Application of Random Search Method for the Optimal Designing of Ribbed Plates

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 9, Issue 10, October 2019)

223 Application of Random Search Method for the Optimal

Designing of Ribbed Plates

George Filatov

1

, Yu. M. Pochtman

2

1,2

Professor, Doctor of Techn. Sciences, Ukrainian State University of Chemical Technology, Ukraine,

Abstract − The paper proposes the use of a random search method for optimal design of plates stiffened by ribs at static load. Adduced an example of optimal designing of the plate, stiffened by two longitudinal ribs and the results of optimization are compared with the results of the solution of this very problem by the method of feasible directions. Is noted, that the use of random search method significantly reduces the losses on search.

Keywords – Optimization, Random Search, Ribbed Plates, Mathematical Programming

I. INTRODUCTION

In the theory of optimal design of continual systems an important place belongs to the optimal designing of ribbed plates. Defining the parameters of such structures that provide the perception of a predetermined load and meet the criteria of optimality, as well as satisfying a certain set of restrictions, is a complex and at the same time quite an urgent problem of applied theory of elasticity. In studies conducted in this area [1],[2] the designing of optimal structures of this type is carried out based on the analytical solution of the corresponding direct problem, under the assumption that restrictions of stiffness are recorded in the form of equalities. This approach involves considerable mathematical difficulties, since the only required solution of the system in most cases is impossible. More natural is the statement of the problem as a problem of nonlinear mathematical programming, which allows you to record restrictions (by strength, rigidity and stability) in the form of inequalities.

Furthermore, in favor of application of mathematical programming methods says also the fact that the minimization of some objective function for continuous systems (as opposed to rod systems) at satisfaction of the conditions of strength, rigidity and stability, of structural and other restrictions, the stresses and strains in individual elements designs are not always explicitly depend on the parameters of the cross sections, which greatly complicates the solution of problems of this type by variational methods and techniques of classical analysis. One of the most effective methods of mathematical programming, used for optimal design of ribbed plates in bending, axial compression and vibrations, is the method of random search.

II. CALCULATION OF BENT PLATES OF MINIMUM

WEIGHT STIFFENED BY RIBS

The questions of theory at optimal designing of plates with taking into account the plastic properties of the material were considered by different researchers. For elastic plates, reinforced by ribs, optimization problem was solved mainly by the method of selection designing [3]. It is also known the use of the method of possible directions [4] at the optimal designing of ribbed plates in bending [5]. Consider the application of a random search for the optimal designing of ribbed bent plates. The results are compared with the results in [5].

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International Journal of Emerging Technology and Advanced Engineering

224

Fig.1 Calculated scheme of ribbed bent plate

Let's apply the method of displacements for the determination of internal forces in elements of plate and ribs and let's decompose displacement of nodal lines in a

trigonometric series [6]. Construct the system of canonical equations for the amplitude of displacements:

   

 

   

 

   

 























0 ...

....

...

0 ...

1 1

1

2 1 21 1

22 1 21

1 1 1 1

12 1 11

k mp k

k mm k

k m k k m

k p k

k k

k k k

k p k k m k

k k k

R

z

r

z

r

z

r

R

z

r

z

r

z

r

R

z

r

z

r

z

r

(1)

This system of equations is solved separately for each harmonic number

k

.

In different selected points of plate and ribs the stresses we get as linear functions of displacements nodes, which are in turn determined by the system reaction matrix:

   







 





1

sin

...

k

k l k il k j k ij i

l

x

k

z







,

(2)

where



_ij

 

k  the stress at the point

i

_from

 

1 

k j

z

;

x

 the abscissa of the point, that is

considered.. The number of harmonics retained in the calculation depends on the required accuracy of the calculation.

Let's vary the thickness of individual elements of the plate and ribs. The task is to determine such distribution material in structure under set conditions of strength and

other restrictions that satisfy to minimum theoretical weight of structure:





n

i i i

b

l

F

1



, (3)

i

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International Journal of Emerging Technology and Advanced Engineering

225 



_j



, (4)



 the calculated resistance of the material of the plate and ribs for corresponding type of deformation.

The main difficulty in minimizing the function of weight (3) consists in that in formula for the stresses the displacements of nodal lines explicitly have a complex dependence on the sought-for thickness of the elements of plate and ribs.

The dependence is more difficult, the higher the order of the matrix coefficients of the system (1). This difficulty can be overcome by considering the problem in the space

n



mp

of variabl

p

− the number harmonics retained in the calculation . We obtain the following nonlinear programming problem: minimize the weight of the structure (3) at the performance of

restrictions

:

 

   

 

   

 

...

   

...

   

...

_...

...

   

...

....

 

₀

0 ...

1 1 1 1 1 2 1 21 1 22 1 21 2 1 1 1 1 12 1 11 1















k mp k k mm k k m k k m k m k p k k k k k k k k p k k m k k k k k

R

z

r

z

r

z

r

g

R

z

r

z

r

z

r

g

R

z

r

z

r

z

r

g

;

(5)

0

1 1

















r s m

g

; (6)



i

n



i



0 ;



1 ,

2 ,....,



, (7)

which determine the existence of a certain area of permissible solutions.

In addition to the difficulties about which has already been mentioned in connection with the minimizing of weight function (3), at using the method of random search there is a problem of satisfying the restrictions in form of equalities. To avoid this difficulty, we can introduce in the weight function (3) an additional penalty function [7]. In our case, the objective function (3) takes the form:









m i i i

g

F

f

1 2



, (8)

i



 fairly great numbers.

Then the restriction in the form of (5) can be represented by the inequality:

 

_

_

 

_

_,

...

_

₀

.

...

0 ,

1









k m k

g

(9)

To solve this problem we apply the algorithm of continuous self-learning with forgetting [8].

To illustrate, consider the problem of finding a minimum weight of plate with two ribs subjected the action of the first harmonic of a uniformly distributed load

q



100

kN / m, applied in the plane of the ribs (Fig. 1, b). The calculated span is

l



7 ,

85

m,

b



1

m, the thickness of the ribs



₁, should be not less than 12

mm, plate thickness



₂ and



₃ 

Material − plate steel, the allowable stress

160 

adm



MPa, Poisson's ratio





0 ,

3

,

The modulus of elasticity

E



2 ,

1 

10

5MPa, the specific gravity of material





78

kN / m3.

Taking as the unknown the functions of the vertical and longitudinal movements of the nodal line and without taking into account the displacement of nodal line in the horizontal plane due to their negligible influence on the results of the calculation, we can write for determining the amplitudes of unknown displacements two equations:

   

 

   

 

1

0 ₀

2 1 2 1 22 1 1 1 21 1 1 1 2 1 12 1 1 1 11









p p

R

z

r

z

r

R

z

r

z

r

(10)

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226

 

_{ }

b

E

b

E

r

1 7 1 1

11

0 ,

0073









_



;    

 

b

E

b

E

r

1 8 1 1 21 1

12

0 ,

0268









_



; ₁ 1





4 



127 ,

4 

q

R

_p ;

 

_{ }



_{ }



b

E

b

E

b

E

b

E

b

E

b

E

r

1 2 3

6 5 3 2 9 1 1

22

2

0 ,

1699

0 ,

1699

0 ,

1699

2 





























,  

0

1 2p



R

, wherе

0 ,

2

85 ,

7

1 1416

,

3

1

2

1 



_



_



l

b





.

Fig. 1.2, shows a points at which the stresses are calculated. Because the normal stresses at points 1,2 and 3 differ from each other by a small amount, the amplitude of longitudinal stresses we define only in the points 1, 4, 5 and 6. We calculate the stress amplitudes using the following expression:

 



_{ }

 



_{ }

 





b

E

z

8 21

1 1 7 1

1





2 















;  



_{ }

 

_{ }

 



b

E

z

₁1 ₃ ₂1

1 1

4

2 



2 









;  

_{ }

 

b

E

z

1 2 3 1

5

2 









;  











 







 

b

E

z

ch

sh

ch

sh

1 2 1 6

1

3

1

3 







_



_

_



























Omitting further the factor

b

E

, since the amplitudes of

the longitudinal stresses do not depend on

it, we get the final expression for the determination of

stresses:



1







0 ,

0022

z

1



0 ,

392 z

2



2 1

4



0 ,

1397

z



0 ,

3206

z



(11)



₅





0 ,

3206

z

₂



6





0 ,

2085

z

2

The theoretical weight of plates equals



₁ ₂ ₃



2 













lb

F

(12)

Introducing the notation

x

₁





₁;

2 2





x

;

x

₃





₃ ;

x

₄



z

₁ ;

x

₅



z

₂ and substituting in equation (10)

-nonlinear programming problem: find a non-negative values of the variables, which minimize the objective function:

 







 2 1 2 1

m m m

f

A

G

X

, (13)

where:

G

₁

 

X



2 

lb



x

₁



x

₂



x

₃



,

and simultaneously satisfy the following inequalities:

g

₁

 

X



0 ,

0073

x

₁

x

₂



0 ,

0268

x

₁

x

₃



127 ,

4 

0 g

₂

 

X





0 ,

0268

x

₁

x

₄





0 ,

1699

x

₁



0 ,

1699

x

₂



0 ,

1529

x

₃



x

₅



0 g

₃

 

X



0 ,

0022

x

₄



0 ,

3920

x

₅



1600



0 g

₄

 

X



0 ,

1397

x

₄



0 ,

3206

x

₅



1600



0

[image:4.612.51.283.364.561.2]

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International Journal of Emerging Technology and Advanced Engineering

227 g

₆

 

X





0 ,

2085

x

₅



1600



0

1 

x

₁



4

см;

1 

x

₂



4

см;

1 

x

₃



4

см;

10000



x

₄



15000

;

[image:5.612.129.487.145.456.2]

500 

x

₅



2000

.

Fig. 2. Curved plate with two ribs: the design scheme (a), scheme of the displacements nodes of plate (b), scheme of arrangement

of points in the nodes of plate, in which the stresses determined (c)

The calculation results are shown in Table 1. The table for comparing these results are adduced the results of

[image:5.612.57.555.545.628.2]

solution of same problem by the method of possible directions [5].

Table 1

The results of calculation of the bent plate

Method

_F

_(кН)

1



(mm)



₂(mm)



₃(mm)

z

₁

z

₂

Possible

directions 45,837 17,430 10,0 10,0 13849,9 1045,3

Random search

45,876 17,437 10,0 10,0 13863,5 1051,97

As can be seen from Table 1.1, the results obtained by the method of random search, is quite close to the results found by the method of possible directions. Necessary to note, that the method of possible directions [4] used in this problem under the assumption of convexity of the objective function and constraints in the form of equalities.

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International Journal of Emerging Technology and Advanced Engineering

228

Introduction restrictions in form of inequalities (9) instead of restrictions in form of equalities (1) and application the method of random search, allows to check the affiliation of selected point permissible area without solving equations (1), which significantly reduces the losses on search.

REFERENCES

[1] Burns A.B. Combined load minimum weight analysis of stiffened plates and shells, “J.Spacecraft and Rockets”, Vol.3, №2, 1966, P. 68-75.

[2] Ginzburg I.N., Kan S. N. On one method of selecting optimal parameters of structures // Proceedings of the VII All-Union Conference on the theory of shells and plates, Rostov, 1971 p.34-35.

[3] Zhu S.J., Prager B. Recent developments in the optimal design of structures .// In the book .; Mechanics (Collection of translations) .- Moscow: IL. - 1969. - № 6. - P.129-143.

[4] Zoytendeyk G. Methods of possible directions, IL, 1963, 278 p. [5] Mitchell R.A., Kaplan J.L. Nonlinear constrained optimization by

a nonrandom complex method // J. Res. Nat. Bur. Stand. - 1968. - С 72. - № 4, p.1022 – 1028.

[6] Smirnov A.F., Alexandrov, A.V., Shaposhnikov N.N. , Laşcencov B.N. Calculation of structures using computers. - M .: Gosstroiizdat, 1964,432 p.

[7] Hadley Dk. The non-linear and dynamic programming.- M .: Mir, 1967.- 506 p.

[8] Filatov G.V/ Pseudo-gradient Algorithms of Adaptation of Random Search with the Accumulation Information for Determination of the Direction of Working Step. International Journal of Emerging Technology & Advanced Engineering, Volume 7, Issue 3, March, 2017, p.121-128.