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Explicit stiffness matrix of the linearly varying strain triangular element
Computers& brucruns. Vol. 8, pp. 311-314. PergamonPress 1918. Printed in Great Britain
EXPLICIT STIFFNESS MATRIX OF THE LINEARLY VARYING STRAIN TRIANGULAR ELEMENT K. MOSERt and G. SWOBODAS School of Civil Engineering, Univeristy Innsbruck, Austria (Received 9 March 1977;received for publication
13 July 1977)
Abstract-An explicit stiffness matrix for a linearly varying strain triangular element is derived. It is suggested that this new stiffness matrix is compotationaly efficient to use.
INTROWCTION
Y
Many simple finite element programs for the solution of plane stress and plane strain problems are still using the Constant Strain Triangular (CST) element. Generally this model is chosen because the stiffness matrix of the CST element is well known and can be easily and explicitly programmed. The accuracy of this element is not very high, which constitutes a serious disadvantage when steep stress gradients are to be descriied. The simplest way to improve results with the same number of unknowns is by using the linearly varying strain triangular (LST) element. Normally stiffness matrix of the LST element is evaluated by a procedure of multiplying the component matrices and numerical integration. This process may take more computer time and on a 16-bit word computer it can be difficult due to the effect of the roundoff errors. If a matrix calculation can be carried out by simple substitution of numerical values for its final algebraic terms, it is an explicit matrix. This method was used here to obtain the stiffness matrix of the LST element. The LST element (Fig. l), was first proposed by De Veubek[ll. Felippa[2] used an interpolation matrix for stresses and strains to evaluate the stiffness matrix. Also Argyris[31 derived this type of element by a method using matrix multiplication and inversion. In the solution proposed in this paper a simple straightforward integration procedure is used.
Fig. 1. LST element with I2 degrees of freedom.
STIFlNES!3 MATRIX
The LST element is a 12 degree of freedom element, with a parabolic shape function being used to describe the displacement inside of the corners. This chosen function gaurantees continuity of displacements with adjacent elements, because three points (see points 1, 4 and 2 in Fig. 2) can fix only one parabolic function. A description of the relevant shape function is given in [4]. Following this approach the area coordinates are: L, = (a, + b,x + c,y)/26 L, = (a2 + b,x + c,y)/2A L3 = (a3 + b,x + c,y)RA
(1)
tProf. of Civil Engineering. $Assistant Professor.
Fig. 2. Unit displacements. 311
312
K.
MOSER and G. SWOBODA
in which [D] is the elasticity matrix for plane stress or strain
where A = la& - a,b,(. The geometrical constants are:
(7)
a, =x2--x3
b, =
a2=x3-x,
b2=~1-~3
aj=x,-x*
bs=y,-y,.
Y3 - Y2
The constants for plane stress of an isotropic material are : d,, = 1, d,, = v, d,,= 1, d,,=$?
If the vector, {S)’ represents the nodal displacements of an element. The displacement II) for any point within the element can be expressed by the general form:
f=j-$
and for plane strain d12=v,
d,,=l-v,
d22=l-v
or for the LST element: For symmetrical anisotropic materials(51 eqn (7) is also acceptable. The matrix multiplication must be executed before the integration of the stiffness matrix eqn (6) is carried out. The coefficients of his product depend on the area coordinates and are typically of the form:
(2)
Ul=
k 11.12 -- 16(d12+d33Xu3b3L,2ta,b3L,L3t
With eqn (2) the strain at any point of an element can be evaluated as 2v
a3b,L,L3
+ a$,L32} k 12.12
-
16((d,,a32+d33b32)L,2+2(d,,a,a3+d33b,b,)L,L3
+(d,,a,2+
d33h2)L321.
For the integration it is useful to use the following expression
t3)
a!b!c! L,“~52~&
The [B] matrix has the general form:
JN,
aN2
-z
z aN,
[Bl=
JN4
aa aN,
JN2
ay
ay 3 i ay
ah5
a(2
aNI aN, aa ay
aN,
aa
ay
aN,
ay
dy
ta
+
aN5
ay ahi,
=
aN,
ay da
aN,
ay
b +
c +
2J!
a&
aa aN,
BY
aN,
da
a(2
I
dN5
ahr,
-6
ay
aN, aa
aN,
ay
alv, aa
(8)
2~
. (4)
Substituting elements IN] of eqn (2) into eqn (4) and considering of the rules for the differentiation of area coordinates, the matrix can be explicitly written as 0
b2(4L2 - 1)
&(4L, - 1)
0
b,(4L, - 1)
“I=
0 a,(4L,-l)
b,(4L,-l)
0 a~(4L2- 1)
0 a,(4L3 - 1) aA4L2 - 1) bA4L2 - 1) a3(4~~(4L3 - 1) b3(4L3 - 1) b3(4L3- 1) 0
4(b,L,+ &L,) 0
0 4(b,Lz + b&3) 0 4&L, + OIL,)
0 4@3L2+ a2L3)
4(b,L, + U,) 0
4(a2Lt+alL2)
4(b,L,+b,L,)
4(b3L2+b2LJ
4(a3L,+a,L3)
4(f13L2+a&
In contrast to the well known [B] matrix of the CST element, the matrix eqn (5) is not independent of the position within the element. A linear function of the strain inside the element is defined within the matrix. The stiffness matrix of any type of element is:
The final coefficients of the stiffness matrix evaluated by formula (8) are:
[k]’ = Ef
k 1.1
k,
k,,z
k2.z
. .
k,,,,
. k2.u
46
r
[k]’ = 1
1
[B1’[Dl[BI d(VOL)
(6)
1 k 12.1 1
ku2.z
k,2,,2 1
(9)
Explicit stiffness matrix of the linearly varying strain triangular element
313
Fig. 3. Limit analysis of the LST element.
50.0
m
Dognor
0.4481
4.11°
ot Frnodom
/
si
1
360
Anglo
Constant
*
Load
-
Fig. 4. Error in the calculated deflection for a cantilever beam.
where t is the constant thickness of the element. The coefficients of the stiffness matrix, eqn (9), depend only on the geometry of the element (a,, a,, as, b,, blr bj) and the elasticity matrix [D], and are listed in the Appendix. TFSTEXAMPLE
A serious problem in the application of an element arises very often due to extreme geometrical dimensions. For instance, if an element is very long or the distance b (Fig. 3) is extremely short, the numerical errors in the stiffness matrix of the element can be critical and give inaccurate results. A cantilever system (Fig. 4) of constant height and different lengths L was tested to study the influence of extreme geometrical dimensions. In Fig. 4 the angle (I lies between 33.69 and 4.11”. For a point at the end of the beam the theoretical deformation and the finite element solution differ only within the range of 0.31 and 0.44%. From the practical point of view, in this example the use of the LST element resulted in solutions with negligible error.
metry of the stiffness matrix can be fully utilized. More calculations with the element would be required to define the critical geometric dimensions.
1. B. M. Fraeijs de Veubeke, Displacement and equilibrium models in finite element method. Stress Analysis, Chap. 9. Wiley, New York (1%5). 2. C. A. Fellippa, Refined Finite Elecent Analysis of Linear and Non-linear
Two-dimensional
Structures. Structural Engineer-
3 ing Laboratory, University of California, Berkeley (1966). J. H. Argyris, Triangular elements with linearly varying strain ’ for the matrix displacement method. J. Royal Aeron. Sot. 69 (1%5).
4. 0. C. Zienkiewicz, 5. 6.
The Finite Element Mefhod in Engineering Science. McGraw-Hill, London (1971). S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body. HoldenaDay, San Francisco (1%3). G. Swoboda, Rissuntersuchungen in Stahlbetonbalken und Seheiben mit Hilfe des LST-Elementes. Eauingenieur 50, p. 465-468 (1975).
CONCJNSION
An explicite stiffness matrix of an LST element has been developed here for plane stress and plane strain with anisotropic elastic symmetry. The usual matrix multiplication is not required and therefore the number of computer operations is a minimum. Also, the sym-
= (d3.3a1bi=1.2.3.2.3+ dl.*bla,-l,2,3.2.3)C1,+I k,.,=2...6.8.,2 k2.,=2.4.~,~.12 = (d2.2a1~,=1.2,X.2.3 f d3,3b,b,=,.2.3.2.3)Cll+l kt.,=3.J.7.1, = (d3.3b1a,=2.3.2.3+ d1.20,=2.3.2.3+ cl,
K. MOSER and G. SWOLIODA
314 k,,,=,.5.7.9
= (d,.,a2a,=2.,.l,,+
d~.~b~b,=z.,,&‘T
k,.r=m,.,o
= (d1,2b2a,=2.,.1.,
t d,,,a2b,=2.,,&2,+~
kwm.,o
= (d2.2a2a,=2.,,,.,
t d,.,bzb,=z.,.~.,)C*,+~
k,,,,
= (d,,2a2b,=~,,,,
1
kw.w
= (d,,2b,a,=,,2,1
k6,,=6.,0.12
=
k,.,2=~[d,,(2a2b,+a,b,+a2bIta,b,)
t d,,,b2a,=,,I,,)C2,
8= (d,,,a,a,=,,2,,
k,.,=mm
+ d,,,b,b,=,,S3,
+d,,(2a,b2+
+ d,,,alb,=,.2,1)C3,,,
(d2.2a,a,=,.2.1
kg,8 =
,.5.7.,,
c*t
=,.x,.9
b, b,) + dz2(a12
+ az2 + a,a,)l
b, + a,b,
t alb2
+
a2b2)l
ks.,o=~ld,,(2b,b,+b,b,+b,b,+b,‘)
3’3’3
+ a,a2
t a2a3 t az2)1
= Lf,; k,,,,
k,,,
a,b,)l
+ a@,+a2b,+a2b2)
+d,,(2a,a, c3,=,,9,,,
+ a,&+
t
t bt
+d2,(6x,
-144
-
i [d,,(b,*
k,,.9=ild,,(2a,b,
=1*-f, =I
a$,
+ d,,b,b,-,.2$3,+l
ke.,r=w 1 = ~d,,2a@,=2,1 + d&a,=&3, ,=I,
2
+d,,(2b2b2+b,b,+b,b2+b,2~l
k~,r=m
Cl
4
=jId,,(2a2b,+ala,+ala2ta,
= d,,a,’
=~ld,,(2a,b2+a,b2+a,b,+a,b,)
t d,,b,” +d,,(2a2b,+a2b,
ksg = a,b,(d12
+a,b,+a,bJl
+ d,,) k8,,2=f[d,,(2bb,+b,b,+b,b,+b,2)
k5,9 =~(d,,a,a,t
d,,b,b,)
k,,,,
+ d,,a,b,)
+d22(2a,a,+a,a2+a,a,+a,2)1 =
i(d,2azb,
k5.,,
4 = j(d,,a,a,
+ d,,b,bh
k,,,,
= i (d,,a,b,
+ d,,a,b,)
k6.6 = d2,a,2
+ d,,b,’
k,,,
= i [d,,(a:
t
k,,,,
= i (d,,
+ d,J2(a2b2
kg.,,
= i [d,,(2a,a2
+ d,,b,a2J
k,,.,,, = i (dz2a,a2
+ d,,b,bJ
k6.,]
= ~(d,s,b,
+d,,b,aJ
k,.,,
4 = J (ds,a,
+ d,,b,bJ
k9.,2 = i [d&a,b, t dr2(2a,b2 k 1o.,o = ; 14,(b22
i[d,,(o,‘t at t
=i(d,,+
+ a&,)
+ a2a3 + ala3
+ b,2 t bzb,)l
+ a2b,
+
a,b21
t a,‘)
+ a,b, + a,62 + b,‘+
t a,b, t 0, b2h)
t a,&) + a,b,)l + .fz2(az2
+ a,‘+
k la.il=~ld,,(~,b2+a,b~+a,b,ta,b,) +d,,(2azb,
k,,,
+ d,,(b2*
+d,,(2b,b2+b2b,+b,b,+b:)1
k6.9 = i (d,,a,b,
k,,, =
at + a,a,)
a,a,)
t
d,,(b,“+
d,,)12(a,b,+a2b2)t
bt+
a,b2+a2b,1
b,b,)l
+ a2b,
+ a,b,
+
a,b,)l
k,o.,z= ; [d,,W,b, + bzb, + 0, + b,‘) +d2,(2a,a2
+ a2a3 + ala,
t a,‘)]
k,,.,,=~[d,,(a12+a,‘+a,a,)td,,(b,2+b,2+b,b,)l k7,9 = i [d,,(2a,a, +d,,(2b,b,
t a,a2
+ a2a,
t a,fI
t b,b,+ b,b, t
b,‘)l
t a,b,
a,bd
k,,,=~Id,,(2a,b,+a,b,ta,b,ia,b,) +d,,(2a,b,
k ,,.,2=~(d,2+d,,H2(a,b,+a,b,)+a,b,+a,b,l
k,2,,2=~[d,,(b,2+b,2+b,b,)td22(a,2+a,2+a,a31. t aA
t
a2a,)1
EXPLICIT STIFFNESS MATRIX OF THE LINEARLY VARYING STRAIN TRIANGULAR ELEMENT K. MOSERt and G. SWOBODAS School of Civil Engineering, Univeristy Innsbruck, Austria (Received 9 March 1977;received for publication
13 July 1977)
Abstract-An explicit stiffness matrix for a linearly varying strain triangular element is derived. It is suggested that this new stiffness matrix is compotationaly efficient to use.
INTROWCTION
Y
Many simple finite element programs for the solution of plane stress and plane strain problems are still using the Constant Strain Triangular (CST) element. Generally this model is chosen because the stiffness matrix of the CST element is well known and can be easily and explicitly programmed. The accuracy of this element is not very high, which constitutes a serious disadvantage when steep stress gradients are to be descriied. The simplest way to improve results with the same number of unknowns is by using the linearly varying strain triangular (LST) element. Normally stiffness matrix of the LST element is evaluated by a procedure of multiplying the component matrices and numerical integration. This process may take more computer time and on a 16-bit word computer it can be difficult due to the effect of the roundoff errors. If a matrix calculation can be carried out by simple substitution of numerical values for its final algebraic terms, it is an explicit matrix. This method was used here to obtain the stiffness matrix of the LST element. The LST element (Fig. l), was first proposed by De Veubek[ll. Felippa[2] used an interpolation matrix for stresses and strains to evaluate the stiffness matrix. Also Argyris[31 derived this type of element by a method using matrix multiplication and inversion. In the solution proposed in this paper a simple straightforward integration procedure is used.
Fig. 1. LST element with I2 degrees of freedom.
STIFlNES!3 MATRIX
The LST element is a 12 degree of freedom element, with a parabolic shape function being used to describe the displacement inside of the corners. This chosen function gaurantees continuity of displacements with adjacent elements, because three points (see points 1, 4 and 2 in Fig. 2) can fix only one parabolic function. A description of the relevant shape function is given in [4]. Following this approach the area coordinates are: L, = (a, + b,x + c,y)/26 L, = (a2 + b,x + c,y)/2A L3 = (a3 + b,x + c,y)RA
(1)
tProf. of Civil Engineering. $Assistant Professor.
Fig. 2. Unit displacements. 311
312
K.
MOSER and G. SWOBODA
in which [D] is the elasticity matrix for plane stress or strain
where A = la& - a,b,(. The geometrical constants are:
(7)
a, =x2--x3
b, =
a2=x3-x,
b2=~1-~3
aj=x,-x*
bs=y,-y,.
Y3 - Y2
The constants for plane stress of an isotropic material are : d,, = 1, d,, = v, d,,= 1, d,,=$?
If the vector, {S)’ represents the nodal displacements of an element. The displacement II) for any point within the element can be expressed by the general form:
f=j-$
and for plane strain d12=v,
d,,=l-v,
d22=l-v
or for the LST element: For symmetrical anisotropic materials(51 eqn (7) is also acceptable. The matrix multiplication must be executed before the integration of the stiffness matrix eqn (6) is carried out. The coefficients of his product depend on the area coordinates and are typically of the form:
(2)
Ul=
k 11.12 -- 16(d12+d33Xu3b3L,2ta,b3L,L3t
With eqn (2) the strain at any point of an element can be evaluated as 2v
a3b,L,L3
+ a$,L32} k 12.12
-
16((d,,a32+d33b32)L,2+2(d,,a,a3+d33b,b,)L,L3
+(d,,a,2+
d33h2)L321.
For the integration it is useful to use the following expression
t3)
a!b!c! L,“~52~&
The [B] matrix has the general form:
JN,
aN2
-z
z aN,
[Bl=
JN4
aa aN,
JN2
ay
ay 3 i ay
ah5
a(2
aNI aN, aa ay
aN,
aa
ay
aN,
ay
dy
ta
+
aN5
ay ahi,
=
aN,
ay da
aN,
ay
b +
c +
2J!
a&
aa aN,
BY
aN,
da
a(2
I
dN5
ahr,
-6
ay
aN, aa
aN,
ay
alv, aa
(8)
2~
. (4)
Substituting elements IN] of eqn (2) into eqn (4) and considering of the rules for the differentiation of area coordinates, the matrix can be explicitly written as 0
b2(4L2 - 1)
&(4L, - 1)
0
b,(4L, - 1)
“I=
0 a,(4L,-l)
b,(4L,-l)
0 a~(4L2- 1)
0 a,(4L3 - 1) aA4L2 - 1) bA4L2 - 1) a3(4~~(4L3 - 1) b3(4L3 - 1) b3(4L3- 1) 0
4(b,L,+ &L,) 0
0 4(b,Lz + b&3) 0 4&L, + OIL,)
0 4@3L2+ a2L3)
4(b,L, + U,) 0
4(a2Lt+alL2)
4(b,L,+b,L,)
4(b3L2+b2LJ
4(a3L,+a,L3)
4(f13L2+a&
In contrast to the well known [B] matrix of the CST element, the matrix eqn (5) is not independent of the position within the element. A linear function of the strain inside the element is defined within the matrix. The stiffness matrix of any type of element is:
The final coefficients of the stiffness matrix evaluated by formula (8) are:
[k]’ = Ef
k 1.1
k,
k,,z
k2.z
. .
k,,,,
. k2.u
46
r
[k]’ = 1
1
[B1’[Dl[BI d(VOL)
(6)
1 k 12.1 1
ku2.z
k,2,,2 1
(9)
Explicit stiffness matrix of the linearly varying strain triangular element
313
Fig. 3. Limit analysis of the LST element.
50.0
m
Dognor
0.4481
4.11°
ot Frnodom
/
si
1
360
Anglo
Constant
*
Load
-
Fig. 4. Error in the calculated deflection for a cantilever beam.
where t is the constant thickness of the element. The coefficients of the stiffness matrix, eqn (9), depend only on the geometry of the element (a,, a,, as, b,, blr bj) and the elasticity matrix [D], and are listed in the Appendix. TFSTEXAMPLE
A serious problem in the application of an element arises very often due to extreme geometrical dimensions. For instance, if an element is very long or the distance b (Fig. 3) is extremely short, the numerical errors in the stiffness matrix of the element can be critical and give inaccurate results. A cantilever system (Fig. 4) of constant height and different lengths L was tested to study the influence of extreme geometrical dimensions. In Fig. 4 the angle (I lies between 33.69 and 4.11”. For a point at the end of the beam the theoretical deformation and the finite element solution differ only within the range of 0.31 and 0.44%. From the practical point of view, in this example the use of the LST element resulted in solutions with negligible error.
metry of the stiffness matrix can be fully utilized. More calculations with the element would be required to define the critical geometric dimensions.
1. B. M. Fraeijs de Veubeke, Displacement and equilibrium models in finite element method. Stress Analysis, Chap. 9. Wiley, New York (1%5). 2. C. A. Fellippa, Refined Finite Elecent Analysis of Linear and Non-linear
Two-dimensional
Structures. Structural Engineer-
3 ing Laboratory, University of California, Berkeley (1966). J. H. Argyris, Triangular elements with linearly varying strain ’ for the matrix displacement method. J. Royal Aeron. Sot. 69 (1%5).
4. 0. C. Zienkiewicz, 5. 6.
The Finite Element Mefhod in Engineering Science. McGraw-Hill, London (1971). S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body. HoldenaDay, San Francisco (1%3). G. Swoboda, Rissuntersuchungen in Stahlbetonbalken und Seheiben mit Hilfe des LST-Elementes. Eauingenieur 50, p. 465-468 (1975).
CONCJNSION
An explicite stiffness matrix of an LST element has been developed here for plane stress and plane strain with anisotropic elastic symmetry. The usual matrix multiplication is not required and therefore the number of computer operations is a minimum. Also, the sym-
= (d3.3a1bi=1.2.3.2.3+ dl.*bla,-l,2,3.2.3)C1,+I k,.,=2...6.8.,2 k2.,=2.4.~,~.12 = (d2.2a1~,=1.2,X.2.3 f d3,3b,b,=,.2.3.2.3)Cll+l kt.,=3.J.7.1, = (d3.3b1a,=2.3.2.3+ d1.20,=2.3.2.3+ cl,
K. MOSER and G. SWOLIODA
314 k,,,=,.5.7.9
= (d,.,a2a,=2.,.l,,+
d~.~b~b,=z.,,&‘T
k,.r=m,.,o
= (d1,2b2a,=2.,.1.,
t d,,,a2b,=2.,,&2,+~
kwm.,o
= (d2.2a2a,=2.,,,.,
t d,.,bzb,=z.,.~.,)C*,+~
k,,,,
= (d,,2a2b,=~,,,,
1
kw.w
= (d,,2b,a,=,,2,1
k6,,=6.,0.12
=
k,.,2=~[d,,(2a2b,+a,b,+a2bIta,b,)
t d,,,b2a,=,,I,,)C2,
8= (d,,,a,a,=,,2,,
k,.,=mm
+ d,,,b,b,=,,S3,
+d,,(2a,b2+
+ d,,,alb,=,.2,1)C3,,,
(d2.2a,a,=,.2.1
kg,8 =
,.5.7.,,
c*t
=,.x,.9
b, b,) + dz2(a12
+ az2 + a,a,)l
b, + a,b,
t alb2
+
a2b2)l
ks.,o=~ld,,(2b,b,+b,b,+b,b,+b,‘)
3’3’3
+ a,a2
t a2a3 t az2)1
= Lf,; k,,,,
k,,,
a,b,)l
+ a@,+a2b,+a2b2)
+d,,(2a,a, c3,=,,9,,,
+ a,&+
t
t bt
+d2,(6x,
-144
-
i [d,,(b,*
k,,.9=ild,,(2a,b,
=1*-f, =I
a$,
+ d,,b,b,-,.2$3,+l
ke.,r=w 1 = ~d,,2a@,=2,1 + d&a,=&3, ,=I,
2
+d,,(2b2b2+b,b,+b,b2+b,2~l
k~,r=m
Cl
4
=jId,,(2a2b,+ala,+ala2ta,
= d,,a,’
=~ld,,(2a,b2+a,b2+a,b,+a,b,)
t d,,b,” +d,,(2a2b,+a2b,
ksg = a,b,(d12
+a,b,+a,bJl
+ d,,) k8,,2=f[d,,(2bb,+b,b,+b,b,+b,2)
k5,9 =~(d,,a,a,t
d,,b,b,)
k,,,,
+ d,,a,b,)
+d22(2a,a,+a,a2+a,a,+a,2)1 =
i(d,2azb,
k5.,,
4 = j(d,,a,a,
+ d,,b,bh
k,,,,
= i (d,,a,b,
+ d,,a,b,)
k6.6 = d2,a,2
+ d,,b,’
k,,,
= i [d,,(a:
t
k,,,,
= i (d,,
+ d,J2(a2b2
kg.,,
= i [d,,(2a,a2
+ d,,b,a2J
k,,.,,, = i (dz2a,a2
+ d,,b,bJ
k6.,]
= ~(d,s,b,
+d,,b,aJ
k,.,,
4 = J (ds,a,
+ d,,b,bJ
k9.,2 = i [d&a,b, t dr2(2a,b2 k 1o.,o = ; 14,(b22
i[d,,(o,‘t at t
=i(d,,+
+ a&,)
+ a2a3 + ala3
+ b,2 t bzb,)l
+ a2b,
+
a,b21
t a,‘)
+ a,b, + a,62 + b,‘+
t a,b, t 0, b2h)
t a,&) + a,b,)l + .fz2(az2
+ a,‘+
k la.il=~ld,,(~,b2+a,b~+a,b,ta,b,) +d,,(2azb,
k,,,
+ d,,(b2*
+d,,(2b,b2+b2b,+b,b,+b:)1
k6.9 = i (d,,a,b,
k,,, =
at + a,a,)
a,a,)
t
d,,(b,“+
d,,)12(a,b,+a2b2)t
bt+
a,b2+a2b,1
b,b,)l
+ a2b,
+ a,b,
+
a,b,)l
k,o.,z= ; [d,,W,b, + bzb, + 0, + b,‘) +d2,(2a,a2
+ a2a3 + ala,
t a,‘)]
k,,.,,=~[d,,(a12+a,‘+a,a,)td,,(b,2+b,2+b,b,)l k7,9 = i [d,,(2a,a, +d,,(2b,b,
t a,a2
+ a2a,
t a,fI
t b,b,+ b,b, t
b,‘)l
t a,b,
a,bd
k,,,=~Id,,(2a,b,+a,b,ta,b,ia,b,) +d,,(2a,b,
k ,,.,2=~(d,2+d,,H2(a,b,+a,b,)+a,b,+a,b,l
k,2,,2=~[d,,(b,2+b,2+b,b,)td22(a,2+a,2+a,a31. t aA
t
a2a,)1