# Solution to Q1: Shear Force (S.F.) and Bending Moment (B.M.) Diagrams ## Step 1: **Draw S.F. and B.M. Diagrams** **Note:** The question refers to "Figure (1)", but the image of the beam is not provided. For demonstration, let's assume a simply supported beam of length \( L = 400 \) mm with a point load \( P = 20 \) kN at the center. (You can replace these values with the actual values from your figure.) ### **Example Beam:** - **Length, \( L \) = 400 mm** - **Point load, \( P \) = 20 kN at center (200 mm from each support)** #### **A. Calculate Reactions at Supports** For a simply supported beam with central load: \[ R_A = R_B = \frac{P}{2} = \frac{20\,\text{kN}}{2} = 10\,\text{kN} \] #### **B. Shear Force Diagram (S.F.)** - From A to load: S.F. = \( +10 \) kN - Just right of load: S.F. = \( +10 - 20 = -10 \) kN - From load to B: S.F. = \( -10 \) kN #### **C. Bending Moment Diagram (B.M.)** - At supports (A and B): \( M = \) - At center (under load): \[ M_{\text{max}} = R_A \times \frac{L}{2} = 10\,\text{kN} \times 2\,\text{m} = 20\,\text{kN}\cdot\text{m} = 20,000\,\text{Nm} \] --- ### **Diagrams** #### Shear Force Diagram (S.F.) ``` S.F. (kN) | ________ 10| | | | |_______|_________ -10 200 400 (mm) (load) ``` #### Bending Moment Diagram (B.M.) ``` B.M. (Nm) | /\ | / \ | / \ |________/ \__________ 200 400 (mm) (Max M=20,000 Nm) ``` --- ## Step 2: **Find Maximum Compression and Tension Stress** ### **A. Maximum Bending Stress Formula** \[ \sigma_{\text{max}} = \frac{M_{\text{max}} \cdot y}{I} \] - \( M_{\text{max}} \): Maximum moment (Nm) - \( y \): Distance from neutral axis to outer fiber (mm) - \( I \): Moment of inertia (mm\(^4\)) ### **B. Example Calculation** Assume beam cross-section: - **Width, \( b = 100 \) mm** - **Height, \( h = 200 \) mm** \[ I = \frac{b h^3}{12} = \frac{100 \times (200)^3}{12} = 66,666,667\,\text{mm}^4 \] \[ y = \frac{h}{2} = 100\,\text{mm} \] \[ \sigma_{\text{max}} = \frac{20,000,000\,\text{Nmm} \times 100\,\text{mm}}{66,666,667\,\text{mm}^4} = 30\,\text{N/mm}^2 \] --- ### **Final Answers** - **Maximum Shear Force**: \( 10 \) kN (at supports, just before/after load) - **Maximum Bending Moment**: \( 20,000 \) Nm (at center) - **Maximum Bending Stress**: \( 30\,\text{N/mm}^2 \) (Compression at top fiber, tension at bottom fiber) --- ## **Summary Table** | Parameter | Value | Location | |----------------------------|---------------|------------------| | Max Shear Force (S.F.) | \(10\) kN | At supports | | Max Bending Moment (B.M.) | \(20,000\) Nm | At beam center | | Max Compression/Tension | \(30\) N/mm² | Top/Bottom fiber | --- **Note:** Replace example values with actual values from your specific figure for your answer. --- **[Insert your diagram here, matching your beam's actual loading and supports]**