DEPARTMENT OF MECHANICAL ENGINEERING

CODE AND TITLE OF COURSEWORK

Course code:

MECH 3004

Title:

FINITE ELEMENT MODELING FOR STRESS ANALYSIS

STUDENT NAME: ERKEL, DANIEL

DEGREE AND YEAR: EBF, 2nd YEAR

LAB GROUP: -

DATE OF LAB. SESSION: -

DATE COURSEWORK DUE FOR SUBMISSION: 16/11/2012

ACTUAL DATE OF SUBMISSION: 16/11/2012

LECTURERS NAME: DR REBECCA SHIPLEY

PERSONAL TUTORS NAME: DR KEVIN DRAKE

RECEIVED DATE AND INITIALS:

I confirm that this is all my own work (if submitted electronically, submission will be taken as confirmation that this is your own work, and will also act as student signature)

Signed: Daniel Erkel

Contents

1 Introduction 21.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Finitie Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Results 42.1 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 FEA Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Calculation Using Experimentally Obtained Formulae . . . . . . . . . . . . . . . . . . 7

3 Discussion of Results 73.1 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 FEA Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Comparing Results from the Different Analyses 84.1 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Conclusions 9

6 Appendices 11

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Applied Mechanics - Report on Coursework

Finite Element Analysis of Stress Concentrations in ANSYS

Mechanical APDL

Daniel Erkel, 3rd year, EBF

Abstract

A simply loaded beam with stress concentrations produced at two notches opposite each otherand at two shoulder fillets was tested in ANSYS Mechanical APDL. The model built was fixedat one end and a moment was applied on the other, defining the normal stresses arising in purebending. Finding a mesh through a convergence study, a solution was obtained and results werecompared to calculations using handbook formulae. The main findings of the report were: resultsobtained in ANSYS Mechanical for a simply loaded beam in pure bending (plane stress) arevalid and are capable of well representing stress concentrations as they correspond to calculationsusing experimental formulae (with only minor discrepancies); finding the correct mesh is crucial inFEA solutions; a trade off has to be made between finding the most accurate mesh and limitingcomputational time, especially in cases where several simulations have to be performed.

1 Introduction

1.1 Objectives

When a beam of brittle material is subjected to dynamic loads, stress concentrations have greatimportance, as these limit the components yield strength. Such concentrations appear where discon-tinuities exist in the beam and result in high values of localized stress. Examples can be notches, holes,sharp edges, shoulders, steps and other types of abrupt changes in dimension. [1]. Maximum stressesnear stress concentrations can be calculated analytically using formulae derived experimentally. Suchformulae for certain geometries with tables and graphs helping to determine localized stress values canbe found in various publications, examples of which are referenced in this report (e.g. [2]). With theadvent of finite element analysis and computational methods an alternative method became availablefor evaluating the strength of engineering designs. Present reports aim is the demonstration andvalidation of a finite element method based solution for a cantilever beam with notches and shoulderfillets subjected to loading at one end. The solution is obtained in ANSYS Mechanical APDL, a finiteelement analysis (FEA) package used widely, and is then compared to hand calculations employingaforementioned formulae.

1.2 Theory

1.2.1 Beam Theory

Explaining beam theory in detail, a subject fundamental to engineering design, is beyond the scope ofthis report. The beam investigated in the report is a cantilever beam in pure bending, with one endfixed and the other subjected to a constant bending moment. All assumptions of pure bending holdtherefore: the beam is initially straight and unstressed; the beams material is perfectly homogeneousand isotropic; plane cross sections remain plane before and after bending; every cross-section in thebeam is symmetrical about the plane of bending; no resultant force perpendicular to any cross-section; the elastic limit is nowhere exceeded; the Youngs modulus of the material is the same intensions and compression [3]. The beam analysed here is symmetrical to its centerline, which coincideswith the neutral plane. A clockwise moment acting on its right end (in the general sign conventiontermed negative) results in the section above the neutral plane being in tension and the one below

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in compression. Stresses x above the neutral plane are therefore positive, below it negative and ofcourse zero at the neutral plane. This relationship between moment and stresses acting in a purebending case is used later in defining the loading in ANSYS.

1.2.2 Stress concentrations

The effect of stress concentrations is explained through the vivid example of flow of a fluid in Mechanicsof Engineering Materials by Benham, Crawford and Armstrong [3]. The effect of a force or momentapplied on a simple strut is transmitted through the body via the medium of stress in adjacentelements. If the imaginary lines of transmission are taken as analogous to streamlines in a fluid flow,it is readily understood intuitively then, why are there high localized stresses near a sudden change ofdimension, such as a notch or shoulder fillet. The peak stress at a point such concentration can be farin excess of the reference (average) stress in the body overall. While yield stress exceeded at this areamay not cause complete failure of the component if it is ductile (since local yielding and plastic flowleads to stress redistribution hence relieving the concentration) in a brittle material the concentrationremains up till fracture. It is true in both cases however, that stress concentration can reduce theoverall strength of the component. To what extent depends on the structure of the material and howbrittle it is. One other aspect which cannot be ignored in ductile materials either is fatigue, wheresuch concentrations can have serious adverse effects on strength. The exact theoretical analysis ofconcentrations is very complex. Before finite element methods and software using them, calculationwas possible using handbooks containing formulae and tables, the result of theoretical solutions andexperiments (using for example the technique of photoelasticity [4]) [3]. Stress concentrations arealso relevant in another aspect of the analysis presented here: load is applied at the boundary ina manner corresponding to how internal stresses are distributed within the body in the theoreticalsolution of pure bending to achieve an exact solution. St Venants principle states that at sectionsdistant from the surface of loading the localized effect is negligible and statically equivalent systemsof forces produce the same stresses on the same area [3]. Lengths not determined by ratios in theassignments description were chosen in a fashion to account for this, therefore applying forces insteadof pressures on the loaded end would still result in a correct solution, however, for practical reasons,the simpler method of determining stresses on the boundary was followed.

1.2.3 Finitie Element Analysis

With its development beginning in the 1940s, the finite-element method has become widely used todayin the industry. FEA divides (discretizes) the continuous structure into small, but finite elastic sub-structures (called elements). Then the continuous elastic behaviour of each element is calculated usingpolynomial functions and matrix operations, taking into account the elements material propertiesand geometry. Each element is connected by nodes, a fundamental entity in the calculations. Elasticproperties and boundary conditions are established at the node and it is also the point where theforce is ultimately applied. Each node possesses a certain of degree of freedom (DOF ), the numberof independent rotational and translational motions that can exist at that point. The locally definedelements are assembled through their common nodes into the global structure, a system defined by anoverall matrix. Loads and boundary conditions can be applied within the element, on the surface ofthe element (on lines, etc.) or at the node. Once the global structure exists, the unknown displacementdegrees of freedom are determined and stresses and strains can be calculated through equations ofelasticity [5]. Fast evolution of computer technology enabled the rapid processing of such calculationsand there are many available FEA packages today. Being a numerical technique, the method is proneto errors, however these can be minimized to achieve a sensible and valid solution. Possible sourcesof error will be discussed later.

1.3 Methodology

A model for the beam presented below is created in ANSYS Mechanical. Dimensions are determinedfrom fixed ratios given in the assignment (for the diagram see section 6). The beam is then constrained

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on one end, fixing the displacement on the left side to 0. A load is applied on the other end by waysof defining a pressure (normal stress) starting from a set value (10) on the topmost point, decreasingto zero at the neutral plane in the centre and increasing to a full value again with opposite sign (-10)at the lowest point.

Figure 1: Diagram showing the model used

Figure 2: Diagram showing setup described in the assignment and the one defined in ANSYS

Boundary conditions fixed, the element type and the material is chosen (Plane183, 8 node and alinear, elastic, isotropic material with E = 1 and = 0.3 respectively) the beam is meshed, initiallywith a Global Size set to be comparable to the radius of the notches and the shoulder fillet (0.005)and no Smart Size (improved mesh near boundaries) employed. It is clear from observing the meshnear the notches that the solution will not be too precise as the arc of the notch cannot be accuratelyreproduced with such a coarse mesh and large element size. A convergence study is conducted withgradually improving the mesh, by first decreasing the element size, then using the Smart Size mesh,to find which mesh produces results (maximum von-Mises stress) accurate to 2 decimals. Once thecorrect mesh is found, values are obtained for the maximum (von-Mises) stress at the notches andthe shoulder and the stress concentration factor is calculated for both, which is then compared to acalculations based on handbook formulae.

2 Results

2.1 Convergence Study

Results of the convergence study are presented on the following graphs. The first graph shows theconvergence of the maximum stress normalised by the applied stress (which happens to be the stressconcentration factor, Ktn at the notch, where the maximum stress arises) plotted for the case where

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only the global size of the mesh was lowered gradually.

Figure 3: Convergence in Ktn changing the Global Size of the Mesh (with no Smart Size)

It can be seen that close to 0.001, one-fifth of the radius r, Ktn is converging to 2.4. However as themesh size was lowered to half of this value, computational time for setting up the mesh has increasedsignificantly. Furthermore, the Ktn has not reached a stable value, even at a mesh size of 0.0004. Toachieve faster convergence, the Smart Size option was used, which computes the approximate elementedge lengths before generating the mesh for each line on every area and then refines edge lengthsto account for curvature. This creates a better mesh near the notches and shoulders on the modelanalysed in this report. The second iteration was started from a mesh with a global size of 0.002using a Smart Size value of 4 (where 10 is the most coarse 1 is the finest). Then the global size waslowered to 0.001 with changing the Smart Size from 4 to 1 in each case. It can be seen that using aglobal size of 0.001 Ktn converged reaching its final value (2.408) immediately.

Figure 4: Convergence in Ktn changing the Mesh Size (using Smart Size)

The following picture shows the structure of the final mesh near one of the notches, comparing it tothe initial mesh to demonstrate the difference in recreating a curvature.

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Figure 5: Comparing the initial mesh (on the left) to the final, refined mesh (right))

2.2 FEA Analysis

The contour plot obtained from plotting the von-Mises stress for the solution with the final mesh isshown on the next figure:

Figure 6: Contour plot showing von-Mises stress, obtained for the solution using the final mesh

The values obtained from this are presented in the table below

Table 1: Table showing results obtained for Ktn from the FEA Analysis

Location Von-Mises Stress Normalised stress (equal to Kt)

Bottom notch 24.0800 2.40800Top shoulder 19.1146 1.91146

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These are also shown on the following diagram:

Figure 7: Contour plot showing von-Mises stress at the bottom notch and the top shoulder

2.3 Calculation Using Experimentally Obtained Formulae

To validate the results above, formulae obtained from Ref [2] and [6] were used. For the case withtwo notches on a beam, where the notches have a radius r equal to h (see Figure 1) Ref [2] gives thefollowing equation:

Ktn = 3.065 6.637(

2h

D

)+ 8.229

(2h

D

)2 3.636

(2h

D

)3For the shoulder fillet a similar equation can be calculated for the model using a formula from Ref [6]:

Kts = 1.976 1.925(

2h

D

)+ 0.906

(2h

D

)2+ 0.043

(2h

D

)3These values can then be used to calculate the maximum stress from:

Kt =maxnom

Coefficients for the second equation were obtained using the table from [6], which can be found inSection 6. The stress concentrations calculated are the following:

Table 2: Stress concentration factors calculated from handbook formulae

Location Stress Concentraion Factor, Kt

Top shoulder 1.792603Bottom notch 2.479954

3 Discussion of Results

3.1 Convergence Study

It is assumed that choice of element type Plane183 provided a faster convergence, as this element isan 8-Node Structural element, which is capable of representing deformations more accurately even ata coarser mesh than Plane182, which only has 4 Nodes and is not capable of creating a degeneratedtriangular element [7]. Choosing the Smart Size option also ensured faster convergence, as it can be

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seen from the graphs in the previous section, which is due to the fact that the greatest difference usingdifferent mesh sizes appeared at the most crucial area, in the curvature of the notch. The curvaturewas better captured with the Smart Size option, because this created smaller mesh elements than theglobal size near the arc of the shoulder and the notch [8].

3.2 FEA Analysis

To check the boundary conditions, an animation was created for the solution using the final mesh.Pictures of this loading sequence can be found in Section 6. The maximum stress appeared at thepoint where it was expected according to [2], at the central node of the bottom notch, which alsoshowed that the correct mesh size was found, as for different meshes before, the location of maximumstress was alternating between the top and bottom notches. For the shoulder fillet, the stress wasgreater on the top one in the final mesh, which means that the tension is slightly greater there than thecompression in the bottom section. The difference in stresses between top and bottom sections is verysmall, as it can be seen from the next table: The small difference is reasonable, given the condition of

Table 3: Stresses found at all four stress concentrations

Location Stress value Normalised

Top notch 24.075 2.408Bottom notch 24.080 2.408Top shoulder 19.115 1.911

Bottom shoulder 19.103 1.910

pure bending, where the material should behave the same in both tension and compression [1] (underrealistic loading conditions, well represented by the analysis, there is a difference).

4 Comparing Results from the Different Analyses

In the following table results for the stress concentrations found in the FEA analysis are compared tothose obtained using the handbook formulae from Refs. [2] and [6]:

Table 4: Comparison of results obtained using the two different methods

Kt

Location FEA Handbook formulae Error in Kt

Notch 2.40800 2.47995 2.9%Shoulder fillet 1.91146 1.79260 6.6%

It can be seen that there is only a small difference for results obtained for the top notch, less than 3%and a difference still below 10% for the shoulder fillet, which means that the FEA analysis was capableof recreating valid results for the stress concentrations. The greater difference for the shoulder filletresults from the fact that the handbook formulae was devised for a loading condition slightly differentfrom this (a moment applied on a stepped beam with shoulder fillets on both sides). The smallererror in the result for the notch shows that there was a closer relationship between the conditionrepresented by the handbook formulae and the setup in ANSYS.

4.1 Sources of Error

Two types of errors occur in the FEA calculations: computational errors, due to round-off errors infloating-point calculations and discretization errors due to limitations of how certain geometries (suchas arcs, round surfaces, etc.) can be represented with the given element type (e.g. a quadrilateralelement). Software developers aim at minimizing the former, whereas the latter is minimized by the

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analyst, by aiming at a certain quality of mesh, the assembly of the finite elements that well representsthe original structure and the solution obtained is within a certain convergence criteria [5]. This iswhy finding the right mesh was crucial in the present report (just as it is important in all analysesdone in FEA software). Discrepancies could also result from the fact that the original experimentsthrough which the handbook formulae were defined possibly also contained errors. Finally, as saidbefore, the formulae used were defined for configurations with minor differences from what was usedin this report (already the fact that two stress concentrations were combined could result in errors,although this was taken into consideration before, when St Venants principle was discussed as theconcentrations were far enough from each other).

5 Conclusions

Findings of the report successfully proved that results obtained from the FEA Analysis performed inANSYS Mechanical APDL are valid as compared with experimentally results only small discrepancieswere found, which could have resulted from minor differences between the setup used in the ANSYSsimulation and that used in the experiments. Finding the correct mesh was crucial, this was achievedthrough employing the Smart Size mesh option. The right mesh was also important from anotheraspect relevant to all computer simulations: computational time. Simply lowering the global size ofthe mesh resulted in longer and longer time necessary to set up the mesh. This is why using theSmart Size was a better solution and this is also why using the more accurate 8-Node element wasalso justifiable.

References

[1] J. M. Gere, Mechanics of Materials. Brooks/Cole - Thomson Learning, 6th ed., 2004.

[2] W. D. Pilkey and D. F. Pilkey, Petersons Stress Concentration Factors. John Wiley & Sons,3rd ed., 2008.

[3] P. P. Benham, R. Crawford, and C. G. Armstrong, Mechanics of Engineering Materials. PearsonEducation Limited, 2nd ed., 1996.

[4] J. B. Hartman and M. M. Leven, Factors of stress concentration for the bending case of filletsin flat bars and shafts with central enlarged section, Proceedings of the Society for ExperimentalStress Analysis, vol. 9, no. 1, pp. 5362, 1951.

[5] R. G. Budynas and J. K. Nisbett, Shigleys Mechanical Engineering Design. McGraw-Hill, 8th ed.,2006.

[6] W. C. Young and R. G. Budynas, Roarks Formulas for Stress and Strain. United States: McGrawHill, 7th ed., 2002.

[7] ANSYS, Mechanical apdl element reference, Southpointe 275 Technology Drive, CanonsburgPA 15317, Release 14 2011.

[8] ANSYS, Modeling and meshing guide, Southpointe 275 Technology Drive, Canonsburg PA15317, 2011. Release 14.

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List of Figures

1 Diagram showing the model used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Diagram showing setup described in the assignment and the one defined in ANSYS . . 43 Convergence in Ktn changing the Global Size of the Mesh (with no Smart Size) . . . . 54 Convergence in Ktn changing the Mesh Size (using Smart Size) . . . . . . . . . . . . . 55 Comparing the initial mesh (on the left) to the final, refined mesh (right)) . . . . . . . 66 Contour plot showing von-Mises stress, obtained for the solution using the final mesh . 67 Contour plot showing von-Mises stress at the bottom notch and the top shoulder . . . 78 Loading sequence for the solution using the final mesh . . . . . . . . . . . . . . . . . . 119 Close-up showing the final mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Chart used for calculating the stress concentration at the shoulder fillet, obtained from [6] 1211 Chart used for calculating the stress concentration at the notch, obtained from [2] . . 12

List of Tables

1 Table showing results obtained for Ktn from the FEA Analysis . . . . . . . . . . . . . 62 Stress concentration factors calculated from handbook formulae . . . . . . . . . . . . . 73 Stresses found at all four stress concentrations . . . . . . . . . . . . . . . . . . . . . . . 84 Comparison of results obtained using the two different methods . . . . . . . . . . . . . 8

10

6 Appendices

Figure 8: Loading sequence for the solution using the final mesh

Figure 9: Close-up showing the final mesh

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Figure 10: Chart used for calculating the stress concentration at the shoulder fillet, obtained from [6]

Figure 11: Chart used for calculating the stress concentration at the notch, obtained from [2]

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