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A new BEM approach for reissner's plate bending
Pergamon
Computers & Swucrures Vol. 54. No. 6. pp. 1085~1090, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + 0.00
0045-7949(94)00402-l
A NEW BEM APPROACH
FOR REISSNER’S Lei
PLATE
BENDING
Xiao-Yan
Department of Mechanics, University of Science and Technology of China, Anhui, Hefei 230026, People’s Republic of China (Received
31 October
1993)
Abstract-A
new boundary element formulation for Reissner’s plate bending is presented. This form of BEM has an advantage in that the bending stresses on the boundary can be calculated directly from the numerical solution, avoiding the use of tangential derivatives of displacement for finding plate bending stresses on the boundary. The effectiveness of the approach is also discussed through some test examples. In the present BEM formulation, the singular orders of the two kernels are the same as those in the standard BEM formulation of a Reissner’s type plate-one of which is logarithmic singular and the other is l/r singular.
1. INTRODUCTION
The direct boundary element method (BEM) has been applied quite successfully to Reissner’s plate bending problems in static analysis [l-3]. In the standard BEM approach of the Reissner’s plate, the boundary variables are generalized displacements and generalized tractions. To calculate the bending stresses on the boundary, the popular approach is to use tangential derivatives of nodal displacements and nodal tractions obtained by the boundary element analysis [ 1,4]. Although this approach is very simple and efficient, the obtained stress components are always discontinuous between the neighboring elements when C” elements are used, and the accuracy of them becomes lower near a corner point and the boundary, where a stress concentration occurs. Cruse and Vanburen [5] proposed a boundary integral equation relating the stress components at the boundary nodal point to the displacements and tractions over the entire boundary for two-dimensional elasticity. This approach is almost the same as that in the computation of stress components at internal points, and should evaluate all the integrals in the Cauchy principal value sense and deal with a strongly singular kernel of l/r’. The present paper discusses Reissner’s plate bending. A new boundary integral equation was formulated with new stress components on the boundary. Combined with the standard BEM [l], the complete bending stress tensors on boundary can be calculated directly from the numerical solution on this approach. It has some advantages over the conventional BEM, where the boundary bending stress components should be derived by differenting the displacement at the boundary with respect to the coordinates. In the present BEM formulation, the singular orders of the two kernels are the same as
those in the standard BEM formulation of a Reissner’s type plate-one of which is logarithmic singular and the other l/r singular. 2. REISSNER’S PLATE THEORY
Considering a plate of uniform thickness h with midplane coordinates x, (LX= 1,2) and thickness coordinate xX, the equilibrium equations of Reissner’s plate model are [l] h4 18.8-
Q, + F, = 0
Qz,,+ 4 + 4 = 0,
(1)
where Fi are equivalent body force distributions and loading on the plate faces; Mss denotes the bending-stress couples and Q, the shear-stress resultants. The notation (u,, u,) will be used for the generalized displacements (&, w), of which 4, are the rotations and w is the deflection. Generalized stress-displacement relations are
q the transversal
I&=-
l-v 2
D
2v u,,,i + up, + I-v
1
u..,‘y6 z/j
+ vqS,p/(l - v)12
Qs = C(u, +
u3.x1,
(2)
where 1 = a/h is a characteristic quantity of Reissner’s model, D = Eh3/12( 1 - v*) is the flexural rigidity and C = OS(1 - v)DL2. Substituting relations (2) into eqns (1) yields the equilibrium equations in terms of the generalized displacements ui (i = 1,2,3):
108.5
A;u,+PI=O,
(3)
Lei Xiao-Yan
1086
where A$ denotes operator
the components
l+v A*)&, + I_V
A$,=!$ D [ (V2 A:,= -C-&=
of the Navier
s* m
1
1
P
-A;,
Substituting A,, in eqn (6) into eqn (10) and integrating it repeatedly by parts, the resulting equation is of the form
I[;
(M,
+ (1 - v2)Dw,)t,u:
1
+ c](s Pz=F,+vq,z/(l
-v)l’,
P,=F,+q.
-
A,,=(-l)‘fl&,,_
4~:
sR
[( - l)(%_,(A,&
+ A,&)
+ u,(A,,u,* + A.33u:)]d0 = 0, (6)
8’
V
/(l - v)A* + u:]qdR
l)‘“‘G-,
(5)
3. NEW WEIGHTED RESIDUAL EQUATION
Let us introduce a new function,
df
dT
(4)
and
1
1
f
A$ = CV2
+ Q,n,u:
r
(11)
where
Ars = D
The superscript (8) only indicates the fi th power of (- 1) and does not mean the summation with subscript 8. Considering relations (2), one can find new expressions of the equilibrium eqns (3) or (1)
(V* - C/D)&,
- q
&
1 A,, = ( - 1)‘“‘C&
= A,, 3
1 -v* A 18.8- Dw,*.~ + e, + F= = 0
BI
I
A,, = - CV*
(12)
V
and Q,,,
+
F3
+
4 =
0
(7) p:=
where %p=I 0%
=
’ kp - up.% 1 (-
lY”‘C(u,_.
p: = c(u:l, +
u3,3_,)
Fz = -( - 1)‘“‘F3 _ 19
(8)
for planar elasticity, where u, are displacement components, o,~ is the so-called infinitesimal rotation tensor and is antisymmetric [6]. The unit normal vector n, and unit tangential vector t, on the boundary have the relation t =(-l)‘“‘n,_ 9
(9)
2
In this paper only the transversal loading q on the plate faces is considered. We take the weighted residual functions u: and UT. Multiplying eqns (7) by uf and integrating it over the domain R give
SK R
1 -v2
A NJ ---a,,,*+ V
-D
B,)u:
+ (Ql,, + q)u:
1
dn = 0.
(10)
u&,-vu,&+--
l-v
2
vu;&
$ 1
- ug,n,).
(13)
It can be seen that the operator components in eqn (12) are different from those in the equilibrium eqn (4). 4. BOUNDARY INTEGRAL FORMULATION AND FUNDAMENTAL SOLUTION
In eqn (10) the weighted residual function u) are defined. We suppose that the fundamental solution U,, is a particular solution satisfied by the following equation
A,,u&
xl = S(C,x)&k>
(14)
where the operator components A,, are defined in eqn (12) S(t, x) is the Dirac 6-function, 5 is the singular source point and x is the field point. According to the Hiirmander operator method [7], the solution may be found in the following form:
c;,(L x1 =“A,k,k(L XL
(15)
BEM
approach for plate bending
1087
1 -( - l)(r)-r.nr.j-z 2
where “Ajk is the matrix of cofactors and 4(<, x) is a scalar function. Substituting eqn (15) into eqn (14) gives
1 (20)
where
in which [det A] = -G
CD2V4(V2- A*).
E(Z) = (4 - l/Z)lZ,
(17)
A (Z) = K,, + 2(K, - l/Z)/Z,
(21)
A particular solution of eqn (17) is 2
__-_~-2&~-4
dJ(L x) = (1 _ “)C x [K,(Z) + In Z + Z2(ln Z - 1)/4],
(18)
where &(Z) and K,(Z) are the modified Bessel functions with argument Z = Ir, and r is the distance between the source point 5 and the field point K. It is found that A(Z) is continuous while E(Z) has a singularity In(Z). The weighted residual statement (11) can then be written as
by using eqns (15), (18) and (13) the displacement and traction kernels of U, and To, respectively, can be obtained: 2 In r + 1 +
lJrs = & [(
&A(Z)
U,, = &
>
r,z.s
+
Tot<,x)+(x) dr(x)
-
s
1
1
( - I)%(2 In r - l)r,, --z = U,,
V,, = &
7 [
In r - r2(ln r - 1) 1
(19)
*4(x)m3K
xl + (-
1)‘“’
and vK, Z
where P, = [M,, + (1 - v2)Do,&,
l+v + r.Bt, 2A(Z) + 2
P, = -Q,n,.
(23)
If < E R, considering the last term of eqn (11) and the Dirac 5 -function in eqn (14), the coefficient C, can be derived:
1+v + &8r,X 2A (Z) - --2--
- r,,r,gr,s(l -v + 2k,Z + 8A(Z))
1
.
(24)
When putting the source point 5 at the boundary, the coefficient C, will be evaluated in Appendix A. In the case of uniform load q(x) = q,, the R-integral is convertible to a r-integral with the functions +
vr.*r.n- ( - 1Yr)rsr,, _ r
I &
( - I)@+‘(4 In r - S)r,, _ I
1 V3=----256nD
r’(2 In r - 3) - 32 sr’(ln
V” =
2 - r.2r.s ‘+ 2 ---A(Z) 1_ v
1,
r - 1)
(25)
Lei Xiao-Yan
1088
Table 1. Clamped circular plate with uniform pressure h/a = 0.2
Number of elements
(O.%SO
O.&O)
(0.%07
0.520)
(0.%79
V?? 0.5000)
4 6 8 IO
0.03698 0.03720 0.03730 0.03736
0.4905 0.4958 0.4976 0.4985
0.03661 0.03685 0.03695 0.03701
0.4905 0.4958 0.4976 0.4985
0.03570 0.03589 0.03596 0.03599
0.4906 0.4958 0.4976 0.4985
which satisfy the Poisson divergence theorem gives
qo
h/a =O.l
h/a = 0.01
[U, + (-
equation
V’V’ = U,, , the
=
S[L I
l)“‘U,,,_,/(l
v
u,,(<, xh&,(x)
- v)l’]dR
5R = qo
[V:*n. + U,,t,/(l
- v)l=]dr.
(26)
sr 5.
BOUNDARY CONDITIONS
In a standard Reissner’s type plate the geometric and static boundary values, respectively, are
where M,, and IV,,, are surface moment components and the third component of the complete plate stress tensors on the boundary is
(28)
M, = Mafi f, lg. The new variables be expressed by
P, on the boundary
in eqn (22) can
P, = [M@ + (1 - v2)Dw,&fl
= 0s
+ t,M,,
(30)
in which w,, =
f(dt~,p~,,
-
aup/ax,)n, tjr
= gau,jax2- au2jax,)= w,2.
6. NUMERICAL RESULTS
(29)
where B,,, = M,, + (1 - v=)Dw,,,
In eqn (32) the boundary variables are u,, B,,,T,M,r,rand Q,,. For the case of known boundary displacements U, and z+ (or P,), the others can be calculated directly from the numerical solution. For the mixed boundary conditions, the unknown generalized displacements and generalized tractions on the boundary can be solved by the standard BEM approach. Then the boundary moment M_ and o,? can be determined by the present eqn (32) or (22). The complete bending-stress tensors M,, on the boundary can be easily obtained by M,,, M,, and M,,. The singular order of the two kernels in the present boundary integral eqn (32) are the same as those in the standard BEM formulation for Reissner’s plate bending [I]; U,, is logarithmic singular and T,, is l/r singular.
(31)
Numerical implementation of eqns (22) or (32) is carried out in standard fashion. The boundary r is discretized into boundary elements. Poisson’s ratio v is 0.3. Numerical results presented are obtained by using straight boundary elements with piecewise linear representation of the boundary variables on these elements. The singular integration with T&s, x) in eqn (22) or (32) has a singularity l/r, those terms are calculated by a rigid motion scheme (Appendix A) and the integrations with a singularity of type In(r)
The very interesting thing is that the present approach can take M, as one of the boundary variables. Substituting eqn (29) into (22) yields : // 5
t++++++++j q/2
/
2a I-
-1
Fig. 1. Clamped circular plate.
1089
BEM approach for plate bending
Fig. 2. Bending stress concentration
are carried out analytically. The rest of the integrations are derived using four integration points of Gaussian quadrature. 6.1. Example pressure
1: clamped circular plate with uniform
A circular plate of radius a and thickness h is clamped at r = a and subjected to a uniform pressure q/2 on the plate faces z = + h/2 (Fig. 1). The boundary conditions were taken as u, = u0 = u) = 0
when r = a.
problem.
M,, which models the plate bending of an infinite plate with a circular hole solved by Reissner [8]. Boundary conditions were taken as y=O
and
Ixl>l;
M_,=O, uJ= 0
x =&lo;
IV,,=
fM,,
y = 10;
IV+, = 0,
M,,=O
L%&= 0
and
and
V?=O
and
V,=O
VP = 0
(33) r=l;
Using quarter symmetry, a quarter of the plate is modeled and is discretized into elements with equal length. The plate bending-stress components M, (or M,,) and shear-stress resultant V,, on the boundary with various mesh numbers of a quarter of the plate are listed in Table 1. These results are very close to the theoretical values [l] displayed in the parentheses of the table. 6.2. Example 2: bending stress concentration Figure 2 indicates a plate with a semicircular groove that is subjected to a uniform end bending
M,,=O,
M,O=O
and
24,(x= _+lO,y = lO)=O.
V,=O (34)
Sixteen elements with equal length are used to model the semicircle and the number of total elements is 52. First, the generalized displacements on the boundary was solved by the standard BEM [1] and then the stress concentration M, was obtained by the present approach. For the various values of a/h, the bending stress concentration factors Kh = M,/M,, at point A are plotted in Fig. 3, they show good agreement with the Reissner’s solution [S]. 7. CONCLUSIONS
3 FT.
i2.5 _:“-ri:
I
Present BEM Thin plate
q
---
2
9
._._._._._._._._._._.~.~.~.~.~.
t
1.51 0
I
0.1
I
I
I
0.2
0.3
0.4
a/h Fig. 3. Bending stress concentration
1
0.5
factor of various a/h.
A new boundary integral formulation for Reissner’s plate bending has been developed with the boundary variables u,, [M,, + (1 - v2)Doz,,]tp (or h4,, + (1 - v2)Do, and M,) and V,, which is different from the standard BEM approach in that the bending stress component MS, should be calculated by differentiating the boundary displacements with respect to the coordinates. The singular order of the two kernels in the present BEM approach are the same as those in the standard BEM formulation [I], Ufl is logarithmic singular and cj is l/r singular. Numerical implementation of the present BEM can be carried out in standard fashion.
1090
Lei Xiao-Yan
For calculating boundary stresses, this approach is advantageous over that proposed by Cruse and Vanburen with l/r* singular integration. The present approach is suitable to solve the stress concentration and fracture mechanics and also can be used in other conventional field mechanics problems. Ackno~~,ledgement-This research has been supported National Education Committee Foundation of China young researchers.
by for
1. F. Vander Ween, Application of the boundary integral equation method to Reissner’s plate model. Int. J. Numer. Mefh. Engng 18, l-10 (1982). 2. H. Antes, On a regular boundary integral equation and a modified trefftz method in Reissner’s plate theory. Engng Annl. l(3). 1499153 (1984). 3. Lei Xiao-yan, Huang Mao-kuang and Wang Xiuxi, Geometrically nonlinear analysis of a Reissner type plate by the boundary element method. Cornput. Struct. 37(6), 911-916 (1990). 4. J. C. F. Telles, Elastostatic problems. In Topics in Boundary Element Research (Edited by C. A. Brebbia), Ch. 9. Springer, Berlin (1987). 5. T. A. Cruse and W. Vanburen, Three-dimensional elastic stress analysis of a fracture specimen with an edge crack. Int. J. Fracture Mech. 7, l-15 (1971). 6. Fung Y. C., Foundations of Solid Mechanics. PrenticeHill, New Jersey (1965). 7. H. Hormander. Linear Parlial Differential Operators. Springer, Berlin (1963). 8. E. Ressner, The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12 (1945).
A
Evaluation qf C,, ,for the new BEM formulation Referring to eqn (22) we consider the source point 5 at the boundary with discontinuous tangents (Fig. Al). Assuming that the body under consideration can be augmented by a small region r(, which is part of a circle of radius t centered at point 5 on the boundary I’, and that the function u,(x) satisfies a Holder condition at 5, we have the coefficient C,, at the boundary:
\
----_,y r \‘.
l
Xf
"1
1
/:_._._._01 4.
Fig. Al.
REFERENCES
APPENDIX
v\ I * .-.-.-._
Discontinuous
tangents
so eqn (Al) can be written
at the boundary.
as
x[-2(1+3v)n,tl,-2(1-v)n,,r,]dB Cj* = rzn,, = 0 +C,,+$(n+H,-0,). When the boundary 0, = t$, then cU is
(A3)
is smooth
at the source
point
5, i.e.
If the source point 5 is located at the boundary, the coefficient C, in the boundary integral eqns (22) and (32) should be replaced by cg, presented in eqn (A3). The matrix C,, may be also evaluated indirectly by the rigid motion scheme. It is noted that the stress-free problem
+ T,,(& x)u,(x) Substituting
the
independent
LO,x,(5)-x,(x)l,
1
dr(z)
= 0.
displacement
(As)
solutions
10,1,x2(5)-x2(x)1 and (%O, 1) into
eqn (AS), one obtains
x ; T,,(L x). r,,(C>x) drht).
(Al)
I When integrating exist:
along the circle r,, the following
r,, = n,,
r,kli = 0
G(C) = -
J[! I
v
T,,(C xl + (x,(5) -x,(x))
relations
W)
646)
Computers & Swucrures Vol. 54. No. 6. pp. 1085~1090, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + 0.00
0045-7949(94)00402-l
A NEW BEM APPROACH
FOR REISSNER’S Lei
PLATE
BENDING
Xiao-Yan
Department of Mechanics, University of Science and Technology of China, Anhui, Hefei 230026, People’s Republic of China (Received
31 October
1993)
Abstract-A
new boundary element formulation for Reissner’s plate bending is presented. This form of BEM has an advantage in that the bending stresses on the boundary can be calculated directly from the numerical solution, avoiding the use of tangential derivatives of displacement for finding plate bending stresses on the boundary. The effectiveness of the approach is also discussed through some test examples. In the present BEM formulation, the singular orders of the two kernels are the same as those in the standard BEM formulation of a Reissner’s type plate-one of which is logarithmic singular and the other is l/r singular.
1. INTRODUCTION
The direct boundary element method (BEM) has been applied quite successfully to Reissner’s plate bending problems in static analysis [l-3]. In the standard BEM approach of the Reissner’s plate, the boundary variables are generalized displacements and generalized tractions. To calculate the bending stresses on the boundary, the popular approach is to use tangential derivatives of nodal displacements and nodal tractions obtained by the boundary element analysis [ 1,4]. Although this approach is very simple and efficient, the obtained stress components are always discontinuous between the neighboring elements when C” elements are used, and the accuracy of them becomes lower near a corner point and the boundary, where a stress concentration occurs. Cruse and Vanburen [5] proposed a boundary integral equation relating the stress components at the boundary nodal point to the displacements and tractions over the entire boundary for two-dimensional elasticity. This approach is almost the same as that in the computation of stress components at internal points, and should evaluate all the integrals in the Cauchy principal value sense and deal with a strongly singular kernel of l/r’. The present paper discusses Reissner’s plate bending. A new boundary integral equation was formulated with new stress components on the boundary. Combined with the standard BEM [l], the complete bending stress tensors on boundary can be calculated directly from the numerical solution on this approach. It has some advantages over the conventional BEM, where the boundary bending stress components should be derived by differenting the displacement at the boundary with respect to the coordinates. In the present BEM formulation, the singular orders of the two kernels are the same as
those in the standard BEM formulation of a Reissner’s type plate-one of which is logarithmic singular and the other l/r singular. 2. REISSNER’S PLATE THEORY
Considering a plate of uniform thickness h with midplane coordinates x, (LX= 1,2) and thickness coordinate xX, the equilibrium equations of Reissner’s plate model are [l] h4 18.8-
Q, + F, = 0
Qz,,+ 4 + 4 = 0,
(1)
where Fi are equivalent body force distributions and loading on the plate faces; Mss denotes the bending-stress couples and Q, the shear-stress resultants. The notation (u,, u,) will be used for the generalized displacements (&, w), of which 4, are the rotations and w is the deflection. Generalized stress-displacement relations are
q the transversal
I&=-
l-v 2
D
2v u,,,i + up, + I-v
1
u..,‘y6 z/j
+ vqS,p/(l - v)12
Qs = C(u, +
u3.x1,
(2)
where 1 = a/h is a characteristic quantity of Reissner’s model, D = Eh3/12( 1 - v*) is the flexural rigidity and C = OS(1 - v)DL2. Substituting relations (2) into eqns (1) yields the equilibrium equations in terms of the generalized displacements ui (i = 1,2,3):
108.5
A;u,+PI=O,
(3)
Lei Xiao-Yan
1086
where A$ denotes operator
the components
l+v A*)&, + I_V
A$,=!$ D [ (V2 A:,= -C-&=
of the Navier
s* m
1
1
P
-A;,
Substituting A,, in eqn (6) into eqn (10) and integrating it repeatedly by parts, the resulting equation is of the form
I[;
(M,
+ (1 - v2)Dw,)t,u:
1
+ c](s Pz=F,+vq,z/(l
-v)l’,
P,=F,+q.
-
A,,=(-l)‘fl&,,_
4~:
sR
[( - l)(%_,(A,&
+ A,&)
+ u,(A,,u,* + A.33u:)]d0 = 0, (6)
8’
V
/(l - v)A* + u:]qdR
l)‘“‘G-,
(5)
3. NEW WEIGHTED RESIDUAL EQUATION
Let us introduce a new function,
df
dT
(4)
and
1
1
f
A$ = CV2
+ Q,n,u:
r
(11)
where
Ars = D
The superscript (8) only indicates the fi th power of (- 1) and does not mean the summation with subscript 8. Considering relations (2), one can find new expressions of the equilibrium eqns (3) or (1)
(V* - C/D)&,
- q
&
1 A,, = ( - 1)‘“‘C&
= A,, 3
1 -v* A 18.8- Dw,*.~ + e, + F= = 0
BI
I
A,, = - CV*
(12)
V
and Q,,,
+
F3
+
4 =
0
(7) p:=
where %p=I 0%
=
’ kp - up.% 1 (-
lY”‘C(u,_.
p: = c(u:l, +
u3,3_,)
Fz = -( - 1)‘“‘F3 _ 19
(8)
for planar elasticity, where u, are displacement components, o,~ is the so-called infinitesimal rotation tensor and is antisymmetric [6]. The unit normal vector n, and unit tangential vector t, on the boundary have the relation t =(-l)‘“‘n,_ 9
(9)
2
In this paper only the transversal loading q on the plate faces is considered. We take the weighted residual functions u: and UT. Multiplying eqns (7) by uf and integrating it over the domain R give
SK R
1 -v2
A NJ ---a,,,*+ V
-D
B,)u:
+ (Ql,, + q)u:
1
dn = 0.
(10)
u&,-vu,&+--
l-v
2
vu;&
$ 1
- ug,n,).
(13)
It can be seen that the operator components in eqn (12) are different from those in the equilibrium eqn (4). 4. BOUNDARY INTEGRAL FORMULATION AND FUNDAMENTAL SOLUTION
In eqn (10) the weighted residual function u) are defined. We suppose that the fundamental solution U,, is a particular solution satisfied by the following equation
A,,u&
xl = S(C,x)&k>
(14)
where the operator components A,, are defined in eqn (12) S(t, x) is the Dirac 6-function, 5 is the singular source point and x is the field point. According to the Hiirmander operator method [7], the solution may be found in the following form:
c;,(L x1 =“A,k,k(L XL
(15)
BEM
approach for plate bending
1087
1 -( - l)(r)-r.nr.j-z 2
where “Ajk is the matrix of cofactors and 4(<, x) is a scalar function. Substituting eqn (15) into eqn (14) gives
1 (20)
where
in which [det A] = -G
CD2V4(V2- A*).
E(Z) = (4 - l/Z)lZ,
(17)
A (Z) = K,, + 2(K, - l/Z)/Z,
(21)
A particular solution of eqn (17) is 2
__-_~-2&~-4
dJ(L x) = (1 _ “)C x [K,(Z) + In Z + Z2(ln Z - 1)/4],
(18)
where &(Z) and K,(Z) are the modified Bessel functions with argument Z = Ir, and r is the distance between the source point 5 and the field point K. It is found that A(Z) is continuous while E(Z) has a singularity In(Z). The weighted residual statement (11) can then be written as
by using eqns (15), (18) and (13) the displacement and traction kernels of U, and To, respectively, can be obtained: 2 In r + 1 +
lJrs = & [(
&A(Z)
U,, = &
>
r,z.s
+
Tot<,x)+(x) dr(x)
-
s
1
1
( - I)%(2 In r - l)r,, --z = U,,
V,, = &
7 [
In r - r2(ln r - 1) 1
(19)
*4(x)m3K
xl + (-
1)‘“’
and vK, Z
where P, = [M,, + (1 - v2)Do,&,
l+v + r.Bt, 2A(Z) + 2
P, = -Q,n,.
(23)
If < E R, considering the last term of eqn (11) and the Dirac 5 -function in eqn (14), the coefficient C, can be derived:
1+v + &8r,X 2A (Z) - --2--
- r,,r,gr,s(l -v + 2k,Z + 8A(Z))
1
.
(24)
When putting the source point 5 at the boundary, the coefficient C, will be evaluated in Appendix A. In the case of uniform load q(x) = q,, the R-integral is convertible to a r-integral with the functions +
vr.*r.n- ( - 1Yr)rsr,, _ r
I &
( - I)@+‘(4 In r - S)r,, _ I
1 V3=----256nD
r’(2 In r - 3) - 32 sr’(ln
V” =
2 - r.2r.s ‘+ 2 ---A(Z) 1_ v
1,
r - 1)
(25)
Lei Xiao-Yan
1088
Table 1. Clamped circular plate with uniform pressure h/a = 0.2
Number of elements
(O.%SO
O.&O)
(0.%07
0.520)
(0.%79
V?? 0.5000)
4 6 8 IO
0.03698 0.03720 0.03730 0.03736
0.4905 0.4958 0.4976 0.4985
0.03661 0.03685 0.03695 0.03701
0.4905 0.4958 0.4976 0.4985
0.03570 0.03589 0.03596 0.03599
0.4906 0.4958 0.4976 0.4985
which satisfy the Poisson divergence theorem gives
qo
h/a =O.l
h/a = 0.01
[U, + (-
equation
V’V’ = U,, , the
=
S[L I
l)“‘U,,,_,/(l
v
u,,(<, xh&,(x)
- v)l’]dR
5R = qo
[V:*n. + U,,t,/(l
- v)l=]dr.
(26)
sr 5.
BOUNDARY CONDITIONS
In a standard Reissner’s type plate the geometric and static boundary values, respectively, are
where M,, and IV,,, are surface moment components and the third component of the complete plate stress tensors on the boundary is
(28)
M, = Mafi f, lg. The new variables be expressed by
P, on the boundary
in eqn (22) can
P, = [M@ + (1 - v2)Dw,&fl
= 0s
+ t,M,,
(30)
in which w,, =
f(dt~,p~,,
-
aup/ax,)n, tjr
= gau,jax2- au2jax,)= w,2.
6. NUMERICAL RESULTS
(29)
where B,,, = M,, + (1 - v=)Dw,,,
In eqn (32) the boundary variables are u,, B,,,T,M,r,rand Q,,. For the case of known boundary displacements U, and z+ (or P,), the others can be calculated directly from the numerical solution. For the mixed boundary conditions, the unknown generalized displacements and generalized tractions on the boundary can be solved by the standard BEM approach. Then the boundary moment M_ and o,? can be determined by the present eqn (32) or (22). The complete bending-stress tensors M,, on the boundary can be easily obtained by M,,, M,, and M,,. The singular order of the two kernels in the present boundary integral eqn (32) are the same as those in the standard BEM formulation for Reissner’s plate bending [I]; U,, is logarithmic singular and T,, is l/r singular.
(31)
Numerical implementation of eqns (22) or (32) is carried out in standard fashion. The boundary r is discretized into boundary elements. Poisson’s ratio v is 0.3. Numerical results presented are obtained by using straight boundary elements with piecewise linear representation of the boundary variables on these elements. The singular integration with T&s, x) in eqn (22) or (32) has a singularity l/r, those terms are calculated by a rigid motion scheme (Appendix A) and the integrations with a singularity of type In(r)
The very interesting thing is that the present approach can take M, as one of the boundary variables. Substituting eqn (29) into (22) yields : // 5
t++++++++j q/2
/
2a I-
-1
Fig. 1. Clamped circular plate.
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BEM approach for plate bending
Fig. 2. Bending stress concentration
are carried out analytically. The rest of the integrations are derived using four integration points of Gaussian quadrature. 6.1. Example pressure
1: clamped circular plate with uniform
A circular plate of radius a and thickness h is clamped at r = a and subjected to a uniform pressure q/2 on the plate faces z = + h/2 (Fig. 1). The boundary conditions were taken as u, = u0 = u) = 0
when r = a.
problem.
M,, which models the plate bending of an infinite plate with a circular hole solved by Reissner [8]. Boundary conditions were taken as y=O
and
Ixl>l;
M_,=O, uJ= 0
x =&lo;
IV,,=
fM,,
y = 10;
IV+, = 0,
M,,=O
L%&= 0
and
and
V?=O
and
V,=O
VP = 0
(33) r=l;
Using quarter symmetry, a quarter of the plate is modeled and is discretized into elements with equal length. The plate bending-stress components M, (or M,,) and shear-stress resultant V,, on the boundary with various mesh numbers of a quarter of the plate are listed in Table 1. These results are very close to the theoretical values [l] displayed in the parentheses of the table. 6.2. Example 2: bending stress concentration Figure 2 indicates a plate with a semicircular groove that is subjected to a uniform end bending
M,,=O,
M,O=O
and
24,(x= _+lO,y = lO)=O.
V,=O (34)
Sixteen elements with equal length are used to model the semicircle and the number of total elements is 52. First, the generalized displacements on the boundary was solved by the standard BEM [1] and then the stress concentration M, was obtained by the present approach. For the various values of a/h, the bending stress concentration factors Kh = M,/M,, at point A are plotted in Fig. 3, they show good agreement with the Reissner’s solution [S]. 7. CONCLUSIONS
3 FT.
i2.5 _:“-ri:
I
Present BEM Thin plate
q
---
2
9
._._._._._._._._._._.~.~.~.~.~.
t
1.51 0
I
0.1
I
I
I
0.2
0.3
0.4
a/h Fig. 3. Bending stress concentration
1
0.5
factor of various a/h.
A new boundary integral formulation for Reissner’s plate bending has been developed with the boundary variables u,, [M,, + (1 - v2)Doz,,]tp (or h4,, + (1 - v2)Do, and M,) and V,, which is different from the standard BEM approach in that the bending stress component MS, should be calculated by differentiating the boundary displacements with respect to the coordinates. The singular order of the two kernels in the present BEM approach are the same as those in the standard BEM formulation [I], Ufl is logarithmic singular and cj is l/r singular. Numerical implementation of the present BEM can be carried out in standard fashion.
1090
Lei Xiao-Yan
For calculating boundary stresses, this approach is advantageous over that proposed by Cruse and Vanburen with l/r* singular integration. The present approach is suitable to solve the stress concentration and fracture mechanics and also can be used in other conventional field mechanics problems. Ackno~~,ledgement-This research has been supported National Education Committee Foundation of China young researchers.
by for
1. F. Vander Ween, Application of the boundary integral equation method to Reissner’s plate model. Int. J. Numer. Mefh. Engng 18, l-10 (1982). 2. H. Antes, On a regular boundary integral equation and a modified trefftz method in Reissner’s plate theory. Engng Annl. l(3). 1499153 (1984). 3. Lei Xiao-yan, Huang Mao-kuang and Wang Xiuxi, Geometrically nonlinear analysis of a Reissner type plate by the boundary element method. Cornput. Struct. 37(6), 911-916 (1990). 4. J. C. F. Telles, Elastostatic problems. In Topics in Boundary Element Research (Edited by C. A. Brebbia), Ch. 9. Springer, Berlin (1987). 5. T. A. Cruse and W. Vanburen, Three-dimensional elastic stress analysis of a fracture specimen with an edge crack. Int. J. Fracture Mech. 7, l-15 (1971). 6. Fung Y. C., Foundations of Solid Mechanics. PrenticeHill, New Jersey (1965). 7. H. Hormander. Linear Parlial Differential Operators. Springer, Berlin (1963). 8. E. Ressner, The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12 (1945).
A
Evaluation qf C,, ,for the new BEM formulation Referring to eqn (22) we consider the source point 5 at the boundary with discontinuous tangents (Fig. Al). Assuming that the body under consideration can be augmented by a small region r(, which is part of a circle of radius t centered at point 5 on the boundary I’, and that the function u,(x) satisfies a Holder condition at 5, we have the coefficient C,, at the boundary:
\
----_,y r \‘.
l
Xf
"1
1
/:_._._._01 4.
Fig. Al.
REFERENCES
APPENDIX
v\ I * .-.-.-._
Discontinuous
tangents
so eqn (Al) can be written
at the boundary.
as
x[-2(1+3v)n,tl,-2(1-v)n,,r,]dB Cj* = rzn,, = 0 +C,,+$(n+H,-0,). When the boundary 0, = t$, then cU is
(A3)
is smooth
at the source
point
5, i.e.
If the source point 5 is located at the boundary, the coefficient C, in the boundary integral eqns (22) and (32) should be replaced by cg, presented in eqn (A3). The matrix C,, may be also evaluated indirectly by the rigid motion scheme. It is noted that the stress-free problem
+ T,,(& x)u,(x) Substituting
the
independent
LO,x,(5)-x,(x)l,
1
dr(z)
= 0.
displacement
(As)
solutions
10,1,x2(5)-x2(x)1 and (%O, 1) into
eqn (AS), one obtains
x ; T,,(L x). r,,(C>x) drht).
(Al)
I When integrating exist:
along the circle r,, the following
r,, = n,,
r,kli = 0
G(C) = -
J[! I
v
T,,(C xl + (x,(5) -x,(x))
relations
W)
646)