Effect of Elastic Constants on Stress Concentration Factor and its Mitigation in Rectangular Plate With Central Circular Hole Under in Plane Loading

Effect of Elastic Constants on Stress Concentration Factor and its Mitigation in Rectangular Plate With Central Circular Hole Under in Plane Loading

Shubhrata Nagpal

Department of Mechanical Engineering Bhilai Institute of Technology

Durg (C.G.), India

Dr. S. Sanyal, Dr. N. K. Jain Department of Mechanical Engineering National Institute of Technology Raipur (C.G.), India

Abstract The effect of different material properties on SCF has been studied. Five models of plate have been considered for analysis. Poisson's ratio and elastic constants EX, EY and GXY have been varied from its original values. All the five models for four materials considered have been analyzed for D/A=0.1 and 0.5, by considering all the variations in material properties. Model 1, is the plate with central circular hole for isotropic and orthotropic materials. All other models are modified design of Model1 for mitigation of SCF.

Keywords Stress concentration factor, mitigation of stress concentration factor, elastic constants.

NOMENCLATURE

A Width of rectangular plate

1. Diameter of main hole

   D Diameter of Auxiliary hole

2. Modulus of elasticity

   Ei Modulus of elasticity in i direction Gxy Modulus of rigidity for XY plane

   SCF Stress concentration factor = max/nom L Length of rectangular plate

   (SCF)M1 SCF in Model 1

   (SCF) VM1 Percentage Variation in SCF in Model 1 (SCF) RM2 Percentage reduction in SCF in Model 2 (SCF) RM3 Percentage reduction in SCF in Model 3 (SCF) RM4 Percentage reduction in SCF in Model 4 (SCF) RM5 Percentage reduction in SCF in Model 5 ()V Percentage Variation in Poissions ratio (EX)v Percentage Variation in EX

   (EX / Ey)V Percentage Variation in EX / Ey (EX / GXy)V Percentage Variation in EX / GXy (EY / GXy)V Percentage Variation in EY / GXy Length of cavity

   o Uniformly distributed load (N)

   max Maximum stress at discontinuity, N/mm2 Poissons Ratio

   Width of Cavit

   1. INTRODUCTION

      Failure in real engineering components almost invariably begins at the root of a geometrical discontinuity. The classic example of discontinuity is rectangular plate with central circular hole. Analysis of stress concentration around discontinuities in plates under various loading conditions has been worked out by various researchers.

      The work carried out by various researchers for analysis of SCF is compiled and presented by Peterson [1]. Shastry and Raj [2] have analyzed the effect of fibre orientation for a unidirectional composite laminate with finite element method by assuming a plane stress problem under in plane static loading.

      Hanus [3] formed parameterized geometry models of orthotropic material subjected to uni-axial tension and studied the interaction between elliptical holes and free edges.

      Rajaiah et. al. [4] proposed hole shape optimization in a finite plate by photo elasticity method. They introduced auxiliary holes around main hole for mitigation of SCF and also optimized the shape of circular holes. The effort is made by experimental determination of reduction in SCF by a) introduction of circular holes b) optimization of shape of main hole c) optimizing the shape of main hole as well as auxiliary holes.

      Stress concentration factors, Kt, for a flat bar with circular-arc or V-shaped notches are considered by Noda and Takase [5] based on the exact solutions for special cases and accurate numerical results. A set of Kt formulas useful for any notch shape is proposed. For the limiting cases of deep

      (d) and shallow (s) notches, the body force method is used to calculate the Kt values and is formulated as Ktd and Kts. The notch shape is classified into several groups according to the notch radius and notch depth. The least squares method is applied for calculation of Kt/Ktd and Kt/Kts. Convenient formulas are proposed that are useful for any notch shape in a flat test specimen.

      Zirka et. al. [6] have analyzed stress concentration around circular hole in a rectangular plate for orthotropic and

      isotropic plates under dynamic and static loading. They have used photo elastic method for analysis.

      Sanyal and Yadav [7] have proposed the optimum distance and size of auxiliary holes for mitigation of SCF in plate with circular hole. Introducing the optimum auxiliary holes in the line of original hole, about 17% mitigation in SCF is achieved by them. They have proposed an optimum distance between the original hole and relief hole and also an optimum size of relief hole by assuming elliptical stress flow lines.

      Kubair [8] numerically investigated the effect of the material property in-homogeneity on the SCF due to a circular hole in functionally graded panels. Functionally graded materials are composites in which the material properties vary continuously as a known function of the spatial position. A parametric study was performed by varying the functional form and the direction of the material property gradation. The results from parametric study showed that the SCF is reduced away from Youngs modules .In exponential functionally graded materials the variation in the in-homogeneity length scale influences the SCF the most.

      Mittal and Jain [9] analyzed the effect of fibre orientation on stress concentration factor in fibrous plate with

      central circular hole under transverse static loading by using two dimension finite element methods.

      Rao et. al.[10] evaluated the stress around square and rectangular cutouts in symmetric laminates. It has been analyzed that the maximum stress and its location is mainly influenced by the type of loading.

      Extensive literature has been published on analysis of stress concentration and mitigation of stress concentration. However research work reported in the area of effect of material properties on SCF is limited.

   2. PROBLEM DESCRIPTION

      The five models of plate have been considered for analysis of effect of elastic constants of different composite materials on Stress Concentration Factor.

      Model1 is the Plate with central circular hole for orthotropic materials, Model2 is Plate with central circular hole and one set of auxiliary holes for orthotropic materials and Model3 is Plate with central circular hole and two set of auxiliary holes for orthotropic materials. Model4 is plate with central cavity of size 0.8*D by 8*D. Model5 is plate with central cavity of size 0.9*D by 8*D. All the models are shown in Fig.1 to Fig.5.

      A

      L

      D

      o

      Fig.1 Model1

      Fig.2 Model2

      Fig.3. Model3

      Fig.4. Model4

      Fig.5. Model5

      The model1 is the basic model of plate and has been optimized for mitigation of SCF by providing auxiliary holes and optimizing the shape of the main hole. The models considered are for optimum size of auxiliary holes and optimum shape of the main hole.

   3. FINITE ELEMENT ANALYSIS Rectangular plate of 400*100 has been considered

      for model1, model2 and model3. Rectangular plate of

      600*100 has been considered for model4 and model5. Element 8node 82solid has been taken and size of element has been taken as 1mm.

      As the modulus of elasticity in x-direction is varied EX/EY and EX/EXY changes keeping Poissons ratio and EY constant.

      Similarly, when modulus of elasticity in y-direction is varied EX/EY and EY/EXY changes keeping Poissons ratio and EX constant.

      The properties of materials has been varied from

      -50% to 50%, at an interval of 10 from the original values. The magnitude of properties has been considered and all the five models have been analyzedfor D/A=0.1 and 0.5.

   4. RESULT AND DISCUSSION

All the models have been analyzed for all the considered variables. The SCF for all the cases has been reported in tabular form and compared. The SCF in model1 for all the materials has been taken as reference for studying the variation in SCF with change in material properties and

reduction in SCF. The variation in SCF has been observed in model1 by changing the material properties of model1. As the Poissons ratio increases the SCF increases, this variation is very less.

The percentage reduction in SCF for all other models as compared to SCF of model1 has been determined and tabulated. The variation in percentage reduction in SCF by varying different material properties for same model is less. The maximum reduction in SCF has been reported for model4 for all the cases considered.

The material properties and variation in these properties has been shown in Table 1 to Table 7.

Table1. Properties of orthotropic materials

Table 2. Percentage variation in Poissons Ratio

Table 3. Percentage variation in EX/EY by varying EY

Table 4. Percentage variation in EX/GXY

Table 5. Percentage variation in EX

Table 6. Percentage variation in EY

Table 7 Percentage variation in GXY

on SCF.

The variation in Poissions ratio has very less effect The ratio Ey /Gxy is varying from 1.3 to 4.02 for all

As the magnitude of Ey varies by keeping all other properties constant, the value of Ex/Ey changes. The magnitude of Ey /Gxy is very less as compared to Ex/Ey and consequently its effect is less on SCF. The magnitude of Ex

the materials considered. Ey /Gxy varying from 1.72 to 45.94. The variation in Ey /Gxy values is maximum from 5 to 60.

varies for, -50% to 50% which changes the magnitude of both Ex/Ey and Ey /Gxy.

Table 8. Variation in SCF due to different Poissons ratio in E-glass Epoxy models for D/A=0.1

Table 9. Variation in SCF due to different EX / Ey in E-glass Epoxy models for D/A=0.1

Table 10. Variation in SCF due to different EX / GXy in Eglass Epoxy models for D/A=0.1

Table 11. Variation in SCF due to different EX in E-glass Epoxy models for D/A=0.1

For E-glass epoxy the percentage variation in SCF in Model1 for D/A=0.1 when changing the magnitude of Poission;s ratio from -50% to 50% is very less . The variation in SCF is very less in model1 when varying poissons ratio in all cases. It is reported as -.0.45% to 0.45% .As the value of Poission;s ratio decreases so does the value of SCF. The effect of reducing SCF is insignificant as it comes to just about 2%. The maximum reduction on SCF has been reported for Model 4.

As Ex/Ey changes the percentage variation in SCF is

-2.4% to 3.4% As the ratio Ex/Ey increases the SCF decreases . The percent reduction in SCF from model 2 to model 5 increases for both the cases of Ex/Ey that is increasing or decreasing the Ex/Ey causes very less percentage change as compared to the actual values of properties of materials .It is 3% to 4 % .

By changing the ratio Ey /Gxy the percentage variation in SCF for Model1 varies from 14.5 % to -4.1%.

The percentage variation in Ex reported in SCF in Model 1 is from -9% to 14.3% .As the Ex decreases the SCF also decreases and as the magnitude of Ex increases the SCF decreases .The reduction in SCF varies from 2% to -2% in Model 2,-17% to 8% in Model 3 , -16% to 9% in Model 4 , – 18% to 10% in Model 5 . This shows that as the Ex decreases

the SCF increases and the percentage reduction in SCF reported is les as compared to the SCF for original material. This percentage deviation in SCF is maximum as compared to the variation due to other material properties.

Table 12. Variation in SCF due to different Poissons ratio For Boron Epoxy

Table 13. Variation in SCF due to different EX / Ey For Boron Epoxy

Table 14. Variation in SCF due to different EX / GXy for Boron Epoxy

Table 15. Variation in SCF due to different EX for Boron Epoxy

Poissions ratio variation in Boron Epoxy Models for D/A=0.1 has negligible effect on SCF. The percentage reduction in SCF is maximum in Model 4.As compared to the other materials it has 46% reduction in SCF.

The effect of percentage reduction in SCF is very less when changes are made in Ex/Ey. The

percentage reduction in SCF varies from 4% to 8% when changing Ex/Gxy from -50% to 50%.

Variation in SCF of Model 1 is 12.8% to -801%

.The percentage reduction in SCF varies from 1% to 4 %, The SCF reduces as Ex decreases and SCF increases as Ex reduces.

Table 16. Variation in SCF due to different Poissons ratio for Boron Aluminium

Table 17. Variation in SCF due to different EX / Ey for Boron Aluminium

Table 18. Variation in SCF due to different EX / GXy for Boron Aluminium

Table 19. Variation in SCF due to different EX for Boron Aluminium

In Boron Aluminium Model 1 the percentage variation in SCF is very less as the Poissions ratio varies , the reduction in SCF reported for Model 2,3,4,and Model 5 also varies very less with changing Poission;s ratio.

The maximum percent reduction in SCF has been reported for Model 4. Ex/Ey ratio has been varied from -50% to

50% by changing Ey and keeping other properties constant. The variation in SCF for Model 1 has been reported as 4% to -3%. the SCF decreases as Ex/Ey effect is maximum in case of Model 5 .The percentage reduction in SCF reported is 2.5% to 30% .The change in Ex /Gxy is 5.7% to -7.7% on SCF in Model 1.The variation in Ex effects the most on SCF.

Table.20. Variation in SCF due to different Poissons ratio for Graphite Epoxy

Table 21.Variation in SCF due to different EX / Ey for Graphite epoxy

Table22. Variation in SCF due to different EX / GXy for Graphite epoxy

Table 23. Variation in SCF due to different EX for Graphite epoxy

The reduction in SCF is 46% in Model 4 .The effective change in Poissions ratio is negligible on SCF.

Effect of variation in Ex is 11% to -7.6%. The reduction in SCF is very less. The reduction in SCF is maximum in Model 4.

All the Models have been analyzed by considering all the variation in material properties for D/A=0.5.The variation in SCF has been reported in tabular form.

Table 24.Variation in SCF due to different Poissons ratio for E-Glass for D/A=0.5

Table 25. Variation in SCF due to different EX / Ey for E-Glass Epoxy for D/A=0.5

Table 26. Variation in SCF due to different EX / GXy for E-Glass Epoxy D/A=0.5

Table 27. Variation in SCF due to different EX for E-Glass Epoxy for D/A=0.5

Table 28. Variation in SCF due to different Poissons ratio for Boron Epoxy for D/A=0.5

Table 29. Variation in SCF due to different EX / Ey for Boron Epoxy for D/A=0.5

Table 30. Variation in SCF due to different EX / GXy for Boron Epoxy for D/A=0.5

Table 31. Variation in SCF due to different EX for Boron Epoxy for D/A=0.5

Table 32. Variation in SCF due to different Poissons ratio for Boron Aluminium for D/A=0.5

\

Table 33. Variation in SCF due to different EX / Ey for Boron Aluminium for D/A=0.5

Table 34. Variation in SCF due to different EX / GXy for Boron Aluminium for D/A=0.5

Table 35. Variation in SCF due to different EX for Boron Aluminium for D/A=0.5

Table.36. Variation in SCF due to different Poissons ratio for Graphite Epoxy for D/A=0.5

Table 37. Variation in SCF due to different EX / Ey for Graphite Epoxy for D/A=0.5

Table 38. Variation in SCF due to different EX / GXy for Graphite Epoxy for D/A=0.5

Table.39. Variation in SCF due to different EX for Graphite Epoxy for D/A=0.5

Variation in SCF is small for D/A=0.5 as compared to D/A=0.1 in all the cases. The reduction in SCF is very less in Model5. In some cases the SCF increases as can be served when comparing the results of Model1 and Model5 and this is due to changing material properties.

V .CONCLUSIONS

The variation of Poissons ratio effects very less on SCF of model1. The reduction in SCF in different models is almost same when varying Poissons ratio.

The variation in elastic constants effects the SCF, the variation in EX effects the most on SCF . We can conclude that the modulus of elasticity of the material in the direction of loading effects the most on SCF. The reduction in SCF is maximum in Model4 in all the cases and minimum in Model5.

The effect of variation in material properties is most in Model4. We can conclude that the proposed Model4 is best for highest mitigation in SCF.

As the size of hole increases the mitigation in SCF is less by the proposed methods. In Model5 the SCF increases as compared to Model1 by varying material properties.

ACKNOWLEDGMENT

The authors acknowledge the institute authorities for supporting the present work.

REFERENCES

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