- Email: [email protected]
Dynamic large deflection analysis of structures by a combined finite element-Riccati transfer matrix method on a microcomputer
@w-7949/91 163.00+ 0.00 0 1991 PergamonPress plc
Cmpwrs & Stmcmes Vol. 39, No. 6, pp. 699-M% 1991 Printed in Gnat Britain.
DYNAMIC LARGE DEFLECTION ANALYSIS OF STRUCTURES BY A COMBINED FINITE ELEMENT-RICCATI TRANSFER MATRIX METHOD ON A MICROCOMPUTER YUHUA C&EN* and HUIYUXUE Department of Physics, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China (Received 4 May 1990)
Abstract-The combined finite element-Riccati transfer matrix method is applied to the transient analysis of the structures with large displacement under various excitations. The Wilson-8 method is employed for time integration and the modified Newton-Raphson method for equilibrium iteration in each time step. In this paper, Riccati traRsfo~tion of state vectors is proposed to avoid the propagation of round-offs errors occurring in recursive multiplications of the transfer and point matrices. A program TNONDL-W based on this method on a IBM PC-AT microcomputer is developed. Finally the numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for transient analysis of the structures with large displacement under various excitations.
1. INTRODUCIION
The analysis of geometrically nonlinear structural problems has been a subject of considerable interest for several decades, while 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. Furthermore, the transient analysis of the structures with large displacement under random excitations by a direct integration method such as the Wilson-8 method on a microcomputer, these disadvantages of the finite element method will become more serious. 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) is one of those. This method was proposed for the first time by Dokainish [l] for the free vibration problems of plates. In this approach, as the size of stiffness and mass matrices was equal to the number of degrees of freedom in only one subsystem, it had the advantage of reducing the size of a matrix to much less than that obtained by the ordinary finite element method. Since the publication of Dokainish’s paper, the approach has been successfully applied to various linear static, free vibration, dynamic and nonlinear static problems by several authors [Z-8]. However, there are little studies on the extension of this method to geometrical nonlinear dynamic problems, especially using a microcomputer,
The purpose of this paper is to present a method of transient analysis of structures with large displacement under random excitations by the combined finite element-Riccati transfer matrix [9] method (FE-RTM). In this paper, the linearizing incremental equations of motion in the finite element are deduced and Wilson-8 method is used for time integration. The modified Newton-Raphson method is employed in equilibrium iterative procedures of each time step, also Riccati transformation of state vectors is proposed to avoid the propagation of round off errors occurring in recursive multiplication of the transfer and point matrices. A program TNONDL-W on a microcomputer is developed. Some numerical examples of nonlinear dynamic problems are given and their results are compared with those obtained by the ordinary finite element method and other methods in this paper. 2. INCREMENTALEQUILIBRIUM EQUATION AND DIRECT INTEGRATION METHOD It is well known that the governing equation for dynamic problem in the finite element method at time t is generally given by
WI b&i+ rcli&}+ {F,)= I&J
in which [M] and [C] are mass and damping matrices, (r&1, I&>, IF,) and (4) are the velocity, acceleration, nodal elastic force and nodal external force vectors at time t, respectively. At time t + r, where 7 is the time incremental interval,
* Author to whom correspondence should be addressed.
(1)
eqn (1) is given repeatedly
by
YUHUACHENand HUIYUXIJE
700
It is obvious that eqn (2) is a nonlinear algebraic equation because the geometrically nonlinear structural problems are concerned in this paper. Hence, a method of linearizing the equations of motion is proposed for geometrically nonlinear structural analysis by using following approximation {F,+J = (4;) + [GI,{Au)
{u,+u) = Iu,> + At@,> + $
(12) It is pointed out that where {Au} is only approximate incremental displacements. 3. FINITR ELEMENT-RICCATI TRANSFER MATRIX METHOD
(3)
in which {Au} is the incremental displacement vectors in time interval 7, with {Au} = {u,+,} - {u,}, [Kr], is the tangent stiffness matrix at time t which has been defined in Ref. [lo]. Substituting eqn (3) into (2) and eqn (2) minus (1), we obtain linearizing incremental equilibrium equations from time t to t + t
({ii,+A,} + 2. {ii,}).
Without losing generality, we consider the plate shown in Fig. 1, that is divided into rn strips, each of strips is subdivided into finite elements. The vertical sides dividing or bordening the strips are called sections. It is apparent that the right of section i is the left of strip i. (a) Transfer and point matrices
WIW} + WA4 + [&I, {Auf= W 1
(4)
in which (AR} = {R,,,) -{R,}. As described previously, the Wilson-B method is used for time integration in this paper. Hence, we assumed that 7 equals 0 ‘At, where At is the time step and 0 is the parameter that can control stability. We gain by series operating
{AC) = &
{Au} - &
{i&j - 3{r7,},
{Ati} = -& {AU} - 3{zi,} -F
{ii,}.
Proceedings as in Ref. [8], which are concerned with transient analysis of linear system, we obtain i strip’s transfer matrices
and i section’s point matrices
(5)
(6) where
Substituting
eqns (5) and (6) into (4), we have
Wl{AuI = {A@,
(7)
where
WI=&
WI + & [Cl + [&I,7
(8)
{AG)= {AR)+ WI & {k> + 3&J) (
+ [Cl(3{ti,} + y
{iir}). (9)
and [H,], [H,], [H‘s] and [HJ are the submatrices of matrix [H] in eqn (8), {AQ} is the incremental node generalized load vector acting on section i, which is evaluated from the general loading function in eqn (9), {Au}:, {Au}R, {AN}; and {AN}? 2n
n
Equation (7) is an equation with unknown variables {Au} only and can be solved. Finally the accelerations, velocities and displacements in time t + At are given by
I&+&} =&
2n-1
n-l
{Au}-~i~,}+(l-~){4}, n+2
2
(10) 1
(11)
2
i-1
I
I+1
m m+1
1
I
n+l
I+1
Fig. 1. Subdivision of structure into strips and finite element.
Dynamic large deflection analysis are the left and right incremental displacement and force vectors of section i from time t to r + BAt. Substituting eqn (14) into (13), we have
701
{AE}, . Next, using eqns (20) and (21), IS] and {AE} are transfered from left to right through all the structure, hence we have Pf
>i+, =[Sl,+,{A~)~+,+{AEJ,+,.
(22)
The known state variables at the right-hand
or {AZ):+ I = IYDIi(
+ lA&)i*
(16)
Equation (16) describes the relation that the incremental state vectors on the left at section i + 1, (AZ):+ 1is related to the incremental state vectors on the left at section i, {AZ);. (b) The Riccati transformation of incremental state vectors In order to minimize the propagation of round off errors occurred in the standard transfer matrix method, in this paper, the Riccati transformation of incremental state vectors is proposed to use as a means of reducing the propagation of round off error. In this method, eqn (16) is rearranged and repartitioned, we obtain
(17) where {Af} contains the half incremental state variables that are known at the left-hand boundary and {AT] contains the respective half complementary state variables. As pointed out in Ref. [9], a generalized Riccati transformation at section i is given by
(Af }F= ]Sl,{A~}F + {AE}i
(18)
in which matrix [S] is the Riccati transfer matrix and {Ag} is vectors concerned with force. From eqns (17) and (18), we obtain @f K+l= [Yi+,IGIf+,
+iAEI,+l,
(19)
where
[Sli+I = (~~,,I~~1 + ~~n1M~TzJlSl+IT&-’ t (20) IA.%+I = V’,,lW~ + W>>i -PI,+ ,([G,l{AEI + @P)>i. (21) Equations (20) and (21) are the general recursive relationships for [S] and {AE}. First, using the left hand boundary conditions, we easily obtain [S], and
boundary are substituted into eqn (22) to determine the unknown state variables in {AZ}, + , . After incremental state vectors {AZ}, + , at right-hand boundary of the total structure is solved, the incremental state vectors at any section i is calculated by the following formulation
U,,Wl+
[T221)r1(I~2JiW+ W>,i
(23)
(c) Iterative procedure It is pointed out that, because of employing the approximation eqn (3), the incremental state vectors solved from the above is only approximation of accurate incremental state vectors. Hence, a modified Newton-Raphson method is applied to iterate for dynamic equilibrium equations. Iterative scheme is as follows. (i) Let initial {Au}(o)be equal to the incremental displacements solved in eqn (7), j = 0. (ii) j =j + 1. (iii) Incremental effective unbalance forces 6 {AR]“-‘) is calculated. (iv) Replacing {AGI in eqn (7) by ?i{AR)“-‘), we obtain [H]G(Au)‘” = 6(AR)“-?
(24)
Using FE-RTM procedure described above, jth correction G{Au}Q is obtained. (v) The correction 6{Au}o is added to {Au}“-‘) to obtain a more nearly correct jth approximate incremental displacements {Au)Gl= {Au)“-“+
6{Au}(Jj.
(25)
(vi) This iterative procedure is continued until the incremental unbalance forces become sufficiently small. Finally using eqns (lO)-(lZ), the new displacements, velocities and accelerations at time r + At is obtained. 4.
NUMERICAL
EXAMPLES
In order to investigate the accuracy as well as the cornputative efficiency of the proposed method, we developed a program TNONDL-W based on this method on the microcomputer IBM PC-AT and some numerical results of the plate and stiffened plate are compared with those obtained by the ordinary finite element method and others.
702
YUHUA CHEN and HIJIYU XUE
-
Bayles o
FE-RTM,FEM
1.5 -3
3.0 Time
4.5 S.O‘h/7.5 (x10-’
set)
t
y stiffeners
Fig. 2. Central displacement response for clamped plate.
Fig. 3. Orthogonally stiffened plate structure.
(i) A clamped square plate subjected to a suddenly applied uniform pressure is analysed in the first example. The plate chosen is 244 x 244 x 0.635 cm with a specific weight of 24.74 kN/m3 E = 6.895 x lo4 MPa, v = 0.23 and subjected to uniform pressures 479 N/m*. In the numerical calculation, a quarter of the plate is divided 6 x 8 elements and time step At = 0.0015 set is used. Figure 2 shows a comparison between the FE-RTM solutions and the finite element solutions by using the ADINA program,* and Bayle’s [ll] results, where FE-RTM and finite element methods are applied to 6 x 8 same mesh pattern. A comparison indicates that the results obtained by the FE-RTM method coincide completely with those obtained by the finite method for same mesh pattern and that there is very good agreement between FE-RTM method solutions and Bayle’s results. Table 1 shows comparisons of average computation time for each time step between the FE-RTM method and finite element method in this example. It may be observed from Table 1 that in computation time the FE-RTM method is less than half as same as the finite element method. (ii) A clamped square orthogonally stiffened plate structure subjected to a suddenly applied uniform pressure (479.0 N/m*) is shown in Fig. 3. The dimensions and material of the plate are identical with that in example (i). There are seven stiffeners and five stiffeners in the x- and y-directions of the plate, respectively. For stiffeners, axial strength EF = 1.1 x lo6 N, flexure strength EJ1= 9 Nm*, EJ, = 40 N m*, A quarter of the stiffened plate structure is divided
into 4 x 6 plate elements and 24 beam elements. Figure 4 shows the dynamic responses of the deflections at central point, where time step At = 0.002 set is used. In Fig. 4, the results obtained by the finite element method is also shown, in which the same mesh pattern, time step and accuracy as those used in the FE-RTM method are employed. Very little difference exist between the results, so that the plots in Fig. 4 are not distinct. The consumed average computation time for each time step is shown in Table 2. It is found that the FE-RTM method has the same accuracy with the finite element method when the same element mesh pattern are employed, but it’s computation efficiency is higher than that one.
* ADINA program has been translated microcomputer in China.
Table 1. Comparison
of computation nlate
for use on a
5. CONCLUSIONS
A combined finite element-transfer matrix method is applied to the transient analysis of geometrical nonlinear structures under various excitations. The Riccati transformation of state vectors is employed to avoid the propagation of round-off errors in ordinary combined finite-transfer matrix method. A microcomputerprogramTNONDL-W basedon thismethod has been developed. Because the memory capacity of the microcomputer is small, many measures are employed to overcome this disadvantage. 3
x
FEM
-zg2 E :
2
time for clamped
Method by applying
Computation time for each time step (xc)
FE-RTM Finite method
346 700
0
2
1
Time
(x10-'
3
4
SeC)
Fig. 4. Central displacement response of clamped orthogonally stiffened plate.
Dynamic large deflection analysis Table 2. Comparison of computation for clamped orthogonally stiffened plate Method hy applying FE-RTM Finite method
Computation time for each time step (see)
3. G. Ghiatti and A. Sestieri, Analysis of static and dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vibr. 67, 35-42 (1979). 4. S. Sankar and
200 365 5.
Some numerical examples presented in this paper show that this method can be sucessfully applied to the transient analysis of the structure with large displacement subjected to various excitations by reducing the size of matrix and relative computation time to less than those obtained by the method based on the ordinary finite element procedures. REFERENCES
703
6.
7.
8.
9.
1. M. A. Dokainish, A new approach for plate vibration:
combination of transfer matrix and finite-element technique. Trans. ASME, J. Engng Ind. 94, 526530 (1972).
10.
2. T. J. McDaniel and K. B. Eversole, A combined finite element-transfer matrix structural analysis method. J. Sound Vibr. 51, 157-169 (1977).
11.
S. V. Hoa, An extended transfer matrix-finite element method for free vibration of plates. J. Sound Vibr. 70, 205-211 (1980). M. Ohga, T. Shigematsu and T. Hara, Structural analysis by a combined finite element-transfer matrix method. Comput. Struct. 17, 321-326 (1983). M. Ohga, T. Shigematsu and T. Hara. A combined finite element-transfer matrix method. J. Engng Mech. Div., Am. Sot. Ciu. Engrs 110, 1335-1349 (1984). E. E. Degen, M. S. Shephard and R. G. Loewy, Combined finite element-transfer matrix method based on a finite mixed formulation Compuf. Sfruct. 20, 173-180 (1985). M. Ohga and T. Shigematsu, Transient analysis of plates by a combined finite element-transfer matrix method. Comput. Struct. 26, 543-549 (1987). G. C. Homer and W. D. Pilkey, The Riccati transfer matrix method. ADA0 19848 (1975). 0. C. Zienkiewicz, The Finite Element Method, 3rd edn. McGraw-Hill (1977). D. J. Bayles, R. L. Lowery and D. E. Boyd, A nonlinear dynamic lumped-parameter model of a rectangular plate. J. Sound Vibr. 21, 329-337 (1972).
Cmpwrs & Stmcmes Vol. 39, No. 6, pp. 699-M% 1991 Printed in Gnat Britain.
DYNAMIC LARGE DEFLECTION ANALYSIS OF STRUCTURES BY A COMBINED FINITE ELEMENT-RICCATI TRANSFER MATRIX METHOD ON A MICROCOMPUTER YUHUA C&EN* and HUIYUXUE Department of Physics, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China (Received 4 May 1990)
Abstract-The combined finite element-Riccati transfer matrix method is applied to the transient analysis of the structures with large displacement under various excitations. The Wilson-8 method is employed for time integration and the modified Newton-Raphson method for equilibrium iteration in each time step. In this paper, Riccati traRsfo~tion of state vectors is proposed to avoid the propagation of round-offs errors occurring in recursive multiplications of the transfer and point matrices. A program TNONDL-W based on this method on a IBM PC-AT microcomputer is developed. Finally the numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for transient analysis of the structures with large displacement under various excitations.
1. INTRODUCIION
The analysis of geometrically nonlinear structural problems has been a subject of considerable interest for several decades, while 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. Furthermore, the transient analysis of the structures with large displacement under random excitations by a direct integration method such as the Wilson-8 method on a microcomputer, these disadvantages of the finite element method will become more serious. 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) is one of those. This method was proposed for the first time by Dokainish [l] for the free vibration problems of plates. In this approach, as the size of stiffness and mass matrices was equal to the number of degrees of freedom in only one subsystem, it had the advantage of reducing the size of a matrix to much less than that obtained by the ordinary finite element method. Since the publication of Dokainish’s paper, the approach has been successfully applied to various linear static, free vibration, dynamic and nonlinear static problems by several authors [Z-8]. However, there are little studies on the extension of this method to geometrical nonlinear dynamic problems, especially using a microcomputer,
The purpose of this paper is to present a method of transient analysis of structures with large displacement under random excitations by the combined finite element-Riccati transfer matrix [9] method (FE-RTM). In this paper, the linearizing incremental equations of motion in the finite element are deduced and Wilson-8 method is used for time integration. The modified Newton-Raphson method is employed in equilibrium iterative procedures of each time step, also Riccati transformation of state vectors is proposed to avoid the propagation of round off errors occurring in recursive multiplication of the transfer and point matrices. A program TNONDL-W on a microcomputer is developed. Some numerical examples of nonlinear dynamic problems are given and their results are compared with those obtained by the ordinary finite element method and other methods in this paper. 2. INCREMENTALEQUILIBRIUM EQUATION AND DIRECT INTEGRATION METHOD It is well known that the governing equation for dynamic problem in the finite element method at time t is generally given by
WI b&i+ rcli&}+ {F,)= I&J
in which [M] and [C] are mass and damping matrices, (r&1, I&>, IF,) and (4) are the velocity, acceleration, nodal elastic force and nodal external force vectors at time t, respectively. At time t + r, where 7 is the time incremental interval,
* Author to whom correspondence should be addressed.
(1)
eqn (1) is given repeatedly
by
YUHUACHENand HUIYUXIJE
700
It is obvious that eqn (2) is a nonlinear algebraic equation because the geometrically nonlinear structural problems are concerned in this paper. Hence, a method of linearizing the equations of motion is proposed for geometrically nonlinear structural analysis by using following approximation {F,+J = (4;) + [GI,{Au)
{u,+u) = Iu,> + At@,> + $
(12) It is pointed out that where {Au} is only approximate incremental displacements. 3. FINITR ELEMENT-RICCATI TRANSFER MATRIX METHOD
(3)
in which {Au} is the incremental displacement vectors in time interval 7, with {Au} = {u,+,} - {u,}, [Kr], is the tangent stiffness matrix at time t which has been defined in Ref. [lo]. Substituting eqn (3) into (2) and eqn (2) minus (1), we obtain linearizing incremental equilibrium equations from time t to t + t
({ii,+A,} + 2. {ii,}).
Without losing generality, we consider the plate shown in Fig. 1, that is divided into rn strips, each of strips is subdivided into finite elements. The vertical sides dividing or bordening the strips are called sections. It is apparent that the right of section i is the left of strip i. (a) Transfer and point matrices
WIW} + WA4 + [&I, {Auf= W 1
(4)
in which (AR} = {R,,,) -{R,}. As described previously, the Wilson-B method is used for time integration in this paper. Hence, we assumed that 7 equals 0 ‘At, where At is the time step and 0 is the parameter that can control stability. We gain by series operating
{AC) = &
{Au} - &
{i&j - 3{r7,},
{Ati} = -& {AU} - 3{zi,} -F
{ii,}.
Proceedings as in Ref. [8], which are concerned with transient analysis of linear system, we obtain i strip’s transfer matrices
and i section’s point matrices
(5)
(6) where
Substituting
eqns (5) and (6) into (4), we have
Wl{AuI = {A@,
(7)
where
WI=&
WI + & [Cl + [&I,7
(8)
{AG)= {AR)+ WI & {k> + 3&J) (
+ [Cl(3{ti,} + y
{iir}). (9)
and [H,], [H,], [H‘s] and [HJ are the submatrices of matrix [H] in eqn (8), {AQ} is the incremental node generalized load vector acting on section i, which is evaluated from the general loading function in eqn (9), {Au}:, {Au}R, {AN}; and {AN}? 2n
n
Equation (7) is an equation with unknown variables {Au} only and can be solved. Finally the accelerations, velocities and displacements in time t + At are given by
I&+&} =&
2n-1
n-l
{Au}-~i~,}+(l-~){4}, n+2
2
(10) 1
(11)
2
i-1
I
I+1
m m+1
1
I
n+l
I+1
Fig. 1. Subdivision of structure into strips and finite element.
Dynamic large deflection analysis are the left and right incremental displacement and force vectors of section i from time t to r + BAt. Substituting eqn (14) into (13), we have
701
{AE}, . Next, using eqns (20) and (21), IS] and {AE} are transfered from left to right through all the structure, hence we have Pf
>i+, =[Sl,+,{A~)~+,+{AEJ,+,.
(22)
The known state variables at the right-hand
or {AZ):+ I = IYDIi(
+ lA&)i*
(16)
Equation (16) describes the relation that the incremental state vectors on the left at section i + 1, (AZ):+ 1is related to the incremental state vectors on the left at section i, {AZ);. (b) The Riccati transformation of incremental state vectors In order to minimize the propagation of round off errors occurred in the standard transfer matrix method, in this paper, the Riccati transformation of incremental state vectors is proposed to use as a means of reducing the propagation of round off error. In this method, eqn (16) is rearranged and repartitioned, we obtain
(17) where {Af} contains the half incremental state variables that are known at the left-hand boundary and {AT] contains the respective half complementary state variables. As pointed out in Ref. [9], a generalized Riccati transformation at section i is given by
(Af }F= ]Sl,{A~}F + {AE}i
(18)
in which matrix [S] is the Riccati transfer matrix and {Ag} is vectors concerned with force. From eqns (17) and (18), we obtain @f K+l= [Yi+,IGIf+,
+iAEI,+l,
(19)
where
[Sli+I = (~~,,I~~1 + ~~n1M~TzJlSl+IT&-’ t (20) IA.%+I = V’,,lW~ + W>>i -PI,+ ,([G,l{AEI + @P)>i. (21) Equations (20) and (21) are the general recursive relationships for [S] and {AE}. First, using the left hand boundary conditions, we easily obtain [S], and
boundary are substituted into eqn (22) to determine the unknown state variables in {AZ}, + , . After incremental state vectors {AZ}, + , at right-hand boundary of the total structure is solved, the incremental state vectors at any section i is calculated by the following formulation
U,,Wl+
[T221)r1(I~2JiW+ W>,i
(23)
(c) Iterative procedure It is pointed out that, because of employing the approximation eqn (3), the incremental state vectors solved from the above is only approximation of accurate incremental state vectors. Hence, a modified Newton-Raphson method is applied to iterate for dynamic equilibrium equations. Iterative scheme is as follows. (i) Let initial {Au}(o)be equal to the incremental displacements solved in eqn (7), j = 0. (ii) j =j + 1. (iii) Incremental effective unbalance forces 6 {AR]“-‘) is calculated. (iv) Replacing {AGI in eqn (7) by ?i{AR)“-‘), we obtain [H]G(Au)‘” = 6(AR)“-?
(24)
Using FE-RTM procedure described above, jth correction G{Au}Q is obtained. (v) The correction 6{Au}o is added to {Au}“-‘) to obtain a more nearly correct jth approximate incremental displacements {Au)Gl= {Au)“-“+
6{Au}(Jj.
(25)
(vi) This iterative procedure is continued until the incremental unbalance forces become sufficiently small. Finally using eqns (lO)-(lZ), the new displacements, velocities and accelerations at time r + At is obtained. 4.
NUMERICAL
EXAMPLES
In order to investigate the accuracy as well as the cornputative efficiency of the proposed method, we developed a program TNONDL-W based on this method on the microcomputer IBM PC-AT and some numerical results of the plate and stiffened plate are compared with those obtained by the ordinary finite element method and others.
702
YUHUA CHEN and HIJIYU XUE
-
Bayles o
FE-RTM,FEM
1.5 -3
3.0 Time
4.5 S.O‘h/7.5 (x10-’
set)
t
y stiffeners
Fig. 2. Central displacement response for clamped plate.
Fig. 3. Orthogonally stiffened plate structure.
(i) A clamped square plate subjected to a suddenly applied uniform pressure is analysed in the first example. The plate chosen is 244 x 244 x 0.635 cm with a specific weight of 24.74 kN/m3 E = 6.895 x lo4 MPa, v = 0.23 and subjected to uniform pressures 479 N/m*. In the numerical calculation, a quarter of the plate is divided 6 x 8 elements and time step At = 0.0015 set is used. Figure 2 shows a comparison between the FE-RTM solutions and the finite element solutions by using the ADINA program,* and Bayle’s [ll] results, where FE-RTM and finite element methods are applied to 6 x 8 same mesh pattern. A comparison indicates that the results obtained by the FE-RTM method coincide completely with those obtained by the finite method for same mesh pattern and that there is very good agreement between FE-RTM method solutions and Bayle’s results. Table 1 shows comparisons of average computation time for each time step between the FE-RTM method and finite element method in this example. It may be observed from Table 1 that in computation time the FE-RTM method is less than half as same as the finite element method. (ii) A clamped square orthogonally stiffened plate structure subjected to a suddenly applied uniform pressure (479.0 N/m*) is shown in Fig. 3. The dimensions and material of the plate are identical with that in example (i). There are seven stiffeners and five stiffeners in the x- and y-directions of the plate, respectively. For stiffeners, axial strength EF = 1.1 x lo6 N, flexure strength EJ1= 9 Nm*, EJ, = 40 N m*, A quarter of the stiffened plate structure is divided
into 4 x 6 plate elements and 24 beam elements. Figure 4 shows the dynamic responses of the deflections at central point, where time step At = 0.002 set is used. In Fig. 4, the results obtained by the finite element method is also shown, in which the same mesh pattern, time step and accuracy as those used in the FE-RTM method are employed. Very little difference exist between the results, so that the plots in Fig. 4 are not distinct. The consumed average computation time for each time step is shown in Table 2. It is found that the FE-RTM method has the same accuracy with the finite element method when the same element mesh pattern are employed, but it’s computation efficiency is higher than that one.
* ADINA program has been translated microcomputer in China.
Table 1. Comparison
of computation nlate
for use on a
5. CONCLUSIONS
A combined finite element-transfer matrix method is applied to the transient analysis of geometrical nonlinear structures under various excitations. The Riccati transformation of state vectors is employed to avoid the propagation of round-off errors in ordinary combined finite-transfer matrix method. A microcomputerprogramTNONDL-W basedon thismethod has been developed. Because the memory capacity of the microcomputer is small, many measures are employed to overcome this disadvantage. 3
x
FEM
-zg2 E :
2
time for clamped
Method by applying
Computation time for each time step (xc)
FE-RTM Finite method
346 700
0
2
1
Time
(x10-'
3
4
SeC)
Fig. 4. Central displacement response of clamped orthogonally stiffened plate.
Dynamic large deflection analysis Table 2. Comparison of computation for clamped orthogonally stiffened plate Method hy applying FE-RTM Finite method
Computation time for each time step (see)
3. G. Ghiatti and A. Sestieri, Analysis of static and dynamic structural problems by a combined finite element-transfer matrix method. J. Sound Vibr. 67, 35-42 (1979). 4. S. Sankar and
200 365 5.
Some numerical examples presented in this paper show that this method can be sucessfully applied to the transient analysis of the structure with large displacement subjected to various excitations by reducing the size of matrix and relative computation time to less than those obtained by the method based on the ordinary finite element procedures. REFERENCES
703
6.
7.
8.
9.
1. M. A. Dokainish, A new approach for plate vibration:
combination of transfer matrix and finite-element technique. Trans. ASME, J. Engng Ind. 94, 526530 (1972).
10.
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