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Finite Element Analysis ofCompression of Thin, High

Modulus, Cylindrical Shells withLow-Modulus CoreRobert S. Joseph

Design Engineering Analysis Corporation, McMurray, PA

ABSTRACTLong, 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 Hookes law. A uniformly distributed axial downward

acting load is applied to the top boundary. The system is

modelled 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 von

Mises stresses and strains. Results can be reported nu-

merically 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 elemen ts in reference 1.

1. INTRODUCTION

The 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 of

the shell, a load is applied in a vertical, downward direc-

tion. 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. Numeri-

cal analysis of mechanical systems has served design

engineers in finding optimal solutions for a long time. Usu-

ally, the system under consideration is described by a set of

higher order, nonlinear, partial differential equations andboundary conditions specific to the system. Exact, closed

solutions of these problems are generally not possible.

Approximations were and still are developed by simplify-

ing, sometimes drastically, the original mathematical

formulations. For a given system the degree of success of

this approach depends largely on the ingenuity of the ana-

lyst. If closed, exactor approximate solutions are not

required, the original problem can be rewritten in form of

difference equations. Using digital computers and observ-

ing the pertinent, system specific precautions, the rewritten

problem can then be solved with reasonable effort by con-

ventional methods.2,3,4,5 For many of todays 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

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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.

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 the

outside,

r = 0.5 w = radius of curvature at top and bottom of the

inside,

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 SYSTEM

Figure 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 procedure

greatly. 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, B

l, and C where selected.

They represent combinations of different geometries andboundary conditions:

B0shell with circular cross section, rigid support at bot-tom, rigid support at both sides, load applied at top.

B1shell 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 of

the 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 and

Raviart. 6The most recent edition of Eshbachs Handbook

of Engineering Fundamentals contains a concise summary

of FEM, supported by selected examples and a brief bibli-

ography. One of the most widely used and accepted FEM

codes in the world today is ANSYS8, introduced nearly 25

years 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 Hookes 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 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 nonlineari-

ties (contact surface elements with sliding and compression

capabilities are employed. Although not used in this study

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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 devel-

oped 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 resultsare 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 de-

scribed in section 2.

Figure 3. Finite element mesh for model C. The model ex-

hibits reflective symmetry relative to the vertical, central

plane through its axis.

section 2. are listed in Table 3. The finite element meshesfor model B

0with circular cross section and model C with

oval 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 more

flexibly. Friction between shells and the rigid supports is

assumed 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 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 pro-gram uses Newton-Raphson equilibrium iterations toachieve convergence to a specified tolerance of 0.1%. The

solution results are saved on the results file and then they

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

exhibits reflective symmetry relative to the vertical, central plotted) in the POST1 general postprocessor. A flow chart

plane through its axis. illustrating the basic ANSYS concepts used in this analysisis shown in Figure 4.

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

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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. Summ ary of computed results for case Bo: Circularcross sect ion, no gap between shel l and side wal ls 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. Summ ary 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; Pside = total Table 5b. Reactions at side boundary of shell; Pside = totalnorm al load at the side boundary. norm al 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.

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Figure 4. Flow chart illustrating ANSYS basic concepts.

Table 6a. Summ ary of computed results for case C: Ovalcross sect ion, no gap between shel l and side wal ls 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.

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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.

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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.

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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.

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, In trod uc tio n to Non lin ea r Dif fere nt ia l an dIntegral Equations, (Dover, New York, 1962), 467-488.

R. W. Hamming, Num er ic al Met ho ds fo r Sc ie nt is ts an dEngineers, 2nd edition, (McGraw-Hill, New York, 1973).

M. E. Goldstein and W. H. Braun,Advanced Methods

for the Solution of Differential Equat ions, (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 inEshbachs Handbook of Engineering Fun-damentals, 4th ed. edited by B. D. Tapley, (Wiley, NewYork, 1990), 2.145-2.168,2.191.

ANSYS Users 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 1971. He

started his professional career at Westinghouse Astro-

nuclear Laboratory where he was employed for seven

years. There he was responsible for static, dynamic, and

stability 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 has

been extensively involved in the application of finite ele-ment 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 deal-

ing 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.

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