DESIGN OPTIMIZATION OF TRIPOD
TRUSS: SLP APPROACH
Goteti Chaitanya
R.V.R&J.C College of Engineering (A),
Andhra pradesh India
E-mail: chaitanyagoteti16@gmail.com
Reddy Sreenivasulu
R.V.R&J.C College of Engineering (A),
Andhra Pradesh India
E-mail: rslu1431@gmail.com
Submission: 23/07/2014
Accept: 06/08/2014
ABSTRACT
The efficiency of sequential linear programming technique in optimizing
nonlinear constrained structural optimization problems is studied in this paper
considering tripod truss structure as a case study. The problem is formulated
for minimum weight considering localized buckling stress, Euler buckling stress
and direct compressive stress as constraints. The axial force in each of the
members of the truss due to payload is estimated using vector mechanics. The
structure is optimized considering mean diameter and payload height as design
variables. The weight of the truss got reduced by 20.51%.The optimum values of
design variables obtained are compared with the values obtained using graphical
method. The optimum values of objective functions obtained using both the
approaches are in reasonable agreement with a mere 5.17% variation.
Keywords: Sequential
Linear programming, mean diameter, height, buckling stress.
1. INTRODUCTION
The
optimization of nonlinear multi variable constrained problems can be broadly
addressed using four approaches. The heuristic search methods (eg: box method),
methods of feasible search direction (Rosen, zoutendijk’s...etc), sequential
linear and quadratic methods, using sequential unconstrained minimization
techniques (Interior, exterior penalty methods and Augmented Lagrange methods).
Rao (2009)
presented in detail various nonlinear constrained optimization techniques,
their relative advantages and limitations. The sequential linear programming
has the following advantages over other methods. Unlike box method, SLP doesn’t
insist that the starting design vector should be a feasible design vector.
The
rate of convergence in most of the methods based on feasible search direction
depend on the choice of initial starting design vector and step length as the
gradient value of the function evaluated at the starting design vector and step
length influences the successive design vector. In case of SLP (Sequential
linear programming), the nonlinear problem is solved as a series of LP (Linear
programming) problems without relying on random search direction and step
length. This ensures faster convergence compared to gradient (feasible
direction) methods.
Penalty
function and Augmented Lagrange approaches cannot be applied independently to
many structural design problems as it is very difficult or sometimes nearly
impossible to express design variables explicitly in terms of penalty
parameters upon partial differentiation. These penalty methods have to be
applied in conjunction with any of the nonlinear unconstrained methods.
This
makes the process complex, highly iterative involving large computational time
and effort. On the other hand SLP (Sequential Linear programming) is computationally
simple requiring less computational time and effort. SLP also known as cutting
plane algorithm was first introduced by Cheney and Goldstein and later improved
by Kelly.
Deb (2009)
presented in detail with examples, the Frank-Wolfe method which is another SLP
technique. It also works on the principle of linearization of objective
function and constraints and solves a sequence of LPPs to arrive at optimum.
However, it relies on the parameter α є
(0,1) for generation of successive points in a unidirectional search approach.
The major limitation of this method is that in highly nonlinear problems, the
search is limited to a small neighborhood of the start point. The present
problem is modeled as a multi variable nonlinear constrained optimization
problem.
Local
buckling stress, Euler’s buckling stress and direct compressive stress are
considered as constraints to the optimization problem. Schafer and Asce (2002)
presented various empirical models for localized buckling of thin walled
columns and struts depending upon end conditions, t/w or t/d ratio and section
geometry.
Mamaghani
(2004) studied the influence of ratio parameter (t/d), slenderness ratio,
residual stress on the ultimate strength of concrete filled thin steel columns.
Bradford, Hy and Uy (2002) established slenderness limits for various circular
thin walled steel tubes by giving the
localized buckling stress its due importance.The problem so
formulated with above mentioned constraints and variables is optimized using Kelly’s SLP approach and
graphical method of optimization.
2. FORMULATION OF THE PROBLEM
A
tripod truss with the following specifications is considered as the case study
problem. Elastic modulus (E)= 207x109 N/m2, Density (ρ)=7800 kg/m3, payload
(p)=111kN and yield stress (σy)=414x106 N/m2 and Poisson’s ratio υ=0.3.The geometry
of the tripod truss is shown in Figure 1. The truss is made of three identical
members of hollow circular section arranged in the manner shown. The coordinate
positions A, B, C and D of the truss are estimated from the geometry of the
figure.
The
axial forces in each of the members of the truss AD, CD and BD are estimated as
follows.
(1)
(2)
(3)
Figure 1: Geometry of tripod truss
The
resultant is formed as algebraic sum of equations 1 to 3 and =111kN. For equilibrium of
system of forces, the resultant R=0
For, each of i, j, k equal to zero
Equating each i, j, k equal to zero we have
i.e.,
The
objective of minimizing weight of truss as function of design variables mean
diameter (d) and height of the truss (h) is expressed as:
(4)
For
The
Euler and local buckling stresses and direct compressive stresses are expressed
as:
(5)
The
localized buckling is given by Schafer’s empirical relation [Ref no:3] as:
(6)
The
direct compressive stress may be expressed as:
(7)
Therefore,
the problem of minimizing the weight of the tripod truss structure for the
given set of design variables subject to various stress constraints stated can
be expressed as follows:
Subject to
3. SEQUENTIAL LINEAR PROGRAMMING TECHNIQUE
The
flow chart shown in Figure 2 illustrates the working of Sequential Linear
programming technique. In case of sequential linear programming technique, the
starting design vector need not be feasible. However, for the present problem,
a feasible design vector is chosen satisfying all the constraints for a
possible reduction in the number of iterations. The starting design vector for
the present problem is (x1=0.9m, x2=0.08m). The linearized objective function
and constraints based on the starting design vector for the formulation of
initial simplex table are given by equations 8 to 11.
(8)
(9)
(10)
(11)
Figure 2: Flow Chart for Sequential Linear Programming
technique
4. RESULTS AND DISCUSSION
The
linearized objective function and constraints given by equations 8 to 11 are
optimized by forming LPP and deploying two phase simplex scheme. Table 1
presents the results of the final optimized simplex. From the table, it can be
seen that the optimum value of the objective function is 469.31 N at x1=0.3
meters and x2=0.0701meters.
A
20.51% reduction in weight of the truss is observed from a starting value of
590N for an initial design vector of X= (0.9m, 0.08m). The values of the
optimum design variables satisfied the original nonlinear constraints and the
need for relinearization of constraints did not arise for the particular
problem as the condition gj(Xi+1)≤ε is satisfied for all j=1 to 3, for the
chosen value of ε=0.001.
This
can be attributed to the fact that the initial design vector X=(0.9m, 0.072m)
chosen is feasible, in spite of the fact that SLP doesn’t insist for a feasible
starting design vector. The SLP program generated a local optimum in the close
neighborhood of the initial design vector. Figure 3 presents the optimum values
of the nonlinear objective function and constraints from graphical approach.
From
the graph shown in Figure 3, Xopt= (0.22m, 0.067m), which yielded a value of
445N to the objective function. The optimum values of objective functions from
SLP and graphical approaches differed by 5.17%. The variation in the optimum
values of objective function obtained using the two approaches can be explained
using Figure 4.
From
Figure 4, it can be observed that the points c, e and f fall outside the
feasible space and point “a” corresponds to the actual optimum lying on the
boundary of feasible region. Each stage of linearization produces only an
approximate linear function which may not satisfy all the constraints given by
gj(X). As seen from Figure 4, to move close to the point “a”, a series of
linearization steps are required which in turn depend upon the order of
nonlinearity, convexity of the function and the chosen value of starting design
vector. Therefore, a small positive quantity ε is chosen as the convergence
criterion to minimize the iterative steps. The value of the parameter “ε”
chosen influenced the variation in the results.
Table
1: Optimum Simplex table from two phase simplex method
Figure
3: Optimum solution for the nonlinear constrained problem using Graphical
approach
Figure
4: Graphical representation of SLP approach:
Source: Rao, 2009
5. CONCLUSIONS
The
following conclusions are drawn from the present work:
·
The efficacy of Sequential linear programming
technique in optimizing nonlinear constrained structural engineering problems
is studied in this paper.
·
A 20.51% reduction in weight of the truss is found
using SLP approach.
·
The design variable x1(Height of the truss
h) predominantly influenced the optimum value of the objective function.
·
The optimum value of design variable x2
(Mean diameter) did not oscillate much from the starting feasible value due to
the linear restriction imposed on the design variable (x2≤0.072).
·
The value of the chosen convergence parameter “ε”, influenced
the variation in results obtained from the two approaches (SLP and graphical).
REFERENCES
BRADFORD, M. A.; HY, L.; UY, B. (2002) Slenderness
limits for circular steel tubes. Journal
of Constructional Steel Research, v. 58, p. 243-252.
DEB, K. (2009) Optimization for Engineering Design:
Algorithms and examples, PHI Pvt ltd.
MAMAGHANI, I. H.
P. (2004) Seismic design and retrofit of thin walled steel tubular columns, 13th world conference on earth
quake Engineering, august, p.1-15.
RAO, S. S.
(2009) Engineering Optimization-Theory
and Practice, 4th edition, John-Wiley & sons.
SCHAFER, B. W.;
ASCE, M. (2002) Local, distortional and Euler buckling of thin walled columns, Journal of structural Engineering, March,
p. 289-299.