ABSTRACT The problem under consideration involves … · circular hole is considered using generalised plane strain elements. The pressure is applied to the surface of a circular

Elastic analysis of a square plate with

circular holes subjected to uniform pressure

J. Caldwell

Department of Mathematics, City Polytechnic of Hong

Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

ABSTRACT

The problem under consideration involves the computation ofstresses and displacements of a square plate subjected to uniformpressure. In the first instance a square section with a centralcircular hole is considered using generalised plane strainelements. The pressure is applied to the surface of a circularhole located at the centre of the section. The problem is thengeneralized to that of a square section with nine holes subjectedto internal pressure. Results involving stresses anddisplacements are obtained for both cases using a typical BoundaryElement software package such as BEASY. These results arevalidated by an accuracy check using Lame thick cylinder theory atselected points. The good agreement obtained gives confidence inthe use of the Boundary Element method for problems of this type.

INTRODUCTION

Elastic analysis of plates with various geometries is important inmany traditional Engineering industries e.g. elastic analysis of aplate with circular holes/cavities in the Food Manufacturingindustry. It is important to be able to deal with the simpliedproblem when pressure is applied to the surface of a circular holelocated at the centre of the section. An obvious extension is toboth orderly and randomly arranged holes/cavities withpressure/shear.

For problems of this type Boundary Element (BE) methods haveadvantages over FE methods. A typical BE package, BEASY (BoundaryElement Analysis System), was selected to carry out the analysis.Useful references for this work include the textbook by Brebbia[11 which discusses the mathematical foundations of the work and areview paper by George [2] which discusses the relative merits ofBE analysis and FE methods.

Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

170 Boundary Element Technology

The Boundary Element Method (BEM) has been successfully usedin the solution of problems in linear elastic static stressanalysis. Results are obtained for the case of a square platewith a central hole subjected to internal pressure. These resultsare then generalized to the case of a square plate with 9 circularholes. The stresses and displacements are computed using BEASYand checked in certain cases using Lame thick cylinder theory.Some important conclusions are then drawn.

FORMULATION OF PROBLEM

Initially the geometry to be considered is a square section with asingle central circular hole. We require an elastic analysis whenthe plate is subjected to uniform pressure. The pressure isapplied to the surface of a circular hole located at the centre ofthe section. By symmetry we need only consider one quarter of thesection for the mathematical model (see Figure 1).

10 .9

SQUARE SECTION WITH HOLE - INTERNAL PRESSUREPLOT a.xaoc.i 11

CMC/SEASYLOAO CASE 0

Figure 1 - Geometry plot with defined elementsfor square section with a holeunder internal pressure


Boundary Element Technology 171

- Assume material behaves elastically with' the followingparameters:

Young's modules E = 50 x 10*Poisson's ratio u = 0.2Section thickness = 1Uniform pressure = 1000

Boundary conditions - Assume zero displacements on the lines ofsymmetry

i.e. u = 0 at x = 0v = 0 at y = 0

The geometry is then generalized to one which contains 9 identicalcircular holes placed symmetrically in the square section. Againby symmetry, we need only consider one quarter of the section (seeFigure 2).

SQUARE SECTION WITH NINE HOLES - INTERNAL PRESSURES cnc/B£flSYPLOT '•***" " LOAD CASE 0

Figure 2 - Geometry plot with defined elementsfor squre section with 9 holesunder internal pressures



BEM FOR ELASTOSTATIC PROBLEMS

The BEM is an ideal tool for the solution of problems in linearelastic static stress analysis. It is preferable to FEM in thatonly the "boundary" of the problem need be modelled. Also BEASYcan be used to estimate the stresses and displacements for a bodysubjected to a prescribed loading.

The principle of virtual displacements for linear elasticanalysis of a problem domain V bounded by the boundary surface Sis given by

(CP. . + b ) 0% dV

p* dS +T (pi - pj u*J c

dS , (1)

where p = boundary "traction" (or pressure),u = displacement,(Tjnj = derivatives of the stress tnesor,

b^ = acting body force,

u% = displacement corresponding to the virtual system,$

Pk = traction corresponding to the virtual system.

The boundary conditions are :

u = u on portion S^ of S,p = p on portion S% of S.

Integrating (1) by parts twice and applying a "fundamental"solution of the type

(r* , + A! = 0 , (2)

where A| is the Dirac delta function representing a unit load inthe "1" direction, leads to

14 p% dS = J Pt ul dS + J b, u* dV . (3)



This equation (3) represents (in the absence of body forces bjthe boundary integral formulation for electrostatic analysis. Itconsists of a set of n equations in 2n unknowns (traction anddisplacement in each direction at each node). However, inpractice, we only have n unknowns since we know the relationshipbetween traction and displacement at every node point.

LAM£ THICK CYLINDER THEORY

Two-dimensional problems in elastostatics arise naturally in twodistinct ways i.e. plane strain and plane stress. Our problemsare plane strain i.e. the body being deformed is a long rightcylinder acted upon by external forces such that the componenet ofthe displacement in the direction of the sylinder axis vanishesand the remaining components remain constant along the length ofthe cylinder. In this work it is advantageous to use polarcoordinates r, 9. Then the equations of motion reduce to (referto Timoshenko and Goodier [3D.

For the problem of a hollow cylinder subjected to uniform pressureon the inner and outer surfaces, the boundary conditions for theradial stress component are

<>V = - Pa at r = a (6)OY = ~ Pb at r = b

where a < b.

If u, v are the radial and tangential components of thedisplacement, respectively, then for plane strain expressions canbe obtained for the radial and tangential components of strain.Using the boundary conditions (6) the stress components are foundto be

2,2, \ 2 ,2<r = a b ( p y - pj ^ a p, - b p*,

/, 2 2, 2 ,22 ^ '(b - a )r b - a



<r = _ aV(pb - p.) a=p, - b*pb , ,8 ,2 2. 2 ,22 ^(b - a )r b - a

In our case, the displacement must be axially symmetric and thisresults in the following expression for the radial component ofdisplacement :

(9)

This means that in Lame thick cylinder theory the stresscomponents o^ and <r are given by equations (7) and (8) and theydisplacement u by equation (9). For comparison purposes withBEASY we use the following data:

a = 1 , b = 5 , p,, = 1000 , p,, = 0 ,

v = 0.2 , E = 50 x 10* .

SQUARE SECTION WITH CENTRAL HOLE

BEASY analysis uses conditions of plane strain. It can also dealwith plane stress. By symmetry it is only necessary to considerone quarter of the body. The geometry plot with defined elementsis shown in Figure 1.

It is important to note that elements are only defined on theactual boundary. Also quadratic elements with 3 node points areused. Three elements are placed along the quadrant representingthe hole and five elements are placed along each (half) edge ofthe block.

A variety of internal points has been defined, namely, twolinear arrays i.e. one horizontal (points 13-22) and one at 45°tothe axis (points 23-32) plus one rectangular array in the corner(16 points). The only loading is the 1000 pressure on the insideof the hole. A negative value is used in the data to denote thatcompression stress is implied.

BEASY is used to calculate the normal and shear stresses atall internal points together with corresponding principalstresses. For the results, a typical stress plot is given inFigure 3. This shows variations along the line of internal points13 to 22 on the horizontal axis. In order to establish confidencein these results comparisons are made with solutions to theproblem of a pressurized hollow cylinder with the data previouslyindicated, found by using Lame thick cylinder theory.



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•I.300C»J z

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"*'**~"— -«

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» 2« » 26 27 S » 10 Jl 2

gg. SQUARE SECTION WITH HOLE - INTERNAL PRESSURE cr-c/SCPSYPLOT 2.*ssc.2 ,i INTERNAL SOLUTION LCPQ CASE 0

Figure 3 - Typical stress plot for square section with holeunder internal pressure

Table 1 gives the normal stresses cr, and cr0 at points

equivalent to internal points 13 to 22 by using BEASY and Lamethick cylinder theory. Agreement is excellent which givesconfidence in results at other internal points produced by BEASY.Table 2 gives the computed values of the displacement u at thesame internal points produced by BEASY and Lame thick cylindertheory. Again there is close agreement with less than 17.discrepancy.



Table 1 - Computed values of normal stresses ov and cr6at internal points 13 to 22 (see Figure 1)using BEASY and Lame thick cylinder theory

InternalPoint

13141516171819202122

Posit i onr

1. 1001.21 11.3221.4331.5441.6561.7671.8781.9892. 100

BEASYcr crr 9

-821 . 2 891.0-670.0 742.0-555.4 628.4-466.3 539.9-395.7 469.8-338.7 413.2-292. 1 366.9-253.4 328.5-221.0 296.4-193.5 269.2

Lame Thick Cylindercr crr 8

-819.2 902.5-668.5 751.8-554.2 637.5-465.4 548.7-395.0 478.4-338.4 421.7-292. 1 375. 4-253.7 337.1-221.7 305.0-194.5 277.9

Table 2 - Computed values of the displacementat internal points 13 to 22 (see Figure 1)using BEASY and Lame thick cylinder theory

InternalPoint

13141516171819202122

Pos it ionr

1. 1001.21 11.3221.4331.5441.6561.7671.8781.9892. 100

BEASY

u ( x 1 0 " ̂ )

2.32542.12381 . 95721 . 81751.69901 . 59741 . 50951.43291 . 36571 . 3066

Lame Thick Cylinderu(xlO^)

2.33872.13691 . 97011.83021.71141 . 60941 .521 11 . 44401 .37631.3165

The actual values of displacement u for all mesh points havebeen calculated using BEASY. Figure 4 shows exaggerateddisplacements indicating the deformed shape and the movement ofinternal points.



I*1 - »

SQUARE SECTION WITH HOLE - INTERNAL PRESSUREPLOT j.3ooe.t ,i OEF. 391S.1S :l

CMC/3EA3YLOAD CASE 0

Figure 4 - Exaggerated displacements indicating the deformed shapeand movement of internal points for a square sectionwith a hole under internal pressure

SQUARE SECTION WITH NINE HOLES

This is a generalisation of the single hole problem in which ninecircular holes are placed symmetrically in the square section.Again, plane strain is assumed and, by symmetry, only one quarterof the section need be considered. The corresponding geometryplot with mesh points is shown in Figure 2. The command file wasproduced by simply editing that for the single hole case. In thisway any distribution of holes, whether uniform or random, canreadily be described.

A complete set of results as for the single hole case can beobtained including

normal and shear stresses at internal pointsprincipal stressesdisplacements at boundarydisplacements internallystress plot for internal points 36 to 45deformed shape



Figure 5 shows the stress plot for internal points 36 to 45and Figure 6 shows the deformed shape.

SUM SQUARE SECTION WITH NINE HOLES - INTERNAL PRESSURES CMC/BEASYrfAXPLOT 3.Q7SC.2 -I INTERNAL SOLUTION LCfiO CASE 0

Figure 5 -Typical stress plot for square sectionwith 9 holes under internal pressures

SQUARE SECTION WITH NINE HOLES - INTERNAL PRESSURES CMC/BEASYPLOT 3.3ooe.i .1 OEF. 331.66:1 LOGO CASE 0

Figure 6 - Exaggerated displacements indicating the deformed shapeand movement of internal points for a square sectionwith 9 holes under internal pressures


Boundary Element Technology

CONCLUDING REMARKS

179

(1) Using the BEASY software package has meant that thedefinition of the computer model is a fairly straightforwardtask. This is an inherent advantage of BE analysis over themore usual FEM. Since elements are defined on the boundaryof the plate and holes only, then changing the number ofelements or the geometry of the boundaries has little impacton the users data file (.GAT). This contrasts with the FEapproach which would imply a totally new mesh to be definedin either of these cases. Similar advantages would beexhibited for three dimensional problems. To illustrate thisan equivalent FE mesh for the single hole case is given inFigure 7.

tmdiu, of Ch.

Figure 7 - Equivalent FE mesh for square sectionwith central hole under internal pressure



(2) BEASY permits free choice of the positions of internal pointswhich are not linked to any meshes as is the case for FEanalysis. However, they should not be placed too close tothe boundary as unpredictable results can sometimes beproduced. To enable contouring of stress to be performed itis necessary to have a significant number of internal pointsevenly spread throughout the geometry.

(3) The results for the single hole case based on linear planestrain analysis are excellent. Comparisons with thickcylinder analysis show good agreement on stresses anddisplacements. This has been backed up by a separateexercise using FE software which is not described here. Thefree choice of position of internal points in BEASY shows anadvantage when a fine description of stress variation isrequired. The stress gradient near the holes is steep and itis clearly advantageous to be able to define the stress insuch regions by having internal points close together.

(4) The problem involving a square section with nine holesdemonstrates the ease with which changes in the model can bemade in BEASY. The basic editing of the user command file(.GAT) required only a few minutes of effort to modify thatfor the single hole case. An excellent description of thestresses and displacements for the nine hole problem isobtained by using a substantial number of freely definedinternal points. In particular, the close definition ofstress variation between two of the holes is easilydemonstrated.

(5) The real significance of this work is that it demonstratesthe powerfulness of the BE software (BEASY) in thecomputation of stresses and displacements of plates/sectionswith orderly or randomly arranged holes/cavities withpressure/shear.

REFERENCES

1. Brebbia, C.A., The Boundary Element Method for Engineers, 2nded., Pentech Press, 1980.

2. George, J., Boundary methods back in favour, EngineeringComputers, Jan. 1985.

3. Timoshenko, S. and Goodier, J.N., Theory of Elasticity,McGraw-Hill, London, 1951.


Documents

ABSTRACT The problem under consideration involves … · circular hole is considered using generalised plane strain elements. The pressure is applied to the surface of a circular