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http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=7&IType=2 Journal Impact Factor (2016): 9.2286 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6340 and ISSN Online: 0976-6359
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DETAILED STUDIES ON STRESS
CONCENTRATION BY CLASSICAL AND
FINITE ELEMENT ANALYSIS
Prof. S. S. Deshpande, P. N. Desai, K. P. Pandey and N. P. Pangarkar Department of Mechanical Engineering,
Keystone School of Engineering, Savitribai Phule Pune University, India
ABSTRACT
Stress concentration is very important aspect in all aspects of mechanical design, it arises due to geometric discontinuities in the structure common examples are openings in pressure vessels and piping it is observed that the maximum stress is much more than the nominal stress .The conventional methods for determining the stress concentration factors are the empirical formulae mentioned in design handbooks such as Roark’s formulae for stress and strain this formulas can also be expressed in terms of graphs the finite element method a regarded as the third dimension in engineering plays very important role in the overall design process, this is mainly because it reduces the dependence on standard available geometries, experimentation and most importantly the time and cost associated with it however it has been observed that the finite element analysis results are depended on mesh quality parameters and this fact has not been studied thoroughly the main aim of the present study is to consider a standard configuration that is a plate with a circular hole in it subjected to axial tension, We present here a number of meshes such as manual(ruled and splined)and automatic and study the impact on the results. Further we also present the comparison of several 2D elements such as triangular, quadrilateral and 3D elements such as hexahedral. We present the FEA solution and compare it with the exact analytical solution, it is hoped that the design engineers and the CAE community will benefit from the present study.
Key words: Finite Element Analysis, Stress Concentation Factors, Cae, Machine Design.
Cite this Article S. S. Deshpande, P. N. Desai, K. P. Pandey and N. P. Pangarkar, Consistent and Lumped Mass Matrices In Dynamics and Their Impact on Finite Element Analysis Results. International Journal of Mechanical Engineering and Technology, 7(2), 2016, pp. 148–167.
1. INTRODUCTION
Finite element Analysis (FEA) plays a very important role in the overall mechanical design process. It has been applied successfully to almost all kinds of problems and the complexity of problem ranges from static to dynamics and multyphysics problems. Today much of the work in CAE (Computer Aided engineering) is done with the FEA softwares such as ANSYS, NASTRAN, ABAQUS etc. Stress concentration in machine elements [1,4,5 ] has been of much academic focus in the textbooks and the data on how much is the rise in the stress level as compared to the nominal stress at the discontinuity is available for some standard configurations in the literature [2,3,6,7,8,9] . It is our experience that the industrial Finite Element Analysis user is least aware of the standard data and most of the times the computations are done by blind use o the selection of an element and good mesh quality. This paper presents a comparison of the stress concentration in a rectangular plate due to a circular hole and we compare here several elements and different types of meshes. We notice that it is very much beneficial for a practical finite element to be aware of this and then they can predict these effects which are of utmost importance in all industry verticals such as automobile, aerospace , pressure vessel, process piping to name a few. Stress concentration is not discussed in depth in any of the Finite Element textbooks [10, 11, 12] in academics and hence it is important for an user on how the element choice and mesh quality affects the results. We believe that the CAE Community will benefit from the present detailed study when it comes to solving practical problems.
2. THE PROBLEM
The problem considered here is a plate of length 100 mm and width 40 mm and with a thick ness is subjected to a tension of 900N and this force is applied as uniformly distributed force. The thickness of the plate in the present problem is taken larger so that both the solid and shell elements of various types can be compared. The performance of solids is not discussed much in the literature and this is the reason we have taken a configuration which can be useful to a variety of problems in Finite Element Analysis. Fig.1 represents the loading and the dimensions.
Figure 1 The Geometry 40mm 100mm Φ = 20mm A A B B
3. THE EXACT SOLUTION
The Exact Solution of these problem has been given in Roark’s Formulas for stress and strain [5] and is given by following set of equations,
(1)
In this formula, W= Width of the Plate D = Diameter of the Hole
(2)
(Nominal stress at hole section)
Where, t= Thickness of the plate
(3)
(Maximum Stress at the hole)
Figure 2 The computations for the stress levels are as follows
The stress away from the hole near the loading (say section A –A) is given by =
= 2.25 MPa
The nominal stress at the section B-B due to the hole cut out is given by
=
= 4.5 MPa
The stress at the point c and d which face the maximum stress levels is given by
=Kt *
=2.25MPa
=4.5MPa
We use the equation (1) to calculate Kt as follows: = 2.158
Thus the exact value of the maximum stress at the hole at point C should be 9.714 MPa.
The representation of the stress levels in the plate at different sections and points is shown in Fig 2.
4. FINITE ELEMENT ANALYSIS RESULTS
We have used several finite elements for analyzing the stress concentration. The software used for finite element modeling i.e. preparation of the mesh is HYPERMESH and we have used a combination of all available algorithms like ruled / mapped meshing , spline mesh, automesh for generating the elements in 2- Dimensions . Three dimensional elements are also used in the analysis. The analysis is done using the MSC NASTRAN solver. The results are post processed in Hypermesh. We have taken only the von-Mises stress plots as they are the most important. Further mesh refinement (approximately the number of elements is doubled) has been carried out wherever it was felt that appropriate (i.e. say refined mesh of linear quadrilateral vs. coarse mesh of second order quadrilateral) to study whether the element gives a higher level of accuracy. The deflection of the hole is not presented here as it would lead to very large number of plots but a comparison table has been given so that the user can get an idea on the accuracy. The question on why two meshes of coarse and fine are compared can be answered as in the industry , where stress concentration is not there , one often uses element sizes ranging from 10 mm to say 40 mm and in the region of stress concentration the size typically used ranges from 3 to 6 mm . In the coarse mesh results presented here average size is about 10 mm and fine mesh has an average size of 5 mm.
4.1 SHELL ELEMENTS
4.1.1 Quadrilateral element (First Order, 4 noded )
This is a shell element in NASTRAN it is identified as CQUAD4. One can see a fantastic mesh quality Fig. 3 and the stress plots are given in Fig. No. 4
Figure 4-a Deformed shape plot
Figure 4-b Displacement plot
Figure 4-c Stress Diagram of coarse Quadrilateral mesh
Mesh refinement study was carried out and the results of which are presented in Figures Fig. Nos. 5 and 6.
Figure 5 Finemesh of 4 noded quadrilateral elements
Figure 6 Stress Diagram of 4 noded quadrilateral
4.1.2 Quadrilateral element (second order, 8 noded)
This is an eight noded shell element of the serendipity family as usually in FE softwares we don’t have Lagrangian elements with which are fully second order accurate with an internal node (i.e. 9 noded) The mesh remains the same as shown in the Fig. No. 3 but only the element order has changed. Stress plots are shown in Fig. 7
Figure 7 Coarse mesh of 2nd order quadrilateral
Figure 8 Stress Diagram of 2nd order coarse quadrilateral
4.1.3 Traingular element (First order / 3 noded )
This is a three noded finite element and in NASTRAN terminology called as CTRIA3 ) . It is not so commonly used as it has been found tp be much stiffer i.e predicts lesser displacement and stress and hence the results can be misleading.
The corresponding mesh and the stress plot is shown in Fig. no. 8 and 9
Figure 10 Mesh of coarse 3 noded triangle
Figure 11 Stress Diagram of coarse 3 noded triangle
Various types of refined meshes are shown in the figures given below. These are dependent on the various algorithms used for mesh splitting. We have found that in one case it leads to a polyhedral type of mesh used very much in Computational fluid dynamics nowadays. Although it is having lots of advantages with respect to computational time by using a Finite Volume Method, we have to keep in mind that such kind of meshes are not used in Structural mechanics. We also wish to keep in mind that there are some studies going on in Structual mechanics on showing that Finite Volume Method can be used for structural mechanics but this is limited only to one dimensions and on the other hand Finite element methods have been widely used
in the area of Computational Fluid dynamics. The mesh and the stress plots of these refinements are shown in Fig Nos. 11 to 14.
Figure 12 Fine Mesh of coarse 3 noded triangle
Figure 13 Stress Diagram of finemesh 3 noded triangle
Figure 15 Stress plot of Another refinement of TRIA mesh Looking like a Polygon
4.1.4 Traingular element (Second order / 6 noded)
This is a 6 noded element in NASTRAN and is identified as CTRIA6. It is a remedial action to remove all the drawbacks of the linear triangular element and is commonly used in the literature.
The corresponding mesh and the stress plot are shown in the figure nos. 15 and 16
Figure 16 Coarse 2nd order Triangular Mesh
4.1.5 Ruled Quad mesh
In this type of mesh also called as mapped mesh, there is a one to one correspondence between the nodes on the hole boundary and the nodes on the outer boundary. This mesh looks good but apparently it has highly distorted quadrilaterals. This type of mesh is typically used in Computational Fluid Dynamics simulations as there one follows an Eulerian approach and the mesh doesn’t deform continuously as in the Structural mechanics problems.
Figure 18 Ruled Qadrilateral Mesh (4 noded)
Figure 19 Stress plot of Ruled Quadrilateral mesh (4 noded )
4.1.6 Automesh results
This mesh algorithm works only when there is a surface and it is difficult to control the mesh quality against that of the manual mesh created meshes. The user gets a feel that he has used all quadrilaterals but the mesh gives some distorted elements as seen in the mesh plot.
Figure 20 Automesh of Quadrilaterals (4 noded )
Figure 21 Stress Plot of Automeshed Quadrilaterals
Figure 23 Stress Plot on Fine Automeshed Quadrilaterals
4.1.7 Solid Elements
The Analysis was done using the same number of QUAD mesh extruded into HEXAHEDRONS Coarse And Fine Mesh results are shown in the figures,23 and 25 . The other elements considered were Tetrahedrons and Pentas .
Figure 24 Coarse Hexahedral mesh
Figure 26 Fine Mesh of Hexahedrons
Figure 27 Stress Diagram for fine hexahedral mesh
The tetrahdraons are of two type. The four noded ones (TETRA4) are first order accurate and the ten noded ones (TETRA10) are second order accurate. It is usual to use second order tetrahedrons as the first order ones are quite stiff in practice. That is they give very less displacement as compared to the exact displacement. Most of the automeshing in structural mechanics is done with second order tetras whereas for fluids most of the meshing is done using first order accurate tetras as the equations involve only first order derivatives.
The coarse and fine meshes for the first order tetrahedral elements are shown in Fig. Nos. 27, 29 and Fig. 31 shows the coarse mesh for a second order accurate tetrahedral element. The corresponding stress plots are given in Fig. Nos. 28, 30 and Fig.No. 32.
Figure 28 Coarse Mesh of 4 noded Tetrahedrons
Figure 29 Stress Plot of coarse 4 noded tetrahedral mesh
Figure 31 Stress Plot of 4 noded tetrahedral fine mesh
Figure 32 Coarse Mesh of 10 noded tetrahedrons
Figure 33 Stress Plot of coarse 10 noded tetrahedral mesh
The Pentas are 6 noded elements commonly called as wedge or prism elements and are not used commonly in structral mechanics as they are very stiff. However
they have to be used as filler elements as complex geometries with hexahedral meshes present lots of problems at the interfaces .
Figure 34 Coarse Mesh of Pentas .
Figure 35 Stress plot of coarse penta mesh
5. RESULTS AND DISCUSSIONS
We now analyse the results obtained on various mesh types as discussed above. We have also taken a note of maximum deflection but the main motive of this paper is to analyse the stresses and hence wherever it is felt necessary we will discuss on the issue of displacement .As already discussed the practical mesh sizes in the industrial Finite element sizes range from coarse (average size is about 10 mm) to fine (average size is about 5 mm). This also depends whether the mesh is for stress analysis, vibration analysis or crash. In crash analysis the user has to go with a coarse size as otherwise the time step becomes to small. In vibration analysis one has to predict the natural frequencies correctly and the representation of the mass becomes more important. one can go with an average size of 10 mm . It is in stress analysis where
the mesh has to be a mix of coarse (away from stress concentration region) and fine ( near the stress concentration ) and hence we have compared two types of meshes for the elements. Deflection is not so important but it depends on the problem one is solving. e,g. If the problem is to determine say the cylinder bore- distortion in an Internal combustion engine in thermal analysis, then the major focus should be on the displacement. In the present paper the main interest is stress which can also be used further for fatigue analysis.
Sr.
No. Element
Mesh and the type
Displacement obtained by FEA
solution in mm
Von-Mises Stress obtained by FEA solution in MPa
1 QUAD4 Coarse Mesh 0.149e-2 6.03
2 QUAD4 Fine Mesh 0.161e-2 7.50
3 QUAD4 Coarse Ruled
Mesh 0.1e-2 5.81
4 QUAD4 Fine Ruled
Mesh 0.11e-2 7.58
5 QUAD4 Coarse
Automesh 0.1e-2 5.10
6 QUAD4 Fine Automesh 0.11e-2 6.40
7 QUAD8 Coarse Mesh 0.165e-2 6.04
8 QUAD8 Fine Mesh 0.175e-2 7.47
9 TRIA3 Coarse Mesh 0.142e-2 7.29
10 TRIA3 Fine Mesh 0.142e-2 7.69
11
QUAD4 obtained from TRIA3
Fine Mesh
polygonal type 0.157e-2 8.20
12 TRIA6 Coarse Mesh 0.159e-2 6.91
13 TRIA6 Fine Mesh 0.163e-2 8.03
14 HEXA8 Coarse Mesh 0.154e-2 5.90
15 HEXA8 Fine Mesh 0.152e-2 7.37
16 TETRA4 Coarse Mesh 0.148e-2 7.68
17 TETRA4 Fine Mesh 0.142e-2 9.11
18 TETRA10 Coarse Mesh 0.162e-2 7.65
The major observations of the finite element analysis based on the table given above are as follows:
1. The ruled mesh of quadrilaterals is unable to predict the stresses due to worst mesh quality. The value of stress precdicted is much smaller than the exact value of 9.714 MPa. Such a kind of mesh should not be used for stress analysis. If a cfd result is available from such a mesh then that result should be appropriately interpolated before it can be used for structural analysis. Even though stress improves on a finer mesh, displacement is not and hence this comment.
2. If one increases the order of the element from QUAD4 to QUAD8 then displacement improves but the stress does not. This means that whether it is QUAD4 or QUAD8, reasonable stress representation is obtained only on a fine mesh with a better performance obtained by QUAD8 as compared to QUAD4.
3. The automesh results are also of the worst quality. Typically there is a blind practice followed by the industry user to use an automesh and make it fine but as one can see from the table, this belief is proven totally wrong. On a fine mesh, one doesn’t get the required level of accuracy and still the stress is represented with a very less value. 4. The TRIA3 performs much better as compared to QUAD4. However the industry and
academics opinion about this element is that it is too stiff. This is true only for bending. The present problem is of pure axial stress and a good performance of the element is seen for such problems. The polygonal looking type of mesh performs much better as compared to conventional refinement and this is not noted in the literature of finite elements.
5. Increasing the order of the element from TRIA 3 to TRIA6 doesn’t necessarily improve the stress. CTRIA 6 performs better on fine mesh with results comparable to fine mesh of TRIA3.
6. HEXA8 performance improves on a fine mesh as compared to coarse mesh. Industry belief is that hexahedral mesh is alsways better than the tetrahedral one is not necessarily true. If one can use the automesh option using TETRA10 then results are much accurate to the exact solution. Further for practical problems the time taken to generate a hexahedal of tetesh is an order of magnitude greater than the tetrahedral mesh. TETRA4 is better for stress analysis but displacement is predicted on the conservative side and further cannot represent bending properly . So in our view a fast algorithm for automesh with TETRA10 elements is much better as compared to HEXA8.
7. The PENTA element does not represent the bending and is too stiff for frequency calculations which is shown in the reference [13] and hence is not to be used for practical finite element analyses where the same kind of mesh is to be used further for vibration. For static stress analysis, results are much lesser than TETRA4 coarse and TETRA10 coarse mesh. This justifies that this element is to be used only as a filler element wherever necessary and should not be used in general.
6. CONCLUSION
We have presented a comparison of several meshes and element types in this paper for analyzing a standard problem of stress concentration in a plate. One can see that there is lot of variation of the results and if the user expects that he will get closer to the exact value by using a fine mesh then it is not true. Using a fine mesh is also not practical always. Thus the practical finite elementer has to keep in mind two things 1. appropriate choice of element and 2. Desired mesh quality with lesser of distorted elements. It is the authors’ observation sometimes that people simply fall in love with their meshes and then don’t worry on the accuracy. This can be misleading. e.g. the ruled mesh look fantastic but has severly distorted quadrilaterals and such types of
meshes should not be used by the CAE community. One should not have any biases in carrying out FEA as what may be appropriate for one type of analysis may not be for another problem. We feel that if the user carries out such kind of studies apriori , then he gets an idea of the accuracy. It is expected conclusions of this paper can help the industrial finite element analysis and also throws a new light into the academic side of the FEA research.
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