a general formulation for the optimum design of …
TRANSCRIPT
..:>~~
~ • ..2.cIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 362
A GENERAL FORMULATION FOR
THE OPTIMUM DESIGN OF
FRAMED STRUCTURES
by
W. J. McCutcheon
S. J. Fenves
A Technical Report of a Research Program
Spo nsored by
THE OFFICE OF NAVAL RESEARCH DEPARTMENT OF THE NAVY
Contract No. N 0014-67-A-0305-001 0 Project NA VY-A-0305-001 0
UNIVERSITY OF I LLI NOIS URBANA, I LLI NOIS
AUGUST, 1970
A GENERAL FORMULATION FOR
TEE OPTIMUM DESIGN OF
FRAr.:IED STRUCTURES
by
'(,V. J. r,;:cCutcheon
S. J. Fenves
A Technical Report of a Research Program
Sponsored by T}-<]!; OFFICE OF N.LI~VAL RESEARCH
DEPAR TMENT OF 'IlKS ~1A VY Contract No. N 0014-67-A-0305-0010
Project NAVY-A-0305-0010
UNIV-BRS ITY OF ILLIr'~OIS URBANA, ILLINOIS
AUGUST, 1970
AC KNOWLEDGENIENTS
This report was prepared as a doctoral dissertation
by I','Ir. William J. McCutcheon and was submi tted to the Gradua te
College of the University of Illinois at Urbana-Champaign in
partial fulfillment of the requireme~ts for the degree of
Doctor of Philosophy in Civil ~ngineering. The work was done
under the supervision of Dr. steven J. Fenves, Professor of
Civil Engineering.
iii
TABLE OF' CONTENTS
Page
ACKNOWLEDGEIvlENTS • . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . vii
Chapter
1 .L
2
INTRODUCTION . . . . . . . . . . . . . . . . 1
1.1 Purpose •••••••••••••••• 1
1.2 Scope ••••••••••••••••• 2
1.3 Assumptions and Limitations ••• • •• 4
1.4 Organization of Report • • • . . . · . . BAS Ie FOR MULli TION • • • • • • • • • • • • • •
2.1 Structural Variables and Matrices • • •
2.2 Yield Constraints • • • • • • • • • • • 2.2.1
2.2.2
Equilibrium constraints • Member force constraints
. . . . • • • •
5
7
7
9 10
11
2.3 Working Stress Constraints. • • • • •• 17 2.3.1 Equilibrium constraints. • • •• 18
2.3.2 Member force constraints • • •• 18 2.3.3 Compatibility constraints. • •• 21
2.4 Optimum Design as a Linear Programming
2.5
Problem • • • • • • • • • • • • • • •• 22 2.4.1 Equilibrium constraints. • • •• 23 2.4.2
2.4.3 2.4.4
2.4.5
Member force constraints • • • • Compatibility constraints.
Objective function
Basic formulation • • • • •
• • • •
• • •
• • •
• • •
Specification of Member Groups • • • • •
iv
24
25 27 27
30
v
Chapter Page
3 GENERALIZA'rION OF FORMULA.TIOr~ FOR ALTERNATIVE
4
5
6
LOADS • • • • • • • • • • • • • • • • • • • • 33
3.1 Independent Loads and Alternative Loading Combinations • • · • • · • • • • 33
3.2 Generalization of Formulation · • • • · 35 3.2.1 Redefinitions • • · • • • • • • • 35 3.2.2 Equilibrium constraints · • • 37 3.2.3 Member force constraints • • • · 38 3.2.4 Compatibility constraints • • · • 40
3.2.5 Generalized formulation · • • • • 42
IMPLEIvrENTATION • • • • • • • • • • • • • • • , 44
4. 1 POST - FORTRAN Program • • • • • • • •• 44
4.2 Linear Programming Solver • • • • • • •
ILLUSTRATIVE EXAMPLES • • • • • • • • • • · •
5.1 Plane Frame Design • • • • • • • • • · • 5.1.1 Effect of load factor and stress
ratio • . • • • • • • • • • • · •
5.1.2 Effect of member force inter-action . • • • • • • • • • • • •
5.1.3 Effect of alternative loadings
5.2 Space Frame Design • • • • • • • • • • •
5.3 Computer Program Efficiency • • · • • •
CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY
6.1 8onclusions • • • • • • • • • • • • • •
6.2 Suggestions for Further Study • • • • •
47
49
49
52
54 55
56
60
63
63
64
LIST OF FIGURES
Figure Page
1 Computer Program Flow Diagram · • • · • · • • 68
2 Two-Story Plane Frame . · · · • · • • • · • • 70
3 Volume vs. Load Factor · · • • • • • • · · · 71
4 Volume vs. Reciprocal of Stress Ratio • • • • 72
5 Independent Loads on Plane Frame · • · • • • 73
6 Staircase . • . • • . . · • • • · • • • · • • 74
7 Staircase Support Structure • • • • • • • • • 75
8 Staircase Dead Plus Live Load (kips) • • • • 76
vii
Chapter 1
INTRODUCTICK
The increased availability and use of digital com
puters has caused rapid changes in practically all areas of
study. In structural engineering, two trends have become
evident. The first of these has manifested itself in the
increased use of general network and matrix formulations in
structural analysis. The second trend has been toward the
use of mathematical programming techniques in the solution
of problems in optimal design. While many types of optimi
zation problems have been treated in recent literature, very
little has been done to present a unified formulation for
the optimum design of framed structures.
1.1 Purnose
The object of this study is to develop a unified
approach for the formulation and solution of problems in
volving the optimum design of structural systems. It is not
the purpose of this study to develop any new methods of
structural analysis or new techniques of mathematical pro
gramming, but rather to demonstrate that a single formulation
may be used to formulate and solve most problems of optimum
structural design.
1
2
1.2 Scone
Methods of plastic analysis and design have tradi
tionally been treated separately from elastic methods, and
the same trend has been followed in studies of structural
optimization. While it is recognized that linear programming
techniques may be applied to optimum design when limit analy
sis or the theory of plastic collapse mechanisms is used [7,
12, 23J,* optimization problems involving elastic behavior
have, in general, been formulated as non-linear programming
problems [2, 14, 26J. Moses introduced the concept of apply
ing linear programming to elastic structures by replacing
the non-linear equations by their first-order Taylor series
term [22J.
While much has been written concerning structural
optimization, the majority of the literature has been directed
toward the solution of particular structural types. Very
little has been done to present a unified approach, based on
a general matrix formulation, to the optimum design of framed
structures.
Fenves and Branin [5J developed the network-topo
logical formulation of elastic structural analysis and
Gonzales and Fenves [8J have presented a general formulation
* Numbers in brackets indicate works in the List of References.
:3
for bo~h the analysis and design of rigid-plastic structures.
The latter work considers only design variables and con
straints which relate to ultimate capacity, and the equations
obtained are linear. Even when stress interaction is consi
dered, reasonable assumptions can be made to linearize the
problem.
This study develops a general formulation for
optimum design which includes the entire range of design
variables and behavior constraints. Elastic behavior, in
the form of working stress limitations, and ultimate capacity
under multiple loading conditions are considered. In addi
tion, provisions are made for designating groups of members
to be made identical.
The formulation is presented in the form of an
iterative set of linear prograwning problems. The design
variables relating to elastic and ultimate behavior are line
arly related and the objective function to be minimized is
expressed as a linear function of the design variables. By
making assumptions about the member properties and sizes,
the constraints which define the problem are also expressed
in linear form.
It would be possible to develop a somewhat more
general formulation in the form of a non-linear programming
problem, but methods now available for solving such problems
4
[2, 30J are not readily applicable, since they involve
selecting a feasible solution, moving the solution until a
constraint is encountered, moving along that constraint
until another constraint is encountered, determining whether
or not the neVI constraint is active, etc., until the optimum
is rea.ched. Because of the large number of constraints
which are possible, such a procedure can be a very laborious
and time-consuming operation, even when carried out on a
large-scale digital computer. Therefore, this study is re
stricted to the linear programming approach only.
1.3 Assumptions and Limitations
The assumptions and limitations used in this study
are:
(a) Structural geometry is known. It is the sizes
of the members, rather than their configuration, which is to
be optimized"
(b) The types of structures to be considered are
plane grids, plane frames, and space frames, since these
types can be represented as networks. It is possible to
extend the formulations which are developed to trusses also,
but portions of the formulations will then degenerate (e.g.,
member force interaction need not be considered).
(c) The material is linearly elastic - perfectly
5
plastic. When yielding occurs, there is no "spread length"
alon~ the member.
(d) All members are straight and prismatic with
cross sections having two orthogonal axes of symmetry.
(e) Buckling is not considered.
(f) Deflections are assumed to be small. There
fore, secondary effects can be ignored.
(g) yield hinges and working stress limits can
occur only at member ends. Therefore:
(h) Loads are restricted to concentrated loads
(forces and couples) at the joints. If it is desired to
place a load elsewhere, a fictitious joint must be inserted
at that point.
1.4 Organization of Report
In Chapter 2, after a brief review of structural
vectors and matrices, a basic formulation for optimum design
is developed. The yield and working stress constraints are
presented separately and then combined to form a linear pro
gramming problem. A method for specifying "member groups"
is also presented.
Chapter 3 generalizes the basic formulation to
include multiple loads and alternative loading combinations.
Chapter 4 discusses a computer program which was
6
written to implement the formulations developed in Chapters
2 and 3.
In Chapter 5, several sample problems are presented.
These are solved using the computer program described in
Chapter 4. The efficiency of the program in solving these
problems is discussed.
Finally, Chapter 6 presents a summary of conclu
sions reached during the course of this study and several
suggestions for further investigation.
Chapter 2
BASIC FORMULATION
This chapter presents a basic formulation for the
optimum design of framed structures. First, some basic con
cepts concerning structural variables and matrices are re
viewed, and then the types of constraint equations and in
equalities necessary to describe the problem are discussed.
Finally, the basic formulation is presented as an iterative
set of linear programming problems, and a method for defining
"member groups" is presented.
2.1 structural Variables and Matrices
Consider a structure having b members and n free
joints. A member is represented as an oriented branch going
from its positive or A-end to ·its negative or B-end. Each
member has its own member coordinate system, with the member
x-axis lying along, and oriented in the same direction as,
the member. Since the members can have completely different
orientations and coordinate systems from one another, it is
necessary to define a single global coordinate system for
the structure as a whole. In this study, only frames and
grids are considered. For such structures, the number of
force and distortion vector components (degrees of freedom)
7
8
in member and global coordinates are the same and is denoted
by f.
The following vectors are needed to define the
forces and distortions of the structure (the notation used
follows that of Fenves and Branin [5J as closely as possible):
P' is a vector of externally applied joint loads,
consisting of n subvectors (one for each free joint), each
of order f. It is expressed in global coordinates.
u' is a vector of joint displacements, also in
global coordinates and of the same order as pt.
R is a vector of member forces, expressed in mem-
ber coordinates at the B-ends of the members and containing
b subvectors of order f.
V is a vector of member distortions, also in mem-
ber coordinates and of the same order as R.
The structural matrices used in this study are the
following:
A is the branch-node incidence matrix, a typical
submatrix, Aij , of which is (Tl, -HIT{, 0) if member i is
(negatively, positively, not) incident on joint j. Hi is a
square translation matrix of order f which transfers the
member force from a negative end force to a positive end
force (i.e., RAi = HiRBi ), both in member coordinates. T. l
is a square rotation matrix of order f which transforms the
9
member force vector at the negative end of member i from
member to global coordinates.
k is the unassembled stiffness matrix, a diagonal
matrix of b submatrices k., where k. is a square matrix of 1. 1.
order I which defines the stiffness of member i (i.e., Ri =
k.V.) for working loads. 1. 1.
other vectors and matrices will be defined as they
are introduced.
The stiffness (or node) method is the "classical"
method for elastic analysis and presents greater assets for
computer implementation than does the flexibility (or mesh)
method. Gonzales and Fenves [8J have shown that the node
method is also more readily adaptable for plastic analysis
and design. For these reasons, the formulations developed
in the following sections are based on the node method.
Therefore, it is not necessary in this study to employ the
node-to-datum path and branch-circuit matrices or the redun-
dant member force vector, since these are needed only for
the mesh method.
2.2 Yield Constraints
The static or lower-bound theorem of plastic analy
sis [9J states that the load-carrying capacity of a struc-
ture is the largest load which corresponds to a statically
10
admissible state of stress. A statically admissible state
of stress is defined as a state in which:
(a) the stresses are in internal equilibrium, as
well as in equilibrium with the external loads; and
(b) the yield limit is not exceeded anywhere in
the structure.
Thus, for plastic analysis, two types of constraints,
corresponding to (a) and (b) above, are requiredl equilibrium
constraints and member yield force constraints.
Gonzales and Fenves [8J have shovm that plastic
design problems can be formulated similarly to plastic analy-
sis problems and that the same types of constraints are re-
quired for design. Thus, equilibrium and member force con-
straints must be imposed for design.
2.2.1 Equilibrium constraints
In order for the structure to be in equilibrium
(both internal and external), the sum of the member forces
at each joint must be equal to the externally applied load(s)
at that joint. In matrix notation:
A tR = P' Y Y
(2-1)
where Ry is a vector of member yield forces at the B-ends of
the members, P' is a vector of externally applied joint yield y
loads, and A is as defined in Section 2.1. The superscript
11
"t" denotes matrix transposition.
2.2.2 l';'Iember force constraints
Recalling that yield hinges can occur only at
member ends, let R' denote a vector of yield forces at both y
ends of the members in the structure. Ry can be expressed
as:
R' =QR (2-2) Y Y
where Q is a diagonal matrix of submatrices Q .• A typical l.
subvector P , . \. . yl.
of R' is: y
R '. = _;t~~ {
R A~ yl. R B. Y l.
(2-3)
Since Rt\. = H. R3.' Eq. 2-2 can be rewritten as: nl l.: 1
R BO
Y l.
where I is an identity matrix of order f.
of the form:
(2-4)
Therefore, Q. is l.
(2-5)
The vector R' contains all the member yield force y
12
components. However, since not all these components are
necessarily used in the definition of yielding, it is neces-
sary to extract from H' the components actually needed. This y
can be accomplished by the premultiplication of Rt by an y
extractor matrix ~ , so that: y
rtf = ~ R' Y Y Y
(2-6)
where the vector Ry contains only those components which are
used to define yielding and ~y is a diagonal matrix of sub
matrices ~ .• y~
If, for example, the structure of interest is a
plane frame, Ry contains six (2f) elements per members namely,
axial force, shear force, and bending moment at each end.
Assuming that axial force and bending moment are used in de
fining the yield criterion, a typical submat~ix ~ . will be yl
of the form:
(2-7)
In general, the combinations of member forces
which define yielding are not described by linear relations.
However, it is possible to approximate the yield surface as
a series of linear segments [13J. To continue with the above
13
example of a planar frame in which axial force and bending
moment are the components of interest, the general form for
one segment of the linearized yield surface at one end of a
typical member i is:
(2-8)
where p and m are the axial force and bending moment at that
end of member i, Pult and IDult are the ultimate values of p
and m (i.e., Pul t is the value of p which will cause plastic
yielding to occur when only axial force is acting on the
member, etc.), and 91 and 92 are constants which define the
yield surface segment.
It is assumed in this study that Pult and mult can
be linearly related to a single member reference yield ca
pacity, Py • Thus, Pul t and mult can be expressed as:
1 - (a) Pult = y Py 1
(2-9) 1 - (b) mult = y Py
2
The yield capacity of member i, p ., can be selected as any y~
section property (e.g., cross-sectional area, plastic modu-
lus, mult ' etc.) which can be linearly related to the sec
tion's ultimate axial force and moment capacities, as in
14
Eqs. 2-9, by constants 1/Y1 and 1/Y2.
Substituting Eqs. 2-9 into 2-8 and rearranging
terms gives:
Eq. 2-10 can be cast in matrix form as:
-p . -yl
(2-10)
> 0 (2-11)
Thus, it can be seen from Eq. 2-11 that a vector R* of memy
ber yield forces, normalized with respect to the member
plastic capacities, p , can be defined as: y
R* = r R' y y y (2-12)
fy isa diagonal matrix of submatrices r . which contain the yl
constants y. Recalling that Eq. 2-11 represents only one end
of member i, fyi for the above example can be written as:
r . = yl.
in which Yl and Y2 are constants defined in Eqs. 2-9.
(2-13)
Finally, ~~ is defined as a vector of linear com
binations of member yield forces, and (again referring to
15
Eq. 2-11) can be written as:
R* = e R* y Y Y
(2-14)
where 8 is a diagonal matrix of submatrices e .. Y yl For example, in the 1970 AlSO Specification, the
linearized yield surface for compression and bending is de-
fined by:
+ m < 1 (a) - mult -(2-15)
---.:E.- + .85 _m_< 1 (b) Pult - mu1t -
Assuming that the same relation holds for combined tension
and bending, a typical submatrix e . will be: Yl
o 1 I , o -1 I
1 .85 : 1 -.85:
-1 .85 I
o
-1 -.8~: -------~l--O---i--
o I 0 -1 : 1 .85 : 1 -.85 I -1 .85 : -1 -.85
(2-16)
Thus, if the yield surface is defined by s linear segments y
(s = 6 above), e ).. will contain 2s rows. y y y
Defining U . as a 2Sy vector of ones, the yield yJ.
16
force constraints for member i can be written as:
u .p . - ~*. > 0 yl yl Yl- (2-17)
and the entire set of member yield force constraints as:
(2-18)
in which Py is a'vector of member yield capacities and Uy is
a diagonal matrix of vectors U .• Since each vector U . is yl y~
of the order 2Sy ' the matrix Uy is therefore of the order
2bs x b. y
Substituting Eqs. 2-14, 2-12, 2-6, and 2-2 into
2-18, the yield force constraints can be written in terms of
the basic variables, namely the forces, R , used in the y .
equilibrium equations (Eq. 2-1) and the unknown plastic ref-
erence capacities, P J as: y
(2-19) .
In summary, the transformations on Ry are, from
right to left, as follows I
Q transforms R , a vector of member B-end yield y
forces, into a vector of forces at both ends of the members;
~y extracts the components which are used to define
yielding;
fy normalizes the force components with respect to
the member reference yield capacities, p ; and y
17
8 y combines the normalized forces as defined by
the linearized yield surface.
IntroducingITy to denote the matrix product 8y fy6yQ,
the yield force constraints can be written as:
Up -DR >0 Y y Y Y -
(2-20)
The constraints of Eq. 2-20 can be represented
graphically as:
(2-21)
where n . represents the matrix product 8 .f .~ .Q .• yl yl yl yl 1
2.3 Working Stress Constraints
In the "classical" elastic solution of a framed
structure, three types of equations must be satisfied. equi-
librium equations, member force-distortion equations, and
structural compatibility equations. Due to the design nature
of the problems considered in this study, it is necessary to
impose a fourth set of constraints, namely member force con
straints, to insure that working stress limits are not ex-
ceeded anywhere in the structure.
It will be shown in the following sections that
18
the equilibrium and member force constraints for working
loads can be formulated in a manner analogous to those for
yield loads, and that the member force-distortion equations
and structural compatibility equations can be combined into
a single set of "compatibility" constraints.
2.).1 Equilibrium constraints
The equilibrium equations for working loads are
exactly analogous to those for yield loads, namely:
A tR = P' w w (2-22)
The subscripts "w" indicate working forces and loads.
2.).2 Member force constraints
The member force constraints for working loads are
also very similar in form to those for yield loads. R~ is a
vector of working forces at both ends of the members and can
be expressed as:
R' = QR w w (2-23)
where Q is exactly the same as defined by Eq. 2-5, since
working stress limits, like yield limits, are specified only
at member ends.
R~ is a vector of member force components which are
used to define working stress limits. It can be written ass
19
H' =L1 Rt w w w (2-24)
~w is an extractor matrix similar to ~y,and extracts those
components which enter into the definition of working stress
limits, vlhich mayor may not be the same components used to
define yielding.
Pw (similar to Py) is defined as a vector of member
working capacities which can be related linearly to the mem
bers' maximum allowable force components by a matrix f (also w
s~milar to its yield counterpart, fy). Thus, R~J a vector of
member force components normalized with respect to Pw' can be
computed by premultiplying R~ by fw:
. R* = r R' w w w (2-25)
R*, a vector of combinations of member force corow
ponents which define working stress limits, is obtained by
premultiplying R* bye: w w
R* = e R* (2-26) w w w
8w is a diagonal matrix of submatrices ewi~ For example,
limiting working stress combinations can be defined by the
inequalities:
+~+ m <1 Pall - mall
(2-27)
20
where p and m are the axial force and bending moment at one
end of a typical member, and Pall and mall are the maximum
allowable values of p and m when they act separately. A
typical submatrix, e .J for such a case will be: Wl
1 1 I I
1 -1 I 0
-1 1 I I
-1 -1 I e . = -------}--r---r (2-28) Wl
t 1 -1
0 , I -1 1 I I -1 -1
Thus, the formulation of the member force con-
straints for working conditions is analogous to that for
yield conditions, and can be written as:
where Uw is a diagonal matrix of unit vectors U ., each of Wl
order 2s , and s is the number of segments defining working' 'N \~
stress limits (equal to 4 in Eqs. 2-27 and 2-28).
Substituting Eqs. 2-26, 2-25, 2-24, and 2-23 into
2-29. the member working force constraints can be written as:
(2-30)
Thus, the form of the working force constraints is
similar to that for the yield force constraints. However,
there is still a large degree of independence between the two
21
sets of force constraints in that:
(a) the number and type of force components used
to define working limits can be different from those defining
yield (~w vs. ~y);
(b) the relationships between the maximum allowable
member forces and the working capacities, Pw' can be entirely
different from those between the ultimate forces and the
yield capacities, Py (fw vs. fy);
(c) the number of linear segments defining working
stress limits and yield limits can be different from one
another (sw vs. Sy); and
(d) the linear combinations of forces which define
the working stress limits can be different from those which
define yield (8w vs. 8 y ).
Using I1w to denote the matrix product 8wfw~Qf the
working force constraints can be rewritten as:
(2-31)
2.3.3 Comnatibility constraints
The compatibility requirements for the structure,
namely that the distortions of each member must be equal to
the difference between the deflections of its end joints, can
be stated in matrix form as:
22
v = Au' (2-32)
where V is a vector of member distortions at working loads
and u' is a vector of joint displacements.
The member compatibility requirements are defined
by the force-distortion equations:
(2-33)
where k is the unassembled stiffness matrix defined in Sec-
tion 2.1.
Since the member distortions, V, are not of primary
interest and need not be solved for, the two sets of equa-
tions above can be combined by substituting Eq. 2-32 into
2-33 and rearranging terms, giving:
kAu' - R = 0 w (2-34)
Eq. 2-34 represents the compatibility constraints as they are
used in this study.
2.4 Optimum Design as a Linear Programming Problem
The relationships of Eqs. 2-1, 2-20, 2-22, 2-31,
and 2-34 represent the entire set of constraints which are
needed to define the problem. As these constraints are formu-
lated, however, it is assumed that the yield variables and
working stress variables are independent of one another. It
23
is therefore necessary to define the relationships between
P' and P' of the equilibrium constraints (Eqs. 2-1 and 2-22) y w
and between pyand Pw of the member force constraints (Eqs.
2-20 and 2-)1) before the yield and working stress constraints
can be combined into a single formulation. It is also neces-
sary to recognize that the stiffness matrix k is not actually
known, since for a design problem the sizes of the members
are not known.
Thus, it is necessary to modify the constraints be
fore an objective function is defined and the basic formula
tion is presented as a linear programming problem.
2.4.1 Eguilibrium constraints
The applied joint loads for yield and working con-
ditions, Py and P~J are not actually independent of one
another. The yield loads can be taken as being equal to the
working loads times a "load factor":
P' = tP' Y w (2-35)
where t is the load factor.
The equilibrium constraints can therefore be re-
written as:
AtR = tP' Y
A tR = P' w
(a)
(b) (2-36)
24
The subscripts By" and "w" can thus be dropped from P' since
it is the only independent vector of joint loads.
2.4.2 Member force constraints
As the member force constraints are formulated in
Eqs. 2-20 and 2-31, the member yield capacities and working
capacities, p- and p , are independent of one another. This, y -w -
of course, is not the case, and it is necessary to take into
account the relationship between the two sets of capacities.
It is possible to define p (no subscript) as a vec-
tor of member reference capacities. These reference capaci-
ties can be selected as any section property of the members
(e.g., area) which can be related to both their yield and
working stress capacities, Py and Pw' by linear relationships
of the form:
= r yp -= rw p
(a)
(b) (2-37)
where rand r are diagonal matrices (of order b) of con-y w
stants r . and r .• The selection of values for ry and rw yl Wl
can involve many considerations (such as the member shape
factors, assumed member proportions, etc.), as will be illus-
trated in the examples of Chapter 5.
Using the above relationships, the member force
constraints can be rewritten in the following form:
Uyryp - I\Ry > 0
uwr wP - I\;Rw > 0
2.4.3 Compatibility constraints
(a)
(b)
25
(2-38)
As the compatibility constraints are presented in
Eq. 2-34, it is assumed that the member stiffnesses, as ex-
pressed by the k matrix, are known. This is not the case,
however, since the stiffness of a member depends on its size.
which is unknown. It is possible, however, to represent the
stiffness matrix of any member as a matrix of constants times
that member's reference capacitys
k. = poke (2-39) ~ ~ ~
where Ki is a "scaled stiffness" matrix of member i. For
example, the stiffness matrix for a member of a plane frame
is:
EA ~ 0 0 p
0 EIz EIz (2-40) k. = 12- -6-
~ p3 22 . EIz EI
0 -6- 4_z p2 2
where E is the modulus of elasticity, A is the cross-secx
tional area, I z is the moment of inertia, and 2 is the length
26
of the member. The scaled stiffness matrix is therefore:
EAx 0 0
f E! Elz k. = 0 12 __ z -6- (2-41)
~ p3 12 EIz EI
0 -6- 4_z 22 2
inhere Ax is the scaled area (equal to Ax/Pi) and I z is the
scaled moment of inertia (equal to Iz/Pi). If, for example,
Pi is selected to represent the member's area, then Ax will
equal 1.0 and I z will equal the (assumed) ratio of the mem
ber's moment of inertia to its area, i.e., the square of the
assumed value of the radius of gyration about the z-axis_
Substituting Eq. 2-39 directly into Eq. 2-34 would
result in non-linear compatibility equations, since the pro
ducts of two sets of unknowns, p and u t, appear in the term
pE~u·. However, by replacing the unknowns Pi by a set of
trial, or assumed, member capacities, Pi' the compatibility
constraints can be rewritten in linear form as:
pKAu' - R = 0 w (2-42)
where K is a diagonal matrix of submatrices k. and p is a ~
diagonal matrix of submatrices which are identity matrices
(of order f) times the trial values Pi-
27
2~4.4 Objective function
-Any set of values for p, R , R , and u t which y w
satisfies the constraints of Eqs. 2-36, 2-38, and 2-42 rep-
resents a feasible design for the structure of interest.
Since the object of this study is to develop a formulation
for optimum design, it is necessary to define an objective
or cost function, which is used as the criterion for deter-
mining which of the feasible designs is optimal.
In this study, it is assumed that the cost per unit
length of each member is proportional to its reference capa
city, Pi. Therefore, the cost of a member is equal to its
length times its reference capacit~ and the cost of the entire
structure is the sum of the member costs. In vector notation,
then, the objective function is:
minimize tL P
where L is a vector of individual member lengths.
(2-43)
If, for example, the member areas are selected as
the reference capacities, Eq. 2-43 will yield a minimum
volume design.
2.4.5 Basic formulation
Using the objective function of Eq. 2-43 and the
constraints of Eqs. 2-36, 2-38, and 2-42, the problem of
optimum design can be cast in the following form:
minimize tL P
subject to:
A tR Y
uyr yP - TIyRy
A. tR . w
U~wP -l\.Rw
= tP'
> 0 -= P'
> 0 -pkAu' - Rw = 0
P > 0
u' unrestricted in sign
28
(a)
(b)
(c)
(d) (2-44)
(e )
(f)
(g)
(h)
Constraints (b), (c), (d), (e), and (f) of Eq. 2-44
can be cast more graphically ass
f At , I = tp' , I I
-----~---~---~--- P I I I Uri -IT I I > 0 - ¥. -¥. - ~ - -~ -:- - - =t -:- - - - -
I I A I = p' (2-45) I I I
~----~---~---~---U I 'I\, I > 0 wrw I I - I -_____ L ____ ~ ___ ~---
I I -I I pKA 0 I I I =
Eqs. 2-44 represent the basic formulation for the
optimum design of framed structures, and is in the form of
a linear programming (L-P) problem, with the values of p, R , Rw' and u' forming the solution vector. y
An optimum solution is reached when the values of
the reference capacities, p, are the same as the trial values
29
-of p. Thus, the problem can be solved as an iterative set of
L-F problems, with the solution values, p, of one iteration
being used for the trial values, ~J of the next iteration.
If no reasonable values for p can be selected for the first
trial, it is possible to avoid this problem by neglecting
the constraints of Eq. 2-44(f) on the first iteration. Con-
vergence criteria (i.e., specifying how close the values of
p and p must be before the solution can be considered to have
converged) can be set to whatever is deemed necessary for the
problem being solved.
The initial selection of constants for the ry , rw'
r y , rw' and K matrices involve assumptions regarding the
member types and sizes. The iterative nature of the solution
process, however, allows revisions to be made in these matri
ces if the initial assumptions prove to be invalid.
The solution vector contains the values for the
member reference capacities (p), member yield forces (R ), y
member working forces (Rw)' and elastic joint deflections
(u'). If desired, the ultimate and allowable member force
component capacities can be obtained by using the relation-
ships of Eqs. 2-9 and 2-37 to give:
1 - (a) Puki = -- r .p. Y k" yl. l. Y l.
(2-46) 1 - (b) Paki = --r wiPi Ywki
)0
where Puki and Paki are, respectively, the ultimate and
allowable capacities for the kth force component of member i,
Y kO and Y kO are the corresponding elements of the rand y 1 W 1 Y
r matrices, and r 0' r 0' and p. are as previously defined. 'IN yl Wl 1
2.5 Specification of Member Groups
When designing framed structures, it is usually
desirable to specify groups of members as being identical.
This type of "member group" designation can be incorporated
very simply into the basic formulation of Eqs. 2-44 by rede
fining the p and L vectors and the Uy and Uw matrices as
follows I
(a) p is a vector of elements -Pk' where -Pk is the
reference capacity of all members in member group ki
(b) L is a vector of elements Lk , where Lk is the
sum of the lengths of all members in member group k;
(c) the unit vectors, Uyi ' Which comprise uy ' are
arranged so that every Uyi which corresponds to a member in
group k is placed in column k. Thus, there are as many col
umns in Uy as there are member groups in the structure; and
(d) Uw is redefined similarly to Uyo
Obviously, the input values ryi' rwi' and Pi will
be the same for all members in the same member group.
As an example of the above redefinitions, consider
31
a structure of three members in which members 1 and 3 are to
be identical (i.e., comprise one member group). L will be
of the forml
(2-47)
-where the subscripts refer to individual members; p will be:
(2-48)
where the subscripts refer to member groups; and Uy and Uw
will appear as I
1 I I
1 I I
• 1
• I • I
I 1 I -...1--
: 1 : 1
U = I • (2-49) I • t • I I 1 _..J.._ I
1 I
1 : I • • I
• I I
1 I I
-Henceforth, L, p, Uy ' and Uw will be assumed to
conform to the redefinitions given in this section. The case
32
where all member capacities are independent of one another
is merely a special form of these redefinitions, with each
member comprising its own member group.
8hapter 3
GENERALIZATION OF FORMULATION FOR ALTERNATIVE LOADS
The basic formulation of Eqs. 2-44 represents a
linear programming formulation for the optimum design of a
framed structure acted upon by a single set of external joint
loads. Alternative loading combinations are not considered.
In this chapter, the basic formulation is general
ized to consider alternative loads.
3.1 Independent Loads and Alternative Loading Combinations
The types of independent loads which can act on a
structure include a dead load; live load; wind load; and
forces due to earthquake, hurricane, or other natural phe-
nomenon.
It is grossly over-conservative in structural de
sign to assume that the maximum values for all the indepen
dent loadings act simultaneously. Therefore, it is neces
sary to define the alternative loading combinations which
can reasonably be expected and which must be considered in
designing the structure.
For example, the 1970 AISC Specification requires
that the following loading combinations be considered in the
plastic design of a continuous frame subject to dead, live,
33
and wind load:
1.85 (dead + live)
1.40 (dead + live + wind)
(a)
(b)
34
(3-1)
The corresponding combinations for working stress design are:
1.00 (dead + live)
.75 (dead + live + wind)
(a)
(b) (3-2)
The alternative loading combinations defined by
Eqs. 3-1 and 3-2 can be expressed in matrix form. If A is y
a matrix which defines the yield combinations, containing
one column for each independent load and one row for each
alternative yield load combination, then for Eqs. 3-1:
A = [1.85
y 1.40
1.85
1.40 (3-3)
Similarly, the working load combinations of Eqs.
3-2 are defined by matrix
A = [1.00 w .75
A : w
1.00
.75 (3-4)
The number of columns in Ay and Aware the same
(equal to the number of independent loadings, which is de
noted n i ), but the number of rows need not be the same.
35
Denoting the number of yield combinations and working stress
combinations as ny and nw' respectively, Ay contains ny rows
and AW contains nw rows.
3.2 Generalization of Formulation
In order to generalize the basic formulation to
consider alternative loading combinations, it is necessary
to redefine some of the structural vectors and matrices
which were introduced in Chapter 2 and to modify the equi-
librium. member force, and compatibility constraints.
3.2.1 Redefinitions
The vector of external joint loads, pt, considers
only one set of loads as defined in Chapter 2. To consider
multiple independent loads, p' for the generalized formula
tion is defined as:
P' = (3-5)
where the subscripts ~, S, etc. indicate the independent
loads.
Since a separate load factor, t, can be associated
with each independent loading, t is redefined as a diagonal
matrix of submatrices which are identity matrices (of order
nf) times constants t , t~, etc. Cl .....
The member force vectors, Ry and Rw' are similar
in form to P' a
(3-6)
with the subvectors Rya' Rwa.' etc. corresponding to the mem
ber yield and working forces for independent loadings a, etc.
It is possible to redefine the vector of joint dis-
placements, u t , similarly to pt, Ry ' and Rw' with each sub
vector corresponding to an independent loading. However,
since it is the joint displacements of the alternative load
ing combinations which are of practical interest to the
designer, u' is redefined as:
(" u' A I
u' B I u' = ) (3-7)
• ( • • I u' j ~)
37
where the subscripts A, B, etc. indicate the alternative
loads.
-The p vector of member reference capacities remains
as previously defined in Section 2.5.
3.2.2 Equilibrium constraints
The equilibrium constraints for a single indepen
dent load case are presented in Eqs. 2-36. In the case of
multiple independent loadings, equations of the same type
are required for each loading:
t taP; (a) ARyo. =
t p' Cb) A Rwa. = a (3-8) t A Ryl3 = t~Ps (c)
t p' Cd) A RwS = S
etc.
Thus, the entire set of equilibrium constraints
can be written aSI
At /'
R ex I ta. P; _"'t_ At
R S ~~:~ ,.} _"'t~ = (3-9) )
• i • • !
t ~''''
and:
38
= (3-10) •
Using At to represent a diagonal matrix of n. sub-1
matrices At (one subrnatrix per independent load), the equi-
librium equations area
At R = tp' (a) - y (3-11) At R = p' (b) - w
where Ry ' Rw' p', and t are as defined in Section 3.2.1.
3.2.3 Member force constraints
Eqs. 2-38 give the member force constraints for a
single independent loading. For multiple independent loads
and alternative loading combinations, it is necessary to
consider combinations of the independent member forces (R , ya
Rwa' etc.), as defined by relationships similar to those of
Eqs. 3-1 and 3-2.
Consider, for example, a structure of two members
with the alternative loadings defined by Eqs. 3-1 and 3-2.
For simplicity, let the dead load and live load be combined
into a single independent loading so thats
r1 • 85 0 l Ay = ~ .40 1.4~ (3-12)
39
The member yield force constraints required for this struc-
ture are:
U r 13 - ~ 1 (1 • 8 5Rya 1 ) > 0 (a) yl yl 1 -
UylrylPl - ~1 (1.40Rya1 + 1.40RyS1 ) > 0 (b) - (3-13) Uy2ry2P2 - f\2 (1. 85Rytl2 ) > 0 (c) -U r -0 - f\2 (1.40nya2 + 1.4oR "2) > 0 (d) y2 y2~2 YP -
where the numerical subscripts refer to the members.
The constraints of Eqs. 3-13 can be expressed more ~
graphically in the form:
> 0
(3-14)
Using the symbol A to denote the matrix which nrey
multiplies the Ry vector in Eq. 3-14, the member yield force
constraints for a general structure can be written in the
form:
Urp-AR >0 y y y y (3-15)
-where rand p are as previously defined, R is as defined y y
by Eq. 3-6, U is as defined in Section 2.5 except that y
there are n subvectors U . per member, and A is a bn x bn. y yl Y Y l
40
matrix of submatrices, each 2s x f, where a typical suby
matrix for member i, independent loading j, and loading
combination q, is:
A = "- II yi (a) Ygh Yqj
where g = n (i - 1) + q (b) y h = b (j - 1) + i (c )
(3-16) i = 1, · . . , b Cd)
j = 1, · . . , n. (e) l.
q = 1, • •• J n (f) y
All other submatrices are zero.
Redefining Uw similarly to Uy (i.e., nw subvectors
U . per member) and A similarly to A J the member working Wl. . W Y
force constraints can be written as:
(3-17)
3.2.4 Compatibility constraints
The compatibility constraints for a single loading
are given by Eq. 2-42. Since u' for multiple loadings is
defined as the joint deflections for the alternative loads
(Eq. 3-7), rather than for the independent loads, it is
necessary to combine the independent member forces of Rw
(Eq. 3-6) in accordance with the alternative loading combina-
tions defined by the ~ matrix. w
Using, for example, the alternative loads defined
by the AW matrix of Eq. 3-12, the compatibility constraints
required are:
pkAUA pkAuB
-- 0
( • 75 Rwa. + • 7 S RWd ) = 0
(a)
(b) (3-18)
where uA and uB are the joint deflections corresponding to
the alternative loads and Rwa and Rw~ are the working member
forces corresponding to the independent loads.
Eqs. 3-18 can be written in the form:
(3-19)
Thus, the compatibility constraints for alternative
loads can be written as!
PKA u' - A * R = 0 w w (3-20)
where pKA denotes a diagonal matrix of nw submatrices pkA and A: is a matrix whose typical submatrix is equal to the
corresponding coefficient of the Aw matrix times an identity
matrix (of order bf).
42
3.2.5 Generalized formulation
A generalized formulation which considers alterna-
tive loading combinations can be assembled from the con-
straints of Eq.s. 3-11, 3-15, 3-17, and 3-20 and the same
objective function (Eq. 2-43) used for the basic formula-
tions
minimize tL P
subject to:
A tR - y
U r p y y - A R y y A tR - w
Uwrwp - A R w w
pkAu' A*R w w -p
= tp'
> 0 -= p'
> 0 -= 0
> 0 -R y' Rw' u' unrestricted
in sign
(a)
(b)
(c)
(d) (3-21)
(e)
( f)
(g)
(h)
Thus, the generalized formulation, like the basic
formulation, is an iterative set of linear programming prob
lems. The final solution vector contains the values of the
member reference capacities, p, the yield and working member
forces, Ry and Rw' corresponding to the independent loads,
and the elastic joint deflections, u', corresponding to the
combined working loads.
In graphical form, the constraints of the general-
43
ized formulation are:
"' ::~~ ('
---P'
~~~ j (3-22)
I A t I I I -I 1
----~----~---_r---Uri -A I I _~_~JL __ ~_L __ %_L __ _
I I 11 I I I A I I 1-' -----,----r----.---
U r' I -A I W W I I W I ----~----~---~---, , -A* I piCA
I I Wi: I I I
= -p >
R ---~ =
Rw >
u' -=
In general, the t matrix in constraint (b) of Eq.
3-21 can be dropped from the generalized formulation since
the load factors are explicitly contained in the A and A y w
matrices.
It should be noted that the basic formulation (Eqs.
2-44) is merely a special case of the generalized formulation
with the number of independent loads and the number of alter
native loading combinations for both yield and working loads
all equal to one (n i = ny = n = 1), and with A = A = 1.00. w y w
Chapter 4
IMPLEMENTATION
A computer program was written to implement the
generalized formulation of Eqs. 3-21. It consists of two
main parts: (1) a POST program with FORTRAN subroutines and
(2) a general-purpose linear programming solver. The pro
gram was wri tten to run as a single job on the IBI'.'i/360
system. One run corresponds to one cycle of the iterative
process. Thus, all intermediate results are available and
changes can be made in the various matrices between itera
tions. The output values of p can be compared to the input
-assumed values of p to determine whether the solution has
converged.
A flow diagram for the program is presented in
Fig. 1. The individual operations within the program are
discussed in the succeeding. paragraphs.
4.1 POST - FORTRAN Program
POST [20J is a computer language very similar to
FORTRAN, but includes implicit matrix operations and dynamic
storage allocation. It is therefore ideally suited for this
study in which large numbers of matrices of various sizes
44
45
must be generated and manipulated.
In the POST program, data which define the struc
tural type (plane frame, plane grid, or space frame), size
(number of members, joints, and supports), geometry (joint
coordinates and~ember incidences), and member group desig
nations are read first (Box 1 in Fig. 1). The program is
written so that a problem may be solved by considering- only
the yield constraints or only the working stress constraints,
as well as by considering the entire set of constraints.
Also, for the purpose of obtaining initial trial values for
p, the compatibility equations may be neglected even when
the other working stress constraints are considered. There
fore, indicators which define the problem type (yield only,
working stress only, or both) and the status o~ the compati
bili ty constraints (enforced 'or not enforced) are also read.
After certain values are initialized (e.g., f is set to 3 for
a plane frame or grid, to 6 for a space frame) in Box 2, the
rest of the data which define the assumed member properties,
yield and working stress limits, independent sets of joint
loads, and alternative loading combinations are read in Box 3.
Information which defines the number and types of constraints
of the L-P problem is then passed to a disk (Box 17), by
means of a FORTRAN subroutine (Box 4), in a format which can
be read by the L-P solver.
46
The objective function, the member IT matrices, and
the submatrices which comprise the A, k, etc. matrices are
computed by considering each member in turn (Boxes 5 through
10). Since much of the information can be passed to the
disk immediately, the amount of storage required is greatly
reduced. However, this sequence of computations causes the
order of the variables (columns) to be changed, since the
L-P matrix (the matrix in Eq. 3-22) must be input to the L-F
solver columnwise. The order in which the variables are
generated by the computer program is: first R, then u', and
finally p, where R combines Ry and Rw in the form:
Ry1J
~!! ~ R =
Ry2 (4-1)
Rw2 i --- ,
~ J where the numerical subscripts indicate the members. As
these computations are made, a FORTRAN subroutine (Box 11)
writes the columns corresponding to R onto the disk. Infor
mation pertaining to the u t and p columns is retained in
primary storage.
In Boxes 12 through 14, the information needed to
47
define the u' columns is computed and written onto the disk,
if compatibility is enforced. Finally, two more FORTRAN
subroutines (Boxes 15 and 16) write information pertaining
to the p columns, including the coefficients of the objective
function, the load vectors, and the bounds (Eq. 3-21(h)) on
the disk.
4.2 Linear Programming Solver
The second part of the program consists of a Mathe
matical Programming System/360 (MPS) routine which solves
the L-P problem defined by the generalized formulation. The
data is read (Box 18) from the disk file (Box 17) which was
created by the POST-FORTRAN part of the program. The problem
is solved in Box 19 and the solution printed i~ Box 20.
The !VIPS language [18, 19J, in addition to its large
capacity (up to 4095 constraints and unlimited number of
variables), allows many options in selecting solution stra
tegies and specifying the type of documentation and output
desired. For example, it is possible to have all the input
data printed out in tabular form and/or to have the input
matrix represented in graphical form.
The MRS output gives the value of the objective
function, the row activities (i.e., an evaluation of the
terms to the left of the = and ~ signs of the constraints),
Chapter 5
ILLUSTRATIVE EXAMPLES
This chapter describes a number of problems which
were solved using the formulations developed in Chapters 2
and 3 and the computer program discussed in Chapter 4.
The first structure considered is a two-story plane
frame. This structure is small enough that it can be solved
for many parameter variations without the computer time re
quired becoming too great. In the second part of the chapter,
a larger structure, a space frame, is considered in order to
determine the applicability of the formulations and computer
program to structures of this type.
In all cases, the results presented consist only of
the value of the objective function and, for the space frame,
the values of the member capacities. The computer output,
however, contained complete information regarding the member
forces, joint deflections, etc.
5.1 Plane Frame Design
The first structure considered is the two-story
plane frame shown in Fig. 2. The beams are divided into two
members each in order to accommodate mid-span loads. The
beams constitute one member group (i.e., are to be made iden-
49
Chapter 5
ILLUSTRATIVE EXAMPLES
This chapter describes a number of problems which
were solved using the formulations developed in Chapters 2
and 3 and the computer program discussed in Chapter 4.
The first structure considered is a two-story plane
frame. This structure is small enough that it can be solved
for many parameter variations without the computer time re
quired becoming too great. In the second part of the chapter,
a larger structure, a space frame, is considered in order to
determine the applicability of the formulations and computer
program to structures of this type.
In all cases, the results presented consist only of
the value of the objective function and, for the space frame,
the values of the member capacities. The computer output,
however, contained complete information regarding the member
forces, joint deflections, etc.
5.1 Plane Frame Design
The first structure considered is the two-story
plane £rame shown in Fig. 2. The beams are divided into two
members each in order to accommodate mid-span loads. The
beams constitute one member group (i.e., are to be made iden-
49
50
tical), the upper story columns a second, and the lower story
columns a third member group. Thus, the structure contains
six free joints, eight members, and three member groups.
As stated in Chapter 2, it is necessary to make
certain assurnntions about the members in deriving the fy' fw'
r , r , and k matrices. First, the member capacities are y w defined as:
-p = A x (a)
(b) (5-1)
(c)
-8hoosing p as A will result in a minimum volume x
design.
Selecting axial force and bending moment as the
member force components which define the yield and working
stress limits, the coefficients which are needed to define
the ryand fw matrices (see Eqs. 2-9 and 2-13) aret
- Z Z ~ mult a = = -X- = A Pult Pult ° A
Y x x (a)
~ = mult 1 (b)
- GaS S Pw mall = =
°aAx = A Pall Pall x
(5-2) (c)
-Pw 1 =
mall (d)
51
where 0 and 0 are the yield and allowable stresses, re-y a
spectively, and Z and S are the plastic and elastic section
moduli.
The coefficients needed to define the ry and rw
matrices are:
- Z ~ m ° = ult -L r = = y P A A x x
(a)
(5-3) -°as Pw mall
rw = = = - A A P x x
(b)
and the values needed to compute Ie are:
A A x 1 = = x -p
!z I z I z = = P A x
(a)
(5-4) (b)
It is assumed that the structure is to be made of
steel wide flange (WF) sections, with the beams 10 inches
deep and the columns (both upper and lower) 8 inches deep_
Selecting an "average" radius of gyration of 4.32
inches and a value of 0.9 for the ratio S/Z, the following
approximate relationships exist for the 10 WF membersz
(4.32)2 = 18.6 (a)
52
S Iz/Ax 18.6 A = ~ s.o = 3.72 (b)
d/2 x (5-5) z
"" .l!.E 4.13 (c ) = A '" .9 x
where d is the depth of the member.
Similarly, for the 8 WF columns, with an assumed
radius of gyration of 3.47 inches:
I z (3.47)2 = 12.0 (a) A ~
x
S 12.0 3.0 (b) (5-6) A ~ --zr.o = x
z l& - 3.33 (c) A ~ .9 -x
Thus, by substituting Eqs. 5-5 and 5-6 and the
values of cr and a into Eqs. 5-2, 5-3, and 5-4, all coef-y w
ficients of f , fw' r , r , and k are known. y y w
5.1.1 Effect of load factor and stress ratio
Initially, the plane frame is designed for the
single loading shown in Fig. 2, with only bending moment
defining the yield and working stress limits, i.e.:
m + --L < 1 (a) - m -ult
(5-7)
+..i< 1 (b) - mall -
53
where my and mw are the bending moments of the yield and
working member forces, respectively.
The ratio pw/Py' denoted r, for this problem is:
-pw
r = _ = n ~y
cr S a = OZ = .9 y
(5-8)
Thus, r is a "stress ratio", since its value depends solely
on the ratio 0 /0 • a y
In this phase of the problem, cr is held constant y
at 36 ksi and r is set to values of 0.45, 0.55, 0.65, and
0.75 by altering the value of Ga. In addition, the load
factor, t, is varied from 1.33 to 2.33 in increments of 0.33.
The results are presented in Figs. 3 and 4.
In Fig. 3, the objective function is plotted against
the load factor for the various values of r. It can be seen
that for low values of t, the curves are horizontal, indi-
cating that only the working stress constraints control the
solution. As the load factor increases, some of the yield
constraints become active, as indicated by the curved sec-
tions, until the inclined straight line is reached. This
line indicates that all the working stress constraints are
inactive and the solution depends only on the yield con-
straints, and the value of the objective function is propor-
tional to the load factor.
54
For a single-story frame, the transition from a
solution based solely on the working stress constraints to
a purely plastic solution was found to be very abrupt. On
the other hand, for a very large structure, the transition
curves can be expected to be much longer than in Fig. 3.
The results for the two-story frame are also pre-
sented in Fig. 4, with the volume plotted against the recip-
rocal of r for the various values of t. In this plot, the
horizontal portions of the curves represent solutions based
on the yield constraints and the inclined straight line indi
cates that only the working stress constraints are active.
5.1.2 Effect of member force interaction
The solutions presented in the previous section
were obtained without considering member force interaction,
with bending moment only considered in the definitions of
the yield and working stress limits.
In order to demonstrate the effect of member force
interaction, the frame with t = 2.00 and r = 0.65 (corre
sponding to 0a = 26 ksi) is redesigned considering the effect
of interaction between bending moment and axial force. The
yield surface defined by Eqs. 2-15 and the working stress
limits of Eq. 2-27 were considered in conjunction with one
another, as well as with the simple limits of Eqs. 5-7. The
volumes of the four structures thus designed are summarized
below:
Yield limits
Eq. 5-7(a)
Eq. 2-15
Working stress limits
Eq. 5-7 (b)
4578 in3
4641 in3
Eq. 2-27
4740 in3
4769 in3
55
As could be predicted, considering interaction in
either the yield or working stress constraints increased the
volume of the structure, and the largest volume was obtained
when interaction was considered in both.
5.1.3 Effect of alternative loadings
For the designs discussed in the previous two sec-
tions, the structure was considered to be acted upon by a
single set of loads, as shown in Fig. 2. In order to demon-
strate the effect of alternative loadings, the same plane
frame is designed for the independent loads shown in Fig. 5.
Thus, the vertical forces represent the dead plus live load,
while the .horizontal forces represent the wind load.
The alternative loading combinations to be consid
ered are defined as follows:
(5-9)
Because the member group designations (Fig. 2) dictate that
the frame be symmetric, reversibility of the. wind load need
56
not be considered.
'rhe frame is first designed wi thout member force
interaction, using the yield and working stress limits de-
fined by Eqs. 5-7. Then, it is designed using the inter-
action relations of Eqs. 2-15 and 2-27. The results are:
without interaction, volume
with interaction, volume
= 3520 in3
= 3689 in3
For the purpose of comparing these results to those
from the previous section, it should be noted that the pre-
vious designs can also be obtained in terms of the independ-
ent loads of Fig. 5, by setting ~y = [2.00 2.00l and A = - w [1.00 1.00J. Thus, considering the less severe alternative
loads of Eq. 5-9 reduced the volume of the frame without
interaction from 4578 to 3520 in3 , and the volume of the
frame with interaction from 4769 to 3689 in3•
5.2 Space Frame Design
In order to determine the applicability of the
formulations and computer program to larger problems, the
staircase of Fig. 6 is considered. The structure which
supports the staircase (Fig. 7) lies along the centerline
and has fixed supports at the upper and lower landings, with
no intermediate supports of any kind. It is designed as a
space frame (f = 6) and made of 12-inch square structural
57
tUbing with 0y = 36 ksi and 0a = 22 ksi.
The design live load is taken as 100 psf and the
dead load as 20 psf. Since loads are restricted to concen-
trated loads at the joints, the dead load plus live load is
expressed as the set of 14 concentrated forces shown in
Fig. 8 and the structure is divided into 15 segments (mem
bers). The load factor, t, is taken as 1.85.
The member force components which enter into the
definitions of the yield and working stress limits are the
torsional moment (m1 ) and the bending moments (m2 and m3).
The member capacities are defined as:
p = A x (a)
- (b) (5-10) py = m2ult = m3ult - (c) Pw = m2a11 = m)all
Selecting 4.66 inches as a typical radius of gyra-
tion and computing the torsional constant, Ix' the values
needed in the K matrix are:
A = 1 (a) x
Ix Ix
~ 35.9 (b) (5-11) = Ax
I lz I z (4.66)2 = 21.7 (c) = = A~ y x
The ultimate and allowable values of m1, m2 , and
m3 are determined by computing their theoretical values [21J
and reducing these by 10% (the same reduction required to
58
bring theoretical values of Ax' I z ' etc. into agreement with
the handbook [lJ values) to obtain:
m1u1t ~ 3.46 ° A (a) y x
m2u1t = m3u1 t ~ 4. 50 ° Y Ax (b) (5-12)
mlal1 ~ 3.46 ° A (c) a x
m2al1 = ffi3a11 ~ 3. 60 0a Ax (d)
Therefore, the coefficients needed for the ryl fw l
r y ' and rw matrices are:
P:i 4.50 a A = Y.. x = 1.30 m1u1t :3 .46 a A y x
(a)
P:y: = P:y: = 1.00 m2u1t m3u1t (b)
- 3.60 0a Ax Pw 1.04 = = m1a11 3.46 O"a Ax (c)
-Pw = = 1.00 m3a11 (d)
P 4.50 ° A r =:..x = Y.. x = 162.0 (e)
y p Ax
3.60 0a Ax
A x
= 79.2 (f)
(5-13)
The yield limits are defined by a fictitious unit
cube in ml - m2 - m3 space J i.e.:
59
+ m1
< 1 - m1ult - (a)
+ m2 < 1 - m2u1t - (b) (5-14)
+ m:2
< 1 - m3ult -(c)
'and the working stress limits are defined similarly (i.e.,
substitute "all" for "ult" in Eqs. 5-14).
Two design cases are considered: (1) all the mem
bers are identical (i.e., comprise a single member group),
and (2) the horizontal members comprise one member group and
the inclined members a second.
The resulting designs are:
1 memo ~ou£ 2 memo grou"Os
horizontal members 15.62 in 2 16.74 . 2 area, In
inclined members 15.62 in 2 5.76 . 2 area, In
total volume 8597 in3 6597 in3
It should be noted that the area of 5.76 in2 com-
puted for the inclined members in the second case is consid-
erably smaller than the area of the smallest 12-inch square
tubing listed in the AISC handbook [lJ.
60
5.3 Computer Program Efficiency
In solving the problems described in the preceding
sections, there was a wide range of computer times required
to obtain solutions. For all problems, a one per cent con
vergence criterion was used, i.e., the solutions were it
erated until the output values, p, were within one per cent
of the input values, p. The plane frames designed considering only bending
moment in the definitions of the yield and working stress
limits (Section 5.1.1) required approximately 10 seconds each
of IBM 360/75 processor time (5 for POST-FORTRAN + 5 for MPS)
per iteration. On the average, five iterations were needed
for the solutions to converge.
Consideration of member force interaction (Section
5.1.2) increased the computer time only slightly to an aver
age of approximately 12 seconds (5 + 7) per iteration. A
drop in the average number of iterations to four can be at
tributed to a better selection of initial values for p.
The inclusion of alternative loadings (Section
5.1.3) caused a large increase in computer time. For the
structure with no member force interaction, iterations of 30
seconds (6 + 24) each were required. For the frame with in
teraction, each iteration required 102 seconds (7 + 95) of
computer time. In both cases, five iterations were needed
61
for convergence.
The space frame designed with all members the same
required 95 seconds (11 + 84). Because all members are iden-
tical, no cycling was required. The frame with two member
groups required four iterations of approximately 106 seconds
(11 + 95) each for convergence.
It can be seen from the above that the time required
for the POST-FORTRAN part of the program increased very slowly
as the size of the problems increased, whereas the IvlPS time
increased very rapidly.
In addition to the structures presented in the pre
ceding sections, it was attempted to obtain solutions to some
larger problems. Solutions were not obtained for one of two
reasonss (1) the structure was so highly constrained that
the MPS L-P solver could not reach a feasible solution, or
(2) the computer time required became so great that the cost
became prohibitive.
As an example of the former, it was attempted to
design the staircase structure of Section 5.2 considering the
yield limits defined by a unit tetrahedron, i.e.:
+ + + < 1 (5-15)
Even for the case with all members identical, a feasible
62
solution could not be obtained by the MPS program.
An example of (2) above occurred when it was at
tempted to design the staircase for alternative loads, con
sidering one reversible wind loading in addition to the dead
plus live load. Allowing the MPS program to run for twenty
minutes, no solution was obtained.
Chapter 6
CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY
6.1 Conclusions
The formulation developed in this study represents
a unified approach to the optimum design of framed struc
tures. The solution process takes the form of an iterative
set of linear programming problems.
The formulation presents the following advantagesz
(a) It is in a general form and can be applied to
framed structures of all types. While developed for frames
and grids, with minor modifications it can also be applied
to the design of trusses.
(b) Both ultimate capacity and worki~g stress
limitations are considered.
(c) The definitions of yield and working stress
limits include the effect of member force interaction.
(d) Multiple loads and alternative loading combi
nations are considered.
(e) Groups of members may be specified as being
identical.
While it is necessary to make various assumptions
about the member sizes and shapes, this restriction does not
appear to be unreasonable. Due to the iterative nature of
64
the solution process, the initial assumptions may be modi-
fied as the solution progresses.
Satisfactory results were obtained for several sam-
pIe problems, as described in Chapter 5. Eowever, solutions
could not be obtained for some larger problems. Apparently,
general-purpose L-P programs, such as MPS, are not well
suited for solving problems of the type defined by the formu
lation which has been developed.
6.2 Suggestions for Further Study
There are many ways in which the formulation and
implementation developed in this study can be extended or
improved %
(a) The objective function used in this study is
somewhat restrictive. It should be generalized to include
functions of all the variables (Ry ' Rw' andu', as well as
p). It is possible, for example, to express the cost of
connections as a function of the member forces [24J.
(b) It seems feasible to take into account second-
ary effects due to displacement. The iterative solution
process is ideally suited for this type of computation.
(c) Buckling constraints for individual members
can be considered either by redefining some of the matrices
or by adding new constraints.
(d) In this study, all loads are assumed to be
known. It is possible, however, to consider loads which
depend upon the member sizes (e.g., dead load due to the
weight of the members) in the following manner:
The loads of any independent loading, P', are
taken as:
P' = pI + P' 1 2 (6-1)
where Pi is known and P2 can be expressed as a function of
the member design capacities, in the form:
P2 = G p (6-2)
where G is a matrix of constants. The equilibrium equations
are:
(6-))
and can be rewritten by substituting Eqs. 6-1 and 6-2 into
Eq. 6-3 and rearranging terms to give:
(6-4)
(e) The formulation, though developed for design,
can be modified for use in analysis. For analysis, the sizes
of the members, as expressed by p, are known. The loads can
be expressed as a vector of constants, P', times an "analysis
66
factor," F. Thus, the equilibrium equations can be written
as:
and the objective function is to maximize F. Since D is
known, the compatibility constraints can be left in the form
of EQ. 2-34 and iteration is not necessary. For multiple
loads, the objective function can be defined as a linear
function of the several analysis factors F1 , F2 , ••• , Fni
•
(f) Imposing deflection constraints can cause spe
cial computational difficulties [17J. While it would appear
that such constraints can be incorporated into the formulation
very simply by adding u'<u' to the constraints of Eqs. 2-44 - ~x
and 3-21, this approach was attempted for a very simple prob-
lem and it was found that the solution oscillated rather than
converged. This phenomenon can be attributed to the fact that
the stiffness matrix, as expressed by the product PK, is a
constant during any iteration and does not take into account
-changes in the member capacities, p.
(g) It has been noted previously that the general
purpose L-P solver was poorly suited for this study. There
fore, special algorithms, such as the decomposition method
[4J, should be investigated for possible use in solving the
L-P problem.
67
(h) As an alternative to, or in addition to, (g)
above, the possibility of employing a partially non-linear
formulation should be investigated. If the values pare
used in the compatibility constraints, rather than p, then
Eq. 2-42 will appear as:
pkAu t - R = 0 w (6-6)
where the first term is non-linear since the products of
variables p and u' appear.
Despite the difficulties which can be anticipated
in implementing such a formulation, it does present two dis-
tinct advantages over the purely linear formulation: (1) so-
lutions can be obtained in one step, without iteration, and
(2) this type of formulation appears to proviqe an answer to
the problem of including deflection constraints in the formu-
lation (see (f) above).
Start
Initialize ob j. func.,
f, etc.
3"- I - . \Read load';::7 \ member - I
4
properties I etc. I
Define types o~
constraints
KEY ----e. Program flow
- - - • Data flow
c=:) Terminal
c=J Compu ta tion
D Read data
c=) FORTRAN subroutine
C~ Loop start
o Loop end
<> Decision
D Printed output
o Disk file
\) Offpage connector
5 Do to a.
--______ ~w\ for \~ = 1 to b
6 __ .1 __ Compute
L., T., H., 1 1 1
(TH)i
7 -- In.crem~nt 1 obJ. func. i
by L. I
8
9
10
11
0-
1 I
yes
co~put~ k. I
1 I
Compute
IIyi ' I\ri
Figure 1: Computer Program Flow Diagram
68
Compute
15k
14 r--------, columns of L-P matrix
1 r--=----L.--. p columns of L-P
atrix (incl ob j. func.)
1~ _______ _
End POST-FORTRAN
-----"
Figure 1 (cont.)
C Start MPS
18 ....-_---' __ .....,
19 .-----&.----.
Solve L-P problem
20 r-----"------.
Print results
End
70
2Sk
10k > CD \V CD
10 WF '------r i I~ ~
0 ~ :;::
® ro co 2Sk
1 :
CD CD N
10k > C"-
10 WF @J
N
~ rx.. G) 3: ~ a> co <l)
rrtrr rr.'m~ ~ .. 2 @ 72" ----- - .. -~-.~
CD, 0, G) = member groups
Figure 2: Two-Story Plane Frame
71
7000
r = .45 • • • •
6000
II ......... C") 5000 ~
-M
---Q)
S ::s r-i 0 :>
4000 .75
3000--------~~------~--------~--------~---------
1.33 1.67 2.00 2.33
load factor (t)
Figure 3: Volume vs. Load Factor
(V"'\
~ .r-f
Q)
s ~
r-f 0 :::-
72
7000
6000
t = 2.33
5000
4000
3000~------~~----~----------~------------~---1
.75 1
:b3 1
.55
reciprocal of stress ratio (l/r)
Figure 4s Volume vs. Reciprocal of Stress Ratio
1 :43
! .-m-;
(a) Dead plus live load
10~ >-----------------~
10l): >
iT/Ti i7TTi
(b) Wind load
Figure 5: Independent Loads on Plane Frame
73
76
y
2.04
2.10 1.98 1.92 X 1,.98 2.10
2.16
1.92 /
2.04/ 1.98 ':J
2.10
z
Figure 8: staircase Dead Plus Live Load (kips)
A
A x A x Ai;
b
d
E
f
H. ~
I
Ix
I y ' I z
Ix' I y '
k
K
L
L. , f 1
m
m1
m2 , m)
n
I z
APPENDIX
NOTATION
branch-node incidence matrix
member cross-sectional area
scaled member area
diagonal matrix of n. submatrices At 1
number of members
depth of member
modulus of elasticity
number of degrees of freedom
translation matrix for member i
identity matrix
torsional constant
moments of inertia about member y- and z-axes
scaled values of Ix' I y ' and I z unassembled stiffness matrix
scaled stiffness matrix
vector of member lengths
member length
bending moment
torsional moment
bending moments
number of free joints
77
P'
p' p' y' w
p
-p
= P
piCA
R
Ry ' Rw
R' R' y' w
R' R' y' w
R*, R* Y w -* r\ Ry ' n: r
r y , rw
S
T. 1.
t
u'
78
number of independent loads
number of yield and working alternative loading combinations
vector of joint loads
vectors of yield and working joint loads
axial force
vector of member reference capacities
member yield and working stress capacities
matrix containing trial member capacities
diagonal matrix of nw submatrices pkA vector of member forces
vectors of yield and working member forces
vectors of forces at both ends of the members
vectors of force components used to define yield and working stress limits
vectors of normalized member forces
vectors of combined member forces
stress ratio pwipy
diagonal matrices of ratios PyiP and pwip elastic section modulus
number of linear segments defining yield and working stress limits
rotation matrix for member i
load factors
matrices used in member force constraints
vector of elastic joint displacements
v
z
fy' fw
Yl' Y2
~y' !J. w 8, y 8 w
9 1 , 92
A y ' Aw
A y' AW
A* w
IIy ' l\;
cry' O"a
Q
Subscri:Qts
A, B
A, B, ... a, all
i
u, ult
w
y
vector of elastic member distortions
plastic section modulus
normalization matrices
coefficients of the fy matrix
extractor matrices
79
matrices defining linear combinations of normalized forces
constants defining yield surface segment
matrices used in member force constraints for alternative loads
matrices which define alternative loading combinations for yield and working loads
matrix used in compatibility constraints for alternative loads
matrix products 8yr~yQ and ewfw~Q
yield and allowable stresses
matrix which transforms vector of member Bend forces into vector of forces at both ends of the members
positive and negative member ends
combined working loads
allowable
typical member i
ultimate
working stress
yield
1 •
2.
4.
5.
6.
8.
10.
11.
LIST OF REFERENCES
American Institute of Steel Construction, Manual of Steel Construction, AISC, 1963.
Brown, D. M. and A. H.-S. Ang, "Structural Optimization by Nonlinear Programming," Journal of the Structural Division, ASCE, Vol. 92, No. ST6, December 1966.
Dantzig, G. B., Linear Programming and Extensions, Princeton University Press, 1963.
Dantzig, G. B. and P. Wolfe, etA Decomposition Principle for Linear Programs," Operations Research, Vol. 8, No.1, January-February 1960.
Fenves, S. J. and F. H. Branin, "Network-Topological Formulation of Structural Analysis," Journal of the structural Division, ASCE, Vol. 89, No. ST4, August 1963.
FORTRAN IV Language (C28-6515-6), IBM Systems Library, IBM, 1966.
Foulkes, J., "Minimum Weight Design and the Theory of Plastic Collapse," Quarterly of Applied Mathematics, Vol. 10, No.4, January 1953. .
Gonzales C., A. and S. J. Fenves, "A Network-Topological Formulation of the Analysis and Design of Rigid-Plastic Framed Structures," Civil Engineering Studies, structural Research Series No. 339, University of Illinois, September 1968.
Greenberg, H.J. and W. Prager, "Limit Design of Beams and Frames," Proceedings of the American Society of Civil Engineers, Vol. 77, Separate No. 59, February 1951.
Hadley, G., Linear Programming, Addison-Wesley Publishing Company, 1962.
Hall, A. S. and R. W. Woodhead, Frame Analysis, John Wiley and Sons, 1961.
81
82
12. Heyman, J., "Plastic Design of Beams and Frames for Minimum Material Consumption," Quarterly of Applied Mathematics, Vol. 8, No.4, January 1951.
13. Hodge, P. G., Plastic Analysis of structures, McGrawHill Book Company, 1959.
14. Klein, B., "Direct Use of Extremal Principles in Solving Certain Optimizing Problems Involving Inequalities," Operations Research, Vol. 3, No.2, May 1955.
15. Livesley, R. Ke, "The Automatic Design of Structural Frames, tI Quarterly Journal of IVlechanics and Applied Mathematics, Vol. 9, Part 3, September 1956.
16. Livesley, R. K., Matrix Methods of structural Analysis, Pergamon Press, 1964.
17. Livesley, R. K., "Automatic Design of Guyed Masts Subject to Deflection Gonstraints,1t International Journal for Numerical Methods in Engineering, Vol.·2, No.1, JanuaryNIarch, 1970.
18.
19.
20. Melin, J. 1,,/., "POST: Problem-Oriented Subroutine Translator," Journal of the Structural Division, ASCE, Vol. 92, No. ST6, December 1966.
21. Morris, G. A. and S. J. Fenves, "A General Procedure for the Analysis of Elastic and Plastic Frameworks," Civil Engineering Studies, Structural Research Series No. 325, University of Illinois, August 1967.
22. Moses, F., "Optimum Structural Design Using Linear Programming," Journal of the Structural Division, ASCE, Vol. 90, No. ST6, December 1964.
23. Prager, W., "Minimum-Weight Design of a Portal Frame," Journal of the Engineering Mechanics Division, ASCE, Vol. 82, No. EM4, October 1956.
24.
25.
26.
28.
30.
8)
Ridha, R. A. and R. N. Wright, "Minimum Cost Design of Frames," Journal of the Structural Division, ASCE Vol. 93, No. ST4, August 1967.
Schmit, L. A., "Structural Synthesis 1959 - 1969, a Decade of Progress," Survey paper presented at the Japan-U. S. Seminar on Matrix Methods of Structural Analysis and Design, Tokyo, August 1969.
Schmit, L. A., T. P. Kicher, and W. M. Morrow, "Structural Synthesis Capability for Integrally Stiffened Waffle Plates," AL~ Journal, Vol. 1, No. 12, December 1963.
Shames, I. H., I'.:Iechanics of Deformable Solids, PrenticeHall, 1964.
Sheu, C. Y. and W. Prager, "Recent Developments in Optimal Structural Design," Applied Mechanics Reviews, Vol. 21, No. 10, October 1968.
Smith, J. O. and O. M. 'Sidebottom, Inelastic Behavior of Load-Carryinp: Members, John Wiley & Sons, 1965.
Wright, R. N., COSD (Constrained Optimum Stee"Dest Descent), Department of civil Engineering, university of Illinois, December 1967.
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Navy (cont'd)
Naval Air Systems Command Dept. of the Navy Wash ington, D. C. 20360 Attn: NAIR 03 Res. & Technology
320 Aero. & Structures 5320 Structures
604 Tech. Library
Naval Facil ities Engineering Command
Dept. of the Navy Washington, D.C. 20360 Attn: NFAC 03 Res. & Development
04 Engineering & Design 04128 Tech. Library
Naval Ship Systems Command Dept. of the Navy Washington, D.C. 20360 Attn: NSHIP 031 Ch. Scientists for R&D
0342 Ship Mats. & Structs. 037 Ship Silencing Div. 052 Shock & Blast Coord.
2052 Tech. Library
Naval Ship Engineering Center Main Navy Building Washington, D.C. 20360 Attn: NSEC 6100 Ship Sys. Engr. & Des. Dept.
61020 Computerated Ship Des. 6105 Ship Protection 6110 Ship Concept Design 6 1 20 H u 1 1 D i v. - J. Na c h t she i m 61200 Hull Div. - J. Vasta 6132 Hull Structs. - (4)
Naval Ordnance Systems Command Dept. of the Navy Washington, D.C. 20360 Attn: NORD 03 Res. & Technology
035 Weapons Dynamics 9132 Tech. Library
Air Force
Commande r WADD Wright-Patterson Air Force Base Dayton, Ohio 45433 Attn: Code WWRMDD
AFFDL (FDDS)
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Wright-Patterson AFB (cont'd) Attn: Structures Division
AFLC (MCEEA) Code WWRC AFML (MAAM)
Commander Chief, Appl ied Mechanics Group U. S. Air Force Inst. of Tech. Wright-Patterson Air Force Base Dayton, Ohio 45433
Chief, Civil Engineering Branch WLRC, Research Divis ion Air Force Weapons Laboratory Kirtland AFB, New Mexico 87117
Air Force Office of Scientific Res. 1400 Wilson Blvd. Arlington, Virginia 22209 Attn: Mechs. Div.
Structures Research Division National Aeronautics & Space Admin. Langley Research Center Langley Station Hampton, Virginia 23365 Attn: Mr. R. R. He1denfels, Chief
National Aeronautic & Space Admin. Associate Administrator for Advanced
Research & Technology Washington, D.C. 20546
Scientific & Tech. Info. Facil ity NASA Rep res e nt a t i ve (S-A KID L) P. O. Box 5700 Bethesda, Maryland 20014
National Aeronautic & Space Admin. Code RV-2 Washington, D.C. 20546
Other Government Activities
Commandant Chief, Testing & Development Div. U. S. Coast Guard 1300 E Street, N. W. Washington, D. C. 20226
Director Marine Corps Landing Force Dev. Cen. Marine Corps Schools Quantico, Virginia 22134
Director National Bureau of Standards Washington, D. C. 20234 Attn: Mr. B. L. Wilson, EM 219
National Science Foundation Engineering Division Washington, D. C. 20550
Science & Tech. Division Library of Congress Washington, D. C. 20540
Director STBS Defense Atomic Support Agency Washington, D. C. 20350
Commander Field Command Defense Atomic Support Agency Sandia Base Albuquerque, New Mexico 87115
Chief, Defense Atomic Support Agency Blast & Shock Division The Pentagon Washington, D. C. 20301
Director, Defense Research & Engr. Technical Library Room 3C-128 The Pentagon Washington, D. C. 20301
Chief, Airframe & Equipment Branch FS-120 Office of Fl ight Standards Federal Aviation Agency Washington, D. C. 20553
Chief, Division of Ship Design Maritime Administration Washington, D. C. 20235
4
Deputy Chief, Office of Ship Constr. Maritime Administration Washington, D. C. 20235 Attn: Mr. U. L. Russo
Mr. Milton Shaw, Director Div. of Reactor Devel. & Technology Atomic Energy Commission Germantown, Md. 20767
Ship Hull Research Committee National Research Council National Academy of Sciences 2101 Constitution Avenue Washington, D. C. 20418 Attn: Mr. A. R. Lytle
PART 2 - CONTRACTORS AND OTHER TECHNICAL COLLABORATORS
Universities
Professor J. R. Rice Division of Engineering Brown University Providence, Rhode Island 02912
Dr. J. Tinsley Oden Dept. of Engr. Mechs. University of Alabama Hunstvi11e, Alabama
Professor M. E. Gurtin Dept. of Mathematics Carnegie Institute of Technology Pittsburgh, Pennsylvania 15213
Professor R. S. Riv1 in Center for the App1 ication of Mathematics Lehigh University Bethlehem, Pennsylvania 18015
Professor Jul ius Mik10witz Division of Engr. & Appl ied Sciences Cal ifornia Institute of Technology Pasadena, California 91109
Professor George Sih Department of Mechanics Lehigh University Bethlehem, Pennsylvania 18015
Dr. Harold Liebowitz, Dean School of Engr. & App1 ied Sciences George Washington University 725 23rd Street Washington, D. C. 20006
Universities (cont1d)
Professor El i Sternberg Div. of Engr. & Appl ied Sciences Cal ifornia Institute of Technulogy Pasadena~ Cal ifornia 91109
Professor Paul M. Naghdi Div. of Appl ied Mechanics Etch ever ry Ha 1 1 University of Cal ifornia Berkeley~ Cal ifornia 94720
Professor Wm. Prager Reve 1 1 e Co 11 ege University of Cal ifornia P. O. Box 109 La Jolla~ Cal ifornia 92037
Professor J. Baltrukonis Mechanics Division The Catholic Univ. of America Washington, D. C. 20017
Professor A. J. Durelli Mechanics Division The Catho1 ic Univ. of America Washington, D. C. 20017
Professor H. H. Bleich Department of Civil Engineering Columbia University Amsterdan & 120th Street New York, New York 10027
Professor R. D. Mindl in Department of Civil Engineering Columbia University S. W. Mudd Building New York, New York 10027
Professor F. L. DiMaggio Department of Civil Engineering Columbia University 616 Mudd Building New York, New York 10027
Professor A. M. Freudenthal Department of Civil Engr. &
Engr. Mechs. Columbia University New York, New York 10027
5
Professor B. A. Boley Dept. of Theor, & Appl. Mechs. Cornell University Ithaca, New York 14850
Professor P. G. Hodge Department of Mechanics III inois Institute of Technology Chicago, III inois 60616
Dr. D. C. D rue ke r Dean of Engineering University of Illinois Urbana, III inois 61803
Professor N. M. Newmark Dept. of Civil Engineering University of Illinois Urbana, III inoi s 61803
Professor A. R. Robinson Dept. of Civil Engineering University of III inois Urbana, III inois 61803
Professor S. Taira Department of Engineering Kyoto University Kyoto, Japan
Professor James Mar Massachusetts Inst. of Tech. Rm. 33-318 Dept. of Aerospace & Astro. 77 Massachusetts Avenue Cambridge, Massachusetts 02139
Professor E. Reissner Dept. of Mathematics Massachusetts Inst. of Tech. Cambridge, Massachusetts 02139
Professor William A. Nash Dept. of Mechs. & Aerospace Engr. University of Massachusetts Amherst, Massachusetts 01002
Library (Code 0384) U. S. Naval Postgraduate School Monterey, Cal ifornia 93940
Professor Arnold Allentuch Department of Mechanical Engineering Newark College of Engineering 323 High Street Newark, New Jersey 07102
Universities (cont'd)
Professor E. L. Reiss Courant Inst. of Math. Sciences New York University 4 Washington Place New York, New York 10003
Professor Bernard W. Shaffer School of Engrg. & Science New York University University Heights New York, New York 10453
Dr. Francis Cozzarel1 i Div. of Interdiscipl inary
Studies and Research School of Engineering State University of New York Buffalo, New York 14214
Professor R. A. Douglas Dept. of Engr. Mechs. North Carol ina St. Univ. Raleigh, North Carol ina 27606
Dr. George Herrmann The Technological Institute Northwestern University Evanston, Illinois 60201
Professor J. D. Achenbach Technological Institute Northwestern University Evanston, Illinois 60201
Director, Ordnance Research Lab. Pennsylvania State University P. O. Box 30 State College, Pennsylvania 16801
Professor Eugene J. Skudrzyk Department of Physics Ordnance Research Lab. Pennsylvania State University P.O. Box 30 State College, Pennsylvania 16801
Dean Oscar Baguio Assoc. of Struc. Engr. of
the Ph j 1 i pp i nes University of Phil ippines Manila, Philippines
6
Professor J. Kempner Dept. of Aero. Engr. & Appl ied Mech. Polytechnic Institute of Brooklyn 333 Jay Street Brooklyn, New York 11201
Professor J. Klosner Polytechnic Institute of Brooklyn 333 Jay Street Brooklyn, New York 11201
Professor A. C. Eringen Dept. of Aerospace & Mech. Sciences Princeton University Princeton, New Jersey 08540
Dr. S. L. Koh School of Aero., Astro. & Engr. Science Purdue University Lafayette, Indiana 47907
Professor R. A. Schapery Purdue University Lafayette, Indiana 47907
Professor E. H. Lee Div. of Engr. Mechanics Stanford University Stanford, Cal ifornia 94305
Dr. Nicholas J. Hoff Dept. of Aero. & Astro. Stanford University Stanford, Cal ifornia 94305
Professor Max An1 iker Dept. of Aero. & Astro. Stanford University Stanford, California 94305
Professor J. N. Goodier Div. of Engr. Mechanics Stanford University Stanford, Cal ifornia 94305
Professor H. W. Liu Dept. of Chemical Engr. & Metal. Syracuse University Syracuse, New York 13210
Professor Markus Reiner Technion R&D Foundation Haifa, Israel
Universities (cont'd)
Professor Tsuyoshi Hayashi Department of Aeronautics Faculty of Engineering University of Tokyo BUNKYO- KU Tokyo, Japan
Professor J. E. Fitzgerald, Ch. Dept. of Civil Engineering Univers ity of Utah Sa 1 t La ke Cit y, Ut a h 84 11 2
Professor R. J. H. Bollard Chairman, Aeronautical Engr. Dept. 207 Guggenheim Hall University of Washington Seattle, Washington 98105
Professor Albert S. Kobayashi Dept. of Mechanical Engr. University of Washington Seattle, Washington 98105
Off ice r- i n- Charge Post Graduate School for Naval Off. Webb Institute of Naval Arch. Crescent Beach Road, Glen Cove Long Is 1 and, New York 11542
Librarian Webb Institute of Naval Arch. Crescent Beach Road, Glen Cove Long Island, New York 11542
Sol id Rocket Struc. Integrity Cen. Dept. of Mechanical Engr. Professor F. Wagner University of Utah Salt Lake City, Utah 84112
Dr. Daniel Frederick Dept. of Engr. Mechs. Virginia Polytechnic Inst. B1acksburgh, Virginia
Industry and Research Institutes
Dr. James H. Wiegand Senior Dept. 4720, Bldg. 0525 Ball istics & Mech. Properties Lab. Aerojet-General Corporation P. O. Box 1947 Sacramento, Cal ifornia 95809
-7-
Mr. Ca r 1 E. Ha r t b owe r Dept. 4620, Bldg. 2019 A2 Aeroject-General Corporation P. O. Box 1947 Sacramento, Cal ifornia 95809
Mr. J. S. Wise Aerospace Corporation P. O. Box 1300 San Bernardino, Cal ifornia 92402
Dr. Vi to Sa 1 e r no Appl ied Technology Assoc., Inc. 29 Church Street Ramsey, New Jersey 07446
Library Services Department Report Section, Bldg. 14-14 Argonne National Laboratory 9700 S. Cass Avenue Argonne, III inois 60440
Dr. M. C. Junger Cambridge Acoustical Associates 129 Mount Auburn Street Cambridge, Massachusetts 02138
Dr. F. R. Sc hwa r z 1 Central Laboratory T.N.O. Schoenmakerstraat 97 Delft, The Netherlands
Research and Development Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340
Supervisor of Shipbuilding, USN, and Naval Insp. of Ordnance
Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340
Dr. L. H. C he n Basic Engineering Electric Boat Division General Dynamics Corporation Groton, Connecticut 06340
Dr. Wendt Valley Forge Space Technology Cen. General Electric Co. Valley Forge, Pennsylvania 10481
Dr. Joshua E. Greenspon J. G. Engr. Research Associates 383 1 Me n 1 0 D r i ve Baltimore, Maryland 21215
Industry & Research Inst. (cont'd)
Dr. Wa 1 t. D. Pi 1 ke y lIT Research Institute lOWe s t 35 S t r e e t Ch i cago, III i no is 60616
Library Newport News Shipbuilding & Dry Dock Company
Newport News, Virginia 23607
Mr. J. I. Gonzalez Engr. Mechs. Lab. Ma r tin Ma r jet t a MP - 233 P. O. Box 5837 Orlando, Florida 32805
Dr. E. A. Ale xa nde r Research Dept. Rocketdyne D.W., NAA 6633 Canoga Avenue Canoga Park, Cal ifornia 91304
Mr. Cezar P. Nuguid Deputy Commissioner Phil ippine Atomic Energy Commission Ma nil a, Phi 1 i p pin e s
Dr. M. L . Me r r itt Divis ion 5412 Sandia Corporation Sand i a Base Albuquerque, New Mexico 87115
Director Ship Research Institute Ministry of Transportation 700, SHINKAWA M i taka Tokyo, Japa n
Dr. H. N. Abramson Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206
Dr. R. C. DeHart Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206
Dr. M. L. Baron Paul Weidlinger, Consulting Engr. 777 Third Ave - 22nd Floor New York, New York 10017
Mr. Roger Weiss High Temp. Structs. & Materials Appl ied Phys ics Lab. 8621 Georgia Ave. Silver Spring, Md.
Mr. Will iam Caywood Code BBE App 1 i ed Phys i cs Lab. 8621 Georgia Ave. Silver Spring, Md.
Mr. M. J. Berg Engineering Mechs. Lab. Bldg. R-1, Rm. l104A TRW Sys tems 1 Space Pa rk Redondo Beach, Cal ifornia 90278
-8-
Unclassified Security etas sification
DOCUMENT CONTROL DATA· R&D
L .5,'curiry cJassi fication of title, body of abs/wct and indexinl5 annolilrion must be entered when tile overall report is cJ,u;sified) '. -~:;';";";";"";;;":;";:';"';:":';';;';~~~~~~~~~ _____ --~-~--t , 1 ORIGINATING ACTIVI TY (Corporate author) 28. REPORT SECURITY CLASSIFICATION
University of Illinois at Urbana-Champaign Department of Civil Engineering
Unclassified 2b. GROUP
3. REPORT TITLE
A GEf\TERAL FORMULATION FOR THE OPTIIVIUM DESIGN OF FRAMED STRUCTURES
4. DESCRIPTIVE NOTES (Type of report and.inclusive dates)
Progress: April 1969 - July 1970 5· AU THOR(S) (First name, middle initial, last name)
William J. McCutcheon Steven J. Fenves
6. REPORT OATE
August 1970 8a. CONTRACT OR GRANT NO.
N 0014-67-A-0305-0010 b. PROJECT NO.
Navy A-0305~0010
78. TOTAL NO. OF PAGES
83 9a. ORIGINATOR'S REPORT NUMBER(S)
Structural Research Series No. SRS 362
c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assi~ned this report)
d.
10. DISTRIBUTION STATEMENT
Qualified requesters may obtain copies of this report from DDC.
11. SUPPLEMENTARY NOTES
13. ABSTRACT
12. SPONSO RING MI LI TARY AC TI VI TY
Office of Naval Research Structural Mechanics Branch Department of the Navy
A unified approach to the optimum design of framed structures based on a general matrix formulation is presented. Both elastic' behavior, in the form'of working stress limitations, and ultimate capacity are considered.
The formulation is presented in the form of an iterative set of linear programming problems. The design .variables relating to elastic and ultimate behavior are linearly related and the objective function is expressed as a linear function of the design variables. By making assumptions about the member properties and sizes, the constraints are also expressed in linear form.
The formulation is initially developed for a structure acted upon by a single set of external loads, and is subsequently generalized to include multiple loads and alternative loading combinations. Yield and working stress limits, which are defined to include the effect of stress interaction, are specified only at member ends. Therefore, loads are restricted to concentrated forces and couples at the joints.
A computer program was written to implement the formulation and was used to solve several sample problems. However, solutions could not be obtained for some large, highly constrained problems. It is expected that the implementation can be improved by using special linear programming algorithms or by employing a partially non-linear formulation.
DO lFN°o~~\51473 (PAGE 1) -Unclassified
SIN 0101-807-6811 Security Classification A-31~08