Finite Element Analysis of Compression of Thin, High Modulus, Cylindrical Shells with Low-Modulus Core

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1 Finite Element Analysis of Compression of Thin, High Modulus, Cylindrical Shells with Low-Modulus Core Robert S. Joseph Design Engineering Analysis Corporation, McMurray, PA ABSTRACT Long, cylindrical shells, of high modulus polymer with low modulus elastomeric core, rest horizontally on the rigid bottom of a groove with rigid side walls. At both sides, gaps ranging from zero to approximately the dimension of the shell thickness are allowed. Shell and core are assumed to obey Hooke s law. A uniformly distributed axial downward acting load is applied to the top boundary. The system is modelled using the ANSYS finite element program, Revision 5.0. The applied vertical load serves as the independent variable. Dependent variables include the top shell boundary reactions (loads and total deformation), reaction at the side of the shell (load), and maximum von Mises stresses and strains. Results can be reported numerically and graphically. The analytical model is described briefly and its application is illustrated by three examples. Purpose of this work is to provide parametric trend data for estimating mechanical response of AMPLIFLEX connector elements in reference INTRODUCTION The behavior of certain elements of the AMPLIFLEX connector was to be studied by the following model. l Cylindrical shells consisting of polyimide foil, an organic polymer with relatively high modulus of elasticity, enclose a core of low modulus silicone rubber. The shells are assumed to be of infinite length, and their cross sections can be circular, oval or polygonal. They rest in a horizontal groove with rigid bottom and side walls as shown schematically in Figure 1. Between the sides of the shells and the side walls of the groove a gap of finite width can exist. At the top, uniformly distributed parallel to the long axis of the shell, a load is applied in a vertical, downward direction. The response to this load, in particular deformations at the top and reactive loads at the top and the sides of the shells, are of interest. To avoid time consuming experimental studies requiring preparation of parts with different shapes and dimensions, the problem was to be modelled mathematically. Numerical analysis of mechanical systems has served design engineers in finding optimal solutions for a long time. Usually, the system under consideration is described by a set of higher order, nonlinear, partial differential equations and boundary conditions specific to the system. Exact, closed solutions of these problems are generally not possible. Approximations were and still are developed by simplifying, sometimes drastically, the original mathematical formulations. For a given system the degree of success of this approach depends largely on the ingenuity of the analyst. If closed, exactor approximate solutions are not required, the original problem can be rewritten in form of difference equations. Using digital computers and observing the pertinent, system specific precautions, the rewritten problem can then be solved with reasonable effort by conventional methods. 2,3,4,5 For many of today s applications even these approaches are unsatisfactory. Difficulties encountered with these earlier conventional procedures led to the development of the finite element method (FEM). An early, fundamental discussion of its Copyright 2004 by Tyco Electronics Corporation. All rights reserved. 16 R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

2 and in Reference 1, the material nonlinearities (viscoelasticity, viscoplasticity, and hyperelasticity with the Mooney- Rivlin strain energy function) are available in ANSYS should it become necessary to include these approximations. Figure 1. Cross section and schematic of support of one of the examples analyzed. Infinite length of the cylinder was assumed. Quantities are measured in conventional U.S. units. Subscript o indicates outside dimensions of the shell, c of the core. h o = height of the shell, w o = width of the shell, h c = height of the core, w c = width of the core, r o = 0.5 w o = radius of curvature at top and bottom of the outside, r c = 0.5 w c = radius of curvature at top and bottom of the inside, t = 0.5 (h o h c) = 0.5 (W o w c ) = shell thickness, g = physical gap between sidewalls of shell and rigid support, P = applied external load in lbs/in. 2. THE SYSTEM Figure 1 shows the cross section of one of the examples used in the study. Their symmetry and the assumption of infinite length of the cylinders simplify the procedure greatly. Three cases termed B0, B l, and C where selected. They represent combinations of different geometries and boundary conditions: B 0 shell with circular cross section, rigid support at bottom, rigid support at both sides, load applied at top. B 1 shell with circular cross section, rigid support at bottom, gap between side walls of shell and support at both sides, load applied at top. C shell with oval cross section, rigid support at bottom, rigid support at both sides, load applied at top. Table 1 gives dimensions of the elements of each of the examples, Table 2 the material constants for shell and core. Justification for use of these constants and the linear materials model are given in reference 1. The effect of a finite gap width between the side walls of the supporting structure and the shell is shown for a shell with circular cross section. Table 1. Dimensions used in the examples. Infinite length of the cylinder was assumed. Definitions of the parameters are given in Figure 1. application to solving a number of non-trivial, specific engineering problems is presented for instance by Girault and Raviart. 6 The most recent edition of Eshbach s Handbook of Engineering Fundamentals contains a concise summary of FEM, supported by selected examples and a brief bibliography. One of the most widely used and accepted FEM codes in the world today is ANSYS 8, introduced nearly 25 years ago by Swanson Analysis Systems, Inc. Revision 5.0 of the ANSYS program provides extensive nonlinear capabilities including geometric nonlinearities, element nonlinearities, and material nonlinearities which are required to solve contact problems of this type. In the study described herein, the geometric nonlinearities (large strain and large deflection effects) and element nonlinearities (contact surface elements with sliding and compression capabilities are employed. Although not used in this study Table 2. Material constants used in the model. Shell and core are assumed to obey Hooke s law. Applied external loads were 0.2, 1.0, 2.0, 4.0, 6.0 lb/in. AMP Journal of Technology Vol. 4 June, 1995 R.S. Joseph 17

3 3. THE FINITE ELEMENT ANALYSIS Revision 5.0 of ANSYS is used to model and perform the analysis of the long cylindrical shells discussed herein. A one-half axial symmetry model of each geometry is developed using 2-D solid plane strain elements and contact surfaces. Since the model exhibits reflective symmetry along the length and the loading is symmetric, a one-half symmetry model is only required for the solution. However, for graphical presentation in section 4., the model results are reflected so that the full model can be used to view the displaced shape and the stress/strain contours. The AN- SYS elements used to model the system described in Table 3. ANSYS elements used to model the system described in section 2. Figure 3. Finite element mesh for model C. The model exhibits reflective symmetry relative to the vertical, central plane through its axis. section 2. are listed in Table 3. The finite element meshes for model B 0 with circular cross section and model C with oval cross section are shown in Figures 2 and 3, respectively. The shell is modelled with one layer of 2-D isoparametric elements (PLANE 42) with extra displacement shapes, which allow the elements to move more flexibly. Friction between shells and the rigid supports is assumed to be zero. For the purpose of the exploratory study in reference 1, the modelling approximations regarding material properties, mesh sizes, friction and plain strain end conditions are satisfactory. Figure 2. Finite element mesh for model B o. The model exhibits reflective symmetry relative to the vertical, central plane through its axis. The ANSYS program uses a frontal solver to solve the set of simultaneous equations generated by the FEM. Since geometric (large strain and large deflection) and element (gaps) nonlinearities are included in the model, the program uses Newton-Raphson equilibrium iterations to achieve convergence to a specified tolerance of 0.1%. The solution results are saved on the results file and then they can be conveniently reviewed (scanned, sorted, tabulated, plotted) in the POST1 general postprocessor. A flow chart illustrating the basic ANSYS concepts used in this analysis is shown in Figure R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

4 4. THE FEM RESULTS placements throughout the cross sections for the three Tables 3 to 5 give summaries of the FEM results of particumodels. Figure 8 illustrates the von Mises strain distribular interest for the three selected models. For a global view tion in shell and core for model C. In addition to these they can also be represented graphically. Such graphs are more or less arbitrarily selected graphs, others can be genof importance if undesirable distribution of local stresses or erated from the ANSYS POST1 general postprocessor. strains are to be identified. Figures 5 to 7 show the dis- Table 4a. Summary of computed results for case B o : Circular cross section, no gap between shell and side walls of groove. P is the load applied at the top of the shell. a) Reactions at top boundary of shell; P/2 = total nodal contact force at top boundary for 1 /2 symmetry model = sum of the terms in the column; = vertical displacement of top of shell. Table 5a. Summary of computed results for case B 1 : Circular cross section, gap of 1 mil between shell and side walls of groove. P is the load applied at the top of the shell. a) Reactions at top boundary of shell; P/2 = total nodal contact force at top boundary for 1 /2 symmetry model = sum of the terms in the column; = vertical displacement of top of shell. Table 4b. Reactions at side boundary of shell; P side = total normal load at the side boundary. Table 5b. Reactions at side boundary of shell; P side = total normal load at the side boundary. Table 4c. Maximum von Mises stress and strain; = von Table 5c. Maximum von Mises stress and strain; = von Mises elastic stress; = von Mises elastic strain. Mises elastic stress; = von Mises elastic strain. AMP Journal of Technology Vol. 4 June, 1995 R.S. Joseph 19

5 Figure 4. Flow chart illustrating ANSYS basic concepts. Table 6a. Summary of computed results for case C: Oval cross section, no gap between shell and side walls of groove. P is the load applied at the top of the shell. a) Reactions at top boundary of shell; P/2 = total nodal contact force at top boundary for 1 /2 symmetry model = sum of the terms in the column; = vertical displacement of top of shell. Table 6b. Reactions at side boundary of shell; P side = total normal load at the side boundary. Table 6c. Maximum von Mises stress and strain; Mises elastic stress; = von Mises elastic strain. = von 20 R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

6 Figure 5. Displacement plot for model B 0, a) for applied load P = 0.2 lb/in, b) for applied load P = 6.0 lb/in. Figure 6. Displacement plot for model B 1, a) for applied load P = 0.2 lb/in, b) for applied load P = 6.0 lb/in.

7 Figure 7. Displacement plot for model C, a) for applied load P = 0.2 lb/in, b) for applied load P = 6.0 lb/in, c) enlargement of upper portion of Figure 7b. 22 R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

8 Figure 8. Plots of von Mises strain for model C at applied load P = 6.0lb/in, a) for the shell, b) for the core. 5. REFERENCES 1. E. W. Deeg, Mechanics of AMPLIFLEX Connector Elements, AMP J. of Technol. 4 (1994), pp 24 to E. G. Keller and R. E. Doherty, Mathematics of Modern Engineering, Volume I, (Wiley, New York, 1936), H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, (Dover, New York, 1962), R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd edition, (McGraw-Hill, New York, 1973). M. E. Goldstein and W. H. Braun, Advanced Methods for the Solution of Differential Equations, (NASA, Washington, D. C., 1973), V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, (Springer, Berlin, 1979), J. N. Reddy in Eshbach s Handbook of Engineering Fundamentals, 4th ed. edited by B. D. Tapley, (Wiley, New York, 1990), , ANSYS User s Manual for Revision 5.0, vol. I to IV. Developed by Swanson Analysis Systems, Inc., Houston, PA. Robert S. Joseph is President and co-founder of Design Engineering Analysis Corporation (DEAC), a professional engineering consulting firm based in McMurray, PA. Mr. Joseph earned his B.S. in Engineering Mechanics from Pennsylvania State University in 1966 and his M.S. in Civil Engineering from the University of Pittsburgh in He started his professional career at Westinghouse Astronuclear Laboratory where he was employed for seven years. There he was responsible for static, dynamic, and stability analysis of various metallic and graphite components of the NERVA nuclear rocket engine. During the past 21 years Mr. Joseph has worked as a consultant to both, industry and government agencies. He has been extensively involved in the application of finite element analysis methods to solve a wide variety of complex engineering problems involving static, dynamic, inelastic, large deflection, and heat transfer analyses in many diverse industries. He has published several technical papers dealing with structural dynamics using finite element methods and has taught short courses on Section VIII, Division 1, of the ASME code. Mr. Joseph is a Registered Professional Engineer in Pennsylvania and a member of the American Society of Mechanical Engineers. AMP Journal of Technology Vol. 4 June, 1995 R.S. Joseph 23