MEE 323 Su21 - Homework #2

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Jan 9, 2024

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Summer 2021B 1 MEE 323 Computer Aided Engineering II Homework #2 Assigned: July 12, 2021 Due Date: July 19, 2021, by 11:59 PM Instructions: C reate a folder named “Homework _2 ” in your U: \ drive. Save all of your model data and a copy of your homework report in this folder. Use this word file as a template for your homework report. Turn in the assignment on the course Gradescope before the deadline. Homework Objectives: Learn to create and validate 2-D surface models in ANSYS Workbench. Learn how to apply symmetry boundary conditions to reduce model size and analysis time as well as cost. Learn about stress concentration, structural error, shape functions, and FE convergence. Reading Assignment: Read Section 3.3 of the textbook it contains details of the 2-D plane elasticity assumptions and operational details of ANSYS Mechanical. ANSYS Exercises: Question 1: Repeat the analysis of the in-class triangular plate problem (the full structure do not use symmetries), but model the structure in 3-D with a thickness of 10mm (you may use the geometry provided in class). Use an Element Size of 4mm for the mesh (same as the in-class model). (1) Show a figure of the deformed shape with the undeformed wireframe displayed. Is the result obtained consistent with the 2-D model analyzed in class? (2) Show the plot of the maximum von Mises stress with elements displayed. How does the value of the stress compare with that obtained from (i) the 2-D full model and (ii) the 2-D sector model analyzed in class? Calculate a percentage difference between the three values. (3) Find the number of nodes and elements in the full 3-D model and compare with (i) the 2-D full model and (ii) the 2-D sector model analyzed in class. What is the percentage reduction in the number of nodes and elements achieved by using the 2-D surface models in class? (4) Carry out an automatic/adaptive convergence analysis (see section 5.3-6 on page 229 of the textbook for step-by-step instructions) on the maximum von Mises stress on both the 3-D and
Summer 2021B 2 2-D full models. Use a convergence criterion of 1% for both. Compare the maximum von Mises stress after convergence and comment. Question 2: Determine the magnitude and location of the maximum principal stress and the magnitude and location of the maximum displacement of the structural steel part shown below (all dimensions given in mm). Model the part as a 2-D structure with appropriate behavior. Use symmetry to simplify the model to a one-quarter section. Use the suggested symmetry simplification below (use frictionless supports if/where appropriate):
Summer 2021B 3 Analyze the part under the three conditions described below. In all three conditions, there is a pressure of 25 MPa acting on the inside walls/surfaces of the part. (Figures shown below are just for your reference and are cut to show features. Please model all parts as 2-D surfaces with the appropriate 2D behavior selected. Use frictionless supports if necessary). For each case, show plots of the total displacement and maximum principal stress and show the location of the maximum displacement and stress on these plots (use the "Max" probe). (1) The part has a thickness of 15mm and is in a state of plane stress.
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Summer 2021B 4 (2) The part is a long underground pipeline, and is thus in a state of plane strain. (3) It is axisymmetric, i.e. it is a sphere with an internal cavity.
Summer 2021B 5 Question 3: A structural steel anchor device is loaded by an axial force as shown below. All of the lower surfaces (excluding the groove) are fixed as indicated. Note that the center of the 5 mm diameter groove is aligned with the outer surfaces of the anchor as shown in the sketch. All dimensions are given in mm.
Summer 2021B 6 Make the model by sketching the cross section given on the X-Y plane as shown and revolving the sketch around the Y axis. You may use the geometry that you created for Homework Assignment #1 for this purpose. Use a mesh element size of 2 mm. Find the magnitude and location of the maximum deflection and maximum von Mises stress. (a) Show the figure of the mesh. (b) Show the figure of the deformed shape + undeformed edge. Include probes showing the locations of the maximum and minimum total deformations.
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Summer 2021B 7 (c) Show a plot of the von Mises stress. Add a probe to show the location of the maximum von Mises stress. Explain in your own words why the stress occurs at this location. The largest stress occurs at this area because it is the part with the smallest area to distribute the load. (d) Find the number of nodes and elements in the model.
Summer 2021B 8 (e) There is a plane of rotational symmetry in the model. Use this to your advantage and model the part as a 2-D axisymmetric body (frictionless supports will not be needed as long as the sketch is made as shown above). Use a mesh size of 2 mm consistent with that of the 3-D model. Show plots/results for the following: (a) Show the final figure of the mesh. (b) Show the contour of the von Mises stress. How does the maximum value of the stress compare with the full 3-D analysis? Calculate a percentage difference. Why do you think that this difference is seen? Maximum value in the axisymmetry analysis is 282.04MPa vs 248.31 in the full 3D analysis, which is a percent difference of 11.96%. This difference comes from the boundary conditions and the fact that axisymmetry has different boundary conditions and assumptions than those of a 3D analysis. (c) Calculate the percentage reduction in the number of nodes and elements compared to the 3-D model. 94.585% reduction in number of nodes and 97.14% reduction in number of elements from using the axisymmetric analysis instead of the full 3D model. Question 4: Consider the concrete overpass structure with dimensions shown in the figure below. 30 kN/m 3 m 4 m
Summer 2021B 9 Analyze the structure in 2-D assuming plane strain conditions. There is a reflective plane of symmetry in the structure use it appropriately and model just half of the structure. Make sure to account for the effect of using symmetry on the loading of the structure. (a) Create a 2-D surface body by creating a sketch on the XY plane. Show a figure of the surface body after having used symmetry. (b) Mesh the model with a mesh size of 0.2 m using quadrilateral elements with linear element order. Show a screenshot of the mesh method control with quadrilateral elements and linear element order. Also show a figure of the mesh. (c) Apply the boundary conditions and loads and determine the maximum total deformation, maximum principal stress, and minimum principal stress. Show a screenshot of the geometry details showing that you are indeed running a plane strain analysis. Show figures for the applied boundary conditions & loads as well as each of the required results.
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Summer 2021B 10 (d) Carry out a convergence analysis on (i) the maximum value of the maximum principal stress and (ii) the minimum value of the minimum principal stress. You may carry out the convergence study for both simultaneously. Choose an allowable change in the stress value of 1% with the maximum number of refinement loops set to 5. Show convergence plots for both (i) and (ii) and explain the results. Why does one value converge while the other one diverges? How can you prevent the divergence (in words only)? The maximum value converges just fine because there are not any stress concentration regions for the areas of interest for finding maximum principal stress.
Summer 2021B 11 The minimum principal stress does not converge because there is a stress concentration region exactly where the minimum principal stress occurs, meaning that the stress will go to infinity unless that region is accounted for in the analysis. (e) Repeat the analysis of (d) but mesh the model with a mesh size of 0.2 m using quadrilateral elements with quadratic element order. Comment on the differences seen. For the maximum principal stress convergence, the solution converges just the same with a much cleaner curve for convergence. For the minimum principal stress convergence analysis, the solution actually converges instead of climbing further and further away from a set value.
Summer 2021B 12 Question 5: Consider the structural steel stepped shaft with the dimensions shown below. The shaft is subject to a uniform axial tensile load of 𝑃 = 7200𝜋 𝑙𝑏𝑓 . Simplify the model as a 2D axisymmetric problem. Carry out the following analyses. Please make any required reasonable assumptions (for e.g., on the geometry, analysis type, mesh parameters, convergence criteria, etc.) with justifications (Hint: take lengths of the two sides of the shaft to be about 1.5 times the larger diameter. Explain why this is a good assumption) . (a) The stepped shaft designed as shown above will show a stress singularity. Where will this be seen and why? Carry out a finite element analysis with a convergence study and show that indeed this is the case. Show a figure of the convergence behavior (including number of nodes and elements) and explain what you see. Set the number of refinement loops in the convergence study judiciously. Stress singularity will be at the step corner where the diameters of the shaft change, because sudden and sharp changes in geometry cause stress singularities. (b) To remove the stress singularity above, add a fillet of radius 0.75 𝑖𝑛 at the appropriate location (duplicate the Static Structural analysis block and make the modification in the duplicate
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Summer 2021B 13 block). Repeat the analysis ensuring that the results have converged. Show results for and explain what you see for: (i) Directional deformation in the direction of loading. (ii) Normal stress in the direction of loading. (iii) Structural error with an explanation of what it tells you. This tells us where and what the analyses error is by highlighting nodes of the mesh and coloring them based on their error, in this case the higher error is within the stress concentration area and has a magnitude of 10^-9 which is acceptable error for this academic analysis. (iv) S afety factor (under “Stress Tool”). What is th e minimum safety factor for the geometry and loading given?
Summer 2021B 14 Minimum safety factor for the geometry and loading given is 6 (v) Convergence plots with number of nodes & elements. (c) Verify the results of your analysis using the stress concentration formulae given below. The given formula is accurate to within 5% - check the difference between what the formula predicts and your converged finite element result and confirm this.
Summer 2021B 15 Question 6: A structural steel plate with an off-center hole is shown in the figure below. Determine how close to the top edge (to the closest whole dimension in mm) the hole can be placed before yielding of the steel occurs (based on the maximum von Mises stress). The applied tensile pressure is 70,000 kPa and the plate thickness is 6 mm. Note that the figure may not be to scale. Analyze the structure in 2-D assuming plane stress conditions (remember to change the analysis settings in the geometry properties to 2D). There is a reflective plane of symmetry in the structure use it appropriately and model just half of the structure. Make sure to account for the effect of using symmetry (if any) on the loading of the structure. (a) Create a 2-D surface body by creating a sketch on the XY plane. Show a figure of the surface body after having used symmetry. (b) Mesh the model with an appropriate mesh size and justify your choice of mesh size. Use quadrilateral elements with quadratic element order. Show a figure of the mesh. Used 0.2 in elements for the large part and used mesh refinement for smaller elements in the area of interest to save resources on computing areas that are not relevant. 70,000 kPa 120 60 33 48 12 mm dia.
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Summer 2021B 16 (c) Apply the boundary conditions and loads and determine the maximum von Mises stress making sure that your result is converged. Show a figure of the von Mises stress result. Modify the design by moving the hole closer to or further away from the top edge until the stress reaches approximately the value of the yield stress. At what distance is the hole from the bottom of the plate when this occurs? Show a figure of the converged von Mises stress after the change in geometry. The structure experiences yielding failure at approximately 32.5 mm (d) If the plate thickness is changed to 12 mm, how does this change the result? Use the same total force as when the plate is 6 mm. Show a figure of the von Mises stress with the changed dimension to justify your answer. It does not change the results when the plate thickness is changed to 12mm from 6mm