Let's break down the **flexibility method** for analyzing the given truss. ### **Step 1: Identify Redundant Member(s)** The truss in the figure is a statically indeterminate truss (since it has more unknown forces than equilibrium equations). Let's select **AC** as the redundant member. ### **Step 2: Remove Redundant Member** Remove member AC and analyze the remaining structure as a statically determinate truss. The external load is 100 N at C. ### **Step 3: Calculate Reactions** For a truss with pin at A and roller at B, calculate reactions at A and B. #### Geometry: - \(AB\) and \(BC\) are both at 60°, so each lower member (AD, DC) = 800 mm. - AC is diagonal. ### **Step 4: Calculate Forces in the Determinate Structure (without AC)** Use the method of joints or sections to determine the forces in members due to external loading (100 N). ### **Step 5: Calculate Deflection at C Due to External Load** Calculate the vertical deflection at C due to the 100 N load using virtual work or unit load method. ### **Step 6: Calculate Deflection at C Due to a Unit Force in Redundant Member (AC)** Apply a unit force in member AC and compute the deflection at C due to this unit force. ### **Step 7: Compatibility Equation** Total deflection at C (sum of deflection due to external load + deflection due to force in AC) must be zero. ### **Step 8: Solve for Force in Redundant Member** Solve the compatibility equation to get the force in AC. --- Let's go through the key calculations step by step: --- ## **Step 1: Geometry and Lengths** All members (AD, BC, DC) = 800 mm Let’s denote the forces: - \(F_{AC}\) = Force in AC (redundant) - \(F_{BD}\), \(F_{AD}\), \(F_{BC}\), \(F_{DC}\) ## **Step 2: Material Properties** - For AC and BD: \(A = 30\,\mathrm{mm}^2\) - Other members: \(A = 20\,\mathrm{mm}^2\) - \(E = 2 \times 10^6\,\mathrm{N/mm}^2\) ## **Step 3: Determinate Truss Analysis (Remove AC)** ### **A. Calculate Support Reactions** Assume both supports are pinned for now. Find vertical reactions by taking moments about A or B. ### **B. Method of Joints** Start at joint C (since 100 N is applied there): #### **Joint C:** Let \(F_{CD}\) and \(F_{CB}\) be the forces in CD and CB. - \(\sum F_y = 0: F_{CB} \sin 60^\circ = 100\) - \(F_{CB} = \frac{100}{\sin 60^\circ} = \frac{100}{0.866} \approx 115.5\,\mathrm{N}\) - \(\sum F_x = 0: F_{CD} = F_{CB} \cos 60^\circ = 115.5 \times 0.5 = 57.75\,\mathrm{N}\) #### **Joint D:** At D, the member CD and DA. Continue similarly for all joints to get internal forces. --- ## **Step 4: Flexibility (Deflection) Calculation** ### **A. Flexibility Coefficient (\(f_{11}\))** The flexibility coefficient for member AC is: \[ f_{11} = \sum \frac{l_i}{A_i E} \] Where \(l_i\) is the length, \(A_i\) is the area, \(E\) is the modulus. ### **B. Compatibility Equation** The total deformation in the direction of AC must be zero: \[ \delta_{C,\,\text{external}} + F_{AC} \cdot f_{11} = 0 \] Solve for \(F_{AC}\): \[ F_{AC} = -\frac{\delta_{C,\,\text{external}}}{f_{11}} \] --- ## **Step 5: Back-calculate All Member Forces** With \(F_{AC}\) known, superpose effects to get all member forces. --- ### **Summary of Steps** 1. **Remove member AC** and analyze the truss under external load. 2. **Calculate deflection at C** in the direction of AC due to 100 N load. 3. **Apply unit load in AC** and compute deflection at C due to this unit load. 4. **Write and solve compatibility equation** to find the force in AC. 5. **Superpose**: Add the effect of force in AC back to the original member forces. --- ### **If you want a detailed numerical solution (step-by-step), please confirm so I can proceed with all calculations.**