Finite Element Analysis ofCompression of Thin, HighModulus, Cylindrical Shells withLow-Modulus Core

Robert S. JosephDesign Engineering Analysis Corporation, McMurray, PA

ABSTRACTLong, cylindrical shells, of high modulus polymer with lowmodulus elastomeric core, rest horizontally on the rigidbottom of a groove with rigid side walls. At both sides, gapsranging from zero to approximately the dimension of theshell thickness are allowed. Shell and core are assumed toobey Hooke’s law. A uniformly distributed axial downwardacting load is applied to the top boundary. The system ismodelled using the ANSYS finite element program, Revi-sion 5.0. The applied vertical load serves as theindependent variable. Dependent variables include the topshell boundary reactions (loads and total deformation),reaction at the side of the shell (load), and maximum vonMises stresses and strains. Results can be reported nu-merically and graphically. The analytical model isdescribed briefly and its application is illustrated by threeexamples. Purpose of this work is to provide parametrictrend data for estimating mechanical response ofAMPLIFLEX connector elements in reference 1.

1. INTRODUCTIONThe behavior of certain elements of the AMPLIFLEX ®

connector was to be studied by the following model.l Cylin-drical shells consisting of polyimide foil, an organicpolymer with relatively high modulus of elasticity, enclose acore of low modulus silicone rubber. The shells are as-sumed to be of infinite length, and their cross sections canbe circular, oval or polygonal. They rest in a horizontalgroove with rigid bottom and side walls as shown schemati-cally in Figure 1. Between the sides of the shells and the

side walls of the groove a gap of finite width can exist. Atthe top, uniformly distributed parallel to the long axis ofthe shell, a load is applied in a vertical, downward direc-tion. The response to this load, in particular deformationsat the top and reactive loads at the top and the sides of theshells, are of interest.

To avoid time consuming experimental studies requiringpreparation of parts with different shapes and dimensions,the problem was to be modelled mathematically. Numeri-cal analysis of mechanical systems has served designengineers in finding optimal solutions for a long time. Usu-ally, the system under consideration is described by a set ofhigher order, nonlinear, partial differential equations andboundary conditions specific to the system. Exact, closedsolutions of these problems are generally not possible.Approximations were and still are developed by simplify-ing, sometimes drastically, the original mathematicalformulations. For a given system the degree of success ofthis approach depends largely on the ingenuity of the ana-lyst. If closed, exactor approximate solutions are notrequired, the original problem can be rewritten in form ofdifference equations. Using digital computers and observ-ing the pertinent, system specific precautions, the rewrittenproblem can then be solved with reasonable effort by con-ventional methods.2,3,4,5 For many of today’s applicationseven these approaches are unsatisfactory.

Difficulties encountered with these earlier conventionalprocedures led to the development of the finite elementmethod (FEM). An early, fundamental discussion of its

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16 R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

Figure 1. Cross section and schematic of support of one ofthe examples analyzed. Infinite length of the cylinder wasassumed. Quantities are measured in conventional U.S.units. Subscript o indicates outside dimensions of the shell, cof the core.ho

o

o o•

•

• •o o

= height of the shell,w = width of the shell,hc

c

c c

c

= height of the core,w = width of the core,r = 0.5 w = radius of curvature at top and bottom of theoutside,r = 0.5 w = radius of curvature at top and bottom of theinside,t = 0.5 (h – h ) = 0.5   (W – wc) = shell thickness,g = physical gap between sidewalls of shell and rigid sup-port,P = applied external load in lbs/in.

2. THE SYSTEMFigure 1 shows the cross section of one of the examplesused in the study. Their symmetry and the assumption ofinfinite length of the cylinders simplify the proceduregreatly. Three cases termed B

and in Reference 1, the material nonlinearities (viscoelas-ticity, viscoplasticity, and hyperelasticity with the Mooney-Rivlin strain energy function) are available in ANSYS

approximations.should it become necessary to include these

0, Bl, and C where selected.They represent combinations of different geometries andboundary conditions:

B0 shell with circular cross section, rigid support at bot-tom, rigid support at both sides, load applied at top.

B1 shell with circular cross section, rigid support at bot-tom, gap between side walls of shell and support at bothsides, 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 theexamples, Table 2 the material constants for shell and core.Justification for use of these constants and the linear mate-rials model are given in reference 1. The effect of a finitegap width between the side walls of the supporting struc-ture and the shell is shown for a shell with circular crosssection.

Table 1. Dimensions used in the examples. Infinite length ofthe cylinder was assumed. Definitions of the parameters aregiven in Figure 1.

application to solving a number of non-trivial, specific engi-neering problems is presented for instance by Girault andRaviart. 6 The most recent edition of Eshbach’s Handbookof Engineering Fundamentals contains a concise summaryof FEM, supported by selected examples and a brief bibli-ography.’ One of the most widely used and accepted FEMcodes in the world today is ANSYS®8, introduced nearly 25years ago by Swanson Analysis Systems, Inc.

Table 2. Material constants used in the model. Shell andRevision 5.0 of the ANSYS program provides extensive core are assumed to obey Hooke’s law. Applied externalnonlinear capabilities including geometric nonlinearities, loads were 0.2, 1.0, 2.0, 4.0, 6.0 lb/in.element nonlinearities, and material nonlinearities whichare required to solve contact problems of this type. In thestudy described herein, the geometric nonlinearities (largestrain and large deflection effects) and element nonlineari-ties (contact surface elements with sliding and compressioncapabilities are employed. Although not used in this study

AMP Journal of Technology Vol. 4 June, 1995 R.S. Joseph 17

3. THE FINITE ELEMENT ANALYSISRevision 5.0 of ANSYS is used to model and perform theanalysis of the long cylindrical shells discussed herein. Aone-half axial symmetry model of each geometry is devel-oped using 2-D solid plane strain elements and contactsurfaces. Since the model exhibits reflective symmetryalong the length and the loading is symmetric, a one-halfsymmetry model is only required for the solution. However,for graphical presentation in section 4., the model resultsare reflected so that the full model can be used to view thedisplaced 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 de-scribed in section 2.

Figure 3. Finite element mesh for model C. The model ex-hibits reflective symmetry relative to the vertical, centralplane through its axis.

section 2. are listed in Table 3. The finite element meshesfor model B0 with circular cross section and model C withoval cross section are shown in Figures 2 and 3, respec-tively. The shell is modelled with one layer of 2-Disoparametric elements (PLANE 42) with extra displace-ment shapes, which allow the elements to move moreflexibly. Friction between shells and the rigid supports isassumed to be zero. For the purpose of the exploratorystudy in reference 1, the modelling approximations regard-ing material properties, mesh sizes, friction and “plainstrain” end conditions are satisfactory.

The ANSYS program uses a frontal solver to solve the setof simultaneous equations generated by the FEM. Sincegeometric (large strain and large deflection) and element(gaps) nonlinearities are included in the model, the pro-gram uses Newton-Raphson equilibrium iterations toachieve convergence to a specified tolerance of 0.1%. Thesolution results are saved on the results file and then they

Figure 2. Finite element mesh for model Bo. The model can be conveniently reviewed (scanned, sorted, tabulated,

exhibits reflective symmetry relative to the vertical, central plotted) in the POST1 general postprocessor. A flow chartplane through its axis. illustrating the basic ANSYS concepts used in this analysis

is shown in Figure 4.

18 R.S. Joseph AMP Journal of Technology Vol. 4 June, 1995

4. THE FEM RESULTS placements throughout the cross sections for the three

Tables 3 to 5 give summaries of the FEM results of particu- models. Figure 8 illustrates the von Mises strain distribu-

lar 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 gen-

of 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 Bo: Circularcross section, no gap between shell and side walls of groove.P is the load applied at the top of the shell. a) Reactions attop boundary of shell; P/2 = total nodal contact force at topboundary for 1/2 symmetry model = sum of the terms in thecolumn; = vertical displacement of top of shell.

Table 5a. Summary of computed results for case B1: Circularcross section, gap of 1 mil between shell and side walls ofgroove. P is the load applied at the top of the shell. a) Reac-tions at top boundary of shell; P/2 = total nodal contactforce at top boundary for 1/2 symmetry model = sum of theterms in the column; vertical displacement of top ofshell.

Table 4b. Reactions at side boundary of shell; P side = total Table 5b. Reactions at side boundary of shell; P side = totalnormal load at the side boundary. normal load at the side boundary.

Table 4c. Maximum von Mises stress and strain; = von Table 5c. Maximum von Mises stress and strain; = vonMises 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

=

Figure 4. Flow chart illustrating ANSYS basic concepts.

Table 6a. Summary of computed results for case C: Ovalcross section, no gap between shell and side walls of groove.P is the load applied at the top of the shell. a) Reactions attop boundary of shell; P/2 = total nodal contact force at topboundary for 1/2 symmetry model = sum of the terms in thecolumn; = vertical displacement of top of shell.

20 R.S. Joseph

Table 6b. Reactions at side boundary of shell; Pside = totalnormal load at the side boundary.

Table 6c. Maximum von Mises stress and strain; = vonMises elastic stress; = von Mises elastic strain.

AMP Journal of Technology Vol. 4 June, 1995

Figure 5. Displacement plot for model B0, a) for applied load P = 0.2 lb/in, b) for applied load P = 6.0 lb/in.

Figure 6. Displacement plot for model B1, a) for applied load P = 0.2 lb/in, b) for applied load P = 6.0 lb/in.

22 R.S. Joseph

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.

AMP Journal of Technology Vol. 4 June, 1995

Figure 8. Plots of von Mises strain for model C at applied load P = 6.0 lb/in, a) for the shell, b) for the core.

5. REFERENCES1.

2.

3.

4.

5.

6.

7.

8.

E. W. Deeg, “Mechanics of AMPLIFLEX ConnectorElements,” AMP J. of Technol. 4 (1994), pp 24 to 40.

E. G. Keller and R. E. Doherty, Mathematics of ModernEngineering, Volume I, (Wiley, New York, 1936), 163-188.

H. T. Davis, Introduction to Nonlinear Differential andIntegral Equations, (Dover, New York, 1962), 467-488.

R. W. Hamming, Numerical Methods for Scientists andEngineers, 2nd edition, (McGraw-Hill, New York, 1973).

M. E. Goldstein and W. H. Braun, Advanced Methodsfor the Solution of Differential Equations, (NASA,Washington, D. C., 1973), 320-345.

V. Girault and P.-A. Raviart, Finite Element Approxima-tion of the Navier-Stokes Equations, (Springer, Berlin,1979), 58-86.

J. N. Reddy in Eshbach’s Handbook of Engineering Fun-damentals, 4th ed. edited by B. D. Tapley, (Wiley, NewYork, 1990), 2.145-2.168,2.191.

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 DesignEngineering Analysis Corporation (DEAC), a professionalengineering consulting firm based in McMurray, PA.

Mr. Joseph earned his B.S. in Engineering Mechanics fromPennsylvania State University in 1966 and his M.S. in CivilEngineering from the University of Pittsburgh in 1971. Hestarted his professional career at Westinghouse Astro-nuclear Laboratory where he was employed for sevenyears. There he was responsible for static, dynamic, andstability analysis of various metallic and graphite compo-nents of the NERVA nuclear rocket engine.

During the past 21 years Mr. Joseph has worked as a con-sultant to both, industry and government agencies. He hasbeen extensively involved in the application of finite ele-ment analysis methods to solve a wide variety of complexengineering problems involving static, dynamic, inelastic,large deflection, and heat transfer analyses in many diverseindustries. He has published several technical papers deal-ing with structural dynamics using finite element methodsand has taught short courses on Section VIII, Division 1, ofthe ASME code. Mr. Joseph is a Registered ProfessionalEngineer in Pennsylvania and a member of the AmericanSociety of Mechanical Engineers.

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