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International Journal of Advanced Structures and Geotechnical Engineering ISSN 2319-5347, Vol. 02, No. 01, January 2013
IJASGE 020105 Copyright © 2012 BASHA RESEARCH CENTRE. All rights reserved.
Optimal Geometry of Pin-Jointed Plane frames using a Hybrid Complex Method
B
ASHIR
A
LAM
1
,
M.
I.
H
AQUE
2
,
Q
AISAR
A
LI
1
,
K
HAN
S
HAHZADA
1
,
S
YED
M
OHAMMAD
A
LI
1
,
A
FZAL KHAN
1
,
M
UHAMMAD
I
BRAHIM
1
1
Department of Civil Engineering, University of Engineering and Technology Peshawar, Pakistan
2
School Engineering and Applied Sciences, George Washington University, Washington DC, USA Email: bashir@gwmail.gwu.edu, mhaque@gwu.edu, drqaisarali@nwfpuet.edu.pk, shah_civil2003@yahoo.com bridge_doctor@yahoo.com, afzal5426@gmail.com, ibrahim_004@hotmail.com
ABSTRACT:
This paper describes a computational procedure of obtaining minimum weight design of pin- jointed plane frames using design variables that specify both the skeletal geometry and member sizes of the structure. The entire design space is decomposed into two subspaces
—
the geometric design space, and the member design space. The variables of the geometric design space contain the coordinates of the variable joints of the structure, while the variables of the member design space contain the cross sectional dimensions of the members. The optimal design vector of the geometric design space is found strictly by the mathematical programming technique, and the complementary part of the design vector in the member design space is found by the fully-stressed optimality criterion. To validate the applicability and proficiency of the proposed procedure a numerical problem has been optimized and the results are compared with the previous results available in the literature. The study demonstrates that the developed algorithm is capable of finding near-optimal design in a some what robust and efficient manner.
Keywords:
Shape Optimization, Pin-Jointed Frames, Box Complex Method, Fully Stressed Design
Introduction:
The goal of structural optimization is to select optimal values of the design variables such that the predefined objective function is minimized (or maximized) and all the restrictions imposed by explicit and implicit constraints are met. The objective function is a function of the design variables, which provides a basis for choice between alternative acceptable designs. It can be the weight or stiffness of the structure, cost of material, or any combination of these or similar factors. The constraints could be explicit such as the restriction on the width or height of the structure, or implicit such as the restrictions on stresses and displacements. Shape or geometrical optimization introduces additional design variables, which allow for boundary movement. The coordinates of joints and the cross-sectional sizes are treated as design variables and optimized simultaneously. In general, the design variables are assumed to be continuous and numerical search algorithm is used to find optimum. In order to overcome the difficulties in obtaining the first order information in shape optimization and shortcomings of local and global optimization search methods, this paper presents a reliable and efficient numerical procedure, referred to as the Hybrid Complex Method. This procedure is developed by combining a modified version of the Complex Method of Box with the fully-stressed optimality criterion which treats problems with continuous design variables. Successive improvements in the design are achieved quite effectively by decomposing the design space into two spaces: the geometric design space and the member design space. The variables of the geometric design space are obtained by a modified version of the Complex Method of Box, while the variables of the member design space are obtained by using the fully stressed design criterion. The design routine allows multiple load cases and design variables linkage. The Complex Method of Box [1] has been applied successfully to relatively small academic nature of structural optimization problems in the past (Fu [3], Haque [5, 6], Lai [7], and Lipson [8-12]), and still accomplishes the goals set for it in this study. The Complex Method is a mathematical programming procedure for finding an optimal solution of non-linear, constrained optimization problems. This method derives its acronym COMPLEX from two words, Constrained and Simplex. The Complex Method was proposed srcinally by M. J. Box in 1965, where he demonstrated efficacy of the method in finding near optimal solution to non-linear, constrained optimization problems. It is a Zero-order method optimization method; that is, it does not require either the gradient of the objective function, or that of the constraints. To solve an optimization problem, a computational methodology is developed consisting of three logically separable phases: the optimization phase, the structural analysis phase, and the design evaluation phase. During the optimization phase, attempts are made to improve by finding feasible points that are successively closer to an optimum. In the structural analysis phase, the structure, provisionally obtained in the optimization phase, is analyzed and, finally, the feasibility of the structure is checked in the design
BASHIR ALAM, M. I. HAQUE, QAISAR ALI, KHAN SHAHZADA, SYED MOHAMMAD ALI, AFZAL KHAN, MUHAMMAD IBRAHIM International Journal of Advanced Structures and Geotechnical Engineering ISSN 2319-5347, Vol. 02, No. 01, January 2013, pp 24-30 evaluation phase. The developed procedure is applied to the optimization of a pin-jointed plane frame used by several previous investigators to study minimum weight designs constrained by allowable stresses, joint coordinates and member sizes. The formulation worked quite well, and generally converged to better solutions than those reported in the literature.
Proposed Modifications and Implementation of the Complex Method
The modifications to the Complex Method as used in this study are summarized as follows.
1) Separation of design space
In the Hybrid Complex Method proposed in this study, the entire design space is decomposed into two subspaces: the geometrical design space and the member design space. The variables of geometrical design space consist of the unknown joint coordinates while variables of the member design space define the cross sectional dimensions of the members. During the Optimization process, the variables of the geometrical design space are strictly obtained by the Complex Method and the variables of the member design space are found by using the fully-stressed optimality criterion.
2) Feasibility of the initial design
An initial point (skeletal geometry) in an n-dimensional geometric design space is chosen. In the srcinal procedure this point was required to be feasible, but the present algorithm has been written in such a way that if the initial chosen point is not feasible it is made feasible by adjusting one or more of the coordinates of the design vector.
3) Satisfying the implicit constraints
In the srcinal procedure proposed by Box the points in the initial complex which violated the implicit constraints were moved halfway back towards the centroid of the remaining, already accepted points. The process of moving halfway in towards the centroid is repeated until the point becomes feasible. In the present method, an attempt is made to satisfy all the implicit constraints for each randomly chosen point during the sizing of the members in the member design space, using the fully-stressed optimality criterion. If it is impossible to satisfy all the implicit constraints by this method, then, the srcinal procedure is used.
4) The improvement procedure
The improvement procedure has been modified in that at every iteration the worst design is reflected through the centroid of the remaining designs in the geometrical design space to a new point. Then, when this new point has been optimally sized, its objective function is evaluated and compared with that of worst design in the complex. If the new point is less, it is accepted as a design improvement and termination criteria are checked; if greater, instead of continuously halving
, it is halved only thrice and then centroid is considered as a candidate for improvement. If centroid is still greater than the worst, then a new point is located at the midpoint of a line joining centroid to the best point in the complex. If the objective function is still greater than the worst, then the worst point is replaced by the best design in the complex.
5) Termination criteria
The procedure in 3 is repeated until a preset termination criterion is reached. The first termination criterion used in this study is based on the objective function values of all
k
points in the complex. This convergence criterion is met if the ratio of the difference between the maximum objective function value and the minimum objective function value to maximum objective function value of the points in the complex is less than or equal to the value of
(a user defined variable). The second criterion that is checked for the convergence of the solution is a measure of the design space spanned by the vertices of complex. Finally, a constraint is placed on the maximum number of iterations that may occur before terminating the optimization. The optimization process is terminated as soon as any of the termination criteria is satisfied.
Sizing of Members
The design procedure used is an iterative technique, and commences with the smallest cross sections of the members whose skeletal geometry has been established by the complex method. The program takes each load condition and calculates the deflections and forces. It determines the maximum axial force for each member of the structure. After completing the final loading condition, it summarizes the condition of forces and deflections, by selecting their highest value from the loading conditions for each member. The cross-sectional design of the members is undertaken in the member design space. During this phase of calculations, the skeletal geometry of the structure is regarded as fixed, while its member cross-sections are determined based on fully-stressed design. It then considers each group of members and selects the maximum value for that group. The program substitutes the modified cross-sectional dimensions for the members, making the necessary alterations to the stiffness matrix. The analysis and design cycle is repeated until the program is unable to modify any group of members. This constitutes the final design for the structure whose skeletal geometry has been established by the complex method.
Example: Eighteen-Bar 2-D Pin-jointed Frame
The example is used to demonstrate the capability of the Hybrid Complex Method to determine the best shape and member sizes of the eighteen-bar 2-D Pin- joint frame, in order to minimize its weight without exceeding the permissible stresses. This structure with identical load condition has been analyzed in previous works by Hansen and Vanderplaats [4], and Felix and Vanderplaats [2].
Optimal Geometry of Pin-Jointed Plane frames using a Hybrid Complex Method International Journal of Advanced Structures and Geotechnical Engineering ISSN 2319-5347, Vol. 02, No. 01, January 2013, pp 24-30 The figure 1 shows initial geometry, loading condition, member and node numbering for the structure. The design data used in this example are given in table 1. All members are solids and circular in cross-sections, thus the dimension of each member is the dimensional design variable. To satisfy the practical construction requirements, the dimensional design variables are further divided into four independent groups as Group A [
1 4 8 12 16
d d d d d
]
,
Group B [
2 6 10 14 18
d d d d d
]
,
Group C [
3 7 11 15
d d d d
] and Group D
[
5 9 13 17
d d d d
]. The diameters of the members included in Group A, Group B, Group C and Group D are denoted by d
a
, d
b
, d
c
and d
d
respectively in Figure 1. The coordinates in the x and y directions of nodes 3, 5, 7, and 9 are assigned as geometrical design variables. There are four independent dimensional design variables and eight independent geometric design variables. During optimization process, the geometric variables are modified by the Complex Method and member sizes by fully-stressed optimality criteria. Eight times the design optimization process has been repeated using eight different seed numbers to generate eight initial complexes. The final weights are presented in Table 2, which establishes the near global solution of the problem. The optimum weight obtained in this study is 3875.85 lb, which is about 0.80% lighter than the design reported by Hansen and Vanderplaats [4], and 1.43% lighter than the optimum design presented by Felix and Vanderplaats [2]. The final configuration is shown in figure 2. The computer run for the best design showed the initial complex, randomly generated, consisted of sixteen designs with total truss weights ranging from 4257.40 to 45925.33 lb. After 356 iterations the stopping criteria was satisfied. The best design has a total truss weight of 3875.82 lb. Even in the initial complex, the best design is about 6% lighter than that of Felix and Vanderplaats [2] and just 9% heavier than that reported by Hansen and Vanderplaats [4]. Figure 3 depicts the progression of maximum and minimum weights of the complex towards the optimum solution. The worst design in the initial complex is about 13 times heavier than the final design. After 50 iterations, the worst design in the complex becomes only 1.3 times heavier than the final design, which shows how the optimization problem with eight geometrical design variables rapidly reaches the optimum. The process of optimization could have been terminated early by setting higher convergence criteria and would still have arrived at roughly the same solution. The histories of geometric design variables and convergence criteria shown are in figures 4 through 6.
Figure 1
Initial geometry of 18-bar 2-D truss
Figure 2
Final configuration of 18-bar 2-D
BASHIR ALAM, M. I. HAQUE, QAISAR ALI, KHAN SHAHZADA, SYED MOHAMMAD ALI, AFZAL KHAN, MUHAMMAD IBRAHIM International Journal of Advanced Structures and Geotechnical Engineering ISSN 2319-5347, Vol. 02, No. 01, January 2013, pp 24-30
05010015020025030035040000.511.522.533.544.55x 10
4
Iteration number
S t r u c t u r e w e i g h t i n c o m p l e x ( l b )
maximum weight in complexminimum weight in complex13.3% heavier than the final design final design (3875.82 lb)(4257.40 lb)(45925.33 lb)
Figure 3
History of max/min weight truss
050100150200250300350400050100150200250300Iteration number
G e o m e t r i c d e s i g n v a r i a b l e s ( i n . )
node 3 y-coordinatenode 5 y-coordinatenode 7 y-coordinatenode 9 y-coordinate
Figure 4
History of geometric design variables (y-coordinates)

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