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Optimization of Trusses-2
Genetic Algorithm Optimization in Structural Mechanics
S.I.Rodin
Department of Material Science and Mechanics

 A genetic algorithm approach with island ring topology is used for optimization of trusses so as to minimize the weight of the structure.

 Standard methods are used to determine cross sections for compressed, tensile and zero steel bars with flexibility limits (compressed bars = 120, tensile bars = 200 and zero bars = 150).

 In all cases weight of the best truss is compared to the weight of base steel standard model.

 This solution is rather general - hinges are scattered at random with 2 bars each to make structure geometrically stable. Mutations are also applied to the position of hinges.

 Loads are shown on the trusses and there may be any number of load combinations - like it's in real construction design.

Two examples (Figure 1 and Figure 2) are shown for a single load - a force is applied to one hinge.

Another example (Figure 3) is shown for a single load - forces are applied to every hinge of lower chord.

On Figure 4 example is shown for a combination of 8 loads: a force is applied in succession to every hinge of lower chord - moving load.

Figure 5 shows an example to get more optimal truss for chosen shape with fixed number of hinges.

Figures 6,7 show shape dependence upon load value. Different shapes of trusses for different values of force are due to influence of compressed bars flexibility and flexibility limits for all bars.

Figure 7,8 shows possibility of implantation of one and two forbidden areas for bars and hinges by modifying fitness function to get necessary restrictions on shapes. It's important to point out - not by changing algorithm - but by changing environment. It's like in living world - changing environment - changing biological species (giraffe, whale, etc.).

Figure 9 shows plane structures upon different load values also with implantation of forbidden area.

Figure 10 shows weight of steel plane structures for different values of flexibility limits for bars. Normal flexibility limits are taken as follows:
       compressed bars = 120, tensile bars = 200 and zero bars = 150.

Sure, not all best structures may be used - there is a possibility to choose among either the best shown or the best in every island. 

 

Examples of Genetic Algorithm Optimization - 2

 

c51.gif (8353 bytes)

Figure 1, 2.
Single load - a force is applied to one hinge.

 

c811.gif (7485 bytes)
c71.gif (3892 bytes)

Figure 3.
Single load - forces are applied to every hinge of lower chord.

 

c10.gif (3520 bytes)
c91.gif (3753 bytes)

Figure 4.
Combination of 8 loads:
a force is applied in succession to every hinge of lower chord - moving load.

 

etalon1_1.gif (3517 bytes)

Figure 5.
Optimal truss for chosen shape with fixed number of hinges.

 

d16.gif (1302 bytes)    d15.gif (1365 bytes)
a. F=10 kN                                                  b. F=1000 kN
Figure 6.
Optimal trusses for different values of force F.
Span and height of force = 5 meters.

 

d121.gif (1756 bytes)    d131.gif (1992 bytes)
a. F=10 kN                                                  b. F=1000 kN
Figure 7.
Optimal trusses for different values of force F.
Modified fitness function - implantation of forbidden area (gray dotted square).
Span and height of force = 5 meters.

Note:
Different shapes of trusses for different values of force are due to influence  of compressed bars flexibility and flexibility limits for all bars.

 

d171.gif (1698 bytes)   d181.gif (1909 bytes)
a. F=10 kN                                                  b. F=1000 kN
Figure 8.
Optimal trusses for different values of force F.
Modified fitness function - implantation of two forbidden areas (gray dotted square).
Span and height of force = 5 meters.

 

e3111.gif (2224 bytes)   e3221.gif (3090 bytes)
a. F=1 kN                                                       b. F=1 kN
e111.gif (2463 bytes)   e511.gif (2230 bytes)
c. F=10 kN                                                  d. F=100 kN
Figure 9.
Optimal plane structures for different values of forces F  (a, c, d).
Modified fitness function (b) - implantation of forbidden area (gray dotted square).
Span and height of the structure = 5 meters.

 


    a
- no flexibility limits
eca11.gif (2602 bytes)eca21.gif (2504 bytes)eca33.gif (2618 bytes)
             a1. P=4.17 kN                                   a2. P=3.88 kN                                  a3. P=4.40 kN

    b - no flexibility limits for tensile bars
ecb12.gif (2539 bytes)ecb24.gif (2434 bytes)ecb31.gif (2485 bytes)
             b1. P=4.92 kN                                   b2. P=4.34 kN                                  b3. P=4.06 kN

    c - normal flexibility limits
ecc16.gif (2416 bytes)ecc22.gif (2291 bytes)ecc32.gif (2328 bytes)
             c1. P=9.75 kN                                   c2. P=4.16 kN                                   c3. P=4.67 kN

    d - cross sections of all bars are the same
ecd13.gif (2453 bytes)ecd22.gif (2495 bytes)10x10.gif (50 bytes)
             d1. P=12.55 kN                                 d2. P=11.12 kN

Figure 10.
Weight of structure P for different values of flexibility limits for bars:

a - no flexibility limits; b - no flexibility limits for tensile bars;
c - normal flexibility limits; d - cross sections of all bars are the same;

left
raw - a1,b1,c1,d1 - fixed positions for two supports;
middle raw - a2,b2,c2,d2 - optimal positions for two supports;
right raw - a3,b3,c3 - optimal are both positions and number of supports;

blue - tensile bars, red - compressed bars.

Load = 8 forces each 100 kN = 800 kN, span and height of the structure = 5 meters.

 

 


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