TMA01 T804 Finite element analysis: basic principles and applications VICTOR DE BLAS GARCIA F539167X TMA01 Contents Index of figures ........................................................................................................................................................... 2 Index of Tables............................................................................................................................................................ 3 INTRODUCTION .......................................................................................................................................................... 4 TASK GOAL .............................................................................................................................................................. 5 STRUCTURAL ANALYSIS ............................................................................................................................................... 5 LOADING................................................................................................................................................................. 5 BOUNDARY CONDITIONS AND ASSUMPTIONS ......................................................................................................... 5 MATERIAL PROPERTIES ........................................................................................................................................... 6 MODEL DESCRIPTION .............................................................................................................................................. 6 2-D MODEL ......................................................................................................................................................... 6 DISCUSSION OF RESULTS ....................................................................................................................................... 13 3-D MODEL ....................................................................................................................................................... 14 DISCUSSION OF RESULTS ................................................................................................................................... 15 THERMAL ANALYSIS .................................................................................................................................................. 16 LOADING............................................................................................................................................................... 16 BOUNDARY CONDITIONS AND ASSUMPTIONS ....................................................................................................... 16 MATERIAL PROPERTIES ......................................................................................................................................... 16 MODEL DESCRIPTION ............................................................................................................................................ 17 3-D MODEL ....................................................................................................................................................... 18 DISCUSSION OF RESULTS ....................................................................................................................................... 24 THERMAL-STRUCTURAL ANALYSIS............................................................................................................................. 25 LOADING............................................................................................................................................................... 25 BOUNDARY CONDITIONS AND ASSUMPTIONS ....................................................................................................... 25 MATERIAL PROPIERTIES ........................................................................................................................................ 25 MODEL DESCRIPTION ............................................................................................................................................ 25 3-D MODEL ....................................................................................................................................................... 26 DISCUSSION OF RESULTS ....................................................................................................................................... 31 FATIGUE.................................................................................................................................................................... 32 Case 1: Structural Analysis..................................................................................................................................... 32 Case 2: Combined Analysis Structural + Thermal, Convection 10 W/m2K ............................................................... 33 Case 3: Combined Analysis Structural + Thermal, Convection 200 W/m2K ............................................................. 34 DISCUSSION OF RESULTS ....................................................................................................................................... 35 CONCLUSIONS .......................................................................................................................................................... 36 References ................................................................................................................................................................ 36 APPENDIX OF FORMULAS ......................................................................................................................................... 37 THE OPEN UNIVERSITY 1 TMA01 Index of figures FIGURE 1: LINK’S SKETCH FIGURE 2; CANTILEVER BEAM FIGURE 3; PLANE 183 REPRESENTATION. FIGURE 4; PLANE 182 REPRESENTATION. FIGURE 5; GEOMETRY DRAWN BY LINES FIGURE 6; DISPLACEMENT RESTRICTIONS AND LOADS FIGURE 7; TOTAL REACTIONS FIGURE 8; LIST RESULT. NODAL SOLUTION & SUM FORCES. FIGURE 9; DEFORMED SHAPE. FIGURE 10; VON MISES STRESS FIGURE 11; SEQV THROUGH THE PATH FIGURE 12; SHEAR STRESS PLANE XY, UNAVERAGED, TOP HALF, AND AVERAGED BOTTOM HALF. FIGURE 13; STRESS PLANE X, UNAVERAGED TOP HALF, AND AVERAGED BOTTOM HALF. FIGURE 14; STRESS PLANE Y, UNAVERAGED TOP HALF, AND AVERAGED BOTTOM HALF. FIGURE 15; SEMICIRCULAR NODE PATH FIGURE 16; SX, SY, SXY & SEQV THOUGH SEMICIRCULAR NODE PATH FIGURE 17; SOLID 186 REPRESENTATION FIGURE 18; SOLID 185 REPRESENTATION FIGURE 19; DEFORMED SHAPE BOTTOM WINDOW AND VON MISES TOP WINDOW. FIGURE 20; SURFACE AREAS OVER VOLUME. SIDE 1 FIGURE 21; SURFACE AREAS OVER VOLUME. SIDE 2 FIGURE 22; SOLID 278 HOMOGENOUS FIGURE 23;SOLID 279 HOMOGENOUS FIGURE 24; SOLID 70 HOMOGENOUS FIGURE 25; GEOMETRY FIGURE 26; LOADING VIEW FIGURE 27; SOLID 279, PYRAMID OPTION FIGURE 28; MESH 279, SIZE 0.001, REFINEMENT 1, PYRAMID FIGURE 29; 10 W/M2K TEMPERATURE, SMX, SMN FIGURE 30; 10 W/M2K TEMPERATURE ZOOM, MX FIGURE 31; 10 W/M2K THERM GRADIENT VECTOR SUM, SMX, SMN FIGURE 32; 10 W/M2K THERM GRADIENT ZOOM VECTOR SUM, MX FIGURE 33;10 W/M2K THERM FLUX VECTOR SUM, SMX,SMN FIGURE 34;10 W/M2K THERM FLUX ZOOM VECTOR SUM, MX FIGURE 35:200 W/M2K TEMPERATURE, SMX, SMN FIGURE 36;200 W/M2K TEMPERATURE ZOOM, MX FIGURE 37;200 W/M2K THERM. GRAD. SUM, SMX, SMN FIGURE 38;200 W/M2K THERM. GRAD. ZOOM VECTOR SUM, MX FIGURE 39;200 W/M2K THERM. FLUX SUM, SMX,SMN FIGURE 40;200 W/M2K THERM. FLUX ZOOM VECTOR SUM, MX FIGURE 41; SOLID186 HOMOGENEOUS STRUCTURAL SOLID GEOMETRY FIGURE 42; SOLID 186, PYRAMID OPTION FIGURE 43; DIFFERENT LOADS APPLIED. FIGURE 44; REACTION SOLUTION. FIGURE 45; DEFORMED SHAPE. CONVECT-10 FIGURE 46;DISPLACEMENT. VECTOR SUM.CONVECT-10 FIGURE 47; VON MISES AVERAGE. CONVECT-10 FIGURE 48; VON MISES UNAVERAGE. CONVECT-10 FIGURE 49;X- STRESS AVERAGE CONVECT-10 FIGURE 50;X- STRESS UNAVERAGE CONVECT-10 FIGURE 51;Y- STRESS AVERAGE CONVECT-10 FIGURE 52;Y- STRESS UNAVERAGE CONVECT-10 FIGURE 53;XY SHEAR STRESS AVERAGE CONVECT-10 FIGURE 54;XY SHEAR STRESS AVERAGE CONVECT-10 FIGURE 55; DEFORMED SHAPE. CONVECT-200 THE OPEN UNIVERSITY 4 5 7 7 7 8 9 9 10 10 11 11 12 12 13 13 14 15 17 17 18 18 18 19 19 21 21 22 22 22 22 22 22 23 23 23 23 23 23 26 26 27 27 27 27 28 28 28 28 28 28 29 29 29 2 TMA01 FIGURE 56;DISPLACEMENT. VECTOR SUM.CONVECT-200 FIGURE 57; VON MISES AVERAGE. CONVECT-200 FIGURE 58; VON MISES UNAVERAGE. CONVECT-200 FIGURE 59;X- STRESS AVERAGE CONVECT-200 FIGURE 60;X- STRESS UNAVERAGE CONVECT-200 FIGURE 61;Y-STRESS AVERAGE CONVECT-200 FIGURE 62;Y- STRESS UNAVERAGE CONVECT-200 FIGURE 63;XY SHEAR STRESS AVERAGE CONVECT-200 FIGURE 64;XY SHEAR STRESS AVERAGE CONVECT-200 FIGURE 65; FATIGUE STUDY AREA FIGURE 66; FATIGUE VON MISES STRESS FIGURE 67; FATIGUE DEFORMED SHAPE X FIGURE 68;FATIGUE DEFORMED SHAPE Y FIGURE 70;FATIGUE DEFORMED SHAPE Y – CONV. 10 FIGURE 73;FATIGUE DEFORMED SHAPE Y – CONV. 200 29 30 30 30 30 30 30 31 31 32 32 33 33 34 35 Index of Tables TABLE 1; MATERIAL PROPERTIES OF THE LINK AND THE SURROUNDING ENVIRONMENT TABLE 2; MESH SELECTION COMPARATIVE. TABLE 3; MESH SELECTION COMPARATIVE TABLE 4; MESH SELECTION COMPARATIVE TABLE 5; THERMAL RESULTS. THE OPEN UNIVERSITY 4 8 14 20 24 3 TMA01 INTRODUCTION Figure 1 shows a slotted link which forms a connection in a particular machine. A pin at B fits in the slot with its longitudinal axis in the z direction and centred at B. This pin imposes a total force of up to 200 N acting towards the inner quadrant surface, C. This force is uniformly distributed as a pressure on the upper inner quadrant surface, as depicted by the arrows. The contact between the pin and the link also causes the temperature at the same surface to rise to 80 °C. At A and parallel to the pin is a shaft firmly fitted to the link, which can transmit side forces in the x, y plane and torques about its longitudinal axis which is parallel to the z-axis, but no heat is transferred through link A. The link material is steel with a modulus of elasticity of 203 GPa and a Poisson’s ratio of 0.29. The link has a constant thickness of 4.0 mm. There are no other components in contact with the link and its weight can be ignored in this instance. The engineering data are given in Table 1 below. Y X Figure 1: LINK’S SKETCH Rate of thermal expansion Modulus of elasticity Poisson's ratio Thermal conductivity Convective heat transfer coefficient (HTC) Ambient temperature 1.0 x 10-5/ºC 203 GPa 0.29 46 W/m·K 10 and 200 W/m2K 27ºC Table 1; Material properties of the link and the surrounding environment THE OPEN UNIVERSITY 4 TMA01 TASK GOAL Resolve the stresses, strains and the temperature distributions along the link, through a Finite Element Analysis, carrying out a report that include a structural analysis, a thermal analysis and one combinate analysis (structural and thermal at the same time). And make a fatigue analysis as well. For achieve the goal, it has to be made all that should be expected on an engineering report, discussing and contrasting assumptions, results and putting the conclusions on the record. STRUCTURAL ANALYSIS The task has been raised before, for the resolution has been utilized the software ANSYS, the analysis has been done, taking like reliable the next data. LOADING Force acting towards the inner quadrant surface, called C in the sketch, with a value of 200 N. This force is being applied acting like a uniformly distributed pressure. The value of this pressure it will be the next: 1. 𝐹 = 𝑁𝑒𝑤𝑡𝑜𝑛𝑠 𝑁 2. 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑃𝑎 = 𝑚2 3. 𝐴𝑟𝑒𝑎 = 4. 𝐴𝑟𝑒𝑎 = 1 1 × 𝐶𝑖𝑟𝑐𝑙𝑒 ′ 𝑠 𝐿𝑒𝑛𝑔𝑡ℎ (𝑚)× 𝐷𝑒𝑒𝑝𝑡ℎ(𝑚) = × 4 4 1 −5 2 × 𝜋 × 0.015 × 0.004 = 4.1238 · 10 𝑚 4 𝑁 𝜋 × 𝐷 × 𝐷𝑒𝑒𝑝𝑡ℎ 200 𝑁 5. 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑚2 = 4.1238·10−5 𝑚2 = 4244131.8 𝑃𝑎 BOUNDARY CONDITIONS AND ASSUMPTIONS 1. At point A on the sketch (Figure 1), there is a shaft which is fitted, acting like a fix end (“Fix End”), along all the internal circle, with centre at point (A). Transmit Forces in X and Y axis directions and bending moment along the Z axis. 2. The straight lines that link both semicircles, one situated at left (Ø36mm) and the other at right (Ø 28mm), are not tangents to that semicircles. Both semicircles are perfects and cover an angle of 180º, connected by two straight lines. 3. It is possible apply the same boundary conditions than usually for a simple cantilevered beam (Figure 2), the boundary conditions are follows: [1] i. w(0)=0 . This boundary condition says that the base of the (at the wall) does not experience any deflection, in the link Figure 2; Cantilever Beam THE OPEN UNIVERSITY apply as beam 5 TMA01 case this happen at the centre of the shaft which pass through the point A at figure 1. ii. w'(0)=0 . We also assume that the beam at the wall is horizontal, so that the derivative of the deflection function is zero at that point. Like just above this apply in this case on point A at figure 1. iii. w''(L)=0 . This boundary condition models the assumption that there is no bending moment at the free end of the cantilever (Node situated on rightest at middle point of Ø28mm semicircle, Figure 1). iv. w'''(L)=0 . This boundary condition models the assumption that there is no shearing force acting at the free end of the beam (Node situated on rightest at middle point of Ø28mm semicircle, Figure 1). 4. The actions are loaded in a quasi-static way, this means that from zero value to the final value, will pass time enough for avoid dynamic effects. 5. All the data are presented according to the International System of Units (SI). 6. The material behaves as lineal, elastic, isotropic and homogenous. 7. The structure always obey the equilibrium equations. MATERIAL PROPERTIES The material properties on which all the calculations have been calculated from, are the Poisson’s ratio and the Young module, already defined on Table 1. MODEL DESCRIPTION For make a structural analysis has been made, a 2-D model using an area in a plane and adding it a thickness, and a 3-D model, that has been carried out through volumes. Only has been analysed on depth in this report, the one that is considered more effective by the author. Has been analysed as well, with a several types of elements, PLANE 183 and PLANE 182, for the 2-D model and for the 3-D model, SOLID 186 and SOLID 185. Furthermore, has been applied different dimensions to the mesh and several refinement levels, mainly on the areas that have contact with circular or rounded areas or lines. 2-D MODEL Preprocessing. It has been used as Element Type: - - PLANE 183 (Figure 3), 8 nodes, it can to adapt very well to modelling irregular meshes and the behaviour against the displacements is quadratic, it has two degrees of freedom in each node: on directions X and Y. This element can be used as plane element as axisymmetric element. Furthermore, PLANE 183 has a very good plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large strain capabilities. Has been chose the PLANE 183 instead of the PLANE 182 (Figure 4), due to the PLANE 182 behaves in more stiff way against deflection. THE OPEN UNIVERSITY 6 TMA01 Figure 3; Plane 183 representation. - Figure 4; Plane 182 representation. This element is a plain element with a thickness of 0.004m added. Drawing the geometry, the geometry has been done through lines, as it is possible see on the Figure 5, just below. From that contour, has been filled the area. For improve the model’s accuracy, has been added the lines L1, L9, L12 and L15 to the design which make the mesh transition smoother from the circular part to the straight part smoother. Figure 5; Geometry drawn by lines Applying the loads, has been applied a pressure of 4244131.8 Pa (N/m2) Figure 6, over the line L13 on Figure 5. Has been applied as well a restriction of movement along the axis X and Y (Figure 6), at lines L6, L5, L7 and L8 on Figure 5. THE OPEN UNIVERSITY 7 TMA01 Figure 6; Displacement restrictions and loads Selecting the mesh, it has been done several times the calculations changing the global size of the mesh and the refinement. Searching a balance among the results accuracy and the computation time. For this purpose, has been carried out several tests as is being reflected on Table 2, has been selected the mesh with the row highlighted on green. The refinement has been applied picking the next lines: L1, L2, L3, L12, L7, L6, L8, L5, L18, L10, L11, L22, L21, L24, L13, L16, L15, L9, L17 and L20 on Figure 5. 183 MESH GENERAL SIZE (meters) 0.004 183 PLANE REFINEMENT DEFORMED(meters) MIN. STRESS (Pa) MAX. STRESS (Pa) NO 0.840 E -4 60068.2 0.728 E +08 0.004 1 0.841 E -4 18001.7 0.754 E +08 183 0.002 2 0.841 E -4 12356.9 0.753 E +08 183 0.001 1 0.841 E -4 6970.12 0.752 E+08 183 0.002 3 0.841 E -4 621.366 0.752 E+08 182 0.001 1 0.836 E -4 19677.9 0.754 E +08 Table 2; Mesh selection comparative. THE OPEN UNIVERSITY 8 TMA01 Postprocessing: Viewing the Results. Has been checked that the Reactions that we take from ANSYS are right and match with the theoretical results. Rx P Figure 7; Total Reactions Ry 1. Force divide by area (Pressure) is converted to force by lineal meter. 4244131.81 𝑁 𝑁 ×0.004 𝑚 = 16976.53 2 𝑚 𝑚 2. That act as a Uniformly Distributed Load, the radius of a quarter of circle is es 0.0075 m. 16976.53 𝑁 × 0.0075 𝑚 = 127.32 𝑁 𝑚 3. The equilibrium equations are solved. ∑ 𝐹𝑥 = 0; 𝐹𝑥 + 𝑅𝑥 = 0 𝑅𝑥 = −127.32 𝑁 ∑ 𝐹𝑦 = 0; ∑ 𝑀𝑧 = 0; 𝐹𝑦 × 𝑑 = 𝑀𝑧 127.32 𝑁 × 0.095 𝑚 = 12.09578 𝑁× 𝑚 = 𝑀𝑧 𝐹𝑦 + 𝑅𝑥 = 0 𝑅𝑦 = −127.32 𝑁 4. The result has been compared, against the results that ANSYS throw (Figure 8). Figure 8; List Result. Nodal Solution & Sum Forces. Deformed Shape, Figure 9, the value obtained is 0.0000841 meters. THE OPEN UNIVERSITY 9 TMA01 Figure 9; Deformed Shape. Von Mises Stress, Figure 10, has been obtained a Maximum Stress value of 75200000 Pa, the area appear on red and the point named like MX, in the right part, and a Minimum Stress value of 618.571 Pa at point MN on the blue part at left. Figure 10; Von Mises Stress THE OPEN UNIVERSITY 10 TMA01 A Path has been created along the X axis, going through the piece on normal direction to the plane Y-Z. Has been plotted the Von Mises Stress along the Path, drawing the nodes for have an idea of the stress behaviour in function of the link’s shape. (Figure 11). Figure 11; Seqv through the path Just below, are showed the next plots. Shear Stress on Plane XY (Figure 12), Stress Sx (Figure 13) and Stress Sy (Figure 14), all of them averaged and unaveraged. Figure 12; Shear Stress Plane XY, unaveraged, top half, and averaged bottom half. THE OPEN UNIVERSITY 11 TMA01 Figure 13; Stress Plane X, unaveraged top half, and averaged bottom half. Figure 14; Stress Plane Y, unaveraged top half, and averaged bottom half. In addiction, has been done a Path through the nodes situated on the right inner semicircle, (semicircle drawn in red, Figure 13), because is the place with more stresses concentration, as has been shown on the Figures above, and has been plotted a Graph (Figure 16), and is showing Stresses at X, Y, Shear Stress XY and the Von Mises. THE OPEN UNIVERSITY 12 TMA01 Figure 15; Semicircular Node Path Figure 16; Sx, Sy, Sxy & Seqv though semicircular node path DISCUSSION OF RESULTS Has been chosen an element edge size intermediate, and a refinement high at the rounded zones, it could have been done with less refinement, but in this case and for the type of analysis at hand, the computation time is not very high, due to be a 2-D plane mesh. And do a high refinement is viable, and throw accurate results. Furthermore, predictably, the maximum stresses are focus on critical areas, like whorl limits, holes and slots on the link. In this model for the task is negligible the difference of results between the averaged results and the unaveraged, this is an indication that the mesh size is proper for make the job, and appear a low number of singularities at the model. If we try to extract a correlation among the pictures, the Path that has be cut through the piece (Figure 11), could mislead us, because does not appear the higher stress near the limits of the internal semicircle, but, if is observed in detail the following graph (Figure 16), that represent the limit nodes through the semicircle (Figure 15). Can be seen that regarding point M that match with the middle point in the semicircle, all lines regards to point M (Sx, Sy, Sxy and Seqv) make a symmetry, Sxy and Seqv, axial symmetry, Sx and Sy, central symmetry, this means that, at this point the stresses are in any way counteracted. An at ¾ parts of axis X (Figure 16), Seqv is coincident with SMX on figure 10. THE OPEN UNIVERSITY 13 TMA01 3-D MODEL Preprocessing. It has been used as Element Type: - - SOLID186 (Figure 17) is a higher order 3-D 20-node solid element that exhibits quadratic displacement behavior. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. The element supports plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials. Se ha optado por utilizar SOLID 186, debido a que con el mismo tamaño de lado de los cubos, y el mismo refinamiento se ve que arroja mejores resultados que el material SOLID 185 (Figure 18) como se ve en la Tabla 3. [2] Figure 17; Solid 186 representation Figure 18; Solid 185 representation Postprocessing: Viewing the Results. Has been viewed the results indicated on Table 3 and on Figure 19, on which appear the strain deformed shape and the Equivalent Stress of Von Misses. BRICK SOLID 186 185 MESH GENERAL SIZE (meters) 0.002 0.002 REFINEMENT DEFORMED(meters) MIN. STRESS (Pa) MAX. STRESS (Pa) 1 0.841 E -4 4947.4 0.761 E +0.8 1 0.841 E -4 24767.7 0.698 E +08 Table 3; Mesh Selection Comparative THE OPEN UNIVERSITY 14 TMA01 Figure 19; Deformed Shape bottom window and Von Mises top window. DISCUSSION OF RESULTS After a glimpse, has been observed that is more convenient do the analysis using the plane elements, adding them a thickness. This is because the computation time is not worthy, due to the results 2-D and 3D are the same. THE OPEN UNIVERSITY 15 TMA01 THERMAL ANALYSIS It will be made a thermal analysis with HTC 10 y 200 W/m2K The problem has been raised before, for the resolution has been used ANSYS, this analysis has been carried out, taking as true the next assumptions. LOADING The PIN contact in the slot of the link, it causes that the upper inner quadrant Surface (A21 at Figure 21) reach a temperature of 80 ° C. In addition, two analyses have been performed, one with natural convection and the other with forced convection. This temperature affects the areas A11, A10, A9, A7, A17, A22, A2, A1, A19, A18, A12, A3, A4, A5 and A6 in Figures 20 and 21. - Natural convection: Heat transfer coefficient 10 W / m2K. - Forced convection: Heat transfer coefficient 200 W / m2K. BOUNDARY CONDITIONS AND ASSUMPTIONS 1. Where the link contact with the shaft A14, A15, A16, A13 (Figures 20 & 21) there is not heat transfer. 2. The ambient temperature is 27ºC, so, it will be the minimum temperature that the link can reach. 27ºC. 3. The temperature that the PIN apply is 80ºC, by instance, is the maximum temperature that the material could have. 80ºC is the superior limit. 4. Homogenous, Isotropic material. 5. The properties of the material do not change strongly with the temperature range of the problem. 6. No contraction or expansion work due to thermal processes. 7. No internal sources of heat. 8. Has been assumed a Isotropic Thermal Conductivity. 9. The convention, is of external flow, the object is not confined 10. In convection, At the point that the fluid contacts with the surface in relation to the surface the speed is null, reason why the heat flow is transmitted by conduction. To characterize this phenomenon the convection coefficient is used (HTC). 11. All the data are presented according to the International System of Units (SI). MATERIAL PROPERTIES The material characteristics on which the thermal analysis calculations are based are the thermal conductivity of the isotropic material of 46 W / m · K (Table 1). THE OPEN UNIVERSITY 16 TMA01 MODEL DESCRIPTION To conduct the analysis has been selected a 3-D model, made with volumes, due to the convection is affecting all the surfaces except for those that are directly in contact with the PIN (A20, A21 on Figures 20 & 21) and the shaft (A15, A14, A16, A13 on Figures 20 & 21). Just below are shown two pictures (Figures 21 & 22), with the areas numbered and highlighted with different colours. Figure 20; Surface Areas Over Volume. Side 1 Figure 21; Surface Areas Over Volume. Side 2 Have been done as well, several analyses with a divers kind of elements such as SOLID 279, SOLID 278 and SOLID 70. Moreover, has been meshed with various dimensions and several levels of refinement, mainly on areas that adjoin with lines or areas with rounded shape. THE OPEN UNIVERSITY 17 TMA01 3-D MODEL Preprocessing Tipos de modelo utilizados para el análisis SOLID 279, SOLID 278 y SOLID 70. [3] - SOLID 278 (Figure 22): Has a 3-D thermal conduction capability. The element has eight nodes with a single degree of freedom, temperature, at each node. The element is applicable to a 3-D, steady-state or transient thermal analysis. If the model containing the conducting solid element is also to be analysed structurally, the element should be replaced by an equivalent structural element. (such as SOLID 185). It is available in two forms, homogenous and layered, here has been used the homogenous. - SOLID 279 (Figure 23): Is a higher order 3-D, 20 node solid element that exhibits quadratic thermal behaviour. The element is defined by 20 nodes with a temperature degree of freedom at each node. It is available in two forms, homogenous and layered, here has been used the homogenous, which is suited to modelling irregular meshes. The element may have any spatial orientation. If the model containing the conducting solid element is also to be analysed structurally, the element should be replaced by an equivalent structural element (such as solid 186). - SOLID 70 (Figure 24): Has a 3-D thermal conduction capability. The element has eight nodes with a single degree of freedom, temperature, at each node. The element is applicable to a 3-D, steady-state or transient thermal analysis. The element also can compensate for mass transport heat flow from a constant velocity field. If the model containing the conducting solid element is also to be analysed structurally, the element should be replaced by an equivalent structural element (such as solid 185). Figure 22; SOLID 278 Homogenous Figure 23;SOLID 279 Homogenous Figure 24; SOLID 70 Homogenous Geometry drawing, the geometry has been carried out through lines, as is shown on the image provided below (Figure 25). From that lines, has been filled an area and then has been extruded 0.004 m. Again, has been drawn small straight lines attached to the semicircles, for utilize them on the mesh refinement later. THE OPEN UNIVERSITY 18 TMA01 Figure 25; Geometry Has been load the link with the loading mentioned before, this means with temperature of 80ºC, on the face that the PIN is doing the pressure. Also, the convection on the parts that are not in contact with the shaft and with the PIN either. The areas have been enumerated before when the model has been described. Figure 26; Loading View Postprocessing : Viewing the Results. For select the proper mesh, has been loaded with the two different convection cases, 10 and 200 W/m 2K, the temperature of 80ºC as well, and has been tried several element types and several sizes of mesh and refinement. But following the same pattern for each element type, with the goal of compare the results. Choosing the mesh, taking in account the times of computation and the accuracy of the results. THE OPEN UNIVERSITY 19 TMA01 ELEMENT TYPES MESH SIZE (m) REFINEMENT CONVECTION W/m2K SOLID 70 0.004 0 10 SOLID 70 0.004 1 (MINIMAL) SOLID 70 0.002 SOLID 70 TEMP SMIN ºc TEMP SMAX ºC TEMP NODE (0.109,0,0.002) ºC 46.4685 80 78.0364893 10 46.246 80 77.6833187 1 (MINIMAL) 10 46.1401 80 77.5166201 0.001 1 (MINIMAL) 10 46.0731 80 77.4114575 SOLID 278 0.004 0 10 46.4685 80 78.0364893 SOLID 278 0.004 1 (MINIMAL) 10 46.246 80 77.6833187 SOLID 278 0.002 1 (MINIMAL) 10 46.1401 80 77.5166201 SOLID 278 0.001 1 (MINIMAL) 10 46.0731 80 77.4114575 SOLID 279 0.004 0 10 46.1181 80 77.4996677 SOLID 279 0.004 1 (MINIMAL) 10 46.0574 80 77.3926188 SOLID 279 0.002 1 (MINIMAL) 10 46.0311 80 77.3357511 SOLID 279 0.001 1 (MINIMAL) 10 46.0206 80 77.3189147 SOLID 70 0.004 0 200 27.0854 80 69.7417942 SOLID 70 0.004 1 (MINIMAL) 200 27.0808 80 68.5468021 SOLID 70 0.002 1 (MINIMAL) 200 27.0787 80 68.0301346 SOLID 70 0.001 1 (MINIMAL) 200 27.0775 80 67.6813069 SOLID 278 0.004 0 200 27.0854 80 69.7417942 SOLID 278 0.004 1 (MINIMAL) 200 27.0808 80 68.5468021 SOLID 278 0.002 1 (MINIMAL) 200 27.0787 80 68.0301346 SOLID 278 0.001 1 (MINIMAL) 200 27.0775 80 67.6813069 SOLID 279 0.004 0 200 27.0782 80 67.97575 SOLID 279 0.004 1 (MINIMAL) 200 27.0772 80 67.623675 SOLID 279 0.002 1 (MINIMAL) 200 27.0767 80 67.4363018 SOLID 279 0.001 1 (MINIMAL) 200 27.0765 80 67.382178 Table 4; Mesh Selection Comparative THE OPEN UNIVERSITY 20 TMA01 On Table 4 has been studied the Maximum temperatures, the Minimum temperature, and the exact temperature on a Node that correspond with the Cartesian coordinates of (X,Y,Z)-(0.109,0,0.002). Also, add that all the meshes have been applied with the same Pyramidal structure (Figure 27). How is observed, the temperature results only variate slightly. But has been preferred use the mesh with 0.001m size of element edge and the refinement 1 at the rounded areas and SOLID 279. Has been chosen that mesh because even if the mesh is not very important for the temperature, the Thermal Gradient and the Thermal Flux are going to be affected by the size and refinement of the mesh. Figure 27; SOLID 279, Pyramid Option If the Table 4 is observed, can be notice, a correlation among the results and the element type's structures (Figures 22, 23 & 24). As can be seen the results of SOLID 278 and SOLID 70 are exactly the same. This is due to the interpolation, which is lineal for both types, and therefore, the equation used for both types is the same and the results as well. Has been chosen the option that add a intermediate node between nodes, giving place to a interpolation based in a quadratic function. And this will give better results on the thermal-structural analysis that will part from this one. Viewing the Results in depth of SOLID 279, SIZE of element edge length 0.001 and REFINEMENT 1 On Figure 28 is shown a mesh image. With refinement 1 at the areas A3, A12, A11, A10, A13, A14, A15, A16, A18, A17, A21, A20, A5, A6, A7 and A8. Figure 28; Mesh 279, SIZE 0.001, REFINEMENT 1, PYRAMID Natural Convection: Heat transfer coefficient 10 W/m2K. THE OPEN UNIVERSITY 21 TMA01 Figure 29; 10 W/m2K Temperature, SMX, SMN Figure 30; 10 W/m2K Temperature Zoom, MX Figure 31; 10 W/m2K Therm Gradient Vector Sum, SMX, SMN Figure 32; 10 W/m2K Therm Gradient Zoom Vector Sum, Mx Figure 33;10 W/m2K Therm Flux Vector Sum, SMX,SMN Figure 34;10 W/m2K Therm Flux Zoom Vector Sum, Mx THE OPEN UNIVERSITY 22 TMA01 Natural Convection: Heat transfer coefficient 200 W/m2K. Figure 35:200 W/m2K Temperature, SMX, SMN Figure 36;200 W/m2K Temperature Zoom, MX Figure 37;200 W/m2K Therm. Grad. Sum, SMX, SMN Figure 38;200 W/m2K Therm. Grad. Zoom Vector Sum, Mx Figure 39;200 W/m2K Therm. Flux Sum, SMX,SMN Figure 40;200 W/m2K Therm. Flux Zoom Vector Sum, Mx THE OPEN UNIVERSITY 23 TMA01 Material Thermal Condutivity Temperature STEEL 46 80 (W/m·K) (ºC) Convection (HTC) (W/m2K) Nodal Temp. Min (ºC) Thermal Gradient Vector Sum (K/m) 10 46.0203 6586.75 200 27.0765 26951.8 (W/m-2) 302990 0.124·107 Thermal Flux Vector Sum Reaction Heat Flow Rate W 1.8354 Table 5; Thermal Results. 8.9731 The temperature is a scalar magnitude and by instance does not have any direction and sense. The Thermal Flux is defined as: 𝑞 = −𝐾𝑋𝑋 · ∇𝑇 The Thermal Flux is related to the Thermal Gradient ∇𝑇, and the thermal Flux has three components that can help to the user with the sense and direction of the flux. It can be represented as magnitude (Figures 39 & 33), or as vectors (Figures 40 & 34). Also, has been obtained the Reaction Heat Flow Rate, shown on Table 5. DISCUSSION OF RESULTS The 3-D model has been the ideal option to perform this exercise because one of the hypotheses in this problem is that the convection also affects faces A1 and A2 (Figures 20 and 21). This can only be done with a 3-D model, otherwise, and doing so with a 2-D model would result in a lot of variation from the one obtained now, here we can see the importance of the correct hypothesis approach. In addition, it is observed the great difference of results in the minimum temperature using natural and forced convection, which directly influences the properties of the material, seeing that with the forced one arrives in its minimum point almost until the ambient temperature. Also, add that mesh size and refinement in this case does not influence too much in getting the temperature, but rather to obtain gradients and thermal flows. In the flow analysis of vector form (Figures 40 and 34), we can see the direction in which the heat travels, being fulfilled that travels from the hottest point to the coldest. THE OPEN UNIVERSITY 24 TMA01 THERMAL-STRUCTURAL ANALYSIS Has been done the thermal-structural analysis, based in the combination of both analysis presented before, now has been made use of all the material properties described on Table 1. LOADING Thermal, exactly the same loads which have been used on the thermal analysis. And with two different convection loads, which split the problem in two solutions, one for natural convection and another for forced convection. Structural, exactly the same that have been used on the structural analysis. Displacement and pressure. BOUNDARY CONDITIONS AND ASSUMPTIONS 1. All the boundary conditions applied in both models before. 2. For do this has been used the file (archive) “.th” from the thermal analysis, for continue with this analysis, we need set up a Reference Temperature, which has been assumed as 27ºC. On this analysis, we consider that at that temperature 27ºC, the material is free of internal stresses, and that temperature on this analysis is coincident with the ambient temperature. In other cases, could have been used as Reference Temperature between 20ºC/22ºC, as well known as “Room Temperature”. MATERIAL PROPIERTIES The material’s characteristics which the calculations come from, are all that have been presented in Table 1, like we said before. This means the same materials properties that have been used each analysis before, have been used all, plus a new one: ´Rate of Thermal Expansion´, which is the tendency of matter to change in shape, area, and volume in response to a change in temperature, through heat transfer [4]. MODEL DESCRIPTION The model as had been said before, come from the thermal previous analysis converting that analysis to structural, it cannot be done from structural to thermal, because that reason has to be used the 3-D model that has been used on the Thermal analysis, after loading on ANSYS the thermal model, have been added the structural properties of the material, applying the loading and restrictions of displacement. The element type will be SOLID 186, and is the type which ANSYS automatically use when comes from at thermal analysis that uses SOLID 279, to structural. THE OPEN UNIVERSITY 25 TMA01 3-D MODEL Preprocessing It has been used as Element Type: SOLID 186. [3] - SOLID186 (Figure 41) is a higher order 3-D 20-node solid element that exhibits quadratic displacement behaviour. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. The element supports plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials. At this study, has been used like a homogenous structural solid element. With the Pyramid option (Figure 42). Figure 42; SOLID 186, Pyramid Option Figure 41; SOLID186 Homogeneous Structural Solid Geometry The geometry is loaded from the thermal analysis. The mesh, has been used the same than on the thermal analysis, size 0.001 m y refinement 1. After mesh, has been load the part regarding to the structural analysis. On Figure 43 can be observed, the pressure on red, on cyan blue the restriction of movements, and with a wide spectrum of colours can be observed the thermal changes, loaded from the “.th” archive. THE OPEN UNIVERSITY 26 TMA01 Figure 43; Different Loads Applied. Postprocessing : Viewing the Results. Has been tested out at first the reactions obtained (Figure 44), for check that the loads have been applied properly, the reactions should be the same that the reactions obtained at the structural analysis. As can be seen the reactions are the same that were at the structural analysis. With Rz because the model now is 3D, and Rz is approximately 0. Figure 44; Reaction Solution. Natural Convection: Heat transfer coefficient 10 W/m2K. The Figures 45, 46, 47, 49, 51 y 53 are averaged, not the rest. Figure 45; Deformed Shape. Convect-10 THE OPEN UNIVERSITY Figure 46;Displacement. Vector Sum.Convect-10 27 TMA01 Figure 47; Von Mises Average. Convect-10 Figure 48; Von Mises Unaverage. Convect-10 Figure 49;X- Stress Average Convect-10 Figure 50;X- Stress Unaverage Convect-10 Figure 51;Y- Stress Average Convect-10 Figure 52;Y- Stress Unaverage Convect-10 THE OPEN UNIVERSITY 28 TMA01 Figure 53;XY Shear Stress Average Convect-10 Figure 54;XY Shear Stress Average Convect-10 Natural Convection: Heat transfer coefficient 200 W/m2K. The Figures 55, 56, 57, 59, 61 y 63 are averaged, not the rest. Figure 55; Deformed Shape. Convect-200 THE OPEN UNIVERSITY Figure 56;Displacement. Vector Sum.Convect-200 29 TMA01 Figure 57; Von Mises Average. Convect-200 Figure 58; Von Mises Unaverage. Convect-200 Figure 59;X- Stress Average Convect-200 Figure 60;X- Stress Unaverage Convect-200 Figure 61;Y-Stress Average Convect-200 Figure 62;Y- Stress Unaverage Convect-200 THE OPEN UNIVERSITY 30 TMA01 Figure 63;XY Shear Stress Average Convect-200 Figure 64;XY Shear Stress Average Convect-200 DISCUSSION OF RESULTS Appreciating the results, we can see the important effect of forced ventilation of the material when it is subjected to thermal loads. When the material is subjected to a convection of 10 W / m2 · K, i.e. A natural ventilation, the PIN pushing and warming the piece to the pressure 4.24MPa and temperature of 80ºC, it ravages the link, since the natural convection is not enough to dissipate the heat. If we compare with the purely structural analysis, the link passes from deforming from 0.841E-4 to 0.894E-4 meters, in addition comparing the equivalent tensions, can be observed that it goes from 0.752E + 8 Pa to 0.121E + 9 Pa, in addition, the Zone in which the maximum tension appears is totally different, and comparing graph by graph we see that shear is where the distribution of stresses change with respect to the structural analysis. Another thing that should be appreciated is that in the 10W / m2 · K convection analysis, the results vary between averaged and unaveraged, them are non-convergent, both the stress on Y and the shear stress analysis affecting the equivalent final result, as well, the analysis becomes more complex to be badly refrigerated. On the other hand, when is analysed the results with forced convection, not only do not all these defects appear, all the maximum and minimum tensions appear in the same zones as when the simple structural analysis is carried out, we see that the tensions only increase a little comparing with the structural, and the strain that the material undergoes is 0.791E-4 meters, is smaller than 0.841E-4, can be appreciated that although in specific zones the tensions can be slightly greater, altogether the material works better. And the results between averaged and unaveraged are the same, that means that the result is accurate. THE OPEN UNIVERSITY 31 TMA01 FATIGUE Has been done an analysis about the material behaviour when it is exposed to fatigue, under the three loading cases studied before: - Case 1: Structural simple analysis. (Figures 66, 67, 68). Case 2: Thermal Structural Analysis with Natural Convection. (Figures 69, 70, 71). Case 3: Thermal Structural Analysis with Forced Convection. (Figures 72, 73, 74). Has been introduced a new material property for this analysis, the density, has been chosen like steel density 7830 kg/m3. The analysis has been harmonic and the link has been exposed to a determined number of cycles, and then has been graphed the deformed shape at X direction and at Y direction, and the Von Mises Stress around the same area of the piece. This has been done with the help of tutorials from the Alberta’s University web [4]. The Graph that are shown just below, have been done with nodes inside of the red circle at the Figure 65. Figure 65; Fatigue Study Area Case 1: Structural Analysis Figure 66; Fatigue Von Mises Stress THE OPEN UNIVERSITY 32 TMA01 Figure 67; Fatigue Deformed Shape X Figure 68;Fatigue Deformed Shape Y This case has been done at 250 cycles, due to is a 2D model and the computational time is not very high, for finish the analysis the computer has needed around half an hour. Case 2: Combined Analysis Structural + Thermal, Convection 10 W/m2K Figure69; Fatigue Von Mises Stress - Conv. 10 THE OPEN UNIVERSITY 33 TMA01 Figure 70; Fatigue Deformed Shape X - Conv.10 Figure 69;Fatigue Deformed Shape Y – Conv. 10 This case has been done at 100 cycles, due to is a 3D model and the computational time is very high, that is because the 3-D model has a higher number of nodes, and for solve the task the computer has needed more than 8 hours. Case 3: Combined Analysis Structural + Thermal, Convection 200 W/m2K Figure 71; Fatigue Von Mises Stress - Conv. 200 THE OPEN UNIVERSITY 34 TMA01 Figure 72; Fatigue Deformed Shape X - Conv.200 Figure 70;Fatigue Deformed Shape Y – Conv. 200 This case has been done at 100 cycles, due to the same reason that has been said just before for the other combined model. And we have the same issue the computational time. DISCUSSION OF RESULTS A harmonic analysis has been used to see the response of the link to fatigue, analysing the behaviour of isolated nodes, within a zone critical to loads, has been corroborated what had seen in the previous analyses, i.e., the fatigue response is quite relieved when the ventilation is forced, a hundred cycles is a very short analysis, but at the graphs, we can notice a series of behaviours in the material, both, strain and the stresses have an increase of exponential form as a function of the cycles, which can lead us to think that to follow infinitely in time would reach the point of failure of the material. In addition, to say that the strain, has a different behaviour in the 2-D plane, maybe because one is acting within the traction zone (Y) and the other within the zone of compression (X), usually the fatigue failures appear with tensile stresses, but compressive loads may result in local tensile stresses. On the other hand, say that for the 3-D analysis, the mesh is the same in both, and the same node has been chosen for the fatigue analysis of the equivalent stress. Although the number of cycles is very small, the one that is subjected to natural convection has had an increase of 4.3 Pa / 100 Cycles and the one that is subjected to forced convection has had an increase of 3.8 Pa / 100 Cycles. The fatigue analysis is strongly influenced by computational time, especially 3-D, it would have been interesting to look for some symmetry in the piece to minimize the number of nodes, this could only have been done with respect to the XY plane (Figure 28). Dividing the number of nodes in half, decreasing the computation time. Finally add, that this test is not conclusive, and can be stated that this test is only for have a small idea for now how works ANSYS, and is not sufficient for be conclusive, because the analysis has fewer than 1000 cycles, the ideal would has been use between 104 and 108 cycles, for have a good idea of the endurance limit. If the material would keep working under the theoretical fatigue limit could be working with continue loading and does not have fatigue failure. On steels this limit could variate between ½ of the ultimate tensile strength, to a maximum of 290 MPa. THE OPEN UNIVERSITY 35 TMA01 CONCLUSIONS The analysis with ANSYS, is very useful, always that the user introduces the proper data and is be able to interpret the result with accuracy, for this specific case: 1. This kind of piece/link could be useful in multiple machines on the real life, for example, industrial machinery in a production line, in a factory of tyres where the machines has to bear high temperatures, or in a car, a link like this could act like support between the motor and the chassis. 2. We have introduced the data, under conservative and generalist assumptions, for every independent case, with standard assumptions which suit e.g. convection, conduction, structural analysis, etc. and the boundary conditions that are dictated by the enunciate of the problem. 3. The structural analysis, and thermal analysis throw some interesting results, that should be contrasted against actual data (experimental) or against database with different characteristics of the material (http://www.splavkharkov.com/en/). 4. The Preprocessing, plays a very important role, and the software can variate a lot the results, task like choose the mesh or the element type, can be very significant for have the appropriate results and the user has to be able to valuates which one use in function of which characteristic of the material need to search or the accuracy of the result needed, e.g., on the thermal analysis if the heat flux or the gradient would not be needed, the temperature was very accurate with a coarse mesh. 5. Always is important try to search symmetries of try to reduce the number of nodes for facilitate complexes analysis to the software, as can be notice on the fatigue analyses. References [1] Inside.mines, "http://inside.mines.edu/," [Online]. Available: http://inside.mines.edu/~apetrell/ENME442/Documents/SOLID186.pdf. [2] A. M. A. .. Reference., Element Reference. [3] Wikipedia. [Online]. Available: https://en.wikipedia.org/wiki/Thermal_expansion. [4] A. University. [Online]. Available: http://www.mece.ualberta.ca/tutorials/ansys/IT/Harmonic/Harmonic.html. [5] U. o. Minesota, “The Geometry Center.,” REF 1. [Online]. Available: http://www.geom.uiuc.edu/education/calcinit/static-beam/boundary.html. THE OPEN UNIVERSITY 36 TMA01 APPENDIX OF FORMULAS Hooke’s Law 1. 𝐹 = 𝑘 · 𝑋; F = Force K = Constant of Stiffness X = Displacement On a Cartesian coordinate system: 𝑘11 𝐹1 2. 𝐹 = [𝐹2 ] = [𝑘21 𝐹3 𝑘31 𝑘12 𝑘22 𝑘32 𝑘13 𝑋1 𝑘23 ] · [𝑋2 ] = 𝑘 · 𝑋 𝑋3 𝑘33 Newton’s Law of cooling 1. ̇ 𝜕𝑄 𝛿𝑡 = ℎ · 𝐴 · (𝑇(𝑡) − 𝑇∞ ) ; Q = Thermal Energy h = is the heat transfer coefficient A = transfer surface area T = temperature of the object surface and interior T∞=is the temperature of the environment ΔT(t) = (T(t) - T∞) = is the time-dependent thermal gradient between environment and object Thermal conductance ⃗⃗T; 1. 𝑞⃗ = −𝑘 · ∇ q = heat flux THE OPEN UNIVERSITY ⃗⃗T = the temperature gradient ∇ 37
