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Macroelement approach to the analysis of cylindrical shells connected by taper-roller bearings
Pergamon
Computers & S/rucrurrs Vol. 54. No. 2, pp. 247-254. 1995 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 S9.50 + 0.00
0045-7949(94)00325-4
MACROELEMENT APPROACH TO THE ANALYSIS OF CYLINDRICAL SHELLS CONNECTED BY TAPER-ROLLER BEARINGS R. Badys and R. Katianauskas Department
of Computer
Assisted
Engineering,
Vilnius
Technical
University,
Vilnius,
Lithuania
(Received 18 October 1993) Abstract-The application of the macroelement technique to non-linear analysis of cylindrical shells connected by taper-roller bearings is presented. Two types of macroelements are proposed. The linear macroelement is based on substructuring technique and describes the shells, while a non-linear one refers
to the taper-rollers. Evaluation of the stiffness matrix of the latter is discussed in detail. The results obtained by solving a numerical example are given to illustrate the model capability. The influence of connection
flexibility
on displacements
and stresses on cylinders
is also studied
and expressed
numerically.
equations representing an approximate model of the complete structure, This technique described by Bathe and Wilson [I] and Argyris and Mlejnek [2] is suitable for solving linear problems, though Refs [3,4] present additional examples of its application to localized nonlinearities. In this case, the part of the structure remains linear while the other part can be treated as non-linear. During the incremental analysis of the complete structure only the stiffness of the non-linear substructures needs to be modified according to stress change. The technique for reducing the DOF, expressed in terms of the generalized variables so that the resulting number is considerably smaller than that of the DOF of the initial discretization, is referred to herein as the reduction technique. The essence of the reduction method is described in Refs [5-l I], where the deformations of the structure are limited by certain modes expressed in terms of basic vectors (physical co-ordinates). Various alternatives for the basic vectors were proposed including Rayleigh-Ritz approximation functions, path derivatives, eigenfreqnency modes, linear buckling modes, etc. Different global discretization methods widely used in the analysis of various structures may be defined as the macroelement technique [12-161, whereby several examples are considered. Basically, the structural members such as beams, plates or shells as well as various connections, supports, stiffeners, holes and other concentrators and/or locations of non-linearity may be considered as macroelements. In the present paper, attention will be focused on the computational strategies for the modelling of cylindrical shells connected by taper-roller bearings. By applying three-dimensional elements and using a rather fine finite element mesh, the taper-roller can be modelled very accurately. However, in practice, the
INTRODUCTION
The accurate numerical simulation of complex engineering structures remains one of the most challenging problems of computational mechanics. Prediction of the response characteristics of a complex structure is very time consuming; so much research has been devoted to the study of solution methodologies which reduce the computational efforts of solving large structural problems. It is a well-recognized fact that a large number of degrees of freedom (DOF) in conventional finite element discretization of a largescaled structure is often dictated by its topology rather than by the expected complexity of the behaviour. The simplest and widely used approach to the analysis of complex structures is to introduce a smaller number of variables (sometimes new generalized variables) to represent a large number of DOF of initial discretization of a structure, thereby reducing the size of the stiffness matrix involved in the calculations while retaining high accuracy. The brief review of various computational strategies presented here focuses on the following techniques: (a) substructuring technique; (b) reduction technique; (c) macroelement technique. The substructuring technique, treating the total structure as an assemblage of substructures, is based on the idea of reducing the number of DOF prior to the element assemblage process. The reduction is not global but is performed only at the substructure level, while the compatibility of displacements and forces at the substructure interface generates the global set of 247
248
R. BauSys and R. KaEianauskas
above analysis of the taper-roller involves the use of about a thousand finite-element equilibrium equations [17]. It means that the non-linear analysis even of a single taper-roller is rather expensive. A macroelement approach involving the essential features of various discretization techniques is proposed. It is based on earlier research performed with the use of a multi-level approach, documented in Ref. [ 181. The proposed methodology assumes a complex structure to be separated into smaller macroelements which may be described by the structural matrices equivalent to the conventional finite element matrices defined by different numerical techniques. In order to simplify the modelling of the taper-roller bearings, a specific macroelement based on the assumptions of a simplified engineering theory of the bearings [19] is derived. The proposed model has serious limitations with regard to the accuracy of predicting local response characteristics such as stresses and strains of a taper-roller, but it is quite suitable for predicting the global response characteristics, i.e. the displacements and the stresses of cylinders. PROBLEM DESCRIPTION
A representative structure consisting of two circular cylindrical shells is investigated. The geometry of the inside cylinder is defined by the radius of the middle surface R, and the length L,, while the geometry of the outside cylinder is determined by the parameters R,, and L,, respectively. Both of the cylinders have constant thickness h and constant material properties, such as Young’s modulus E and Poisson’s ratio v. The outside cylinder is subjected to
a set of transverse concentrated loads F. Due to the symmetry of the geometry, the loading and the boundary conditions, only half of the structure shown in Fig. 1 is considered. The inside cylinder is clamped in the plane z = L,. The cylinders are connected by the taper-roller bearings, also shown schematically in this figure. The bearing connection consists of two (inside and outside) rings and several rollers. The cross-section of the structure with a bearing is illustrated in Fig. 2, while the detailed geometry and the most important parameters of an individual conical roller are shown in Fig. 3. Numerical analysis of the structure is based on the following assumptions: (1) the behaviour of the materials of which the cylinders and the taper-roller bearings are composed is linear-elastic; (2) the loading is of a static, monotonically increasing character, linearly depending on time t; (3) structural elements are assumed to be geometritally linear and all of the characteristics are referred to the undeformed state of structure. In this paper, the modelling of taper-roller bearings is based on engineering theory, details of which may be found in Ref. [19]. The basic assumptions are as follows: (1) all of the sliding and frictional neglected; (2) the contact surface is continuous; (3) the length of the initial contact bearing elements is equal to the length roller.
effects
are
surfaces of of a taper-
Analysis of cylindrical shells
249
t
_----__-------_-_--
Ri
Fig. 2. Connection
of cylinders
Within the framework of the aforementioned modelling strategy, two types of macroelements are used in the analysis: (a) macroelements
(b) macroelements
of cylinders; of taper-rollers.
Hence, the formulation of the macroelements is based on two different approaches. For the formulation of a cylinder macroelement a standard linear substructuring technique is used. This macroelement describes the linear mechanical behaviour of one cylindrical shell and two rings of the bearing connection. The linear four-node thick shell element with 24 DOF proposed by Sacharov [20], which is similar to a thin shell element, is used in shell discretization. The thick shell element is derived on the basis of more soft
I-
1
III-
-I
=
b
s
by taper-roller
bearings.
assumptions than that usually provided by thin shell theory, and shows a good convergence by decreasing the size of mesh. The finite element model of the structure under consideration is shown in Fig. 4. The rectangular finite element meshes for both cylinders have a regular division in the direction of the circumstantial co-ordinate and a non-regular division of the longitudinal co-ordinate, where the appropriate lines in the mesh of both cylinders are shifted one to another by the distance e shown in Fig. 3. The nodes of such meshes must coincide with the connection points of the rollers of the bearings. The rings of the rollerare described by special bearing connection eccentrically aided beam elements, having a nonsymmetric cross-section. After the condensation of internal DOF, the macro model of a complex ROLLER
MACROELEMENTS
I I
-I
Fig. 3. Geometry of individual roller.
Fig. 4. Discretization
scheme.
250
R. BauSys and R. Kacianauskas
structure consists of 2n, which is the number of rollers in the bearing. A non-linear macroelement connecting two cylinders is proposed to describe the taper-roller. This element reflects the ring-roller contact deformations and allows for two types of non-linearity. The first one reflects non-linear behaviour of contact deformations, while the second indicates the unilateral character of a bearing when the disconnection of cylinders occurs. This one-dimensional element involving basic features of reduction and macroelement techniques behaves as a simple spring, for which a natural beam element proposed in Ref. [2] may be used as an explicit mechanical model. The macroelement approach is based on the idea of multilevel discretization, where the global model consists of both linear and non-linear macroelements. The linear cylinder-type macroelements reduce the linear region and the size of the problem by numerical computations on the macroelement level only, while the suggested spring-type elements additionally simplify the model relations, avoiding the solution of the complex contact problem. The global relationship is transformed into a relatively small non-linear problem, retaining generalized DOF in which only the simplest one-dimensional non-linear elements will need to be considered. The solution of a non-linear problem requires an iterative/incremental procedure with a number of loading steps, finally reaching the total applied load. The basic approach in an incremental step-by-step algorithm is to assume that the solution of a nonlinear problem for the discrete time t + AZ is required, where At is a time increment. As a result of some manipulations, the solution of the non-linear problem is approximated by the solution of the non-linear equations
KT(u(t )) Au(At) = AF(At),
K, = A,k,Af, where K, and k, are the global (Cartesian) and internal (natural) stiffness matrices and A, is the transformation matrix relating the natural forces/ displacements to the global forces/displacements. The deformed state of the taper-roller is described by the displacements of points i, andj, (Fig. 3). These points are the middle points of the initial contact surfaces of the taper-roller. We consider only the deformations of the i-j element, while the joint elements i-i, and j-i, are taken to be rigid. The transformation is performed in three steps:
A, = L,H,T,,
where matrix T, transforms the internal forces into external ones, matrix H, describes the effects of eccentricities of a deformable element i-j,, and matrix L, refers to the rotation from the local coordinate system to the global co-ordinate system. For details of the construction of these transformation matrices one should refer to Ref. [21]. In a general three-dimensional case, the beam element has six internal DOF related to six deformation modes, but in our case the proposed taperroller element has only two natural DOF, rotation and translation in a direction perpendicular to the contact surface. So if the following vectors of internal forces and displacements are introduced as
the internal stiffness matrix k, = k,(u,), which is the tangential stiffness matrix of non-linear deformation process (1) accordingly takes the form
(2)
The exact displacements u(t + At) at time t + At correspond to the applied load F(t ) + AF(At ). Having the considered approximation of displacements, we can obtain the approximation of the stresses and corresponding nodal forces at time t + At.
The entries of the internal stiffness matrix may be obtained by differentiating the nodal force vector f, with respect to the nodal displacement vector II, and may be expressed as
k,,,,,(f)+
7
rm u,r, = L,,
MODELLING
(4)
(1)
where KT(u(t )) is the tangent stiffness matrix, which corresponds to the equilibrium conditions at time t, while the vector Au(At ) is the incremental nodal point displacement vector. The displacements at the time t + At can be calculated by an approximation: u(r + At ) = u(t ) + Au(At ).
element with 12 DOF [21], and reflects the unilateral stiffness characteristics of the taper-roller. A general expression of the stiffness matrix of the proposed element denoted by r may be written in the matrix form as
(m,n = IA
(5)
OF TAPER-ROLLERS
As stated above, the taper-rollers of a bearing are described by a special spring-type macroelement, which is based on the model of a general beam
where uf is the value of displacement at the appropriate loading level. In order to obtain the numerical characteristics of this stiffness matrix for element Y, it is necessary to
Analysis of cylindrical shells consider the equilibrium state of the taper-roller. Within the framework of the theory of simplified engineering analysis, calculations of static equivalent contact forces with respect to the displacements of the taper-roller and the rings of a bearing consist of solving a set of governing equilibrium equations. The directions of acting forces and moments are shown in Fig. 5. In constructing the stiffness matrix, the response characteristics of point j are considered as internal variables, thus F s F,, M 3 M, and u z ui, 4 = $,, respectively. The equilibrium state of a taperroller is described by the following set of equations: F;cos
cp;-~coscp,+F,coscp,=O,
6 sin cp, - F, sin ‘p, + F, sin cp, = 0,
(6)
251
where dF is the contact force, c refers to the characteristics that include the effects of the contact surface geometry and material response and u(z) is a relative displacement of the narrow cylinder of a taper-roller to the narrow cylinder of the ring. The relationship (7) is based on a simplified theory of engineering analysis [19] and is obtained with respect to the function of relative displacement which can be expressed as follows: u(z)=u,+4z.
Here u is the relative displacement of the middle point of the taper-roller to the ring and Q, refers to the relative rotation of the taper-roller to the ring. Therefore, the expressions of all acting contact forces and moments will take the following form
-M,+M,+M,=O.
i+
To obtain these equations the constitutive relationships between contact forces and displacements that are acting in the contact areas between the taperroller and rings should be known. Taking into account the assumption made for small contact areas, aforementioned relationships can be obtained in the framework of the Hertz theory. Then, this contact area has an oblong form, while the length of the contact area would be of the same order as the length of the taper-roller. In deriving the formulae for a governing relationship of contact force/displacement in the taper-roller element, we shall resort to a so-called flat-sections approach [22], which is based on the assumption that contact pressure changes more slowly in the longitudinal direction of the contact area than perpendicularly to it. This assumption makes it possible to idealize the contact bodies as an assemblage of narrow cylinders, which are obtained by the subdivision of these bodies through a set of parallel planes. Using this approach a contact problem comprising forces acting in a single plane is solved for each pair of these narrow cylinders. Now the contact force/displacement relationship for each pair of cylinders is written as follows: dF = cu (z)‘0’9dz,
(8)
(7)
F,=
c, ui (z)“‘~ dz s L_
(9)
:+ F,=
c, a/(~)“‘~ dz s Z_ F, = c,u:”
M, =
L+ c,ui(z)‘o’9zdz 5i-
(10) (11)
(12)
2+ M, =
c,u~(z)‘~‘~zdz s L_ M, = c,F,z,
(13)
(14)
where c,, c,, c, refer to the characteristics that describe the material properties and the geometry of the contact surfaces of the taper-rollers, respectively, u,(z) is a relative displacement of the outside ring to the taper-roller, u,(z) is a relative displacement of the inside ring to the taper-roller, u,(z) is a relative displacement of the taper-roller to the edge of the ring, and z+ and z - are the co-ordinates which describe the contact area between the taper-roller and the rings. The values of internal forces are numerically established by solving a set of non-linear equilibrium equations (6). A linearized form of eqn (6) must be solved by iterations until the force imbalance is reduced to below or within a fixed tolerance. Finally, the entries of the internal stiffness matrix k, are computed by numerical differentiation of eqn (5). SOLUTION PROCEDURE
Fig. 5. Forces acting on the roller.
The solution procedure of the macroelement approach used is basically a three-step technique, and consists of solving a group of subproblems, each having an individual subdomain with a set of DOF, data and individual equations to be solved. The first step involves finite element discretization of two
R. BauSys and R. Kacianauskas
252
cylinders and computation of the macroelement stiffness matrices as well as the load vectors, by forward substitution procedures of the substructuring technique. It should be recognized that the substructuring technique is in fact Gauss elimination. Therefore, substructuring computations involve the whole range of all necessary procedures used in the linear analysis. They are the initialization of substructure data, generation of stiffness matrices for individual elements, assemblage of substructure stiffness block-matrices and load vectors, as well as solution of a set of linear equations. The second and the most important step is the non-linear analysis of a global structure. An incremental-iterative approach is used to solve the non-linear equation (1). For each step of the control parameter (load factor) a modified Newton-Raphson method is applied as an iterative strategy. The initialization of data for the global model and the assembly of the linear part of the stiffness matrix are carried out in the non-repeatable part, while the other procedures, such as the generation of non-linear stiffness matrices for roller type macroelements, the final assemblage of the global non-linear stiffness matrix of the structure, its factorization and checking of contact conditions of rollers as well as the NewtonRaphson iterations, are implemented in the cyclic running of the iteration process. The third step is devoted to the implementation of the backward substitution procedure of substructuring and involves the calculation of the internal displacements and stresses applied for both of the cylinders separately. The solution procedure mentioned above is implemented by the modular program package BEMS adapted for PCs. Through successive refinement, a computation process at a certain level is broken down into less complex sub-processes (modules). According to these concepts, the software for FE analysis is designed as a hierarchical tree-structure program package composed of separate modules. Modules share a common database and software tools written in the form of utility subroutines. NUMERICAL
LJ.L._
0
0.1
0.2
0.3
0.4
0.1
0.0
~- Rlgld 6Iom6nt + Bmrlng
Fig. 6. History
0.7
0.6
0.0
1
.I6mml
of characteristic displacement cylinder.
of the outside
bearing connection (see Fig. 3) is defined by the following values: a = 35 mm, b = 40 mm, c = 34 mm, e =4mm, s =42.5mm, d,= 18mm, Z,=27.5mm and CI= 15”. The isotropic material of shells is characterized by the following material data: E = 21,000 MPa and v = 0.3. The external loading reaches its final value F = 2.5 kN in 10 time steps. In order to avoid the disjunction of the taper-rollers, they are initially prestressed. In Figure 6 the characteristic displacement u at the point (x = 0, y = R,, z = 0.8 L,) of the outside cylindrical shell is reported as a function of the load factor J The figure shows clearly that the displacement under consideration in both examples varies almost linearly and the flexibility of bearings causes consistently (slightly) higher values in comparison with a rigid element connection. The influence of the type of the connection element on the displacements of internal cylinder is negligible. Figures 7-10 present the distribution of stress intensities c of both cylinders with the values of the load factor f = 0.5 and f = 1 .O. The stress intensities in the end cross-section (x = 0, y = R) of the cylinders in the longitudinal direction are defined as the functions of a normalized co-ordinate z/L, and are
EXAMPLE
Numerical examples are considered in order to demonstrate the technique proposed and to study the behaviour of the structure highlighted previously. A structure with a taper-roller bearing connection is solved in the first example. For the sake of comparison the second example is selected to analyse the same structure, in which rigid connection elements are used. The cylindrical shells used in the calculations have the following dimensions: the outside cylinder has the length L, = 160 mm and the radius of the middle surface R, = 80 mm, while the internal cylinder’s length Li = 200 mm and the radius of the middle surface Ri = 35. The thickness of both cylinders is taken as h = 10 mm. The geometry of the taper-roller
’
40 20,;;
-- RIgId 61. f = 1.0
i
866hlg
e1. f = 1.0
*
* b6d1’~
al. i = 0.6
FOgld 61. I = 0.s
Fig. 7. Longitudinal distribution of stress intensities inner surface of the outside cylinder.
on the
Analysis of cylindrical shells
ter. This trend is further verified for stress intensities in the characteristic radial cross-section of the cylinders (z = 0.8 L,). The distribution of c in the angular direction 9 is plotted in Figs 9 and 10. Numerical investigations enable us to conclude that the consideration of contact deformations may influence the distribution of displacements and stresses of cylinders, especially at higher loading levels.
60
207-
,l 0
0.1
_--_-_-
0.2
0.3
‘RIgkid. *
Rlgld
al.
0.4
0.5
LL__
0.6
fL1.0
+Burlng.l.
f = 0.6
*
0.7
Iz/L
2___
0.6
0.6
1
111.0
BewIng
.I.
f = 0.6
I,
Fig. 8. Longitudinal distribution of stress intensities on the outer surface of the inside cylinder. 0
10 0
253
YPrn
L 0
20
40
60
60
100
120
140
160
-‘RIgkId.
t-1.0
+~BearlnQel.
1=1.0
*
t = 0.6
*
I = 0.5
Rigid
Fig. 9. Radial
.I.
Boaring
@I.
160
distribution of stress intensities surface of the outside cylinder.
on the inner
_
__.I__L
20
40
--RIgkid. *
RIgId
Fig. 10. Radial
6g
60
f -
120
+l3.wing*I.
111.0 .I.
1Do
0.6
*
marirng
140
16g
1
160
79
f = 1.0 41.
f = 0.6
distribution of stress intensities surface of the inside cylinder.
The macroelement approach to the analysis of cylindrical shells connected by the taper-roller bearings is applied. A special spring-type macroelement for the description of an individual roller is proposed and the stiffness matrix of this element is derived. Since the macroelement has been formulated in the framework of a simplified engineering theory, it cannot capture the accurate prediction of local response characteristics. The model capability is illustrated by the solution of numerical examples. The proposed model allows study of the character of the bearing flexibility, which is compared with that of a rigid spring connection. The approach considered is expected to be very economical compared to the detailed discretization techniques. Thus, the most desirable area of applications is where the analysis of complex structures with the taper-roller bearing connections is carried out. REFERENCES
plotted in Figures 7 and 8. The values for the taper-roller bearing connection are compared with those obtained with a rigid element connection. It is important to realize that stresses increase in the outside cylinder and decrease in the inside cylinder when the proposed taper-roller connection element is used. The longitudinal stress distribution in other cross-sections of both cylinders is of the same charac-
0
CONCLUSIONS
on the outer
1. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1976). 2. J. Argyris and H.-P. Mlejnek, Die Methode der jnifen Elemente in der elementaren Strukturmechanik, Band I, Verschiebungsmethode in der Statik. Wieweg, Braunschweig/Wiesbaden (1986). The Cholecky 3. D. Hitchings and K. Balasubramamiam, method in substructuring with application to fracture mechanics. Comput. Struct. 18, 417424 (1984). substructuring 4. T.-Y. Han and J. F. Abel, Adaptive techniques in elasto-plastic finite element analysis. Comput. Struct. 20, 181-192 (1985). 5. A. K. Noor, Recent advances in reduction methods for non-linear problems. Comput. Struct. 13, 31-44 (1981). 6. A. K. Noor, On making large non-linear problem small. Comput. Meth. appl. mech. Engng 34, 955-985 (1982). 7. Ch. U. A. Bucher, A modal synthesis method employing physical co-ordinates, free component modes and residual flexibilities. Comput. Struct. 22, 559-564 (1986). non-linear 8. D. A. Nagy and M. Konig. Geometrically finite element behaviour using buckling mode superposition. Comput. Meth. appl. mech. Engng 19, 447484 (1979). 9. A. K. Noor, J. M. Peters and C. M. Andersen, Mixed models and reduction techniques for large-rotation nonlinear problems. Comput. Meth. appl. mech. Engng 44, 67-89 (1984). 10. A. K. Noor, C. D. Balch and M. A. Shibut, Reduction methods for non-linear steady-state thermal analysis. Inc. J. numer. Meth. engng 20, 1323-1348 (1984). 11. S. Utku and J. Clemente and M. Salama, Errors in reduction methods. Compur. Struct. 21, 1153-I 157 (1985).
254
R. BauSys and R. Kacianauskas
12. J. C. Taig, Implicit modelling and mesh refinement for complex structures. Int. J. numer. Meth. Engng 22, 17-38 (1986). 13. H. V. S. G. Rao and H. J. Farran, Macroelement analysis of skewed and triangular orthotropic thin plates with beam boundaries. Compuf. Struct. 22, 399404 (1986). 14. F. W. Williams and M. S. Anderson, Inclusion of elastically connected members in exact buckling and frequency calculations. Comput. Strucl. 22, 395-397 (1986). 15. F. Arbabi and F. Li, Macroelements for variable section beams. Cornput. Strucf. 37, 5533559 (1990). 16. R. Kacianauskas. The integral characteristics of complex macroelements for linear equilibrium equations. Bull. Lith. Insl. Sci. In$ N2153-Li I-18 (1988) (in Russian).
17. M. J. Hartnett, The analysis of contact stresses in rolling element of bearings, Lubric. Technol. ASME 101, 105-109 (1979). 18. R. Kacianauskas, Basic modelling principles of complex engineering structures. Bull. Lilh, Inst. Sci. Inj N 1941Li I-50 (1988) (in Russian). 19. R. D. Beizelman, B. V. Cypkin and L. J. Perel, Rolling Bearings. Mashinostroenie, Moscow (1975) (in Russian). 20. Die Merhode der finiten Elemente in der FestkorDer Mechanik (Edited by J. Altenbach and A. S. Sachardv). VEB Fachrichtunn. Leiozie (1982). 21. H. A. Balmer, General’beim for’ASKA. ISD Report No. 75, Stuttgart (1969). 22. V. M. Berezinskij, Influence of misalignment of taperroller bearings on fatigue life. Trans. Bearing Industry Insr. 2(104), 63-79 (1980) (in Russian).
Computers & S/rucrurrs Vol. 54. No. 2, pp. 247-254. 1995 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 S9.50 + 0.00
0045-7949(94)00325-4
MACROELEMENT APPROACH TO THE ANALYSIS OF CYLINDRICAL SHELLS CONNECTED BY TAPER-ROLLER BEARINGS R. Badys and R. Katianauskas Department
of Computer
Assisted
Engineering,
Vilnius
Technical
University,
Vilnius,
Lithuania
(Received 18 October 1993) Abstract-The application of the macroelement technique to non-linear analysis of cylindrical shells connected by taper-roller bearings is presented. Two types of macroelements are proposed. The linear macroelement is based on substructuring technique and describes the shells, while a non-linear one refers
to the taper-rollers. Evaluation of the stiffness matrix of the latter is discussed in detail. The results obtained by solving a numerical example are given to illustrate the model capability. The influence of connection
flexibility
on displacements
and stresses on cylinders
is also studied
and expressed
numerically.
equations representing an approximate model of the complete structure, This technique described by Bathe and Wilson [I] and Argyris and Mlejnek [2] is suitable for solving linear problems, though Refs [3,4] present additional examples of its application to localized nonlinearities. In this case, the part of the structure remains linear while the other part can be treated as non-linear. During the incremental analysis of the complete structure only the stiffness of the non-linear substructures needs to be modified according to stress change. The technique for reducing the DOF, expressed in terms of the generalized variables so that the resulting number is considerably smaller than that of the DOF of the initial discretization, is referred to herein as the reduction technique. The essence of the reduction method is described in Refs [5-l I], where the deformations of the structure are limited by certain modes expressed in terms of basic vectors (physical co-ordinates). Various alternatives for the basic vectors were proposed including Rayleigh-Ritz approximation functions, path derivatives, eigenfreqnency modes, linear buckling modes, etc. Different global discretization methods widely used in the analysis of various structures may be defined as the macroelement technique [12-161, whereby several examples are considered. Basically, the structural members such as beams, plates or shells as well as various connections, supports, stiffeners, holes and other concentrators and/or locations of non-linearity may be considered as macroelements. In the present paper, attention will be focused on the computational strategies for the modelling of cylindrical shells connected by taper-roller bearings. By applying three-dimensional elements and using a rather fine finite element mesh, the taper-roller can be modelled very accurately. However, in practice, the
INTRODUCTION
The accurate numerical simulation of complex engineering structures remains one of the most challenging problems of computational mechanics. Prediction of the response characteristics of a complex structure is very time consuming; so much research has been devoted to the study of solution methodologies which reduce the computational efforts of solving large structural problems. It is a well-recognized fact that a large number of degrees of freedom (DOF) in conventional finite element discretization of a largescaled structure is often dictated by its topology rather than by the expected complexity of the behaviour. The simplest and widely used approach to the analysis of complex structures is to introduce a smaller number of variables (sometimes new generalized variables) to represent a large number of DOF of initial discretization of a structure, thereby reducing the size of the stiffness matrix involved in the calculations while retaining high accuracy. The brief review of various computational strategies presented here focuses on the following techniques: (a) substructuring technique; (b) reduction technique; (c) macroelement technique. The substructuring technique, treating the total structure as an assemblage of substructures, is based on the idea of reducing the number of DOF prior to the element assemblage process. The reduction is not global but is performed only at the substructure level, while the compatibility of displacements and forces at the substructure interface generates the global set of 247
248
R. BauSys and R. KaEianauskas
above analysis of the taper-roller involves the use of about a thousand finite-element equilibrium equations [17]. It means that the non-linear analysis even of a single taper-roller is rather expensive. A macroelement approach involving the essential features of various discretization techniques is proposed. It is based on earlier research performed with the use of a multi-level approach, documented in Ref. [ 181. The proposed methodology assumes a complex structure to be separated into smaller macroelements which may be described by the structural matrices equivalent to the conventional finite element matrices defined by different numerical techniques. In order to simplify the modelling of the taper-roller bearings, a specific macroelement based on the assumptions of a simplified engineering theory of the bearings [19] is derived. The proposed model has serious limitations with regard to the accuracy of predicting local response characteristics such as stresses and strains of a taper-roller, but it is quite suitable for predicting the global response characteristics, i.e. the displacements and the stresses of cylinders. PROBLEM DESCRIPTION
A representative structure consisting of two circular cylindrical shells is investigated. The geometry of the inside cylinder is defined by the radius of the middle surface R, and the length L,, while the geometry of the outside cylinder is determined by the parameters R,, and L,, respectively. Both of the cylinders have constant thickness h and constant material properties, such as Young’s modulus E and Poisson’s ratio v. The outside cylinder is subjected to
a set of transverse concentrated loads F. Due to the symmetry of the geometry, the loading and the boundary conditions, only half of the structure shown in Fig. 1 is considered. The inside cylinder is clamped in the plane z = L,. The cylinders are connected by the taper-roller bearings, also shown schematically in this figure. The bearing connection consists of two (inside and outside) rings and several rollers. The cross-section of the structure with a bearing is illustrated in Fig. 2, while the detailed geometry and the most important parameters of an individual conical roller are shown in Fig. 3. Numerical analysis of the structure is based on the following assumptions: (1) the behaviour of the materials of which the cylinders and the taper-roller bearings are composed is linear-elastic; (2) the loading is of a static, monotonically increasing character, linearly depending on time t; (3) structural elements are assumed to be geometritally linear and all of the characteristics are referred to the undeformed state of structure. In this paper, the modelling of taper-roller bearings is based on engineering theory, details of which may be found in Ref. [19]. The basic assumptions are as follows: (1) all of the sliding and frictional neglected; (2) the contact surface is continuous; (3) the length of the initial contact bearing elements is equal to the length roller.
effects
are
surfaces of of a taper-
Analysis of cylindrical shells
249
t
_----__-------_-_--
Ri
Fig. 2. Connection
of cylinders
Within the framework of the aforementioned modelling strategy, two types of macroelements are used in the analysis: (a) macroelements
(b) macroelements
of cylinders; of taper-rollers.
Hence, the formulation of the macroelements is based on two different approaches. For the formulation of a cylinder macroelement a standard linear substructuring technique is used. This macroelement describes the linear mechanical behaviour of one cylindrical shell and two rings of the bearing connection. The linear four-node thick shell element with 24 DOF proposed by Sacharov [20], which is similar to a thin shell element, is used in shell discretization. The thick shell element is derived on the basis of more soft
I-
1
III-
-I
=
b
s
by taper-roller
bearings.
assumptions than that usually provided by thin shell theory, and shows a good convergence by decreasing the size of mesh. The finite element model of the structure under consideration is shown in Fig. 4. The rectangular finite element meshes for both cylinders have a regular division in the direction of the circumstantial co-ordinate and a non-regular division of the longitudinal co-ordinate, where the appropriate lines in the mesh of both cylinders are shifted one to another by the distance e shown in Fig. 3. The nodes of such meshes must coincide with the connection points of the rollers of the bearings. The rings of the rollerare described by special bearing connection eccentrically aided beam elements, having a nonsymmetric cross-section. After the condensation of internal DOF, the macro model of a complex ROLLER
MACROELEMENTS
I I
-I
Fig. 3. Geometry of individual roller.
Fig. 4. Discretization
scheme.
250
R. BauSys and R. Kacianauskas
structure consists of 2n, which is the number of rollers in the bearing. A non-linear macroelement connecting two cylinders is proposed to describe the taper-roller. This element reflects the ring-roller contact deformations and allows for two types of non-linearity. The first one reflects non-linear behaviour of contact deformations, while the second indicates the unilateral character of a bearing when the disconnection of cylinders occurs. This one-dimensional element involving basic features of reduction and macroelement techniques behaves as a simple spring, for which a natural beam element proposed in Ref. [2] may be used as an explicit mechanical model. The macroelement approach is based on the idea of multilevel discretization, where the global model consists of both linear and non-linear macroelements. The linear cylinder-type macroelements reduce the linear region and the size of the problem by numerical computations on the macroelement level only, while the suggested spring-type elements additionally simplify the model relations, avoiding the solution of the complex contact problem. The global relationship is transformed into a relatively small non-linear problem, retaining generalized DOF in which only the simplest one-dimensional non-linear elements will need to be considered. The solution of a non-linear problem requires an iterative/incremental procedure with a number of loading steps, finally reaching the total applied load. The basic approach in an incremental step-by-step algorithm is to assume that the solution of a nonlinear problem for the discrete time t + AZ is required, where At is a time increment. As a result of some manipulations, the solution of the non-linear problem is approximated by the solution of the non-linear equations
KT(u(t )) Au(At) = AF(At),
K, = A,k,Af, where K, and k, are the global (Cartesian) and internal (natural) stiffness matrices and A, is the transformation matrix relating the natural forces/ displacements to the global forces/displacements. The deformed state of the taper-roller is described by the displacements of points i, andj, (Fig. 3). These points are the middle points of the initial contact surfaces of the taper-roller. We consider only the deformations of the i-j element, while the joint elements i-i, and j-i, are taken to be rigid. The transformation is performed in three steps:
A, = L,H,T,,
where matrix T, transforms the internal forces into external ones, matrix H, describes the effects of eccentricities of a deformable element i-j,, and matrix L, refers to the rotation from the local coordinate system to the global co-ordinate system. For details of the construction of these transformation matrices one should refer to Ref. [21]. In a general three-dimensional case, the beam element has six internal DOF related to six deformation modes, but in our case the proposed taperroller element has only two natural DOF, rotation and translation in a direction perpendicular to the contact surface. So if the following vectors of internal forces and displacements are introduced as
the internal stiffness matrix k, = k,(u,), which is the tangential stiffness matrix of non-linear deformation process (1) accordingly takes the form
(2)
The exact displacements u(t + At) at time t + At correspond to the applied load F(t ) + AF(At ). Having the considered approximation of displacements, we can obtain the approximation of the stresses and corresponding nodal forces at time t + At.
The entries of the internal stiffness matrix may be obtained by differentiating the nodal force vector f, with respect to the nodal displacement vector II, and may be expressed as
k,,,,,(f)+
7
rm u,r, = L,,
MODELLING
(4)
(1)
where KT(u(t )) is the tangent stiffness matrix, which corresponds to the equilibrium conditions at time t, while the vector Au(At ) is the incremental nodal point displacement vector. The displacements at the time t + At can be calculated by an approximation: u(r + At ) = u(t ) + Au(At ).
element with 12 DOF [21], and reflects the unilateral stiffness characteristics of the taper-roller. A general expression of the stiffness matrix of the proposed element denoted by r may be written in the matrix form as
(m,n = IA
(5)
OF TAPER-ROLLERS
As stated above, the taper-rollers of a bearing are described by a special spring-type macroelement, which is based on the model of a general beam
where uf is the value of displacement at the appropriate loading level. In order to obtain the numerical characteristics of this stiffness matrix for element Y, it is necessary to
Analysis of cylindrical shells consider the equilibrium state of the taper-roller. Within the framework of the theory of simplified engineering analysis, calculations of static equivalent contact forces with respect to the displacements of the taper-roller and the rings of a bearing consist of solving a set of governing equilibrium equations. The directions of acting forces and moments are shown in Fig. 5. In constructing the stiffness matrix, the response characteristics of point j are considered as internal variables, thus F s F,, M 3 M, and u z ui, 4 = $,, respectively. The equilibrium state of a taperroller is described by the following set of equations: F;cos
cp;-~coscp,+F,coscp,=O,
6 sin cp, - F, sin ‘p, + F, sin cp, = 0,
(6)
251
where dF is the contact force, c refers to the characteristics that include the effects of the contact surface geometry and material response and u(z) is a relative displacement of the narrow cylinder of a taper-roller to the narrow cylinder of the ring. The relationship (7) is based on a simplified theory of engineering analysis [19] and is obtained with respect to the function of relative displacement which can be expressed as follows: u(z)=u,+4z.
Here u is the relative displacement of the middle point of the taper-roller to the ring and Q, refers to the relative rotation of the taper-roller to the ring. Therefore, the expressions of all acting contact forces and moments will take the following form
-M,+M,+M,=O.
i+
To obtain these equations the constitutive relationships between contact forces and displacements that are acting in the contact areas between the taperroller and rings should be known. Taking into account the assumption made for small contact areas, aforementioned relationships can be obtained in the framework of the Hertz theory. Then, this contact area has an oblong form, while the length of the contact area would be of the same order as the length of the taper-roller. In deriving the formulae for a governing relationship of contact force/displacement in the taper-roller element, we shall resort to a so-called flat-sections approach [22], which is based on the assumption that contact pressure changes more slowly in the longitudinal direction of the contact area than perpendicularly to it. This assumption makes it possible to idealize the contact bodies as an assemblage of narrow cylinders, which are obtained by the subdivision of these bodies through a set of parallel planes. Using this approach a contact problem comprising forces acting in a single plane is solved for each pair of these narrow cylinders. Now the contact force/displacement relationship for each pair of cylinders is written as follows: dF = cu (z)‘0’9dz,
(8)
(7)
F,=
c, ui (z)“‘~ dz s L_
(9)
:+ F,=
c, a/(~)“‘~ dz s Z_ F, = c,u:”
M, =
L+ c,ui(z)‘o’9zdz 5i-
(10) (11)
(12)
2+ M, =
c,u~(z)‘~‘~zdz s L_ M, = c,F,z,
(13)
(14)
where c,, c,, c, refer to the characteristics that describe the material properties and the geometry of the contact surfaces of the taper-rollers, respectively, u,(z) is a relative displacement of the outside ring to the taper-roller, u,(z) is a relative displacement of the inside ring to the taper-roller, u,(z) is a relative displacement of the taper-roller to the edge of the ring, and z+ and z - are the co-ordinates which describe the contact area between the taper-roller and the rings. The values of internal forces are numerically established by solving a set of non-linear equilibrium equations (6). A linearized form of eqn (6) must be solved by iterations until the force imbalance is reduced to below or within a fixed tolerance. Finally, the entries of the internal stiffness matrix k, are computed by numerical differentiation of eqn (5). SOLUTION PROCEDURE
Fig. 5. Forces acting on the roller.
The solution procedure of the macroelement approach used is basically a three-step technique, and consists of solving a group of subproblems, each having an individual subdomain with a set of DOF, data and individual equations to be solved. The first step involves finite element discretization of two
R. BauSys and R. Kacianauskas
252
cylinders and computation of the macroelement stiffness matrices as well as the load vectors, by forward substitution procedures of the substructuring technique. It should be recognized that the substructuring technique is in fact Gauss elimination. Therefore, substructuring computations involve the whole range of all necessary procedures used in the linear analysis. They are the initialization of substructure data, generation of stiffness matrices for individual elements, assemblage of substructure stiffness block-matrices and load vectors, as well as solution of a set of linear equations. The second and the most important step is the non-linear analysis of a global structure. An incremental-iterative approach is used to solve the non-linear equation (1). For each step of the control parameter (load factor) a modified Newton-Raphson method is applied as an iterative strategy. The initialization of data for the global model and the assembly of the linear part of the stiffness matrix are carried out in the non-repeatable part, while the other procedures, such as the generation of non-linear stiffness matrices for roller type macroelements, the final assemblage of the global non-linear stiffness matrix of the structure, its factorization and checking of contact conditions of rollers as well as the NewtonRaphson iterations, are implemented in the cyclic running of the iteration process. The third step is devoted to the implementation of the backward substitution procedure of substructuring and involves the calculation of the internal displacements and stresses applied for both of the cylinders separately. The solution procedure mentioned above is implemented by the modular program package BEMS adapted for PCs. Through successive refinement, a computation process at a certain level is broken down into less complex sub-processes (modules). According to these concepts, the software for FE analysis is designed as a hierarchical tree-structure program package composed of separate modules. Modules share a common database and software tools written in the form of utility subroutines. NUMERICAL
LJ.L._
0
0.1
0.2
0.3
0.4
0.1
0.0
~- Rlgld 6Iom6nt + Bmrlng
Fig. 6. History
0.7
0.6
0.0
1
.I6mml
of characteristic displacement cylinder.
of the outside
bearing connection (see Fig. 3) is defined by the following values: a = 35 mm, b = 40 mm, c = 34 mm, e =4mm, s =42.5mm, d,= 18mm, Z,=27.5mm and CI= 15”. The isotropic material of shells is characterized by the following material data: E = 21,000 MPa and v = 0.3. The external loading reaches its final value F = 2.5 kN in 10 time steps. In order to avoid the disjunction of the taper-rollers, they are initially prestressed. In Figure 6 the characteristic displacement u at the point (x = 0, y = R,, z = 0.8 L,) of the outside cylindrical shell is reported as a function of the load factor J The figure shows clearly that the displacement under consideration in both examples varies almost linearly and the flexibility of bearings causes consistently (slightly) higher values in comparison with a rigid element connection. The influence of the type of the connection element on the displacements of internal cylinder is negligible. Figures 7-10 present the distribution of stress intensities c of both cylinders with the values of the load factor f = 0.5 and f = 1 .O. The stress intensities in the end cross-section (x = 0, y = R) of the cylinders in the longitudinal direction are defined as the functions of a normalized co-ordinate z/L, and are
EXAMPLE
Numerical examples are considered in order to demonstrate the technique proposed and to study the behaviour of the structure highlighted previously. A structure with a taper-roller bearing connection is solved in the first example. For the sake of comparison the second example is selected to analyse the same structure, in which rigid connection elements are used. The cylindrical shells used in the calculations have the following dimensions: the outside cylinder has the length L, = 160 mm and the radius of the middle surface R, = 80 mm, while the internal cylinder’s length Li = 200 mm and the radius of the middle surface Ri = 35. The thickness of both cylinders is taken as h = 10 mm. The geometry of the taper-roller
’
40 20,;;
-- RIgId 61. f = 1.0
i
866hlg
e1. f = 1.0
*
* b6d1’~
al. i = 0.6
FOgld 61. I = 0.s
Fig. 7. Longitudinal distribution of stress intensities inner surface of the outside cylinder.
on the
Analysis of cylindrical shells
ter. This trend is further verified for stress intensities in the characteristic radial cross-section of the cylinders (z = 0.8 L,). The distribution of c in the angular direction 9 is plotted in Figs 9 and 10. Numerical investigations enable us to conclude that the consideration of contact deformations may influence the distribution of displacements and stresses of cylinders, especially at higher loading levels.
60
207-
,l 0
0.1
_--_-_-
0.2
0.3
‘RIgkid. *
Rlgld
al.
0.4
0.5
LL__
0.6
fL1.0
+Burlng.l.
f = 0.6
*
0.7
Iz/L
2___
0.6
0.6
1
111.0
BewIng
.I.
f = 0.6
I,
Fig. 8. Longitudinal distribution of stress intensities on the outer surface of the inside cylinder. 0
10 0
253
YPrn
L 0
20
40
60
60
100
120
140
160
-‘RIgkId.
t-1.0
+~BearlnQel.
1=1.0
*
t = 0.6
*
I = 0.5
Rigid
Fig. 9. Radial
.I.
Boaring
@I.
160
distribution of stress intensities surface of the outside cylinder.
on the inner
_
__.I__L
20
40
--RIgkid. *
RIgId
Fig. 10. Radial
6g
60
f -
120
+l3.wing*I.
111.0 .I.
1Do
0.6
*
marirng
140
16g
1
160
79
f = 1.0 41.
f = 0.6
distribution of stress intensities surface of the inside cylinder.
The macroelement approach to the analysis of cylindrical shells connected by the taper-roller bearings is applied. A special spring-type macroelement for the description of an individual roller is proposed and the stiffness matrix of this element is derived. Since the macroelement has been formulated in the framework of a simplified engineering theory, it cannot capture the accurate prediction of local response characteristics. The model capability is illustrated by the solution of numerical examples. The proposed model allows study of the character of the bearing flexibility, which is compared with that of a rigid spring connection. The approach considered is expected to be very economical compared to the detailed discretization techniques. Thus, the most desirable area of applications is where the analysis of complex structures with the taper-roller bearing connections is carried out. REFERENCES
plotted in Figures 7 and 8. The values for the taper-roller bearing connection are compared with those obtained with a rigid element connection. It is important to realize that stresses increase in the outside cylinder and decrease in the inside cylinder when the proposed taper-roller connection element is used. The longitudinal stress distribution in other cross-sections of both cylinders is of the same charac-
0
CONCLUSIONS
on the outer
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R. BauSys and R. Kacianauskas
12. J. C. Taig, Implicit modelling and mesh refinement for complex structures. Int. J. numer. Meth. Engng 22, 17-38 (1986). 13. H. V. S. G. Rao and H. J. Farran, Macroelement analysis of skewed and triangular orthotropic thin plates with beam boundaries. Compuf. Struct. 22, 399404 (1986). 14. F. W. Williams and M. S. Anderson, Inclusion of elastically connected members in exact buckling and frequency calculations. Comput. Strucl. 22, 395-397 (1986). 15. F. Arbabi and F. Li, Macroelements for variable section beams. Cornput. Strucf. 37, 5533559 (1990). 16. R. Kacianauskas. The integral characteristics of complex macroelements for linear equilibrium equations. Bull. Lith. Insl. Sci. In$ N2153-Li I-18 (1988) (in Russian).
17. M. J. Hartnett, The analysis of contact stresses in rolling element of bearings, Lubric. Technol. ASME 101, 105-109 (1979). 18. R. Kacianauskas, Basic modelling principles of complex engineering structures. Bull. Lilh, Inst. Sci. Inj N 1941Li I-50 (1988) (in Russian). 19. R. D. Beizelman, B. V. Cypkin and L. J. Perel, Rolling Bearings. Mashinostroenie, Moscow (1975) (in Russian). 20. Die Merhode der finiten Elemente in der FestkorDer Mechanik (Edited by J. Altenbach and A. S. Sachardv). VEB Fachrichtunn. Leiozie (1982). 21. H. A. Balmer, General’beim for’ASKA. ISD Report No. 75, Stuttgart (1969). 22. V. M. Berezinskij, Influence of misalignment of taperroller bearings on fatigue life. Trans. Bearing Industry Insr. 2(104), 63-79 (1980) (in Russian).