SPRING 2018 STATEMENT CANTILEVER BEAMS Page 1/6 TERM PROJECT: THEORETICAL/FE BEAM ANALYSES  CANTILEVER BEAMS PART I: AXIAL TENSION Date: 03-30-2018 The main goal of this term project is to compare solutions obtained by the 1-D Mechanics-of-Materials (MoM) approach, which are approximate yet more practical, with the 3-D finite element (FE) solutions. Whenever the exact and rigorous Elasticity solutions are available, they should also be included for comparison. All FE simulations must be performed using SolidWorks 2016 version. Students are strongly encouraged to use the computer stations in ME CAD Room (Steinman Room ST-213). Once executed successfully, be sure to save all key results (e.g., stresses and displacements in ASCII/text, MS Excel and/or graphic formats), which are necessary for comparison with the theoretical MoM/Elasticity predictions. When presenting results graphically, students are suggested to use the deformed geometry. If possible, also superimpose them the undeformed meshes. Otherwise, juxtaposing the deformed and undeformed geometries for easy comparison. It should be noted that SolidWorks, just like almost all commercial general-purpose FE software, allows users to adjust scale factors to control Deformation Shape and Stress Fringes for better viewing. In the term project, each student is assigned with a different set of load values, although all students will work on the same geometries and boundary conditions of the 3-D bars. All student are expected to add his/her own proper loading and boundary conditions, material properties and other information/data, if so required by SolidWorks and abstract all needed results (graphical, texted and/or tabulated) for the report. In order to compare the SolidWorks FE simulations with theoretical MoM/Elasticity results numerically and/or graphically, the analytical solutions should first be formulated, derived and/or simply cited with proper references. When an analytical solution is adopted from a know source, the equation(s) needed for comparison with the FE results should only be included with concise explanation in the main text of the report and the formula can be simply, yet clearly, referred to the textbook or class notes (e.g., Eq. so and so in Page so and so, Section so and so, etc.). If a equation for comparison is not available for lifting off directly from the textbook or class notes, the student should derive it by him/herself. Details of the formulation/derivation should be given in the Appendix of the report. PART I: AXIAL TENSION I.1 Cantilever Beams of Circular Solid Sections (Required for all students) As shown in Fig I.1, a prismatic cantilever beam of length L and circular solid cross-section of radius R0 is fixed at one end (A) while loaded axially at the other (B). The x-axis is through the centroidal axis (C.A.) of the axial bar, which is made of a linearly elastic material of Young’s modulus E and Poisson’s ratio . The bar is loaded axially at the free end by (a) a uniform tension  0 and (b) a concentrated axial force P0   0 A   R02 0 through the centroid of the end cross-section. In this term project, each student is assigned with a different  0 value (thus, of course a different P0 ) although all students will work on the same geometries, boundary conditions and material properties. Refer to the file: Term-Project Case Assignment (2018-02-27) for the load values you need to work on and Table I.1 for the material and dimensional data. ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS TERM PROJECT: THEORETICAL/FE BEAM ANALYSES SPRING 2018 STATEMENT xL 2 A C CANTILEVER BEAMS Page 2/6 xL 2 0 B x A z C R0 L B P0 x P0 z R0 L y y y y (a) uniform axial tension (b) concentrated axial force through C.A. FIGURE I.1 An axially loaded circular cylindrical bar. TABLE I.1 Material and dimensional data of bars of circular cross-section. E GPa   - m m m m 2 0.25 0.2 6R0 0.5R0 1.5R0 R0 R1 L R2 The afore-mentioned bar geometry has been prepared in a SolidWorks Assembly file: Assembly 2_1.SLDASM, which represents a quarter of the circular bar in the 1st octant  x  0, y  0, z  0  of the Cartesian x-y-z coordinate system. Each student is expected to add his/her proper loading and boundary conditions, material properties and other information/data, if so required by SolidWorks. For Case (a): Uniform Axial Tension, the following boundary and loading conditions should be applied and the mesh size should be chosen:  Roller/Slider option should be imposed in SolidWorks “Fixtures” to the restrained end face  x  0  , the horizontal  z  0  and vertical  y  0  symmetric/central planes.  A uniform tension of  0 value should be applied on the free end face  x  L  through the Pressure option of SolidWorks “External Loads”.  An element size of 0.01m with a tolerance of 0.0005m should be chosen to create the meshes (approximate 202,000 elements) through the Create Mesh option of SolidWorks “Mesh”. Similarly, for Case (b): Concentrated Tensile Force, the following boundary and loading conditions should be applied and the mesh size should be chosen:  Fixed Geometry option should be imposed in SolidWorks “Fixtures” to the restrained end face  x  0  whereas, again, the Roller/Slider option should be applied in SolidWorks “Fixtures” to the horizontal  z  0  and vertical  y  0  symmetric/central planes.  A concentrated tensile axial force of P0 value should be applied at the center of on the free-end  x, y, z    L,0,0 m .  Once again, an element size of 0.01m with a tolerance of 0.0005m should be chosen to create the meshes (approximate 202,000 elements). ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS TERM PROJECT: THEORETICAL/FE BEAM ANALYSES SPRING 2018 STATEMENT CANTILEVER BEAMS Page 3/6 Report: (minimum items to be included) 1. Summarize the MoM solutions of the problem, including the two field variables: stress tensor  ij  i, j  1, 2,3 and displacement vector ui  i  1, 2,3 . 2. For each FE case, show the mesh with loading and restraint B.C.’s. Be sure to include the coordinate system used in the FE simulation. 3. Compare the MoM solutions with the two FEM results for the following cases: a) In order to demonstrate how the data were abstracted, using the SolidWorks Probe option in Plot Tools to probe the FE results along the C.A. approximately every 0.1m. On a separate page, plot on the same graph the axial extensions u in [mm] from both FE simulations and MoM prediction along the C.A.  0  x  L; y  0; z  0 as well as the near-field u-displacement 2-D Flamant normal & 3-D Bossinesq Elasticity solutions given in Eqs (I.2 & I.4) below for comparison. b) Plot on the same graph the transverse contraction v in [mm] from both FE simulations and MoM L prediction along the vertical diameter of the mid-section  x  ;0  y  R0 ; z  0  . In a separate plot, 2   demonstrate how the data were abstracted using the SolidWorks Probe option in Plot Tools. Probe the FE results along the diameter approximately every 0.05m. c) Display 3 normal stress contour plots for  x ,  y ,  z  , respectively, on the mid-span cross-section L   x   on the same page from the Axial-Tension Case. Repeat the same for the Concentrated-Force 2  Case. L d) Display 3 shear stress contour plots for  xy , yx , zx , respectively, on the mid-span cross-section  x   2  on the same page from the Axial-Tension Case. Repeat the same for the Concentrated-Force Case. Compare the results with the MoM predictions. R R e) Consider the area:  L  0  x  L;0  y  0 ; z  0  , which is a vicinity within the x-y plane around the 2 2   loading point, i.e.,  x, y, z    L,0,0  . Display 3  x stress contour plots in the area from the   Concentrated-Force Case FE result and two Elasticity solutions based on the 2-D Flamant and 3-D Boussinesq normal-force solutions, Eqs (I.1 & I.3), shown below. Hint: The MATLAB command meshgrid can be used to defined the contour plot area whereas, as the name implies, the command contour is for contour plots. A. 2-D HALF-PLANE NORMAL FORCE (FLAMANT SOLUTION) 3  2 P0  L  x   x  2 2    L  x   y 2     2  2 P0  L  x  y 2-D stress field:  y  2  2 2   L  x  y       2 2 P0  L  x  y   xy   2 2   L  x   y 2      ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS (I.1) TERM PROJECT: THEORETICAL/FE BEAM ANALYSES SPRING 2018 STATEMENT CANTILEVER BEAMS Page 4/6 2 2  L  x  y2   P0   L  x  u   1   ln   2 G  x12  x22 L   2-D displacement field:    L  x  y  1  2    tan 1 L  x   P0  v        2 2 2  G 2 y L  x  y         (I.2)   force where the out-of-(x-y plane line-load P0 has a unit of   . For this term project,  out-of-plane thickness  simply choose P0  P0 . B. 3-D HALF-SPACE NORMAL FORCE (BOUSSINESQ SOLUTION) 3-D stress field: 3  3P0  L  x   x  2 R 5  2 2  P 3 L  x y 1  2   2 R  L  x  y 2   2  y  0 3    R  R L  x  L  x       2 2 R  R RLx  RLx     2 2  P0  3  L  x  z 1  2   2 R  L  x  z 2   2   R  R L  x  L  x       z    3 2 2 R  R RLx  RLx     2  3P  L  x  z  xz  0 2 R 5  2  3P0  L  x  y  xy   2 R 5    P0 yz  3  L  x   1  2  2 R  L  x    yz 2 R 5  R 2 RLx  2   P0   L  x  u   2 1     2 4 GR  R    P y  Lx 1  2  3-D displacement field: v   0  2    4 GR  R RLx   w   P0 z  L  x  1  2     4 GR  R 2 RLx where R   L  x 2  y2  z2 (I.3) (I.4) (I.5) f) Repeat e) for  y ,  xy ,  max ,  min ,  vM and  Tr (called Stress Intensity in SolidWorks), respectively. R R g) Consider the area:  0  x  0 ; 0  y  R0 ; z  0  , which is a vicinity within the x-y plane around the 2 2   bottom edge point of the fixed end, i.e.,  x, y, z    0, R0 ,0  . Display the stress contour plots of  x ,  y , ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS TERM PROJECT: THEORETICAL/FE BEAM ANALYSES SPRING 2018 STATEMENT CANTILEVER BEAMS Page 5/6  xy ,  max ,  min ,  vM and  Tr in the area from the Concentrated-Force Case FE result. h) (Option) Compare the FE results in g) with available Elasticity solutions, e.g., Sinclair, GB “Stress singularities in classical elasticity - I. Removal, Interpretation and Analysis and II. Asymptotic identification,” Applied Mechanics Review, Vol. 57, No. 4, pp. 251-297 and No. 5, pp. 385-439, 2004. I.2 Truncated Conical Cantilever Beam (Required for all students) Figure I.2 shows a truncated conical bar of length L and end radii  R1 , R2  is fixed as a roller/slider boundary at R02  0 at the free end (B). In the figure R12 the x-axis is through the centroidal axis (C.A.) of the bar, which is made of a linearly elastic material of Young’s modulus E and Poisson’s ratio . the restrained end (A) while loaded axially by a uniform tension:  1  R2 xL 2 A C R1 B x z 1 L y y FIGURE I.2 A truncated conical bar loaded by uniform axial tension. Refer to the file: Term-Project Case Assignment (2018-02-27) for the load values you need to work on and Table I.1 for the material and dimensional data. Finally, the following boundary and loading conditions should be applied and the mesh size should be chosen:  Roller/Slider option should be imposed in SolidWorks “Fixtures” to the restrained end face  x  0  , the horizontal  z  0  and vertical  y  0  symmetric/central planes.  A uniform tension of  1 value should be applied on the free end face  x  L  through the Pressure option of SolidWorks “External Loads”.  An element size of 0.01m with a tolerance of 0.0005m should be chosen to create the meshes (approximate 215,000 elements) through the Create Mesh option of SolidWorks “Mesh”. Report: (minimum items to be included) 1. Show the mesh with loading and restraint B.C.’s. Be sure to include the coordinate system used in the FE simulation. 2. Find the MoM solution of the axial normal stress  x  distribution. Hint: Force balance. 3. Find the MoM solution of the displacement field ui  i  1, 2,3 .  Note: The governing eq of the axial displacement u   u1  of this axially loaded bar with variable cross-section is E du  x, t   d   A x 0 dx  dx  (I.6) 4. Due to its axi-symmetrical nature, the Basic MoM approach, i.e., ignoring the taper effect, concludes this ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS TERM PROJECT: THEORETICAL/FE BEAM ANALYSES SPRING 2018 STATEMENT CANTILEVER BEAMS Page 6/6 axially loaded circular object will not have transverse shear stresses  xy , xz  , hence  xr , x  , where  r , , x  constitutes a cylindrical coordinate system. Inspecting the equilibrium equations in cylindrical coordinate system, Eq (1.8-C3), to prove this conclusion is wrong. Indeed, an Advanced MoM approach taking into account the tapered-beam effect, does prove the existence of transverse shears. The results are summarized in the file: Transverse Shear Stresses in Tapered Beams (2017-12-06) . 5. Compare the above Basic MoM/Advanced MoM solutions with the FEM results for the following cases: a) Plot on the same graph the axial extensions u in [mm] from the FE simulation and MoM prediction along the C.A.  0  x  L; y  0; z  0  . In a separate plot, demonstrate how the data were abstracted using the SolidWorks Probe option in Plot Tools. Probe the FE results along the C.A. approximately every 0.1m. b) Plot on the same graph the transverse contraction v in [mm] from the FE simulation and MoM prediction R  R2 L  along the vertical diameter of the mid-section  x  ;0  y  1 ; z  0  . In a separate plot, 2 2   demonstrate how the data were abstracted using the SolidWorks Probe option in Plot Tools. Probe the FE results along the diameter approximately every 0.05m. c) Plot on the same graph the axial normal stress  x in [MPa] from the FE simulation and MoM prediction along the C.A.  0  x  L; y  0; z  0  . In a separate plot, demonstrate how the data were abstracted using the SolidWorks Probe option in Plot Tools. Probe the FE results along the C.A. approximately every 0.1m. d) Plot on the same graph the transverse shear stress  xy in [MPa] from the FE simulation and MoM R  R2 L  prediction along the vertical diameter of the mid-section  x  ;0  y  1 ; z  0  . In a separate 2 2   plot, demonstrate how the data were abstracted using the SolidWorks Probe option in Plot Tools. Probe the FE results along the diameter approximately every 0.05m. e) Plot the transverse normal stress  y in [MPa] from the FE simulation along the vertical diameter of the R  R2 L  mid-section  x  ;0  y  1 ; z  0  . In a separate plot, demonstrate how the data were abstracted 2 2   using the SolidWorks Probe option in Plot Tools. Probe the FE results along the diameter approximately every 0.05m. ME 54100: ADVANCED STRESS ANALYSIS ME I4200: APPLIED STRESS ANALYSIS TERM PROJECT: THEORETICAL/FE BEAM ANALYSES
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