SAID_KAOTHAR_ISLAM TEAM LAB_1 REPORT FEA MEEN 610

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Lab Assignment 1 - Plane Stress Analysis Applied Finite Element Analysis (MEEN 610) Fall 2023 Thursday, November 16, 2023 Students name: Student ID No. Signature 1. Kaouthar Mohammad Al Taher Madi 100062611 Kaouthar Madi 2. Said Mohamad El Turk 10004972 Said El Turk 3. Islam Ahmed Elsayed 100060581 Islam Elsayed Write the largest student ID number of your team. P1 P2 P3 P4 1 0 0 0 6 2 6 1 1

Contents 1. Problem Description: ............................................................................................................... 3 2. Given Data: .............................................................................................................................. 3 3. Pre-processing: ........................................................................................................................ 4 4. Solution: ................................................................................................................................... 5 5. Results: .................................................................................................................................... 5 1. Structured Coarse Mesh: ...................................................................................................... 6 2. Structured Intermediate Mesh Size: ..................................................................................... 7 3. Refining Effect on the Structured Mesh: ............................................................................. 7 4. Very Fine Mesh (Mesh Independence): ............................................................................... 8 Stress Concentration Factor for Every Case Study: .................................................................... 9 Results Summary: ..................................................................................................................... 10 6. Conclusions: .......................................................................................................................... 13

1. Problem Description: A normal uniform stress 𝜎 𝑎?? is applied to a double-notched plate as shown below. The double-notched plate is made of tool steel and can be considered as an elastic solid with Young's modulus 𝐸 = 210GPa and Poisson's ratio 𝑣 = 0.3 . Perform a plane stress finite element analysis using Abaqus. (a) Determine and report the contour plot for Von-Mises stresses (𝜎 𝑣𝑀 ) in the plate (b) Calculate and report the maximum stress concentration factor (e.g. 𝐾 𝑡? = 𝜎 max 𝑣𝑀 /𝜎 𝑎?? (c) Make a mesh convergence analysis, starting with a mesh size of 𝑟/5 until the 𝐾 𝑡? value converges within 5% of the exact value as calculated from the diagram below. Provide a plot of 𝐾 𝑡? vs. element size, showing the convergence of the FEA results in comparison to the exact value. 2. Given Data: A double-notched plate as shown below with the following dimensions: Poisson's ratio 𝑣 = 0.3 Young's modulus 𝐸 = 210GPa

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3. Pre-processing: Due to horizontal and vertical symmetries, we can simplify the simulation to a simpler shape that has been drawn using ABAQUS. Symmetry around X Symmetry around Y

The geometry is defined in ABAQUS using Sketch tools which specify various principal coordinates to define the 2D plane stress dimensions. A total of 5 lines defining straight edges in plane stress and a quarter circle of radius r. The element type used is a 2D element. The geometric properties are defined as per the given data. The material is isotropic linear elastic with elastic constant value modulus. 4. Solution: Symmetric (Roller, Fixed) boundary conditions are applied at the symmetric boundaries. The bottom left corner point is constrained in the x- and y-direction and point 2 (along x-direction with P1) in the y-direction. Distributed along X-axis forces are applied on the vertical edge on the right of the 2D plane stress problem. 5. Results: Nodal displacement and deformation shape for the 2D plane stress ( u and v ) for different mesh sizes and mesh refinement. Symmetry Boundary Symmetry Boundary Distributed axial loading.

1. Structured Coarse Mesh: The magnification factor for this analysis is 148 and as we can see maximum stress happens at the notch based on Von-Mises criterion.

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2. Structured Intermediate Mesh Size: The magnification factor for this study was around 148 and as we can see maximum stress ooccurs at the notch based on Von-Mises criterion. 3. Refining Effect on the Structured Mesh: To better visualize the deformation and stress concentrations on the notch, mesh refinement is essential. Mesh refinement until the results converge to certain values with a 0.005 iteration error was acceptable in this case. The following figures show the effect mesh refinement in getting accurate results and better contours.

4. Very Fine Mesh (Mesh Independence): Inflation layers were applied at the desired edges and elements in this case to show the stress and deformation variations better at these edges. The following figures describe deformation at a very fine mesh applied to the plane stress problem.

Stress Concentration Factor for Every Case Study: Case Mesh type 𝝈 ?𝒂𝒙 𝐾 𝑡? = 𝜎 ?𝑎𝑥 𝑣𝑀 𝜎 ??? 1 r/5=2.4 341.6 1.7333 2 2 353.8 1.79518

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3 1.5 364.6 1.8498 4 1 375.8 1.9068 5 1 (0.2refinement) 381 1.9332 Results Summary: 𝑃 = 𝜎 × 𝐵 × 𝐻 = 110 × 10 × 86 = 94600? 𝜎 ??? = 𝑃 𝐵 × 𝑑 = 94600 10 × (86 − 2 × 19) = 197.083?𝑃𝑎 𝐾 𝑡? = 𝜎 ?𝑎𝑥 𝑣𝑀 𝜎 ??? 𝐾 𝑡?1 = 341.6 197.083 = 1.7333 Mesh 𝝈 ?𝒂𝒙 𝑲 𝒕? Number of elements r/5=2.4 341.6 1.7333 510 2 353.8 1.79518 760 1.5 364.6 1.8498 1326 1 375.8 1.9068 6015 1 (0.2refinement) 381 1.9332 5155

Mesh Size against Stress Conc. Factor: Mesh size (no. of elements) Ktn 510 1.7333 760 1.79518 1326 1.8498 3015 1.9068 5155 1.9332

Element size K tn 2.4 1.7333 2 1.79518 1.5 1.8498 1 1.9068 y = -0.0395x 3 + 0.1824x 2 - 0.3825x + 2.1464 R² = 1 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 0 0.5 1 1.5 2 2.5 3 Ktn Element Size Ktn VS Element Size

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6. Conclusions: Applying Symmetry contributed to a very simple case study creation. Instead of drafting the whole axially loaded problem. We made use of vertical and horizontal symmetry to draw just a quarter. Maximum stresses occurred at the bottom part of the notch which is connected to the fixed edge because of symmetry boundary application. Overall, material selection can identify whether the plate is safe or not based on Young’s modulus and maximum allowable deformation at every node. From the results, we conclude that element size is inversely proportional to the stress concentration factor, a the smallest Ktn was around 1.73 at an element size of approximately 2.4. The stress concentration factor is directly proportional to the number of elements used in the case study showing a maximum value when the no. of elements exceeded the 5150 elements. Refinement is necessary to capture detailed and significant stress concentration regions in different case studies. Inflation layers and adaptable mesh, not necessarily to be structured, are also crucial to capture curved and irregular edge shapes in some studies.