Post on 19-Mar-2020
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.
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3. Branin, F. Ho, Machine Analysis of Networks and Its Ape1 ications, Technical Report 8SS» IBM Development Laboratory, Poughkeepsie, New York.
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145
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16. Hall, A. S., and R. W. Woodhead, Frame Analys is, John Wi ley and Sons, Inc., New Yo rk, J 96 L
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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 )
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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
Shipyard T.8chnical Library Building 746, Code 303TL Mare Island Naval Shipyard Vallejo,9 California 94592
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
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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
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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
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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