I. INTRODUCTION
A. GENERAL REMARKS
Normally a structure is designed to serve
certain functional requirements with an ade-
quate factor of safety against failure and,
usually, with a minimum of cost. Functional
requirements determine in large degree the
geometry and loads. Factors of safety, ade-
quate or otherwise, are usually prescribed by
codes and ordinances.
While the determination of the maximum
load-carrying capacity of a given frame is a
problem for which only one answer exists, many
feasible designs for a given geometry and
loading may exist. Only one solution, however,
provides the minimum-weight design.
The determinate structure of given geome-
try can be designed for minimum weight without
resorting to a trial procedure, because the
distribution over the structure of internal
moments and forces is not dependent on member
sizes. On the other hand, the indeterminate
structure, when designed on the basis of elas-
tic analysis, requires a method of trial to
approach the minimum-weight design. This fact
is manifested by the presence of the stiffness
or flexibility factor in the matrix of coef-
ficients which relate the redundant moments in
an indeterminate problem. Before an elastic
analysis can be made, therefore, an estimate
of the individual sizes must be made. Deter-
mination of the redundant moments usually
indicates that different member sizes are re-
quired, and that further analysis is necessary.
With the development of the ultimate load
method of analysis, however, it is possible to
determine an admissible distribution of
moments over an indeterminate structure with-
out a previous estimate of the member sizes.
This simplification is possible because the
matrix of coefficients which relate the criti-
cal moments are functions of the geometry of
the structure and of the hinge positions, and
are independent of member sizes.
Since member sizes do not have to be
estimated before an analysis is made, it
should be possible to determine a distribution
of moments for a given structure which will
yield a minimum-weight solution. In order
that this may be accomplished, suitable rela-
tionships between weight of member and moment
capacity of both beams and columns must be
established. In addition, a method of proceed-
ing efficiently from one distribution of
moments to another which will yield a lighter
structure must be developed.
B. PREVIOUS WORK
A number of researchers have interested
themselves in the minimum-weight design of
steel structures designed by the "plastic" or
"ultimate load" method.(3) (6) (7) (8) (11)
(12) (14) (16) (19) (20) (24)* For the most
Numbers in parentheses refer to entries in
the Bibliography.
part, the approach has been of a theoretical
nature in that a continuous spectrum of struc-
tural shapes is assumed available. Many of
the researchers have further limited the prob-
lem by specifying a linear relationship be-
tween the plastic moment capacities and
weights of this continuous spectrum of shapes.
Others have assumed a nonlinear relationship
of the type g = K(M )n, where g is the
weight per foot of the section, M is the
plastic moment of the section, and K and
n are constants. In none of the papers re-
viewed has the investigator considered the
reduction in plastic moment capacity in the
presence of axial force or the stability of
the beam-column as they affect the minimum-
weight problem. In any case, when the designer
attempts to apply these methods to the minimum-
weight design of structures other than very
simple ones, he is faced with a formidable
array of calculations which may or may not
lead to the minimum-weight design of a real
structure. Since the majority of structures
for which plastic analysis is presently con-
sidered to be appropriate are constructed of
available standard structural shapes, a method
which purports to be useful to the designer
must, in its final application, relate the
minimum-weight problem to the properties of
these shapes rather than to a continuous
spectrum of shapes. This requirement assumes
even greater significance when one notes the
relationships among properties of those stand-
ard shapes which are important to minimum
weight. These relationships do not plot as
well-defined curves, but exhibit some scatter.
At present, the designer must resort to a
comparison between the weights of succeeding
trial designs in an attempt to arrive at a
minimum-weight solution. He has no assurance
of having found a minimum-weight design when
he finally accepts his best solution, nor has
he a way of knowing how close he is to the
minimum-weight design. Even if an electronic
computer is used to make comparisons of trial
designs for a structure consisting of, say,
ten or more members, the time involved in con-
sidering all possible solutions becomes pro-
hibitive.
C. OBJECTIVE AND SCOPE
The objective of this study is to develop
a method for the minimum-weight design of steel
structures, based on plastic analysis, which
satisfies the following requirements:
(1) The method will embrace the problems
of axial compression as well as flexural load-
ing, lateral displacement (sidesway) of the
structure, and the nonlinear relationship be-
tween unit weight of members and their moment
capacities.
(2) The method will allow the determina-
tion of minimum weight for the structure which
is designed for standard structural shapes as
well as the structure for which a continuous
spectrum of shapes may be available.
(5) The method will converge to the min-
imum-weight design by consideration of only a
small percentage of the feasible solutions.
D. NOTATION
A = augmented matrix of the array of
linear restrictions
a. = constant (Figure 7)
al = constant (Figure 7)
= coefficients in the array of linear
restrictions
a.i = coefficients in the array of basic
equations for equation generation
= constant (Figure 6)
B = constant for a given L/r
(Equation 3.5)
= constant (Figure 7)
b! = constant (Figure 7)
1
bk = constant (Figure 6)
c. = length of i column in feet
1
c. = coefficients of variables in the
objective function
ck = length of k beam in feet
D. = weight per foot of i column
1
D: = weight per foot of i column when
subject to sidesway
Dk = weight per foot of k beam
d. = constant corresponding to variables
in array of linear restrictions each
of which limits a single variable
F. = sum of products of coefficients of
solution basis and corresponding
scalar multipliers in jth column of
tableau
f = objective function
F = frame weight
F' = frame weight minus a constant
w
G = constant for a given L/r
(Equation 3.6) x
L = column length
M = pure-bending fully plastic moment
M = moment capacity of a column of given
length in conjunction with a given
axial load
m = number of linear restrictions
expressed by Equation 2.5
n = number of structural variables
nb = number of beams in frame
n = number of columns in frame
c
P = applied axial load
P. = i vector in solution
1
=. = j column vector in the array of
linear restrictions
P0 = column vector of elements represent-
ing the external work for the corre-
sponding restrictions
P = product of the cross sectional area
y and the yield stress of the steel
p = number of beam equations
q = number of panel equations
r = number of joint equations
r = radius of gyration with respect to
x-x axis
X.. = scalar multipliers of the vectors
in solution
X. = structural variables
3
X n+i = slack variables Equation 2.5
Xn+m+j = slack variables Equation 2.6
0 = mechanism angle
Q = criterion by which vector going out
of solution is determined
( = minimum-weight solution consisting
of n sections for the n members
in the frame
II. METHOD OF SOLUTION
A. PLASTIC DESIGN
One of the assumptions underlying plastic
design of steel structures is that the ultimate
moment-resisting capacity at a cross section is
reached when the stress distribution in pure
bending is that shown in Figure 1. Since this
distribution represents complete plastification
of the cross section, rotation of the member at
a point where such a distribution develops pro-
ceeds with no increase in moment, provided, as
is customary, that strain hardening is neg-
lected. In effect, then, with respect to load-
ing beyond that which produces a fully plastic
moment at some cross section, a structural
member behaves as though there were a hinge at
that cross section. Such a fictitious hinge
is called a plastic hinge. The corresponding
moment-rotation diagram is shown in Figure 2.
The formation of one plastic hinge in a
statically determinate structure results in
complete collapse: i.e., the ultimate load has
been reached. For complete collapse of a
statically indeterminate structure, N + 1
plastic hinges are required, where N is the
degree of indeterminacy. Partial collapse
may result from the formation of only one
hinge. This would occur, for example, in a
member cantilevered from the remainder of the
structure. It is possible, then, for partial
collapse of an indeterminate structure to re-
sult from the formation of from one to N
plastic hinges. In addition, an indeterminate
structure will exhibit a number of different
modes of collapse, the critical mode being that
mode which results in the least value of the
ultimate load. The system that results when
the structure is in a state of collapse is
called a mechanism.
An equilibrium equation involving the ex-
ternal loads and the moments at the hinges de-
fining the collapse mechanism may be obtained
for each mode of collapse by using the princi-
ple of virtual displacements. There will be
as many such equations as there are modes of
collapse. In the general case, only one of
these equilibrium equations represents a safe
design, i.e., only one equilibrium equation
will correspond to a distribution of moments
throughout the structure for which nowhere in
a (prismatic) member will the moment exceed
the plastic moment of resistance prescribed
for that member.
Since only one of the n equations of
equilibrium corresponding to the n mechanisms
of collapse represents a safe design, all
equilibrium equations other than that one may
be written as inequalities in which the work
done by the internal moments during a virtual
displacement exceeds the work done by the ex-
ternal loads. Consideration of the inequali-
ties representing all possible modes of col-
lapse will satisfy all the criteria which are
necessary and sufficient to determine the solu-
tion. These criteria are:
(1) A collapse mechanism must be
obtained.
(2) The structure must be in equilib-
rium at the collapse load.
(5) The moment at any point in a pris-
matic member may not exceed the ultimate mom-
ent capacity of that member. This criterion
is called the yield criterion.
Since each inequality represents a par-
ticular mode of collapse, the first criterion
is satisfied. Both the second and third con-
ditions are satisfied, provided all possible
collapse mechanisms are considered, because an
equilibrium equation is obtained if the ine-
quality becomes an equality. This condition
can result only if the moments at the mechan-
ism hinges are reduced.
B. THE MINIMIZATION PROBLEM
For a given load system and structure
geometry, many feasible designs may be deter-
mined. Mathematically this fact is expressed
by the existence of more unknowns than equa-
tions which relate the unknowns. Furthermore,
the distribution of moments over the structure
at the ultimate load is influenced by the
relative moment capacities of the various
members. This fact becomes evident if the
structure is observed during the last stages
of loading leading to the ultimate load. As
the ultimate load is approached, each succeed-
ing hinge brings about a redistribution of
moment which differs from the moment distribu-
tion that would have prevailed had the struc-
ture remained elastic.
It is not feasible to determine the min-
imum-weight solution for a frame of any size
strictly by trial even on a high-speed elec-
tronic computer. Using a trial procedure, the
7090 I.B.M. computer required twenty-seven
minutes to obtain the minimum-weight solution
for a five-member, two-bay, one-story frame.
Over 60,000 designs were considered. If one
more member had been added to this frame the
time required for solution would have been
increased by a factor of about twenty. At
present this five-member frame, then, repre-
sents about the upper limit for trial proce-
dures.
The desired method should proceed from
the first solution to the minimum-weight
solution with the consideration of only a very
small percentage of the possible solutions.
Furthermore, it should proceed from one solu-
tion to the next without having to restart
the solution process. Finally, a criterion to
identify the minimum-weight solution must be
available. Sucha method exists. This method,
known as linear programming, (10) (21) (25) was
first developed by George B. Dantzig, Marshall
Wood, and their associates. Use of the linear
programming method for the solution of the
minimum-weight problem has been suggested by
Charnes and Greenberg.()
C. LINEAR PROGRAMMING
A function may be either maximized or min-
imized by the linear programming method. Since
we are concerned with minimization of weight,
only minimization is considered. The method
described is known as the Simplex Method.
Let it be required to minimize
n
f = c.X.
j=1
subject to
n
>a..X. > b.
Za ijX 1
j=l
X. > d.
J J
(2.1)
i = 1,2,.....m
j = 1,2,..... n
(2.2)
(2.5)
where d. > 0
J -
Equation (2.1) is known as the objective
function, and Equations (2.2) and (2.5) are
the linear restrictions or side conditions.
The simplex method requires both the objective
function and the side conditions to be linear.
In order to apply formal systematic solution
procedures to this problem the above inequal-
ities must be expressed as equalities as
follows:
Minimize
n m n
f = c X+ OXn + YOXn4.m+j (2.4)
j=l i=l j=l
subject to
n
L a. .X. - X
j =1n+i
j=l
= b. i = 1,2,....m
An example will illustrate the above
formulation and the subsequent solution.
Minimize
f = 2X1 + 3X2 (2.7)
subject to
XI + X2 a 6
XI+ 2X 8
1
where X > 01
and X >
2
(2.8)
(2.9)
X. - X . = d. j = 1,2,....n (2.6)
j n+m+j j
where
d. > 0
j -
Xn+i > 0
n+m+j >
X.
J
X . and X
n+i n+m+j
a. .,b.,c.,d.
1j 1 J j
= structural variables
= surplus variables
= constants
Expressing Equation (2.8) as equalities of the
form of Equation (2.5), Equations (2.7) and
(2.8) become
f = 2X1 + 3X2 + OX + OX4 (2.10)
X, + X 2- X
X1 + 2X2
where X3 > 0
and X4 > 0
(2.11)
(2.12)
- X 4 8
n = number of structural
variables
m = number of linear restric-
tions expressed by Equa-
tion (2.5)
The dimensions of the augmented matrix
are (m+n) x (m+2n). Although these dimensions
define a matrix of (m+n) equations for (m+2n)
variables., there is a condition under which all
these equations and variables need not be form-
ally stated even though they will always exist.
The simplex method guarantees that all solutions
are composed of only non-negative values for
all variables. Therefore, if d. = 0, Equation
J
(2.6) for the jth variable is redundant and
need not be included in the linear restrictions.
For this reason, the number of equations that
need be stated will vary from m to m+n, and
the number of variables that need be consid-
ered will vary from m+n to m+2n. However, in
the discussions that follow it will be consid-
ered that the augmented matrix consists of
m+n equations and m+2n variables.
In this case it is not necessary to write Equa-
tions (2.9) as equalities, because the lower
limits for the variables X and X are zero, as
was discussed above.
Equations (2.11) and (2.12) may be repre-
sented geometrically in two-dimensional space
as shown in Figure 3. Lines AB and CD repre-
sent Equations (2.12) and (2.11), respectively,
for the case X3 = X = 0. Line EF represents
one possible location for the objective func-
tion. Any point to the right of the vertical
axis, above and to the right of line AOD, and
above the horizontal axis,represents a feasible
or admissible solution which satisfies Equations
(2.11) and (2.12). This space is known as the
solution space and is shown shaded in Figure 5.
Solutions involving negative variables, which
would be represented by points to the left of
the vertical axis and/or below the horizontal
axis, are not admissible in the simplex proce-
dure. In order that the procedure yield a
minimum solution the solution space must be a
(2.5)
convex set. A convex set exists if lines
drawn between all pairs of arbitrarily selected
points within the solution space are entirely
contained within the solution space. The
apexes on the boundary of the solution space
are known as extreme points. These extreme
points represent solutions which are defined
as basic feasible solutions. Points A, 0, and
D in Figure 5 represent basic feasible solu-
tions. It has been shown that the minimum so-
lution will always be a basic feasible solu-
tion.(10)
Primarily, the object of the simplex pro-
cedure is to generate a new basic feasible
solution from the preceding one such that the
new solution is always closer to the minimum
solution. Further, the minimum solution must
be recognized when it has been reached. This
operation may be visualized as the sweeping
out of the solution space in Figure 5 by the
objective function.
If the augmented matrix is designated [A],
Equations (2.4), (2.5), and (2.6) may be
expressed in matrix notation as
f = {} {x}
[A] {X} = {P 0
where {C} is a row vector and
column vectors, and
{Po =0 =
b
m+n
Furthermore, because [A] may be re]
as a set of column vectors
[A] = [Pl' 2...j*- m+2n]
(2.15)
(2.14)
{X} and {Po} are
(2.15)
presented
(2.16)
Equation (2.14) may be written
XlPl+X2F2...+ X ..+ Xm+2nm+2n (2.17)
where each X is a scalar.
Equations (2.11) and (2.12) for the example at
hand may now be expressed as
1 1 + X+ X P + X = o (2.18)
or
X + +2 1 + X4] = [86] (2.19)
Because the simplex procedure generates
one basic feasible solution from another, the
solution process must begin with a basic feas-
ible solution. A basic feasible solution is
characterized by having m+n variables with
values greater than zero. In the vector repre-
sentation, m+n linearly independent vectors
will have multipliers greater than zero. These
vectors are said to be in solution, or are said
to be the basis of solution. The number of
vectors in solution will always equal the num-
ber of linear restrictions. In the particular
case when d. = 0, j = 1, 2,....n in Equation
J
(2.6) only m linearly independent vectors
will be in solution.
The general tableau for the simplex
procedure is shown at the top of the next
page.
For any one cycle the tableau represents
a particular basic feasible solution. The
particular solution shown on the next page
has as its basis vectors P1 through P+n
The vectors making up a solution basis may be
any m+n of the m+2n vectors. They need not,
and generally will not, occur in sequence as
shown below. A new basic feasible solution is
obtained by the replacement of any vector
already in solution by any vector that is not
in solution.
C2
P2"
C . . .
j
m+2n
Pm+2n
X X . .. XIj. . . X X
11 X12 l j. 1,m+2n X10
X21 X22. . . X2j . . 2,nm+2n X20
X1 Xi2 ij. . i,m+2n XiO
m+n,l1 m+n,2 m+n,j''' m+n,m+2n m+n,0
(F. C.)
where
C. = coefficient of the jth variable of the
objective function
P. = column vector corresponding to the jth
a variable, taken from the original array
of linear restrictions
X1 through Xm+n j = scalar multipliers which,
when multiplied by the corresponding
vectors in solution, will yield the
vector P.
J
C. = coefficient of the variable in the objec-
tive function corresponding to the ith
vector in solution
P. = ith vector in solution
1
P = column vector defined by the column of
constants on the right side of the
original array of linear restrictions
S= Xi0/Xij, where j is
incoming vector. The
the basis for the del
outgoing vector
F = [CI, C2...Ci...Cm+n ]
the index
column of
termination
X1j
xij
X2j
X.
m+n,j
of the
O's is
of the
(2.21)
(2.22)
F.-C. = the criterion by which the incoming
J J vector is determined. This is also
the criterion used to identify the
optimum solution
The operation of the simplex procedure
will now be demonstrated by solving the example
given in Equations (2.7), (2.8), and (2.9).
For a first basic feasible solution let X =
X4 = 0. Equations (2.18) are solved next to
determine the values of X1 and X .
X 1 ^ 2+ X P + 0 P
or [il + 1O] +X + ]
yielding XI = 8 and X3 = 2.
The basis for this solution is, then, P
and P , and
P - P
+ 8 [i] + 2 [ - [']
3 ] 0
or X10 = 8, X10 = 2
The next step is to calculate the scalar
multipliers for the solution basis required to
produce the vectors P2 and P .
S[2] - 12 [X1 + 352
from which X12 = 2 and X32 = 1
A systematic method of determining an initial
basic feasible solution is discussed in
Section IV B.
Basis
C P
c1 12
C2 2
C.
1
C
m+n
m+n
(2.20)
0.
0
m+n
= PO
P4 =l = 14 [ij + 54 [0
from which X 14 = -1 and X = -1
The first tableau, with symbolic scalar multi-
pliers, is
2 5 0 0
Basis P P2 P3
2 1 X11 X12 X15 X14
0 P X51 X52 X55 X4
The first tableau, with numerical
pliers, is
x0
XISO
scalar multi-30
scalar multi-
2 5 0 0
Basis 1 2 P5 4 P0
2 P1 1 2 0 -1 8 4
0 P 0 1 -1 2 2
(F.-C.) 0 1 0 -2 16
Any vector P!' P 2 P2 P 4 PO may now be
expressed as a product of the first basic
feasible solution basis P and P3 and the
appropriate scalar multipliers. For example,
- - - _ 1 ] =
Note that the scalar multipliers X10 and
X30 under P are the values of the variables
XI and X for the first basic feasible solu-
tion. In the general case, then, the values
of the variables corresponding to the vectors
in solution are to be found in the column of
scalar multipliers under the P0 vector for any
basic feasible solution.
It should also be noted that the scalar
multipliers in the columns under the vectors
in solution will always form an identity
matrix. This follows from the fact that in
order to represent a vector in solution as a
function of the vectors in solution the
vector is equated to unity times itself.
The calculation of the (F.-C.)'s become
3 J
- 2 = 0
(F-C 1) = 2, [C11
(F2-C2) = [2,0] ][2]
(F5-C ) = [2,0] [0]
(y4-c4) = [2Io] 1
(F -C ) = [2,0] [1]
0F-O 0 2 ]0
- 0= 0
- 0 = -2
- 0 = 16
where F -C0 is the value of the objective
function for the first basic feasible solu-
tion.
In order for the solution to be optimum,
all F -C., with the exception of Fo-Co, must
be equal to or less than zero. But since
F 2-C2 is positive this initial solution is
not optimum. It is now required to generate
a new basic feasible solution. Because only
two vectors may be in solution, one vector
must go out of solution as a new vector comes
in. Furthermore, a new solution will result
in a decrease in the objective function only
if F.-C. > 0 for the incoming vector. In
J J
general, the vector with the greatest value
of F.-C. should be chosen as the vector com-
J J
ing into solution. In this example vector
P2 is the only one which will improve the
value of the objective function, so it comes
into solution. To determine the vector going
out of solution the 9.'s are calculated:
1
1 = X 0/12 = 8/2 = 4
8 = X0/X52 = 2/1 = 2
The vector in solution with the smallest
positive value of 0 must go out of solution.
This is necessary because the presence of a
negative 0 would result in the incoming
variable having a negative value, while the
presence of a positive 0 greater than the
smallest positive 0 would result in a nega-
tive value for one of the variables remaining
in solution. In tnis example, then, 2 re-
places P3. The intersection of the incoming
and outgoing vectors defines the pivotal ele-
ment, which is shown encircled.
To generate a new basic feasible solution
a new identity matrix involving vectors P2 and
P is required. In general, this matrix is
constructed by first dividing the elements in
the row containing the pivotal element, includ-
ing the P0 element, by the pivotal element.
This modified row is now placed in a position
in the new tableau corresponding to its posi-
tion in the old tableau. Next, multiples of
this row are added to or subtracted from each
of the remaining rows in the old tableau so as
to produce zero elements in the column contain-
ing the pivotal element. As the new rows are
generated, they are placed in the new tableau
in positions corresponding to their positions
in the old tableau. In this example, the first
operation does not alter the row containing the
pivotal element, since the pivotal element is
unity. Next, adding minus one times the second
row to the first row results in the new iden-
tity matrix involving P2 and Pl, and yields
the following new tableau:
2 3
Basis P P
1 2
2 P 1 0 -2
5 2 0 1 1
(F.-C.) 0 0 -1
Since all F.-C. are
J J
zero this is the optimum
0
4 T0
1
-l
-l
equal to or less than
solution. Therefore,
X = 4 X = 2, X3 = X4 = 0
and f = (FO-CO) = 14
which corresponds to point 0 in Figure 3.
* * *
III. APPLICATION TO DESIGN PROBLEMS
A. GENERAL REMARKS
In order to apply the method of linear
programming to the problem of determining the
minimum weight of steel frames, a number of
important relationships must be studied. In
general, these relationships are not linear.
On the other hand, the method of linear pro-
gramming requires both the objective function
and the side conditions to be linear. This
contradiction will be discussed in Section
III C.
B. ASSUMPTIONS AND LIMITATIONS
1. This study is limited to frames in
one plane, composed of rigidly jointed mem-
bers which are braced normal to their plane
of action.
2. Only prismatic steel members whose
cross sections have an axis of symmetry lying
in the plane of the structure are considered.
3. All loads act in the plane of the
structure.
4. Lateral loads act only at the joints.
5. Beams are assumed to be subjected
only to bending (effects of axial force
neglected).
6. Further restrictions applying to
particular problems will be discussed as they
arise.
C. THE OBJECTIVE FUNCTION
The objective function which is to be
minimized in order to determine a minimum-
weight steel frame is that function which
expresses the total weight of the frame:
nb nc
F = Ck Dk+ C.i D.i
k=l i=l
(3.1)
where F = weight of frame
wth
Ck = length of k beam in feet
k th
D = weight per foot of k beam
nb = number of beams in frame
C. = length of ith column in feet
1
D. = weight per foot of ith column
1
n = number of columns in frame
c
Although Equation (3.1) is exact, it
cannot be used in its present form because the
side conditions provided by the mechanism equa-
tions relate the moment capacities of the
individual members and not their weights. In
order to effect compatibility between the ob-
jective function and the side conditions the
weights of the members must be expressed as
functions of their moment capacities. These
relationships will be considered separately
for beams and columns.
In Figure 5 the weight per foot, D, of
wide-flange and other I-shapes, as given in
Reference 18, are shown plotted against their
plastic moment capacities M . The solid curve
p
is drawn as a best fit of the "economy" sec-
tions. The economy section is the lightest
standard shape which furnishes a plastic re-
sisting moment which equals (or exceeds) the
required value. In the investigation of
design for minimum weight, only the economy
sections need be considered for beams. The
curve of best fit for these sections has been
determined in Reference 20 to be
D = 1.2 M2/3 (3.2)
Since this equation is nonlinear it cannot be
substituted into the objective function.
Fortunately, the range in M from the smallest
shape that could be used to the largest shape
which probably would be used for a particular
member under a particular loading is limited.
For uniform load this range would normally be
from WL2 /16 to WL 2/8, although in the case
of a short span sandwiched between two longer
spans the upper limit could be greater. A
number of trials showed that a straight line
gives a good fit to the plot of economy sec-
tions for particular conditions of geometry
and loading and for a reasonable range of M .
Line AB of Figure 6 is typical. Therefore,
we may write
k = a + bkMpk (
where M = plastic moment capacity of the
th p
k beam.
Further complications arise in the case
of a similar representation for beam-columns.
Not only is the relationship between weight
per foot and moment capacity nonlinear, but
further complications of axial loading and
possible side-sway of the structure are added.
As has been pointed out, changes in size of
members in proceeding from one basic feasible
design to another affect the distribution of
moments over the structure. To a lesser ex-
tent, the distribution of axial loads is also
affected. However, it will be assumed here,
as is usually done, that axial loads remain
constant with change in member sizes.
In the minimum-weight problem the moment
capacities of the individual members are the
capacity of a beam-column as that moment M0
which can be carried by the section in con-
junction with the axial load P, a suitable re-
lationship may be determined. In using such a
relationship, it must be understood that the
minimum-weight solution then yields the value
of M required in conjunction with the given
axial load for a particular beam-column. On
the other hand, the minimum-weight solution
yields for beams required values of the fully
plastic moment M .
p
It now becomes necessary to develop
weight-moment equations for a given beam-
column of I shape. This column may be subject
to a wide range in loading and end support
conditions.
Massonet(17) has proposed for plastic
design an adaptation of the interaction form-
ula for elastic behavior of the beam-column.
In order to use his equation the magnitude and
sense of the moment at each end of the beam-
column must be known. Although these could be
determined at any stage of the simplex proce-
dure, they would certainly vary from one basic
feasible solution to the next. This would re-
quire that a new objective (weight) function
be established for each new basic feasible
solution. While this could be done, it would
significantly increase the time required to
obtain an optimum design. It is the author's
opinion that, should Massonet's interaction
formula be used, the slight gain in precision
over the method discussed next does not war-
rant the resulting increase in solution time
and complexity.
Galambos and Ketter'9) developed inter-
action formulas relating moment capacity M0
and axial compression P for the following
three specific cases:
Case I. Columns bent in double curvature,
for which M = M0 at each end,
Case II. Columns for which M = M0 at one
unknowns which are sought. Defining the moment
end and M = 0 at the other, and
Case III. Columns bent in single curva-
ture, for which M = M0 at each end.
Case I is the most favorable in respect
to the value of M0 that can be developed in
conjunction with a given value of P. Case II
follows, while Case III columns develop the
smallest M for a given value of P. There-
fore, the difficulty mentioned above can be
met by bracketing beam columns according to
the Galambos-Ketter equations. Thus, a beam-
column bent in double curvature for which M =
M0 at one end, while 0 < M < M0 at the other,
will be classified as Case II, which is the
nearer less favorable case. Similarly, a
beam-column bent in single curvature for which
M = M0 at one end, while 0 < M < M0 at the
other, will be classified as Case III, the
nearer less favorable case. Although this
would appear to result in some loss in preci-
sion, it can be justified on two counts.
First, Massonet's equation is approximate
(since it is a simple extension of the inter-
action formula for elastic behavior), while
the Galambos-Ketter equations are "exact" (for
the particular shape, 8 WF 51, and for the
particular distribution of cooling residual
stress which was assumed). Secondly, the
specification of the American Institute of
Steel Construction, which is the most widely
accepted code for plastic design of steel
frames, incorporates the Galambos-Ketter
formulas.
In Figure 7 the weights per foot D of
wide flange and other I shapes are shown
plotted against the allowable moment capacity
M0 for a particular load P and a particular
beam-column of height L. The points repre-
sented in this figure were plotted in accord-
ance with the equation below for Case II
columns:
(max.) [B-G(P/P)] (.4)
0(max.) p [ y
L/rx (L/r)2
where B = 1.15 + -- + - ,
L/r (L/r )2 (L/rx)3
G = 1.11 + - + L
190 9,000 720,000
(5.5)
(5.6)
P = product of the cross-sectional area and
the yield stress of the steel
r = radius of gyration with respect to the
x-x axis
A similar plot may be made for Case I and
Case III columns. However, the majority of
columns in building frames exhibit end condi-
tions which classify them as Case II columns.
In other words, nearly all such columns are
bent in double curvature, with M = M0 at one
end and 0 < M < M0 at the other. Case II is
on the safe side in all such cases, as has
been explained previously. For the rare case
in which a minimum-weight design contains a
column or columns which exhibit single curva-
ture, with M = M0 at one end and < M<M0 at
the other, the designer must check the ade-
quacy of such columns against the Case III
formula. This is because the Case II formula
is unsafe for columns in single curvature ex-
cept when M = 0 at one end.
As in the case of beams, only the economy
sections need be considered for columns for the
minimum-weight problem. This is because the
economy sections always equal or exceed the
non-economy sections of equal weight with re-
spect to the section properties rx, M , and
P . Because the frame is considered braced
y
normal to its plane of action, the radius of
gyration for the weak axis, ry , need not be
considered. These section properties, to-
gether with the given axial load, determine
the allowable moment capacity M (Equation
5.4). The curve of best fit is again nonlin-
ear. But, as for beams, the range of M0 re-
quired for a particular member is limited to
the extent that a straight line provides a
reasonably good fit over this range. The
range for the column will normally extend from
M = 0 to the maximum M0 the column would re-
ceive from adjacent beams. Line C-D, Figure 7,
is a typical best-fit straight line. The
equation is
D. = a. + b M0i (5.7)
where M0i = that moment capa ity which can be
carried by the i column in con-
junction with its axial load.
Failure of a frame may result from over-
all instability involving sidesway at an ulti-
mate load less than that which would be carried
if the frame were braced to prevent sidesway.
At the present time the ultimate load with re-
spect to this form of instability cannot be
predicted precisely. However, in Reference (5)
the following expression is suggested for
columns subject to sidesway:
2P/Py + L/70rx : 1 (5.8)
This relation is conservative for frames of
proportions likely to be found in practice.
It has been adopted by the AISC Specification
Committee as an interim provision.
For a limitation of the above type the
relationship between D. and M0 must be modified
slightly. Essentially this limitation may re-
quire that the lower limit for M0 be greater
than zero. The lower limit is calculated by
solving Equation (3.8) for Py , selecting the
minimum-weight section to provide the required
P , and then using Equation (3.4) to determine
M O. A different straight line segment is re-
quired to fit the more restricted range of M0.
Line E-F in Figure 7 represents this new rela-
tionship. The more general equation is
written
D! = a! + b!M (5.9)
1 1 1 Oi
where, for the special case when Mo(min.) is
zero, D' = D . The objective function, Equa-
tion (3.1), is now expressed as
nb nc
F = (Ck ak + Ckb Mpk) + (Ca! + CbMoi
k=l i=l
(5.10)
when substitutions are made for D, and D!'.
k . 1
Because all terms Ck ak and C. a', are
k k 1 1
constants for a particular problem, these terms
may be dropped in determining the values of
Mpk and MOi which yield minimum frame weight.
In other words, the value of F is not sought,
but rather the values of Mpk and Moi which
yield the minimum F . Therefore, the objective
function may be written
nb nc
F = C bk Mpk + Cb MOi (5.11)
k=l i=l
where F' = frame weight minus a constant. In
w
order to more conveniently express the objec-
tive function in the linear programming tab-
leau, Equation (5.11) will be written
n
F' = C.X.
j=l
where C.
C.
X.
xj
X.
n
= Ckbk if
= C.b! if
= M if
= Moi if
= ib + nc
(5.12)
th
j member is a beam
jth member is a column
j member is a beam
.th member is a column
j member is a column
D. LINEAR RESTRICTIONS
The linear restrictions or side conditions
originate from the mechanism inequalities (Sec-
tion II A), from sidesway requirements, and,
possibly, from other design requirements. It
was shown in Section III C that sidesway of
the frame requires a greater-than-zero lower
limit for the M0 capacity of a column. This
restriction must, therefore, be imposed in the
form of an additional inequality for each
column in the frame subject to sidesway, and
added to the mechanism inequalities. Further
restrictions could arise in the form of arbi-
trary limitations set by the designer. For
example, it may be desirable to limit either
the maximum or minimum moment capacity, or
both, of one or more members, in which case
additional inequalities are required. However,
since these inequalities do not originate from
the physical problem they should be checked
against the mechanism and sidesway inequalities
for inconsistencies.
To insure compliance with the criteria of
yield and equilibrium (Section II A), ine-
qualities representing all possible modes of
failure should be included in the formulation
of the problem. If, however, an attempt is
made to provide rigorously for this require-
ment in a frame with, say, ten or more members,
a very large system of equations (of the order
of 10 ) will result. Not only will the time
required to generate these equations become
excessive, but the storage capacity of even a
large computer will be exceeded.
The possibilities of making admissible
reductions in the number of inequalities will
now be considered. The solution of interest
is a frame of n structural variables involv-
ing (m+n) hyperplanes. For illustration, how-
ever, a two-dimensional space will be consid-
ered (Figure 4). The line segnents AB, BC,
CD, and DE form the lower bound to the solution
space. Line FG, the objective function, is
shown positioned at the minimal solution. The
bases for these minimal solutions are the equa-
tions represented by lines CD and DE. If
either of these two equations is omitted from
the problem formulation, the correct minimal
solution will not be obtained. For example, if
the equation represented by line CD is inad-
vertently omitted, the minimal solution is
erroneously placed at point H, as shown by line
F'G'. But suppose that instead of the equations
represented by CD or DE, some other equation,
or equations, is omitted. For example, consider
the equation represented by line AB to be
omitted. If the first basic feasible solution
is initiated on the vertical axis, a different
route to the minimal solution is followed,
namely ICD, but the solution is still correct-
ly placed at D.
It is obvious from the discussion above
that no error arises from the omission of those
equations which are not a part of the basis of
solution. Although the omission of an equation
which is a part of the basis of solution re-
sults in an erroneous (unsafe) design, it
should be noted that this solution results in
a structure whose weight is less than the true
minimum. This conclusion follows from the
convex shape of the solution space. Therefore,
it may be concluded that the optimal solution
obtained with a partial set of equations is a
lower bound on the minimum-weight frame.
Furthermore, if a statically admissible moment
diagram for the entire frame can be found for
which the moment nowhere exceeds the values of
the plastic moments prescribed for each member,
the minimum-weight solution is also an upper
bound, and is therefore the optimal solution.
This suggests that if a designer can, by exper-
ience, select the dominant equations and obtain
a minimal solution, he can check compliance with
the yield criterion by the statical method or
by moment balancing techniques.(l) Should the
yield criterion be satisfied, he has indeed
found the minimum-weight solution without re-
sorting to a large system of equations. If the
yield criterion is not satisfied, he has two
choices. He might try to include the dominant
mode or modes of collapse initially omitted.
He would be materially assisted in this effort
by a study of the positions in the frame where
the moment capacities are exceeded. On the
other hand, he might elect simply to increase
the moment capacities of the members whose
capacities are exceeded. The weight of this
frame would then form an upper bound to the
minimum-weight solution, and if the upper and
lower bounds are deemed to be sufficiently
close the upper bound can be accepted as the
final solution.
Finally, the system of equations might be
reduced by solving the problem in parts. It
is well known that the flexure induced by a
particular load diminishes rapidly in members
of a frame at some distance from the point of
loading. Therefore, as is usually done in
elastic design, a tall frame might be solved
story by story. This idea has been suggested
by Boulton.(2)
Although it may often be advisable to
consider reducing the number of linear restric-
tions, it may be possible in some situations to
include all of them. It should be recalled
that consideration of all possible modes of
collapse eliminates the operation of checking
the solution against the yield criterion.
Therefore, a systematic generation of the mech-
anism inequalities which will insure the inclu-
sion of all possible modes of collapse is de-
sirable. The basic modes of collapse of a
frame by beam, panel, and joint mechanism are
shown in Figures 8, 9, and 10. All other
possible modes of collapse may be generated by
combinations of one or more of these basic
mechanisms.
The mechanism inequalities were represented
in a two-dimensional array in Section II C for
the purpose of linear programming, the first
subscript denoting the equation number and the
second the member number. This notation allows
the summation of the rotations of all the hinges
in one member to be represented by a single co-
efficient, and is adequate for a simplex-proce-
dure solution process. However, in order to
formulate a basis for generating equations from
the basic equations, the individual plastic
hinge positions must be designated. This
specification is necessary because the various
mechanisms are identified by the positions of
their hinges along with the magnitudes of the
hinge rotations. Therefore, a third subscript,
k, is added to the above notation to denote
the hinge position. The A matrix of the
linear restrictions,
[a..] X. > bi} i 1,2,....m (5.15)
j = 1,2,....n
is expanded by letting
r+l
ij aijk
k=l
(5.14)
The symbol r in Equation (5.14) denotes the
number of joints in the frame. The summation
is taken to r+1l in order to include the
intermediate beam-hinge. The resulting equa-
tions with expanded A matrix are shown atop
the next page.
All elements a.. of the A matrix of the
13
mechanism inequalities are positive because,
regardless of the direction of rotation of the
corresponding hinge, positive work results.
But for the purpose of equation generation the
direction of rotation must be identified,
since the algebraic summation of corresponding
elements a. of different equations must
ijk
allow for the increase, decrease, or elimina-
tion of rotation at a hinge position. (The
hinge intermediate between joints is an excep-
tion. Because such a hinge is common to only
one basic equation, the corresponding matrix
element will always be positive regardless of
the direction of rotation.) Hinges at a joint
are considered positive if they result from
clockwise rotation of the joint.
To assure generation of the equations
representing all modes of failure, all possi-
ble combinations of the basic equations will
be made, after which redundant combinations
will be eliminated.
A beam equation may enter a particular
combination only with a positive sign. On
the other hand, a panel equation or a joint
equation may enter a particular combination
with either a positive or a negative sign.
This difference occurs because beams are con-
sidered to be subjected only to gravity loads,
while lateral loads may act from either left
or right, and joint rotations may be either
clockwise or counterclockwise.
Accordingly, in determining the number of
combinations of the basic equations, we have
two possibilities for each beam mechanism
(enter, not enter) and three for each panel
mechanism and each joint mechanism (enter
positively, enter negatively, not enter).
Therefore, the number of basic equations and
all their combinations is
(2p . q+r - 1) (3.16)
where p = number of beam equations
q = number of panel equations
r = number of joint equations
Each resulting combination is subjected
to the following criteria for elimination:
(1) Any equation which has as many
hinges at a joint as there are members connect-
ed by the joint is eliminated. While these
equations would be admissible in the simplex
procedure, they are redundant because the pro-
cedure guarantees that all variables will be
xl
(5.15)
greater than or equal to zero.
(2) If a beam mechanism forms in which
the required three hinges occur in the beam
itself, or in which the hinge at either end
(or both) is superseded by hinges at that
joint in all members framing into it, and as
many as one other hinge exists in the struc-
ture, the equation is eliminated. Since such
an equation represents a system with two or
more degrees of freedom, it is redundant.
This redundancy results because failure is
already considered with respect to all single-
degree-of-freedom systems. As an example, the
equations representing the modes of failure
shown in Figures 11 and 12 would be eliminated.
(5) In a multistory building, if any two
or more panel mechanisms do not have at least
one column continuous without hinges across
the joint or joints separating them, then the
equation is eliminated. This equation also
represents a system with two or more degrees
of freedom. As an example, the equation rep-
resenting the mode of failure shown in Figure
15 would be eliminated.
(4) When the A matrix is transformed
from the aik form to the a.. form, each
ijk 13
equation is checked against all previously
generated equations for proportionality of
terms in the left member, i.e., the terms rep-
resenting the internal work during the virtual
displacement. If these terms are found to be
proportional to the corresponding terms of any
(all + all2 ..+a allr+l1)(ja121+a1221' ..+a12r+1)" '... (alnl +aln2 ..+alnr+l )'
(a2111 + a212 +a21r+i1 )( 2211+ a2221' .+a22r+1 )' ( 2nl a2nll + a2n2 ..+ la2nr+ll)
. am ll+ aml2 ..+ lamlr+ll)( am21 +am22 I'.+am2r+1 )" (amnl + amn2 I'.+I mnr+l1)
previously generated equation, the more re-
strictive equation is retained and the other
is eliminated.
It should be noted that this procedure
is intended primarily for machine computation.
A simple example will be considered (Fig-
ure 14). The formulation of the equation for
the beam mechanism is
p1X1l + -1,1 + 2XI > - L3W - = b
This equation is, of course, equivalent to
6X1 ? b . However, for the purpose of gener-
ating equations by combining elementary mech-
anisms, the energy terms corresponding to the
various hinge rotations must remain separate.
The remaining basic equations are formulated
similarly. The symbolic representation of the
nonzero elements of the basic equations are
shown in the expanded A matrix:
(all ll+ all21 + a1131
Ia212
)( 1a2221
Ia 3131 )
a4221
Ia3531
Ia4331 + a 4341).
The numerical values of the elements of the A matrix are
53 -1 2
1
-0.5
-0.5_
The equations in this matrix correspond,
respectively, to the following elementary
mechanisms:
Equation (A) beam mechanism
Equation (B) joint 2 mechanism
Equation (C) joint 5 mechanism
Equation (D) panel mechanism
For the problem at hand, only sidesway
right is considered. Therefore, the panel-
mechanism equation will act negatively in com-
bination. Thus the number of elementary mech-
anisms and their combinations is
(2p+q * 3r - 1) = (21+1 " 32 - 1) = 55
Only a representative number of these
mechanisms will be considered. In consider-
ing the elimination of a particular combined
mechanism by one or more of the first three
criteria, only the left side of the equation
need be studied.
Beam and panel combination: Equation
(A) + Equation (D)
(5 + 1-.+2+0)X1+(0+ l-l|+0+0)X2+(o+0+ 1-.51 +1-.I)X3
which is eliminated by either criterion (1) or
criterion (2).
Beam mechanism combination: Equation (A)
(5+ 1-l +2+0)X1
This satisfies the first three criteria,
and is transferred to the a.. equation array
as
2>
b2
6X, > bt
Joint 2 and panel combination: Equation (B)
+ Equation (D)
(O+1+0+0)X1 + (0+0+ I-.51 + 1-.51 )X3
This satisfies the first three criteria,
and is transferred to a.. equation array as
10
X1 + X > b4 (F)
Combination of beam, panel, and both
joints: Equation (A) + Equation (B)
- 2 Equation (C) + Equation (D)
(5+0+0+0)X1 + (0+0+ 1-2.51 + -. 51 )X
This satisfies the first three criteria,
and is transferred to a .. equation array as
3 + X3 > b1 + b = 4b
x + X3 > 44/5 (G)
Eamination of the three combinations
represented by Equations (E), (F), and (G),
according to criterion (4), results in the
elimination of Equation (F).
After all thirty-five combinations have
been made, the following array survives the
criteria for elimination:
1 2
2
5 1
1 1
0.5
53
1 0.5
0.5
Xli
X2
X3
b +
b4
bl+b 4
bl+b 4
b 4
b4
A program which has been prepared for the
7090 computer takes the equations corresponding
to the basic mechanisms as input and makes all
possible combinations of them, rejecting those
which are revealed to be redundant in accord-
ance with the criteria established above.
E. ADJUSTMENT OF INITIAL SOLUTION TO ACTUAL
SECTIONS
Although the objective function as present-
ly formulated is related to the properties of
the standard sections, it is still based upon a
continuity of these properties. Generally, the
minimum-weight solution for a particular struc-
ture will place the required moments of the
various members intermediate between the moment
capacities of standard sections. It should
also be recalled that the weight-moment capac-
ity relationships are best-fit straight lines.
For this discussion, the initial solution
will be defined as the solution composed of the
lightest members all of whose moment capacities
equal or exceed the theoretical moments obtained
from the linear-programming solution. This
initial solution is usually not the minimum-
weight solution. Suppose these next-largest
moment-capacity sections are selected for an
arbitrary number of members in a multi-member
structure. Then it is quite possible that, for
the remainder of the members, the moment capac-
ities of economy sections lighter than those
adequate to satisfy the original array of linear
restrictions may suffice. This possibility be-
comes more evident if a particular example is
investigated. Consider a structure composed of
two members for which the minimum-weight solu-
tion places the theoretical moments slightly
above the available moment capacities of the
required standard sections. Suppose further
that one member is in a region of sections with
large intervals between moment capacities and
section weights, while the other is in a region
of sections with smaller moment capacities and
smaller intervals between moment capacities
and section weights. This is not an uncommon
condition. The solution consisting of the
sections with moment capacities next below
the theoretical moments must be ruled out,
since the linear programming solution gives a
moment distribution which just satisfies the
given linear restrictions. However, if the
next larger moment capacity section is
selected for the larger of the two members, it
is possible that the resulting reduced moment
required for the smaller member can be fur-
nished by a section which may be lighter than
the corresponding section in the initial
solution.
The most important consideration in de-
termining the minimum-weight solution, once
the initial solution is in hand, is the band-
width of sections which must be surveyed.
Here the bandwidth is defined as the number of
consecutive standard sections which are to be
considered for a given member. The bandwidth
may be constant for all members or it may
vary. As the number of members in the frame
increases, the bandwidth becomes more critical
with respect to the feasibility of obtaining
the minimum-weight solution. For a constant
bandwidth of four members for a three-member
frame, only 64 combinations need be consid-
ered. For a constant bandwidth of four mem-
bers for a ten member frame, however, over one
million combinations would be possible.
A definitive method of determining the
bandwidth which will assure the minimum-weight
solution has not been found. However, exper-
ience which may provide a guide has been ob-
tained in this investigation. The selection
of the bandwidth which will probably include
the minimum-weight solution is governed by the
following considerations:
(1) The position of the theoretical mo-
ments with respect to the adjacent moment
capacities of standard sections, and
(2) The position of the theoretical
moments with respect to the more efficient
sections within the range of sections con-
sidered. As would be expected, the closer
the theoretical moments are to the next larger
available moment capacities, the narrower may
be the bandwidth which probably includes the
minimum-weight solution.
Sections represented by points below the
best-fit lines in Figures 6 and 7 are the more
efficient sections. These have moment capaci-
ties greater than those predicted by the best-
fit straight lines. The closer the theoreti-
cal moments are to the moment capacities of
these more efficient sections, the narrower
may be the bandwidth which probably includes
the minimum-weight solution. Bandwidths are
discussed further in Section V A.
It is proposed here to use the linear
programming method formulated in Sections III
B, III C, and III D to determine an initial
solution to the minimum-weight problem. This
initial solution will be in the immediate
vicinity of the minimum-weight solution. A
suitable bandwidth is then chosen for each
member in the structure. Although these band-
widths define a finite number of solutions,
only those which satisfy the given linear re-
strictions are feasible solutions. These
feasible solutions are then compared to obtain
the minimum-weight solution.
Except for structures composed of very
few members, the electronic computer must be
used in obtaining the solution outlined above.
F. DISTRIBUTION OF LOADS
As has been mentioned in Section III B,
lateral loads are applied only at the joints.
On the other hand, the distribution of loading
on the beams is arbitrary. Therefore, it is
incumbent on the designer to so locate the
intermediate plastic hinges that the moment
capacities of the resulting minimum-weight
f =f
c y
I ---I
-J
L1__I' -±'
ft y
FIGURE 1. ASSUMED STRESS DISTRIBUTION
Curvature
FIGURE 2. IDEALIZED MOMENT CURVATURE DIAGRAM
(0,6,o0,4
(8,0,2,0) X1
NUMERICAL PROBLEM FOR SIMPLEX METHOD
m-
FIGURE 3.
A
I
FIGURE 4. STUDY OF INCOMPLETE SYSTEM OF LINEAR RESTRICTIONS
250
W 200
15O
100
50
Plastic Moment Capacity (in Ft-Kips)
FIGURE 5. WEIGHT PER FOOT VS PLASTIC MOMENT CAPACITY OF ECONOMY SECTIONS
0
0 -
0
,0
0
40
- 0
'II
0
B
b1bk
A s
Probable Range
I in M for Kt
Beaml'
FIGURE 6.
RELATIONSHIP BETWEEN Dk
D
F
b-
./ b
Pr
SMoa
Co
- Allowable Moment on
I Minimum Size Column
I* Allowed with Sidesway
o for ith Column
C I
AND M p FOR BEAMS
pk
obable Max.
ment from
nnecting Beams
r ith Column
FIGURE 7. RELATIONSHIP BETWEEN D. AND M FOR BEAM COLUMNS
1 Oi
S0*
to-
FIGURE 8. BEAM MECHANISM
FIGURE 9. PANEL MECHANISM
FIGURE 10.
JOINT MECHANISMS
FIGURE 11. FIGURE 12. FIGURE 13.
EXAMPLES OF TWO-DEGREE-OF-FREEDOM SYSTEMS
b1 = 2/5 LW1
b = b = 0
b = HW2
b = 5b4
(Encircled numbers represent
hinge positions.)
FIGURE 14. GIVEN CONDITIONS FOR EXAMPLE OF EQUATION GENERATION
EQ. (A) EQ. (B)
FIGURE 15. MODES
EQ. (C) EQ. (D)
OF FAILURE FOR BASIC EQUATIONS
ad)
Example 1 - Frame Braced
Against Sidesway
Example 2 - Frame Unbraced
FIGURE 16. GIVEN CONDITIONS FOR EXAMPLES 1 AND 2.
FIGURE 16. GIVEN CONDITIONS FOR EXAMPLES 1 AND 2.
H
4 5 0 0 0 100 100 100
P 2 P3 P4 5 6 7 8 0 e
-100 P6 ) 0 -1 0 0 1 0 0 1000 250
100 P 2 2 0 -1 0 0 1 0 1000 500
100 P8 0 1 0 0 -1 0 0 1 54 °
(F.-C.) 594 295 -100 -100 -100 0 0 0
3 3
4 P1 1 0 -1/4 0 0 1/4 0 0 250 °
100 P 0 2 1/2 -1 0 -1/2 1 0 500 250
-100 P8 o ( 0 0 -1 0 0 1 54 54
(F.-c.) 0 295 49 -100 -lo00 -149 0 0
33 1
4 P1 1 0 -1/4 0 0 1/4 0 0 250 0
-100 P7 0 0 1/2 -1 (2) -1/2 1 -2 392 196
5 P2 0 1 0 0 -1 0 0 1 54 -54
(F.-c.) 0 0 49 -100 195 -149 0 -295
33 4
4 P1 1 0 -1/4 0 0 1/4 o 0 250 -1000
- P5 0 0o -1/2 1 -1/4 1/2 -1 196 784
5 9P 0 1 1/4 -1/2 0 -1/4 1/2 0 250 1000
(F.-c.) 0 0 0.25 -2.5 0 -100.25 -97.5 -100
t
4 P1 1 0 0 -1/2 1 0 1/2 -1 446
0 P 0 0 1 -2 4 -1 2 -4 784
5 P2 0 1 0 0 -1 0 0 1 54
(F -C ) 0 0 0 -2 -1 -100 -98 -99 Optimum
FIGURE 17. SOLUTION FOR EXAMPLE 2
(ultimate loads)
1- 48'
FIGURE 18.
GIVEN CONDITIONS FOR EXAMPLE 3
Max. Positive Moment = 387.7k
at 3.12' left of 4
Max. Positive Moment = 388'k
396 k
FIGURE 19. ADMISSIBLE MOMENT DIAGRAM AND COLUMN SHEARS FOR EXAMPLE 3
1 1
1
No.
(1)
(2)
(5)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
1
1
1
2
1
2
1
1
1
2
1
2 1
1
1
1 1
4
1 5
3
1 2
1 4
2 4
1 2
2 2
4 2
4 4
FIGURE 20. LINEAR RESTRICTIONS FOR EXAMPLE 3
. -. . 50 Ib/sa.ft x 20' x 1.85
Roofr Load = k 1000
1000
5 = 1.85 /lin. ft.
100 lb./sq.ft. x 20o x 1.85
Floor Load = 1000
= 3.70k/lln. ft.
Frames are 20' on center
Load Factor = 1.85
FIGURE 21. GIVEN CONDITIONS FOR EXAMPLE 4
1
500
500
500
500
500
1152
1152
1152
1452
1452
1452
1452
1452
1452
1568
1568
1568
1568
1868
1868
1868
1868
5020
5020
70.6
138.1
92.8
1.85 /lin. ft.
1
FIGURE 22. MECHANISMS FOR EXAMPLE 4
1.4 k/lin. ft.
185' k
185'k 185'k
1k45 'k 145 k
160 k 345160 k
FIGURE 23. MOMENT DIAGRAM FOR EXAMPLE 4
50 lb./sq.ft.x20'xl.4 k ,
Roof Load = b./s.ft.x20xl.4 = 1.4 /lin.ft.
1000
Floor Load 100 lb./sq.ft.x20'xl.4
1000
= 2.8k/lin.ft.
Wind Velocity 108 MPH By A.S.C.E. Wind Load =
35.7 lb./sq.ft.
Wind Force Roof = x 20x5'xl. = 5
Wind Force 2nd Floor = 15.7 x 20 x 12.5 x 1.4
1000
= 12.5k
Frames are 20' on center
Load Factor = 1.4
FIGURE 24. GIVEN CONDITIONS FOR EXAMPLE 5
115 ' 150
Max. Positive Moment=505.2
135
12 25 k
Max. Positive Moment = 147.8lk
at 0.62' left of
100' k
122. 'k
'Ik ' k
235 275
k
1.5
FIGURE 25. MOMENT DIAGRAM AND COLUMN SHEARS FOR EXAMPLE 5
015
30k
125 K
Z 115' k
16k
FIGURE 26. MECHANISMS FOR DIAGRAM 5
TABLE 1.
INPUT TO COMPUTER PROGRAM:
Nominal
Depth
6.00
7.00
8.oo
10.00
12.00
10.00
12.00
14.00
12.00
12.00
14.00
14.00
16.oo00
16.oo
16.oo
18.00
18.00
18.00
21.00
21.00
21.00
24.00
24.00o
27.00
27.00
30.00
30.00
30.00
53.oo00
55.00
36.oo00
56.oo
36.o00
56.oo
36.oo
53.oo00
56.00
56.0oo
36.oo
36.00
56.00
Weight
Per Ft.
4.40
5.50
6.50
9.00
11.80
15.00
16.50
17.20
22.00
27.00
50.00
34.oo
56.oo00
40o.00
45.00
50.00
55.00
60.oo00
62.00
68.oo
75.00
76.00o
84.00
94.00
102.00
108. oo
116.oo
124.oo
130.00
141.00
150.00
16o.oo
170.00
182.00
194.00
220.00
250.00
245.00
260.00
280.00
300.00
PROPERTIES OF SECTIONS
M
p
7.80
11.10
15.00
25.40o
59.10
45.90
56.70
67.10
80.70
104.40
129.50
149.90
175.70
200.00
226.00
277.00
507.00
557.00
396.oo
459.00
475.00
550.00
616.00
764.00
837.00
950.00
1058.00
1120.00
1282.00
1411.oo
1594.00
1714.00
1855.00
1972.00
2110.00
2300.00
2590.00
2770.00
2960.oo
3210.00
35450.00
Area
1.30
1.61
1.92
2.64
3.45
4.40
4.86
5.05
6.47
7.97
8,81
10.00
10.59
11.77
13.24
14.71
16.19
17.64
18.25
20.02
21.46
22.57
24.71
27.65
30.01
51.77
354.13
36.45
58.26
41.51
44.16
47.09
49.98
53.54
57.11
64.75
67.73
72.05
76.56
82.32
88.17
r
x
2.57
2.74
3.12
5.85
4.57
5.95
4.65
5.40
4.91
5.06
5.75
5.85
6.49
6.62
6.64
7.58
7.41
7.47
8.55
8.59
8.64
9.68
9.78
10.87
10.96
11.85
12.00
12.11
13.25
15.59
14.29
14.58
14.47
14.52
14.56
13.79
14.88
14.95
15.00
15.12
15.17
design will allow for a moment diagram which
nowhere exceeds the moment capacities of the
members. But even the experienced designer
would know the location of the intermediate
hinges for only a very limited number of load
distributions and end moment combinations.
Two courses of action are available to
meet this problem. First, the probable region
of hinge location for each beam might be
bracketed by including two (or even more)
mechanism equations in the array of linear
restrictions. However, for each additional
basic beam equation considered, the total pos-
sible number of combinations of basic equations
would be doubled. Because of this fact, then,
this course of action is not recommended except
for frames composed of only a few members.
The second course of action takes account
of the fact that the moment capacities provided
by the minimum-weight design using standard
sections will usually exceed slightly the pre-
cise moments dictated by the loads. Therefore,
only one basic equation is included in the
linear restrictions for each beam in the frame,
with the intermediate hinge positioned by ex-
perience. After the minimum-weight design has
been determined, each beam is checked to deter-
mine whether or not its moment capacity is
exceeded at any point. It is believed that in
the majority of problems the moment capacities
of the minimum-weight design will not be ex-
ceeded. However, if the moment capacity of
any member is exceeded, a procedure similar to
that outlined in Section III D should be fol-
lowed. This course of action is illustrated
in Examples 3 and 5.
G. MINIMUM-WEIGHT PROBLEM WITH RESPECT TO TWO
OR MORE LOAD SYSTEMS
In most situations the designer must pro-
vide a structure capable of withstanding more
than one system of loading. Such situations
may arise in the case of "checkerboard" load-
ing or where, as in the case of wind and
other lateral forces, reduced load factors are
usually prescribed for combinations of such
iorces with gravity loads.
The frame which is to be designed for two
different load systems will now be considered.
Let ¢I and (II, each consisting of n sections
for the n members in the frame, be the mini-
mum-weight solutions for load systems I and II,
respectively. These solutions satisfy, respec-
tively, the requirements imposed by column
axial forces in load systems I and II, and,
respectively, the moment requirements expres-
sed by the following systems of inequalities:
[AI] {XN} _> {NI
[AII] {XI} > {bIl}
(5.17)
(5.18)
In many cases [A = A] . However, in the
general case A [A .]
Let FW and FW be the frame weights,
respectively, for the two minimum-weight
solutions.
The following possibilities must be
considered:
ij = llj
Ij ( IIj
DIj IIj
I i I( lli
Czj S ®lIj
Ilk - 0 II
Ik IIk
j = 1,2,....n (3.19)
j = 1,2,....n (5.20)
j = 1,2,....n (5.21)
i = 1,2,....r
j = r+l,r+2,...n
i = 1,2,....r
j = r+l,r+2,...s
k = s+l,s+2,...
(5.22)
(5.25)
If Equation (3.19) is satisfied, all members
of (I are identical to the corresponding mem-
bers of 0II, and the minimum-weight solution
is determined. If Equation (5.20) is satis-
fied, all members of (I exceed in size their
counterparts in 0II, and the minimum-weight
solution is 0 . Likewise, if Equation (5.21)
is satisfied, the minimum-weight solution is
SII. If Equation (5.22) is satisfied, r
members of (I and (II are identical, while
(n-r) members of I exceed in size their
counterparts in 0 . In this case, the mini-
mum-weight solution is .II" Of course, the
minimum-weight solution is oI if the sign of
the inequality is reversed.
If Equation (3.25) is satisfied, r mem-
bers of $I exceed in size the corresponding
members of II, (s-r) members of oII exceed
in size the corresponding members of I , and
(n-s) members are identical. In this case, a
minimum-weight solution has not necessarily
been determined. Suppose FWI > FWI . FWI,
then, becomes a lower bound to the minimum-
weight solution. It is quite possible, and
not uncommon, for the moment capacities repre-
sented by $I to satisfy both Equations (5.17)
and (5.18) and the most restrictive axial load-
ing from both systems. Should this happen, I
is also an upper bound, and the minimum-weight
solution is 0I. More generally, an upper
bound to the minimum-weight solution is ob-
tained with a new design I 1I= (i + (I . +
,Ik whose weight is FWIII. If the difference
in weight between FWIII and the lower bound
FWI is considered to be small enough, 0III may
be selected as the final solution. If this
difference is not acceptable, the following
procedure may be initiated. A suitable band-
width for the members of (I is determined.
The lightest design within this bandwidth
which satisfies both Equation (5.17) with
axial loads from load system I and Equation
(3.18) with axial loads from load system II
becomes the minimum-weight solution, within
the limitations imposed by the prescribed
bandwidths.
Should the structure be subjected to N
load systems, the minimum-weight design with
respect to two load systems would be compared
with the minimum-weight design for a third
load system. This process would be continued
until all N load systems had been included
in the analysis. It would, of course, be
expedient to make an effort to select the two
most dominant load systems for the first
comparison.
The minimum-weight problem with respect
to two or more load systems may be approached
in another manner by constructing a solution
space bounded by the most restrictive equations
from each load-system set. However, if the
most restrictive axial loads are taken for the
columns, and these restrictions are used in
conjunction with the composite solution space,
an upper bound to the minimum weight will be
obtained. This is because, in some members,
we may be combining the axial load P from
one load system with the bending moment M0
from another. This upper bound occasionally
may differ appreciably from the minimum-weight
solution, but it is the author's opinion that
for most practical load systems it will be a
close approximation.
H. ADDITIONAL CONSIDERATIONS
The accommodation of the bending moments
and the axial forces induced in a frame by
applied loading usually serves as the primary
criterion for the design of the structure.
After the individual members have been selected,
problems of deflection, incremental collapse,
cyclic loading, connections, clearance, etc.,
may need to be considered. Upon checking the
adequacy of the minimum-weight design against
these so-called secondary criteria, it may be
found necessary to change one or more members.
In this case, the designer may consider the
changed members to be constants in the solution,
and repeat the process as follows: The corre-
sponding variables are dropped from the objec-
tive function, and the moment capacities M
p
(and M0) of the changed members are substituted
for the corresponding variables in the array
of linear restrictions. This change results
in fewer variables, and, usually, a reduction
in the number of equations. Of course, the
new design will be found to have a greater
weight.
Another problem of this type arises when,
either initially or subsequent to the first
solution, practical considerations suggest
continuity of size in two or more members
adjacent to one another. This situation pre-
sents very little difficulty in the case of
the beam. All that need be done is to repre-
sent the M of all such sections by the
same variable X in the array of linear re-
strictions. Of course, the total length of
all members joined under one variable must be
used in determining the corresponding objec-
tive function coefficient. The section deter-
mined by the solution process for this varia-
ble is then used for all members joined under
the variable.
A similar provision for a column that is
to be continuous in size across one or more
joints presents much more of a problem. Be-
cause there may be significant differences in
axial loads and lengths of adjacent-story
columns as well as in the probable range of
MO, the weight-moment relationships could be
quite different. Therefore, the definition
of the corresponding objective function coef-
ficient becomes arbitrary. Furthermore, the
rotations of the hinges of these members must
be separated in all linear restrictions for
the trial portion of the solution because, for
a given section, each of these joined members
will generally have a different capacity MO.
The solution process should be initiated with
the columns to be joined represented by dif-
ferent variables. After the so-called secon-
dary criteria mentioned above have been sat-
isfied for all members, a section is selected
for the joined columns. The allowable value
of M0 is now calculated for each column.
The M0 values are substituted for the corre-
sponding variables in the array of linear re-
strictions, and the solution is completed as
outlined in the first part of this section.
In addition to the primary requirements
for columns discussed in Section III C, the
AISC Specification imposes the following
restrictions:
L/rx < 120
P/P 0.6
y
(3.24)
(5.25)
Furthermore, premature buckling of the webs
of I and WF shapes must be precluded in the
region of a plastic hinge. This requirement
is met if the following equation is satisfied:
d/w < TO - 100 P/Py (3.26)
except that d/w need not be less than 45.
In this equation, d = depth of section and
w = thickness of web. Equation 3.26 also
applies to beams. However, since d/vw 70
for all standard rolled shapes and the axial
compression P is usually relatively small
for beams, in this paper the restriction is
omitted for beams.
A column which does not satisfy Equation
5.26 may be used if its web is reinforced with
longitudinal stiffeners in the plastic-hinge
zones. Therefore, the minimum-weight problem
can be approached in either of two ways.
Equation 5.26 can be used at the outset to
establish a minimum section for each column;
this procedure determines a lower limit on
M just as in the case of Equation 5.8. Alter-
natively, this step may be omitted and the loca-
tions, if any, of required stiffening determined
in the resulting minimum-weight design.
Since Equation 5.26 tends to require sections
with relatively stocky webs where axial loads
are relatively large, occasionally the minimum-
weight design may contain columns which are not
economy shapes. This possibility was not con-
sidered in this paper. 0
IV. EXAMPLES
A. GENERAL REMARKS
The examples in this section are based
on A7 Steel (a = 33 ksi). Provisions of
the AISC Specification discussed in Section
III C and D have been adhered to, except that
Equation 3.26 was not enforced. Therefore,
some of the resulting frames contain columns
which would require longitudinal web stiffen-
ers in the plastic-hinge zones. Of course,
Equation 3.26 could be imposed as an addi-
tional constraint to obtain a frame that
would require no stiffeners, so that the
relative economy of the two alternatives
could be determined. The computer program
described in Section IV C was used for
Examples 3, 4, and 5, using the IBM 7090.
B. EXAMPLES
1. The given conditions for this example
are shown in Figure 16. The example will be
solved first by the linear programming method,
but without making allowance for column ac-
tion. As has been done by a number of inves-
tigators, a single linear relationship between
weight per foot and M will be assumed to hold
for the entire range of standard sections.
The objective function coefficient, then, will
consist of only the length of the correspond-
ing members. This solution will subsequently
be compared with the more precise solution
outlined in this study.
A more general method of determining the
first basic feasible solution is also described
in this example. It will be recalled from
Section II C that, for a basic feasible solu-
tion to exist, m+n linearly independent
vectors must be in solution. This requirement
is satisfied if the m+n vectors of the solu-
tion form an identity matrix. Instead of
solving for an initial basic feasible solution,
we may assume an entirely artificial one. To
do this, we simply add to our augmented matrix
an identity matrix consisting of the same num-
ber of new variables as we have equations.
These new variables must be included in the
objective function. However, we assign them
such arbitrarily large coefficients as to drive
them from solution. The final solution is not
valid unless all these artificial variables are
absent. Further explanation and proof of
validity of this technique may be found in
Reference 10.
The statement of the problem is as
follows:
Minimize
F = 40X1 + 32X2
subject to
41X > 1000
2X1 + 2X2 > 1000
where X1 = plastic moment required for beam and
X = plastic moment required for each
column
The first equation is the objective
(weight) function. The first inequality de-
rives from the beam mechanism in which all
three hinges are in the beam, while the second
derives from the same source, but with one
hinge in the beam and one in each column top.
This completes the list of mechanisms for this
frame.
The addition of slack variables X and
X4 produces the following result:
Minimize
F' = 4OX + 52X2 + OX + OX4
w 1 2 3 4
subject to
- X
2X1 + 2X2
= 1000
- X4 = 1000
Using the technique mentioned above for obtain-
ing the first basic feasible solution, we in-
troduce the artificial variables X and X6,
incorporating them in the objective function
with arbitrary (large) coefficients of 500.
Minimize
Subject to
F' = 4OX + 32X + OX
w 1 2
- X
2X1 + 2X2
In tableau form, this is written
+ OX, + 500X + 500X,
+X5
- X 4
4O 32 0 0 500 500
Basis P P2 P 4 P5 P6
0nn 4 k n -i n0 1 0
2 2 0 -1
0 1 1000
(F-c .)
Values of F.-C. are determined next.
J J
Thu
F -C1 = 500 x 4 +
After this step, the tableau appears thus:
40 52 0
Basis P1 2 P
500 P 4 0 -1
500 P6 2 2 0
(F.-C.) 2960 968 -500
This solution is not optimum because
not all F.-C. < 0. Therefore, we compute
values J by dividing each element in the
values of o by dividing each element in the
Basis
500 P5
500 P6
(F.-c.)
33J
40 52
P1 2
0 0
s, to get F -C1, we have
500 x 2 - 40 = 2960
0
P4
0
-1
-500
1 0 1000
0 1 1000
0 0
PO column by the element which is in its row
and in the column containing the largest F.-C..
J J
The tableau appears then in the following form:
500 500
P5 6 0
0 1 0 1000 250
2 2 0 -1 0 1 1000 500
2960 968 -500 -500 0 0
= 1000
= 1000
500 P6
10
1000
'+ D
^^^ J
0
Since vector P5 corresponds to the
smallest positive 0 , it goes out of solution
while P1, corresponding to the largest F.-C.,
comes in. This transfer is accomplished by
40
500
40 52
Basis P1 P2
T 1 0
P6 °0
0
P3
-1/4
1/2
(F.-C.) 0 968 2
Again, not all F.-C. < 0. The re-
J J
placement of P6 by P2 is accomplished by
4o0
dividing the first row by four, and then add-
ing minus two times the modified first row to
the second row. The result, after computation
of F.-C. and 0 , is
J J
6 0
0 250
-1 -1/2
-500 -740
1 500 250
0
dividing the second row by 2. The new tableau,
with values of F -C. computed, is
J J
4O 52 0
Basis P P2 P
40 P 1 0 -1/4
52 P- 0 1 1/4
(F j-C .)
J 3
0 0 -2 -16 -498 -484
Since all F.-C. < 0, we have ob-
J J-
tained the minimum-weight solution, for which
XI = X2 = 250 foot-kips. The initial design
selected on the basis of these theoretical
moments is an 18 WF50 for both beam and
columns. The selection results in a frame
weight of 5600 lbs.
The following designs were deter-
mined by increasing the beam one section in the
economy table for each succeeding design and
selecting the smallest section for the column
which satisfies the mechanism inequalities.
All columns also satisfy the Case II column
formula.
M (foot-kips)
277
507
557
596
475
550
M (foot-kips)
226
200
175
10o4
54
2
Column
16WF45
16WF4O
16WF56
12WF27
12Jr. 1.8
6Jr.4.4
Frame Weight (lbs.)
5440
5480
5552
5544
5298
5181
Note that six additional trials were required
to determine the minimum-weight design.
This example will now be solved using
the more realistic moment-weight relation-
ships and taking into account the effect of
axial loads on the columns.
The range in M which must be considered
for the beam is M (min.) = PL/8 = 250 foot-
kips to M (max.) = PL/4 = 500 foot-kips. The
method of averages was used for all economy
sections within this range to determine the
linear relationship
-1/2 -1/4
1/2 250
Beam
18WF50
18WF55
18WF60
21WF62
21WF75
24WF76
D2 = 5.5 + 0.178M02
The possible range in M which must be
considered for the columns is M (min.) = 0 to
M0(max.) = 250 foot-kips. Since the axial
force in the column is 25 kips (Figure 16),
values of M0 for an axial load 25 kips and a
column height 16 feet were calculated in
accordance with Equation (3.4) for all economy
sections within this range. The method of
averages was used to determine the linear
relationship
4
100 P5 (D
100 P6 2
(Fj-Cj)
Replace P5 by
Replace P by Pl
The objective function is now written in accord-
ance with Equation (3.12)
Fw = (4o)(0.098)X1 + (16+16)(0.178)X2
4X 2 + 6X2 (4.3)
In general, these coefficients should not be
rounded.
The first basic feasible solution in tab-
leau form with coefficients of the artificial
variables taken to be 100 is
100 100
P5 P6
0
1 0 1000 250
2 0 -1 0 1 1000 500
596 194 -100 -100 0 0
4 PI
100 P6
(F.j-c.)
Replace P6 by P2:
4 6
1 2
1 0
0 ©
0 194
0 0
35 4
-1/4 0
1/2 -1
49 -100
100
1/4
-1/2
-149
0 250
1 500 250
4 P
6 P2
(Fj-Cj)
Replace P2 by P :
4 6 o
S1 2 35
1 0 -1/4
0 1 1/2
0 0 1/2
0
P4
0
-1/2
-3
100 100
P5 6
1/4 0
-1/4 1/2
-100.5 -97
4 6 0 0 100 100
PI 2 3 4 P5 6 0 e
1 1 0 -1/2 0 1/2 500
0 4 1 -2 -1 2 1000
0 -2 0 -2 -100 -98
-250/4
250 250/4
4 P
0 P3
(F.-C.)
a a
D = 24.8 + O.098Mp
(4.1)
(4.2)
P
This is the theoretical minimum-weight
solution, for which X = 500 foot-kips and
X2 = 0. The initial design, selected on the
basis of these theoretical moments and an
axial load of 25 kips for the columns, is a
24WF76 for the beam and a 6 Jr.4.4 for the
columns, which agrees with the solution found
previously by trial. Of course, this is not
a practical solution, but it does suggest that
the column be no larger than practical consid-
erations require.
2. Example 2 will be defined by removing
the brace against sidesway in Example 1, but
without making any changes in loading or
geometry. We must add an inequality to pro-
vide for the restriction imposed by sidesway
of the frame. For an axial load of 25 kips
and a column height of 16 feet, Equation (3.8)
dictates a 12B16.5 as the minimum section.
Equation (3.4) is now used to calculate the
maximum value of M0 that may be used in con-
junction with the 25-kip axial load for this
section. The result is MO(min.) = 54 foot-
kips. Therefore, the following inequality is
added to the two mechanism inequalities:
X2 > 54 (4.4)
The possible range in M0 which must be
considered for the columns is- M0(min.) = 54
foot-kips to Mo(max.) = 250 foot-kips. The
method of averages is used to determine the
linear relationship for all economy sections
within this range:
D2 = 9.1 + 0.155 M02 (4.5)
Note that the constant term in this equation
is greater than the constant term in Equation
(4.2), while the slope is less, because this
line segment does not include the steeper por-
tion of the weight-moment plot near the origin
that was included in the determination of
Equation (4.2). Equation (4.1) for the rela-
tionship between weight and moment for the
beam remains the same.
The objective function, without slack and
artificial variables, is
F = (40)(0.098)X1 + (16+16)(0.155)X,
4x1 + 5x2 (4.6)
The solution is shown in Figure 17. The
theoretical minimum-weight solution yields
X1 = 446, x2 = 54.
The initial design, selected on the basis
of these theoretical moments and an axial load
of 25 kips on the columns, is a 21WF75 for the
beam and a 12B16.5 for the columns. This
design results in a frame weight of 3448 lbs.
Further trials in this region of theoretical
moments show that a 21WF68 for the beam and a
14B17.2 for the columns is a feasible solution.
The resulting frame weights 3270 lbs., which
was found to be the minimum weight for this
example. It is to be noted that the sections
comprising the minimum weight design are only
one section removed (Table 1) from the sections
comprising the initial solution. The differ-
ence in weight is ab'out five per cent.
C. DESCRIPTION OF COMPUTER PROGRAM
The computer program which was developed
to determine the minimum-weight design of
frames is described here briefly. The flow
chart is shown in the Appendix.
1. The input consists of the following
data:
a. The nominal depth and weight per foot,
the plastic moment capacity M , the area,
and the radius of gyration r of the
standard "economy" sections (Table 1).
b. The array of mechanism inequalities.
c. The axial load for each column.
d. The lengths of all members.
e. The range in M to be considered for
each beam and the range in M to be
considered for each column.
f. The bandwidth to be surveyed for each
member for the trial portion of the program.
g. The condition of the frame (braced or
unbraced) with respect to sidesway.
2. The solution is accomplished in the
following steps:
a. Linear relationships between M and
p
the weight per foot for each beam, and
between M and the weight per foot for
each column, are determined. With these
relationships and the individual member
lengths, the objective function is com-
piled.
b. If the frame is subject to sidesway, a
linear restriction is added for each
column specifying the proper lower limit
for MO.
c. The inequalities are augmented by the
slack and artificial variables to form
the first basic feasible solution as an
array of equalities.
d. The solution is effected by the simplex
method, yielding the theoretical minimum-
weight moments.
e. From these moments and the axial-load
input for columns, the lightest section
is selected for each member from the list
furnished in the input data. This selec-
tion becomes the initial solution.
f. The input bandwidth is positioned with
respect to the initial solution, and the
feasibility of all designs within this
bandwidth are determined. The feasible
designs are compared to determine the
least-weight design.
g. Each inequality is converted to an equal-
ity to determine the smallest proportion-
ate increase in load required to produce
collapse. The ratio of this smallest
proportionate increase to the factored
service load is called the load-increase
factor.
5. The output consists of the following
information:
a. All input data are output for problem
identification and reference.
b. Linear relationships between weight per
foot and M for beams and M0 for
P 0
columns. The average residual and maxi-
mum residual between the linear relation-
ships and the true values is also output
as an indication of the reliability of
the linear relationships.
c. The objective function.
d. The theoretical moments and the initial
design.
e. The least-weight design within the band-
widths specified.
f. The load-increase factor and the equation
of the corresponding mechanism of
collapse.
g. The number of cycles required by the
simplex procedure.
D. ADDITIONAL EXAMPLES
5. The given conditions for this example
are shown in Figure 18. The sections and sec-
tion-properties input are shown in Table 1.
The array of inequalities input is shown in
Figure 20. The axial loads and the range in
M and M0 input for each member are shown
at the top of the next page.
The bandwidth was set wide enough to in-
clude the full range of probable values for
M and MO. This range can be covered with
a problem this limited in size, thus insuring
the determination of the minimum-weight solu-
tion.
The linear relationships between weight
per foot and moment capacity, with average and
maximum residuals, expressed in the form of
Equations (5.5) and (3.9), are shown on the
next page.
The resulting objective function, without
slack and artificial variables, is
F' = 4.54X + 5.25X + 2.02X + 2.05X, + 2.06
Mp(min.)
288
592
p(max.)
576
784
MO(min.)
Axial Load.
0
0
48
104
56
Member
1
2
3
4
5
The theoretical moments, in
initial design are
Member
1
2
3
4
5
foot-kips, from the simplex solution and the corresponding
Theoretical
Moment
565
592
70.6
158.1
392
Investigation during the trial portion
of the program of all possible variations of
the initial solution, over the bandwidth
specified, showed that the initial solution
is, in fact, the minimum-weight design.
Equation (25) was found to be satisfied
exactly by the initial (minimum-weight) solu-
tion. Therefore the load-increase factor is
zero.
Eight cycles of the simplex method were
required in this solution. Twenty-four seconds
of machine time were required for the complete
solution.
*Since the frame is unbraced, these limits are
changed by the computer to those shown in
Equations (25), (26), and (27) of Figure 20.
**Galambos and Ketter 9)have shown that if
P/P < 0.15, MO is negligibly smaller than the
fully plastic moment M .
p
Initial
Design
21WF62
21WF62
12B22
16WF56
21WF62
Actual Moment
M M0
596
596
80.7
175.7
396
70.6
158.1
596**
In the formulation of the basic beam equa-
tions the intermediate hinges were assumed to
form in the center of each beam. Because these
hinges will form off center for certain com-
bined mechanisms, the minimum-weight solution
is checked against the yield criterion (Section
III F). An admissible moment diagram is shown
in Figure 19.
4. The two-story symmetrical frame shown
in Figure 21 is solved in this example. The
frame is subjected to only gravity loading,
and is considered to be unbraced against side-
sway. The combination of wind and reduced
gravity load will be considered in Example 5.
The inequalities to be satisfied consist of
the mechanism inequalities and the require-
ments of Equation (5.8):
o*
O*
0*
a. b! Average
i i Residual
1.22
1.70
15.8 0.155 0.97
18.0 0.155 0.89
15.7 0.157 0.82
MO(max.)
288
288
592
Maximum
Residual
2.50
5.72
2.51
1.84
1.95
Member
1
2
3
ak
26.5
26.4
bk
0.095
0.093
4
4
2 2
2 2 2
1
The first two equations are basic beam equa-
tions. The next two are combinations of beam
and joint equations (Figure 22a and b, respec-
tively), while the last two are the lower
limits of M0 dictated by the sidesway condi-
tion for the columns. It should be noted that
Member
1
2
35
Axial Load Mp(min.)
Again, the bandwidth was set wide enough to
include the full range of probable M and MO.
The linear relationships between weight
the panel mechanism was not used in any com-
bination, because the frame and loading are
symmetrical. Thus, the complete array of
inequalities has been established.
The axial loads and the range in M and
M0 input for each member are
Mp(max.)
MO(max.)
per foot and moment capacity, with average
and maximum residuals, expressed in the form
of Equations (3.3) and (5.9), are
. bk
9.0 0.155
29.0 0.089
8.1 0.175
24.8 0.107
The resulting objective function, without
slack and artificial variables, is
F' = 6.14x1 + 3.55X2 + 5.50X + 5.22X4
Member Theoretical
Moment
1 185
2 570
5 210
4 16o
The theoretical moments in foot-kips,
from the simplex solution, and the corres-
ponding initial design are
Initial
Design
16WF4)
21WF62
16WF45
16WF40
Actual Moment
M M0
p 0
74o
740
1480
36.8
16o.o
Member
1
2
5
4
Average
Residual
1.16
1.50
1.06
1.4?
Maximum
Residual
1.92
2.12
3.29
2.70
MO(min.)
The minimum-weight design determined in
the trial solution was the same as the initial
design, except that member three was reduced
to a 16WF40.
Since Equation 6 is satisfied with no
increase in loading, the load-increase factor
is zero.
Seven cycles of the simplex method were
used in this solution. Twelve seconds of mach-
ine time were used for the complete solution.
The distribution of moments given in Fig-
ure 25 shows that the yield criterion is sat-
isfied. However, this check is not required
since the minimum-weight solution was based on
the complete array of restrictions.
5. This example illustrates the deter-
mination of a minimum-weight frame based on a
reduced array of inequalities, as discussed in
Section IIID, together with the subsequent
check of the yield criterion and procedures
which may be used for revising the design when
the criterion is not satisfied.
The geometry of the frame (Figure 24) is
the same as that of Example 4. The loads to
be supported are the gravity loads of that
example plus a wind load of 56 psf. The re-
quired load factor is 1.4. The solution will
be based on the following partial array of
Member
1
2
3
4
Axial Load
0
0
Mp(min.)
140
280
inequalities:
(1)
(2)
(5)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
4 2
2 4
2 2 2
4 2 2 4
4 4 2
4 2 2
56o
1120
50
265
610
610
1120
1995
1995
1995
55.2
125.6
Equations (1) and (2) are beam equations.
Equations (5) and (4) are panel equations,
while Equations (5) through (10) are combina-
tions of beam, joint, and panel equations
(Figure 26 a, b, c, d, e, and f, respectively).
Equations (11) and (12) are the lower limits
on MO, required by the sidesway restrictions,
for the columns.
The remaining input and output data are
similar in format to those of Example 4, and
are given below without further explanation.
p(max.)
280
560
MO(min.)
MO(max.)
140
280
ak bk a' b! Average
k I 1 Residual
10.2 0.151 1.08
26.8 0.094 1.28
5.7 0.201 1.05
15.6 0.140 0.89
F' = 6.05X + 5.75X2 + 4.02X + 4.20X
w 1 2 3 4
Member
1
2
Maximum
Residual
2.07
2.45
3.03
1.80
The theoretical moments in foot-kips from the simplex solution and the initial design, are
Theoretical
Moment
14o
321
82.5
156.5
Initial
Design
14WF34
18WF60
12WF27
16WF4o
Actual Moment
M M0
p 0
The minimum-weight design determined in
the trial solution is the same as the initial
design except that member 2 is reduced to an
18WF55, and member 4 to a 16WF36. The weight
of the frame based upon this design is 5180
lbs.
Because a reduced array of linear
restrictions was considered, the moments pro-
vided by the minimum-weight design must be
checked against the yield criterion. An
attempt to determine a closure of the moment
diagram revealed that either the moment at
the center of the roof beam or the moments
at the tops of the upper-story columns ex-
ceeded the moments furnished by the minimum-
weight design for these members.
As was discussed in Section III D there
are two ways in which the solution may be
corrected. First, consideration is given to
increasing the size of the roof beam or the
upper-story columns or both, so as to satisfy
the yield criterion. A few trials indicated
an increase in the upper-story columns (member
three) to 14WF34 sections as the solution
which results in the least increase in weight.
This design satisfies the yield criterion
with the intermediate beam hinges of the col-
lapse mechanism correctly located left of
center. The weight of the frame based upon
Member
1
2
3
Theoretical
Moment
140
285.5
14o
this design is 5320 lbs.
The true minimum-weight solution is now
known to lie between the lower bound, 5180 lbs.,
and the upper bound, 5320 lbs. This is a dif-
ference of only 2.7 per cent, so that the
upper-bound design might be accepted as the
solution.
The second way in which the solution may
be completed is to increase the array of linear
restrictions, with the intention of including
restrictions which form the basis of the true
minimum-weight solution and which were omitted
in the original array. Because the yield cri-
terion was not satisfied in the region of the
roof beam and its included joints, mechanisms
in this region will be investigated. The most
obvious restriction missing in the original
array is the combination of beam and joint
mechanisms (Figure 26g) which results in the
inequality
2X1 + 2X3 > 560
Although additional restrictions may be
sought and added to the array of inequalities,
the example will now be solved with the addi-
tion of the single inequality above.
The new theoretical moments in foot-kips,
from the simplex solution and the new initial
design, are
Initial
Design
14WF34
18WF55
14WF34
16WF36
Actual Moments
Mp M
150
307
150 150
175 150
Member
1
2
3
4
The new minimum-weight design determined
in the trial solution is the same as the new
initial design, except that member four is
reduced to a 14FW34. The weight of the frame
based upon this design is 5260 lbs., which is,
of course, between the upper (5320 lbs.) and
lower (5180 lbs.) bounds determined previously.
Again, this design must be checked against
the yield criterion because the solution is
based upon a reduced array of linear restric-
tions. The distribution of moments given in
Figure 25 shows the yield criterion to be
satisfied. The reduced array, then, must con-
tain the basis of solution, and the minimum
weight has been determined. The following
comparison of the designs obtained by the two
methods of correcting for the (inadvertent)
omission of a controlling mechanism in the
original array of inequalities is of interest:
Member First Revision Second Revision
1
2
3
4
Weight
14WF354
18WF55
14WF34
16WF36
5320 lbs.
14WF54
18WF55
14WF34
14WF54
5260 lbs.
V. SUMMARY
Methods for finding the minimum weight
of plastically designed steel frames which
are to be composed of standard sections have
been based on the assumption that the weight-
moment relationships of the spectrum of stand-
ard sections can be approximated by a contin-
uous function. Some investigators have used
a best-fit, nonlinear function, others a best-
fit linear function. The first method of so-
lution of the frame in Example 1 illustrates
the latter method. It is interesting to note
that the result is a frame which is 13 per
cent heavier than that found in the same ex-
ample using methods proposed in this thesis.
The plastic moment which can be carried
by a beam-column is less than the fully plas-
tic moment in pure bending, owing to the axial
compressive force. This reduction depends on
the relative magnitude of the axial force and,
because of stability, on the slenderness of
the member. No previous investigator has
taken these factors into account. As it turns
out, the weight-moment relationship is altered
significantly for the beam-column with rela-
tively large axial compression. This effect
is taken into account in the author's method.
Instability of the frame which is un-
braced (i.e., free to deflect laterally under
gravity load) develops at loads which may be
considerably smaller than those for the braced
frame. No completely satisfactory practicable
design procedure for treating this complex
inelastic stability problem has been developed.
An interim approximation, which is believed
to be conservative for frames of proportions
likely to be found in practice, has been
proposed(5) and adopted in the American Insti-
tute of Steel Construction specifications for
plastic design.(22) Example 2 shows how this
interim provision can be taken into account in
the optimization of weight for the unbraced
frame. Although the initial solution does not
coincide with the minimum-weight solution
which was found by trial modifications of the
initial solution, the latter is only one sec-
tion removed in the table of standard sections
(Table 1) from the former for both beam and
columns.
Examples 3, 4, and 5 illustrate optimiza-
tion of frames by means of a program written
for the IBM 7090 computer. This program deter-
mines the best-fit straight-line weight-moment
relationship for each member of the frame,
based on an appropriate range of moment capac-
ity, and obtains an initial solution by the
simplex method of linear programming. The
trial sequence of the program investigates
variations in the initial solution to allow
for the fact that even the best-fit straight-
line weight-moment relationship for a limited
range of standard sections results in some
scatter, and, more importantly, the fact that
a finite number, rather than a continuous
spectrum of sections, exists.
Example 3 illustrates the solution of a
two-bay, one-story frame, subjected to sidesway,
with the attendant array of linear restric-
tions. The increase in size of the array
over that of Example 2 should be noted. The
initial solution is found to be the minimum-
weight solution.
In the solution of Example 4, only one
member of the minimum-weight frame differed
from the corresponding member of the initial
solution, and the two were adjacent in Table
1. Because of symmetry of frame and loading,
there were only six inequalities to be satis-
fied. Thus, the solution was based on the
complete- array, so that it was unnecessary to
check the yield criterion.
The particular reduced array of linear
restrictions used in Example 5 did not produce
the correct theoretical moments because one of
the equations which is part of the basis of
solution was omitted. Two different ways of
completing the solution, both of which yielded
satisfactory results, were then carried out.
This example demonstrates the feasibility of
solutions based upon reduced or incomplete
arrays of restrictions.
Examples 4 and 5 also illustrate another
aspect of minimum-weight design which was dis-
cussed in Section III G, namely, design for
two or more load systems. The minimum-weight
designs for gravity loads (Example 4), and for
the combination of gravity load with wind
(Example 5) satisfy Equation (3.20). There-
fore, the minimum-weight structure is that of
Example 4.
In all the examples, the members of the
minimum-weight frame are either identical with
or only one section removed (in the table of
economy sections) from those of the initial
solution. Of course, situations may arise in
which the theoretical moments place some mem-
bers of the initial frame two or even three
sections from those of the minimum-weight
frame. While this deviation is not serious
for the small problem, the required computation
time to examine all variations of the initial
solution within even a bandwidth of, say, four
for a larger problem may be prohibitive (Sec-
tion III E). However, the author's experience
with his method of solution points to some
restrictions that may be placed on the band-
width. In the first place, if the theoretical
moment for any member, determined by the sim-
plex solution, is found to lie at either the
high or the low limit of the feasible range of
moments, the bandwidth need be extended in only
one direction. Secondly, in the cases enumer-
ated below, bandwidths for particular members
may be restricted even further.
(1) It will generally be found that, for
the minimum-weight frame, moments in the inter-
ior columns lie at the lower limit of the feas-
ible range of moments.
(2) In exterior columns of one-story
frames braced against sidesway, moments for the
minimum-weight solution will lie at the lower
limit of the feasible range of moments if the
ratio of column height to beam span is about
0.25 or greater, but at the upper limit of the
range if the ratio is about 0.2 or smaller.
(5) Theoretical moments in exterior
columns in the bottom story of a two-story
frame braced against sidesway will generally
lie at the lower limit of the feasible range
of moments.
If the simplex solution places the theoretical
moments at any of the limits indicated above,
the designer should consider restricting the
bandwidths in the trial solution for the mem-
bers affected to a narrower width than is used
for the remainder of the members in the struc-
ture. In fact, consideration should be given
to a bandwidth of one, which amounts to holding
these members to the sections of the initial
solution.
Another possibility for the restriction of
bandwidth occurs when the theoretical moment
for any member falls close to the moment capac-
ity of one of the more efficient sections.
These more efficient sections are character-
ized by having a significant increase in M -
p
to-weight ratio over those of adjacent economy
sections of less weight. Reference to the
partial list below will show this to be the
case for the 14WF50, 16WF56, 18WF50, and
21WF62. It should be noted that each of these
is the lightest economy section for its nom-
inal depth.
Section M /Weight Per Cent
Increase
21WF62 6.4 14.5
18WF60 5.6 0
18WF55 5.6 1.8
l1WF50 5.5 10.0
16WF45 5.0 0
16WF40 5.0 2.0
16WF56 4.9 11.5
14WF354 4.4 2.35
14WF50 4.5 10.0
12WF27 5.9 --
When the theoretical moment for a particular
member falls near the moment capacity of one
of these sections, the minimum-weight solution
is quite likely to include it. Thus, if the
initial solution contains such a member, it
will probably be safe to hold it constant in
the subsequent exploration of variations in
the initial design.
For members other than those discussed
above, bandwidths should be set as wide as
computational running-time limitations allow,
if a high degree of precision in optimization
is desired. However, it should be remembered
that if bands are not wide enough to pick up
the minimum-weight design, even the initial
design will usually be only slightly heavier
than the true minimum-weight design.
It was found that if the objective func-
tion was generated from best-fit straight
lines (for the weight-moment relationships)
whose average residuals were within about 15
per cent of ak (Equation 5.5) or a! (Equation
5.9), the simplex solution generally yielded
excellent results.
* * *
VI. CONCLUSIONS
The method of optimization developed in
this report allows for the determination of
the minimum-weight design of steel frames
within the restrictions imposed in Section
III B. The method includes the effects of
axial loading, overall frame instability due
to sidesway, and the nonlinear relationship
between weight and moment capacity of standard
sections.
Although standard sections are used in
the frames of the examples, frames using
built-up sections can also be optimized, pro-
vided that a linear weight-moment equation
for the range of proposed built-up sections
is determined. Furthermore, it should be
noted that, with the built-up section, a
virtually continuous spectrum of moment capac-
ities exists, eliminating the need for the
trial portion of the computer program.
Although gable and other nonorthogonal
frames are not considered in the examples,
they can be optimized by the method proposed
and accommodated by the computer program which
was developed.
If all the attendant independent linear
restrictions are considered, a two-story, two-
bay, unsymmetrical frame subject to sidesway
represents about the upper limit to the
problems that it is feasible to solve by this
method on the 7090 computer. However, if a
reduced array is used, and the minimum-weight
solution is checked against the yield criterion,
as was done in Examples 4 and 5, most build-
ing frames which the engineer is likely to
meet can be accommodated.
* * *