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very extensive and some of the recent work can be

found in [9].

In this study, topology and geometry optimization

of space structures are performed. Double layer grids

and space pyramids are optimized topologically and

single layer domes are optimized geometrically. To

achieve this aim, a modified genetic algorithm (MGA)is employed. In the MGA, a new mutation and

modified elitist selection are proposed. These operators

cause the MGA algorithm converges quickly and the

probability of achieving the global optimization would

be increased.

For topology optimization, a structure with specific

divisions and a maximum number of members, which

is called the ground structure, is considered and its

perimeter joints are connected to rigid end columns.

Design variables for the optimization problem are the

presence of joints and cross-sectional area ofmembers. In order to reduce the computational weight

of optimization, symmetry properties of the structure

are considered for the elimination of joints.

For geometry optimization, single layer domes with

constant rise and span is considered. In this optimization

problem, the design variables are the cross-sectional

area of members, equation of the dome curve and

coordinates of joints. A proper selection of a curve for

dome leads to a suitable placement of joints.

Consequently, this suitable placement optimizes the

load bearing area of joints and configuration of thestructure.

Optimum shape design of space structures under

different static loading conditions is studied considering

stress, slenderness ratio and displacement constraints.

In the optimization process, the weight of the structure

is considered as the objective function.

2. OPTIMUM TOPOLOGYAND GEOMETRYIn space structures topology optimization, geometry of

the structure and coordinates of joints are kept

constant while the presence or absence of joints and

also cross-sectional areas are selected as design

variables. The goal is to find the most efficient parts of

the structure which can transmit the applied loads to

the base without violating the constraints.

In this study, the symmetry properties of the structure

are used for the tabulation of joints, which leads to a

reduction of chromosome length. Presence or absence

of a joint group is identified by a one bit-gene. A zero

indicates that a joint group should not be considered

during structural analysis. In double layer grids, joints

are eliminated in groups of 8, 4 or 1 joints. For example

in the structure shown in Fig. 1 number of joints withsimilar geometry situations is tabulated in Table 1. Their

presence or absence is governed by one gene.

Therefore, in this structure eight genes are needed to

express the variability of any joint groups. In order to

achieve a practical structure, existence of perimeter

nodes in top and bottom grids of space structure will not

be encoded in the final optimum structure.

In geometry optimization, design variables are

coordinates of joints and cross-sectional area of

members. In the process of optimization, the

coordinates of joints change in a way that the structure

46 International Journal of Space Structures Vol. 24 No. 1 2009

Optimum Shape Design of Space Structures by Genetic Algorithm

5

13 24 35 48 9 23 34 44

8 43

7 42

6 19 30 41

L

Hs

Hc

12 22 33 47

11 21 32 46

10 20 31 45

18 29 40 53

4 17 28 39 52

3 16 27 38 51

2 15 26 37 50

1 14 25 36 49

Figure 1. Joint numbers of top and bottom grids and supports.

Table 1. Joint groups considering symmetry

Group

number Joints in each group Position of joints

1 6, 9, 44, 41 Support

2 7, 19, 8, 23, 34, 43, Support

42, 30

3 10, 13, 48, 45 Bottom layer4 12, 24, 35, 47, 46, 31, Bottom layer

20, 11

5 22, 33, 32, 21 Bottom layer

6 17, 39, 37, 15 Top layer

7 16, 28, 38, 26 Top layer

8 27 Top layer

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E. Salajegheh, M. Mashayekhi, M. Khatibinia and M. Kaykha

gains the most effective state against the applied loads.

This situation optimizes the location of joints, their

load bearing area and the configuration of structure.In this research, optimization of single layer domes

with constant rise and span is studied. Since domes are

formed by revolving a curve about a vertical axis, the

best equation for the curve and location of joints on

this curve are considered as the variables of the

optimization problem. For this purpose, first the

equation of the curve is determined and then by

decoding the radius of each ring of the dome and the

use of curve equation, the height of joints on each ring

is calculated.

With regards to the geometry of domes, theequation of curve should have the following properties

in the interval [A, B] (Fig. 2).

1. The curve should have its maximum at A.

i.e.

2. The curve should be descending in the interval

[A, B]. i.e.

wherez and rare the height of joints on each ring and

radius of each ring, respectively.

According to above conditions, the curve is chosen

to be a polynomial of order n:

(1)

By applying the boundary conditions in Fig. 2 and

calculating constants (a0 and a1), the equation takes

the form:

(2)

whereHand S are the rise and the span of the dome,respectively.

A and B( , ) ( / , )0 2 02

1H S z H S

r

n

n = +

z a a r n= +0 1