Let's solve each part step by step: --- ## a) **Overall Free Body Diagram (FBD)** **Step 1: Draw the FBD** - **At A (Pin Support):** - Two reaction components: \(A_x\) and \(A_y\). - **At B (Pin Support):** - Two reaction components: \(B_x\) and \(B_y\). - **At C (Pin Joint):** - Internal forces between members (but for overall FBD, don't show internal forces). - **External Loads:** - Vertical force \(P\) at distance \(a\) from A. - Vertical force \(Q\) at distance \(b\) from B. **Diagram:** ```markdown - Node A: Reaction forces \(A_x\), \(A_y\) - Node B: Reaction forces \(B_x\), \(B_y\) - Node C: External forces \(P\) (down), \(Q\) (down) ``` **Step 2: Number of Unknown Reactions** - \(A_x, A_y, B_x, B_y\) = **4 unknowns** **Step 3: Number of Equilibrium Equations** - 2D system ⇒ \( \sum F_x = \), \( \sum F_y = \), \( \sum M = \) (**3 equations**) --- ## b) **FBDs for Members AC and CB** ### **Member AC** - Isolate AC. - At A: Pin support → \(A_x, A_y\) - At C: Pin joint → Reaction forces from CB (let's call them \(C_{AC,x}, C_{AC,y}\)) **Unknowns for AC:** - \(A_x, A_y, C_{AC,x}, C_{AC,y}\) (**4 unknowns**) **Equilibrium Equations:** - \( \sum F_x = \) - \( \sum F_y = \) - \( \sum M_A = \) (**3 equations**) --- ### **Member CB** - At B: Pin support → \(B_x, B_y\) - At C: Pin joint → Reaction forces from AC (opposite to those on AC, by Newton's 3rd law) **Unknowns for CB:** - \(B_x, B_y, C_{CB,x}, C_{CB,y}\) (**4 unknowns**; \(C_{CB,x} = -C_{AC,x}\), \(C_{CB,y} = -C_{AC,y}\)) **Equilibrium Equations:** - \( \sum F_x = \) - \( \sum F_y = \) - \( \sum M_B = \) (**3 equations**) --- ## c) **Write Out All Equilibrium Equations** ### **For the Whole Structure** \[ \begin{align*} \sum F_x &: A_x + B_x = \\ \sum F_y &: A_y + B_y - P - Q = \\ \sum M_A &: -P \cdot a - Q \cdot (L_{AC} \cos\theta_A + b) + B_y \cdot (L_{AC} \cos\theta_A + L_{CB} \cos\theta_B) - B_x \cdot (L_{AC} \sin\theta_A + L_{CB} \sin\theta_B) = \\ \end{align*} \] (*You may need to adjust the moment arms based on the exact geometry; the form above illustrates the general approach.*) --- ### **For Member AC** \[ \begin{align*} \sum F_x &: A_x + C_{AC,x} = \\ \sum F_y &: A_y + C_{AC,y} - P = \\ \sum M_A &: -P \cdot a + C_{AC,x} \cdot L_{AC} \sin\theta_A + C_{AC,y} \cdot L_{AC} \cos\theta_A = \\ \end{align*} \] --- ### **For Member CB** \[ \begin{align*} \sum F_x &: B_x + C_{CB,x} = \\ \sum F_y &: B_y + C_{CB,y} - Q = \\ \sum M_B &: -Q \cdot b + C_{CB,x} \cdot L_{CB} \sin\theta_B + C_{CB,y} \cdot L_{CB} \cos\theta_B = \\ \end{align*} \] --- ## **Summary Table** | Part | Unknowns | Equilibrium Equations | |--------------|----------|----------------------| | Whole Struct | 4 | 3 | | AC | 4 | 3 | | CB | 4 | 3 | --- ## **Final Answers (Concise):** ### a) - **Draw the FBD** (as described above). - **Unknown reactions:** 4 (\(A_x, A_y, B_x, B_y\)) - **Equilibrium Equations:** 3 ### b) - **Draw FBDs for AC and CB** (as described). - **Unknown reactions per member:** 4 - **Equilibrium Equations per member:** 3 ### c) - **Equilibrium equations:** (as written above for the whole structure and each member). --- **Let me know if you need the actual FBD sketches!**

# Solution to Structural Analysis Problem ## a) Overall Free Body Diagram (FBD) ### Step 1: Draw the FBD - **At Point A (Pin Support):** - Reaction forces: \(A_x\) (horizontal), \(A_y\) (vertical). - **At Point B (Pin Support):** - Reaction forces: \(B_x\) (horizontal), \(B_y\) (vertical). - **At Point C (Pin Joint):** - Internal forces between members (not detailed in overall FBD). - **External Loads:** - Vertical force \(P\) acting downwards at distance \(a\) from A. - Vertical force \(Q\) acting downwards at distance \(b\) from B. ### Diagram Representation ```plaintext - Node A: Reaction forces \(A_x, A_y\) - Node B: Reaction forces \(B_x, B_y\) - Node C: External forces \(P\) (down), \(Q\) (down) ``` ### Step 2: Number of Unknown Reactions - Total unknowns: \(A_x, A_y, B_x, B_y\) = **4 unknowns** ### Step 3: Number of Equilibrium Equations - For a 2D system: 3 equilibrium equations available: - \( \sum F_x = 0 \) - \( \sum F_y = 0 \) - \( \sum M = 0 \) --- ## b) Free Body Diagrams for Members AC and CB ### Member AC - **At Point A:** - Reaction forces: \(A_x, A_y\) - **At Point C:** - Internal forces from member CB (denote as \(C_{AC,x}, C_{AC,y}\)). #### Unknowns for AC: - Unknowns: \(A_x, A_y, C_{AC,x}, C_{AC,y}\) = **4 unknowns** #### Equilibrium Equations: - For Member AC: - \( \sum F_x = 0 \) - \( \sum F_y = 0 \) - \( \sum M_A = 0 \) --- ### Member CB - **At Point B:** - Reaction forces: \(B_x, B_y\) - **At Point C:** - Internal forces from member AC (opposite to those on AC). #### Unknowns for CB: - Unknowns: \(B_x, B_y, C_{CB,x}, C_{CB,y}\) = **4 unknowns** (with \(C_{CB,x} = -C_{AC,x}\) and \(C_{CB,y} = -C_{AC,y}\)) #### Equilibrium Equations: - For Member CB: - \( \sum F_x = 0 \) - \( \sum F_y = 0 \) - \( \sum M_B = 0 \) --- ## c) Write Out All Equilibrium Equations ### For the Whole Structure \[ \begin{align*} \sum F_x &: A_x + B_x = 0 \\ \sum F_y &: A_y + B_y - P - Q = 0 \\ \sum M_A &: -P \cdot a - Q \cdot d + B_y \cdot L - B_x \cdot h = 0 \\ \end{align*} \] (*Where \(d\), \(L\), and \(h\) depend on the geometry of the structure.*) --- ### For Member AC \[ \begin{align*} \sum F_x &: A_x + C_{AC,x} = 0 \\ \sum F_y &: A_y + C_{AC,y} - P = 0 \\ \sum M_A &: -P \cdot a + C_{AC,x} \cdot L_{AC} \sin(\theta_A) + C_{AC,y} \cdot L_{AC} \cos(\theta_A) = 0 \\ \end{align*} \] --- ### For Member CB \[ \begin{align*} \sum F_x &: B_x + C_{CB,x} = 0 \\ \sum F_y &: B_y + C_{CB,y} - Q = 0 \\ \sum M_B &: -Q \cdot b + C_{CB,x} \cdot L_{CB} \sin(\theta_B) + C_{CB,y} \cdot L_{CB} \cos(\theta_B) = 0 \\ \end{align*} \] --- ## Summary Table | Part | Unknowns | Equilibrium Equations | |--------------|----------|-----------------------| | Whole Struct | 4 | 3 | | AC | 4 | 3 | | CB | 4 | 3 | --- ## Final Answers ### a) - **Overall FBD:** Drawn as described. - **Unknown reactions:** 4 (\(A_x, A_y, B_x, B_y\)) - **Equilibrium Equations:** 3 ### b) - **Member AC FBD:** Drawn as described. - **Unknown reactions per member:** 4 - **Equilibrium Equations per member:** 3 ### c) - **Equilibrium equations:** Written above for the whole structure and each member. --- Let me know if you need further clarification or additional details!