# Assignment 1 – Solution Guidance Let's break down and answer both questions step by step. --- ## **Given Data** - **Modulus of Elasticity (E):** 160 GPa = \(160 \times 10^9\) Pa - **Cross-sectional Area (A):** \(10,000 \text{ mm}^2 = 10,000 \times 10^{-6} \text{ m}^2 = .01 \text{ m}^2\) - **Poisson's Ratio (ν):** .3 (for reference, not directly needed for an axial truss problem) - **Truss geometry:** See figure (nodes A, B, C, D, E; loads at B and E) - **Loads:** 6 kN (downward at B), 3 kN (rightward at E) --- ## **Part 1: Finite Element Method (FEM) for Axial Forces and Displacements** ### **Step 1: Number and Label the Nodes and Members** - **Nodes:** A (fixed), B (at top middle), C (bottom left), D (bottom right), E (right top) - **Members:** AB, AC, BC, BD, CD, DE, BE ### **Step 2: Set Up the Global Coordinate System** - Place A at (, ), C at (, -.9), D at (1.2, -.9), E at (2.4, ), B at (1.2, ) ### **Step 3: Find Lengths and Directions of Members** - For each member, calculate length and direction cosines. - **AB:** from (,) to (1.2,): length = 1.2 m - **AC:** from (,) to (,-.9): length = .9 m - **BC:** from (1.2,) to (,-.9): length = \(\sqrt{(1.2)^2+(.9)^2} = 1.5 \text{ m}\) - **BD:** from (1.2,) to (1.2,-.9): length = .9 m - **CD:** from (,-.9) to (1.2,-.9): length = 1.2 m - **DE:** from (1.2,-.9) to (2.4,): length = \(\sqrt{(1.2)^2+(.9)^2} = 1.5 \text{ m}\) - **BE:** from (1.2,) to (2.4,): length = 1.2 m ### **Step 4: Assemble the Stiffness Matrices** For each member: \[ k = \frac{EA}{L} \] The local stiffness matrix for each member in global coordinates: \[ k_{member} = \frac{EA}{L} \begin{bmatrix} c^2 & cs & -c^2 & -cs \\ cs & s^2 & -cs & -s^2 \\ -c^2 & -cs & c^2 & cs \\ -cs & -s^2 & cs & s^2 \end{bmatrix} \] Where \(c = \cos \theta\), \(s = \sin \theta\) ### **Step 5: Assemble the Global Stiffness Matrix** - Map each member's DOFs into the global system (each node has 2 DOFs: x and y) - Assemble the full \(10 \times 10\) stiffness matrix ### **Step 6: Apply Boundary Conditions and Loads** - **Fixed supports:** At nodes A and C (DOFs 1,2,5,6 are zero displacement) - Apply loads at B (node 2, y-dir: -6 kN) and E (node 5, x-dir: +3 kN) ### **Step 7: Solve for Nodal Displacements** - Partition the matrix, solve for unknown displacements at free nodes (B, D, E) ### **Step 8: Compute Axial Forces** - For each member, use the nodal displacements to compute the change in length and thus the force: \[ F = k (u_j - u_i) \] - Report positive for tension, negative for compression. --- #### **Note:** This is a standard truss FEM procedure. The calculations are lengthy but systematic. You may use structural analysis software or MATLAB for the matrix assembly and solution. --- ## **Sample Calculation for One Member (AB)** - \(E = 160 \times 10^9\) Pa - \(A = .01\) m² - \(L_{AB} = 1.2\) m - Direction: purely horizontal (\(c=1, s=\)) - Stiffness: \[ k_{AB} = \frac{EA}{L} = \frac{160 \times 10^9 \times .01}{1.2} = 1.333 \times 10^9 \text{ N/m} \] - Local stiffness in global coordinates: \[ k_{AB} = 1.333 \times 10^9 \begin{bmatrix} 1 & & -1 & \\ & & & \\ -1 & & 1 & \\ & & & \end{bmatrix} \] --- ## **Part 2: Method of Joints (Statics)** ### **Step 1: Draw Free-Body Diagrams at Each Joint** - At each joint, use equilibrium: - \(\sum F_x = \) - \(\sum F_y = \) ### **Step 2: Solve for Axial Forces** - Start at joints with known forces and the least unknowns (often pinned supports or loaded joints). - Work joint by joint: - Solve for unknown member forces using equilibrium equations. **Example:** - At joint B (after calculating support reactions at A and C): - \(\sum F_y = \): forces in AB, BD, and BC - \(\sum F_x = \): forces in AB, BE Continue this for all joints. ### **Step 3: Compare Results** - The axial forces found by statics should match those from the FEM analysis in Part 1. --- ## **Summary Table for Results** | Member | FEM Force (kN) | Statics Force (kN) | Tension (+) or Compression (-) | |--------|----------------|--------------------|-------------------------------| | AB | | | | | AC | | | | | BC | | | | | BD | | | | | CD | | | | | DE | | | | | BE | | | | --- ## **Tips** - For the truss, symmetry and zero-force members may simplify your work. - Use correct units throughout. - Clearly indicate directions (arrows) in your free-body diagrams. - Check for equilibrium at each joint. --- ## **Conclusion** - **Part 1:** Use FEM to assemble matrices, apply BCs, solve for displacements and forces. - **Part 2:** Use method of joints to manually verify the forces. If you need explicit numbers for each force/displacement or help with the MATLAB code, let me know!