application of the boundary element method in determining the critical buckling load of perforated...

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APPLICATION OF THE BOUNDARY ELEMENT METHOD IN

DETERMINING THE CRITICAL BUCKLING LOAD OF PERFORATED

PLATES CONSIDERING THE EFFECT OF SHEAR DEFORMATION

***This is a translation, this translation is not published, the original paper is published in brasilian portuguese.

For citation please refer to

“APLICAÇÃO DO MÉTODO DOS ELEMENTOS DE CONTORNO NA DETERMINAÇÃO DE CARGAS

CRÍTICAS DE FLAMBAGEM DE PLACAS PERFURADAS CONSIDERANDO O EFEITO DA

DEFORMAÇÃO POR CORTANTE”

Romildo Aparecido Soares Junior, [email protected]

Leandro Palermo Junior, [email protected] Universidade Estadual de Campinas, Rua Saturnino de Brito, 224 - Cidade Universitária Zeferino Vaz, Campinas - São Paulo

Resumo. The boundary element method (BEM) is used in this work to obtain the critical buckling parameters of

perforated plates. The plates have square geometry with central square hole. Compression is unidirectional and

uniformly applied at opposite edges. The values for buckling parameters are obtained for many values of thickness.

The shear deformation effect is included in the bending of plates using the isotropic model. The effect of geometric

nonlinearity is introduced by adding two integral in the formulation of the BEM: one is applied in the domain and the

other in the contour. The boundary integral can be related to one of the natural conditions according to the boundary

value problem. Quadratic continuous and discontinuous boundary elements were used. The source points were

positioned on the boundary. The singularity subtraction technique and the transformation of variables were used for

the Cauchy and weak type of singularities, respectively, when the integration is performed in elements containing the

source point. Rectangular cells were used to discretize the domain integral related to the geometric nonlinearity effect.

The results were compared with other authors.

Keywords: Perforated Plates, Critical Buckling Load, Boundary Element Method

1. INTRODUCTION

This work aims to show the influence of the hole size in the critical buckling load in perforated plates. It is known

that the instability analysis of plates is important in many engineering problems, for example the analysis of thin

structures and structural elements used in aerospace engineering, mechanical, civil and others. The influence of the plate

thickness is also measured and for this purpose it has been used the theory that takes into account the shear deformation

effect shown by REISSNER (1945) and MINDLIN (1951). Square plates are evaluated with central square hole with

unidirectional and uniformly applied load at opposite edges, the plates are thin or moderately thick. The effect of

geometric nonlinearity due to the load on the plate plane is taken into account according to the work of SOARES JR.

(2015) using two additional integrals, one is calculated in the domain and the other is calculated on the boundary. The

eigenvalue problem is then solved using the inverse iteration process and the Rayleigh quotient method. The

methodology of the boundary element method used in this work is direct, the source points are positioned on the

boundary and integration is singular. The singularity subtraction and the transformation of variables to integrals of

Cauchy and weak type singularities, respectively are applied. The numerical method used in this work is described in

more detail in SOARES JR. (2015).

Studies show that the critical buckling loads for perforated plates are few when compared with non-perforated

plates. Conducting a literature review there are some researchers who are dedicated to solving problems of perforated

plates. CHOW AND NARAYANAN (1984) presented solutions to problems of plates with holes with different types of

loads. The problem of instability of a square plate with a central hole with bending, shear and compression loads was

analyzed by BROWN AND YETTRAM (1986). SABIR AND CHOW (1986) solved problems with eccentric holes

with respect to the center of the plate. RONG CHANG-JUN (1996) solved perforated plates using the boundary element

method. SHANMUGAM et al. (1999) proposed a formula for the design of perforated plates with uniform loads. EL-

SAWY AND NAZMY (2001) solved plates with circular or square hole of various sizes and in various positions within

the domain, using the finite element method. JAYASHANKARBABU AND KARISIDDAPPA (2013) analyzed the

problem of square plate with circular or square holes with different thicknesses. DOVAL et al (2013) solved problems

of square plates with rectangular holes. KOMUR AND SONMEZ (2015) solved problems with partial uniform loads.

2. APPLICATION OF LOAD - TWO-DIMENSIONAL ELASTICITY PROBLEM




On a square plate without holes with uniaxial uniform load the distribution of the internal stress is constant,

however, on a plate with holes the stress distribution is replaced by wide variations mainly near the edge of the holes.

Figure 1 shows an example of problem in which a perforated plate is subjected to a uniaxial uniformly distributed load:

Figure 1. Plate with square hole - Uniaxial load

This shows the necessity of using a method to calculate two-dimensional elasticity stresses along the domain area.

This paper uses the direct treatment of the problem variables with the boundary element method, according to BECKER

(1992) and KANE (1994). The values of the stresses in the boundary of the plate were found with the integral equations

of the BEM to the plane stress and plane strain. Tensions were obtained from Eq (2) whose values in the contour were

found with Eq (1), FOLTRAN (1999).:

* *( ') ( ) ( ', ) ( ) ( ', )ij iC u x u x T x x d t x U x x d

(1)

* *

, , , ...ab abik j ji k j abjC u x T x d x T x x d x

(2)

The fundamental solutions are:

* 1

1 2 2 1 24 1

dr dr dr dr drT v v n n

v r dx dx dn dx dx

(3)

* 1

3 4 ln8 1

dr drU v r

v dx dx

(4)

2

1abik ai bk ak bi ab ikC

(5)

*

, , , , , , , ,1

2 1 24

1,1 `

2ikl i k l ik l il k kl i i k lr r r r r rr

(6)

Tensions in the perforated domain are used in applying the effect of geometric nonlinearity on the plate.

2. PLATES THEORY THAT TAKES INTO ACCOUNT THE EFFECT OF SHEAR DEFORMATION

According TIMOSHENKO (1959), for moderately thick plates, the classical theory has a greater deviation from the

exact values, especially those with holes in the order of plate thickness, it shows the need for an improved theory.

Theories of REISSNER (1945) and MINDLIN (1951) takes into account the effect of shear deformation and therefore




the results are closer to the elasticity problem in three dimensions. The first applications of the boundary element

method in Reissner plates were made by WEEËN (1982), he deduced the fundamental solutions and the boundary

integral equation to apply them in the calculation of the boundary element method, the integral equation is given by Eq.

(7):

* *

* *

3 ,2

( ') ( ) ( ', ) ( ', ) ( )

( ', ) ( ', )(1 )

ij i i ij ij i

i i

C u x t x U x x d T x x u x d

vq U x X U x X d

v

(7)

Equation (7) is used in the boundary element method for calculation of displacement and forces in a plate when

considering the effect of shear deformation. The fundamental solutions for the displacements found in WEEËN (1982)

or Palermo JR. (2000) are given by:

, ,

18 ( ) (1 )(2ln 1) 8 ( ) 2(1 )

8 (1 )U B z z A z r r

D

(8)

3 3 ,

1(2ln 1)

8U U z rr

D

(9)

2

33 2

1(1 ) (ln 1) 8ln

8 (1 )U z z z

D

(10)

D is flexural rigidity of the plate given by Eq. (11):

3

212(1 )

EhD

v

(11)

E is the Young modulus, v is the Poisson ratio, h is the plate thickness e λ is the Reissner/Mindlin coefficient. The

fundamental solutions for tractions are given by:

1 , ,

, 1

14 ( ) 2 ( ) 1

4

4 ( ) 1 2(8 ( ) 2 ( ) 1 ) , , ,

n

n

T A z zK z r r nr

A z r n A z zK z r r r

(12)

2

3 ( ) ( ) , ,2

nT B z n A z r r

(13)

3

(1 ) (1 )2 ln 1 2 , ,

8 (1 )nT z n r r

(14)

33

1,

2nT r

r

(15)

Onde:

0 1 0 1

2 1 1 1; ( ) ( ) ( ) ; ( ) ( ) ( )z r A z K z K z B z K z K z

z z z z

(16)

The functions K0(z) and K1(z) are modified Bessel functions, found in ABRAMOVITZ e STEGUN (1965).




3. ADDITIONAL INTEGRALS FOR THE GEOMETRIC NON-LINEARITY EFFECT

For the treatment of the problem of instability, it is necessary to obtain a boundary integral equation that includes the

effects of geometric nonlinearity. The deduction of this equation and the natural conditions required for the analysis of

the problem, using the variational calculus, were presented in SOARES JR. (2015) and summarized in PALERMO JR.

and SOARES JR. (2015). The final equation is shown below with the notation of WEEËN (1982):

* * * 33

1( ') ( ', ) ( ) ( ', ) ( ) ( ', )

2ij i ij i ij j i

uC u x T x x u x d U x x t x d U x X q N d

X X

(17)

Applying the divergence theorem in the domain integral on the right side of Eq. (17):

* 33

* *

3, 3 3, 3,

( ', )

( ) ( ) ( ) ( ', ) ( ) ( ) ( ', )

i

i i

uU x X q N d

X X

n x N x u x U x x d N x u x U x x d

(18)

Using Eq. (18) and substituting in the Eq. (17) one can obtain Eq. (19):

* * *

3

* *

3, 3 3, 3,

1( ') ( ', ) ( ) ( ', ) ( ) ( ', )

2

( ) ( ) ( ) ( ', ) ( ) ( ) ( ', )

ij i ij i ij j i

i i

C u x T x x u x d U x x t x d U x X qd

n x N x u x U x x d N x u x U x x d

(19)

Integration in the domain is made using rectangular domain cells. In the present work were used constant cells and

domain integral can be transformed into an integral calculated in the cell borders, ie:

* *

3, 3, 3, 3

1

( ) ( ) ( ', ) ( ) ( ) ( ) ( ', )Ncel

i i

k k

N X u X U x X d n x N x u x U x x d

(20)

The boundary integral equation to the problem of instability of moderately thick plates when taken into account the

effect of geometric nonlinearity becomes given by Eq. (21):

* * *

3

* *

3, 3 3, 3

1

1( ') ( ', ) ( ) ( ', ) ( ) ( ', )

2

( ) ( ) ( ) ( ', ) ( ) ( ) ( ) ( ', )

ij i ij i ij j i

Ncel

i i

k k

C u x T x x u x d U x x t x d U x X qd

n x N x u x U x x d n x N x u x U x x d

(21)

Where the derivative of the transversal displacement is given by Eq. (22):

3, 3 3 3

3 33 3 33

*

3, 3 3,

( ') ( ) [ ( ', )] ( ) ( ) [ ( ', )] ( )

[ ( ', )] ( ) [ ( ', )] ( ) [ ( ', )]

( ) ( ) ( ) [ ( ', )] ( ) ( ) [i i

u X n x M X x u x d n x Q X x u x dx x

U X x t x d U X x t x d U X X qdx x x

n x N x u x U X x d N X u x Ux x

*

3, ( ', )]X X d

(22)




Where x 'is a source point belonging to the contour, x is a field point belonging to the contour, X' is a source point

belonging to the domain and X is a field point belonging to the domain. Taking also into account the natural problem

conditions when the displacements are prescribed in the contour (condition simply supported, for example), the

integrations are made only on the sides of the cells within the area, excluding the sides present in the boundary, and

when the displacements are not prescribed in the contour (as a free side, for example) the integrations are made at the

sides of the cells in the area and those belonging to the free edge. The critical buckling loads are calculated using the

quotient of Rayleigh, numerical method discussed in detail in PALERMO JR. (1985):

( 1)

( 1)

( 1) ( 1)

( , )

( , )

k k

k k

k k k

Ax x

x x

x x

(23)

4. NUMERICAL RESULTS

The results for the critical buckling parameter given by Eq (24) are shown.

2

2

critNk L

D (24)

Where k = value of the critical buckling parameter, L = size of the side of the plate, D = flexural rigidity and Ncrit =

Critical Load found. The analyzed plates are square with side L = 0.5m compressed uniaxially similarly to Figure 1, it is

also evaluated various sizes of thickness h of the plate which can be very thin to moderately thick. Are tested several

sizes of holes aspect of D / L = 0.1 to 0.7 where d is the hole side size. The plane strain tensions are taken from the

center of gravity of the domain cells. Figure 2 shows a mesh with 224 boundary elements and 336 domain cells (left)

and a mesh with 448 boundary elements and 1344 domain cells (right).

Figure 2. Example of two of the meshes used for the numerical results

A code was developed in Fortran for numerical processing of critical loads, this program was described in the

dissertation of SOARES JR. (2015). Table 1 shows the convergence of the critical buckling parameters for the plates

analyzed in all aspects of hole size (d / L) and thickness (h / L), where it was found that the results undergo little change

as you increases the number of boundary elements and domain cells. To check this convergence was adopted a uniform

distribution of the boundary element and also the domain cells as the mesh refinement was being done. The refined

meshes are structured, this was possible due to the development of a program also in Fortran that creates meshes

structured for plates with holes. The relationship between the boundary elements and the adopted domain is that each

cell has two boundary elements.




Table 1. Critical Buckling Parameters for hole sizes and thicknesses analyzed

d/L N° Elem. N° Cells h/L=0.001 0.01 0.05 0.1 0.15 0.2

176 396 3.8085 3.7999 3.7364 3.5865 3.3699 3.1088

0.1 352 1584 3.8024 3.7945 3.7323 3.5828 3.3664 3.1053

528 3564 3.8010 3.7935 3.7316 3.5821 3.3657 3.1046

192 384 3.4646 3.4535 3.3859 3.2462 3.0537 2.8226

0.2 384 1536 3.4546 3.4441 3.3774 3.2373 3.0439 2.8103

576 3456 3.4522 3.4420 3.3754 3.2352 3.0414 2.8068

208 364 3.2131 3.2003 3.1258 2.9825 2.7902 2.5547

0.3 416 1456 3.1961 3.1836 3.1090 2.9640 2.7687 2.5262

624 3276 3.1921 3.1797 3.1050 2.9595 2.7632 2.5179

224 336 3.0698 3.0533 2.9606 2.7938 2.5751 2.2970

0.4 448 1344 3.0411 3.0244 2.9305 2.7607 2.5368 2.2490

672 3024 3.0316 3.0179 2.9234 2.7527 2.5271 2.2342

240 300 2.9968 2.9739 2.8494 2.6396 2.3698 2.1021

0.5 480 1200 2.9481 2.9246 2.7980 2.5847 2.3104 2.0158

720 2700 2.9344 2.9137 2.7864 2.5721 2.2959 1.8552

256 256 2.9781 2.9457 2.7751 2.5023 2.2413 1.7089

0.6 512 1024 2.8933 2.8599 2.6880 2.4148 2.1519 1.4332

768 2304 2.8725 2.8419 2.6695 2.3958 2.1265 1.2794

272 204 3.0601 3.0130 2.7719 2.4806 1.7783 1.0240

0.7 544 816 2.8912 2.8448 2.6070 2.3146 1.3997 0.7966

816 1836 2.8585 2.8119 2.5747 2.2834 1.3565 0.7926

It is expected that the values for plates with h / L = 0.001 approach the EL-Sawy and NAZMY (2001) values since

these authors used the classical theory that does not take into account the effect of shear deformation.

Table 2 shows the comparison of the results of this work with other authors.

Table 2. Comparison with other authors to the aspect of thickness h / L = 0.001

d/L

Present

Work

h/L = 0.001

Present

Work

h/L = 0.01

EL-SAWY

And

NAZMY

(2001)

Difference

(%) (1)

Difference

(%) (2)

0 4.0128 4.0105 4.0000 0.3200 0.2625

0.1 3.8010 3.7935 3.7973 0.0974 -0.1001

0.2 3.4522 3.4420 3.4449 0.2119 -0.0842

0.3 3.1921 3.1797 3.1790 0.4121 0.0220

0.4 3.0316 3.0179 3.0236 0.2646 -0.1885

0.5 2.9344 2.9137 2.9256 0.3008 -0.4068

0.6 2.8725 2.8419 2.8649 0.2667 -0.8014

0.7 2.8585 2.8119 2.8449 0.4780 -1.1600 1 Difference with h/L = 0.001 to EL-SAWY e NAZMY (2001)

2 Difference with h/L = 0.01 to EL-SAWY e NAZMY (2001)

A difference in average of 0.5% was obtained for h / L = 0.001 when comparing the results for the critical buckling

parameter with EL-Sawy and NAZMY (2001), showing that the boundary element method gave similar results to the

finite element method, with less effort for numerical modeling. The comparison of reference for the critical parameters

calculated buckling went to the plates with h / L = 0.001 because due to thickness is too small, the effect of shear

deformation is very small and the result starts to converge to the classical theory. The plates with h / L = 0.01 converged

to lower critical buckling parameters, this is also expected for this thick plate is beginning to show the effect of shear




deformation. Figure 3 shows the critical buckling parameters for each thickness aspect (h / L) as it increases the hole

size (d / L).

Figure 3. Critical buckling parameters as it is increased the hole size

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