optimal hole profile in a finite plate under uniaxial stress by finite element simulation of...

Finite Elements in Analysis and Design 32 (1999) 1—20

Optimal hole profile in a finite plate under uniaxial stress by finiteelement simulation of Durelli’s photoelastic stress minimization

method

Danh Tran*, Vu NguyenDepartment of Mechanical Engineering, Victoria University of Technology, P.O. Box 14428, MCMC, Vic. 8001, Australia

Received 17 November 1998; accepted 17 November 1998

Abstract

Finite element method is used to simulate the photoelastic stress minimization (PSM) method developed by Durelli tosolve the elementary and yet challenging problem of finding the optimum hole profile of a finite plate under uniaxialstress. Durelli’s method of removing material slowly in low stress region of the design domain, so as to make the sectionsof the hole boundary become isochromatics of same Tresca stress, is simulated by iterative element removal within theexternal layer of elements on the hole boundary followed by smoothing and remeshing before proceeding to the nextlayer. The criterion used for element removal is selecting only the number of elements having the lowest stress amongthose that are in the design domain and on the current hole boundary. This number is set by a removal rate andcontrolled to be larger than a minimum number which serves to signify the convergence to the optimum profile. Thealgorithm has successfully found optimum hole profiles for the ratio of hole diameter over plate width D/¼ from 0.140 to0.837 studied by Durelli. The simulations show similar results as those obtained by Durelli with further improvement inmaterial removal, especially in the case of extremely small or extremely large D/¼. The FEM simulation of Durelli’sPSM is easily effected by FEM analysts requiring modest hardware and software, and it can be further extended to morecomplex cases of geometry, material properties, loading and objective function. ( 1999 Elsevier Science B.V. All rightsreserved.

Keywords: Finite element simulation; Optimum profile; Structural optimization; Stress concentration; Shape optimiza-tion; Stress minimization

Nomenclature

D diameter of the holeK

5'gross stress concentration factor based on gross stress ("p

.!9/p)

K5/

net stress concentration factor based on net stress ("p.!9

/p/)

¸ length of plate

*Corresponding author. E-mail: [email protected].

0168-874X/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reservedPII: S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 7 2 - 9

P axial load¹ thickness of plateTresca stress the maximum shear stress at a material point¼ width of platep stress based on gross (unotched) cross section ("P/(¼¹))p/

net stress based on net cross section ("P/(¼!D)¹)p.!9

maximum stress in the design domain

1. Introduction

A plate with a circular hole has been a classic example of stress concentration. Plates of finitewidth with holes are common structural elements in many engineering applications. The presenceof holes, while induces undesirable stress concentration, is however necessary for many otherfunctions, such as providing access to cables, ducting works, reducing weight or even for aestheticsreasons. The analytical solution for the stress distribution of an infinite plate with circular holeunder uniaxial stress has been available [1]. This solution can be used for a plate of finite width andwith the ratio of hole diameter to width D/¼ smaller than 0.5 [2]. For finite plates of higher D/¼,the determination of the stress concentration factor has to rely on experimental techniques likephotoelasticity or on computational methods, mainly finite element method (FEM). Because of itsinherent complex stress field, finding the optimum profile of a plate under uniaxial stress can beelusive and has remained a fundamental and challenging benchmark test for any shape optimiza-tion method.

The more general problem of finding the optimal profile of a stress raiser to minimize the stressconcentration factor has attracted a number of researchers [3—28]. Excellent reviews on shapeoptimization research can be found in [7—11]. In two-dimensional structures that have maximumshear stress on the boundary and that require minimization of the stress concentration effect, anelegant method has long been developed by photoelasticians, notably Heywood [4] and Durelli[5,6] to find the optimal profile of the stress raiser. The availability of fast and affordablecomputers has made it possible to investigate computational methods to solve shape optimizationproblems [7—27]. Some early techniques treated shape optimization as a non-linear programmingproblem using nodal coordinates as variables, in conjunction with sensitivity analysis or optimalitycriterion [12,14,16,17], or adaptive meshing [19—22]. More recent methods are molding [11],biological growth [23], evolutionary optimization [21], homogenization [10] and genetic algo-rithm [26]. Most share the basic notion of Durelli’s PSM that “inefficient” material should beslowly removed from the initial structure so as to converge to the optimum layout. However fora finite plate with hole under uniaxial stress, computational solutions available so far have not beenas comprehensive and explicit as those given by Durelli and Rajaiah whose results have been usedfor experimental validation by many researchers [17,18,24,25].

Theoretical treatment of the optimality criteria of plane stress problem relevant to Durelli’smethod has been provided [13,15,17].

The objective of this paper is to use FEM to simulate Durelli’s PSM method to find optimal holeprofiles in finite plates under uniaxial stress.

2 D. ¹ran, ». Nguyen /Finite Elements in Analysis and Design 32 (1999) 1—20

2. The optimization problem

The structure investigated is a thin plate of constant thickness ¹, of finite width ¼ witha symmetrical hole of diameter D, under uniform axial stress as shown in Fig. 1. Its length ¸ issupposed to be sufficiently long with respect to D so that the upper and lower edge boundaryconditions do not affect the stress distribution around the hole and hence the optimal profile.A typical Tresca stress distribution pattern generated by FEM is shown in Fig. 2. The stressconcentration effect can be expressed in terms of K

5'or K

5/. These notations are adopted from

Pilkey [2] where K5'

is the stress concentration factor based on the gross cross section and K5/

isthe stress concentration factor based on the minimum net cross section. A graph of K

5'and

K5/

(based on FEM results) versus D/¼ is shown in Fig. 3, where it can be seen that the highestvalue of K

5/is 3 for infinite plate, the limiting value of K

5/when D/¼ tends to 1 has been confirmed

to be 2 [28].The problem is to find the optimal hole profile to reduce K

5/for an initial circular hole of various

D/¼ ratios, between 0.140 to 0.837. The objective function is minimizing the global maximumstress occurring at points on the hole boundary like point 1 in Fig. 1. As this is a free boundary,Tresca criterion or Von Mises criterion leads to the same equivalent stress. The design domain of

Fig. 1. Plate geometry.

D. ¹ran, ». Nguyen /Finite Elements in Analysis and Design 32 (1999) 1—20 3

Fig. 2. Contours of Tresca stress distribution in a plate of finite width (D/¼"0.775).

Durelli’s problem is the region around the hole boundary limited to within a square of side D, asshown in Fig. 1. To facilitate the comparison of results of different cases of D/¼, the width ¼ isfixed and the resultant load, uniformly distributed on the upper and lower edges, is also of fixedvalue P, resulting in constant nominal gross average stress p (Fig. 1). It should be noted that thechoice of the design domain by Durelli was probably due to practical requirement, the optimumprofiles do depend on the length of the domain along the plate axis [5]. Durelli did not considerany internal cutout, i.e. material removal has to proceed from the hole boundary. The specificationof the design domain and the objective function should not be overlooked as there exist otherpossibilities for optimization, some that have been looked at are changing profile of the verticaledges [25], changing thickness [25], or allowing internal cutouts [11].

3. Durelli’s photoelastic stress minimization method

Let us review procedures of Durelli’s PSM method of shape optimization: a photoelastic modelof similar geometry and boundary conditions to those of the structure under investigation is loaded


Fig. 3. Variation of K5'

and K5/

with D/¼ ratio by FEM.

in a polariscope and its contours of Tresca stress (isochromatics fringes) are observed. Material inlow stress region is slowly removed in such a way that the boundary tends to have the sameisochromatics fringe order. Furthermore, the cut should provide a smooth transition betweenconsecutive profiles. Initially rough cuts can be made by using a coping saw or a router but at laterstages material should be removed very slowly by filing using sharp files which allow material to beremoved in layers as thin as desired. The fringe pattern is continually observed to guide theremoval of material and the optimal profile of varying curvatures would eventually be reached,regardless of the way in which different analysts take different initial cuts and different subsequentcuts.

It is proposed to call this method the photoelastic stress minimization method (PSM). PSM isintuitively appealing to designers but it is intrinsically linked to the Tresca criterion and ispractically limited to elastic plane problems of simple geometry, loading and boundary conditions.Durelli and his associates have applied the PSM method to a number of interesting plane stressproblems [5,6], most notably the case of a plate under uniaxial stress initially having a centralcircular hole of varying D/¼ ratio (0.140, 0.337, 0.518, 0.775, 0.837). Durelli’s results showeda reduction of stress concentration factor varying between 16% and 43% and increased percentage


hole area from 16% to 26% compared to those of initial circular holes; the shape of optimum holeprofiles changes with D/¼ ratio, from a more rounded hole at small D/¼ ratio, to a squarehole with rounded corners at intermediate D/¼ and to a hole with almost vertical straight edgesand dimpled horizontal edges at large D/¼.

4. A simple finite element algorithm for shape optimization

Next, let us look at a simple FEM algorithm for shape optimization that has been adopted bymany researchers [11,21,27]. It consists of the following main steps:

1. creating an FEM model of the structure with same loading and boundary conditions ofrestraints,

2. specifying the design objective, the design domain,3. specifying criterion to be used for the element removal process,4. analyzing the structure response,5. removing elements in the design domain that satisfy removal criterion,6. monitoring the objective function to see whether it is optimum, i.e. satisfying optimality criteria,7. the new structure with some elements removed is subject to the optimization cycle (steps 4—6)

iteratively until the optimum criteria are satisfied, at which point the required results are outputand the optimization cycle stops.

This simple algorithm can be presented by a flowchart as shown in Fig. 4. It can be seen thatPSM can be simulated iteratively and manually by most FEA software albeit in a tedious andtime-consuming fashion. Only features of steps 3 and 5 are briefly discussed here.

4.1. Criterion to select elements for removal

The concept of gradual removal of material of PSM is universally adopted by all researchers inshape optimization. The essential difference among various methods is in specifying the criterion toremove elements in step 3. To select elements for removal in lowly stressed region, most researchersuse stress bounds as the criterion, i.e. removing those elements that have stress lying within somelower and upper bound stress levels that are evaluated according to a certain rule.

4.2. Mechanism of element removal

The element removal in FEM is effected by actually deleting elements, or by removing them“numerically”, i.e. by simply not incorporating these elements’ stiffness matrix into the structuralstiffness matrix. Once some finite elements are removed, besides having a jagged boundary, theresulting structure may suffer the “notch effect” [27], because of the topology and finite size ofelements, giving “pseudo-maximum stress” and “pseudo-minimum stress”. This notch effect can bevery serious, especially in high stress region, unless some suitable measures like reducing elementsize or smoothing is taken. To avoid over-cutting and introducing internal cutout inadvertently,apart from the first few cuts, elements should be removed only in layers. As the size of an FEM cut


Fig. 4. Flowchart of a simple FEM algorithm for shape optimization.

is controlled by the size of those elements currently on the outermost layer, it is desirable to use assmall element size as feasible by hardware, software and other constraints.

While this simple algorithm can be successful in a number of cases, the problem of finite plateunder uniaxial stress requires further features to simulate the subtlety of Durelli’s PSM.

5. FEM simulation of PSM for finite plate under uniaxial stress

Most researchers on shape optimization have used the case of a plate under biaxial stress asa benchmark for their algorithm because of the availability of its theoretical solution. The case of


a plate under uniaxial stress, even if commonly found in practice, is however more challengingbecause of its complex characteristics as shown by the Tresca stress pattern in Fig. 2, where it canbe seen that:

a. The maximum stress occurs at points like point 1 in Fig. 1 with a very steep stress gradientaround it.

b. Points like point 2 in Fig. 1 are positions of zero stress. They are also isotropic points separatinga tensile stress section of the hole boundary from a compressive stress one, they are alwayspresent in the course of optimization of this problem. It should be noted that isotropicpoints of zero stress are confirmed by theoretical solution and photoelasticity, but due to thefinite element size used in FEM, exactly zero stress is not found by FEM at points like point 2.For example, with ¼"0.1 m, D/¼"0.25, p"10 MPa, for size of element of 0.0002 m,maximum stress at point 1 is found to be 32.939 MPa, the minimum stress found at point 2 is0.366 MPa.

c. Besides these isotropic points, there are nearby regions of relative low stress values (points likepoint 3, Fig. 1). This implies that if rough cutting is used, either by too large element or too highstress bound, there is a possibility of encroaching on regions around points like point 3, resultingin an unstable sudden “avalanche” of removed elements.

Because of these complex characteristics, the FEM simulation of the PSM’s material removalrequires a more elaborate algorithm than the one presented above in Section 4.

5.1. Simulating the process of PSM material removal

The objective is to simulate the element removal of PSM and to ensure that the material removalstarts from the hole boundary without introducing any internal cutout. In PSM, material isphysically removed, either in a big chunk at the beginning or by thin layers in the fine tuning stage.The PSM relies on global observation of isochromatics, which are contours of Tresca stress. Thisobservation is related to both field values and the gradient. The cut is further guided by thedirection of the boundary and that of its nearby isochromatics. The PSM mechanism is removingmaterial in a smooth, continuous cut in a step-by-step fashion. Thus it does not merely mechan-ically focus on regions of low stress confined between isochromatics but, if necessary, makes cutthrough isochromatics with the objective of making the boundary slowly become an isochromatic.This is very different from mechanically removing material from low stress regions based on lowerand upper bound stress levels as presented in Section 4. An illustration is given in Fig. 5 showingthe difference between the two criteria. The combined effect of taking both field values anddirections of PSM would act as a control mechanism that promotes not only minimization of themaximum stress but also ensuring the optimality condition of uniformity of stress on sections of theboundary [13,15,17].

5.2. Strategy of element removal proposed

To ensure that no over-cutting happens, element size should be small, e.g. compared tomaximum possible radial width of the design domain, here R(J2!1); only elements on the


Fig. 5. Mechanism of PSM material removal.

current hole boundary are considered for removal; and element removal should be confined withinone layer only each time. The scheme to simulate the PSM material removal adopted here is toallow the element removal to proceed iteratively within the outermost layer until further elementremoval would make the cumulative cut too rough (hence increasing the maximum stress) beforeproceeding to the next layer. This mechanism would simulate the PSM better than the one basedmerely on stress bounds. Furthermore it would alleviate the notch effect, as illustrated in Fig. 6. Inthis problem there always exist the genuine isotropic points of zero stress, but the element removalwould also introduce “pseudo-minimum” elements of very small or in fact zero stress. Unlesssmoothing is effected after every step, this effect would result in quick false spreading of thesepseudo-minimum elements [27]. If element removal is based simply on the stress bounds and isallowed to proceed straightaway into the next layer in the subsequent iteration step, the notchwould dig deeper and deeper into the structure, as illustrated in Fig. 6.

5.3. Smoothing and remeshing

It has been found that the notch effect is reduced by using regular triangular meshing of smallelement size and iterative element removal within one layer before proceeding to the next [27]. Thenotch effect can be further minimized by using a smoothing subroutine when the intra-layerremoval cycle is completed before entering the inter-layer removal cycle for the next layer followedby remeshing of only a small region around the hole boundary.


Fig. 6. Reducing notch effect.

5.4. Algorithm for FEM simulation

The algorithm is shown by a flowchart in Fig. 7.Comparing to the simple flowchart shown in Fig. 4 the following points have been included:

f Interactive modeling, solving and processing results to improve meshing and to provide data forlater control of the element removal process.

f The design domain and design objective are then specified, regular triangular meshing of smallsize elements should be used in this domain.

f Removal rate, remrate, between 0 and 1 is specified. This rate dictates how many elements out ofall external elements on the current hole boundary are selected for removal.

f Setting the minimum number of elements to be removed, nmin, nmin should be small (should beat least four in this problem if a full plate structure is used). In the first iteration, if the number ofelements selected for removal is smaller than this number, it is considered that no cut has beeneffected, the chosen remrate would have to be increased. But in all other iteration cycles thatfollow, this would be considered as convergence to the optimum profile. Besides this function,nmin also reduces the notch effect by ruling out the case of removing too few elements.

f To ensure that element removal starts from the current boundary, only extelem, the set ofelements in the design domain and currently on the hole boundary are selected, the number ofelements of this set is counted and called nextelem.


Fig. 7. Flowchart of FEM simulation for finite plate under uniaxial stress.

f The number of elements to be removed in the set extelem is determined by nrem, which is theinteger of the product of nextelem and remrate.

f nrem is then compared with nmin. If nrem(nmin, the number of elements to be removed is toosmall in the first cycle, remrate should be increased to give a bigger cut; in other cycles this signalspossible conversion to the optimum profile. Results are then output.

f If nrem'nmin, it is ready to enter the actual element removal, which is first confined within thecurrent external layer (by using only the current extelem set selected above). The elementremoval in the external layer is repeated, the model is updated every time, until an over-cuttingstage is reached that makes the maximum stress increase, then the flow would proceed to the


next layer by redefining the set extelem based on the currently updated model. In the case ofover-cutting, the model is not updated and is passed straight to the next interlayer cycle.

f There is a minor control in the intra-layer cycle to cater for the case of the first iteration whenK"0, the direction of the flow chart should be going straight to the element removal toinitialize the maximum stress value for further comparison.

f Controlling over-cutting: removing too many elements would result in increase of the globalmaximum stress and eventual divergence from the optimum solution in the first iteration.Over-cutting is checked by monitoring the global maximum stress at points like point 1, aroundwhich there exists a very steep stress gradient. These points should remain to be the position ofmaximum stress throughout the optimization process. If over-cutting happens in the firstiteration step (i.e. no updating of model has occurred), by setting inadvertently too high a valuefor remrate, then remrate should be reduced before moving to the removal process.

f Smoothing and remeshing: a simple smoothing subroutine using B-spline to generate a new holeboundary through the midpoints of the jagged boundary obtained after completing an intra-layer removal loop is invoked before proceeding to the next interlayer removal cycle.

f A number of minor checks and on-line output of results in the form of stress plots, element plots,maximum stress values, accumulative material removal, etc. were also included but not shown inthe flowchart.

6. Results and discussions

The optimization was carried out on a plate of dimensions ¼"0.1 m, ¸"0.3 m, ¹"0.001 m,for the D/¼ ratio of 0.140, 0.250, 0.377, 0.518, 0.650, 0.775, and 0.837. The load is assumed to beuniform tension of 10 MPa. The smallest element size used varied between 0.00012 m (forD/¼"0.140) and 0.0003 m (for 0.837). The error in stress values was estimated to be between 0.22and 0.55 MPa. On a Unix DEC-ALPHA 200 using ANSYS 5.3 FEM software, the optimizationprocess takes up to 5 h using a full plate model of 18 000 elements. Obviously, the computing timeis reduced by reducing the number of elements, for example halved by using only 6000 elements.Actually for this problem, quarter plate models were used, further cutting computing time down toabout 1 h. The number of iterative steps to converge to the optimal solution is between 50 and 100steps.

The results are shown in Figs. 8—12.

f Fig. 8 shows the changing hole profiles in the course of optimization (quarter model) at iterationsteps: 0 (initial), 30, 60, 94(optimal) for D/¼"0.837.

f Fig. 9a—Fig. 9g shows the full optimum profiles for various D/¼ studied (0.140, 0.250, 0.377,0.518, 0.650, 0.775, 0.837, which include all cases studied by Durelli plus two intermediate casesof 0.250 and 0.650. It can be seen that compared to the optimum profiles obtained by Durelli,those obtained in this paper have sharper corners, especially at D/¼"0.140. At larger values of0.775 and 0.837, the profiles found by FEM simulation can be described as a square hole withrounded corners, they do not show the dimpled horizontal curves as given by Durelli, notingthat the profiles are to be bounded by the design domain and that the optimization objective isminimizing the global maximum stress. Comparing the profile obtained by the FEM simulation


Fig. 8. Changing hole profiles in the course of optimization for D/¼"0.837 at iteration step 0, 30, 60, 94 (optimal).

here for D/¼"0.837 to that obtained by Durelli [5], it can be seen that FEM simulationremoves more material, i.e. FEM goes further than Durelli in optimization, with the percentagehole area increase of 27%, compared to 20% given by Durelli. The reduction in stressconcentration factor is similar to or better than those obtained by Durelli. This is confirmed bythe criterion proposed by Schnack [17]: the profiles obtained by FEM are more “optimal” thanthose given by Durelli, as they are of longer length.

f Fig. 10 shows the variation of maximum stress and percent increase in hole area (a measure ofweight reduction) with iteration number. It should be noted that the rate of reduction of stress isslow in the first steps, then faster but becomes very slow in the fine tuning stages; while the rate ofmaterial removal is large at first and becomes increasingly much smaller in the fine tuning stages.If the emphasis is to reduce weight, the slowing down when converging to the optimum solutionmay be viewed as inefficient, but this feature of PSM ensures convergence in a divergence-proneproblems like this one, and this “inefficiency” is diminishing with increasing speed of computers.

f Fig. 11 shows the variation of K5/

for the original circular hole and for the optimum profileobtained by FEM simulation.It can be seen that the reduction in the net stress concentrationfactor is quite remarkable, about 43% for D/¼"0.837.

f Fig. 12 shows the percent improvement due to the optimum profiles in reducing K5/

andincreasing in hole area. Durelli’s results indicate that the percentage increase in hole area reaches


Fig. 9a—g. Optimum hole profiles and Tresca stress distribution for D/¼"0.14, 0.250, 0.377, 0.518, 0.650, 0.775, 0.837.


Fig. 9. Continued.


Fig. 9. Continued.

a maximum at D/¼ of about 0.60 and then decreases afterwards [5], whereas the results byFEM show a continual increase.

It can be seen that in the FEM simulation, the size of the element controls how fine a cut can be.The element size, of course, depends on many constraints like software, hardware, time limit, etc.On the other hand, PSM can remove material in layers as thin as desired. In reality due to difficultyin manufacturing photoelastic model, using PSM to optimize plates with very small holes can bequite a challenging task, this is reflected in the comparatively more rounded shapes obtained byDurelli. Furthermore, it is usually not possible to experiment freely at will and to venture too farwith cuts in the photoelastic model, as models cannot be “recut” while the simulation algorithm,once successfully developed, can be used as an “experimental” tool, permitting the studying ofeffects of varying a number of parameters like loading, boundary conditions, the fineness orroughness of the cut, element size, etc. It should be added that FEM simulation easily allowsmaterial to be “added” to places where stress is high, experimenting like this is not easy, or evenimpossible, to carry out in experimental techniques like Photoelasticity.


Fig. 10. Variation of maximum stress and percent increase in hole area with iteration steps for D/¼"0.837.

Fig. 11. Variation of K5/

for circular hole and optimum profile with D/¼ by FEM simulation.


Fig. 12. Percent improvement in reduction of K5/

and in increase of hole area of optimum profiles.

7. Conclusion

It is shown that Durelli’s method of photoelastic stress minimization can be successfullysimulated by FEM to solve the classic problem of finding the optimal profile of a hole in a finiteplate under uniaxial stress. The process of material removal is simulated by removing elements inthe external layer of the boundary iteratively before proceeding to the next layer. The criterion forselecting element for removal is to choose a designated number of elements of lowest stress amongthese external elements rather than using stress lower and upper bounds criterion commonlyadopted by other researchers. This scheme simulates more truthfully the PSM mechanism ofmaterial removal. It also minimizes the notch effect which is often overlooked by most shapeoptimization techniques. The algorithm is easily carried out using commands available on mostFEM software and requiring modest computer hardware. Like PSM, its FEM simulation present-ed here is easily understood and modeled, it does not require any more novel concepts than thoserequired by FEM modeling techniques. As mentioned previously, PSM requires experimental skillsnot readily available and its scope is practically limited to elastic plane problems using Trescacriterion. With the prospect of increasing capability at decreasing cost of computer software andhardware, the FEM simulation of PSM can be easily extended to solve shape optimizationproblems involving complicated geometry, loading, boundary conditions using complex materialsand objective functions.


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