. Since the

velocity associated with the general ized reference load is equal to

unity, the solution gives the value of the load-carrying capacity.

Finally, the values of wI and w2

which appear in the optimal solution

define the mode of collapse of the structure. The relations expressed

kinematic theorem can be obtained by writing the dual of the node

formulation of the static theorem given by Eq .. (3.20) .. It can be

written as

t minimize Mp(w 1+w2) . t t

subject to Au'+n 6 (w 1-w2) = 0

pltu l 2! 1

u' unrestricted in sign

W > 0 wI' 2

In the abOve formulation:

(a)

(b)

(c)

(d)

(3.48)

(i)· The vector u' represents the joint displacements (or

velocities). u' is a vector of the same order as pl.

50

(ii) Constraints (b) and (c) define the kinematically admissible

systems of velocities, constraint (b) specifying continuity while con­

straint (c) defines a unit velocity given to the reference load such that

the power of the external loads is positive.

(I i i) The objective function (a) is the same as in Eq .. (3.47),

i.e., the dissipated internal power which is to be minimizedo

The relations expressed by Eq. (3.48) are represented in the

diagram of Fig. 3.5(b).

As in Eq. (3.47), the solution to the problem yields the value

of the load-carrying capacity of the structure. The values of WI' w2 and

u' given by the optimal solution define the mode of col1apse of the

s true tu reo

In Fig. (3.6) is represented the combined Roth's diagram,

combining the mesh and the node formulations. This diagram summarizes

all pertinent relations between node, mesh and branch variables and may

be used to expl~in tr~nsformation of variables such as the one performed

in Section 2.3.

It should also be noted that:

(i) While in the elastic case the relation between member

forces and member distortions, i .ee, Hooke's Jaw, permits the solution

of the analysis problem, in the present case no relation expressible

by a mathematical function exists between R' and w. The problem can be

solved by using either actions alone or distortions alone. However,

dual ity guarantees that both manners of establ ishing the problem yield

the same solution.

51

(ii) A set of w's (hinge distortions) cannot be directly

obtained from a set of values for u' (joint displacements). In

Appendix A a procedure for generating mechanisms is presented in which

after certain assumptions, the values of w can be obtained for a given

setofui's.

3.8 Network Formulation of the Minimum Weight Design Problem

The formulations establ ished for the rigid-plastic analysis

problem using the static theorem can be easily modified for the minimum

weight design problem as follows:

a) In the case of analysis the objective function is the

reference load F which is to be maximized, while in the case of design

the objective function is the weight of the structure which should be

minimized.

b) The constraints for both- problems are the same, i.e.,

equilibrium constraints specifying that the stresses should not exceed

the plastic capacities at the potential hinge locations. However, while

in the analysis problem the plastic capacities of the structural members

were known and the values of the appl ied loads were unknown, in the case

of design the external loads are completely defined and the plastic

capacities of the members are to be determined. Hence, it is necessary

to rearrange the constraints represented by Eq .. (3.18) and (3.20) in

such a manner' that the unknowns appear on the left s ide of the

inequal ities and the known quantities appear on the right side.

The mesh formulation of the static theorem for _the minimum

weight design problem Is as follows:

minimize

subj ec t to M "'Cpl 2: B'F p

52

M +Cp I ?:"'S IF (3.49) p

M > 0 P

pi unrestricted in sign

In this formulation Lt = (£1' e •• , lb) is a vector of member lengths

used as the price vector for the linear programming problem and the

remaining quantities are as defined in,Section 3.3. The solution to the

problem yields:

(i) the values of the plastic moment capacities of the

members which supply a minimum weight for the structure;

(ii) the values of the redundant forces pi corresponding to

the optimal solutione

The minimum weight design problem can also be established by

usingthek inematictheorem. The formulation is glven- by the duaJ of

the mesh formulation expressed by Eq. (3.49). The probiem can be

wr i tten as fo 110ws

maximize ... t ... t

(a) FB' w ... FB' w 1 2

subj ec t to w1+w2 ~ L (b) (3.50 )

-t ... t 0 (c) -c w1+C w2 =

WI J w2 2: 0

In this formulation, wT

and w2 correspond to the distortions or, more

precisely, rates of distortion at the potential hinge locations and

the remaining matrices are as defined above. The problem calls for the

determination of a set of velocities, w1 and w2, which satisfy the

continuity conditions at the releases {constraints (c» and the

inequal ities wJ+W2 ~ L {constraints (b» and make the external power

as large as possible.

The node formulation of the static theorem for the minimum

weight design problem can be written as follows:

minimize ltM p

subject to AtR:::: PiF

M -mR :> 0 p .....

M +t11R > 0 p -

M :> 0 p .....

R unrestricted in sign

All the quantities in this formulation have been previously definede

The solution to the prob1em yields:

53

(i) the values of the p1astic moment capacities of the members

which supply a structure of minimum weight:

If! \ ,8 I J the values of the member end

are given by.Mp.

TheproiHem can also be solved by us Ing the dual of Eq. (3.51)

which is equivalent to the appJ ication of the kinematic theorem. The

p rob .. ) em becomes

maximize PltFul

subject to t t Aul .. n!:::. (w t ""'W2

) :::: 0

wl-tw2 ~ L (3.52 )

WI' w2 2: 0

u' unrestricted in sign

The problem calls for the maximization of the external power subject to

a set of constraints specifying kinematically admissible system of

velocities.

CHAPTER 4

GENERALIZATION OF FORMULATION

In the preceding chapters it was assumed that the yielding of

a structural member at a given section is determined by only one com-

ponent of the action vector. However, this is a rough assumption since

in general several action components may enter into play when a section

yields. This i~teraction may be accounted for by the concept of a

yield surface. In this chapter the yield surface is initially discussed

and then the effect of stress interaction is included in the "network

formulations previously developed. For simpl icity, only the formulations

corresponding to the static theorem will be general ized to include the

effect of stress interaction.

4, 1 The Yield Surface

The definition of yielding at a given section can be repre ...

sented in terms of a series of functions of the stress components at the

section which constitute the yield surface. As it has been shown else­

where !2']j the yield surface presents the following characteristics:

a) If the stress vector 1 ies inside of the surface, no plastic

flow can occur. When the stress vector reaches the yield surface the

member can undergo unrestricted plastic flow. States of stress for

which the stress vector 1 ies outside of the yield surface are "not

possible ..

b) The plastic strain increment vector corresponding to a

stress vector on the yield surface is defined, up to a mUltipl icative

constant, by the normal to the surface at the corresponding point.

54

55

c) The yield surface is convex.

An extensive amount of ] iterature deal ing with the derivation

of the yield surface for different types of sections exists.(19,27)

Consider a situation in which only two stress components enter into the

yield condition. For example, the axial force and the bending moment

are the components of interest for the case of. a planar frame. In

Fig. 4q 1 the yield surface is represented by the curves with equations

~] (R1

) = c 1 and ~2(RI) = c2 " Two possible situations of the stress

vector RI are represented by RII and RI2 . Stress vector RII I ies inside

the yield surface and therefore the section cannot undergo any plastic

distortions. On the other hand, vector RI2 1 ies on the yield surface

and hence it indicates that a yield hinge has been formed at the section

which can now undergo unrestricted plastic flow.

It is convenient to represent the force vector at a given

section and the equations defining the yield surface in terms of

normal ized stress components a A normal ized stress component is the value

obtained by dividing the actual value of the component by that value of

the same component which would cause p1astic flow at the section if it

were acting alone. Thus the equations of the yield surface become of

the form Ii (r I ) = 1, where r I is the vector consisting of the normal fzed

stress components at section 10

4.2 General ization of the Network-Topological Formulation

Based upon the previous considerations~ the formulations

developed in Section 3.3 for the calculation of the load-carrying

capacity will be extended to include the effect of stress interaction.

The general ization of the mesh formulation given by Eq. (3.18)

will be considered first. As it was shown in Section 3.3, the vector

of forces at both ends of the structural members is given by

R'= nR

56

(3.6)

Since it is more convenient to work with normal ized stresses, each

component of R' is divided by the corresponding fully plastic value.

Let the vector so obtained be denoted by R*. R* is obtained by pre­

multiplying RI by a diagonal matrix, N, containing the inverses of the

fully plastic values. Then R* can be written as

(4. 1 )

In general, the equations for the yield surface are non m l inear, but can

be adequately represented by equations of the second degree. A vector

< R* >, the elements of which are the squares of the elements of R*, is

defined. The brackets around R* indicate that the squares of the elements

of R* are to be considered. The yield restrictions can be represented

by J inear combinations of the elements of the vector R* and of the vector

< R* >, as follows:

(4.2)

in which 1 represents a sum vector and 6 1 and ~ are defined similarly

to the matrix 6 in Section 3.3, but instead of being only extractor

matrices, i.e., composed of ones and zeroes, they define 1 inear com­

binations of the elements of R* and < R* > for a given yield condition.

The typical matrix 6 1 i or ~i of ~J or ~ has 2s rows and 2f columns,

where s is the number of different equations defining the yield surface

segments and f is as defined previously. It shou1d be recalled that

yield restrictions must be written for both member ends.

The. mathematical programming formulation of Eq .. (3 .. l8) can be

genera] ized to include the effect of stress interaction by replacing the

constraints (b) of Eq. (3.12)$ defining yield restrictions in terms of

single components of the stress vectors, by Eq .. (4.2) defining yield

restrictions for the yield surfaces ..

The probJ~m becomes

maximize F

subject to R = Bpl+Cp'

57

.61 R*+~ <R*> ~

F ~ 0

(a)

(b) (4.3)

pI unrestricted in sign

Using Eqo (4 .. 1) the equilibrium conditions (a) of Eqo (4 .. 3)

can be replaced into the yield restrictions (b) of Eq. (4.3) and the

problem reduces to

maximize F

subject to .61Nn(BPIF+Cp8)+~ <Nn(BP'F+Cp'» ~

F 2: 0

pI unrestricted in sign

(4.4)

The formulation given by Eq. (4.4) wil) be called the "

genera) ized mesh formulation of plastic analysis .. The problem is in

general a non1inear programming problem. The solution to the problem

yields both the value of the reference load F corresponding to the load

carrying capacity as well as the values of the redundants at the

releases. Finally, the stresses at any desired cross section of a

member can be computed by statics.

The node formulation given by Eq .. (3 .. 20) is considered next ..

The e'qu i J ibrium constrai nts (a) of Eqo (3.20) wi 11 be the same for the

general ized formulation" On the other hand, the yield restrictions (b)

of Eq .. (3.20) are replaced by the yield restrictions given by Eq. (4.2).

58

The formulation becomes

maximize F

subj ect to

(4.5)

Runrestricted in sign

In this formulation R* is replaced by the value given by Eq.

(4.1), but no further replacements for ,the stress vectors are carried

out since they appear expl icitly in the formulation. Again pi is

replaced by pi times the reference load Fe The formulation reduces to

maximize F

subject to t ...

A R-P'F = 0

6 1 NnR+~ <NnR> S 1

F 2= 0

R unrestricted in sign

(4.6)

The fo rmu 1 at ion represented by Eq.. (4.6) wi 11 be ca 11 ed the

generalized node formulation of plastic analysis. The solution to this

problem yields the value of the reference load F corresponding to the

load-carrying capacity as well as the corresponding member forces.

The optimal solution to both formulations, i.e .. , Eq .. (4 .. 4) and

(4.6), give also information pertaining to the location of the yie1d

hinges at collapse .. This information Is obtained from the yield

restrictions which are fulfilled as strict equal~ties in the optimal

solution, i.e., the stress vectors which reach the yield surfaces.

4 .. 3 Role of a Typical Member In the General 'zed FonmuJatlon

In the present section the concepts developed in Section 3.4

will be extended to show the role of a member in the general ized

59

formulation" For simp1 icitYll a member of a planar frame is again con'"

sidered. At any given section the member is acted upon by three stress

components: the axial force p , the shear force p and the bending x y

moment m • z

The corresponding fully plastic values are denoted by PI'

P2 and m3 respectively. If the effect of the shear force on the

definition of the yield condition is neglected, (19) the yield surface

is given by the following equations, plotted in Fig# 4.2{a):

2

[:~J m

+ J.. ::

m3

2

[:~J m ~ ::

m3

It was shown in Section 3.4 that the contribution of a member

vector RI is given by the vector

R~ == I l

'Rs< I Hi~Bj

(4.7)

to the

(3.7)

Now the vector R* is obtained by premultiplying R' by N. The vector R*

is of the form

and R~ is given by I

R•• .. -.. -. R~ • I

R~ = N.R, I I

where the matrix N. is a diagonal matrix of the following form I

... ) 0 P2

.... 1 m3

... 1 0 P1

-1

L P2

_1

m3j

(4.8)

(4.9)

(4.10)

Thus the subvector R~ is of the form I

R~ = N. I I

60

(4 .. 11)

The contribution of member i to the vector < R .... > is given by the vector

"i': < R. > : I

<R~> = I

(4. 12)

Finally, the matrices bot and ~ are of the form I ~

o

~1 == ~1 i

o (4 .. i 3)

o

o

where

o 0 1 100 0

o 0 -1 I 0 0 0 - - - - ~ - - - -o 00 1 0 0

I ---_ .. _. __ ._------_ ... __ .-_. __ ._-_._" ..•.. _ ... ------- ---_.-.. _-. __ ._---.-_._._ ... _-_ .. - ------_._ .. _---_._- -"--- ._---

o 0 0 I 0 O-J

r~ 0 0 0

o 01 0 0 I 0 am

0 =- ~

~i :::: r "" 0 0

~ ~J L~ 0 o I

The y ie 1 d restrictions for the member can therefore be written as

and the expansion of the last

or

m {:~r ~ ~1 m3

lp +rn +[:~2 ~ z ~ m3

61

(4. 14)

(4. 15)

1

1.J

(4.16)

(4. J 7)

62

The four inequal ities correspond to the yield restrictions for both

member ends.. The general matrix inequal i ty of Eq .. (4.2) defines simi lar

ield restrictions for all member ends in the structure. ___________________ L ___________ _

As an alternate representation, consider the case in which the

yield surface is linearized. For this case Eq. (4.7) is replaced by

the following 1 inear equations, (19) plotted in Figo 4.2(b):

1 Px + m ... -!. = +

2 P .. m ... . I j (4. 18)

Px 1-m

+ ~ :::I I Pl 3 m3

+

All the matrices remain the same as above, with the exception

of ~J i and ~ j. _ ~ i becomes a null matrix since the non1 inear terms

disappe€!r .. ·. Now, since Eq~ (4 .. 18) defines eight yield restrictions

for each member end, 6, i becomes af 16 by 6 matrix as given by

1/2 0

1/2 0 ... 1

-1/2 0 1

... 1/2 0 -1 0

0 213 I

0 -2/3 I

.. 1 0 213 1

.6) i -1 0 -2/3 1

(4. 19) ::: ....... GIll

GO _

... - ... 1 ...

I 1/2 0

1/2 0 ... 1 1

0 I ... 1/2 0 1 I

I -1/2 0 .... 1

1 0 2/3

0 ... 2/3

I eo1 0 2/3

.. 1 0 -2/3

63

can now be written as The yield restrictions for member

* ~l·R. ~ I I (4.20)

in which R~ is as given by Eqo (4.11) 0 The expans ion of Eq. (4.20) is I

straightforward and will not be carried out here.

4.4 General ization of the Network Form~lation for Minimum Weight Design

The ideas stated in Section 3.8 can be general ized to include

the effect of stress interaction as follows:

(i) The weight of a structural member is assumed to be a

function ~ of the fully plastic values of the stress components. The

weight of the structure is to be minimized.

(if) The constraints include the effect of stress interaction

and are the same as the constraints in Eq. ,(4.4) or in Eq .. (4.6), as the

case may be. However, in the minimum weight design problem the external

loads are specified while the fully plastic values of the stress com-

-----------~ne&t~~e-~~d~t~~L~~d-.----------------------------------------------________ _

The design problem corresponding to the mesh formulation given

by Eq. (4 .. 4) for analys is can be written as

min i m i ze i: cp. (p.) I !

subject to ~JNn(BpgF+Cpj)+~ <: Nn(SP'F+Cp') > :::;

p • .2: 0 I

i = 1, ... , b

pi unrestricted

In the above formulation p. is the vector of unknown fully plastic I

(4.21 )

capacities for member i and the remaining quantities are as defined

previously.

The design problem can also be written by using a node formula-

tion which corresponds to that given by Eq. (406) for analysis, as follows

64

minimize L co. (p.) I I

subject to AtR = P'F

6 1 NnR+~ <NnR> ~

Pi 2: 0

(4.22)

R unrestricted in sign

In the two formulations presented above, the values of the

elements of the matrix N, i.e., the inverses of the fully plastic

capacities, and the values of the redundant forces pi in Eq .. (4 .. 21) or

the member forces R in Eqe (4.22) are unknown. As can be seen this

fact makes the constraints highly non) inear. Furthermore, they are In a

form very 1 ittle appropriate for implementation. To s impl ify the

problem in such a way that the formulations obtained be more convenient

fo r imp 1 ementat i on the fa 1 Jaw i n9 assumpt ions are made:

(I) The values of the unknown fully plastic capacities for

a given member are expressed as a 1 inear function of one of them, which

is taken' as a reference.

(ii) The yield surface for a given member is normalized in

terms of the reference plastic capacity for the member.

(iii) The yield surface is linearized.

(Iv) The we ight of a member is assumed to be a 1 inear

function of the reference plastic capacity for the member.

For the purposes of the formulations to be developed it is

necessary to make the following definitions:

(I) let the matrix l be defined as

-6. I

(4.23)

CHAPTER 5

1M PLEMENTATIO N

In this chapter the relative advantages of the mesh and the

node formulations of the static theorem (Eqo (4.4) and (406) respectively)

will be discussed from the point of view of digital computer implemen­

tation.. The actual implementation' is then described.

5.1 Introduction

Initially it is assumed that both the mesh and node formula­

tions can be reduced to I inear programming problems by linearizing the

yield surface. The solution of the plastic analysis problem by means

of 1 inear programming procedures will occupy the main part of the

discussion in the present chapter. As will be seen in Chapter 6, the

results obtained through the 1 inear programming formulations are

sufficiently accurate for practical purposes. Furthenno're, a computer

program using 1 inear programming methods can be implemented much more

expediently than one employing nonl inear programming procedures.

However, in order to gain generality, a computer program which uses

nonl inear programming was also developed for the present study and will

be discussed in Section 5.5.

After I inea,rizing the yield surface the mesh fonnulation given

by Eq. (4.4) reduces to

maximize F

subject to ~lNnBPIF + ~lNnCp! ~

F 2: 0

p~ unrestricted in sign

67

(5. 1)

and the node formulation given by Eq., (4.6) reduces to

maximize F

subject to AtR-P'F = 0

.61 NnR .s

F 2: 0

R unrestricted in sign

- 68

(a)

(b) (5.2)

The discussion will be first carried out with regard to the available

general purpose computer programs for solving 1 inear programming

problems. These programs treat every 1 inear programming problem in the

same manner, regardless of the specific arrangement of the elements

in the constraints. Then a discussion will be made of the usefulness of

special algorithms appl led to the solution of the rigid-plastic analysis

problem by using either one of the formulations given by Eq. (5.1) or

(5.2). Accordingly, it is necessary to study the constraints corres-

pending to each formuiation in order to decide whether or not it is

worthwhi1eto implement special algorithms for the present case.

5.2 Use of General Purpose Linear Programming Routines

a) Case of the mesh formulation as given by Eq. (5.1).

Although the sparseness or banded characteristics of the con­

straints are dependent on the manner in which the releases are

introduced, the size of the problem is the basic factor when general

purpose 1 inear programming routines are used. The global size of the

problem is as follows:

(i) Number of Constraints. The number of constraints is a

function of the number of members in the str~cture and of the number of

segments which are utilized to approximate the yield surface. Thus the

number of constraints is equal to the number-of segments, s, times

minimize l: <:OJ (Pi)

subject to AtR = P'F

64

61 NnR+~ <NnR> ~ (4.22)

P• ,. 0 1-

R unrestricted in sign

In the two formulations presented above, the values of the

e1ementsof the matrix N, I.e., the inverses of the fully plastic

capac i ties, and the va 1 ues 0 f the redundant fo rces pi in Eq.. (4 .. 21) 0 r

the member forces R in Eq. (4.22) are unknown. As can be seen this

fact makes the constraints highly nonlinear. Furthermore, they are in a

form very 1 i ttle appropriate for implementation. To s imp1 i fy the

prob1em in such a way that the formulations obtained be more convenient

for implementation the following assumptions are made:

(I) The values of the unknown fully plastic capacities for

a given member are expressed as a J inear function of one of them, which

is taken as a reference.

(ii) The yield surface for a given member is normal ized in

terms of the reference plastic capacity for the member.

(iii) The yield surface is linearized.

(Iv) The weight of a member is assumed to be a 1 inear

function of the reference plastic capacity for the member.

For the purposes of the formu1ations to be developed it is

necessary to make the following definitions:

(i) Let the matrix l be defined as

... 1:::..

I (4.23)

,"

65

in which the matrix 6. is a 25 x 2f matrix of the coefficients defining I

the 1 inear combinations of the stress components for the yield surface

of member i.

(ii) Let Q be a vector defined as

Q =: Q. I

in which Q. is a 2s vector containing the element representing the I

reference fully plastic value p. for member i. I

(ii i) Let k be a vector defined as

k T k. I

in which k. is a constant relating the weight of member I "

to the

reference fully plastic value for member i.

(4.24)

(4.25)

Using the above assumptions and definitions the genera] mesh

formulation given by Eq. (4 .. 21) is simplified as follows:

min i m i ze E k. P . I I

subject to Q-ln(BP'F+Cp') ~ 0

pi unrestricted in sign

The constraints of Eq. (4 .. 26) can be rearranged as follows:

minimize ~ k.p. I I

subj ec t to Q-tI!Cp B ?; Lfl.sp-, F

Pi 2: 0

pi unrestricted in sign

(4026)

(4.27)

66

The solution to Eq. (4.27) yields the values of the reference

fully plastic values, p., for each member as well as the values of the I

redundants corresponding to the specified loading.

In the same manner, the node formulation of Eq; (4Q22) is

simpl ified as follows:

-min i m i ze L k. p . I I

subject to AtR = PIF

Pi ~ 0

~' unrestricted in sign

(a)

(b) (4.28)

The solution to Eq. (4.28) yields the values of the reference

fully plastic values for each of the members as wel1 as the values of

the member forces correspondi.ng to the specified loading ..

69

twice the number of members, b, since the yield restrictions are written

for both member ends. Then

Number of Constraints - 2xsxb (5. 3)

(ii) Number of Columnss This formulation requires a column

and f columns per complete release.

If r complete releases are introduced the number of columns is given by

Number of Columns = l+fxr (5.4)

Since the value of s increases rapidlY,as the yield surface is more

closely approximated by linear segments, while the other quantities are

fixed for a given structure, it can be concluded that the number of

constraints is in general much larger than the number of variables.

Now, since the solution time for a 1 inear programming problem depends

much more on the number of constraints than in the number of variabJes~~)

it is convenient to solve the problem by means of the dual formulation

of Eq. (5.1).

b) Case of the node formulation as given by Eq. (5.2).

In this formulation it is necessary to consider two types of

constraints as indicated by Eq.(S.2(a» and (5.2(b». The size of the

problem is as follows:

(I) Number of constraints in Eq. (5.2 (a». These con ...

straints are equil ibrium equations corresponding to the free joints of

the structure. There are f constraints per joint and n l free joints.

Thus the number of constraints in Eq. (S.2(a» i~ given by

Number of Constraints] = fxn' (5.5)

(ii) Number of constraints in Eq. (5.2(b». In (S.2(b», as

in Eq. (5.1), there are s J inear segments per yield surface and 2xb

member ends. Thus the number of constraints corresponding to the yield

70

restrictions is given by

Number of Constraints 2 = 2xsxb (5.6)

(iii) Number of Columns. There is one column corresponding

to the externally appJ ied loads, although it does not enter in

Eq. (S.2(b), and f columns for each member. Thus the total number of

columns is given by

Number of Columns = l+bxf (5. 7)

From Eq. (5.5), (5.6) and (5.7) it can,be seen that the total size of

the problem is therefore of fxn'+2xsxb constraints and l+bxf columns.

Similar remarks as those stated for the previous case can be made here

with regard to the relative number of constraints and columns. It can

be concluded that the number,of constraints is much larger than the

number of columns and hence it is convenient to solve the dual problem

of Eq. (5.2).

Comparing the two formulations it is seen that the sizes of

the problems are

Mesh Formulation as given by Eq. (5.'])

Node Formulation as given by Eq. (5.2)

No. of Constraints No. of Columns

2xsxb I+fxr

2X5xb+fxn l l+fxb

It is therefore obvious that the first formulation yields a smaller

problem and hence it is more useful when a general purpose 'routine is

to be ut i I ized.

5.3 Use of Special Algorithms

The decomposition principle developed by Dantzig and Wolfe(8)'

can be appl ied to reduce the solution of a large 1 inear programming

problem to the solution of a series of smaller problems. Although the

71

principle can be appl ied to any 1 inear programming problem it presents

great advantages when appl ied to certain J inear programming problems

presenting special characteristics. The decomposition principle is

specially appl icable to problems presenting one of the fol1owing two

fa rms:

(a) Angular s ys terns of the form

maximize f c.x. j =1 J J

subject to

~1 ~2 . . . . A Xl bO r - ~ - ~ '!'" :- -B1 x

2 b1

B2 0 (5.8)

0 B x b r r r

x. ?: 0 j = i , ... , r J

In Eq. (5.8), A j is a mOx Ilj ma t r ix, B j is a m j x n j rna t r i x j b j is a

m. vector, and x. and c. are n. vectors. Eq. (5.8) can be interpreted J J J J

as a series of independent problems coupled together by the first

co n s t r a i n t .

(b) Staircase systems of the form r

maximize lo c.x. j =1 J J

subj ect to

o 0

o 0

o 0

o o

j = 1, •• ., r

(5.9)

In this problem A. is a m.xn. matrix, ~. is a m.xn. • I I I 1'1"1 I

rna t r i x 1\ b. i s a i

m. vector, and c. and x. are n. vectors~ This problem is also cal Jed I I I I

a mul tistage problem since it is solved by reducing it to a nested

sequence of problems of the type represented by Eq. (5.8).

72

As it was already pointed out, the decomposition principle may

in general be appl ied to any 1 inear programming problem. In particular,

if a portion of a given 1 inear programming problem presents special

characteristics such as those represen~ed by Eq" (5.8) or (5 .. 9), then

the problem may be decomposed and the subproblem presenting special

characteristics may also be soived by means of decomposition~ Hence

several levels of decomposition may be uti) ized for solving a given

problem. However, as it is usually the case for special ized procedures,

the decomposition principle will produce longer calcuLations when

app1 ied to problems not presenting the special characteristics described

above.

5.3.1 Analysis of the General ized Formulations from the Point of View

of the Decomposition PrincIple

In this section the special characteristics of the con-

stralnts corresponding to the formulations given by Eq" (5 .. J) and (5 .. 2)

will be anaJyzed and the convenience of applying the decomposition

principle to the solution of the rigid-piastic analysis problem will

be discussede

(11) Mesh formulation as given by Eq. (5~n ..

In Section 5.2 this formulation was discussed from the point

of view of a general 1 inear programming routine. As it will be seen

below the chara"cteristics of the constraints corresponding to this

73

formulation depend upon the manner of introduction of releases as well

as on the manner in which the joints and members of the structure are

numbered. As an illustration, consider the structure shown in Fig.

5.~'01i)~' For this structure, t\e'lO different systems of releases are

described in Fig. 5.J(b) and 5.1(d). In Figs. 5.1(b) and 5.1(c) are

shown two sequences of numbering the joints and members. It should be

noted that for the present discussion the location of the externa1 loads

is of no special significance. The expansion of the formulation of

Eq. (5.1) for the structure of Fig .. 5.1 (b) is given in Table 1. As can

be seen from the table, the equations do not fit the pattern of either

Eq. (5 .. 8) or (5.9). However the decomposition principle could be

appl ied to the problem as follows:

(i) obtain the dual of the problem.

(Ii) Decompose the problem into two subproblems: the first

subproblem corresponds to the row associated with the reference load F

and the second corresponds to the matrix associated with the redundants.

(iii) The second problem is similar to the problem represented

t t by Eq .. (5 .. 9), wi th the matrices A42 , A73

, etc. taking the place of the

A. matrices, and hence it may be solved by means of the decomposition I

principle ..

A different view of the same problem is obtained if the con-

straints corr'~sponding to the redundants are written last, as . shown in

Table 2. It could be said that this formulation.corresponds to the

formulation given by Eq. (5.8) by associating the matrices above the

partition shown in the table with the A matrices in Eq. (5.8) and the

matrices below the partition with the B matrices. However, since the

size of the upper part of the constraint matrix is much larger than the

74

size of the lower part, not much is gained by applying the decomposition

principle.

As a third illustration, if the members are renumbered in the

manner indicated by Fig. 5.1(c), then the formulation of Eq., (5.1)

becomes as shown in Table.3. It is obvious that this formulation does

not follow the pattern of either Eqa (5.8) or (5 .. 9). However, by a

simple rearrangement of the constra ints, it can be reduced to the

formulation given in Table 1.

Finally, consider the same structure as before but with the

system of releases represented in Fig. 5. J (d). The corresponding

formulation is shown in Tab1e 4. In this case, as in Table 2, no

advantage is obtained by applying the decomposition principle, since

the upper part of the system is much larger than the lower part, as

indicated by the partition. Furthermore, in contrast to the system of

releases given in Fig. 5.1 (b), the formulation corresponding to the

present system of releases cannot be rearranged to yield a useful

staircase pattern.

It can thus be concluded that the characteristics of the con­

straints of the general formulation given by Eq. (5.1) do not follow a

definite pattern, and hence the decomposition principle cannot be

conveniently applied to a structure having an arbitrary system of

releases. Therefore, the automation of a method using the decomposition

principle and based in the formulation of Eq. (5",1) does not seem

prom is i ng ..

b) Node formulation as given by Eq. (5.2).

For the s,tructure represented in Figo 501 (a) the formulation

of Eq .. (5 .. 2) is shown in Table 5 .. In this formulation, the B matrices

75

contain, in general, more rows than the A matrices. Also, in general,

the number of members is much larger than the number of free joints.

Thus, the node formulation fits the pattern of Eq. (5.8) and the

decomposition principle can be appl ied. Furthermore, for any structure

the constraints follow the pattern shown in the table and do not

present the variations caused by member numbering and redundant

selection encountered in the mesh formulation of Eq. (5.1). Hence, if

a program for the solution of very large structures were to be developed,

the decomposition principle could be conveniently app1 ied to the

implementation of the node formulation.

In conclusion, it is to be noted that, as in the case of the

elastic analysis proble~, the node formulation results in a regular and

uniform pattern well suited for efficient computer implementation.

5.4 Implementation of Computer Program

In this section the implementation of the computer programs

corresponding to the mesh and node formulations are described.

a) Mesh Formulation.

The computer program based on the mesh formulation can be

described as follows: An index indicating whether plastic analysis or

minimum weight design is desired is first read. Next, the type of the

structure (plane frame, plane grid or space frame), the topological and

geometric chcfracteristics of the structure and the external loading

(defined by a reference load for the case of analysis) are read. A

tree is automatically selected as the primary structure and the

topological node-to-datum path matrix BT is calculated. From this

information the vector Bpi (BP'F for design) and the mesh matrix Care

76

obtainedo Then purely by static cornsiderations the vector nsP' (nSP'F

for design) and the matrix nC are generated. If analysis is desired

the values of the fully plastic capacities of the members are read and

the vector NnSps and the matrix Nne are obtainedc Now the coefficients

of the components of the stress vector defining the equations of the

yield surface are read and the vector 6 1NnsP' and the matrix ~lNnC

(~BPIF and ~C for design) are calculated. Since for the case of

design the manner in which plastic cap~cities are to be assigned to the

members has to be known, i.e., which members are to have equal plastic

capacities, a vector providing this information is read. At this stage

all the information required for the solution of the optimization

problem is available and the next stage consists of the appl ication of

a J inear programming routine to the solution of the problem.

The computer program was implemented to be run as a single

job on the IBM/360 system and was written in two steps:

{nTh is step cons i s fs of a p rag ram w r it ten- i "-POST and a

1 inking subroutine written in FORTRAN IV. POST, (26) is a FORTRAN-l ike

language with impl ied matrix operations, dynamic storage allocation and

dynamic array dimensioning~ It is very convenient for the solution of

problems in which, I ike in the present one, a Jarge number of matrices

of varying size is to be manipulated. The POST program covers the

process from the beginning until the generation of the matrices ~lNnBPI

and ~lNne (~BPIF and ~C for design)s Then the ,FORTRAN subroutine

takes these matrices, generates the remaining information required (see

Eq. (5 .. 1) for ana Jys is or Eq. (4 .. 27) for des ign) and sends all the

information to a disk in a manner suitable for handl ing by the

optimization routines.

77

(i 1) Th iss tep cons is ts 0 f a MPS/360 prog ram wh i ch takes its

input data from the disk as it was left in the. previous step. The MPS

language(23,24) is appropriate for solving 1 inear and separable pro-

gramming problems. Since the number of constraints of the problems to

be solved is generally much larger than the number of variables, a

dual procedure is util ized to obtain feasibil ity and then a primal

procedure is appl ied until optimality is reached.

The solution to the MPS program yields:

(i) For analysis: Load factor corresponding to the load-

carryin·g capacity of the structure .. For design: value of the minimum

weight of the structure and corresponding design values for the fully

plastic capacities of the members.

(ii) The values of the redundants at the releases at

collapse.

(iii) location of yield hinges, Le .. , yield restrictions which _. - - - -

are fulfilled as strict equal ities in the opt-imaf- so-lution.

(iv) Magnitude of the plastic strain increment vector, i.e.,

the dual variables.

b) Node Formulation.

The computer program for the node formulation can be described

as follows: the required data are initially read, as in the mesh

formulation. From the topological and geometric description of the

structure, the matrix At is obtained. The loading condition defines

the vector pi (PIF for design). This portion of the process defines

the matrices corresponding to constraints (5.2 (a» for analysis or

(4.28(a» for design. From the geometry of the structure it is possible.

to obtain the matrix n. Finally the matrix 6,Nn (~ for design) is

78

obtained as in the previous case. This matrix defines constraints

(5.2(b) for the case of analysis. In the case of design, the manner

in which plastic capacities are to be assigned is read and constraints

(4.28(b») are generated. The final stage consists of the solution of

the optimization problem.

The computer program was also implemented to be run as a

single job on the IBM/360 system and was also written in two steps,

namely, first a POST program for the g~neration of the required matrices

and a FORTRAN IV subroutine to send this information to the disk in a

manner suitable for the optimization routine, and, second, an MPS

program which also uses a dual-primal technique to find the optimal

solution to the problem.

The solution to the MPS program yields:

(i) For analysis: Load factor corresponding to the Joad­

carrying capacity of the structure. For design: value of the minimum

weight of the structure and corresponding design values for the fully

plastic capacities of the members.

(ii) Member forces at collapse.

(iii) Location of yield hingese

(iv) Joint displacements and plastic strain increment vectors,

Le., the dual variables. These values are given up to a mUltiplicative

cons tant.

As can be seen the program for generat~ng the constraints for

the node formulation is much simpler and shorter than the corresponding

program for the mesh formulat ion. Furthe-rmore, the resul ts of the node

method give somewhat more complete information than those of the mesh

method. When stress interaction is considered it is not possible to draw

79

the exact collapse mode from the results given by the mesh method, since

the dual variables define the strain increment vectors which include

all relevant distortion components.

5.5 Nonl inear Programming in Plastic Analysis

When the equations defining the yield surface are expl icitly

included in the formulations given by Eq. (4.4) or (4.6) they are

usually nonl inear and the resulting problem is of the nonl inear pro-

gramming type.

As it was mentioned in Section 2.2, several methods have been

developed for the solution of certain type of nonl inear programming

problems. For the present study a non1 inear programming routine

(Constrained Optimum Steepest Descent) deveJoped by wright(35) was

uti1 ized. The routine is based on a gradient method of optimization

which uses an iterative procedure. The main features of the process

can be described as follows:

(i) From a feasible point move in the "admissible" direction

of steepest descent until a constraint becomes active.

(if) The lIadmissibJe 'l direction of steepest descent is

defined as either the direction opposite to that of the gradient of ~

the objective function at a point where no constraints are active (or

do not restrict movement), or the vector obtained by sweeping out of the

negative grad'ient of the objective function those components in the

direction of the gradient of the active constraints which constrain

movement.

(i ii) The process continues until a minimum is found, i.e.,

there does not exist an admissible direction of steepest descent.

80

The following observations should be made with regard to the

application of the previous method to the plastic analysis problem:

(i) If the starting point is very far from the optimal

solution then the global optimum may not be reached or convergence may

be very slow. This is especially true for the present problem in which

the number of constraints is much larger than the number of variables.

(ii) It is a difficult task to implement the generation of

the constraints when nonl inear terms are involved.

(I i i) In the node formulation, the equal ity constraints, Le.,

the equil ibrium conditions, must be replaced by inequaJ ity constraints

and therefore the size of the problem increases considerably.

The above drawbacks for the use of nonl inear programming in the

present study, as well as the accurate results obtained through 1 inear

programming, have led to the decision of basing the largest portion of

th~s study on 1 inear programming procedures.

CHAPTER 6

ILLUSTRATIVE EXAMPLES

In this chapter, several illustrative examp1es, obtained by

the use of the computer programs, are described. The results reported

represent the different points developed in this study, including:

plastic analysis and minimum weight design, mesh and node formulations,

effect of stress interaction, and various structural types (pJane

frame, plane grid and space frame).

In all examples, loads and forces are expressed in kips,

moments in inch-kips, 1 inear dimensions in inches, displacements in

inches and rotations in radians. The yield stress of the material is

assumed to be 36 k.so i.

6.1 Comparison of Mesh and Node Solutions

The structure considered in this section is the planar frame

shown in Fig. 6.1, which has the same geometry as the one in Fig. 5.1.

Comparisons are made between the results obtained by the mesh and node

formulationso For the mesh solution, the effect of different release

-systems was also investigated. Two systems of releases were chosen so

as to represent the frbest" and "worst,r primary structures, with

releases restricted to be complete cuts.

In every case, results were obtained without stress inter-

action, i.e., only bending moment governing, and,considering interaction

between bending moment and axial force. When stress interaction was

included, the yield restrictions of the AISC specifications (1) were

util ized as follows:

81

82

(6. 1 )

In summary, the structure was anaJyzed in the fol1owing

manners

Mesh Method

No stress interaction (

System of 5. 1 (b)

Sys tem of . \.. 5.1 (d)

releases of Fig.

releases of Fig.

Sys tern of releases of Fig. S. J (b)

Stress interact ion Sys tern of releases of Fig. 5.1 (d)

No s t res s j n t era c t ion

Node Method

Stress interact inn

The main results are summarized as follows:

(I) When no stress interaction was included, the load factor

F obtalned was of 5.08. For the case of stress interaction the load

factor was 4.88. Both of these values are for the nodal so1ution.

(Ii) The value of the load factor given by the two systems

of releases in the mesh solution was practically the same, i.e., less

than 0.01 percent of difference. This vaiue was also equaJ to that

given by the nodal solution. Certain minor differences in the stress

distribution were observed for the two systems of releases with the

"better" primary structure yielding values closer to those given by the

nodal solution.

83

(ii i) The collapse configurations for the cases of no stress

interaction and stress interaction are shown in Fig. 6.2 and 6.4,

respectively. The points where the stress vector reaches the yield

surface are also indicated. It can be seen that for the case of no

stress interaction a combined mechanism of the upper story occurs while

for the case of stress interaction collapse takes place as a paneJ

mechanism of the three lower stories as well as some beam mechanisms.

(iv) The normal ized distribution of moments for the cases of

no stress interaction and stress interaction are presented in Fig. 6.3

and 6.5 respectively.

The following conclusions can be drawn from the results

obtained:

(i) Since in the present problem two completely different

systems of releases were chosen, it can be concluded that the load

factor is independent of the release system selected. This is in sharp

contrast to the mesh (flexibil ity) method for elastic analysis, where

a "bad' ! selection of a primary structure may lead to inaccurate results.

The differences in stress distribution obtained by using the two systems

of releases indicate alternate admissible states of stress at co1Japse.

(ii) The effect of stress interaction on the load factor was

not very high, i.e., only 4 percento This is due to the fact that the

majority of the members do not carry heavy axial loads. However, the

right-hand columns are subjected to axial 10ads ~1 ightly larger than 15

percent of the fu11y plastic values, and this accounts for the differences

in the load factors, as well as the mode of collapse.

84

6.2 Effect of Stress Interaction

In this section three structures are considered in order to

investigate the effect of stress interaction on the calculated 10ad-

carrying capacity.

a) Planar Frameo The structure of Fig. 6~6 was· studied under

a loading condition which causes large axial forces~ so that the effect

of stress interaction could be more clearly observed. This structure

was first studied by Morris and Fenves(27) by means of an eJastic-

plastic analysis procedure which considers the exact nool inear yield

surfaces.

In the present study the analysis was first performed by using

approximate 1 inearizations for the yield surface and then by using a

nonlinear yield surfaceo

In the 1 inear analysis, four different yie1d surfaces were

considered:

Case 1. Linear yield restrictions specified by AISC (Eq.

(6.1».

Case 2. Linear approximation to the second degree yield

surface for rectangular sections developed by Hodge. (19) The equations

for this 1 inearized surface are:

(6.2)

Case 3. Lower bound surface corresponding to a 45 degree line

joining the normal ized values of the fully plastic capacities for axial

85

load and bending moment, as given by the equations:

(6.3)

Case 40 Simple plastic theory (no interaction).

The main results obtained from the analysis may be summarized

as fo flows:

(i) The values obtained for the vertica1 load F appl ied at

the center of the beam at collapse are· as follows:

Case 1.

Case 2.

Case 3.

18300 K

18.27 K

16.82 K

19098 K

(it) In all cases the same collapse mode is obtained$ ioe.,

the yield hinges are located at the points indicated in Fig. 6.6.

(iii) The values of the normal ized member forces at collapse

are represented in Fig. 6.7, together with the i ioes defining the

normal ized yield surfaces. In the figure all stress components have

been transferred to the first quadrant.

The following conclusions can be drawn from the previous

resuits:

(i) Cases 1 and 2 gave very close resu]ts$ It should be

noted from Fi.g. 6.7 that the yield surfaces for these two cases are

very close in the control I Ing region.

(ii) As could be expected, cases 3 and 4 gave extreme results

forthe load factor, but they provide a good indication of the 1 imits

between which the load factor may vary.

86

(i ii) In their non1 inear analysis Morris and Fenves obtained

values for the vertical load at the center of the beam of 19.03 K for a

spherical yield surface and of 17.60 K for the exact yield surface for

an I"'shap~d section. These two values bracket the values obtained in

cases 1 and 2 of the present study.

(iv) It is worthwhile to note in Fig. 6.7 that while the

bending moments at coJlapse change according to the yield surface

util ized, as does the load-carrying capacity, the values of the axial

loads at collapse are practica11y the same for all cases.

For the non1 inear case the structure was analyzed by using

the nonl inear programming routine described in Section 5.5. A second

degree YIeld surface given by Hodge was util ized. The equations of the

surface are as follows:

(6.4)

This yield surface is plotted in Fig. 6.8, together with the

spherical surface and the exact yield surface for an I-shaped section

presented in the study mentioned. The values of the calculated

normal ized stress components at collapse are plotted for that study and

for the present one.

In the present case a load of 18.31 K at the center of the

beam was obtained at collapse. This value is practically equal to the

average of the values obtained in the study mentioned,. as the present

yieid surface i iss in between the two surfaces considered in that study.

The results reported were obtained after 90 iterations of the

nonl inear routine without reaching the optimum value and beginning with

87

an initial value of 15K for the load and proportional va'lues:for the

remaining unknowns.

In summary, the value of the reference load F obtained by using

by 1 ineariz ing the yield sur'face and applying I inear programming.

Accordingly, if such close results are obtained even when the axial

loads are quite large, it can be concluded that the 1 inear programming

approximation is sufficiently accurate for most practical problems.

b) Gable Frame. In order to re1ate the theory developed in

this study to practical results a gable frame (Fig. 6.9(a)) presenting

, (29) similar characteristics to the one tested by Nelson et alo was also

considered. The study was carried out with the same four yield surfaces

as used in the previous example.

The results obtained are as follows:

(i) The value obtained for the reference load Fare:

Case 1.

Case 2.

Case 30

Case 4.

16 .. 33K (AISC)

15.44K (Hodge)

14,64K (Lower Bound line)

1 6 • 33 K (No i n t era c t ion)

(ii) The mode of collapse, as given by the dual variables in

the nodal solution, was the same for all cases and is represented in

Fig. 6.9(b)o However, the stress vector also reaches the yield surface

at the negative end of member 5.

The following conclusions can be drawn from the previous

results and from those obtained in the experiment:

(i) The value of F reported in the paper mentioned was 16.25K,

which is in close agreement with the results obtained for cases J and 4.

88

The value of F is the same in cases 1 and 4 since for all members the

axial load is less than 15 percent of the fully plastic value.

(ii) The mode of collapse obtained by the analysis, shown in

Fig. 6.9(b), is somewhat different from the one obtained in the test

mentioned, ioe., in the present case rotation occurs at the negative end

of member 7 while in the test mentioned rotation takes place at the

negative end of member 5. All other hinges are as indicated in Fig.

6.9(b)0 However, the fact that in the present study the stress vector

reaches the yield surface at the negative end of member 5 indicates that

that mode of collapse is also in agreement with the results obtained in

this studyo

c) Plane Grid. In the previous examples only planar frames

were considered, where the components of the stress vector defining

yielding are the axial force and the bending momentG In the present

example a plane grid is considered and the stress components defiriing .................... _ ........................... _ ..................... _ ...................... -.. __ ................. -.. - .............................. _ ....... _ .............. _ ................. -.- ............. _._ ...... _ .•........ _ ..... _..... ..................... _ ............. _ .. -_ .... _._ .............................. _ .. __ ..... _- . __ ........... . __ ._ ....... __ ....... _ ...... .

yielding are the bending moment with respect to the principal axis of

the member and the torsional momento

The planar grid of Figs 6010 is similar to that presented by

Hodge(19) and the yield surface utilized was the 1 inearized lower bound

surface for this type of interaction which is also given by Hodge. The

equations of the 1 inearized surface are:

m m x z + -- + 0.414 -- = - m - m 1 3

(6.5)

Hodge makes a study of this structure for the case in which the external

load is located at any point along members 1 or 2 and for a wide range

89

of ratios between the fu11y plastic values for the torsion moment and

the bending moment. For the present study the fully plastic values

were: 60 inch-kips for the torsion moment and 100 inch~kip5 for the

bending moment.

The results obtained were as follows:

(i) the value of F at collapse was of 21.89K.

(ii) the location of the yield hinges at collapse are given

in Fig. 6.10 and the norma) ized stress vectors, as transferred to the

first quadrant, are given in Fig. 6~11o

The value of F at collapse agrees with that value calculated

by the formulas given by Hodge and the mode of collapse is a1so the same

as the one presented in that book.

6.3 General Case of Interaction: A Space Frame

In order to study the interaction of forces for a space frame,

segments, was analyzed. The structure is made up of a square

structural tube with a nominal size of 6x6 in. and a wall thickness

of OQ375 ino The yield stress was taken as 36 Klin2•

For this type of structure the predominant stress components

defining yielding are the torsion moment and the two bending moments.

The fully plastic value of the torsion moment is 561 inch-kips and

those of the 'bending moments are equal to 729 inch-kips. These values

were computed from the formulas given in Table 1 'of the study of Morris

and Fenves. (27)

A yield po1yhedron for combined bending and torsion for a

square sec t ion is deve loped in the book of Hodge (J 9) and was uti 1 j zed

90

here. The equations for the yield surface are as follows:

(6 .. 6)

m m m + ~ + 0.207 -l + 0.414 ~ = - ~1 m2 - m3

The value obtained for the load F at collapse was 3.72K. The

distribution of the normal ized value~ for the stress components and

stress vector for the unfolded helix are represented in Fig. 6.13. It

can be seen in the figure that:

(i) The distribution of moments and the value of the stress

vector present symmetric characteristics with respect to the center of

the unfolded helix.

(ii) Collapse is obtained by yield hinges of the members near

the supports, with the torsion moment being the predominant stress com-

ponent.

(iii) A reasonable efficient utilization of the material is

achieved as the smallest value of the normal ized stress vector is 0.77.

6.4 Minimum Weight Design

As an example of minimum weight design, the structure with the

geometry described in Section 6.1 was designed by using the formulation

presented ih Section 4.4. The design was carried out for the loading

specified in Fig. 6.1 with F = lK. The equations for the yield surface

91

were those given by the AISC Specifications.

In order to relate the fully plastic values of the axial load

and bending moment, as proposed in Section 4.4, the following simp) ifi-

cation was madeo For the purpose of calculating the plastic moment

capacity, the area is assumed to be concentrated in the top and bottom

of the section, i.e., in the flanges for a wide-flange section. Hence

the fully plastic values are given by

m ;::: 3

F A Y x

d F A 2 Y x

(6. 7)

in which F is the yield stress of the material, A the area of the y x

section and d the depth of the section. For the present example, a

trial depth of 8 inches was selected for all members and therefore

Three design cases or strategies were considered. These

three cases differ in the manner in which plastic capacities were

assigned to the members, or, more specifically, the manner in which

members were grouped into design variables, where each design variable

represents all members having equal plastic capacities. The three

cases are:

Case 1. Six design variables: one for aii beams, and one

for each pair of columns in a story.

Case 2. Four design variables: one for all beams, one for

the columns of the first and second stories, one for the columns of the

third and fourth stories, and one for the columns of the fifth story.

Case 3. Three design variables: one for aJJ beams, one for

the columns of the first, second and third stories, and one for the

92

columns of the fourth and fifth stories.

The weight of the members was assumed to be proportional to

the product of the length times the plastic moment capacity.

Based upon the former assumptions the minimum weight design

was carried out using the computer program through the nodal solution.

The results are as follows;

Plastic Moment Ca~acities! inch .... ki 2s

Weight Columns Beams (G Iven up to a cons tant) S to r):: 1 S to r~ 2 Storl: ~ Storl 4 Stor~ 2

386788 431072 231 .. 20 230.64 163.56 107. 16 304.36

2 405424 293.76 293.76 208.36 208.36 63 .. 24 349.60

3 406772 305.84 305.84 305.84 1 J 5.60 115.60 335.20

The distribution of moments and the modes of collapse for the

three design cases appear in Figures 6.14 to 6.19.

The design corresponding to Case 2 was carried to completion.

The sections chosen and the corresponding fully plastic values and

depths were as follows:

Columns

Stories 1 & 2 S to r i es ~ & 4

Section 8 WF 40 8 WF 28

m3

, inch-kips 1436.4 975.6

p 1 ' kips 423.36 296.28

d, inches 8.25 8.0

The following comments are in order:

Stor~ ~

6 B 12

298.8

127.08

6.0

Beams

10 WF 39

1692 .. 0

413.28

10.0

(i) The members were selected such that the actual values of

the fully plastic moments would keep the ratios of the computed values

given above, but would supply a larger load factor, Joe., sections

having larger plastic capacities than those required for minimum weight.

93

(ii) The depths of the members were chosen as near to 8 inches

as practical, so as to fulfill the initial assumption given by Eq.

(6.B). The resulting fully plastic values for axial loads are larger

than one fourth of the fully plastic moments as required by Eq. (6.8).

This result is on the conservative side.

The resulting structure was the one analyzed in Section 6.1.

From the results of the analysis it can be said that:

(i) A well balanced design was achieved, as can be seen from

the comparison of the modes of collapse represented in Figs. 6.4 and

6.17. The two cases correspond to the yield surface specified by the

AISC Specifications. The same mode of collapse was attained but the

design case presents more yield hinges since the analysis was made on a

structure made up of available sections.

(ii) An additional indication of how closely the structure was

designed can be seen from Fig. 6.2, i.e., using no interaction~ an

entirely different mode of collapse was obtained, but only 4 percent

of difference in the load factor was observed.

CHAPTER 7

CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY

7 ,,1 Cone 1 us ions

It has been shown in this study that the analysis and design

of rigid-plastic structures subject to proportional loading presents

al) the characteristics of a network problem and as such can be

formulated and solved by using network-topological concepts. However,

while in the ~]astic or elastic-plastic case the problem is estab1 ished

as a set of simultaneous equations, in the present case the problem is

established as a mathematical programming problem in accordance with

the concepts of the plastic analysis theorems. This difference is due

to the fact that in the present case the load-deformation characteristics

of the material are not expressible by mathematical functionso The

problem can be solved by using either an action approach or a distortion

approach and dual ity guarantees that both methods yield the same

solutiona Thus, it can be seen that four methods or formulations have

been presented rather than the two existing for the elastic case.

There are several ana10gies between the formulations corres­

ponding to the elastic and to the plastic analysis of structures which

should be pointed out:

0) The mesh and node formulations of plastic analysis,

the counterpa·rt of which are the flexibility and the stiffness methods

for elastic structure~ have been developed and it is shown that, as in

the elastic case, the node formulation presents greater assets for

computer implementation in spite of the fact that the size of the

problem obtained by using the mesh method is usual~y smaller.

94

(ii) In order to study the convenience of applying the

decomposition principle of 1 inear programming to the solution of the

problem a study of the arrangement of the elements of the constraints

was carried out and it was found that the node formulation suppJies a

problem having appropriate characteristics, i.e., an angular system,

for such application. On the other hand, the mesh method yields a

prob1em with constraints of unpredictable arrangement, depending upon

the manner of introduction of releases and upon the numbering scheme

for the structural members. Thus, as in the elastic case, the node

formulation of plastic analysis is more adaptable to special ized

algorithms.

95

(iii) A general formulation of the purely kinematical approach,

i.e., the use of collapse mechanisms, presents the same complexities

as the elastic analysis with constrained distortionse

The formulations developed in this study present the fo11owing

advantages:

(i) They are completely general and can be appl ied to

framed structures regardless of the particular type of structure being

considered.

(ii) The formulations are initially developed for the simple

plastic case and the net~rork-topo1ogical concepts provide an excellent

tool for their immediate general ization to include the effect of stress

interaction.

(iii) The formulations developed for plastic analysis are

easily converted to cover the minimum weight design case. The computer

programs first developed for analysis were extended for design with

very i ittie additional programming.

Finally, it was also found that the results obtained by

1 inearizing the yieJd surface are sufficiently accurate for practical

purposes. Furthermore, a computer program using 1 inear programming

methods can be implemented much more expediently than one using nonl inear

programming procedures and thus much emphasis is placed on the use of

1 j n ear p ro g r amm i n 9 •

7.2 Suggestions for Further Study

In the following several extensions to the formulations

developed above are suggested which will enhance the capabil ities of the

fo rmu 1 at ions.

1. The decomposition principle as appl ied to the node formula-

tion should be j1npJemented and the efficiency of the resulting program

should be compared to that obtained with the program using a general

1 inear programming routine. Further efficiency can be gained by

separating the member force vector R into RJ

and R2 in which Rl contains

all the action components entering into the definition of yielding and

R2 those components not entering into that definition. The corresponding

expansion of Eq. (5.2) for a given structure would supply a linear

programming problem of the form:

maximize F

subj ect to

rAIl • I I

1

• . Alb' I. A? 1 • I ..! ...... .:" _ ...

== [oJ o

o r ~ (7. I)

~U

o

97

This formulation presents definite advantages for the imple-

mentation of the decomposition principle.

2. Because the formulations presented c]arify the relations

between the primal and dual variables of the plastic analysis and

design problems and relate the same to the corresponding elastic

variables, it appears feasible to formulate a large class of analysis

and design problems involving both plastic and elastic relations and

constraints. An approach to such problem c~uld be as follows:

Consider the nodal formulation of the minimum weight design problem

given by Eq. (4.28). Next, define the following quantities:

(i) Let the vec to r Ry be the member fa rces at col lapse.

(i i) Let the vec to r Rw be the member fa rces at working loads.

(i i i) Let f be the flex i b i J i t y matr ix of the structure.

(i v) Let t denote the factor of safety.

(v) Let u' be a specified vector of maximum joint displace .... s

ments at work i ng loads.

The formulation of the minimum weight design problem can be

written as follows:

minimize Ek.p. I I

subject to AtRy :I P'F (a)

AtRw = l/tP'F (b)

Qa .6nRY > 0 (c) (7.2)

Q -dlRw > 0 , (d)

StfRw < u· - s (e)

P! ~O - I

Ry, Rw unrestricted in sign

In the above formulation:

(i) Constraints (a) a'nd (b) specify joint equilibrium at

collapse and working loads, respectively.

98

(ii) Constraints (c) and (d) specify the yieJd restrictions

which must be fulfil led at collapse and at working loads, respectively.

(i ii) Constraint (e) specifies that the joint displacements

at working loads must not exceed the maximum specified values.

The problem given by Eq. (70,2) is nonlinear due to

constraint (e). However, following an argument similar to the one used

in Section 4.4, it is possible to 1 inearize the constraints by assuming

the fJexibil ity of a member to be a 1 inear function of its reference

plastic capacity.

3. It seems feas ible to include within the formulation

buckl ing constraints for the individual structural members either by

redefinition of some of the matrices or by addition of constraints.

4. If the reJations~ip between the weight of a member and its

reference plastic capacity is not simply I inearized but piecewise

1 inearized then the minimum weight design problem can be treated as a

parametric programming probiem.

5. The formulations presented appear to Jend themselves to

the study of the case "in which the loads appl ied on the structure

present probabiJ istic characteristics. The probJem becomes a stochastic

programming problem.

6. The network formulation may prove useful for a more clear

and unified treatment of the problem of alternative loads.

99

2 3

FIG. 2.1 SYSTEM OF COLLINEAR SPRINGS

2 3V ·4 ClIF r aF Im~ m.., m ... l~ I ,~

I ~ .I.. .I.. I I I

I I h

1 1 m

1 m1 m' I z Px

I- ~ (a) (b)

FIG. 2.2 PLANAR FRAME

(a) Meehan ism 1

(b) Meehan ism 2

n Fl -II I I II-+-F I , I I LJ

FIG .. 2.3 COLLAPSE MECHANISMS FOR CO lLINEAR SPRINGS

100

101

l (a) Basic Mechanism (b) Basic Mechanism 2

(e) Comb i ned Meehan ism

FIG. 2.4 COLLAPSE MECHANISMS FOR PLANAR FRAME

102

y

x

y Coordinate

A

x

Global Coordinate System

z

(a)

y

\ x

y

t _--B­~--

A Z

..1----............ X

z

(b)

FIG. 3.1 TYPICAL STRUCTURAL MEMBER

(a)

2 3

Redundant

(b)

FIG. 3.2 LINEAR GRAPH AND PRIMARY STRUCTURE FOR SYSTEM OF COLLINEAR SPRINGS

103

2

2

I

3

(a)

3 •

(b)

4

4

I. Redundant Member

FIG. 3.3 LINEAR GRAPH AND PRIMARY STRUCTURE FOR PLANAR FRAME

104

pi

0 ==

C B R pi

dl

R' <M - P

a) Static (primal) Formulation

Min M t • P

nt6

t

I Ct

iIfI

w

" I

. I I 8

t u' U

b) Kinematic (dual) Formulation

pi ... ....

pit

FIG. 3.4 ROTH'S DIAGRAM FOR MESH FORMULATION

..

105

Max F

>

t Min M · p

106

At pi R pi

,It

I I

a) S ta tic (p rima 1) Fo rmu 1 at ion

y ... -.. ---........ ---- .. -------.-----------.------------ .--------.- .. ---- .--... --.--------- ~~t,~J -_ .. -._-._---_._-- --_ .. ----.-- .. __ ._------ --._._-----_ .. _--_ .. _-_._---_ ... -.-

t I u .·It-""'1!'C!--_A;...;....", ......................... u .... ' ; __ p_l

_

t_. -111,,- >

t t (n t:, w-Au 1=0)

b) Kinematic (dual) Formulation

FIG. 3. 5 RO T HIS D I A G RAM FO R NO DE FO RM U LA i ION

C B p' R pI ..

-'" At

l{)

R' < M - p

a) .Static (primal) Fonnulation

.. o = ..

u' A

b) Kinematic (dual) Formulation

FIG. 3.6 COMBINED ROTH'S DIAGRAM

pi Max F

107

... _........_.- .... _--_ .. __ .... __ .... _-_ ....... __ ._ ...... _._---_.... . .. __ .. _-_ ......... _._ ..... _ ..• _ .. .

108

R"

f

FIG. 4.1 YIELD SURFACE

'OJ

(a) Exact Curve

(p 12p,)+(m 1m3) = , x I Z

(b) Lower Bound Surface (Linear Approximation)

FIG. 4.2 YIELD SURFACES FOR RECTANGULAR SECTION

1

3

5

7

9

11 7r.","

n l ¥ '} .

.J 5

12

:..1 J

19

~

P'~'" 5

(a)

14

10

6

3

(c)

2

3

4

6

6

9'

8

l~

1 o pI? 4

,.., ~

B

5

7

2

1 3

~""

15

PI~ 5

9

7":

5.

-3

1

4

5 - •.

'7

8

II)

1

13

1 4

"., '" (b)

pi ~l

f2 6

pi 3

l!t P4

(d)

FIG. 5.1 MULTISTORY FRAME AND SYSTEMS OF RELEASES

110

111

10F

2.5 F if "', - -- .. 1-" ._"

.. 10WF39

72" 6812 lOF 68)2

2.5F I~ do

721: 8WF2 8 8WF28 lOF

2.5F -It

dO

72 ' : 8WF2 8 8WF28 lOF

2.5F ,

do

72 II 8WF4 0 8WF40 lOF

II 2. 5F , .. ~ do

72" 8WF4 0 8WF40

~ 144'1 1

FIG. 6.1 MULTISTORY FRAME

1 1 FIGo 6.2 COLLAPSE MECHANISM FOR MULTISTORY FRAME

(NO INTERACTIO N)

112

FIG. 6.3 NORMALIZED MOMENT DIAGRAM AT COLLAPSE FOR MULTI­STORY FRAME (NO INTERACTION)

113

FIG .. 6.4 COLLAPSE MECHANISM FOR MULTISTORY FRAME (INTERACTION INCLUDED)

114

115

FIG. 6.5 NORMALIZED MOMENT DIAGRAM AT CO LLAPSE FO R HUL TI ... STORY FRAME (INTERACTION INCLUDED)

J 16

3.2F

0.24F ! 2

4 120'

I 240'1 I .... _ .................................. _ .... _ ........... _ ............. _............... . .... _ ................... _ ...... ~_~._. . .. _ ...... _ ... __ .......... __ ............ __ ..... _ ....... _ ...... _._ ... _ .. _ ..... _ . ...:. .. _._. __ ........ _ ..... _.:=)110 ............. _ .. _.................................. ..

PI = 212.4K.

m3 = 532.84 in.-K.

FIG. 6.6 PLANE FRAME UNDER HEAVY AXIAL LOADING

Member + end

Membe r ... end

Member +end

e .........

N E

2

3

, .. 0

0.6

+.I c: 0.4

Member 3 ... end

Member 2 -end

Member 3 + end

Memb r 1 Simple Both ends Pl as tic

Member + end

Bound

----- ------- .--.-----.----... -----.--.----------~--------.- .. ---------.-.------------------ 1------------_·_---------_· __ ·_····_·_----------+ --------------···---------------1

o :::c "'C ~ N

• 0.2 ~--------~--_u--~~~----~--~--------~~~~----~ e b. o

Z

o

• o 0 .. 2 0.4

Member 4 - end

0.6 0 .. 8 1.0

FIG. 6. 7 NORMALIZED FORCE COMPONENTS AT COLLAPSE FOR PLANAR F8AUE (I T~~AD rAS~\

I'ViIn e.. ..... e..n" V" ... J

117

(

'Member - end

Member + end

E "-N E ... c w E 0

::r: "'0 Q) N

~ s-o :z

2

0 .. 8 3

0.6

0.4

0.2

o

+ end

o 0.2

Member 4 ... end

0.4 0.6

Member ... end

Normal ized Axial Force px/p ]

I-shaped

0.8 1.0

FIG·" 6 .. 8 NORMALIZED FORCE COMPONENTS AT CO LLAPSE FOR PLANAR FRAME (NONLINEAR CASE)

118

40 '1

3211

32"

,~

F

240 11

All members 8WF17

PI= 239.6K m3 = 746.6 in.-K.

(a)

(b)

F

FIG. 6.9 GABLE FRAME AND MODE OF COLLAPSE

119

10

10"

F

m1

== 60 in .... K.

m3 == 100 ina-KG

FIG. 6.10 PLANAR GRID

120

/

1 .0

0 .. 8

f'I"'I E .......

N E 0.6

oI..J c I)

5 :c O'l c :c 0.4 c

CD co "'0 tU N

f 0 .. 2 ~ .

o ·0 0 .. 2 0.4

Meaber + end

Member - end

Member 2 + end

ember 2 - end

Member 3 . + end

0.6 0 .. 8

Member 3 - end

1.0

FIG. 6.' 1 NORMALIZED FORCE COMPONENTS AT COLLAPSE FOR PLANE GRID

J 21

40'

m = 561 in. -K. x

m = m = 729 in.-K. y z

All loads o. 1 F

y

z

13.2'

FIG. 6. 12 CIRCULAR HELIX

X

Global Coord i n"ate Sys tern

122

To

Bot tom

123

... 1 0 ... 1 0 -1 0 0 . 1

Tors ion Bending Bending Stress Moment Moment Moment Vector

mx/ml myim2 mzlm3

FIG. 6.13 NORMALIZED VALUES FOR THE STRESS COMPONENTS AND STRESS VECTOR FOR UNFO LDED HELIX

)24

----~~ --------- I

I ~~ ----- ~

1/ I

I [I .~~ "- / ;, I

[I ~ rr -.L ~ ~ I

v 1/ I

.,/ 1 .0 500,··in .... K • I

- ~ II

I I '/1 . , I

I

FIG. 6.14 MOMENT DIAGRAM FOR MINIMUM WEIGHT DESIGN (CASE 1)

125

FIG. 6.15 COLLAPSE MECHANISM FOR MINIMUM WEIGHT DESIGN (CASE 1)

126

r========:= ~

~

- ~ r /

~ r !-----~ 7

II I ~ 7

J" _. -_ .. _.-

/ _ ... _. __ ..... _ .. _ ...... _-------------/

V /

I / 1 o 500 in ..... K. ! I

I / 1/ v

V I FIG. 6.16 MOMENT DIAGRAM FOR MINIMUM WEIGHT DESIGN (CASE 2)

127

FIG. 6. 17 COLLAPSE MECHANISM FOR MINIMUM WEIGHT DESIGN (CASE '2)

128

--t-_

- ----------II

L

I

~ ~

H -------~ 1/ I

1/

~~ I ~

I '/ ~

/

j ~

17 I

j ~ tl I-( ~~

V / ~ / I

o 500 in. -K. I I

FIG. 6.18 MOMENT DIAGRAM FOR MINIMUM WEIGHT DESIGN (CASE 3)

129

,."-,,,.

FIG. 6. 19 COLLAPSE MECHANISM FOR MINIMUM WEIGHT DESIGN (CASE 3)

TABLE 1. EXPANDED LINEAR PROGRAMMING FORMULATION ... MESH METHOD

maximize F

subj ect to

Membe-i

a 1 All pI

1

2 012 A21

pI 2

4 a4 A41 A42 pi 3

S as A52 pI

4

7 a 7 A72 A73 pi 5

8 as A83

10 a lO A103 A104

11 all F+ A114

13 a 13 A, 34 AJ35

14 Cl 14 A14~ ... "" - ... IIIilII _ I11II\I .. .. .. .. .. ..

3 0 A31

6 0 A62

9 0 A93

12 0 A124

15 0 A155

F ~ 0

pi unrestricted in s)gn

TABLE 2. EXPANDED LINEAR PROGRAMMING FORMULATION -MESH METHOD (REORDERED CONSTRAINTS)

13 J

~

maximize F

subject to

Member

a 1 All pi

I

2 132 A24 p'

2

~ «33 A34 A35

pi .., '3

4 0 A45 P4 5 as AS2

pi 5

6 a6 A63 . A64

7 a7 A73

8 aa F+ A8)

9 0 A94

10 a lO A102 AI03

11 0 A1l3

12 0 A122

13 CD 13 A135

14 <3 14 A141 A142

15 l 0 A151 J F > 0

p' unrestricted in sign

TABLE 3. EXPANDED LINEAR PROGRAMMING FORMULATION -MESH MET~D (MEMBERS RENUMBERED)

132

~

Ll

max imize F

subject to

Meraber

a 1 AU A12 A13 AJ4 A,S pi 1

2 d 2 A21 A22 A23 A24 A2S pI 2

3 Ci 3 A31 A32 A33 A34 pi 3

4 d4 A41 A42 'A43 A44 Pit 5 as ASl A52 AS3

pi 5

6 a6 A61 A62 A63

7 d7 A71 A72

8 as F+ A8] A82 ~

9 a9 A91

10 dl O A10I .......... - - - - - - - ~ - - -11 0 Al11

12 0 A122

13 0 A133

14 0 A144

IS 0 A15S

F 2: 0

p' unrestricted in sign

TABLE 4. EXPANDED LINEAR PROGRAMMING FORMULATION ... MESH METrlJ D (ALTERNATE SYSTEM 0 F RELEASES)

133

134

maximize F

subject to

Node

AJ I A13 Rl pl 1

0

2 A21 A22 R2 pI

2

3 A338 R3 pI

3 :::::

8 pI 8

P.

9 A913 A915 p' 9

10 A1013 A)014 R13 Pia 0

Member - - - - ~ - - - - ~ - - - - - - - - R14

B) RJ5

2 82 0

3 83 oS

13 0

8 13

14 B14

15 D

°15 J. L'J

F ~ 0

R unrestricted in sign

TABLE 5. EXPANDED LINEAR PROGRAMMING FORMULATION - NODE METHOD

APPENDIX A

NETWORK FORMULATION OF PLASTIC ANALYSIS BY USING THE COLLAPSE MECHANISMS OF THE STRUCTURE

The first problem that arises when using the collapse

mechanisms for the calculations of the ioad-carrying capacity is the

generation of a set of basic mechanisms. After generating the basic

mechanisms it is necessary to find the combined mechanisms. Each one

of the basic and combined mechanisms defines a possible mode of

collapse of the structure and yields a value for the reference load F.

According to the kinematic theorem the load carrying capacity is the

smallest of the values so obtained.

A.l Method for Generating Basic Mechanisms

In this section the method for generating basic mechanisms

is developed for the case of a planar frame. For a plane grid or a

space frame the derivations follow similar patterns.

a) Let the vector of member distortions in member coordinates

be denoted by u. ~u is a vector containing b subvectors of order f.

b) let the vector of joint displacements in global coordinates

be denoted by u i •. u i Is a vector containing n l subvectors of order f.

c) The vectors u and u l are related by the equation.

U :: Au· (A. 1 )

This is a continuity equation relating Joint displacements and member

distortions. It can also be seen from the node formulation of the

kinematic theorem that the continuity requirements can be written in

terms of the hinge distortions and member displacements (Eq. (3.48(b»)

as follows:

135

From Eqe (A"l) and (A.2) u can be written as

U := ntb, tw

l36

(A.2)

(A. 3)

Let the elements of u be rearranged in such a way that the

axial distortions u of all members appear first, the shear distortions x

u" appear next and the angular rotations u7

appear last. Let this y -

rearrangement of the elements of u be denoted by Ue The vector u is

obtained from u by premultiplying u by a permutatIon matrix f1

, given

by:

r l1

r 1 == r 12 (A.4)

r 13

Then u is given by

r rill r::l u == rlu == r 12 u ==

Lr13J Luz J (A.5)

Let the elements of u' be rearranged in such a way that the horizontal

displacements u~ of the Joints appears first, the vertical displacements

u' appear next and the joint rotations u' appear last. let the vector y z

so obtained be denoted by Ul. The vectors u' and lj' are related through

(A.6)

such that

u' :::: r u' = 2

Equations (3.53) can now be rewritten as

U ::: r Ar lil 1 2

Let the matrix AI be defined as

Tnis matrix can be pa rt it ioned as fo J lows

I All A12

A' ::: A21 A22

L A31 A32

and Eq .. (A.8) rewr it ten as

Abl A23

AhJ

u A I i A12 Abl x

U ::: A21 A22 A23 Y

" il' A32 il' L-z J L"31 "33 J

u' y

u' z

rU1

x

u'

u~ L zJ

J 37

(A.7)

(A.8)

(A.9)

(A. 10)

(A. 11 )

From the nature of the rotation and translation matrices it can be shown

that

A'3 :::: 0

A31 :::: 0 (A. 12)

A32 ::: 0

The. method for generating basic mechanisms can be described as

follows. Since the structural members cannot u~dergo axial distortions

the subvector u is set equal to Zero. Equation (A. II) can be rewritten x

as

138

0 All A1:z 0 u· x

u = A21 A22 A23 u' (A. 13) Y Y

u 0 0 A33 u' z z

Let the co 1 umns of AI be rearranged into a matr ix All given by

r A" A" :~J 11 '12

All = All A" (A. 14) l /1 22 ~> J

0 A'l 33

In such a manner that the matr ix AI' 11 be nonsingular and

A'l 23 = A' 23

A'l = AI (A. 15) 33 33

This Is possible whenever the axial distortions of the member

can be constrained to be zero in an independent manner. For the most

general case a procedure slmi1ar to that described by Mauch:and ;Fenves(25)

can be used. It shoul d be n'o ted that:

(i) it is only necessary to rearrange the first 2n l columns

of AI;

(i i) wh i 1 e the • Al AI A' matrIces 11' 12' 21 and A22 are of order

b x n B , the matr ices A" 11 and All

21 are of order b x b and the matr ices

A'l 12

and All

22 are of order b x (2n ' -b).

To th is rearrangement of the co 1 umns of A I there co rresponds

a rearrangement 5· of the elements of u~ ~'can be written as

(A" J6)

139

in which u1 is a b-vector, u~ is a (2n'-b) vector and u~ is an nl-vector.

After these remarks Eq. (A.13) can be rewritten as

I:J I A" A"

:bl I:~ l 11 12

::: A" A" (A .. 17) 21 22

LUzJ 1° 0 AhJ Lu~J The upper part of Eq. (A. 17) yields

(A. J 8)

and

(A. 19)

Replacing the value of ui as given by Eq. (A.19) into the lower part of

Eq. (A.17) it is possible to write

in which

~2:t_

A33

A /I ::: - A II (A II ) ... 1 A f f + A If 22 21 J1 12 22

u' 2

u' z

From Eq. (A.3) and (A.5) resu1 ts the fo11owing express ion

and from Eqo (A.20) and (A.22)

r12l

r13 J let the matrix l be defined as

l :::

o

u y

u z

u' Z

(A. 2 1)

(A.22)

(As 23)

(A.24)

140

From Eq. (A. 23) and (A Q 24) resu 1 ts

A" 22 A23 u l

2 -1 w = L (A.25)

0 A33 u l

z

F i na 11 y if the matrix E is defined as

A" A23 -1 22

E = [E1 E2 ] = L

0 A33 (A. 26)

then Eq .. (A e 25) reduces to

u l

2 u I

2 w = E = [ £1 E2 ]

u B u' (A.27)

Z z

The matrix E defined by Eq. (A.26) thus provides a direct

relationship between the member h-inge rotations w and the independent

joint displacements. Each column of the matrix E} defines a basic

col1apse mechanism for the structure. The non-zero entries of each

column define the rotations at the member ends constituting a mechanism~

The columns of E2 represent patterns of member hinge rotations compatible

with a unit rotation of each joint of the structure. The joint

mechanisms are therefore defined by the non-zero entries at each co1umn

A.2 Generation of Combined Mechanisms.

Several methods have been suggested for the generation of com'"

bined mechanisms in rigid-plastic ana1ysis. In general, all of them

follow the original ideas presented by Neal and Symmonds. (28) It is

shown that the number of combined mechanisms increases astronomically as

the number of members in the structure do. Hence, in a method which is

141

based on]y on the use of collapse mechanisms there always exists the

possibil ity of overlooking one or more combined mechanisms.

Heyman and Prager(18) have presented a method (for minimum

weight design) which simultaneously uses the static and the kinematic

theorems and thus the problem of neglecting combined mechanisms is

avoided.

This section will be 1 imited to present a mathematical pro-

gramming formulation for the problem of. finding the critica1 collapse

mechanism. A combined mechanism represented by a vector w is given by

w = E u' + E u' J 2 2 z (A.28)

i"e., it is a linear combination of the basic and joint mechanismse

Let the columns of E, be denoted by Ell' .0"' Elk and the corresponding

elements ofu2 by u2J , ... J u2k"

Let the columns of E2 be denoted by E21 , ••• , E2nl and the

corresponding elements of u· by u'l' •• Oll U' I. Z Z zn

Then the expansion of Eqe (A.28) yields

(A.29)

For this combined mechanism the work of the internal forces can be

written as

M~H = M~ I Ell u21 + ••• + Elku2k + E21u~1 + .•• + E2n,u~nll

in which H is the vector of plastic capacities. p

(A. 30)

The joint displacements corresponding to ware given by

t t t -t u' = B n ~ w ~ B w

and the work of the external forces is given by

(A.31)

142

(A.32)

Now, using Eq. (A.29), the work of the external forces becomes

For each one of the col1apse mechanisms the equal ity of external and

internal power supplies a value ofF, i .. e.»

F = (A.34)

and the load carrying capacity is the smallest of all the F's obtained,

provided that the system of velocities be kinematically admissible,

i.e., the external power is positive.

Hence, the problem of finding the critical load or,

equivalently, the critical collapse mechanism can be formulated as an

optimization problem as follows

minimize F

subject to external power> 0 (A .. 35)

The last condition, i.e., external power greater than zero, is more con-

veniently written as in the formulation of the kinematic theorem,

namely, the external power is restricted to be greater than or equal to

one.

The expansion of Eq. (A.35)

H: IEIIU~l + .,. + Elku~k + E21u~1 + ..• + E2nlu~nll minimize -

'It~t (El1u~1 + ..• + Elku~k + E21u~1 + ••• + E2n,U~n')

(A. 36)

subject to

u' is unrestricted in sign

The solution to this problem yields the value of the load-carrying

capacity as well as the coefficients u21 , ... , u2k, u'z1' ••• J u~nl

which define critical collapse mechanism.

143

The following remarks can be made with respect to the previous

deve lopments:

(i) Let the ith column of E1, define a basic mechanism and

the jth columns of E2 define a joint mechanism. Then if E1 (k, i) =

a ~ 0 and E2 (k,j) = b ~ 0 then the vector

(A. 37)

also defines a basic mechanism. This is equivalent to relocating a

hinge at a joint in a basic mechanism.

(i i) In Eq. (A.29) not every set of u i IS defines a mechanism

(basic of combined). I However, a set of u2 s, not all of them zero,

define a combined mechanism w given by

(A.38)

if there exists some k such that wk = 0 in Eq. (A.38) and there exists

some i such that E, (k, i) F o.

To the author's knowledge there does not exist a mathematical

programming method suitable for the solution of this type of problem.

Thus, the genera) ization of the method of mechanisms through a mathe-

matical programming formulation does not seem to have an easy

implementation. Finally, it should be pointed out that the usual methods

based on the collapse mechanisms take into account the specific nature

of the loads to eliminate some of the mechanisms while the procedure

presented here is load-independent.

LIS T 0 F REFER ENe ES

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

2. Bigelow, R. H., and E .. H" Gaylord, "Design of Steel Frames for Minimum Weight,' Journal of the Structural Division, ASCE, Vol 93 , No ;; S r6 :; De c., 1 967.

3. Branin, F. Ho, Machine Analysis of Networks and Its Ape1 ications, Technical Report 8SS» IBM Development Laboratory, Poughkeepsie, New York.

4. Bruinette J K. E., and S. Je Fenves, "A Genera) Formulation of the Elastic-Plastic Analys is of Space Frameworks,'i International Conference on Space Structures, University of Surrey, 1966.

5. Busacker, R. Go, and T. L. Saaty, Finite Graphs and Networks: An Introduction with Ape] ications, McGraw-HilI Book Co., New York, 1965.

6. Charnes, A", and H. J. Greenberg, IIPlastic Collapse and linear Programming ' (abstract), Bulletin of the American Mathematical Society 57, Oct., 19518

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

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9. Darn, W. 5., and H. Ja Greenberg, "Linear Programming and Plastic Limit Analysis of Structures,ll Quarterly of Applied Mathematics XV, No.2, July, 1957.

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11. Fenves, S. J., Logcher, R. D., and Mauch, S. P., STRESS'" A User1s Manual, MIT Press, Cambridge, Mass., 1964.

12. Foulkes, Jo D., liThe Minimum-Weight Design of Structured Frames," Proceedings of the Royal Society, London, Series A, Vol. 223, i954.

13. Greenberg, H. J., and W. Prager, "Limit Design of Beams and Frames,'! Proceedings of the American Society of Civil Engineers 77, Feb., J 951 .

145

14. Hadley, Ge, Linear Programming, Addison~WesJey Publ ishing Co., Inc., 1962.

15. Hadley, Go» Non} inear and Dynamic Programming, Addison-Wesley PubJ ishing Co., Inc., 1964.

16. Hall, A. S., and R. W. Woodhead, Frame Analys is, John Wi ley and Sons, Inc., New Yo rk, J 96 L

17. Heyman, J., ':Plastic Design of Beams and Plane Frames for Minimum Material Consumption,' Quarterly Journal of Mechanics and Appl ied Mathematics, Vol. 8, No.4, 19510

18. Heyman, J., and W. Prager, "Automatic Minimum Weight Design of Steel Frames," Journal of the Franklin.Institute, Vo1. 226, No. 5, Nov., J 958.

19. Hodge, P. G., Plastic Analysis of Structures, McGraw-Hill Book Co .. , New York, 1959.

20. Lehigh University, Plastic Design of Multi-Story Frames, Lehigh Univ., Report No. 273.20, 24, Aug. 1965.

21. Livesley, R. K., Matrix Methods of Structural Analysis, Pergamon Press, New York, 1964.

22. Hassonet, c. E., and M. A. Save, Plastic Analysis and Design, Blaisdell Pub; ishing Co., New York, 1965.

146

23. Control Lan uaae User's Manual (H20-0290-1, ,

Yo r k .., 1 967. IBM, White Plains, New

24. Se arable IBM Appl ication Program,

25. Mauch. S~ Pq and S. iJ. Fenves, "Releases and Constraints in Structural Networks," Journal of the Structural Division, ASCE, Vo J. 93 , No • S T5, 0 ct., 1963 •

26. Mel in, J. We, "POST - Problem Oriented Subroutine Translator," Journal of the Structural Division, ASCE, Vol. 92, No. ST6, Dec., 1966.

27. Morris, G. A. and S. J. Fenves, JlA General Procedure for the Analys is of Elastic and Plastic Frameworks,'1 Civi 1 Engineering Studies, Structural Research Series No. 325, University of I 11 i no is, Aug., 1967.

28. Neal, B. G., and P. S. Symonds, "The Rapid Calculation of the Plastic Collapse Load for a Framed Structure,': Proceedings of the Institut·ion of Civil Engineers, Vol. 1, Part 3, April, 1952.

147

29. Nelson, H. M., et a1., "Demonstration of Plastic Behavior of Steel Frames,' Journal of Engineering Mechanics, ASCE, Vol. 83, No. EM4, Oct., 1957.

30. Prager, W., "Minimum Weight Design of a Portal Frame," Journal of the Engineering Mechanics Division, ASCE, Oct., 1956.

31. Prager, W., I Lineare Ungleichungen in der Baustatik,' S c hwe i z e r i s c he B au z e i tun 9 , 80 , J 962 •

32. Prager, W., I'Mathematical Programming and Theory of Structures,'1 Journal of the Society for Industrial and Appl ied Mathematics, Vol. 13, No. J, March ll 1965.

3 3 • P rag e r , W., "a p t i mum P 1 as tic De s i 9 n 0 f a Po r tal F r a me for A J t ern ate Loads,' Journal of App) ied Mechanics, Transactions of the ASHE, Vol. 34, Series E, No.3, Sept., 1967.

34. Ridha ,l R. A., and R. N. Wright, ':Minimum Cost of Frames,' Journal of the Structural Division, ASCE, Vo1. 93, No. ST4, Aug., 1967.

35. Wright, R. N., COSD (Constrained Optimum Steepest Descent), Department of Civil Engineering, University of III innis, Dec., 1967.

PART 1 - GOVERNMENT

Administrative & Liaison Activities

Chief of Naval Research Department of the Navy Washi.ngton, Do C" 20360 At.tn: Code 423

Director

439 468

ONR Branch Office 495 Summer Street Boston., Massachusetts 02210

Director ONR Branch Office 219 S. Dearborn Street Chicago, Illinois 60604

Commanding Offi.cer ONR Branch Office Box 39, Navy 100 c/o Fleet Post Office New York, New York 09510

Commanding Officer ONR Branch Office 207 West 24th Street New York, New York 10011

Director ONR Branch Office 1030 Eo Green Street Pasadena, California 91101

U .. So Naval Research Laboratory Attn: Technical Informat'ion Di v.

(2 )

(5 )

WaShington, Do C. 20390 (6)

Defense Documentation Center Cameron Station Alexandria, Virginia 22314

Commanding Officer U a S .. Army Research Ofr 0 Durham Attn: Mre J. J. Murray CRD-M~IP

Box CM, Duke Station Durham, North Carolina 27706

."l:> ....

(20 )

-1-

Commanding Officer AMXMR=ATL Attn: Mr. J .. Blubm U. S" :Army Materials Res. Agcy" Watertown, Massachusetts 02172

Watervliet Arsenal MAGGS Research Center Watervliet, New' York Attn: Director of Research

Redstone Scientific Info. Center Chief, Document Section U. So Army Missile Corr~and Redstone Arsenal., Alabama 35809

Army R&D Center Fort Belvoir, Virginia 22060

Technical Library Aberdeen Proving Ground Aberdeen, Maryland 21005

Commanding Officer and Director Naval Ship Research & Developmer..t Center Washington,_ D" C. 20007 Attn: Code 042 (Tech. Lib 0 B.r.)

700 (Struc. Mech. Lab,,) 720 725 800 (Appl" Mathe Lab e )

012e2(Mre We J. Sette) 901 (Dr9 M. Strassberg) 941 (Dr .. R .. Liebowitz) 945 (Dr., W" S" CrBlner,) 960 (Mre Eo F., Noonan) 962 (Drs Eo Buchmann):

Aeronautical Structures Lab­Naval Air Engineering Center Naval Base, Philadelphia, Pa. 19112

Naval Weapons LaboratorJr Dahlgren,. Virginia 22448

Naval Research Laborato17 Washington, D .. Ce 20390 Attn: Code 8hoo

8430 8440

NayY (cont 1 d)

Undersea Explosion Research Div. Naval Ship R&D Center Norfolk Naval Shipyard Portsmouth, Virginia 23709 Attn: Nro Dc So Cohen Code 780

Nav G Ship R&D Center Annapolis Division Cotie 257;J Library Annapolis, M~land 21402

Technical Library Naval Underwater Weapons Center Pasadena .Annex 3202 Ea Foothill Blvd. Pasadena~ California 91107

U" So Naval Weapons Center China Lake,9 California 93557 Attn: Code 4062 Mr.. Wo Werback

4520 Mra Ken Bischel

Naval Research Laboratory Washington,9 Do C. 20390 Attn: Code 8400 Ocean Tech. Div.

8440 Ocean Structures 6300 Metallurgy Div. 6305 Dr e J II Krafft

Commanding Offic.eX' U9 S. Naval Civil Engr. Lab. Code L31 Port. Hueneme.9 California 93041

Shipyard Technical Library Cod~- 242 L Portsmouth Naval Shipyard Portsmouth, New Hampshire 03804

U.. S., Naval Electronics Laboratory Attn: Dr. Re Jo Christensen San Diego, California 92152

U" S,. Naval Ordna,nce Laboratory Mechanics Division RFD 1, White Oak Silver Spring, Maryland 20910

U 0 S .. Naval Ordnance. Laboratory Attn: Mr. He As Perry, Jr. Non-Metallic Materials Division Silver Spring, Mar,yland 20910 -2-

Supervisor of Shipbuilding U. S. Navy Newport News, Virginia 23607

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Ue S. Navy Underwater Sound Ref. Lab. Office of Naval Research P. 0 .. Box 8337 Orlando,9 Florida 32806

Technical Libr~ U. $e Naval Ordnance Station Indian Head,9 Maryland 20640

U" S" Naval Ordn..ance Station Attn~ Mr" Garet Bornstein Research & Development Division Indian Head, Maryland 20640

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

Op-07T

Special Projects Office (CNM-PM-1) (MUN) Department of the Navy Washington, D. ... 0.".20360 Attn: NSP-001 Dr. J" Pa Craven

Deep SUbmergence Sys. Project (CNM-PM=11 ) 6900 Wisconsin Ave .. Chevy Chase, Md ... 20015 Attn~ PM-1120 So Hersh

Ue So Naval Applied Science Lab. Code 9832 Technical Library Buildi.ng 291 J Naval· Base Brooklyn;! New,York '125'1

Direc,.tor Aeronautical Materials Lab 9

Naval Aj,r Engineering Center Naval Base Philadelphia,,, Pennsylvania 19112

Naval Air Systems Command Dept. of the Navy Washington, D Q Ce. 20360 Attn: NAIR 03 Res" & Technology

320 Aero. & Structures 5320 Structures

604 TechQ Library

Naval Facilities Engineering Command

Dept .. of the Navy Washington" Dc> C.. 2-0360 Attn: N"lfAG 03 Res .. & Development

04.EngineeriI1..g &- Design 04128 Tech. Library

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

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

2052 Tech .. Librar,y

Naval Ship Engineering Center Main Navy Building. 1/IJashingto.n D .. C .. 20]60 Attn: NSEC 6100 Ship SYSe. Engr .. & Des" Dept e

6102C Computerated Ship Des" 6105 Ship Protec.tion 611.0 Ship Concept Design 6120 Hull Div., - J., Nachtsheim 6120D Hull Div,,,, -SIT Vasta 61 32 Hull Strncts.b - (4 )

Naval Ordnance Syste~ Command Dept<!l of the Nav;r "Was.bingt on, D,,, .C. 2.0360 Attn: NORD 03 Res ... & Technology

0-35 We.apons Dynamics 9132 Techo LibraFj

Air Force

C onnnander" WmD Wright-Patterson Air Force Base Dayton, Ohio 45433 .: Attn: .Code WWIlMDD

AFFDL (FDDS) -3-

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

AFLC (MCEEA) Code WWRC

AFML (MAAM) Code WCLSY

SEG (SEFSD, Mro Lakin)

Commander Chief, Applied Mecharri..cs Group U. Sf{ Air Force lnst." of Techo Wright-Patterson Air Force Base Dayton, Ohio 45433

Chief., Ci v:D- Engineering Branch WLRC, Research Division Air Force Weapons Laborator,y Kirtland ArB, New Mexico 87117

Air Force Office of Scientific ReSB 1400 Wilson Blvde Arlington, Virginia 22209 Attn: Mechs" Div.

NASA

structures Research Division National Aeronautics & Sp~ce Admin" La.ngley Res e.arch Center Laruzlev Stati.on HamPto~, Virginia. 2336, Attn: Mro Ro Ro Heldenfels, C~ief

National Aeronautic & Space Admin" Associate Adwinistrator for Advanced

Research & Technology Washington, Do C. 20546

Scientific & Tech-o Info..,. Fac.ility NASA Representative (S~AK/DL)' Fe 0" Box 5700 Bethesda, Maryland 20014

National. Aeronautic & Space Admin" Code RV:"2 Washington, De Co 20546

Othe~ Government Activities

Commandant Chief, Testing & Development Dive U9 S. Coast Guard 1300 E Street, N. We Washington, Do C. 20226

Direc.tor Marine Corps Landing Force Devel. Cene Marine Corps Schools Qu~ntico; Virginia 22134

Direct.or Nati.onal Bureau of Standards Washington, D'6 G •. 20234 Attn~ :Mr a ,Ba, L", Wilson, EM 219

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

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

Di.rector STBS' Defense Atorrie Support Agency Washington, D .. C .. 20350 .

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

Chief, Defense Atomic Support Agcye Blast & Shock Division The Pentagon Wasbington, D. C., 20301

Director Defense P~seareh & Engre Technical Library Room 3C-128 The Pentagon Washington, D C. 20301

Chief, Airframe & Equipment Branch __ X~=t~_____ _

Office of Flight Standards Federal Aviation Agency Washington, De C. 20553

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

·-4-

Deputy Chief, Office of Ship Constr. Maritime Administration Washingt.on, D.. c.. 20235 Attn~ Mr .. U .. La Russo

Mr Milton Shaw, Director DiVe of Reactor Devel. & Technology Atomic Energy Commission Ger'11lantoW!1,9 Mda 20767

Ship Hull Research Committee National Research Counci+ National Academy of Sciences 2101 Constitution Avenue Washington, Do C. 20418 Attn~ Mr. A" Re Lytle

PART 2 - CONTlL4..CTORS .A.L'\ID OTHER TECHNICAL COLLABORATORS

Universities

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

Dr.. J.. Tinsley Oden Dept. of Engrm Mechs~

University of Alabama Huntsville, Alabama

Professor Mo E~ ~xrtin Depte of Mathematics Carnegie Institute of Technology Pittsburgh, Pennsylvania 15213

Professor R" S. Rivlin Center for the Application of Mathematics Lehigh Uni vers·i ty Bethlebem, Pennsylvania 18015

Professor Julius Miklowitz Division of Engrc & Applied Sciences California Institute of Technology Pasadena, Cali~ornia 91109

Professor George Sih D_ap..a.r:t111ent.-nf-.Machru1ic.s Lehigh University Bethlehem, Pennsylvania 18015

DreHarold Liebowitz, Dean School of EI"l...gr..Q,· & Applied Science George Washington University 725 23rd Street Washington, Do Co 20006

Universities (cont'd)

Professor Eli Sternberg DiVa of Engr. & Applied Sciences California Institute of Technology Pasadena, California 91109

Professor Paul M. Naghdi DiVe of Applied Mechanics Etcheverry Hall University of California Berkeley, California 94720

\

Professor'WITl.e Prager Revelle College U~iver8ity of C~lifornia P:..O .. Box 109 La Jolla, California 92037

Professor Je Baltrukonis Mechanics Division The Catholic Univ. of America Washington, De Co 20017

Professor A. J. Durelli Mechanics Division , The Catholic Uni v" of America Washington, D. O. 20017

Professor H. Ho Bleich Department of Civil Engr. Columbia University Amsterdam & 120th street New York, New York 10027

Professor Ro D. Mindlin Dep~tment of Civil Engro Columbia University S .. 1-[4 Mudd Building New York, New York 10027

Professor F. 1. DiMaggio Department of Civil Engr. Columbia University 616 Mudd Building New York, New York 10027

-------

Professor As M. Freudenthal Department of Civil Engr" &

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

Professor Bo A. Boley Department of Theora & Appl. Mech. Cornell University Ithaca, New York 14850

Professor Pe Go Hodge Department of Mechanics Illinois Institute of Technology Chicago, Illinois 60616

Dr .. D. C. Drucker Dean of Engineering University of Illinois Urbana, Illinois 61803

Professor N .. ;MOl Newmark Depto of Civil Engineering University.of Illinois Urbana, Il~inois 6180)

Professor L Ro Robinson Department of Civil Engro University of Illinois Urbana, Illinois 61803

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

Professor James Mar Massachusetts Inst.. of Tech. Rm" 33-318 Dept. of Aerospace & fl~tro9 77 Massachusetts Ave .. Cambridge, MaSSa 02139

Professor Eo Reissner Dept& of ~~thew~tics Massachusetts lust. of Tech. Cambridge, Mass. 02139

Professor William A. N~h Dept a. of Mechs.. & Aerospace Engr. University of Mass.

____ .A.rnl?-erst" Mass. 0._1_,0_0_2 __

-5-

Library (Code 0384) U .. 8. .. Naval Postgraduate School Monterey, California 93940

Professor Arnold Alleptuch Dept G of Mechanic.a1 ~ngineering . Newark .College of Engineering 323 High Street Newark, New Jersey 07102

Universitiet~ (cont'd)

Professor E. La Reiss Courant Insto of Matha Sciences New York University 4 Washington Place New York, New York 10003

Professor Bernard Wo Shaffer School of Engrgo & Science New York University University Heights

, New York, New York 10453

Dr. Francis Cozzarelli Div~ of Interdisciplinary Studies and Research

School of Engineering State Univo of No Ye Buffalo, New York 14214

Professor Re Ao Douglas Depte of Engro Mechso North Carolina State University

Raleigh, North Carolina 27607

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

Professor Jo Do Achenbach Technological Institute Northwestern University Evanston,. Illinois 60201

Director, Ordnance Research Lab. Pennsylvania state University Po 00 Box 30 State College, ·Pennsylvania 16801

Professor Eugene J~ Skudrzyk Department of Physics Ordnance Eesearch Labo Pennsylvania State University Pe 0 .. Box 30 . State College, Pennsylvania 16801

Dean Oscar Baguio Assoc .. of Struc 0 Engr. of. the Philippines

University of Philippines Manila, Philippines

-6-

Professor J~ Kempner Depts of Aero. Engr. & Applied 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 Ao C, Eringen Dept. of Aerospace & Meche Sciences Princeton University Princeton, New Jersey 08540

Dr. S .. L. Koh Sohool of Aeroe, ABtro. & Engr_ Sc. Purdue University Lafayette, Indiana 47907

Professor Re A. Schapery Purdue University Lafayette, Indiana 47907

Professor E. H. Lee Dive of Engr. Mechanics Stanford University Stanford, California 94305

Dr. Nicholas J .. Hoff Dept. of Aero. & Astro .. Star~ord University Stanford, California 94305

Professor Max Anliker Dept. of Aero. & Astra. Stanford University Stanford, California 94305

Professor J, No Goodier Dive of EngrQ Mechanics Stanford University stanford, California 94305

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

Professor Marku~ Reiner Technion R&D Foundation Haifa, Israel

Universities (cont'd)

Professor Tsuyoshi Hayashi Department of Aeronautics Faculty of Engineering University of Tokyo BUNKYO~KU

Tok-yo , Japan

Professor j'. Eo Fitzgerald, Ch .. Depto of Civ~l Engineering University of Utah Salt Lake- City, Utah 84112

Profes s or R~ J'" H. Bollard Chainnan, Aeronautical Engr. Dept .. 207 Guggenheim Hall University of Washington Seattle, Washington 9810.5

Professor Albert Sa Kobayashi Depto of Mechanical Engr. University of Washington Seattle, Washington 9810.5

Officer-in-Charge Post Graduate School for Naval Off .. Webb Institute of Naval Arch. Crescent Beach Road, Glen Cove Long Island J New York 11.542

Librarian Webb Institute of Naval P~chG Crescent Beach Road, Glen Cove Long Island, New York 11.542

Solid Rocket Struc~ Integrity Cen .. Dept ... of Mechanical Engr .. Professor F~ Wagner Univ~rsity of Utah Salt Lake City, Utah 84112

Dro Daniel Frederick Depte of Engro Mechs. Virginia Polytechnic lnst. Blacksburgh, Virginia

Industry and Research Institutes

Dr. James Hq Wiegand Senior Dept .. 4720, Bldge 052.5 Ballistics & Mech.. Properties Lab .. Aerojet-General Corporation P .. 0 .. Box 1947 - . ~ 1· f . 9~or.u~_ Q bacramento, v~l~Ornla _J /

-7-

Mr .. Carl E. Hartbower Dept. 4620, Bldg. 2019 A2 Aerojet-General Corporation Po O. Box 1947 Sacramento, California 95809

:M"r., J. S~ Wise Aerospace Corporation Po 0.. Box 1 300 San Bernardino~ California 92402

Dr .. Vi to Salerno Applied Technology Assoc .. , Inc. 29 Church Street Ramsey, New Jersey 07446

LibraFJ Services Department Report Section, Bldg. 14-14 Argonne National Laboratory 9700 Se Cass Avenue Argonne, Illinois 60440

Dr. M. C9 Junger Cambridge Acoustical APsociates 129 Mount Auburn Street Cambridge, Massachusetts 02138

Dr. F. R. Schwarzl Central Laboratory TQN~OQ Schoemnakerstraat 97 Delft, The Netherlands

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

Supervisor of Shipbuilding, USN, and Naval Insp6 of Ordnance

Elect,ric Boat Division General Dynaw~cs Corporation Groton, Connecticut 06340

Dr. Lo H. Chen Basic Engineering Electric Boat Div~sion General Dynamics Corporation Groton, Connecticut 06340

Dro Wendt Valley Forge Space Technology Cen. General Electric CO e

Valley Forge, Pennsylvania 10481

Drg joshua E~ Greenspon J. Go Engr. P£search Associates 3831 Menlo Drive Baltimore, }1aryland 2121.5

Industry & Research Instc (cont'd)

Dr Q • Wal to D" Pilkey lIT Research Institute 10 West 35 Street Chic.ago:; Illinois 60616

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

Mr" Roger Weiss High Temp" Structs. & Materials

------Library--Ne-wport---News--Ship bui-J:ding-----·-----------Appl-re-d-Physrc-s-1-ab-;-·--------·--------------.. - .. -.-.. -------& Dry Dock Company 8621 Georgia Ave"

Newport News, Virginia 23607 Silver Spring, Md ..

MTc Jo 10 Gonzalez ,Engro Mechso Lab 0

Martin Marietta MP = 233 Po 00 Box 5837 Orlando, Florida 32805

Dro Eo Ao Alexander Research Depto Rocketdyne DoW 0 ~ NAA 6633 Canoga Avenue Canoga Park, California 91304

Mr 0 Cezar P D Nuguid Deputy Commi.ssi.oner Philippine Atomic Energy

Co:mmission Manila, Philippines

Dro Mo Lo Merritt Division 5412 Sandia Corporation Sandia Base Albuquerque, New Mexico 87115

Director Ship Research Institute Ministry of Transportation 700, SHINKAWA Mit aka Tokyo, Japan

Dr. H.o No Abramson Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206

Dr 0 Ro C" DeHart Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206

- 8 -

Mr. William Caywood Code BBE Applied Physics Lab. 8621 Georgia Ave. Silver Spring, Md.

Mr .. ' MEr J II Berg Engineering Mechso Lab 0

Bldg .. R-1, Bro." 1104A TRW: . Sys tema 1 Space Park Redondb. Beach, California 90278

Spcuritv Classification

DOCUMENT CONTROL DATA· R&D ;St'C'tlrity classifiC'atian of tit/c, body of abstwC't iltld indexing annotation must /Je entered wlwn thl' uverilll rt'[Jort is C'iilssifiecl)

1 . .oRIGINATING ACTIVITY (Corporate author) 2<1. REP.oRT SECURITY CLASSIFICATI.oN

University of III inois, Urbana, III inois Department of Civil Engineering

Unclassified r-2-h.-G-R-.o-U-P-------·-------·-·-

3. REP.oRT TITLE

----'--A-NETWORK;;,.-rO-Plrcotlt7rc--Fo-RfrorATID-f[-UF--TRl:-AN7·\[YSTS--ANU--rJ'ESTGrrU'F-RI'C;-I-D';;'-P t7XST"I-C---' "-"'-,,-­FRAMED STRUCTURES

4. ,DESCRIPTIVE N.oTES (Type of report and.inclush·e dates)

Progress; September 1967 - August 1968 5. AU TH.oR(S) (First name, middle initial, last name)

Gonzalez-Caro, Alfonso Fenves, Steven J.

6· REP.oRT DATE 7a. T.oTAL N.o . .oF PAGES

September, 1968 147 8a. C.oNTRACT .oR GRANT N.o. 9a • .oRIGINAT.oR'S REP.oRT NUMBER(S)

N00014-67-A-0305-00l0 b. PR.oJECT N.o.

Structural Research Series No. SRS 339

Navy 4-0305-0010 c.

d.

10. DISTRIBUTI.oN STATEMENT

Qualified requesters may obtain copies of this report from DDC.

11. SUPPLEMENTARY N.oTES

13. ABSTRACT

12. SP.oNS.oRING MILITARY ACTIVITY

Structural Mechanics Branch, Office of Naval Research, Department of the Navy

Mesh and the node formulations of the plastic analysis and design of rigid-plastic framed structures are developed. The formulations are initially developed for the simple plastic case and the network-topological concepts are then used to obtain a general ization including the effect of stress interaction. In gene ra I, the p rob 1 ems obta i ned a re non linea r p rog ramm i ng p rob 1 ems due to the nonl inearities in the yield surface. By I inearizing the yield surface the problems become I inear programming problems. The results obtained through this I inearization are sufficiently accurate for practical purposes. Furthermore, a computer program using 1 inear programming can be implemented much more expediently· than one us ing nonl inear programming. In contrast to the elastic case in which the analysis and design problems follow different patterns in the plastic case the formulation of the two problems is quite similar. By establ ishing approximate relations between the fully plastic values of the action components for the structural members, the design problem can be readily extended to include stress interaction into the definition of yielding.

o D /NOOR:6514 73 (PAGE 1) Unclassified

SIN 0101-807-6811 Se.curity Classifica tion .d_ '11 A (\ 0

Unclassified Security Classification . -

LIN K A LIN K B LIN K C 14,

KEY WORDS ROLE WT ROLE WT ROLE WT

Structural Analysis Structural Design Network theory Netwo rk flows To po logy Mathemati<;:al P rag ramm i ng Compu te r Programming

J

,

. _ . .... .. -" ... - .. _ .. - - _ .... - .. -_.

"

Unclassified SIN 0101-807-6821 Security CIa s sifica tion A-31409