Let's solve the **Finite Element Analysis (FEA)** problem step by step for the structure shown in Figure 1, using the mesh models in Figures 2 (triangular elements) and 3 (rectangular elements). We are given: - **Elastic modulus (E):** 70 GPa = 70,000 MPa - **Poisson's ratio (ν):** 0.33 - **No friction** - **A = 78 mm** - **F = 18,000 N** - **Thickness (t) = 1 unit (assume 1 mm for simplicity)** ### **Step 1: Calculate the Displacements at Each Node** #### **Approach** - Use the **Finite Element Method (FEM)**. - For each element (triangular in Case 1, rectangular in Case 2), assemble the global stiffness matrix. - Apply boundary conditions and solve the system to get the nodal displacements. #### **Explanation** - For each element, use shape functions to derive the element's stiffness matrix. - Assemble these into a global matrix for each mesh. - Apply the force (F) at the loaded node, and fix the nodes as per the support in Fig. 1. - Solve for displacements using: \[ \mathbf{K} \mathbf{u} = \mathbf{F} \] where \(\mathbf{K}\) is the global stiffness matrix, \(\mathbf{u}\) is the nodal displacement vector, and \(\mathbf{F}\) is the nodal force vector. #### **Result** - **Case 1 (triangular):** More nodes, potentially more accurate, but more computational effort. - **Case 2 (rectangular):** Fewer nodes, possibly less accurate, but easier to compute. *(Actual computation requires the coordinates of nodes and detailed element assembly, which is typically done using FEA software.)* --- ### **Step 2: Calculate the Strains at Each Node** #### **Approach** - Use the **nodal displacements** from Step 1. - For each element, calculate the strain using the strain-displacement relationship. #### **Explanation** - For 2D elements, the strain vector is given by: \[ \boldsymbol{\epsilon} = \mathbf{B} \mathbf{u}_e \] where \(\mathbf{B}\) is the strain-displacement matrix, and \(\mathbf{u}_e\) is the vector of nodal displacements for the element. - Calculate strains at each node by averaging the strains of elements sharing the node. #### **Result** - Strain values at each node, which tell how much the structure stretches or compresses at those points. --- ### **Step 3: Calculate the Stresses at Each Node** #### **Approach** - Use **Hooke's law** with the strains from Step 2. #### **Explanation** - For plane stress: \[ \boldsymbol{\sigma} = \mathbf{D} \boldsymbol{\epsilon} \] where \(\mathbf{D}\) is the material stiffness matrix (depends on E and ν). - Calculate stresses at each node by applying the above for each strain vector at the node. #### **Result** - Stress values at each node, indicating the internal forces per unit area. --- ### **Step 4: Summing Reaction Forces & Validation** #### **Approach** - Sum the reaction forces at the fixed/support nodes (nodes 1, 2, 3 in both meshes). - Compare the total reaction force to the applied force F. #### **Explanation** - By equilibrium, the sum of the reaction forces at the supports should equal the applied force (F = 18,000 N). - If there is a mismatch, it indicates an error in assembly, boundary conditions, or calculation. #### **Result** - The closer the summed reaction forces are to the applied force, the better the model's accuracy. - **Triangular elements (more mesh density):** Typically yield more accurate results due to better geometric approximation. - **Rectangular elements (coarser mesh):** May have larger errors due to less precise modeling of the structure. --- ## **Final Answer Summary** **1. Displacements:** - Calculated by solving the global stiffness matrix equation for each mesh. - Triangular mesh (Case 1) gives more accurate, detailed displacement fields. **2. Strains:** - Obtained from displacement gradients within each element. - More accurate with finer mesh (triangular elements). **3. Stresses:** - Computed from strains using Hooke’s law. - More reliable values with more refined mesh. **4. Reaction Forces & Validation:** - Summed reaction forces at supports should match the applied force (18,000 N). - Closer agreement in the triangular mesh case due to better model fidelity. **Better Results Explanation:** - **Mesh refinement (triangular elements):** Better captures the geometry and stress distribution, leading to more accurate displacement, strain, stress, and reaction force calculations. - **Validation:** The sum of the reaction forces closely matching the applied force validates the FEA model. --- ### *Note:* For actual node-by-node numerical results, the coordinates and element connectivity are needed to assemble and solve the matrices, which is typically done via FEA software like ANSYS, Abaqus, or MATLAB. If you need a sample calculation for a single element, let me know!