Design of a Defence Hole System for a Shear-loaded Plate

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The Journal of Strain Analysis for Engineering

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DOI: 10.1243/030932403770735872

2003 38: 507The Journal of Strain Analysis for Engineering Design S. N Akour, J. F Nayfeh and D. W Nicholson

Design of a defence hole system for a shear-loaded plate

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Design of a defence hole system for a shear-loadedplate

S N Akour1 * , J F Nayfeh

2 and D W Nicholson2

1 Mechanical Engineering Department, University of Jordan, Amman, Jordan2 Mechanical, Material and Aerospace Engineering Department, University of Central Florida, Orlando, Florida, USA

Abstract: Stress concentrations associated with circular holes in pure shear-loaded plates can bereduced by up to 13.5 per cent by introducing elliptical auxiliary holes along the principal stressdirections. These holes are introduced in the areas of low stresses near the main circular hole in orderto smooth the principal stress trajectories.

A systematic study based on univariate search optimization method is undertaken by using niteelement analysis (FEA) to determine the optimum size and location for an auxiliary defence holesystem. The results are validated using RGB (redgreenblue) photoelasticity.

Keywords: shear stress, defence hole system, stress reduction, RGB photoelasticity, optimizationand design

1 INTRODUCTION

To obtain efcient, economical and reliable designs, it isimportant to devise techniques for the reduction of stress concentrations around geometrical discontinu-ities. The defence hole system yields a signicantreduction of maximum stress in the structure.

There are a few related studies focusing on reducingstress concentration by introducing other geometricdiscontinuities (see Table 1). Most of the work that hasbeen done so far deals with uniaxial loading. Heywood[1] and Erickson and Riley [ 2] investigated the effect of the defence holes on the stress concentration around theoriginal hole using two-dimensional photoelasticity.Jindal [ 3, 4] examined the reduction of stress concentra-tion around circular and oblong holes using the niteelement method (FEM) and photoelasticity analysis.Meguid [ 5, 6] studied the reduction of stress concentra-tion in a uniaxially loaded plate with two coaxial holes

using nite element analysis (FEA). Rajaiah and Naik[7] investigated hole-shape optimization in a nite platein the presence of auxiliary holes using the two-dimensional photoelastic method. Naik et a l. [8]optimized the hole shapes in beams under pure bending.Ulrich and Moslehy [ 9] and Providakis et al. [10 ] usedboundary element methods to reduce stress concentra-tion in plates by introducing optimal auxiliary holes.

The introduction of auxiliary holes along the principalstress directions reduces the stress concentration andthus increases the fatigue life of the structure. Thecurrent investigation is carried out by FEA andvalidated by enhanced RGB (redgreenblue) photo-elasticity [ 13 , 14].

2 PROBLEM DESCRIPTION

In the pure shear case, four areas of low stresses (black)near the main circular hole (white) are shown in Fig. 1a.These four areas are along + 458 axes relative to theplate axes. The low stress areas for the tensile loadedplate are shown in Fig. 1b. The optimum number of auxiliary holes for the shear case is four holes.

Figure 2 represents the model that is used in thecurrent investigation, where S is the distance betweenthe centres of the main hole and the auxiliary holes, Ddenotes the diameter of the main hole and the major andminor axes of any of the elliptical auxiliary holes aredenoted by a and b respectively. Circular and ellipticaldefence hole systems are investigated. To reduce theplate width and edges effects, the auxiliary hole system isrotated 45 8 and a biaxial stress (tensioncompression) isapplied. Because of the symmetry, a quarter of the platehas been modelled using the FEM by applyingsymmetry boundary conditions along the axes of symmetry. Moreover, the D/W ratio, where W is theplate width, is constrained to be less than 0.02.

3B2 Version Number 7.51a/W (May 2 2001) j:/Jobsin/M11069/S01303.3d Date: 28/10/03 Time 09:12am Page 507 of 518

The MS was received on 5 February 2003 and was acceptedafter revision for publication on 21 July 2003.* Corresponding author: University of Jordan, PO Box 13512, Amman11942, Jordan.

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The position of stresses s 1 , s 2 and s 3 are shown inFig. 2. These stresses are the maximum stresses in thedenoted positions. These positions are found to be thecritical stress spots, which have been monitored andrecorded, while the main objective is to minimize thestress over the model. The univariate search optimiza-tion method is used to achieve the objective of the design[15].

3 FINITE ELEMENT MODELLING

SDRC I-DEAS is used in the FEA simulations [ 16]. Themesh is developed using linear plane stress elements.Map meshing is used to control the number of elementsin the positions of high stress concentration. A ne meshis utilized in the areas where the stress concentration isexpected to be high. Figure 3 illustrates the mesh thathas been used in the analysis and the way it is loaded ispresented in Fig. 2. An illustration of the nite element

mesh in the areas close to the holes is shown in Figs 3aand b. The area close to each hole is partitioned by asquare [ 16]. The mesh is produced for each partition byhaving a contour shape similar to the hole shape andchanging gradually to a rectangular shape as itapproaches the partition boundary. Radial lines areproduced starting from the hole centre to form the mesh(Fig. 3).

This research is intended to investigate large plates.All FEM models investigated have a hole diameter plate width ratio D=W 4 0:02 (see Fig. 4) [ 17 ]. Sym-metry boundary conditions are applied along the X andY axes (Fig. 3) [ 18 ]. This means better utilization of the maximum number of elements allowed by the soft-ware in modelling the problem, i.e. a larger size platethat could be accepted as a presentation of the inniteplate with a very small amount of error. To ensureconvergence and to obtain accurate results, a gradualchange in the element size is employed. Therefore,smaller elements are used near the points of stressconcentrations while larger elements are utilized far


Table 1 Summary of some of the previous studies

Author Type of load Specimen Stress reduction

Erickson and Riley [ 2] Tension 18 %

Durelli et al. [11 ] Tension 25 %

Dhir [ 12] Tension and shear Tension 17.7 % Shear 23.25 %

Jindal [ 3, 4] Tension 25 %

Rajaiah and Niaik [ 7] Tension 30 %

Meguid [ 5, 6] Tension First specimen 7.4 %Second specimen 9.4 %Third specimen 10.9 %

Ulrich and Moslehy [ 9] Tension Reduces the SCF(stress concentration factor)up to 3.0 for the ellipse

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from the high stress areas. Also, the size of elements isdecreased until changes in the results of well-knowncases are less than 1 per cent.

4 EXPERIMENTAL MODEL AND RGBPHOTOELASTICITY

4.1 RGB photoelasticity

For circular polariscopes using monochromatic light of wavelength l 0 , the light intensities for the dark eld can

be represented by

I I 0 sin 2 pD

l 0 1where l 0 is the wavelength, D is the retardation (fringeorder) and I 0 is the basic light intensity. The equationthat describes the relation between the principal stressand retardation (fringe order) is given by

D C 0ts 1 s 2 2where C 0 is the photoelastic coefcient of the material,


Fig. 1 Low stress (Von Mises) regions for (a) a shear-loaded plate of size 10 in 6 10in (254mm 6 254mm)and (b) a tensile-loaded plate of size 8 in 6 10in (203mm 6 254mm)

Fig. 2 The three stress regions used in the criterion of optimization

DESIGN OF A DEFENCE HOLE SYSTEM FOR A SHEAR-LOADED PLATE 509



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t is the thickness of the model and s 1 , s 2 are theprincipal stresses.Using the full visible spectrum allows complete

determination of retardation (fringe order). Whenbirefringent materials are observed with polarized light,

a fringe pattern appears as a series of successive andcontiguous bands of different colours. Each bandrepresents a different degree of birefringence corre-sponding to the strain in the material. Thus, the colourbands uniquely identify the birefringence (strain level,


Fig. 3 (a) Illustration of the mesh near the holes used in FEA. (b) Zoom shot for the mesh enclosed by therectangle in (a)




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i.e. stress level) everywhere along that band. Colourbands (fringes) are closely spaced to identify a steepstrain gradient.

The RGB photoelasticity method depends on theintensity of three basic colours: red, green and blue(RGB). Each colour has a band of wavelengths. As afunction of the wavelength, the intensity of each colourcan be determined by averaging over the wavelengthband representing the intensity of each colour. Theequation that represents the intensity of colours is

I i 1D l i

l 2i

l 1i F l i f l i dl i 3

Here I i is the intensity of RGB colours deducted by thecamera and stored in the image, F l i is the cameraspectral response, f l i is the intensity emerging from adark eld circular polariscope and the subscript i represents red, green or blue colour. The functionf l i for ideal optics can be written as

f l i I 0l i sin pD

l i 4The intensities of the basic RGB colours are recorded asa digital image by means of a stationary digital camera.

Each combination of RGB colour intensities representsa fringe order, i.e. represents the principal stressdifference [ 13 , 14].

4.2 Experimental model

The RGB colours appear when a birefringent loadedmodel is being put in a white-light circular polariscope.A coloured digital image of the loaded model is taken,so this digital image contains the colour intensities (red,green and blue) of the model under investigation. In

order to analyse the image and convert it into a surfacemesh that represents the fringe order of each pixel in themodel, a calibration specimen is made. This calibrationspecimen should have a linear distribution of the fringeorder along one of its axes (i.e. linear stress variation).By comparing the colour intensities of the calibrationspecimen with those of the model, the pixels that haveequal RGB intensities in both the model and thecalibration specimen have an equal fringe order [ 13 ].

All the experimental specimens are manufacturedusing a stereolithography machine (SLA250). Thismachine cuts computer aided design (CAD) shapesinto plastic models using a laser beam to draw the

geometry in successive two-dimensional layers. Theaccuracy of the models made using this machine is+ 0:01 cm + 0:004in . These models are free of resi-dual stresses. The physical properties of the models canbe controlled by the curing time, i.e. by exposing themodel to ultraviolet (UV) light. The longer the materialis exposed to UV light, the closer it will be to brittlematerial behaviour. The material used in building themodels is Cibatool SL 5170. This material has beencured for 1 h in an UV oven after it has been made. Themaximum size of the objects that can be made usingthis machine (SLA250) is 254 mm 6 254 mm 6 254 mm

10in6

10in6

10in . The models were 203 mm6

203 mm 6 6:35 mm in size.

5 NUMERICAL RESULTS

The computational FEM procedures involved numerousiterations that are based on the univariate searchoptimization method [ 15]. The optimization schemefocused on the variables: size, shape and placement of the defence hole system. Intermediate computations arenot represented in this paper due to the limited space.


Fig. 4 Stress concentration factor (SCF) versus the hole diameterplate width ratio D /W for a plate undertension load [ 17]




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Only a summary of the results in tabular and graphicformats are presented.

There are four variables or parameters dening theshape. These parameters are y, S =D , a=

2D

and b=

2D

(see Fig. 2); the optimum value of y is assumed to be+ 45 relative to the plate axes. The range for theoptimum value for S /D is expected to be not very farfrom the optimum value for the uniaxial tension case [ 2].The expected range is S =D 1 1:4. A univariate searchoptimization procedure is applied for each value of S /D in order to nd the optimum values of a/2D andb/2D.The FEA results are shown in Figs 5 to 14. Threecritical stress values or regions labelled as s 1=s 0 , s 2=s 0and s 3=s 0 are highlighted, where s 0 is the stress in aplate of equal size and load but without any stress-

reducing holes in it. Those stress points or regions arepresented in Fig. 2.

For the elliptical defence hole system, the variationsof the stress ratios with the major diameter a=

2D

of

the auxiliary hole for S =D 1:0 to S =D 1:4 are pre-sented in Figs 8 to 11. A summary of the experimentaland analytical optimization of the current work forS =D 1:0 to 1.4 are demonstrated in Figs 12 to 14.

6 EXPERIMENTAL RESULTS

Sample results of the experimental investigation areillustrated in Figs 15 to 19. Figure 15 represents aphotoelastic image [ 19 ] for the plate without the defence


Fig. 5 Stress ratio with respect to the diameter of the circular defence system for S /D 1

Fig. 6 Stress ratio with respect to the diameter of the circular defence system for S /D 1.1

Fig. 7 Stress ratio with respect to the diameter of the circular defence system for s 3=s 0




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Fig. 8 Optimum situation for S /D 1.0



Fig. 11 Optimum situation for S /D

1.4




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hole system and Fig. 16 depicts the fringe distributionnear the main hole for the enclosed area in Fig. 15.Figures 17 to 19 exhibit the plate with the defence holesystem. Good agreement has been obtained betweenexperimental and FEA models. By visually examining

Figs 15 and 17 it is possible to recognize the stressreduction; i.e. fringe order 2 starts to appear on thecircular hole edges in Fig. 15. However, for the sameload for a plate with the optimum defence hole system

S =D

1:2

, fringe order 2 does not yet appear (Fig. 17).

In Fig. 19 S =D 1:4, fringe order 2 starts to appearvery close to the edge of the main hole. The stressreduction is evaluated experimentally and plotted onFigs 12 to 14. The fringe distribution for the enclosedareas in Figs 15 and 17 is demonstrated in Figs 16 and18 respectively.

The difference between the experimental measure-ment and the FEA evaluation of the stresses is due tomany sources of error. The plates that are used in theexperimental investigation are of a limited size; i.e. theratio of the diameter of the hole to the width of the plate


Fig. 12 Optimal values of S /D

Fig. 13 Optimal values of a /(2 D)

Fig. 14 Optimal values of S /D eq

Fig. 15 RGB photoelastic image (isocromatic) for a plate without the defence system




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is D=W 1=12. This ratio is a reasonably goodapproximation, which is obvious from Fig. 4. Calcula-tion of the stress concentration factor is based on thecross-section W and the thickness of the plate t . Another

source of error concerns the uniformity of the loaddistribution; i.e. the load is not 100 per cent uniform.This non-uniformity in the stress is due to variations inthe load application. Despite all these sources of error


Fig. 16 Fringe distribution for the area enclosed in Fig. 15 (1 full fringe 20 divisions)

Fig. 17 RGB photoelastic image (isocromatic) for a plate with a defence hole system of S /D 1.2

Table 2 Summary of all optimal cases and maximum stress reduction s avg s 1=s 0 s 2=s 0 s 3 =s 0=3,where s 0 is the stress in a plate of equal size and load but without any stress-reducing holes in it)

S /D a /(2 D) b/(2 D) s 1/s 0 s 2/s 0 s 3 /s 0 s avg /s 0 Deq S /Deq Stress reduction ( % )

1 0.315 0.205 0.887 0.889 0.880 0.885 0.531 1.88 11.481.1 0.29 0.195 0.867 0.867 0.863 0.866 0.494 2.23 13.411.2 0.28 0.195 0.860 0.870 0.870 0.867 0.482 2.49 13.331.4 0.26 0.2 0.884 0.875 0.880 0.880 0.463 3.02 12.05




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the absolute value of the errors does not exceed 20 percent. This error has been calculated with reference to thenite element model assuming the FEA is the accurateresult.

7 DISCUSSION

Introducing a circular defence hole system reduces thestress on the main hole but the stresss is still high inregion 3 s 3=s 0 and this is obvious in Figs 5 to 7. Tomake the stress equal in the regions discussed, the shapeof the defence holes is modied to be ellipse. Thedirection of the ellipse is chosen so that the stress inregion 2 s 2=s 0 increases and the stress in region 3s 3=s 0 decreases. There are two pairs of ellipses.A representative sample of the elliptical defence holesystem size and shape optimization is illustrated in Figs

8 to 11. As the major axis increases a=2D, the stressin region 2

s 2=s 0

increases and the stress in regions 1

s 1=s 0 and 3 s 3=s 0 decreases. Figures 8 to 11represent how the optimum situation for S =D 1:0 toS =D 1:4 that takes place at b=2D 0:2 is beingapproached. For example, it may be seen thats 1=s 0% s 2=s 0% s 3=s 0 and the reduction of the stress is13.41 and 12.05 per cent for S =D 1:1 and S =D 1:4respectively.

Figures 12 to 14 illustrate the maximum reductionthat can be achieved by using such a defence system fora pure shear-loaded plate. The maximum reduction isachieved at S =D 1:13, a=2D 0:278 and S =D eq 3:328 (where Deq

2p 6

a2

b2p ) for an innite plate.

8 CONCLUSIONS

FEA and experimental study for a shear-loaded platewith and without the defence hole system was con-ducted. The two methods show good agreement. Thisstudy is the baseline data for designing an optimumdefence hole system for a shear-loaded plate (see Table 2and Figs 12 to 14). For example, if there is a restrictionon the position of the defence system, the table providesa range for S /D 11.4. If the value lies within the rangeof S /D then interpolation may be used to nd the othercorresponding parameters. If there is no restriction onS /D , then the placement that gives the best reductionshould be used.

Introduction of a circular defence hole system is notbenecial for the pure shear-loaded case. However, theelliptical defence system shows good results. Themaximum stress reduction achieved by the ellipticaldefence system is 13.41 per cent.

Large at plates loaded in pure shear are rarely usedin engineering structures so further research on narrow


Fig. 18 Fringe distribution for the area enclosed in Fig. 17 (1 full fringe 20 divisions)

Fig. 19 RGB photoelastic image (cropped image) for a platewith a defence hole system of S /D 1.4




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plates and generalized loading are now taking place. Thecurrent study is the rst step in a series of future work tocome.

REFERENCES

1 Heywood, R. B. Designing by Photoelasticity , 1952,pp. 296298 (Chapman and Hall, London).

2 Erickson, P. E. and Riley, W. F. Minimizing stressconcentration around circular holes in uniaxially loadedplates. Expl Mechanics , 1978, 18 , 97100.

3 Jindal, U. C. Stress distribution around an oblong hole in auniaxially loaded plate. J. Inst. Engrs India, Part ME , 1983,64 , 6670.

4 Jindal, U. C. Reduction of stress concentration around ahole in uniaxially loaded plate. J. Strain Analysis , 1983, 18 ,135141.

5 Meguid, S. A. Finite element analysis of defense holesystems for the reduction of stress concentration in auniaxially loaded plate with two coaxial holes. EngngFracture Mechanics , 1986, 25 (4), 403413.

6 Meguid, S. A. Engineering Fracture Mechanics , 1989(Elsevier, Oxford).

7 Rajaiah, K. and Naik, N. K. Hole shape optimization in anite plate in the presence of auxiliary holes. Expl Mechanics , 1984, 24 (2), 157161.

8 Naik, N. K., Ramesh Kumar, R. and Rajaiah, K. Optimumhole shapes in beams under pure bending. J. EngngMechanics , 1986, 112 (4), 407411.

9 Ulrich, T. W. and Moslehy, F. A. Boundary elementmethod for stress reduction by optimal auxiliary holes.Engng Analysis with Boundary Elements , 1995, 15 (3), 219 223.

10 Providakis, C. P., Sotiropoulos, D. A. and Beskos, D. E.BEM analysis of reduced dynamic stress concentration bymultiple holes. Communications in Numer. Meth. Engng ,1993, 9(11), 917924.

11 Durelli, A. J., Brown, K. and Yee, P. Optimization of geometric discontinuities in stress elds. Expl Mechanics ,1978, 18 (8), 303308.

12 Dhir, S. K. Optimization in a class of hole shapes in platestructures. J. Appl. Mechanics , 1981, 48 (4), 905909.

13 Akour, S., Nayfeh, J. and Nicholson, D. Enhanced RGB-photoelasticity technique for fringe orders 05. J. Engng ,2001, 11(2), 155166 (T.B. Center for Science andTechnology, India).

14 Ajovalastit, B. S. and Petrucci, G. Towards RGB photo-elasticity: full-eld automated photoelasticity in whitelight. Expl Mechanics , 1995, 35(3), 193200.

15 Chapra, S. C. and Canal, R. P. Numerical Methods forEngineers , 3rd edition, 1998, pp. 358360 (McGraw-Hill,New York).

16 Lawry, M. H. I-DEAS Master Series, Students Guide , 1998,pp. 280284 (Structural Dynamics Research Corporation,Milford, Ohio).

17 Fuchs, H. O. and Stephens, R. I. Metal Fatigue inEngineering , 1980, pp. 115118 (John Wiley, Chichester).

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Design of a Defence Hole System for a Shear-loaded Plate