EFFICIENT
OPTIMIZATION
OF SPACE TRUSSES
HOJJATADELI? and OSAMAKAMAL:
Department of Civil Engineering, The Ohio State University, Columbus, Ohio 43210, U.S.A.
(Rcwivd
I2 Dawmhrr
1985)
efficient and robust algorithm is developed for minimum weight design of space trusses with
fixed geometry employing the general geometric programming (GGP) technique. The nonlinear projamming (NLP) problem is formulated based on the virtual work method of structural analysis. The
objective function is linear, subjected to linear size and stress constraints and nonlinear displacement
constraints.
Based on an arbitrary starting point, the signomiais are condensed into ~synomials and su~uentiy
into monomials. Next, the monomials are linearized through logarithmic transformation, The resulting
linear programming (LP) problem is solved iteratively using a dual simplex algorithm until the optimal
feasible solution eventually coincides with a local optimum of the approximated problem. This solution
point is then used to formulate a new approximated problem and the same procedure is automatically
repeated until the solution of the original problem is found. Five examples are presented to illustrate the
application of the algorithm presented in this paper.
Abstract-An
GENERAL GEOMETRIC
PROGRAMMING
(GGP)
subject to
In general, the generalized or signomial (positive and
negative term constraints)
geometric programming
(GGP) problem can be expressed in the following
C,(x)
m=O,l,...,M,
----<I
~,W
form.
(5)
where C,,,(x) and D,(x) are both posynomial functions of the form:
Minimize
P,(x)
11)
4&x) =
subject to
,$,
4,Ax) = C cm,fj x2*
_
II-0
I
P,,,(x)bqm
m=1,2,...,M,
m =O,i,...,M
(6)
(2)
C,(x) = same form.
where
We will make use of the generalized arithmeticgeometric mean inequality [I] which takes the
following form:
P,(x)= f &C", fixp
n-l
r-i
m =0,1,2,...,M.
13)
In this formulation, x is the vector of design
variables, P,(x) represents a signomiaf objective function, P,(x) are m signomial constraints, qm= f 1,
qmr= f 1 and c,,,, are positive constants, u,,,,~are
arbitrary real constants, M is the number of constraints, N is the number of design variables and T,
is the number of terms in the mth constraint.
The previous GGP can be written in the following
posynomial (positive term ~onstr~nts) form through
straightforward algebraic mtnipulations [2]:
L
This inequality applies whenever U, > 0 and the w,
are non-negative ‘weights’ such that:
cup
t Associate Professor.
$ Graduate student.
1.
We will identify the terms d,(x) in eqn (6) with the
U, terms in eqn (7). The d,, terms obviously satisfy
the necessary condition, since for any vector x,
d,,(x) > 0.
minimize
x0
I
D,(x>=~d,,(x)=~U,.
(4)
,
I
(9)
Next, consider the posynomial function D,(x)
evaluated at a particular vector, X. At x = R, we
501
502
HOJJ~TADELIand OSAMAKAMAL
define the ‘weight’ of the tth term in the denominator
of the mth constraint by:
plex method, we can proceed from one iteration to
the next at only a modest amount of computation.
OJ’TJMAL
Application of the arithmetic-geometric
mean inequality of eqn (7) results in the following condensed
posynomial:
dJx)
D,(x) 2 D,,,(x,2) = l-‘j
I
w,,
[
1
W*l
.
(11)
The denominator of eqn (5) can now be substituted
by a single-term posynomial ~monomja~). Thus the
problem is expressed as a standard posynomial problem through condensation around an arbitrary starting vector. Similarly, the posynomials are condensed
again into monomials. This second condensation reduces all the constraints into single-term constraints.
Although this second condensation can be executed
at any particular vector, it will bc convenient to define
this vector as the same vector at which the first
condensation
took place. Finally,
logarithmic
transformation yields the monomials into a linear
programming form.
Following
Dembo [4], we use the following
iterative solution technique (GGP algorithm):
(1) Based on an arbitrary starting solution vector, jz,
transform the primal problem into a linear programming problem through condensing twice
and subsequently linearizing. Set the ‘No. of
cuts’ counter, k, equal to zero.
(2) Solve the resulting LP problem for an optimal
solution vector and evaluate the corresponding
x: .
(3) Evaluate the primal constraints of eqn (5) at x,+.
(a) If the constraints of the original problem
are all satisfied, then the solution xz is
optimal and terminate the algorithm.
(b) If one or more of the constraints are violated, condense the most violated constraint
twice at the solution, x$, then linearize it
through the use of logarithmic transformation.
(41 Add this new constraint (usually called cut
constraint) to the initial LP problem, increment
k by I, and return to step 2.
This algorithmic procedure is based on the successive solution of the LP problem. Step 4 adds constraints iteratively to the initial simplex tableau, each
of which reduces the allowable feasible region of
the LP problem until the optimal feasible soiution
eventually coincides with that of the original
problem.
Further, it should be noted that the LP problem at
a ‘cut’ k f I differs from the one at ‘cut’ k only in that
it has an additional constraint. Using the dual sim-
DESIGN
OF SPACE
TRUSSES
The cross-sectional design variables in optimal
design of truss systems are usually assumed to be
continuous, even if in practice only discrete values
may be available. The members are assumed to be
prismatic and each member is described by a single
design variable represented by its cross-sectional
area, xi.
The problem can be stated in the following nonhnear programming form: Find the vector of design
variables x = {xi, . . . , x,,,} such that:
W=y
i&x,--yL’x+min
(12)
i-l
subject to
design constraints
xL<x<xU
(13)
displacement constraints
r’<r<r”
(14)
aLCage”,
(19
stress constraints
where xL, XC; rL, r”, uL and u” are lower and upper
bounds on the design variables x, displa~ments r and
stresses u, respectively, and L is the vector of member
lengths. The superscript T in eqn (12) indicates the
transpose of the L vector and y is the specific weight
of the truss material.
For any given value of x, the corresponding t and
e can be computed using either the force or the
displacement method of analysis. The displacement
method of analysis is most suitable for general formulation and computer applications. Therefore, it will
be employed for analyzing the structure in the present
formulation. For a given value of x, the joint degrees
of freedom I are computed by solving the set of
simultaneous linear equilibrium equations:
Kr=R,
(16)
where K is the structure stiffness matrix and R is the
nodal load vector. If the loads are applied at the
joints, the stress vector a can be computed from the
following equation:
u =Sr,
jn which S is the structure
matrix.
(17)
stress transfo~ation
Efficient optimization
In this formulation, the elements of S are independent of the design variables. The elements of R are
constants. The elements of K arc linear functions of
the cross-sectional areas. For an explicit value of
the design variables x (cross-sectional areas), the
displacements of eqn (16) and the stresses of eqn (17)
are used to formulate the problem based on the
virtual work theory.
Let us re-state our goal before deriving the
necessary equations for solving the problem. The aim
here is to find the minimum weight design of a space
truss (fixed geometry) subjected to a set of loads
(multiple independent loading conditions). The truss
is subjected to lower-bound and upper-bound limits
placed on the member stresses, joint displacements
and member sizes (cross-sectional areas of members).
Let us consider the case of a truss with N members each of length Li and cross-sectional area x,
(i=l,.
. ., N) and subjected to a typical load case j.
If member stresses are limited to a specified tension
upper limit r~,”or compression lower limit a:, there
will be N constraints of the form:
503
of space trusses
Examining the stress constraints, eqn (18), and the
fabricational constraints, eqn (19), we realize similarity in their basic form. Thcrcfore, these equations can
be written as:
c,<x,<x:
i= 1,2 ,...,
N,
where C, is the greatest value occurring in the set
comprising x: and the values of F,,/(aF or CJ,‘)for all
load cases j. Thus, there is a total of N constraints
covering the stress and size limits, each of which has
only one term.
The nonlinear programming problem (NLP) can
now be expressed as:
W-y
:L,x,+min,
,-I
(23)
subject to
C,,<xi,<x,U
i=1,2
,...,
km.
i=l,2,.
..,N,
(18)
where Fj, is the force in member i due to the loading
casej, uu and c” are the maximum permitted tensile
and compressive stresses in member i, respectively,
and x,” is the upper-size limit of the member i.
In addition, fabricational limits may require the
following N constraints on member sizes (crosssectional areas):
x:<x,<x,~
i=1,2
,...,
N,
(19)
where xF is a strictly positive lower bound on the size
of member i.
Using the virtual work method, the deflection r, of
the joint k under the action of a typical load case j
can be expressed using the following formula:
(20)
If this displacement at joint k is to be limited to
maximum and minimum values of +rlrm, then the
constraint equation takes the following form:
(21)
In this formulation fkiis virtual force in member i
caused by the application of unit load at joint k in the
direction in which the displacement rk is restricted,
and Ei is the modulus of elasticity of member i.
One constraint is present for each constrained joint
displacement and for each loading case.
(22)
N
(24)
(25)
The nonlinearity in the previous formulation arises
from eqn (25) since the design variables are in the
denominator
of these constraints. Equation (24)
represents N design variable constraints and eqn
(25) represents the displacement constraints. Also,
it should be noted that this formulation has the
capability of handling all cases of loading at the same
time, thus eliminating the need for solving for each
loading condition separately.
Furthermore, eqn (23) is in the form of eqn (1).
Equation (25) can be readily substituted by two
equations of the following form:
N
c QlCIXl-’ < 1,
i-l
where q, = f 1 and c, are non-negative constants.
Equation (26) is in the form of eqn (2). Now the
problem is in the GGP form and can be solved using
the GGP algorithm presented earlier.
Let us now discuss some of the merits of this
formulation before presenting an algorithm for solving the truss optimization problem. Regarding the
case of statically determinate structures, the values of
4, and fki are constant and independent of the design
variables x,. This is not the case in statically indeterminate structures, where these values are dependent
on the design variables x,. The method of obtaining
a solution to the problem would be to select a starting
solution vector for variables x, (i = 1,2,. . ., N), to
analyze the structure using eqns (16) and (17), and to
find the values of 4, and hi using the selected set of
variables xi. Then, the optimization problem represented by eqns (23x25) is solved assuming that F,,
and fk,
do not vary. Next, the resulting solutions are
504
HOBAT ADELI and OSAMAKAMAI.
used to re-analyze the structure, find new values of F,,
and& and repeat the solution until the value of the
objective function is practically constant. This design
process is one of analysis/optimization/analysis/optimization, etc. This process will converge to at least a
local minimum of the problem. This convergence is
guaranteed provided that 4, andhi are not especially
sensitive to the changes in member sizes x,, which is
usually the case in this type of problem.
On the other hand, since the solution time is
directly related to the number of variables and constraints, any reduction in these numbers will speed up
the solution process considerably. In many truss
problems it is required that some members he identicat in their cross-sectional areas such as in cases
of symmetry. In this case, the number of variables
is reduced from number of truss members to the
number of different types of cross-sections; the same
thing happens with the number of eqn (24).
Also, it can be shown that if any of eqn (24) is not
assigned an up~r-bound
limit, it can be dropped in
the final formulation of the problem [2]. In cases
where none of the design variables requires an upperbound, which is a characteristic of many truss problems, the number of constraints is independent of the
number of the members or joints of the truss. It is
only dependent on the number of constraint
displacements and cases of loading, which is not large
in most problems. This characteristic enables us to
optimize large structures through solving a relatively
small number of constraints. In fact, the procedure
will be to solve the probem with only lower bounds
on the solution variables hoping that the solution
space is compact. If this condition is not satisfied and
the convergence to optimal is slow, upper bounds will
be imposed on the variables in order to provide a
compact solution space. Finally, if all the terms in
eqn (25) are positive, the negative lower bound will
be satisfied automaticaIIy. Similarly, if all the terms
are negative, the positive upper bound could be
ignored during the solution of the problem (trivial
rejection.)
Based on the aforementioned
discussion, the
problem parameters are now defined as follows:
No. of variables
= No. of different cross-sections f 1 (objective
function)
No. of constraints
= 1 (objective function) + No. of variables with
upper-bounds (if any)
+ (No. of loading conditions x 2) x (No. of
constrained displacements)
- Trivially rejected displacement constraints
+ ‘cuts’ constraints.
Finally, this method requires all the design variables to be strictly positive, thus preventing any of
them from being exactly zero even if the actual
situation at optimality requires that. In these cases,
a lower bound close to zero (practically zero, e.g.
0.01) can be assigned as a lower bound to the design
variables.
Based on the previous formulation and comments,
the following algorithm is adopted for solving the
truss optimization problem.
The alprirhm
(1) Choose any starting point (different types of
cross-sectional areas). Set the iteration counter
of the GGP algo~thm, NGGP, equal to 1.
(2) Solve the problem for the values of 4, and f;,
using eqns (16) and (17). Set up the objective
function and the constraints using eqns (23)(25).
(3) Solve for the design variables using the GGP
algorithm presented earlier (using the current
solution vector) until all the primal constraints
are satisfied with an allowable tolerance of 1%.
(4) If NGGP is equal to 1, go to step 5. If not, check
if the stopping criteria are satisfied. If these
stopping conditions are met, terminate the algorithm. The current value of the objective function
is the desired optimal solution. Otherwise, go to
step 5.
(5) Increment NGGP by 1. Define the new starting
point as the solution point of step 3.
If the truss is statically determinate, go to step 3.
If the truss is statically indeterminate, go to
step 2.
The aforementioned algorithm is based on successive use of the GGP algo~thm presented earlier. The
underlying concept of this algorithmic procedure is to
iteratively reformulate the approximate optimization
problem (based on the current solution vector)
until the value of the objective function becomes
practically constant. Thus, once the starting point has
been fed into the computer, a set of solution vectors
to the approximated
problems is automatically
generated until the optimum design is achieved.
The stopping criterion used in the following
examples is that the relative change in two successive
values of the objective function should not exceed
0.1% and all the original primal constraints shouid be
satisfied with an allowable tolerance less than 1%.
APPLICATIONS
The algorithm presented in this paper is applied to
several space trusses in order to find the minimum
weight design for each case. The machine used for
these applications was an IBM 3081-D mainframe
computer.
For each problem, a uniform structure (all members having equal cross-sectional areas) was used as
the starting point. The program automatically generates successive solution vectors to the approximated
problems until the optimum solution to the original
problem is found.
505
Efficient optimization of space trusses
Table I. Design history for example 1
Variable (in’)
Iteration
1
2
3
4
Start
2.5000
1.5770
1.4959
1.2671
1.3794
1.0380
0.8831
0.5800
0.5377
0.3399
0.1000
0.1000
2.5000
3.3215
3.8950
3.5323
3.5957
3.8972
3.9812
4.0968
3.9960
4.2669
3.4942
4.0108
2.5000
2.2070
2.0050
2.2066
I.7941
1.7063
1.4699
1.3888
1.0828
0.8092
0.7316
0.7409
2.5000
1.4958
1.3558
1.5674
I .8238
I .9037
2.1829
2.2639
2.4996
2.5152
2.8148
2.4314
I
2
3
4
5
6
7
8
9
IO
II
Fig. 1. Four-bar pyramid truss.
All the examples presented in this section are
statically indeterminate structures except example 3,
which is a statically determinate one. The dimensions
on the figures are all in inches.
Example
1 C/our-bar pyramid truss)
The four-bar space truss shown in Fig. 1 has
been solved by Morris [8] using the dual geometric
programming approach. It has the following preassigned parameters:
loads:
P, = 10 kips
P2 = 20 kips
P,= -60kips
modulus of elasticity = 10,000 ksi
specific weight = 0.1 lb/in’.
176.0
0’
I
2.00
I
4.00
I
6.00
I
6.00
174.296
143.449
143.071
142.666
143.038
141.488
140.680
138.243
135.419
130.989
121.521
120.554
Note: I in* = 6.452cm’; I lb = 4.45 N.
respectively. Using a uniform structure (all members
having equal cross-sectional areas) as the starting
point, a minimum weight of 120.554 lb is achieved in
11 interations. The active constraints at optimality
are the compression stress in member 3, the zdisplacement of joint 5, and the minimum crosssectional area of member 1. Table 1 and Fig. 2 show
the design history and the final design. It is of interest
to mention that the minimum weight for this problem
reported by Morris [8] is 130.7 lb. This value is 8%
greater than the minimum weight value obtained in
this investigation.
Example 2 your -bar pyramid 1ru.w)
The members are subjected to the stress limitations
of 225 ksi. Also, the zidisplacement of joint 5 is
limited to f0.3 in., and the cross-sectional areas of
the members should not be less than 0.1 in*.
We define the variables for this problem as follows:
x,, xz, x3 and xq are the areas (in*) of members l-4
1121)
Weight
(lb)
I
IO.00
I
12.00
ftrration
Fig. 2. Design history for example 1.
I
14.00
The four-bar space truss shown in Fig. 1 has also
been solved by Gellatly et al. [5] using the optimality
criteria approach and the following pre-assigned parameters:
load condition 1: P, = 5 kips, P2 = P, = 0
load condition 2: P, = 5 kips, P, = P, = 0
load condition 3: P, = 7.5 kips, P, = Pz = 0
modulus of elasticity = 10,000 ksi
specific weight = 0. I lb/in).
The members are subjected to the stress limitations
of 225 ksi. Also, the jl-displacement of joint 5 is
limited to kO.3 in., the z-displacement of joint 5 is
limited to f 0.4 in. and the members’ cross-sectional
areas should not be less than 0.1 in*. The variables are
defined in the same manner as in example I.
Using a uniform structure (all members having
equal cross-sectional areas) as the starting point, a
minimum weight of 14.238 lb was obtained after
seven iterations. The active constraints at optimality
are the y- and z-displacements of joint 5, and the
minimum cross-sectional area of member 4. Table 2
lists the final design and the design history. Figure 3
presents a comparison between the results obtained
by Gellatly et al. [5] and those obtained in this
investigation. It is of interest to mention that the
minimum weight for this problem reported by Gellatly et al. [S] is 14.2834 lb, which is slightly larger
than the value obtained in this research. They re-
HOJJATADELI and OSAMAKAMAL
506
Table 2. Design history for example 2
Variable (in*)
2
3
Iteration
I
start
:
0.2270
0.2241
0.2582
0.2270
0.3012
0.3091
3
4
5
6
7
0.2516
0.2326
0.2475
0.2388
0.2320
0.3211
0.2895
0.3148
0.2751
0.3294
Note: 1 in*
=
4
Weight
(lb)
0.2270
0.2039
0.1622
0.2270
0.1277
0.1375
15.826
14.251
14.344
0.1746
0.2202
0.1890
0.2340
0.2073
0.1189
0.1158
0.1133
0.1112
0.1000
14.235
14.287
14.235
14.357
14.238
6.452 cm’; 1 lb = 4.45 N.
ported the following cross-sectional
mality (in*):
areas at opti-
x, = 0.2134, x1 = 0.3363,
Plan
I
x,= 0.1508.
x3=0.164,
X
5
6
3
2
Y&7
6
7
Example 3 (12-bar space rruss)
Fig. 4. Twelve-bar space truss
The
algorithm presented in this paper has been
applied to the ACOSS-FOUR (Active Control Of
Space Structures) finite element model [7) shown in
Fig. 4. For this problem, the determinancy option
was used to avoid analyzing the structure more than
once.
The dimensions (in.) are given in Table 3. The truss
has the following pre-assigned parameters:
load condition 1: PI
load condition 2: P,
modulus of elasticity
specific weight
=
=
=
=
-60 kips, P, = 0
15 kips, P, = 0
10,000 ksi
0.1 lb/in’.
IS.00
r
The allowable stresses for the members are 20 ksi
in tension and 15 ksi in compression. The z- and
y-displacements of joint 1 are not allowed to exceed
0.25 and 0.4 in., respectively. The cross-sectional
areas of the members are not allowed to be less than
0.1 in2.
Due to symmetry of the topological configuration,
the truss is required to be symmetric in its element
cross-sectional areas. This condition groups the members in three different groups, shown in Table 4.
Using a uniform structure (all members having the
same cross-sectional areas) as a starting point and
assigning a relative change less than 0.1% for two
successive values of the objective function as the
stopping criterion, the program performed 18 iterations without stopping. Choosing a stopping criterion
of l%, an optimum weight of 143.64 lb was obtained
after only 2 iterations. The active constraints at
optimality are the y- and :-displacements of joint 1.
Table 4 and Fig. 5 summarize the iteration history
and the final design.
Table 3. Coordinates of the joints of example 3
Node
14.2014.00
I
1.00
1
2.00
1
2.00
I
4.00
I
6I)o
I
6.00
I
T.00
Itrrotion
Fig. 3. Design history for example 2.
,
0.00
X (in.)
1
2
3
4
5
6
7
;
00.00
-60.00
60.00
00.00
- 72.00
-48.00
48.00
-24.00
72.00
IO
24.00
Note: 1 in. = 2.54 cm.
Y (in.)
oo.ooo
- 34.644
- 34.644
69.282
- 13.856
- 55.426
- 55.426
- 13.856
69.282
69.282
Z (in.)
121.98
24.00
24.00
24.00
00.00
00.00
00.00
00.00
00.00
00.00
Efficient optimization
of
Table 5. Loading conditions for example 4
Table 4. Design history for example 3
Variable
Member
I
I. 3. 4
2
3
2; 5: 6
7, 8, 9,
10, II, 12
Weight (lb)
Start
(in2)
Iteration I
(in2)
Iteration 2
(in’)
Loading
condition
Loaded
node
X
3.0
3.0
3.0
2.3214
0.970 1
1.2147
2.2282
0.9455
1.4431
I
I
1.0
277.1
143.23
143.64
2
2
3
6
:
0.0
0.5
0.5
0.0
0.0
Note: I in* = 6.452 cm’; 1 lb = 4.45 N.
’
’
’
10.0
10.0
0.0
0.0
20.0
-20.0
Z
-5.0
-5.0
0.0
0.0
- 5.0
-5.0
Table 6. Allowable stresses for example 4
The 2%bar transmission tower shown in Fig. 6 has
been solved by Venkayya et al. 1121,Gellatiy ef 01. [S]
and Khan and Willmert [6] using the optimality
criteria approach, Tempelman and Winterbottom
[ 1I] using the dual geometric programming approach,
Schmit and Farshi [9] using the inscribed hyperspheres technique, Schmit and Miura [IO] using both
the CONstrained function MINimization (CONMIN)
and NEW Unconstrained Sequential Minimization
Technique (NEWSUMT) and Chao et of. (31 using
the reduced quadratic programming technique.
This space truss is subjected to the two independent
cases of loadings shown in Table 5. The structure is
required to be doubly symme&ric about the X and Y
axes. This condition groups all the truss members
into eight groups, given in Table 6.
In addition, the maximum displa~ments at nodes
I and 2 are not allowed to exceed _+0.35 in. in the X
and Y directions. The maximum allowable compressive and tensile stresses in the truss members are
given in Table 6. The compressive stresses are based
upon allowable buckling in thin wall circular tubes.
The cross-sectional areas of the members are not
allowed to be less than 0.01 in’. The modulus of
elasticity is assumed to be 10,000 ksi and the specific
weight is 0.1 Ib/in3.
o>’
Load (kips)
Y
Note: I kip = 4.45 N.
Example 4 (2S-bar space truss)
IS00
507
space trusses
’
’
’
’
Members
Compression
(ksi)
Tension
(ksi)
I
I
2
3
4
5
6
7
8
2, 3, 4, 5
6. 7, 8, 9
10, II
12, 13
14. 15, 16, 17
18, 19, 20, 21
22, 23, 24, 25
35.092
11.590
17.305
35.092
35.092
6.759
6.959
11.082
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
Variable
Note: 1ksi = 6.89 MPa.
Using a uniform structure (all members having
equal cross-sectional areas) as a starting point, the
program was allowed to perform nine iterations in
order to compare the results with those reported in
the literature. An optimal solution of 545.66lb is
obtained after six iterations. Table 7 summarizes the
design history and the final design. It is found that
the active ~nstrain~ at optimality are they-displacement of joints 1 and 2, the compression stress in
members 19 and 20, and the minimum cross-sectional
.oQ
Itoration
Fig. 5. Design history for example 3.
Fig. 6. Twenty-five-bar transmission tower.
508
HOJJAT ADELI
and
&AMA
KAMAL
Table 7. Design history for example 4
lteratlon
wt (lb)
I
Start
734.20
2
543.39
563 76
3
545.76
4
544.71
5
544.54
6
545.66
Note: 1lb = 4.45 N
area of members 1 and 10-13. Figure 7 shows a
compdrison
between the results obtained
by
Gellatly [5], Templeman [I I] and the results obtained
in this investigation. Table 8 lists a comparison
between the optimal solutions reported in the literature and the present work. Note that number of
iterations in the present approach is less than other
methods used for optimization of space trusses.
Example
5 (72-bar
Finally. the cross-sectional arca\ of thz mcmbcr\ are
not allowed to be less than 0.1 in’. The modulus of
elasticity is assumed to be 10,000 ksi and the specific
weight is 0.1 lb/in’.
Using a uniform structure (all members having
equal cross-sectional areas) as the starting point, the
program was allowed to perform eight iterations in
space Iruss)
The 72-bar space truss shown in Fig. 8 has been
solved by Venkayya et al. 1121,Gellatly et al. [S] and
Khan and Willmert [6] using the optimality criteria
approach, Schmit and Farshi [9] using the inscribed
hyperspheres technique, Schmit and Miura [IO]
using both the CONstrained function MINimization
(CONMIN) and NEW Unconstrained
Sequential
Minimization Technique (NEWSUMT) and Chao
et al. [3] using the reduced quadratic programming
technique.
This space truss is subjected to the two independent
cases of loadings shown in Table 9. The structure is
required to be doubly symmetric about the A’ and Y
axes. This condition groups all the truss members
into 16 groups, shown in Table 11.
In addition, the maximum displacements of the
uppermost nodes are not allowed to exceed + 0.25 in.
in the X and Y directions. The maximum allowable
stress limits are &25 ksi in all the truss members.
Typicat storey
Fig. 8. Seventy-two-bar space truss.
750.0r
680
0
r
5oo.oo
’
1.00
I
2.w
1
3.00
I
4.00
0
Rd.
5
.
Rd.
II
.
Thlr Work
I
3.00
I
s.00
Itrrotion
Fig. 7. Design history for example 4
I
1.00
J
1.00
330.01
0
1
, 00
’
2.00
I
3.00
I
4.00
I
5.00
t
6.00
Itoration
Fig, 9. Design history for example 5.
L
7.00
,
1.00
Efficient optimization of space trusses
509
Table 9. Loading conditions for example 5
Loading
condition
Loaded
node
:
17
17
18
x
5.0
0.0
0.0
0.0
0.0
:‘o
Load (kips)
Y
5.0
0.0
0.0
0.0
0.0
Z
-5.0
-5.0
- 5.0
-5.0
-5.0
Note: I kip = 4.45 N.
fible
IO. Design history for
example 5
lleratlon
Weight (lb)
Start
656.879
403.776
379.609
379.306
1
2
3
Note: 1 lb = 4.45 N.
order to compare the results with those reported
in literature. An optimal sot&ion of 379.306lb is
obtained after three iterations. Table 10 summarizes
the design history and the final design. It is found that
the active constraints at optimality are the x-displacement of joint 17, the y-displacement of joint 17, the
z-displacement
of joints 17-20, the compression
stress in members 55-58 and the minimum crosssectional areas of members 13-18, 31-36 and 49-54.
Figure 9 shows a comparison between the results
obtained by Gellatfy et al. [S] and the results obtained
in this work. Table 11 lists a comparison between the
optimal designs reported in the literature and present
work.
The number of iterations for convergence to the
optimum solution is three in the present approach
while number of iterations in the other methods
varies from 8 to 22.
CONCLUSIONS
An efficient and robust algorithm and computer
program is developed for optimization of space
trusses (fixed geometry) using the general geometric
programming technique. The program is capable of
handling multiple loading conditions, initial stresses
in members and settlement of supports. Compared to
other optimization techniques used for solving the
truss optimization problem, this method proves to be
very efficient in terms of computation time and
storage.
In addition, general geometric programming technique can be coded in a very general manner since
it does not suffer from the shortcomings associated
with many other optimi~tion
techniques, such as
the concern about the status of the constraints at
optimality (loose or binding).
Furthermore, the present formulation of the probIem based on the principle of virtual work makes the
GGP algorithm more attractive. In this formulation,
I. 2, 3, 4
5, 6, 7, 8, 9, IO, II, 12
13, 14, IS. 16
17, 18
19. 20, 21, 22
23, 24, 25, 26, 27, 28, 29, 30
31, 32, 33, 34
35, 36
37, 38, 39, 40
41, 42, 43, 44, 45, 46. 47, 48
49. 50, 51, 52
53, 54
55, 56, 57, 58
59, 60, 61, 62, 63, 64, 65, 66
67, 68, 69, 70
71, 72
I
2
3
4
5
6
7
8
9
10
11
12
13
I4
15
16
Note: I in* = 6.452 cm*; I lb = 4.45 N.
Weight (lb)
Iterations
Members
Variable
381.2
12
1.818
0.524
0.100
0.100
I .246
0.524
0.100
0.100
0.61 I
0.532
0,100
0.100
0.161
0.557
0.377
0.506
Venkayya
395.97
8
I .4636
0.5207
O.IiWO
0.1000
1.0235
0.5421
O.IODO
0.1000
0.5521
0.6084
0.1000
0.1000
0.1492
0.1133
0.4534
0.3417
Gellatly
388.63
22
2.0780
0.5030
0.1000
0.1000
1.1070
0.5790
0.1000
0.1000
0.2640
0.5480
0.1000
0.1510
0.1580
0.5940
0.3410
0.6080
379.64
9
I .8850
0.5125
0.1000
0.1000
I .2670
0.5118
0.1000
0.1000
0.5233
0.5173
0.1000
0.1000
0.1565
0.5458
0.4105
0.5699
379.79
8
I .8850
0.5118
0.1000
0.1000
I .2680
0.511 I
0.1000
0.1000
0.5228
0.5161
0.1000
0.1133
0.1558
0.5484
0.4105
0.5614
Optimal areas (in*)
Schmit and Miura
NEWSUMT
CONMIN
results for example 5
Schmit and
Farshi
Table 11. Comparative
387.67
IO
I .8589
0.5259
0.1000
0.1000
I .2526
0.5244
0.1000
0.1000
0.5814
0.5273
0.1000
0.1583
0.1519
0.5614
0.4378
0.5317
Khan and
Willmert
379.62
8
1.8321
0.5119
0.1000
0.1000
I .252 I
0.5241
0.1000
0.1000
0.5127
0.5289
0.1000
0.1000
0.1565
0.5493
0.4061
0.5550
Chao
et al.
379.31
3
2.0259
0.5332
0.1000
0.1000
1.1567
0.5689
0.1000
0.1000
0.5137
0.479 1
0.1000
0.1000
0.1579
0.5501
0.3449
0.4984
This
work
Efficient optimization
the number of constraints in the problem is not
dependent on the size of the problem (number of
joints and members). Instead, it is dependent on the
number of loading conditions and constrained displacements, which are not large in most practical
problems. This feature allows one to solve large truss
problems by considering a fairly small number of
constraints.
of space trusses
6.
7.
8.
REFERENCES
1. C. S. Beightler, D. T. Phillips and D. J. Wilde, Foundulions of Optimizations.Prentice-Hall, Englewood ClilTs,
NJ (1979).
2. C. S. Beightler and D. T. Phillips, Applied Geomerric
Programming. John Wiley, New York (1976).
3. N. H. Chao, S. J. Fenves and A. W. Westerberg,
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0p:imum Structural Design (Edited by E. Atrek, R. H.
Gallagher, K. M. Radgsdell and 0. C. Zienkiewicz).
John Wiley, New York (1984).
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5. R. A. Gellatly, L. Berke and W. Gibson, The use
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9.
10.
II.
12.
511
AFFDL, Proceedinas of 3rd conference on matrix
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