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Basic Relations

Figure 1 The Forces Action on the Shell r(x) – distance of rotation axis

h(x)-thickness of shel (design variable)

r

1

- meridional and r

2

- circumferential curvature

q-internal pressure R – resultant load

(2)

2

Basic Relations

(c)2006 http://www.ipmnet.ru/

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Prospect Vernadskogo 101, 117526 Moscow, Russia

(3)

Basic Relations

Figure 2 Geometric Terminology for the Shell

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4

Basic Relations

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

Ν

θ

Ν

φ

− normal membrane forces per unit length

σ

φ

σ

θ

− Normal membrane stresses

(5)

Fatigue Crack Growth under Cyclic Loading

Safety condition, K1-stress intensity K1C-given quasi-brittle strength constant

n

cr

-number of cycles, l

cr

- critical crack length before fracture

(6)

6 (c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

Safety Condition n

*

- given number of cycles

Optimized Functional (volume of material)

geometrical constraints

Shape Optimization under Structural Longevity

(7)

Analysis of the Longevity Constraint

Integration

Minimum of n

cr

with respect to l

i

Minimum of n

cr

with respect to α

where

Transformed longivity constraint

(8)

8

Analysis of the Longevity Constraint

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

n

cr

is the root of the equation

if

than

For m=4 we have solution

(9)

Analysis of the Longevity Constraint Non-dimensional functional and constraints

parameter

Figure 4 The Shell Loaded by Axisimmetrical Forces

Applied to the Free Ends

(10)

Genetic Algorithm

Modification:

cyclic probability of crossover and mutation, nodes fixation for process axelleration

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(11)

Dependence Shell of Iteration Step

Computational Results with Genetic Algorithm

(12)

Types of Optimal Shapes

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(13)

D 1 =2 D 2 =0.1 L=5

Computational Results with Genetic Algorithm

(14)

D 1 =2 D 2 =1 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(15)

D 1 =2 D 2 =2 L=5

Computational Results with Genetic Algorithm

(16)

D 1 =1.5 D 2 =0.1 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(17)

D 1 =1.5 D 2 =0.75 L=5

Computational Results with Genetic Algorithm

(18)

D 1 =1.5 D 2 =1.5 D 1 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(19)

D 1 =1 D 2 =0.1 L=5

Computational Results with Genetic Algorithm

(20)

D 1 =1 D 2 =0.5 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(21)

D 1 =1 D 2 =1 L=5

Computational Results with Genetic Algorithm

(22)

D 1 =0.5 D 2 =0.1 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

(23)

D 1 =0.5 D 2 =0.25 L=5

Computational Results with Genetic Algorithm

(24)

D 1 =0.5 D 2 =0.5 L=5

Computational Results with Genetic Algorithm

(c)2006 http://www.ipmnet.ru/

[email protected]@ipmnet.ru [email protected]@ipmnet.ru [email protected] [email protected] [email protected]

Prospect Vernadskogo 101, 117526 Moscow, Russia

Figure

Figure 1 The Forces Action on the Shellr(x) – distance of rotation axis
Figure 2 Geometric Terminology for the Shell
Figure 4 The Shell Loaded by Axisimmetrical Forces Applied to the Free Ends

References

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