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Stress analysis around holes in orthotropic plates by the subregion mixed finite element method
Cornputcrs & Slnrerures Vol. 41, No. Printed in Great Britain.
1. pp. 105-108,
1991
Ml45-7949/91 53.00 + 0.00 Pcrgamon Rcss plc
STRESS ANALYSIS AROUND HOLES IN ORTHOTROPIC PLATES BY THE SUBREGION MIXED FINITE ELEMENT METHOD ZHAO JINPING
and SHANHIJIZU
Beijing University of Aeronautics and Astronautics, Beijing 10083, People’s Republic of China (Receioed 5 July IWO) Abstract-The subregion mixed generalized variational principle is applied in this paper. In the region near a hole, the special element of stress type based on stress analytical solution is adopted. In the exterior region far from the hole, standard finite elements of displacement type are used. The Lagrange multiplier term is adopted so as to satisfy the stress boundary condition on the hole. In order to formulate the special element around the hole, the general expressions of stress and the stress boundary condition of the orthotropic plate with a hole are deduced.
INTRODUCTION
GENERAL EXPRRSSIONS OF STRE% AND STRRSS BOUNDARY CONDITION OF ORTHOTROPIC PLATE WITH HOLE
The
stress analysis around a hole in an infinite orthotropic plate has been studied extensively by Lekhnitskii and Savin et al. [l-3]. But, nowadays the same problem in finite orthotropic plates is mainly solved by adoption of finite elements of displacement types. However, as a result of concentration of stress around the hole, not only is it necessary for the grid to be divided into small parts, but also the solution of stress around a hole has a large error. Recently, Chinese scholars Chien Wei-zang, Hu Haichang and Long Yu-chiu et al. have had great achievements in the study of the subregion general variational principle [4-6]. The principle is applied to orthotropic finite plates with holes. In this paper in the vicinity of the hole with great stress, special elements of stress type based on the analytic solution is used, while in the exterior region far from the hole, standard Unite elements of displacement type are adopted. The stress boundary condition around the hole is satisfied by adoption of the Lagrange multiple term, then a subregion mixed finite element method is established. In order to obtain the special element around a hole, we deduce the general expressions of stress and stress boundary conditions of the orthotropic finite plate with a hole, which are suited for holes of all shapes, for example circular holes, ellipse, crack and so on. As mentioned above, the special element of stress type in the vicinity of the hole is based on analytical solution. As long as there are enough terms in the stress expression of the series, the area of the special element can be larger, for example the size of the special element can be 10 times as large as the size of the hole. So the solution obtained by the method introduced in the paper not only has greater precision, but also greater efficiency and a reduced time of calculation. CM 4,/l-”
For an orthotropic plate, the governing equation is as follows:
a9 az2s
-
202,
aw + (2a,, + axsay
-
aw axzay*
a&5)-
g-$+a,,$=o,
-2a16
(1)
where aij is the elastic constant of the material, U is the stress function. The characteristic equation of eqn (1) is a,,s4-2a,,s3+(2a,2+a,)s2-2a,,s+a,=0.
(2)
It has been proved by Lekhnitskii that the roots s, of eqn (2) are only complex or pure imaginary, the general expressions for the stress components are of the form: (i = 1,2,3,4)
aw
cX= aye = ZRe[s: 4 ‘(2,) + s: # ‘(~~11.
a, =
$ =ZReM’(z,)
aw Tm= -- axay=
+ I,+‘(z*)]
-ZRe[s,b’(z,)
+ s&‘(z2)]
(3)
and the free stress boundary condition can be shown as: ReMi)
+ S(z*)l = 0
Relrl~(z,)+s2~[z211=0.
105
(4)
106
ZHAOJINPINGand SHANHIJIZIJ
where z I = x + s,y, z, = x + s2y, $J(z,) and Jl(zJ are arbitrary analytical functions of z1 and z2, respectively. In order to analyse the stresses around the hole, we use the following transformation:
c,-f(q)
- h,b)
(j = 1,2)
(6)
V’(q)
=
-
(i=1,2)
2q’(z,).
C, = CZ= efi into eqn (8) and
.iM
[(A, + CJcos n0 - (B, + D,)sin no] = 0
.iM
{[Re(W,
- Im(si)B, + Re(s,)C,
- Im(s2)D,]cos n6 - [Im(s + Im(s2)C, + Re(s,)D,]sin
for a circle hole with radius r,, c, = [z, + J-J/(/
ci$=
(5) substituting into eqn (4), we can obtain
According to that transformation, the regions around the hole in the planes z,, z, and z are transformed into the region in the unit circle in the plane C. For the ellipse with semi-axes (I and b, respectively, c, = [z, + ,/w]/(u
2$f’(z,)
‘31
Substituting
(f = 192).
CIJ=
- is/)rlj (j = 1,2).
(7)
According to Lekhnitskii theory, when the principal vector of a load along the hole surface is zero, the stress analytic functions can be expressed as:
+ Re(s,)B, no} = 0.
Merging same kinds of terms of cosp6, sinpe[p=1,2,..., nil, respectively, and setting coefficient of every term zero, then we can obtain the stress boundary condition around hole as follows:
vwq~ = 0.
(10)
THE FUNCTIONAL OF SUBBEGION MIXED SYSTEM IN ORTHOTBOPIC PLATE (8)
where A,, , b, , C, and D, are unknown real coefficients to be determined later. Substituting eqn (8) and eqn (5) into eqn (3), we obtain
As shown in Fig. 1, the region I near the hole is the complementary energy region, in which a special element of stress type based on the analytic solution is used, and the stress variables {q} are the basic unknowns. The region II beyond the region I is the potential energy region, in which the finite elements of displacement type are used, and the nodal displacements {w} are basic unknowns. S is the interface between region I and region II. According to the subregion mixed variational principle, the energy functional of the plate is n=l-Ip-rI,+Hs+H,,
(9)
where, II, is the potential energy of region II, II, is the complementary energy of region I, H, is the additional energy on the interface S of region I and region II, and HL is Lagrange multiplier term so as to satisfy the stress boundary condition on the boundary of hole.
where {A} = [A-M, . . . , A_,, A-,, A,,
A29
{B}= [B-M,. . . , B-2, B-,, 4, B2,. {C}= [C_,, . . . , c-2, c_,, c,, c2,
(_i= L2)
4vlr
hlr * * *, CNIT
. .v
&IT . . . ,?q-‘, . . . ,Nc:-‘l
{D} = [D-M,. . . , D-2, D-I, 4,
(4) = [-q-M-‘,
. . .,
D2,.
(11)
. .I
Fig. 1.
107
Stressanalysisof holes in orthotropic plates The potential energy of region II is % = d {WYKliW) - WP%
(12)
Substituting eqns (12), (13), (18) and (20) into eqn (1I), the energy functional of the subregion mixed system in an orthotropic plate with a hole is given by
where [K] is the stiffness matrix, [P] is the equivalent load vector of nodes in region II. The complementary energy of region I is
where [E] is the elastic matrix of the orthotropic material. Substituting eqn (9) into the equation above, we can obtain (13)
BASIC EQUATION OF THE SUBREGION MIXED FINITE ~~ METHOD OF OR~O~O~C PLATE WlTH HOLE
Based on the subregion mixed generalized variational principle, the actual values of basic cowl must make the energy functional mentioned above stational value, i.e.
where
$-i,iot
-=
an - (01, .$ ’ a{q)
(0).
(22)
Substituting eqn (11) into the equations above, we have
The additional energy on the interface S is
where u, IIare displacements on the interface S of the potential energy region, TX, T,, are boundary forces on the interface S of complementary energy region. The boundary forces on the interface S of complementary energy region are
where ($1 is the nodal displacement vector in region II except those in interface S. From eqn (23), we have
where, I, m are the direction cosines of the outer normal to the interface S. The displacements on interface of potential energy region are 0 N,
N2
O...N,,0
0 Nz...O
N,, 1
where
{#ii)= Pw>t (17)
where I, and ($1 are the number and the nodal ~spla~ment vector, respectively, Nt are shape functions, which express the nodal displacements on the interface S induced by the unit displacement of node i in same direction. Substituting eqn (16) and eqn (17) into eqn (IS), we can obtain
PI = [Ql-’ - tQl-‘f~lT(i~l~Ql-i~~lT~-‘I~l~Ql-*. (26) Substituting eqns (24) and (25) into eqn (23), one has the basic equation of mixed finite element method in an orthotropic plate with a hole
(18) where where
VI =
J*
Wl”W1do.
(19)
The Lagrange multiplier term HL can be expressed by stress parameters {q) and Lagran@ multiplier (A). From eqn (IO), one has HL = {~YWq).
(20)
Kl=ra+
M[SlJGl’ 0 o
o
*
I
(28)
Once the displacements {w} are known, we can obtain the stress parameters (q} from eqn (25), and the stresses in the region I can then be determined. After the nodal displacements (w ) having been fixed, we will obtain the stresses in region II by standard way.
Zr~o JINPINGand
108
SHAN
HUIZU
4.23.6 -
----
......
Present method Analytic sol. FEM
3.4 -
3.0 s Ft
Fig. 2.
2.6 -
b” 2.2 -
LO0.6 0.6
I 1.2
I I .6
I 2.0 7
Fig. 3.
M=N=l a-(MPa) 14!(10-~mm)
4.1864 -0.9688
RESULTS
i 2.4
I 2.6
I 3.2
(mm)
Fig. 4.
M=N=3 4.2014 -0.9690
M=N=5
Table 1 M=N=7
4.2036 -0.9690
AND CONCLUSIONS
In order to illustrate the correctness and the convergence of the theories mentioned above, we have carried out some calculations on orthotropic plates with holes. Here we show only one example with a circular hole. An orthotropic plate, as shown in Fig. 2, has elastic constants E, = 111.7 GPa, E, = 20.43 GPa, vi2 = 0.6566, Gn = 16.95 GPa, and is loaded by p = 1 MPa. The mesh is given in Fig. 3. The region with shadow in the figure represents the element of stress type, and others are standard quadratic isoparametric elements. The stress a, on the section A-B are shown in Fig. 4. Comparing this with the solution of standard FEM, the solution of mixed finite element method is closer to the analytic solution. Besides, when the external diameter of the element of stress type is 10 times the diameter of its interior hole, Table 1 shows that the maximum tangential stress a, around the hold and vertical displacement a,” of node B vary with the number M, N of term of series selected. The table shows: when M = N = 3, the solutions of 6, and u,” are precise enough (error
4.2039 -0.9690
M=N=9
M=N=ll
4.2040 -0.9690
4.2040 - 0.9690
&f=N=15 4.2040 -0.9690
< 0.1%); while M = N = 7, a,, and u: trend to be stable. So the solutions are of good convergence. The calculations show the preparing and calculating time requested by the proposed method is much less than by the standard FEM. In conclusion, the proposed method for obtaining the stresses of an orthotropic plate with a hole not only has high accuracy and good convergence, but also high efficiency. The proposed method is also suitable for an orthotropic plate with multiple holes. REFERENCES S. G. Lekhnitskii, Anisotropic Plates, 2nd Edn. Gostekhizdat, Moscow (1957). G. N. Savin, StressDistribution Around Holes. Naukova
Dumka, Kiev (1968). A. E. Green and W. Zema, Theoretical Elasticity. Oxford University Press, London (1954). Chien Wei-zang, Studies on Generalized Variational Principles in Elasticity and Their Applications in Finite Element Calculations. Mech. Practice, Nos 1 and 2 (1979). 5. Hu Hai-chang, Variational Principles in Elasticity and Its Applications. Science Press, Peking, China (1981). 6. Long Yu-chiu, Piecewise Generalized Variation Principles in Elasticity. Shanghai J. Mech. 2, NO. 2 (1981).
1. pp. 105-108,
1991
Ml45-7949/91 53.00 + 0.00 Pcrgamon Rcss plc
STRESS ANALYSIS AROUND HOLES IN ORTHOTROPIC PLATES BY THE SUBREGION MIXED FINITE ELEMENT METHOD ZHAO JINPING
and SHANHIJIZU
Beijing University of Aeronautics and Astronautics, Beijing 10083, People’s Republic of China (Receioed 5 July IWO) Abstract-The subregion mixed generalized variational principle is applied in this paper. In the region near a hole, the special element of stress type based on stress analytical solution is adopted. In the exterior region far from the hole, standard finite elements of displacement type are used. The Lagrange multiplier term is adopted so as to satisfy the stress boundary condition on the hole. In order to formulate the special element around the hole, the general expressions of stress and the stress boundary condition of the orthotropic plate with a hole are deduced.
INTRODUCTION
GENERAL EXPRRSSIONS OF STRE% AND STRRSS BOUNDARY CONDITION OF ORTHOTROPIC PLATE WITH HOLE
The
stress analysis around a hole in an infinite orthotropic plate has been studied extensively by Lekhnitskii and Savin et al. [l-3]. But, nowadays the same problem in finite orthotropic plates is mainly solved by adoption of finite elements of displacement types. However, as a result of concentration of stress around the hole, not only is it necessary for the grid to be divided into small parts, but also the solution of stress around a hole has a large error. Recently, Chinese scholars Chien Wei-zang, Hu Haichang and Long Yu-chiu et al. have had great achievements in the study of the subregion general variational principle [4-6]. The principle is applied to orthotropic finite plates with holes. In this paper in the vicinity of the hole with great stress, special elements of stress type based on the analytic solution is used, while in the exterior region far from the hole, standard Unite elements of displacement type are adopted. The stress boundary condition around the hole is satisfied by adoption of the Lagrange multiple term, then a subregion mixed finite element method is established. In order to obtain the special element around a hole, we deduce the general expressions of stress and stress boundary conditions of the orthotropic finite plate with a hole, which are suited for holes of all shapes, for example circular holes, ellipse, crack and so on. As mentioned above, the special element of stress type in the vicinity of the hole is based on analytical solution. As long as there are enough terms in the stress expression of the series, the area of the special element can be larger, for example the size of the special element can be 10 times as large as the size of the hole. So the solution obtained by the method introduced in the paper not only has greater precision, but also greater efficiency and a reduced time of calculation. CM 4,/l-”
For an orthotropic plate, the governing equation is as follows:
a9 az2s
-
202,
aw + (2a,, + axsay
-
aw axzay*
a&5)-
g-$+a,,$=o,
-2a16
(1)
where aij is the elastic constant of the material, U is the stress function. The characteristic equation of eqn (1) is a,,s4-2a,,s3+(2a,2+a,)s2-2a,,s+a,=0.
(2)
It has been proved by Lekhnitskii that the roots s, of eqn (2) are only complex or pure imaginary, the general expressions for the stress components are of the form: (i = 1,2,3,4)
aw
cX= aye = ZRe[s: 4 ‘(2,) + s: # ‘(~~11.
a, =
$ =ZReM’(z,)
aw Tm= -- axay=
+ I,+‘(z*)]
-ZRe[s,b’(z,)
+ s&‘(z2)]
(3)
and the free stress boundary condition can be shown as: ReMi)
+ S(z*)l = 0
Relrl~(z,)+s2~[z211=0.
105
(4)
106
ZHAOJINPINGand SHANHIJIZIJ
where z I = x + s,y, z, = x + s2y, $J(z,) and Jl(zJ are arbitrary analytical functions of z1 and z2, respectively. In order to analyse the stresses around the hole, we use the following transformation:
c,-f(q)
- h,b)
(j = 1,2)
(6)
V’(q)
=
-
(i=1,2)
2q’(z,).
C, = CZ= efi into eqn (8) and
.iM
[(A, + CJcos n0 - (B, + D,)sin no] = 0
.iM
{[Re(W,
- Im(si)B, + Re(s,)C,
- Im(s2)D,]cos n6 - [Im(s + Im(s2)C, + Re(s,)D,]sin
for a circle hole with radius r,, c, = [z, + J-J/(/
ci$=
(5) substituting into eqn (4), we can obtain
According to that transformation, the regions around the hole in the planes z,, z, and z are transformed into the region in the unit circle in the plane C. For the ellipse with semi-axes (I and b, respectively, c, = [z, + ,/w]/(u
2$f’(z,)
‘31
Substituting
(f = 192).
CIJ=
- is/)rlj (j = 1,2).
(7)
According to Lekhnitskii theory, when the principal vector of a load along the hole surface is zero, the stress analytic functions can be expressed as:
+ Re(s,)B, no} = 0.
Merging same kinds of terms of cosp6, sinpe[p=1,2,..., nil, respectively, and setting coefficient of every term zero, then we can obtain the stress boundary condition around hole as follows:
vwq~ = 0.
(10)
THE FUNCTIONAL OF SUBBEGION MIXED SYSTEM IN ORTHOTBOPIC PLATE (8)
where A,, , b, , C, and D, are unknown real coefficients to be determined later. Substituting eqn (8) and eqn (5) into eqn (3), we obtain
As shown in Fig. 1, the region I near the hole is the complementary energy region, in which a special element of stress type based on the analytic solution is used, and the stress variables {q} are the basic unknowns. The region II beyond the region I is the potential energy region, in which the finite elements of displacement type are used, and the nodal displacements {w} are basic unknowns. S is the interface between region I and region II. According to the subregion mixed variational principle, the energy functional of the plate is n=l-Ip-rI,+Hs+H,,
(9)
where, II, is the potential energy of region II, II, is the complementary energy of region I, H, is the additional energy on the interface S of region I and region II, and HL is Lagrange multiplier term so as to satisfy the stress boundary condition on the boundary of hole.
where {A} = [A-M, . . . , A_,, A-,, A,,
A29
{B}= [B-M,. . . , B-2, B-,, 4, B2,. {C}= [C_,, . . . , c-2, c_,, c,, c2,
(_i= L2)
4vlr
hlr * * *, CNIT
. .v
&IT . . . ,?q-‘, . . . ,Nc:-‘l
{D} = [D-M,. . . , D-2, D-I, 4,
(4) = [-q-M-‘,
. . .,
D2,.
(11)
. .I
Fig. 1.
107
Stressanalysisof holes in orthotropic plates The potential energy of region II is % = d {WYKliW) - WP%
(12)
Substituting eqns (12), (13), (18) and (20) into eqn (1I), the energy functional of the subregion mixed system in an orthotropic plate with a hole is given by
where [K] is the stiffness matrix, [P] is the equivalent load vector of nodes in region II. The complementary energy of region I is
where [E] is the elastic matrix of the orthotropic material. Substituting eqn (9) into the equation above, we can obtain (13)
BASIC EQUATION OF THE SUBREGION MIXED FINITE ~~ METHOD OF OR~O~O~C PLATE WlTH HOLE
Based on the subregion mixed generalized variational principle, the actual values of basic cowl must make the energy functional mentioned above stational value, i.e.
where
$-i,iot
-=
an - (01, .$ ’ a{q)
(0).
(22)
Substituting eqn (11) into the equations above, we have
The additional energy on the interface S is
where u, IIare displacements on the interface S of the potential energy region, TX, T,, are boundary forces on the interface S of complementary energy region. The boundary forces on the interface S of complementary energy region are
where ($1 is the nodal displacement vector in region II except those in interface S. From eqn (23), we have
where, I, m are the direction cosines of the outer normal to the interface S. The displacements on interface of potential energy region are 0 N,
N2
O...N,,0
0 Nz...O
N,, 1
where
{#ii)= Pw>t (17)
where I, and ($1 are the number and the nodal ~spla~ment vector, respectively, Nt are shape functions, which express the nodal displacements on the interface S induced by the unit displacement of node i in same direction. Substituting eqn (16) and eqn (17) into eqn (IS), we can obtain
PI = [Ql-’ - tQl-‘f~lT(i~l~Ql-i~~lT~-‘I~l~Ql-*. (26) Substituting eqns (24) and (25) into eqn (23), one has the basic equation of mixed finite element method in an orthotropic plate with a hole
(18) where where
VI =
J*
Wl”W1do.
(19)
The Lagrange multiplier term HL can be expressed by stress parameters {q) and Lagran@ multiplier (A). From eqn (IO), one has HL = {~YWq).
(20)
Kl=ra+
M[SlJGl’ 0 o
o
*
I
(28)
Once the displacements {w} are known, we can obtain the stress parameters (q} from eqn (25), and the stresses in the region I can then be determined. After the nodal displacements (w ) having been fixed, we will obtain the stresses in region II by standard way.
Zr~o JINPINGand
108
SHAN
HUIZU
4.23.6 -
----
......
Present method Analytic sol. FEM
3.4 -
3.0 s Ft
Fig. 2.
2.6 -
b” 2.2 -
LO0.6 0.6
I 1.2
I I .6
I 2.0 7
Fig. 3.
M=N=l a-(MPa) 14!(10-~mm)
4.1864 -0.9688
RESULTS
i 2.4
I 2.6
I 3.2
(mm)
Fig. 4.
M=N=3 4.2014 -0.9690
M=N=5
Table 1 M=N=7
4.2036 -0.9690
AND CONCLUSIONS
In order to illustrate the correctness and the convergence of the theories mentioned above, we have carried out some calculations on orthotropic plates with holes. Here we show only one example with a circular hole. An orthotropic plate, as shown in Fig. 2, has elastic constants E, = 111.7 GPa, E, = 20.43 GPa, vi2 = 0.6566, Gn = 16.95 GPa, and is loaded by p = 1 MPa. The mesh is given in Fig. 3. The region with shadow in the figure represents the element of stress type, and others are standard quadratic isoparametric elements. The stress a, on the section A-B are shown in Fig. 4. Comparing this with the solution of standard FEM, the solution of mixed finite element method is closer to the analytic solution. Besides, when the external diameter of the element of stress type is 10 times the diameter of its interior hole, Table 1 shows that the maximum tangential stress a, around the hold and vertical displacement a,” of node B vary with the number M, N of term of series selected. The table shows: when M = N = 3, the solutions of 6, and u,” are precise enough (error
4.2039 -0.9690
M=N=9
M=N=ll
4.2040 -0.9690
4.2040 - 0.9690
&f=N=15 4.2040 -0.9690
< 0.1%); while M = N = 7, a,, and u: trend to be stable. So the solutions are of good convergence. The calculations show the preparing and calculating time requested by the proposed method is much less than by the standard FEM. In conclusion, the proposed method for obtaining the stresses of an orthotropic plate with a hole not only has high accuracy and good convergence, but also high efficiency. The proposed method is also suitable for an orthotropic plate with multiple holes. REFERENCES S. G. Lekhnitskii, Anisotropic Plates, 2nd Edn. Gostekhizdat, Moscow (1957). G. N. Savin, StressDistribution Around Holes. Naukova
Dumka, Kiev (1968). A. E. Green and W. Zema, Theoretical Elasticity. Oxford University Press, London (1954). Chien Wei-zang, Studies on Generalized Variational Principles in Elasticity and Their Applications in Finite Element Calculations. Mech. Practice, Nos 1 and 2 (1979). 5. Hu Hai-chang, Variational Principles in Elasticity and Its Applications. Science Press, Peking, China (1981). 6. Long Yu-chiu, Piecewise Generalized Variation Principles in Elasticity. Shanghai J. Mech. 2, NO. 2 (1981).