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Analysis of static and dynamic structural problems by a combined finite element-transfer matrix method
Journal of Sound and Oration
(1979) 67(l), 3542
ANALYSIS OF STATIC AND DYNAMIC STRUCTURAL BY A COMBINED
FINITE ELEMENT-TRANSFER
PROBLEMS MATRIX
METHOD G. CHIATTI AND
A. SESTIERI
Istituto di Macchine e Tecnologie Meccaniche, Universit&di Roma, Rome, Italy (Received 24 October 1978, and in revisedform 18 May 1979)
The combined use of ‘the finite element and transfer matrix techniques (FETM) for the study of dynamic problems was proposed a few years ago, in order to overcome the large amount of computer storage and long computation time that the finite element technique often requires. In this paper some interesting applications are emphasized for both static and dynamic problems of structures. A great deal of attention has been paid to the use of shell isoparametric elements for very thin structures, where the usual numerical integration by a two-by-two Gaussian quadrature of the stiffness matrix leads to an ineffective increase of stiffness in the structure. Particularly appealing seems to be the use of quadratic shell elements in the FETM method, because even with a reduction in the total number of elements of the structure it is possible to increase the accuracy of results. Computation time is appreciably reduced by this method, because of the notable lowering of the final matrix order, the manipulation of which gives the solution of the problem. Some results for natural frequencies of a thin plate are finally presented, showing a favourable agreement with those obtained by other proposed methods.
1. INTRODUCTION The technique of finite elements (FE) nowadays is probably the most satisfactory in the analysis of dynamic problems and particularly in evaluating natural frequencies of vibration of complex structures. However, in order to be able to describe a complex strticture with a good accuracy, it is necessary to divide up the structure and consider a large number of nodal points (i.e., degrees of freedom). Unfortunately this produces stiffness and mass matrices of large dimensions and consequently the solution requires long computation. In fact the order of the matrix obtained is equal to the number of degrees of freedom of the whole structure and this is equivalent to the total number of nodes multiplied by the degrees of freedom of each node. For this reason, and for some time now, various techniques have been suggested to reduce the order of the matrix: i.e., the number of degrees of freedom of the whole structure. In general, however, these methods can be successful only for particular structures (finite strip analysis, use of symmetry), or involve considerable complication of the formulation of the problem (master and slave variables). Another remarkable numerical technique successfully applied to simple structures in which a geometric dimension is predominant with respect to the others (beams, shafts) is the transfer matrix (TM) method. It allows the analysis of the structure to be effected by 35 0022460X/79/210035+08
$02.00/O
0 1979 Academic Press Inc. (London) Limited
36
G. CHIATTI
AND A. SESTIERI
the subsequent translation of the characteristic section state variables (shear, moment, displacement, rotation) from the initial section of the system to the final one. As this translation takes place by subsequent multiplication of “point” and “field” matrices, the order of each being equal to twice the number of degrees of freedom of each section, the method leads to a final matrix of the same order as that of an individual section. This order is low and, therefore, the matrix is easy to deal with and manipulate. Unfortunately, as for the other techniques mentioned, the method is successful only for certain, unidimensional structures, and it becomes very approximate for structures such as plates and shells, in which at least two of the principal dimensions appear. As early as 1962 Leckie tried to apply the TM method to the study of plate vibration [l]. He used a model proposed by Hrennikoff, which consisted of dividing the structure into a system of equivalent beams [2]. However interesting, the method has obvious limitations due to the choice of the model itself. Following up this work, Dokainish in 1972 published a very interesting paper in which the FE and TM techniques were linked [3]. It is from this model that we have drawn our inspiration, and the aim of the present paper is to present our later developments of FETM techniques. Appropriate combination of the very reliable theories by which the methods of FE and TM have been previously separately validated [4, 51 can now allow one to analyze very complicated cases and situations. Some cases are considered below. The computational work was executed on a big computer (1100/20 UNIVAC of the University of Rome), but without ever resorting to mass storages or to particular mathematical algorithms.
2. THE FETM METHOD The essence of the combined finite element-transfer matrix method (FETM) was thoroughly described by Dokainish in his original paper [3]. To deal with simple plates, he divided the structure into rectangular strips (see Figure 1). In practice any suitable shape of strips may be considered. Each of these strips is then partitioned into finite elements (plate elements, as in Dokainish’s work, or any other type of element). The vertical sides dividing or bordering the strips are called sections (1,2, . ..) whilst the horizontal boundaries are the edges. Once the degrees of freedom of each of the nodes necessary to describe the behaviour of the structure, and the corresponding force vector acting on each node, have been defined, expressions for (6) and {F}, the vector of the nodal displacement and the corresponding force vector, respectively, can be formed. If non-conforming plate elements are used, the vector of nodal displacements and the corresponding force vector can be written as
Figure
1. Sub-division
of a plate into rectangular
strips.
THE FE-TM
METHOD
37
whilst for shell elements, which will be used later, they are defined as (61 = cu, 0, w, a, PIT,
(f-1 = C~E’~y>~Z,~,,q!lT,
where u, u, w, are the displacements along the three co-ordinate axes and CIand /I are the rotations of a segment, normal to the middle surface in the general node, about the x and y axes. Proceeding according to the FE technique, one determines the stiffness matrix K and the mass matrix M (necessary only in dynamic problems) for each element. It is then possible to assemble the K and M matrices for each strip, obtaining the “strip matrix”. At this stage it is necessary to consider the boundary conditions at the edges, and hence to omit rows and columns corresponding to the zero displacements of (6) in the K and M strip matrices. For the ith strip one has [K-w’M]
{S}i = {F}i,
(2.1)
where [K - m2M] is the strip matrix. The transfer matrix relating the left and right displacements and forces of the ith strip may be obtained by suitably transforming the strip matrix into four sub-matrices A, B, C and D:
(2.2) a,, 6,) FL and FR are the left and right displacements and forces of the ith strip respectively. By simply operating on the last expression, and relating the characteristic state variables of one section to those of its neighbour by means of the force equilibrium condition (where the external forces are zero because a free vibration problem is considered) one finally obtains
relating two contiguous strips. Passing along from the left of the first section (where the structure begins) to the left of the last one (where it finishes), one can write (2.4) By imposing the boundary conditions at the initial and final sections (some components of the state vector of these sections are required to be zero), it is possible to determine, in the problem of free vibrations, a sub-matrix whose determinant, when equal to zero, supplies all the natural frequencies and consequently all the modes of vibration of the structure.
3. SOME DEVELOPMENTS OF THE METHOD The above brief exposition, even if schematically sufficient to show the effectiveness of the method, obscures some important details concerning further developments to which it can give rise. With the two basic techniques taken as a starting point, some specific cases will now be discussed.
38
G. CHIATTI AND A. SESTIERI
3.1. STATIC ANALYSIS OF STRUCTURES
Neither FE nor TM is limited in application to the study of dynamic problems; consequently it is possible, with the present method, to perform a static analysis of the structure. Equation (2.3) has to be written in terms of only the K matrix, and with addition of the effects of external forces. These may be applied to the nodes, or lumped at some points which do not coincide with the nodes, or be distributed. In any case it is first necessary to consider the equivalent forces on the nodes by defining the shape functions of the elements. When the equivalent forces have been computed one has
or, in a more compact and useful form, -B-‘/l
B-1
DB-IA-C -DB-’ _____________~-e?L 0 0
I
O
1F
(3.2)
I 1
By solving this system the static quantities of interest can be determined.
3.2.
THE USE OF BILINEAR AND QUADRATIC SHELL ELEMENTS IN THE FETM METHOD:
THEIR
APPLICATION TO THIN STRUCTURES
In order to apply the FETM method to more complex geometries, conforming bilinear and quadratic isoparametric shell elements have been introduced. They are both more deformable than the plate elements used by Dokainish [3] and can be successfully adapted to very complex and distorted geometries, with a rather small number of elements and nodes. The rectangular shell element results from the three dimensional cuboid element, with one dimension smaller than the other two. In this element the nodes are placed on the middle surface. Five degrees of freedom instead of three are assigned to each of the nodes [4]. It is worth pointing out that for the isoparametric shell element one does not require the middle surface to be flat as for the plate element. The middle surface, in fact, can be any deformed surface and the possibility of describing such a surface requires the definition of the nodes’ co-ordinates and the relative shape functions. Consequently there are live degrees of freedom for every node that must be transferred from one section to another. With thin shell elements, which are suitable for studying many types of structures, the deformability of thickness hypothesis typical of bilinear shell elements leads to an ineffective increase of the element’s stiffness. The imposed geometric scheme in fact produces some distortions, which are by no means realistic and improve the shear terms in the element strain energy [6, 71. To overcome this phenomenon the numerical integration for the evaluation of the stiffness matrix of the element must be effected by considering the shear terms at some points where such fictitious distortions do not appear. Such effects are not so disturbing if quadratic or higher order shell elements are used. Then, in fact, the element may deflect nearer to the actual deformation. Consequently the danger of unwanted stiffness ties, linked to the type of element considered, is really limited. Therefore, when considering elements with nodes not only on the corners but also, for instance, on the midpoints of the edges (quadratic or second order shell elements), the numerical integration can be effected by employing, as usual, a two-by-two Gaussian
THE FE-TM
39
METHOD
quadrature for both the bending and the shear energy terms. The use of these quadratic elements presents, however, some problems. In the previous sections in fact it was assumed that all the nodes were along the borders of the single strip, whilst now they are inside the strip too. Thus when following the procedure of section 2, in which the state vector was transferred from one section to the next, it is not possible to take into account the degrees of freedom deriving from the internal nodes (see Figure 2). It is therefore necessary to devise a new procedure. The strip matrix can be written as follows: (3.3) Displacements and forces are here divided into two groups, the former (6, and 6,, FL and FR) with the usual meaning, and the latter (din,, Fin,), referring to the nodes inside the strip itself. With regard to forces it is necessary to consider the following. In the free vibration problem the resultant external forces acting on the nodes of the structure are zero. This
Figure 2. Sketch showing
internal
nodes.
does not mean that FL and FR are zero too, because they do not balance the external forces alone, but also the forces of the adjacent strips. On the contrary the Fin, forces, by themselves, balance the external forces acting on the internal nodes and therefore, in this specific case, they are zero. Consequently equation (3.3) may be written for this problem as (3.4) that is, AS, + BS, + ESi,, = FL,
CS, + DS, + FSi,, = FR,
GS, + HS, + LSint = 0,
which implies that dint = -L- ‘(Cd, + H6,).
(3.5)
By substituting the last of equations (3.5) into the first two, one obtains, in matrix form, (3.6) This strip matrix, in the case of internal nodes, corresponds to expression (2.2) and therefore it is susceptible to the same manipulation and transformation.
40
G. CHIATTI AND A. SESTIERI 4.
RESULTS
In Table 1 the non-dimensional natural frequencies (independent of plate dimensions) of a clamped-free square plate, as obtained by different methods, are compared. The dimensions of the plate were 0.6 x 0.6 x 0.005 m. By means of the FETM technique, the following cases have been analyzed :
(4 non-conforming plate elements: structure divided into 6 x 6 rectangular elements; i.e., 147 degrees of freedom for the whole plate; the matrix size for the FE method is 126 x 126, whilst for the FETM method the order of the matrix is only 21 x 21; (W non-conforming plate elements: structure divided into 2 x 2 rectangular elements, i.e., 27 degrees of freedom; FE matrix order 18 x 18, FETM matrix order 9 x 9; (4 bilinear shell elements: structure divided into 3 x 3 rectangular elements, i.e., 80 degrees of freedom, numerical integration performed by a two point Gaussian quadrature on the transverse direction for both shear and bending terms; FE matrix order 60 x 60, FETM matrix order 20 x 20; bilinear shell elements: structure divided into 3 x 3 rectangular elements, i.e., 80 (4 degrees of freedom; numerical integration performed by a one point Gaussian quadrature for shear terms and 2x2 quadrature for bending terms; FE matrix order 60 x 60, FETM matrix order 20 x 20; (4 quadratic shell elements (additional nodes at the centre of the edges): structure divided into 2 x 2 rectangular elements; i.e., 105 degrees of freedom; numerical integration performed by a 2 x 2 Gaussian quadrature; FE matrix order 80 x 80, FETM matrix order 25 x 25. In Table 1 comparisons are made with results obtained by Ritz energy method [S] and by a finite element method. In the latter case the structure was divided into 50 triangular TABLE 1
Non-dimensional natural frequencies of a clamped-free square plate (o,,)t. Comparisons among different methods. Dimension of the plate 0.6 x 0.6 x 0.005 m
Method Ritz method FE method
2nd Mode
3rd Mode
4th Mode
5th Mode
3.494
8.547
21.44
27.46
31.17
3.469
8.535
21.45
27.06
3.51 0.4% 3.53
21.80 1.7%
27.30 0.6%
30.80 1.2%
22.10 3.1%
26.80 2.4%
29.00 6.9%
3.53
8.56 0.2% 8.80 2.9% 8.40
1.O%
1.7%
25.80 20.3%
29.90 9.0%
33.70 8.1%
3.53 1.0%
8.46 1.0%
26.25 22.4%
31.90 16.0%
33.50 7.0%
3.48 0.4%
8.60 0.6%
22.30 4.0%
29.70 8.1%
33.30 6.8%
1st [8]
Mode
1.0%
t oaa = CO”/J-/Et3 = non-dimensional natural frequency, where e = side of the square plate [ml, p = mass per unit volume [kg/m3], t = thickness [ml, E = Young’s modulus [N/m’], Y = Poisson’s ratio, OJ”= natural frequency [rad/s]. $ Differences (A) are evaluated with respect to the Ritz method results.
41
THE FE-TM METHOD
shell elements: i.e., 90 degrees of freedom [4]. It is worth pointing out that Table 1 shows only the application of the FETM method to a square plate with different kinds of finite elements and different numbers of degrees of freedom. Comparisons of the results with those obtained by the two other methods show a good agreement for almost all frequencies, even in those cases where few degrees of freedom are considered. The cases presented here, which refer to a flat plate, do not allow one to show the actual effectiveness of the isoparametric (linear or quadratic) shell elements, when dealing with complex structures. Recently, however, these have been successfully applied with the FETM method to compute the natural frequencies of tapered, pretwisted turbomachinery blades [9], giving very accurate results as compared with the experimental data. A brief TABLE 2
Natural frequencies of an axial compressor blade (rad/s) Mode ‘1 Experimental data FETM results
1822 1822
2
3
4
5
1527 8080
8947 9172
14450
20 527 20487
6‘ 22 870 23 987
summary of that work is presented in Table 2 for an axial pretwisted rotor compressor blade (AN-200 produced by Nuovo Pignone) divided into four isoparametric shell elements per strip and into four strips. The computed results are compared with the experimental ones.
5.
CONCLUSIONS
The most apparent advantage of the FETM method as described, over the FE technique, consists in the remarkable reduction of the matrix size. If q is the number of degrees of freedom of a section in both FETM and FE methods, and n, the number of strips into which the structure is divided, the size of the final matrix is, in the FETM method, q x q, but in the FE method is qn, x qn, . For these sizes account has been taken of the boundary conditions for both methods. This reduction of the matrix size very often implies lower computing time and computer storage, with an equal accuracy of results. Nevertheless, although for the computer storage it is possible to speak about a reduction proportional to the actual lowering of the matrix order, as far as the computing time is concerned it is not possible to consider an equal proportionality. When using the FE method, in fact, it is necessary to solve an eigenvalue problem and the computing time. depends either on the size of the matrix or on the computer solution method adopted [lo]. With the FETM method, on the contrary, the problem is solved through an iterative computation of a determinant, which vanishes at the correct value of natural frequencies. If an iterative eigenvalue computation, analogous to the procedure used in the FETM method, is employed, a rough computation of the elementary operations (multiplications and divisions) shows that the FETM method becomes convenient with respect to the FE technique when the number of operations in the FE method overtakes that in the FETM method. For a single iterative step, the above condition can be expressed as +{qn,(q’n,2-4)+3}
> ${q3[12(n,--l)+l]-493-3):
G. CHIATTI AND A. SESTIERI
42 that is 3
4, 49
n,-
1
n,3--12ns+ll
(valid for n, > 1).
When n, = 1, both FE and FETM methods (see the first relationship) involve the same number of elementary operations. This formula takes account of the number of operations to effect matrix multiplications (in the FETM method) and to compute a determinant, with consideration, in both cases, of the reduction of degrees of freedom due to the boundary conditions. A worthwhile comparison can be made between this method and the finite strip (FS) method [ll]. Because of the division into strips, the two methods could, at a superficial glance, perhaps be confused. The approach, indeed, is completely different, and, whilst for those problems in which the geometry and material properties do not vary in one coordinate direction (e.g., an isotropic plate) the FS analysis is undoubtedly very effective and probably more suitable than the FETM method (but it depends on the boundary conditions), the FETM method is actually more flexible and more easily appliable to complex structures. Furthermore the reduction of degrees of freedom, and consequently of the matrix size, in the FS method depends on several factors (boundary conditions, degree of approximation, number of modes computed), so that it is not possible to generalize the real advantage of the method. This problem of course does not exist with the FETM method, because the matrix order depends only on the type of elements and the kind of divisions proposed. To deal finally with complex structures, the introduction of quadratic shell elements appears to be particularly appealing. They in fact increase the accuracy of results even though the number of elements is reduced. This is of course a well known result in FE but its transfer to the proposed method appeared, at a first analysis, remarkably complex, due to the existence of nodal points inside the strips. REFERENCES 1. F. A. LECKIE 1962 Zngenieur-Archiv 32, 100-111. The application of transfer matrices to plate vibrations. 2. A. HRENNIKOFF 1941 Journal of Applied Mechanics 63, 169-175. Solution of problems of elasticity by the framework method. 3. M. A. DOKMNISH 1972 Journal of Engineeringfor Industry, Transactions of the American Society of Mechanical Engineers 94,526530. A new approach for plate vibrations: combination of transfer matrix and finite element technique. 4. 0. C. ZIENKIEWICZ 1971 The Finite Element Method in Engineering Science. London: McGrawHill Book Company. 5. E. C. F%STELand F. A. LECKIE1963 Matrix Methods in Elastomechanics. New York: McGraw-Hill Book Company. 6. 0. C. ZIENKIEWICZ,R. L. TAYLORand J. M. TOO 1971 International Journalfor NumericalMethods in Engineering 3,275-290. Reduced integration technique in general analysis of plates and shells. 1977 International Journalfor Numerical 7. T. J. R. HUGHES,R. L. TAYLORand W. KANOKNUKULCHA Methods in Engineering 11, 1529-1543. A simple and efficient finite element for plate bending. 8. D. YOUNG1950 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 72,448-453. Vibration of rectangular plates by the Ritz method. 9. G. CH~ATTIand A. SESTIW 1979 Fifth World Congress on the Theory of Machines and Mechanisms, Montreal, Canada, July. A combined finite element-transfer matrix method for the evaluation of free vibration of pre-twisted turbomachinery blades. 10. J. H. WILKINSON1965 The Algebraic Eigenvalue Problem. Oxford University Press. 11. Y. K. CHJXJN~1976 Finite Strip Method in Structure Analysis. Oxford: Pergamon Press.
(1979) 67(l), 3542
ANALYSIS OF STATIC AND DYNAMIC STRUCTURAL BY A COMBINED
FINITE ELEMENT-TRANSFER
PROBLEMS MATRIX
METHOD G. CHIATTI AND
A. SESTIERI
Istituto di Macchine e Tecnologie Meccaniche, Universit&di Roma, Rome, Italy (Received 24 October 1978, and in revisedform 18 May 1979)
The combined use of ‘the finite element and transfer matrix techniques (FETM) for the study of dynamic problems was proposed a few years ago, in order to overcome the large amount of computer storage and long computation time that the finite element technique often requires. In this paper some interesting applications are emphasized for both static and dynamic problems of structures. A great deal of attention has been paid to the use of shell isoparametric elements for very thin structures, where the usual numerical integration by a two-by-two Gaussian quadrature of the stiffness matrix leads to an ineffective increase of stiffness in the structure. Particularly appealing seems to be the use of quadratic shell elements in the FETM method, because even with a reduction in the total number of elements of the structure it is possible to increase the accuracy of results. Computation time is appreciably reduced by this method, because of the notable lowering of the final matrix order, the manipulation of which gives the solution of the problem. Some results for natural frequencies of a thin plate are finally presented, showing a favourable agreement with those obtained by other proposed methods.
1. INTRODUCTION The technique of finite elements (FE) nowadays is probably the most satisfactory in the analysis of dynamic problems and particularly in evaluating natural frequencies of vibration of complex structures. However, in order to be able to describe a complex strticture with a good accuracy, it is necessary to divide up the structure and consider a large number of nodal points (i.e., degrees of freedom). Unfortunately this produces stiffness and mass matrices of large dimensions and consequently the solution requires long computation. In fact the order of the matrix obtained is equal to the number of degrees of freedom of the whole structure and this is equivalent to the total number of nodes multiplied by the degrees of freedom of each node. For this reason, and for some time now, various techniques have been suggested to reduce the order of the matrix: i.e., the number of degrees of freedom of the whole structure. In general, however, these methods can be successful only for particular structures (finite strip analysis, use of symmetry), or involve considerable complication of the formulation of the problem (master and slave variables). Another remarkable numerical technique successfully applied to simple structures in which a geometric dimension is predominant with respect to the others (beams, shafts) is the transfer matrix (TM) method. It allows the analysis of the structure to be effected by 35 0022460X/79/210035+08
$02.00/O
0 1979 Academic Press Inc. (London) Limited
36
G. CHIATTI
AND A. SESTIERI
the subsequent translation of the characteristic section state variables (shear, moment, displacement, rotation) from the initial section of the system to the final one. As this translation takes place by subsequent multiplication of “point” and “field” matrices, the order of each being equal to twice the number of degrees of freedom of each section, the method leads to a final matrix of the same order as that of an individual section. This order is low and, therefore, the matrix is easy to deal with and manipulate. Unfortunately, as for the other techniques mentioned, the method is successful only for certain, unidimensional structures, and it becomes very approximate for structures such as plates and shells, in which at least two of the principal dimensions appear. As early as 1962 Leckie tried to apply the TM method to the study of plate vibration [l]. He used a model proposed by Hrennikoff, which consisted of dividing the structure into a system of equivalent beams [2]. However interesting, the method has obvious limitations due to the choice of the model itself. Following up this work, Dokainish in 1972 published a very interesting paper in which the FE and TM techniques were linked [3]. It is from this model that we have drawn our inspiration, and the aim of the present paper is to present our later developments of FETM techniques. Appropriate combination of the very reliable theories by which the methods of FE and TM have been previously separately validated [4, 51 can now allow one to analyze very complicated cases and situations. Some cases are considered below. The computational work was executed on a big computer (1100/20 UNIVAC of the University of Rome), but without ever resorting to mass storages or to particular mathematical algorithms.
2. THE FETM METHOD The essence of the combined finite element-transfer matrix method (FETM) was thoroughly described by Dokainish in his original paper [3]. To deal with simple plates, he divided the structure into rectangular strips (see Figure 1). In practice any suitable shape of strips may be considered. Each of these strips is then partitioned into finite elements (plate elements, as in Dokainish’s work, or any other type of element). The vertical sides dividing or bordering the strips are called sections (1,2, . ..) whilst the horizontal boundaries are the edges. Once the degrees of freedom of each of the nodes necessary to describe the behaviour of the structure, and the corresponding force vector acting on each node, have been defined, expressions for (6) and {F}, the vector of the nodal displacement and the corresponding force vector, respectively, can be formed. If non-conforming plate elements are used, the vector of nodal displacements and the corresponding force vector can be written as
Figure
1. Sub-division
of a plate into rectangular
strips.
THE FE-TM
METHOD
37
whilst for shell elements, which will be used later, they are defined as (61 = cu, 0, w, a, PIT,
(f-1 = C~E’~y>~Z,~,,q!lT,
where u, u, w, are the displacements along the three co-ordinate axes and CIand /I are the rotations of a segment, normal to the middle surface in the general node, about the x and y axes. Proceeding according to the FE technique, one determines the stiffness matrix K and the mass matrix M (necessary only in dynamic problems) for each element. It is then possible to assemble the K and M matrices for each strip, obtaining the “strip matrix”. At this stage it is necessary to consider the boundary conditions at the edges, and hence to omit rows and columns corresponding to the zero displacements of (6) in the K and M strip matrices. For the ith strip one has [K-w’M]
{S}i = {F}i,
(2.1)
where [K - m2M] is the strip matrix. The transfer matrix relating the left and right displacements and forces of the ith strip may be obtained by suitably transforming the strip matrix into four sub-matrices A, B, C and D:
(2.2) a,, 6,) FL and FR are the left and right displacements and forces of the ith strip respectively. By simply operating on the last expression, and relating the characteristic state variables of one section to those of its neighbour by means of the force equilibrium condition (where the external forces are zero because a free vibration problem is considered) one finally obtains
relating two contiguous strips. Passing along from the left of the first section (where the structure begins) to the left of the last one (where it finishes), one can write (2.4) By imposing the boundary conditions at the initial and final sections (some components of the state vector of these sections are required to be zero), it is possible to determine, in the problem of free vibrations, a sub-matrix whose determinant, when equal to zero, supplies all the natural frequencies and consequently all the modes of vibration of the structure.
3. SOME DEVELOPMENTS OF THE METHOD The above brief exposition, even if schematically sufficient to show the effectiveness of the method, obscures some important details concerning further developments to which it can give rise. With the two basic techniques taken as a starting point, some specific cases will now be discussed.
38
G. CHIATTI AND A. SESTIERI
3.1. STATIC ANALYSIS OF STRUCTURES
Neither FE nor TM is limited in application to the study of dynamic problems; consequently it is possible, with the present method, to perform a static analysis of the structure. Equation (2.3) has to be written in terms of only the K matrix, and with addition of the effects of external forces. These may be applied to the nodes, or lumped at some points which do not coincide with the nodes, or be distributed. In any case it is first necessary to consider the equivalent forces on the nodes by defining the shape functions of the elements. When the equivalent forces have been computed one has
or, in a more compact and useful form, -B-‘/l
B-1
DB-IA-C -DB-’ _____________~-e?L 0 0
I
O
1F
(3.2)
I 1
By solving this system the static quantities of interest can be determined.
3.2.
THE USE OF BILINEAR AND QUADRATIC SHELL ELEMENTS IN THE FETM METHOD:
THEIR
APPLICATION TO THIN STRUCTURES
In order to apply the FETM method to more complex geometries, conforming bilinear and quadratic isoparametric shell elements have been introduced. They are both more deformable than the plate elements used by Dokainish [3] and can be successfully adapted to very complex and distorted geometries, with a rather small number of elements and nodes. The rectangular shell element results from the three dimensional cuboid element, with one dimension smaller than the other two. In this element the nodes are placed on the middle surface. Five degrees of freedom instead of three are assigned to each of the nodes [4]. It is worth pointing out that for the isoparametric shell element one does not require the middle surface to be flat as for the plate element. The middle surface, in fact, can be any deformed surface and the possibility of describing such a surface requires the definition of the nodes’ co-ordinates and the relative shape functions. Consequently there are live degrees of freedom for every node that must be transferred from one section to another. With thin shell elements, which are suitable for studying many types of structures, the deformability of thickness hypothesis typical of bilinear shell elements leads to an ineffective increase of the element’s stiffness. The imposed geometric scheme in fact produces some distortions, which are by no means realistic and improve the shear terms in the element strain energy [6, 71. To overcome this phenomenon the numerical integration for the evaluation of the stiffness matrix of the element must be effected by considering the shear terms at some points where such fictitious distortions do not appear. Such effects are not so disturbing if quadratic or higher order shell elements are used. Then, in fact, the element may deflect nearer to the actual deformation. Consequently the danger of unwanted stiffness ties, linked to the type of element considered, is really limited. Therefore, when considering elements with nodes not only on the corners but also, for instance, on the midpoints of the edges (quadratic or second order shell elements), the numerical integration can be effected by employing, as usual, a two-by-two Gaussian
THE FE-TM
39
METHOD
quadrature for both the bending and the shear energy terms. The use of these quadratic elements presents, however, some problems. In the previous sections in fact it was assumed that all the nodes were along the borders of the single strip, whilst now they are inside the strip too. Thus when following the procedure of section 2, in which the state vector was transferred from one section to the next, it is not possible to take into account the degrees of freedom deriving from the internal nodes (see Figure 2). It is therefore necessary to devise a new procedure. The strip matrix can be written as follows: (3.3) Displacements and forces are here divided into two groups, the former (6, and 6,, FL and FR) with the usual meaning, and the latter (din,, Fin,), referring to the nodes inside the strip itself. With regard to forces it is necessary to consider the following. In the free vibration problem the resultant external forces acting on the nodes of the structure are zero. This
Figure 2. Sketch showing
internal
nodes.
does not mean that FL and FR are zero too, because they do not balance the external forces alone, but also the forces of the adjacent strips. On the contrary the Fin, forces, by themselves, balance the external forces acting on the internal nodes and therefore, in this specific case, they are zero. Consequently equation (3.3) may be written for this problem as (3.4) that is, AS, + BS, + ESi,, = FL,
CS, + DS, + FSi,, = FR,
GS, + HS, + LSint = 0,
which implies that dint = -L- ‘(Cd, + H6,).
(3.5)
By substituting the last of equations (3.5) into the first two, one obtains, in matrix form, (3.6) This strip matrix, in the case of internal nodes, corresponds to expression (2.2) and therefore it is susceptible to the same manipulation and transformation.
40
G. CHIATTI AND A. SESTIERI 4.
RESULTS
In Table 1 the non-dimensional natural frequencies (independent of plate dimensions) of a clamped-free square plate, as obtained by different methods, are compared. The dimensions of the plate were 0.6 x 0.6 x 0.005 m. By means of the FETM technique, the following cases have been analyzed :
(4 non-conforming plate elements: structure divided into 6 x 6 rectangular elements; i.e., 147 degrees of freedom for the whole plate; the matrix size for the FE method is 126 x 126, whilst for the FETM method the order of the matrix is only 21 x 21; (W non-conforming plate elements: structure divided into 2 x 2 rectangular elements, i.e., 27 degrees of freedom; FE matrix order 18 x 18, FETM matrix order 9 x 9; (4 bilinear shell elements: structure divided into 3 x 3 rectangular elements, i.e., 80 degrees of freedom, numerical integration performed by a two point Gaussian quadrature on the transverse direction for both shear and bending terms; FE matrix order 60 x 60, FETM matrix order 20 x 20; bilinear shell elements: structure divided into 3 x 3 rectangular elements, i.e., 80 (4 degrees of freedom; numerical integration performed by a one point Gaussian quadrature for shear terms and 2x2 quadrature for bending terms; FE matrix order 60 x 60, FETM matrix order 20 x 20; (4 quadratic shell elements (additional nodes at the centre of the edges): structure divided into 2 x 2 rectangular elements; i.e., 105 degrees of freedom; numerical integration performed by a 2 x 2 Gaussian quadrature; FE matrix order 80 x 80, FETM matrix order 25 x 25. In Table 1 comparisons are made with results obtained by Ritz energy method [S] and by a finite element method. In the latter case the structure was divided into 50 triangular TABLE 1
Non-dimensional natural frequencies of a clamped-free square plate (o,,)t. Comparisons among different methods. Dimension of the plate 0.6 x 0.6 x 0.005 m
Method Ritz method FE method
2nd Mode
3rd Mode
4th Mode
5th Mode
3.494
8.547
21.44
27.46
31.17
3.469
8.535
21.45
27.06
3.51 0.4% 3.53
21.80 1.7%
27.30 0.6%
30.80 1.2%
22.10 3.1%
26.80 2.4%
29.00 6.9%
3.53
8.56 0.2% 8.80 2.9% 8.40
1.O%
1.7%
25.80 20.3%
29.90 9.0%
33.70 8.1%
3.53 1.0%
8.46 1.0%
26.25 22.4%
31.90 16.0%
33.50 7.0%
3.48 0.4%
8.60 0.6%
22.30 4.0%
29.70 8.1%
33.30 6.8%
1st [8]
Mode
1.0%
t oaa = CO”/J-/Et3 = non-dimensional natural frequency, where e = side of the square plate [ml, p = mass per unit volume [kg/m3], t = thickness [ml, E = Young’s modulus [N/m’], Y = Poisson’s ratio, OJ”= natural frequency [rad/s]. $ Differences (A) are evaluated with respect to the Ritz method results.
41
THE FE-TM METHOD
shell elements: i.e., 90 degrees of freedom [4]. It is worth pointing out that Table 1 shows only the application of the FETM method to a square plate with different kinds of finite elements and different numbers of degrees of freedom. Comparisons of the results with those obtained by the two other methods show a good agreement for almost all frequencies, even in those cases where few degrees of freedom are considered. The cases presented here, which refer to a flat plate, do not allow one to show the actual effectiveness of the isoparametric (linear or quadratic) shell elements, when dealing with complex structures. Recently, however, these have been successfully applied with the FETM method to compute the natural frequencies of tapered, pretwisted turbomachinery blades [9], giving very accurate results as compared with the experimental data. A brief TABLE 2
Natural frequencies of an axial compressor blade (rad/s) Mode ‘1 Experimental data FETM results
1822 1822
2
3
4
5
1527 8080
8947 9172
14450
20 527 20487
6‘ 22 870 23 987
summary of that work is presented in Table 2 for an axial pretwisted rotor compressor blade (AN-200 produced by Nuovo Pignone) divided into four isoparametric shell elements per strip and into four strips. The computed results are compared with the experimental ones.
5.
CONCLUSIONS
The most apparent advantage of the FETM method as described, over the FE technique, consists in the remarkable reduction of the matrix size. If q is the number of degrees of freedom of a section in both FETM and FE methods, and n, the number of strips into which the structure is divided, the size of the final matrix is, in the FETM method, q x q, but in the FE method is qn, x qn, . For these sizes account has been taken of the boundary conditions for both methods. This reduction of the matrix size very often implies lower computing time and computer storage, with an equal accuracy of results. Nevertheless, although for the computer storage it is possible to speak about a reduction proportional to the actual lowering of the matrix order, as far as the computing time is concerned it is not possible to consider an equal proportionality. When using the FE method, in fact, it is necessary to solve an eigenvalue problem and the computing time. depends either on the size of the matrix or on the computer solution method adopted [lo]. With the FETM method, on the contrary, the problem is solved through an iterative computation of a determinant, which vanishes at the correct value of natural frequencies. If an iterative eigenvalue computation, analogous to the procedure used in the FETM method, is employed, a rough computation of the elementary operations (multiplications and divisions) shows that the FETM method becomes convenient with respect to the FE technique when the number of operations in the FE method overtakes that in the FETM method. For a single iterative step, the above condition can be expressed as +{qn,(q’n,2-4)+3}
> ${q3[12(n,--l)+l]-493-3):
G. CHIATTI AND A. SESTIERI
42 that is 3
4, 49
n,-
1
n,3--12ns+ll
(valid for n, > 1).
When n, = 1, both FE and FETM methods (see the first relationship) involve the same number of elementary operations. This formula takes account of the number of operations to effect matrix multiplications (in the FETM method) and to compute a determinant, with consideration, in both cases, of the reduction of degrees of freedom due to the boundary conditions. A worthwhile comparison can be made between this method and the finite strip (FS) method [ll]. Because of the division into strips, the two methods could, at a superficial glance, perhaps be confused. The approach, indeed, is completely different, and, whilst for those problems in which the geometry and material properties do not vary in one coordinate direction (e.g., an isotropic plate) the FS analysis is undoubtedly very effective and probably more suitable than the FETM method (but it depends on the boundary conditions), the FETM method is actually more flexible and more easily appliable to complex structures. Furthermore the reduction of degrees of freedom, and consequently of the matrix size, in the FS method depends on several factors (boundary conditions, degree of approximation, number of modes computed), so that it is not possible to generalize the real advantage of the method. This problem of course does not exist with the FETM method, because the matrix order depends only on the type of elements and the kind of divisions proposed. To deal finally with complex structures, the introduction of quadratic shell elements appears to be particularly appealing. They in fact increase the accuracy of results even though the number of elements is reduced. This is of course a well known result in FE but its transfer to the proposed method appeared, at a first analysis, remarkably complex, due to the existence of nodal points inside the strips. REFERENCES 1. F. A. LECKIE 1962 Zngenieur-Archiv 32, 100-111. The application of transfer matrices to plate vibrations. 2. A. HRENNIKOFF 1941 Journal of Applied Mechanics 63, 169-175. Solution of problems of elasticity by the framework method. 3. M. A. DOKMNISH 1972 Journal of Engineeringfor Industry, Transactions of the American Society of Mechanical Engineers 94,526530. A new approach for plate vibrations: combination of transfer matrix and finite element technique. 4. 0. C. ZIENKIEWICZ 1971 The Finite Element Method in Engineering Science. London: McGrawHill Book Company. 5. E. C. F%STELand F. A. LECKIE1963 Matrix Methods in Elastomechanics. New York: McGraw-Hill Book Company. 6. 0. C. ZIENKIEWICZ,R. L. TAYLORand J. M. TOO 1971 International Journalfor NumericalMethods in Engineering 3,275-290. Reduced integration technique in general analysis of plates and shells. 1977 International Journalfor Numerical 7. T. J. R. HUGHES,R. L. TAYLORand W. KANOKNUKULCHA Methods in Engineering 11, 1529-1543. A simple and efficient finite element for plate bending. 8. D. YOUNG1950 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 72,448-453. Vibration of rectangular plates by the Ritz method. 9. G. CH~ATTIand A. SESTIW 1979 Fifth World Congress on the Theory of Machines and Mechanisms, Montreal, Canada, July. A combined finite element-transfer matrix method for the evaluation of free vibration of pre-twisted turbomachinery blades. 10. J. H. WILKINSON1965 The Algebraic Eigenvalue Problem. Oxford University Press. 11. Y. K. CHJXJN~1976 Finite Strip Method in Structure Analysis. Oxford: Pergamon Press.