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Large deflection analysis of structures by an improved combined finite element-transfer matrix method
> Pergamon
0045-7949(94MO414-5 . ,
Comwters & Structures Vol. 55. No. I, DD. 167-171. 1995 Copyright c 1995 Elh.&er Scmce Ltd Printed in Great Bntain. All riehts reserved 0045.7949195i9.50 + 0.00
LARGE DEFLECTION ANALYSIS OF STRUCTURES BY AN IMPROVED COMBINED FINITE ELEMENT-TRANSFER MATRIX METHOD Chen Yuhua Department
of Physics,
Suzhou
University,
Suzhou,
Jiangsu
215006,
People’s
Republic
of China
(Received 6 April 1993) Abstract-An improved combined finite element-transfer matrix (IFE-TM) method is applied to the static analysis of structures with large displacement. In the present method, the transference of the state vectors from the left to the right in the combined finite element-transfer matrix method is changed into the transference of incremental stiffness equations of every section from the left to the right, in order to overcome the fault that the FE-TM method is generally applied to structures with a regular boundary. Furthermore, in the present method, the propagation of round-off errors occurring in recursive multiplications of the transfer and point matrices is avoided. A load incremental method is combined with modified Newton-Ranhson method for the non-linear eouations. A program ITNONDL-J based on this method is developed on the microcomputer IBM-PC/A?. Finally, numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for geometrically non-linear static analysis of structures with a random boundary.
1. INTRODUCTION The
analysis
problems
of geometrically
has
been
a subject
non-linear of considerable
structural interest
(FEM) is the most widely used and powerful tool for structural analysis. However, the disadvantage of the FEM is that, in the case of complex structures, it is necessary to use a large number of nodes, resulting in very large matrices which require a large computer for their management and regulation. On the other hand, the transfer matrix method of structural analysis is also applied to many structural problems, and has the advantage that the order of the final matrix is the same as those of transfer and point matrices and much less than that obtained by the ordinary FEM. The transfer matrix method is generally a solution procedure for one-dimensional problems as well as two-dimensional problems of simple structures with particular boundary conditions. For this reason various techniques have been suggested to reduce the order of the matrix, for example, the combined finite element-transfer matrix method (FE-TM). This method was proposed for the first time by Dokainish [I] for the free vibration problems of plates. This method has the advantage of reducing the size of a stiffness matrix to much less than that obtained by the ordinary FEM and has been successfully applied to various linear and nonlinear problems by several authors [2-lo]. However, the FE-TM method is generally a solution procedure CAS55,1--L for several
decades.
The finite element
method
for chain-structures as well as structures with a regular boundary. In the FE-TM method, it is required that the submatrix [&I, is a square matrix to derive the inverse matrix of submatrix [K& of the stiffness matrix [K], for the substructure i. Therefore, for the substructure i, it is required that the number of degrees of freedom on the left boundary is the same as on the right boundary. In addition to this, the transformation of state vectors is employed to avoid the propagation of round-off errors occurring in recursive multiplications of the transfer and point matrices [7, 81. For these reasons, various techniques [3,4, lo] for treating the more complicated structures have been presented, but research on the problem is not complete. The purpose of this paper is to present an improved combined finite element-transfer matrix method (IFE-TM) for geometrically non-linear static analysis of structures with a random boundary. In the present method, because the transference of the incremental state vectors from the left to the right in the FE-TM method is changed into the transference of incremental stiffness equations in every section from the left to the right, the inverse matrix of submatrix [&I in the FE-TM method becomes the inverse matrix of submatrix [Z&I in the IFE-TM method. It is well known that [&I is always a square matrix whether the structures are regular or not. As the numerical solution of a two-point boundary value problem in the FE-TM method is converted into the numerical solution of an initial value problem in the present method, the propagation of round-off errors
168
Chen Yuhua
occurring in recursive multiplications of the transfer and point matrices is avoided. An improved solution technique for non-linear problems is presented in that the load incremental method is combined with the modified Newton-Raphson method. A program ITNONDL-J based on this method is developed on a microcomputer. Some numerical examples of non-linear static problems are given and their results are compared with those obtained by the ordinary FEM and other methods in this paper. 2.
IMPROVED
FINITE ELEMENT-TRANSFER METHOD
MATRIX
Without losing generality, we consider the plate shown in Fig. 1 that is divided into m strips, and each of the strips is subdivided into finite elements. The vertical sides dividing or bordering the strips are called sections. It is apparent that the right of section i is the left of strip i. Let {AU}!, {AF}: and {AU}:, {AF}: be the left and right incremental displacement and force vectors of section i from the jth load incremental step to the (j + 1)th load incremental step. We assume that the incremental stiffness equations which relate the incremental force vectors to the incremental displacement vectors on the left of section i are given by
{AF}f = [Gl,{AU}F + (AE},
(i 2 2).
Equation (4) describes the relation between the incremental internal force vectors and the incremental displacement vectors on the right of section i. (2) Transfer in substructure
i
It is well known that the govering equations for non-linear static problems in the FEM by load incremental method are generally given by
Klr{A~),
in which [Krlr is the tangent stiffness matrix of substructure i and {AP}$ is the incremental external load vector of substructure i. Equation (5) is rearranged and repartitioned, and we obtain
in which [K,,], [K,,], [&,I and [KZ2] are the submatrices of matrix [Kr] in eqn (5); (AQ} and {AQ} are, respectively, the generalized incremental external force vectors on the left and the right of substructure i, which are obtained from {AP} in eqn (5). By expanding eqn (6) we obtain
(1)
K,l,lAW
+ K,li~AWf;,
= {AW + {AQL
(1) Transfer at section i The deflections that we obtain
(7)
are continuous
across section i, so
{AU}: = {AU};.
(2)
Without losing generality, we suppose that there are no concentrated external loads acting on the section i. By the continuity of force at section i, we obtain (AF}; = - (AF};. Substituting
(5)
= tAf+, + {Af’Io
- {AE},.
eqn (4) into eqn (7) we obtain
{Au>: = --([&,I + [GlK’K,,l,{AWf;,
+ ([K,, I + [GIL UAQ 1 - {A@), .
(3)
eqns (2) and (3) into eqn (1), we obtain {AF}; = -[G],{AU}P
Substituting
Substituting
eqn (9) into eqn (8), we obtain
{AF):+ I = [Gli,, {AW+,
(4)
(9)
+ {AEJ,+, >
(10)
x ({AQ) - {AE),i - {AQ,i.
(12)
where
(AEJ,+ I = Kxli(K,,l+ 2 1
II+2 n+1 I
m+l
IGIK’
it1
nl
Fig. 1. Subdivision of structure into strips and finite element.
Equation (10) is the relationship for the incremental internal force vectors and the incremental displacement vectors on the left of section i + 1.
Large deflection analysis of structures (3) Transfer of entire structure
Using eqns (11) and (12), [G] and {AE} are transferred from the left of the second section to the right of the total structure, hence we have {A%, + I =[Gl,+,{AU}~+,+~AE},+,.
(13)
By considering boundary conditions, the known incremental force or displacement variables on the right hand boundary of the total structure are substituted into eqn (13) to determine the unknown incremental force or displacement variables. After the incremental force and displacement vectors on the right hand boundary of the total structure are solved, the incremental force and displacement vectors at any section i are calculated by eqns (9) and (4). (4) The method of determining [G]* and {AE}, For substructure
1, by expanding eqn (6), we have
[&,I, {AUP + Wul, WE
= WI? + {AQI,
V&,11 PWP + [&,I, {AW = iA% + W,.
(15)
(a) Displacement boundary condition. It is obvious that {Au}: is known in the displacement boundary condition, hence by eqn (IS), we obtain
KJ,
{A% = &I, W>P - WI>, .
Expanding eqn (20) and solving relations for {AF}f and {AU}f, we obtain
El, = P&l - W~,1W,,l-‘W,~1 W% = P&,lWJ’(W~
+ WQ’>)+ [&I@$
-[H311[H,,l-‘[H,2l(A~} - {AC%. (5) Solution
(21)
(22)
procedures
This paper presents a solution strategy for non-linear static analysis, which is based on the modified load incremental method and modified Newton-Raphson method to improve computational efficiency. The solution procedures are described in the following. (a) Modijied load incremental method.
(al) Select a few load steps and let j = 0. (14)
It is obvious that {Au}; and {AF): may be determined by using the left hand boundary conditions of the total structure.
[Glr =
169
(16) (17)
(b) Force boundary condition. It is obvious that {AF}; is known in the force boundary condition,
hence {AV}f is obtained from eqn (14). Substituting {Au}; into eqn (IS), we have
(c) Mixed boundary condition. In the mixed boundary condition, we suppose {AU}; = [{Ad}, {Aa}]r and the corresponding {AF}; = [{AR}, {AW. If {Ad} is unknown and {AZ’} is known, the corresponding {AR} is known and {A4 is unknown. For substructure 1, eqn (6) is rearranged and repartitioned, and we have
(a2) j = j + 1, the current load increment {AP}m is calculated. (a3) The current tangent stiffness matrix [Kr]Q and unbalance force vectors 6 {F}“- ‘1 are calculated by using displacement vectors { V}u- I). (a4) A modified load increment (AP30 in the jth load step is given by {Aqo’=
{AP}t”++{F}“-I’.
(23)
(as) Replacing {AP} in eqn (5) by {Ano and using the IFE-TM method described above, we obtain incremental displacement vectors {AU}O. (a6)
{V)O’ = (V)+
‘) + {AUjo.
(24)
(a7) This solution procedure is continued until the final step. It is pointed out that, because the load increment in the above solution procedure is not small enough, the displacements obtained by the modified load incremental method are only a good approximation. Hence, the modified Newton-Raphson method is applied to iterate for equilibrium equations. (b) Modijied Newton-Raphson
method.
(bl) Let initial {CJ}(‘)be equal to the displacements solved by the modified load incremental method, and tangent stiffness matrix [Kr] is calculated by the displacement vectors {U}t’). (b2) k = 0. (b3) k=k+l. (b4) Unbalance
force vector 6 {R}(@ is calculated.
(b5) Replacing {AP} in eqn (5) by c~{R}(~), using the IFE-TM method described above, the kth correction 6 {UjCk)is obtained.
Chen Yuhua
170
Table
I. Comparison
of computation loadt
times
Computation
Et?
Fig. 2. Comparison
of central rectangular
displacement plate.
for clamped
(b6) The correction 6 { U}(k) is added to { U}(k _ ‘) to obtain a more nearly correct kth approximate displacement.
{U}(k)= {U}‘k-1)+6(rJ}(k).
(25)
(b7) This iterative procedure is continued until the unbalance forces become sufficiently small.
3. NUMERICAL
EXAMPLES
In order to investigate the accuracy as well as the computational efficiency of the proposed method, we developed a computer program ITNONDL-J based on the program TNONDL-W [9] on a microcomputer IBM-PC-AT and numerical results of the examples are compared with those obtained by the oridinary FEM and others. (1) A rectangular plate under the uniform load q,, with all edges clamped is analysed in the first The example. plate chosen is 160 cm x 240 cm x 1 cm, p = 0.33, E = 7.06 x lo4 MPa. In the numerical calculation, a quarter of the plate is divided into six stripes as shown in Fig. 2. Figure 2 also shows a comparison between the IFE-TM solutions and the FEM solutions obtained by using the ADINAt program and Chein and Yeh’s [l I] results, where the IFE-TM and FE methods are applied to a 6 x 8 same mesh pattern. A comparison indicates that very close agreement is obtained between the results by the IFE-TM method and those by the FEM. Although the deflection of the IFE-TM method is a little greater than that of Chien and Yeh’s method, a good agreement exists between the two sets of results. It may be observed from Table 1 that, in computation time, the IFE-TM method takes less than half of the FEM. (2) A clamped elliptical plate with half major axis b = 150 cm, half minor axis a = 100 cm and thickness
time (set)
Method
Rectangular plate qo(5.54 kN/m2)
Elliptical plate qo(I 1 kN/m’)
IFE-TM FEM
445 920
820 1350
t Microcomputer
2z!z
for a given
AST-386
is used.
h = 1 cm, modules of elasticity E = 2.0 x IO’ MPa and Poisson ratio p = 0.3, is subjected to a uniform pressure qo. It is shown in Fig. 3 that a quarter of the elliptical plate is divided into six substructures which are divided into many triangular plate elements. In Fig. 3, the results obtained by the IFE-TM method as well as those of FEM are shown, in which the same mesh pattern and accuracy are employed. Very little difference exists between the results, so that the plots in Fig. 3 are not distinct and there is very good agreement between the results of IFE-TM method and Weil and Newmark’s solution [12]. The computation time for a given q. = 1 I kN/m’ is shown in Table 1. It is found that the IFE-TM method has the same accuracy as the FEM if the same element mesh pattern is employed, but the computational efficiency of the IFE-TM method is higher than that of the FEM. The two numerical examples described above demonstrate the accuracy and the capability of the present method for geometrically non-linear static analysis of regular and non-regular structures. 4. CONCLUSIONS
An improved combined finite element-transfer matrix method is applied to the static analysis of geometrically non-linear structures with various boundaries. The present method overcomes many of the faults of the FE-TM method which are described in the Introduction, and therefore the present method has a more widely applied scope than does the FE-TM method. A solution strategy for non-linear
1.5 -K h
1.0
Q.5
0.0
4.0
8.0
12.0
16.Q
%.. El-t
t The ADINA program microcomputer in China.
has been translated
for use on a
Fig.
3. Comparison
of central displacement elliptical plate.
for clamped
Large
deflection analysis of structures
static analysis which is based on modified load incremental method and modified Newton-Raphson method is employed to improve computational efficiency. A microcomputer program ITNONDL-J based on this method has been developed. Some numerical examples presented in this paper show that the proposed method can be successfully applied to static analysis of large deformation plates with random boundaries. Like the FE-TM method, the present method has the same advantage of reducing the size of the matrix to less than that obtained by the ordinary FEM. However, the present method has a more widely applied scope than the FE-TM method. Without any difficulty, the present method may be expanded for dynamic analysis of structures.
REFERENCES A new approach for plate vibration: 1. M. A. Dokainish, combination of transfer matrix and finite element technique. Trans. ASME, J. Engng Ind. 94, 526-530 (1972). 2. T. J. Mcdaniel and K. B. Eversole, A combined finite element transfer matrix structural analysis method. J. Sound Vibr. 51, 157-169 (1977).
171
and A. Sestieri, Analysis of static and 3. G. Chiatti dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vibr. 67, 35-42 (1979). 4. S. Sankar and S. V. Hoa, An extended transfer
matrix-finite element method for free vibration plates. J. Sound Vibr. 70, 205-211 (1980).
of
and T. Hara, Structural 5. M. Ohga, T. Shigematsu analysis by a combined finite element-transfer matrix method. Comput. Sfruct. 17, 321-328 (1983). and T. Hara, A combined 6. M. Ohga, T. Shigematsu finite element transfer matrix method. J. Engng Mech. Div.. ASCE 110. 1335-1349 (1984). Transient analysis of plate 7. M. 6hga and T. Shigematsu, by a combined finite element-transfer matrix method. Comput. Struct. 26, 543-549 (1987). and R. G. Loewy, 8. E. E. Degen, M. S. Shephard Combined finite element-transfer matrix method based on a finite mixed formulation. Comput. Struct. 20, 173-180 (1985). 9. Chen YuHua and Xue HuiYu, Dynamic large deflection of structures by a combined finite analysis element-Riccati transfer matrix method on a microcomputer. Comput. Struct. 39, 699-703 (1991). 10. H. V. Muctno and V. Pavelic, An exact condensation procedure for chain-like structures using a finite element-transfer matrix approach. Trans. ASME, J. Mech. Des. 80-c2/DET-123, l-9 (1980). 11. W. Z. Chien and K. Y. Yeh, On the large deflection of rectangular plate. Proc. 9th Int. Congr. Appl. Mech. Vol. 6, pp. 4033412 (1957). 12. N. A. Weil and N. M. Newark, Large deflection of elliptical plates. J. appl. Mech. 23, 21-26 (1956).
0045-7949(94MO414-5 . ,
Comwters & Structures Vol. 55. No. I, DD. 167-171. 1995 Copyright c 1995 Elh.&er Scmce Ltd Printed in Great Bntain. All riehts reserved 0045.7949195i9.50 + 0.00
LARGE DEFLECTION ANALYSIS OF STRUCTURES BY AN IMPROVED COMBINED FINITE ELEMENT-TRANSFER MATRIX METHOD Chen Yuhua Department
of Physics,
Suzhou
University,
Suzhou,
Jiangsu
215006,
People’s
Republic
of China
(Received 6 April 1993) Abstract-An improved combined finite element-transfer matrix (IFE-TM) method is applied to the static analysis of structures with large displacement. In the present method, the transference of the state vectors from the left to the right in the combined finite element-transfer matrix method is changed into the transference of incremental stiffness equations of every section from the left to the right, in order to overcome the fault that the FE-TM method is generally applied to structures with a regular boundary. Furthermore, in the present method, the propagation of round-off errors occurring in recursive multiplications of the transfer and point matrices is avoided. A load incremental method is combined with modified Newton-Ranhson method for the non-linear eouations. A program ITNONDL-J based on this method is developed on the microcomputer IBM-PC/A?. Finally, numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for geometrically non-linear static analysis of structures with a random boundary.
1. INTRODUCTION The
analysis
problems
of geometrically
has
been
a subject
non-linear of considerable
structural interest
(FEM) is the most widely used and powerful tool for structural analysis. However, the disadvantage of the FEM is that, in the case of complex structures, it is necessary to use a large number of nodes, resulting in very large matrices which require a large computer for their management and regulation. On the other hand, the transfer matrix method of structural analysis is also applied to many structural problems, and has the advantage that the order of the final matrix is the same as those of transfer and point matrices and much less than that obtained by the ordinary FEM. The transfer matrix method is generally a solution procedure for one-dimensional problems as well as two-dimensional problems of simple structures with particular boundary conditions. For this reason various techniques have been suggested to reduce the order of the matrix, for example, the combined finite element-transfer matrix method (FE-TM). This method was proposed for the first time by Dokainish [I] for the free vibration problems of plates. This method has the advantage of reducing the size of a stiffness matrix to much less than that obtained by the ordinary FEM and has been successfully applied to various linear and nonlinear problems by several authors [2-lo]. However, the FE-TM method is generally a solution procedure CAS55,1--L for several
decades.
The finite element
method
for chain-structures as well as structures with a regular boundary. In the FE-TM method, it is required that the submatrix [&I, is a square matrix to derive the inverse matrix of submatrix [K& of the stiffness matrix [K], for the substructure i. Therefore, for the substructure i, it is required that the number of degrees of freedom on the left boundary is the same as on the right boundary. In addition to this, the transformation of state vectors is employed to avoid the propagation of round-off errors occurring in recursive multiplications of the transfer and point matrices [7, 81. For these reasons, various techniques [3,4, lo] for treating the more complicated structures have been presented, but research on the problem is not complete. The purpose of this paper is to present an improved combined finite element-transfer matrix method (IFE-TM) for geometrically non-linear static analysis of structures with a random boundary. In the present method, because the transference of the incremental state vectors from the left to the right in the FE-TM method is changed into the transference of incremental stiffness equations in every section from the left to the right, the inverse matrix of submatrix [&I in the FE-TM method becomes the inverse matrix of submatrix [Z&I in the IFE-TM method. It is well known that [&I is always a square matrix whether the structures are regular or not. As the numerical solution of a two-point boundary value problem in the FE-TM method is converted into the numerical solution of an initial value problem in the present method, the propagation of round-off errors
168
Chen Yuhua
occurring in recursive multiplications of the transfer and point matrices is avoided. An improved solution technique for non-linear problems is presented in that the load incremental method is combined with the modified Newton-Raphson method. A program ITNONDL-J based on this method is developed on a microcomputer. Some numerical examples of non-linear static problems are given and their results are compared with those obtained by the ordinary FEM and other methods in this paper. 2.
IMPROVED
FINITE ELEMENT-TRANSFER METHOD
MATRIX
Without losing generality, we consider the plate shown in Fig. 1 that is divided into m strips, and each of the strips is subdivided into finite elements. The vertical sides dividing or bordering the strips are called sections. It is apparent that the right of section i is the left of strip i. Let {AU}!, {AF}: and {AU}:, {AF}: be the left and right incremental displacement and force vectors of section i from the jth load incremental step to the (j + 1)th load incremental step. We assume that the incremental stiffness equations which relate the incremental force vectors to the incremental displacement vectors on the left of section i are given by
{AF}f = [Gl,{AU}F + (AE},
(i 2 2).
Equation (4) describes the relation between the incremental internal force vectors and the incremental displacement vectors on the right of section i. (2) Transfer in substructure
i
It is well known that the govering equations for non-linear static problems in the FEM by load incremental method are generally given by
Klr{A~),
in which [Krlr is the tangent stiffness matrix of substructure i and {AP}$ is the incremental external load vector of substructure i. Equation (5) is rearranged and repartitioned, and we obtain
in which [K,,], [K,,], [&,I and [KZ2] are the submatrices of matrix [Kr] in eqn (5); (AQ} and {AQ} are, respectively, the generalized incremental external force vectors on the left and the right of substructure i, which are obtained from {AP} in eqn (5). By expanding eqn (6) we obtain
(1)
K,l,lAW
+ K,li~AWf;,
= {AW + {AQL
(1) Transfer at section i The deflections that we obtain
(7)
are continuous
across section i, so
{AU}: = {AU};.
(2)
Without losing generality, we suppose that there are no concentrated external loads acting on the section i. By the continuity of force at section i, we obtain (AF}; = - (AF};. Substituting
(5)
= tAf+, + {Af’Io
- {AE},.
eqn (4) into eqn (7) we obtain
{Au>: = --([&,I + [GlK’K,,l,{AWf;,
+ ([K,, I + [GIL UAQ 1 - {A@), .
(3)
eqns (2) and (3) into eqn (1), we obtain {AF}; = -[G],{AU}P
Substituting
Substituting
eqn (9) into eqn (8), we obtain
{AF):+ I = [Gli,, {AW+,
(4)
(9)
+ {AEJ,+, >
(10)
x ({AQ) - {AE),i - {AQ,i.
(12)
where
(AEJ,+ I = Kxli(K,,l+ 2 1
II+2 n+1 I
m+l
IGIK’
it1
nl
Fig. 1. Subdivision of structure into strips and finite element.
Equation (10) is the relationship for the incremental internal force vectors and the incremental displacement vectors on the left of section i + 1.
Large deflection analysis of structures (3) Transfer of entire structure
Using eqns (11) and (12), [G] and {AE} are transferred from the left of the second section to the right of the total structure, hence we have {A%, + I =[Gl,+,{AU}~+,+~AE},+,.
(13)
By considering boundary conditions, the known incremental force or displacement variables on the right hand boundary of the total structure are substituted into eqn (13) to determine the unknown incremental force or displacement variables. After the incremental force and displacement vectors on the right hand boundary of the total structure are solved, the incremental force and displacement vectors at any section i are calculated by eqns (9) and (4). (4) The method of determining [G]* and {AE}, For substructure
1, by expanding eqn (6), we have
[&,I, {AUP + Wul, WE
= WI? + {AQI,
V&,11 PWP + [&,I, {AW = iA% + W,.
(15)
(a) Displacement boundary condition. It is obvious that {Au}: is known in the displacement boundary condition, hence by eqn (IS), we obtain
KJ,
{A% = &I, W>P - WI>, .
Expanding eqn (20) and solving relations for {AF}f and {AU}f, we obtain
El, = P&l - W~,1W,,l-‘W,~1 W% = P&,lWJ’(W~
+ WQ’>)+ [&I@$
-[H311[H,,l-‘[H,2l(A~} - {AC%. (5) Solution
(21)
(22)
procedures
This paper presents a solution strategy for non-linear static analysis, which is based on the modified load incremental method and modified Newton-Raphson method to improve computational efficiency. The solution procedures are described in the following. (a) Modijied load incremental method.
(al) Select a few load steps and let j = 0. (14)
It is obvious that {Au}; and {AF): may be determined by using the left hand boundary conditions of the total structure.
[Glr =
169
(16) (17)
(b) Force boundary condition. It is obvious that {AF}; is known in the force boundary condition,
hence {AV}f is obtained from eqn (14). Substituting {Au}; into eqn (IS), we have
(c) Mixed boundary condition. In the mixed boundary condition, we suppose {AU}; = [{Ad}, {Aa}]r and the corresponding {AF}; = [{AR}, {AW. If {Ad} is unknown and {AZ’} is known, the corresponding {AR} is known and {A4 is unknown. For substructure 1, eqn (6) is rearranged and repartitioned, and we have
(a2) j = j + 1, the current load increment {AP}m is calculated. (a3) The current tangent stiffness matrix [Kr]Q and unbalance force vectors 6 {F}“- ‘1 are calculated by using displacement vectors { V}u- I). (a4) A modified load increment (AP30 in the jth load step is given by {Aqo’=
{AP}t”++{F}“-I’.
(23)
(as) Replacing {AP} in eqn (5) by {Ano and using the IFE-TM method described above, we obtain incremental displacement vectors {AU}O. (a6)
{V)O’ = (V)+
‘) + {AUjo.
(24)
(a7) This solution procedure is continued until the final step. It is pointed out that, because the load increment in the above solution procedure is not small enough, the displacements obtained by the modified load incremental method are only a good approximation. Hence, the modified Newton-Raphson method is applied to iterate for equilibrium equations. (b) Modijied Newton-Raphson
method.
(bl) Let initial {CJ}(‘)be equal to the displacements solved by the modified load incremental method, and tangent stiffness matrix [Kr] is calculated by the displacement vectors {U}t’). (b2) k = 0. (b3) k=k+l. (b4) Unbalance
force vector 6 {R}(@ is calculated.
(b5) Replacing {AP} in eqn (5) by c~{R}(~), using the IFE-TM method described above, the kth correction 6 {UjCk)is obtained.
Chen Yuhua
170
Table
I. Comparison
of computation loadt
times
Computation
Et?
Fig. 2. Comparison
of central rectangular
displacement plate.
for clamped
(b6) The correction 6 { U}(k) is added to { U}(k _ ‘) to obtain a more nearly correct kth approximate displacement.
{U}(k)= {U}‘k-1)+6(rJ}(k).
(25)
(b7) This iterative procedure is continued until the unbalance forces become sufficiently small.
3. NUMERICAL
EXAMPLES
In order to investigate the accuracy as well as the computational efficiency of the proposed method, we developed a computer program ITNONDL-J based on the program TNONDL-W [9] on a microcomputer IBM-PC-AT and numerical results of the examples are compared with those obtained by the oridinary FEM and others. (1) A rectangular plate under the uniform load q,, with all edges clamped is analysed in the first The example. plate chosen is 160 cm x 240 cm x 1 cm, p = 0.33, E = 7.06 x lo4 MPa. In the numerical calculation, a quarter of the plate is divided into six stripes as shown in Fig. 2. Figure 2 also shows a comparison between the IFE-TM solutions and the FEM solutions obtained by using the ADINAt program and Chein and Yeh’s [l I] results, where the IFE-TM and FE methods are applied to a 6 x 8 same mesh pattern. A comparison indicates that very close agreement is obtained between the results by the IFE-TM method and those by the FEM. Although the deflection of the IFE-TM method is a little greater than that of Chien and Yeh’s method, a good agreement exists between the two sets of results. It may be observed from Table 1 that, in computation time, the IFE-TM method takes less than half of the FEM. (2) A clamped elliptical plate with half major axis b = 150 cm, half minor axis a = 100 cm and thickness
time (set)
Method
Rectangular plate qo(5.54 kN/m2)
Elliptical plate qo(I 1 kN/m’)
IFE-TM FEM
445 920
820 1350
t Microcomputer
2z!z
for a given
AST-386
is used.
h = 1 cm, modules of elasticity E = 2.0 x IO’ MPa and Poisson ratio p = 0.3, is subjected to a uniform pressure qo. It is shown in Fig. 3 that a quarter of the elliptical plate is divided into six substructures which are divided into many triangular plate elements. In Fig. 3, the results obtained by the IFE-TM method as well as those of FEM are shown, in which the same mesh pattern and accuracy are employed. Very little difference exists between the results, so that the plots in Fig. 3 are not distinct and there is very good agreement between the results of IFE-TM method and Weil and Newmark’s solution [12]. The computation time for a given q. = 1 I kN/m’ is shown in Table 1. It is found that the IFE-TM method has the same accuracy as the FEM if the same element mesh pattern is employed, but the computational efficiency of the IFE-TM method is higher than that of the FEM. The two numerical examples described above demonstrate the accuracy and the capability of the present method for geometrically non-linear static analysis of regular and non-regular structures. 4. CONCLUSIONS
An improved combined finite element-transfer matrix method is applied to the static analysis of geometrically non-linear structures with various boundaries. The present method overcomes many of the faults of the FE-TM method which are described in the Introduction, and therefore the present method has a more widely applied scope than does the FE-TM method. A solution strategy for non-linear
1.5 -K h
1.0
Q.5
0.0
4.0
8.0
12.0
16.Q
%.. El-t
t The ADINA program microcomputer in China.
has been translated
for use on a
Fig.
3. Comparison
of central displacement elliptical plate.
for clamped
Large
deflection analysis of structures
static analysis which is based on modified load incremental method and modified Newton-Raphson method is employed to improve computational efficiency. A microcomputer program ITNONDL-J based on this method has been developed. Some numerical examples presented in this paper show that the proposed method can be successfully applied to static analysis of large deformation plates with random boundaries. Like the FE-TM method, the present method has the same advantage of reducing the size of the matrix to less than that obtained by the ordinary FEM. However, the present method has a more widely applied scope than the FE-TM method. Without any difficulty, the present method may be expanded for dynamic analysis of structures.
REFERENCES A new approach for plate vibration: 1. M. A. Dokainish, combination of transfer matrix and finite element technique. Trans. ASME, J. Engng Ind. 94, 526-530 (1972). 2. T. J. Mcdaniel and K. B. Eversole, A combined finite element transfer matrix structural analysis method. J. Sound Vibr. 51, 157-169 (1977).
171
and A. Sestieri, Analysis of static and 3. G. Chiatti dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vibr. 67, 35-42 (1979). 4. S. Sankar and S. V. Hoa, An extended transfer
matrix-finite element method for free vibration plates. J. Sound Vibr. 70, 205-211 (1980).
of
and T. Hara, Structural 5. M. Ohga, T. Shigematsu analysis by a combined finite element-transfer matrix method. Comput. Sfruct. 17, 321-328 (1983). and T. Hara, A combined 6. M. Ohga, T. Shigematsu finite element transfer matrix method. J. Engng Mech. Div.. ASCE 110. 1335-1349 (1984). Transient analysis of plate 7. M. 6hga and T. Shigematsu, by a combined finite element-transfer matrix method. Comput. Struct. 26, 543-549 (1987). and R. G. Loewy, 8. E. E. Degen, M. S. Shephard Combined finite element-transfer matrix method based on a finite mixed formulation. Comput. Struct. 20, 173-180 (1985). 9. Chen YuHua and Xue HuiYu, Dynamic large deflection of structures by a combined finite analysis element-Riccati transfer matrix method on a microcomputer. Comput. Struct. 39, 699-703 (1991). 10. H. V. Muctno and V. Pavelic, An exact condensation procedure for chain-like structures using a finite element-transfer matrix approach. Trans. ASME, J. Mech. Des. 80-c2/DET-123, l-9 (1980). 11. W. Z. Chien and K. Y. Yeh, On the large deflection of rectangular plate. Proc. 9th Int. Congr. Appl. Mech. Vol. 6, pp. 4033412 (1957). 12. N. A. Weil and N. M. Newark, Large deflection of elliptical plates. J. appl. Mech. 23, 21-26 (1956).