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Structural analysis by a combined finite element-transfer matrix method
KLl54949/831090321-06SO3.00/0 Pergamon Press Ltd.
Computers & Structuns Vol. 17, No. 3. pp. 321-326. 1983 Printed in Great Britain
STRUCTURAL ANALYSIS BY A COMBINED FINITE ELEMENT-TRANSFER MATRIX METHOD M. OHGA Department of Civil Engineering, Ehime University, Matsuyama, Japan and T. SHIGEMATSU and T. HARA Tokuyama Technical College, Tokuyama, Japan (Receioed 5 August 1982;received for publication 25 September 1982) Abstract-The combined finite element-transfer matrix method has the advantage of reducing the size of a matrix to less than that obtained by the ordinary finite element method. The analytical procedure in this method for bending and buckling problems are described, and techniques for treating the structure with intermediate conditions are proposed. Various numerical examples of these problems are shown to demonstrate the efficiency and accuracy of this method. The results from these examples agree well with those obtained by the finite element method and others. NOTATION
A, a b D E E,
{F) I, [K_li [K] k [PtJ [P,] [T]
t {z) g y {S}
cross section area of rib dimension of finite element or plate dimension of finite element or plate flexural rigidity of plate (Et’/12(1- v*)) modulus of elasticity of plate modulus of elasticity of rib force vector moment of inertia of rib stiffness matrix of strip i stiffness matrix of rib buckling coefficient (Pb*/a*D) point matrix for elastic column point matrix for rib transfer matrix thickness of plate state vector ratio of area (A,/bt) ratio of rigidity (E&/D) displacement vector 1. INTRODUCTlON
The finite element method is the most widely used and
powerful tool for structural analysis. However, the disadvantage of this method is that, in the case of a complex structure, it is necessary to use a large number of nodes, resulting in very large matrices which require large computers for their management and regulation. In order to reduce the size of the matrices in the ordinary finite element method, some techniques have been proposed (condensation, substructing method) [ 11. One numerical technique for reducing matrix size in the ordinary finite element method is the use of finite strips (FSM) suggested by Cheung[2]. Another is the transfer matrix technique (TMM) which was applied to two dimensional problems for the first time by Pestel and Leckie[3]. Plate vibration problems at that time were formulated by using the Hrennikoff model. The above techniques (FSM, TMM) can be successfully applied only for simple structures with particular boundary conditions; otherwise considerable complications arise in the formulation of problems. Dokainish used the combined finite element-transfer matrix (FETE) method in the study of the dynamics of tapered or rectangular plates[S]. Since the size of
stiffness and mass matrices, in his method, was equal to the number of degrees of freedom of one strip, the frequency determinant for a clamped-clamped plate considered by Dokainish was 18x 18 by the FETM method compared to a 108x 108 matrix eigenvalue problem obtained using the standard finite element method with the same number of nodes. McDaniel and Eversole have proposed a similar approach in treating a stiffened plate structures along with some numerical values that warrant consideration [6]. In dealing with complex structures, Chiatti and Sestieri introduced isoparametric shell elements, taking into consideration elements with nodes situated not only on corners but also on the midpoints of edges[7]. Sankar and Hoa offer an approach, in which an extended transfer matrix relating the state vectors which consist of state variables (displacements and forces) and their derivatives with respect to frequency was used[B]. In this method, a Newton-Raphson iterative technique is used to determine natural frequencies. Mucino and Paveric, as a further generalization of the FETM method, have proposed a method in which structures are modeled by means of substructures connected in a chain-like manner. For each of these substructures, a transfer matrix was derived[9]. Application of the FETM method is generally found in literature concerned with vibration problems of structure. This paper shows a successful application of the FETM method to other structural fields, especially to bending and buckling problems. Also, various techniques for treating the more complicated structures, especially those with the intermediate conditions are presented, Some numerical examples of bending and buckling problems are proposed and their results are compared with those obtained by the ordinary finite element method and others.
2. trrm’rEnt.nMEW-TRANSFER MATRM MElliOD A detailed description of the combined finite elementtransfer matrix may be found in Ref. 151.Figure 1 shows a plate divided into m strips and each of which subdivided into finite elements. The vertical sides dividing or bordering the strips are called sections, while the 321
M. OHGA
n
or
a
LT
1
2
i
It1
AB
01
2
i-l
Equation (7) can be recognized as the transfer matrix relating the state vectors {Z,} and {Z,} which consist of the displacements and forces. After continuous multiplications of the transfer matrix [T], we obtain the relation between the state vectors at two ends of the structure:
m
C
i
it1
et al.
m-l
m
i-l
i
{Z),,, = [ UHZ),,
(8)
Fig. I. Subdivision of plate into strips and finite elements.
horizontal boundaries are the edges. Thus BE is the left section of strip i + 1 and the right section of i. There are a total of 2n nodes on strip i with n nodes on the left section AD and n nodes on the right section BE. To derive the transfer matrix relating the left and right state variables (displacements and forces) of the strip i, it is required first to determine the stiffness matrix [K], of strip i: we obtain
[KliISh = {Fh
(1)
where [K], is the stiffness matrix of strip i, {S}i,{F}, are the displacements and forces of strip i, respectively. Equation (1) holds well for bending problems, but in buckling problems, the matrix [Kj, in eqn (1) becomes:
[Kli = [[Kbli- P[Kmlil
(2)
where [Kbli and [Km]i are the bending stiffness matrix and the modified stiffness matrix of strip i, respectively; P is the in-plane load. Matrix [K]i is partitioned into four sub-matrices. Equation (1) then becomes:
where [U]=[T],[T],.., . ..[T].. In bending problems, on considering the left and right boundary conditions of the structure, simultanious equations are obtained from eqn (8). The number of these equations is as same as that of the unknown state variables in {Z},. Thus, we can evaluate the unknown elements in {Z}, by solving these equations. On the other hand, in buckling problems, it is essential that the determinant of a portion [*U] of the matrix [U] be zero: detI*Uj=0.
(9)
Now, the matrix [*U] is obtained from the matrix [U] by deleting the columns corresponding to zero elements of {Z}, and the rows corresponding to the nonzero elements of {ZLl. 3. TECHNIQUJLS FOR INTERMEDIATECONDITIONS
Point matrix for elastic columns
Point matrix for treating structure with elastic columns at the intermediate section, as shown in Fig. 2, is obtained by taking the elastic support restoring forces into consideration. Consider, for example, elastic columns attached to nodes I and m of section i. The relations of the shearing forces to the left and right of the section i are then,
(3) where {S},,{S),, {F}, and {F}, are the left and right displacements and forces of strip i, respectively. By expanding expression (3) and solving for {S,}i and {Fr}i in terms of {Soi and {Fl}i, the following equations can be obtained;
where k, and k, are the elastic column stiffness. Since other elements of the state vector are continuous throughout section i, the following identity exists:
(da) and IF,}, = UK,,1 - [K,,IIK,,I~l[K,,II{S,}, + K,I[KJ’U%
WI
which, when arranged in matrix form, become:
On simplifying the notation, we obtain:
(6)
Fig. 2. Intermediate elastic columns
323
A combined finite element-transfer matrix method In matrix notation, eqns (10) and (11) become:
(12)
Setting wil = wim= 0 from the rigid conditions at the node I and m, we obtain the following two equations:
or
{Zh’ = [Pkl{Zh’.
(13) {U&
Equation (13) relates the left state vectors of the section which has some elastic columns, to the right state vectors. Consequently the matrix [A] is referred to as the point matrix for the elastic column. Intermediate rigid conditions Point matrices for elastic columns break down when elastic columns become infinitely stiff. In this case, when the deflections at the intermediate rigidcolumns are zero, the initial unknowns corresponding to the constrained displacements can be eliminated and introduced new unknownns. For example, consider the structure, shown in Fig. 3, which has rigid columns at nodes 1 and m of section i, with its left boundary simply supported. The equation relating the left state vector of the section i {Z},’ to the initial unknowns {Z}, is
. {Oy, Q, M.xhT = Wil = 0
{ ULt}t . IBy, Q, MxhT = Wm = 0.
(16)
where {Vi}, and {U&h are the I-th, m-th row of the matrix [U’],, respectively, Wirand Wi, are the deflection at nodes I and m of section i. Solving eqn (16) for QOr and Qo,,,, we eliminate these two shearing forces from the initial state vector {z),. Because of the reactions at the rigid columns, the shearing forces at these points are discontinuous. Introducing the new unknown Vrrand Vi, instead of Qor and Qom just above eliminated alone, the right state vectors of section i are expressed as, W, tL’, e;,
Q’, M,‘, Ms’h= = WI&
Q’, MA=
(17) where,
LZ)i’= 1U’l{Zo}.
(14)
From the. left boundary condition {w}~= {0,}, = {My}o = 0, the elementary form of eqn (14) is
{w’,B,l,0,‘, Q’, M,‘, M;]i= = [WI.
{e,, Q, Mx}o=. (15)
By the above technique, the transfer procedure can be performed throughout a section having intermediate rigid columns. The structure which has the intermediate simple support as shown in Fig. 4 can be treated as previously described. In this case, the deflections and rotations about the x-axis are constrained at the intermediate
___ ;.----r-__ ,L__ -a.-__, ___ -*-_&__ -__ I --7---
simple support
H
i-l
Fig. 3. Intermediate rigid columns.
i
it1
Fig. 4. Intermediatesimplesupport.
324
M. OHGA
simple support. By eliminating the initial shear forces and moments about the x-axis, which correspond to the constrained displacements, from the initial state vector {rd. the new unknown discontinouus shears and moments can be introduced to the state vector (z}. 4. APPLICATION TO NON-LINEAR PROBLEMS
So far, our discussion has been limited to linear problems, but the FETM method may also be applied to non-linear problems of structure. The computer storage and time required for analysis of non-linear problems are usually more than those involved in linear problems. Thus, in this regard, the advantages attainable through matrix size reduction in the FETM method will become more evident. The FETM method may be easily extended to nonlinear problems of structure, since the same incremental procedures used in the finite element method can be applied except the evaluation of incremental displacements for each specified incremental load. Furthermore, incremental calculations can be carried out in the same manner as that for linear problems described in this paper. 5. FZAMPLE Bending analysis of a plate structure The rectangular element with three degrees of freedom per one node is used in example; the deflection w is assumed to have the form,
where ff = xla, 5 = y/b and a,, . . , aI2 are unknown coefficients. A partially loaded and all edges clamped rectangular slab with intermediate ribs, shown in Fig. 5, was analysed. In this example, the slab was divided into 6 and 18 strips and each strip into 6 finite elements for both mesh types, as shown in Fig. 6. The point matrices for the rib [P,] proposed by McDaniel and Eversole[6] were used in considering the ribs. The transformation procedure was performed by multiplications of not only the transfer matrices IT] but
eta/
also the point matrices [P,], for the 6 strips mesh pattern:
In Fig. 7, the deflections along the symmetric line obtained by the FETM method are compared with those by the finite element method, in which the same element as that used in the FETM method was employed, for the 18 strips mesh pattern. It can be seen that the results obtained by both methods are in complete agreement with each other. The matrix to be considered in the finite element method is, if the banded matrix is used? 147x 27 for 6 strips mesh pattern and 399 x 27 for 18 strips. Thus the matrix size for latter mesh pattern is 2.7 times larger than that for the former pattern. On the other side, the matrix to be considered in the FETM method is 42 x 42 for both mesh patterns, since the matrix size in the FETM method is dependent on the number of degrees of freedom for one strip in contrast with the finite element method which depends on that for the entire structure. The deflections for EJ, = = are also shown in Fig. 7. In this case, the transformation procedure can be performed in a simple schematic manner by using the technique for intermediate simple support. The deflections by the FETM method agree well with those by the finite element method. To illustrate the efficiency of the technique for an intermediate rigid column and the point matrix for an elastic one, a bridge deck with four intermediate columns acted upon by partially distributed loads, shown in Fig. 8, was analysed. It is divided into 16 strips and each strip into 8 elements. In Fig. 9, the deflections in the case of intermediate rigid columns by the FETM method are compared with those by the finite element method. It can be seen that these results agree well with each other. In this example.
0
9 x(m)
6
3
-0.4 Y
=
0. 0.4 -FETM(6-10)
0.8 E
I
a=1t/m-
," d
/' rib
Fig. 5. AI1edges clamped slab with intermediate ribs
1.2 H (cm)
m
FETM(6-6)
0
FEM
(6-18)
Fig. 7. Deflections along the symmetric line.
ErIr/tl
325
A combined finite element-transfer matrix method 5m
10m
10m
I
10m
I
5m
I I
---
Ref.
10
-FETM --
0
FEM
m2
(6-12) (6-10)
Fig. 8. Simplysupportedbridgedeck with intermediate columns.
_4k
D.lF5
0;2ioj!~Tx;
---((6(6.
6)
(6.
4)
0;5 0
0
8)
FEM
1
2
a/b
3
Fig. 10. Buckling coefficients of all edges clamped plate.
O-
k
4-
---Ref. -
10
F E T M (6-6)
6
Fig. 9. Deflections at line of columns.
4
the matrix size in the finite element method is 459 X 33 for banded matrix, while in the FETM method the order of the matrix is only 54 x 54. The deflections for the intermediate elastic columns are shown together in Fig. 9. In this case, the transformation procedure was performed by introducing the point matrix for the elastic column [&I:
Y = 6 = Ar,'bt = 0.1
I.% = [Tls[Tl,[Pkl6[Tk~. . ~Pkl2mt[Tl1IZ}o.
(20) Buckling
analysis
I
0
of a plate structure
The element and degree of freedom in bending problems are used in buckling problems. A uniformly compressed rectangular plate with four clamped elges was analysed. The plate was divided into 4, 6, 8, 10 and 12 strips along the direction of compresssion and each strip into 6 elements. The buckling coefficients obtained by the FETM method and the finite element method are indicated in Fig. 10. It is seen that although the results by two methods agree as well as those in bending problems, the accuracy of these results decreases as the buckling mode increases. But it is also seen that the accuracv increases as ~.
D
1
2
a/b
3
Fig. 1I. Buckling coefficients of stiffened plate.
the number of strips increases. In the FETM method, the matrix size is 42 x 42 for any mesh pattern used in this example, while in the finite element method, if banded matrix is used, it is 105X 27 for the 4 strips pattern and 273 x 27 for the 12 strips pattern. As the second buclking problem example, a uniformly compressed rectangular plate clamped along two opposite sides perpendicular to the direction of compression and having reinforced free edges by ribs along
3?6
M. OHCA et al.
the other two sides was analysed. The plate was divided into 6 and 12 strips, and each strip as in the previous example, into 6 elements. As shown in Fig. 11, close agreement in the results by the FETM method and the finite element method was obtained. The buckling coefficients for the plate clamped along two opposite sides perpendicular to the direction of compression and simply supported along the other two sides are given simultaneously in Fig. 11,to provide the upper limit of this plate. 6. CONCLU~ONS In this paper, the procedures of the combined finite element-transfer matrix method for bending and buckling problems are described and the techniques for treating the structure with intermediate conditions are proposed. Furthermore, the analysis of non-linear probrems by the FETM method is given. Some numerical examples presented in this paper show that the FETM method can be successfully applied to bending and buckling structural problems by reducing the size of the matrix relative to less than that obtained by the finite element method. The techniques for intermediate conditions make possible the application of this method to the more complicated structures.
REFERENCES
I. W. McGuire and R. H. Gallagher, Mrltris S’trlcc,frrrai .?.cI~I~~\!. Wiley, New York (1979). 2. Y. K. Cheung, Finite Sfrip ~fet~~)din S~r~~~~ur~~~ &t~i\z~k.
Pergamon Press, Oxford ( 197hl. 3. E. C. Pestel and F. A. Leckie. Mutris Muthodr iti E&I’-
tomechanics. McGraw-Hill, New York (1963). 4. B. Nath, Fundamentals of Finite El~mer~tsfor Engineers. The
Athlone Press (1974).
5 M. A. Dokainish, A new approach for plate vibration>: .
Combination of transfer matrix and finite-element technique. Trans. ASME, J. ~~~ineer;n~ ~nd~~.sfr~, 35-530 (1W). 6 T. J. MeDaniel and K. B Eversole. -1 combined finite tiement-transfer marix structural analysis method. .I. So~rntl Vib. 51, 157-169(1977). I. G. Chiatti and A. Sestieri. Analysis of static and dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vib. 67, 3-42 (1979). 8. S. Sankar and S. V. Hoa, An extended transfer matrix-finite element method for free vibration of plates. J. Sound Vib. 70. 205-211(19801. 9. H. V. Mucino and \‘. Pa\rlic. ,AnL’UCIcondencation procedure for chain-like structures using a finite element-transfer matrix approach. Trans. ASME, L~Mech. Design, 1-9 (1980). 10.A. Pfliiger, Stabilitiitsprohleme der Elnstostatit-. SpringerVerlag, Berlin (197.5).
Computers & Structuns Vol. 17, No. 3. pp. 321-326. 1983 Printed in Great Britain
STRUCTURAL ANALYSIS BY A COMBINED FINITE ELEMENT-TRANSFER MATRIX METHOD M. OHGA Department of Civil Engineering, Ehime University, Matsuyama, Japan and T. SHIGEMATSU and T. HARA Tokuyama Technical College, Tokuyama, Japan (Receioed 5 August 1982;received for publication 25 September 1982) Abstract-The combined finite element-transfer matrix method has the advantage of reducing the size of a matrix to less than that obtained by the ordinary finite element method. The analytical procedure in this method for bending and buckling problems are described, and techniques for treating the structure with intermediate conditions are proposed. Various numerical examples of these problems are shown to demonstrate the efficiency and accuracy of this method. The results from these examples agree well with those obtained by the finite element method and others. NOTATION
A, a b D E E,
{F) I, [K_li [K] k [PtJ [P,] [T]
t {z) g y {S}
cross section area of rib dimension of finite element or plate dimension of finite element or plate flexural rigidity of plate (Et’/12(1- v*)) modulus of elasticity of plate modulus of elasticity of rib force vector moment of inertia of rib stiffness matrix of strip i stiffness matrix of rib buckling coefficient (Pb*/a*D) point matrix for elastic column point matrix for rib transfer matrix thickness of plate state vector ratio of area (A,/bt) ratio of rigidity (E&/D) displacement vector 1. INTRODUCTlON
The finite element method is the most widely used and
powerful tool for structural analysis. However, the disadvantage of this method is that, in the case of a complex structure, it is necessary to use a large number of nodes, resulting in very large matrices which require large computers for their management and regulation. In order to reduce the size of the matrices in the ordinary finite element method, some techniques have been proposed (condensation, substructing method) [ 11. One numerical technique for reducing matrix size in the ordinary finite element method is the use of finite strips (FSM) suggested by Cheung[2]. Another is the transfer matrix technique (TMM) which was applied to two dimensional problems for the first time by Pestel and Leckie[3]. Plate vibration problems at that time were formulated by using the Hrennikoff model. The above techniques (FSM, TMM) can be successfully applied only for simple structures with particular boundary conditions; otherwise considerable complications arise in the formulation of problems. Dokainish used the combined finite element-transfer matrix (FETE) method in the study of the dynamics of tapered or rectangular plates[S]. Since the size of
stiffness and mass matrices, in his method, was equal to the number of degrees of freedom of one strip, the frequency determinant for a clamped-clamped plate considered by Dokainish was 18x 18 by the FETM method compared to a 108x 108 matrix eigenvalue problem obtained using the standard finite element method with the same number of nodes. McDaniel and Eversole have proposed a similar approach in treating a stiffened plate structures along with some numerical values that warrant consideration [6]. In dealing with complex structures, Chiatti and Sestieri introduced isoparametric shell elements, taking into consideration elements with nodes situated not only on corners but also on the midpoints of edges[7]. Sankar and Hoa offer an approach, in which an extended transfer matrix relating the state vectors which consist of state variables (displacements and forces) and their derivatives with respect to frequency was used[B]. In this method, a Newton-Raphson iterative technique is used to determine natural frequencies. Mucino and Paveric, as a further generalization of the FETM method, have proposed a method in which structures are modeled by means of substructures connected in a chain-like manner. For each of these substructures, a transfer matrix was derived[9]. Application of the FETM method is generally found in literature concerned with vibration problems of structure. This paper shows a successful application of the FETM method to other structural fields, especially to bending and buckling problems. Also, various techniques for treating the more complicated structures, especially those with the intermediate conditions are presented, Some numerical examples of bending and buckling problems are proposed and their results are compared with those obtained by the ordinary finite element method and others.
2. trrm’rEnt.nMEW-TRANSFER MATRM MElliOD A detailed description of the combined finite elementtransfer matrix may be found in Ref. 151.Figure 1 shows a plate divided into m strips and each of which subdivided into finite elements. The vertical sides dividing or bordering the strips are called sections, while the 321
M. OHGA
n
or
a
LT
1
2
i
It1
AB
01
2
i-l
Equation (7) can be recognized as the transfer matrix relating the state vectors {Z,} and {Z,} which consist of the displacements and forces. After continuous multiplications of the transfer matrix [T], we obtain the relation between the state vectors at two ends of the structure:
m
C
i
it1
et al.
m-l
m
i-l
i
{Z),,, = [ UHZ),,
(8)
Fig. I. Subdivision of plate into strips and finite elements.
horizontal boundaries are the edges. Thus BE is the left section of strip i + 1 and the right section of i. There are a total of 2n nodes on strip i with n nodes on the left section AD and n nodes on the right section BE. To derive the transfer matrix relating the left and right state variables (displacements and forces) of the strip i, it is required first to determine the stiffness matrix [K], of strip i: we obtain
[KliISh = {Fh
(1)
where [K], is the stiffness matrix of strip i, {S}i,{F}, are the displacements and forces of strip i, respectively. Equation (1) holds well for bending problems, but in buckling problems, the matrix [Kj, in eqn (1) becomes:
[Kli = [[Kbli- P[Kmlil
(2)
where [Kbli and [Km]i are the bending stiffness matrix and the modified stiffness matrix of strip i, respectively; P is the in-plane load. Matrix [K]i is partitioned into four sub-matrices. Equation (1) then becomes:
where [U]=[T],[T],.., . ..[T].. In bending problems, on considering the left and right boundary conditions of the structure, simultanious equations are obtained from eqn (8). The number of these equations is as same as that of the unknown state variables in {Z},. Thus, we can evaluate the unknown elements in {Z}, by solving these equations. On the other hand, in buckling problems, it is essential that the determinant of a portion [*U] of the matrix [U] be zero: detI*Uj=0.
(9)
Now, the matrix [*U] is obtained from the matrix [U] by deleting the columns corresponding to zero elements of {Z}, and the rows corresponding to the nonzero elements of {ZLl. 3. TECHNIQUJLS FOR INTERMEDIATECONDITIONS
Point matrix for elastic columns
Point matrix for treating structure with elastic columns at the intermediate section, as shown in Fig. 2, is obtained by taking the elastic support restoring forces into consideration. Consider, for example, elastic columns attached to nodes I and m of section i. The relations of the shearing forces to the left and right of the section i are then,
(3) where {S},,{S),, {F}, and {F}, are the left and right displacements and forces of strip i, respectively. By expanding expression (3) and solving for {S,}i and {Fr}i in terms of {Soi and {Fl}i, the following equations can be obtained;
where k, and k, are the elastic column stiffness. Since other elements of the state vector are continuous throughout section i, the following identity exists:
(da) and IF,}, = UK,,1 - [K,,IIK,,I~l[K,,II{S,}, + K,I[KJ’U%
WI
which, when arranged in matrix form, become:
On simplifying the notation, we obtain:
(6)
Fig. 2. Intermediate elastic columns
323
A combined finite element-transfer matrix method In matrix notation, eqns (10) and (11) become:
(12)
Setting wil = wim= 0 from the rigid conditions at the node I and m, we obtain the following two equations:
or
{Zh’ = [Pkl{Zh’.
(13) {U&
Equation (13) relates the left state vectors of the section which has some elastic columns, to the right state vectors. Consequently the matrix [A] is referred to as the point matrix for the elastic column. Intermediate rigid conditions Point matrices for elastic columns break down when elastic columns become infinitely stiff. In this case, when the deflections at the intermediate rigidcolumns are zero, the initial unknowns corresponding to the constrained displacements can be eliminated and introduced new unknownns. For example, consider the structure, shown in Fig. 3, which has rigid columns at nodes 1 and m of section i, with its left boundary simply supported. The equation relating the left state vector of the section i {Z},’ to the initial unknowns {Z}, is
. {Oy, Q, M.xhT = Wil = 0
{ ULt}t . IBy, Q, MxhT = Wm = 0.
(16)
where {Vi}, and {U&h are the I-th, m-th row of the matrix [U’],, respectively, Wirand Wi, are the deflection at nodes I and m of section i. Solving eqn (16) for QOr and Qo,,,, we eliminate these two shearing forces from the initial state vector {z),. Because of the reactions at the rigid columns, the shearing forces at these points are discontinuous. Introducing the new unknown Vrrand Vi, instead of Qor and Qom just above eliminated alone, the right state vectors of section i are expressed as, W, tL’, e;,
Q’, M,‘, Ms’h= = WI&
Q’, MA=
(17) where,
LZ)i’= 1U’l{Zo}.
(14)
From the. left boundary condition {w}~= {0,}, = {My}o = 0, the elementary form of eqn (14) is
{w’,B,l,0,‘, Q’, M,‘, M;]i= = [WI.
{e,, Q, Mx}o=. (15)
By the above technique, the transfer procedure can be performed throughout a section having intermediate rigid columns. The structure which has the intermediate simple support as shown in Fig. 4 can be treated as previously described. In this case, the deflections and rotations about the x-axis are constrained at the intermediate
___ ;.----r-__ ,L__ -a.-__, ___ -*-_&__ -__ I --7---
simple support
H
i-l
Fig. 3. Intermediate rigid columns.
i
it1
Fig. 4. Intermediatesimplesupport.
324
M. OHGA
simple support. By eliminating the initial shear forces and moments about the x-axis, which correspond to the constrained displacements, from the initial state vector {rd. the new unknown discontinouus shears and moments can be introduced to the state vector (z}. 4. APPLICATION TO NON-LINEAR PROBLEMS
So far, our discussion has been limited to linear problems, but the FETM method may also be applied to non-linear problems of structure. The computer storage and time required for analysis of non-linear problems are usually more than those involved in linear problems. Thus, in this regard, the advantages attainable through matrix size reduction in the FETM method will become more evident. The FETM method may be easily extended to nonlinear problems of structure, since the same incremental procedures used in the finite element method can be applied except the evaluation of incremental displacements for each specified incremental load. Furthermore, incremental calculations can be carried out in the same manner as that for linear problems described in this paper. 5. FZAMPLE Bending analysis of a plate structure The rectangular element with three degrees of freedom per one node is used in example; the deflection w is assumed to have the form,
where ff = xla, 5 = y/b and a,, . . , aI2 are unknown coefficients. A partially loaded and all edges clamped rectangular slab with intermediate ribs, shown in Fig. 5, was analysed. In this example, the slab was divided into 6 and 18 strips and each strip into 6 finite elements for both mesh types, as shown in Fig. 6. The point matrices for the rib [P,] proposed by McDaniel and Eversole[6] were used in considering the ribs. The transformation procedure was performed by multiplications of not only the transfer matrices IT] but
eta/
also the point matrices [P,], for the 6 strips mesh pattern:
In Fig. 7, the deflections along the symmetric line obtained by the FETM method are compared with those by the finite element method, in which the same element as that used in the FETM method was employed, for the 18 strips mesh pattern. It can be seen that the results obtained by both methods are in complete agreement with each other. The matrix to be considered in the finite element method is, if the banded matrix is used? 147x 27 for 6 strips mesh pattern and 399 x 27 for 18 strips. Thus the matrix size for latter mesh pattern is 2.7 times larger than that for the former pattern. On the other side, the matrix to be considered in the FETM method is 42 x 42 for both mesh patterns, since the matrix size in the FETM method is dependent on the number of degrees of freedom for one strip in contrast with the finite element method which depends on that for the entire structure. The deflections for EJ, = = are also shown in Fig. 7. In this case, the transformation procedure can be performed in a simple schematic manner by using the technique for intermediate simple support. The deflections by the FETM method agree well with those by the finite element method. To illustrate the efficiency of the technique for an intermediate rigid column and the point matrix for an elastic one, a bridge deck with four intermediate columns acted upon by partially distributed loads, shown in Fig. 8, was analysed. It is divided into 16 strips and each strip into 8 elements. In Fig. 9, the deflections in the case of intermediate rigid columns by the FETM method are compared with those by the finite element method. It can be seen that these results agree well with each other. In this example.
0
9 x(m)
6
3
-0.4 Y
=
0. 0.4 -FETM(6-10)
0.8 E
I
a=1t/m-
," d
/' rib
Fig. 5. AI1edges clamped slab with intermediate ribs
1.2 H (cm)
m
FETM(6-6)
0
FEM
(6-18)
Fig. 7. Deflections along the symmetric line.
ErIr/tl
325
A combined finite element-transfer matrix method 5m
10m
10m
I
10m
I
5m
I I
---
Ref.
10
-FETM --
0
FEM
m2
(6-12) (6-10)
Fig. 8. Simplysupportedbridgedeck with intermediate columns.
_4k
D.lF5
0;2ioj!~Tx;
---((6(6.
6)
(6.
4)
0;5 0
0
8)
FEM
1
2
a/b
3
Fig. 10. Buckling coefficients of all edges clamped plate.
O-
k
4-
---Ref. -
10
F E T M (6-6)
6
Fig. 9. Deflections at line of columns.
4
the matrix size in the finite element method is 459 X 33 for banded matrix, while in the FETM method the order of the matrix is only 54 x 54. The deflections for the intermediate elastic columns are shown together in Fig. 9. In this case, the transformation procedure was performed by introducing the point matrix for the elastic column [&I:
Y = 6 = Ar,'bt = 0.1
I.% = [Tls[Tl,[Pkl6[Tk~. . ~Pkl2mt[Tl1IZ}o.
(20) Buckling
analysis
I
0
of a plate structure
The element and degree of freedom in bending problems are used in buckling problems. A uniformly compressed rectangular plate with four clamped elges was analysed. The plate was divided into 4, 6, 8, 10 and 12 strips along the direction of compresssion and each strip into 6 elements. The buckling coefficients obtained by the FETM method and the finite element method are indicated in Fig. 10. It is seen that although the results by two methods agree as well as those in bending problems, the accuracy of these results decreases as the buckling mode increases. But it is also seen that the accuracv increases as ~.
D
1
2
a/b
3
Fig. 1I. Buckling coefficients of stiffened plate.
the number of strips increases. In the FETM method, the matrix size is 42 x 42 for any mesh pattern used in this example, while in the finite element method, if banded matrix is used, it is 105X 27 for the 4 strips pattern and 273 x 27 for the 12 strips pattern. As the second buclking problem example, a uniformly compressed rectangular plate clamped along two opposite sides perpendicular to the direction of compression and having reinforced free edges by ribs along
3?6
M. OHCA et al.
the other two sides was analysed. The plate was divided into 6 and 12 strips, and each strip as in the previous example, into 6 elements. As shown in Fig. 11, close agreement in the results by the FETM method and the finite element method was obtained. The buckling coefficients for the plate clamped along two opposite sides perpendicular to the direction of compression and simply supported along the other two sides are given simultaneously in Fig. 11,to provide the upper limit of this plate. 6. CONCLU~ONS In this paper, the procedures of the combined finite element-transfer matrix method for bending and buckling problems are described and the techniques for treating the structure with intermediate conditions are proposed. Furthermore, the analysis of non-linear probrems by the FETM method is given. Some numerical examples presented in this paper show that the FETM method can be successfully applied to bending and buckling structural problems by reducing the size of the matrix relative to less than that obtained by the finite element method. The techniques for intermediate conditions make possible the application of this method to the more complicated structures.
REFERENCES
I. W. McGuire and R. H. Gallagher, Mrltris S’trlcc,frrrai .?.cI~I~~\!. Wiley, New York (1979). 2. Y. K. Cheung, Finite Sfrip ~fet~~)din S~r~~~~ur~~~ &t~i\z~k.
Pergamon Press, Oxford ( 197hl. 3. E. C. Pestel and F. A. Leckie. Mutris Muthodr iti E&I’-
tomechanics. McGraw-Hill, New York (1963). 4. B. Nath, Fundamentals of Finite El~mer~tsfor Engineers. The
Athlone Press (1974).
5 M. A. Dokainish, A new approach for plate vibration>: .
Combination of transfer matrix and finite-element technique. Trans. ASME, J. ~~~ineer;n~ ~nd~~.sfr~, 35-530 (1W). 6 T. J. MeDaniel and K. B Eversole. -1 combined finite tiement-transfer marix structural analysis method. .I. So~rntl Vib. 51, 157-169(1977). I. G. Chiatti and A. Sestieri. Analysis of static and dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vib. 67, 3-42 (1979). 8. S. Sankar and S. V. Hoa, An extended transfer matrix-finite element method for free vibration of plates. J. Sound Vib. 70. 205-211(19801. 9. H. V. Mucino and \‘. Pa\rlic. ,AnL’UCIcondencation procedure for chain-like structures using a finite element-transfer matrix approach. Trans. ASME, L~Mech. Design, 1-9 (1980). 10.A. Pfliiger, Stabilitiitsprohleme der Elnstostatit-. SpringerVerlag, Berlin (197.5).