Let's break down your question about shear force and moment diagrams, and calculation of flexural strength for 3-notched and 4-notched (more commonly called 3-point and 4-point bending) prismatic beams. ## 1. **Understanding the Problem** - **Beam dimensions:** 100 mm (width) × 100 mm (height) × 500 mm (length) - **3-notched bending:** Usually refers to the **3-point bending** test (a single point load at midspan). - **4-notched bending:** Usually refers to **4-point bending** (two equal loads symmetrically placed). We are to: 1. Draw shear force and bending moment diagrams for both loadings. 2. Explain how to calculate flexural strength (modulus of rupture) for both. --- ## 2. **Shear Force and Bending Moment Diagrams** ### **A. 3-Point Bending (3-Notched) Diagram** #### **Beam Setup:** - Simply supported beam, length \( L = 500\,\text{mm} \) - Single load \( P \) at midspan (\( x = L/2 \)) #### **Shear Force Diagram (SFD):** - **Left Support to Load (\( x < L/2 \)):** - Shear force, \( V = +P/2 \) (constant) - **At Load (\( x = L/2 \)):** - Shear drops by \( P \) - **Right of Load (\( x > L/2 \)):** - Shear force, \( V = -P/2 \) (constant) #### **Bending Moment Diagram (BMD):** - **From support to midspan:** - \( M(x) = (P/2)x \) (linear increase) - **At midspan (maximum moment):** - \( M_{max} = (P/2) \times (L/2) = PL/4 \) - **From midspan to right support:** - \( M(x) = (P/2)(L-x) \) (linear decrease) **Diagrams:** ``` Shear Force: Bending Moment: |-----+ +-----| | /\ | | | | | / \ | | | | | / \ | | | | | / \ | | | | | / \ |-----+-----+-----| |----------- +P/2 -P/2 0 PL/4 ``` --- ### **B. 4-Point Bending (4-Notched) Diagram** #### **Beam Setup:** - Simply supported beam, length \( L = 500\,\text{mm} \) - Two equal loads \( P/2 \) each, symmetrically placed at \( a \) from each support. - The region between the loads is pure bending (constant moment, zero shear). Let’s assume the loads are at \( a = L/3 \) from each support, so the central region is \( L - 2a = 500 - 2(167) = 166\,\text{mm} \). #### **Shear Force Diagram (SFD):** - **Left Support to First Load (\( x < a \)):** - Shear force, \( V = +P/2 \) - **Between Loads (\( a < x < L-a \)):** - Shear force, \( V = 0 \) - **After Second Load (\( x > L-a \)):** - Shear force, \( V = -P/2 \) #### **Bending Moment Diagram (BMD):** - **From support to first load:** - \( M(x) = (P/2)x \) - **Between loads:** - \( M = (P/2)a \) (constant) - **After second load:** - \( M(x) = (P/2)(L-x) \) **Diagrams:** ``` Shear Force: Bending Moment: |-------+ +------| | ______ | | | | | / \ | | | | | / \ | | | | |___/ \___ | | | | |-------+-----+------| +P/2 0 -P/2 0 (P/2)a ``` --- ## 3. **Calculation of Flexural Strength (Modulus of Rupture)** ### **A. 3-Point Bending (Single Point Load at Midspan):** Maximum moment at midspan: \[ M_{max} = \frac{PL}{4} \] Flexural strength (modulus of rupture) is: \[ \sigma_f = \frac{M_{max} \cdot y_{max}}{I} \] Where: - \( y_{max} = h/2 = 50\,\text{mm} \) - \( I = \frac{bh^3}{12} \) (for rectangular section, \( b = h = 100\,\text{mm} \)) Plug in: \[ \sigma_{f,3pt} = \frac{\frac{PL}{4} \cdot \frac{h}{2}}{I} = \frac{PLh}{8I} \] Or, using standard formula for 3-point bend modulus of rupture: \[ \sigma_{f,3pt} = \frac{3PL}{2bh^2} \] --- ### **B. 4-Point Bending (Two Equal Loads, Symmetrical):** Maximum moment (between the loads): \[ M_{max} = \frac{P a}{2} \] (where \( a \) is the distance from support to the load) Flexural strength: \[ \sigma_{f,4pt} = \frac{M_{max} \cdot y_{max}}{I} = \frac{P a h}{4I} \] Or, using standard formula for 4-point bend modulus of rupture: \[ \sigma_{f,4pt} = \frac{3P a}{2 b h^2} \] --- ## 4. **Summary Table** | Test | Max Moment | Flexural Strength (\(\sigma_f\)) | |------------|-----------------|------------------------------------------| | 3-Point | \(PL/4\) | \(\frac{3PL}{2bh^2}\) | | 4-Point | \(P a/2\) | \(\frac{3P a}{2bh^2}\) | --- ## 5. **Final Answers** ### **A. Shear Force and Moment Diagrams** - **3-Point Bending:** Shear is \(+P/2\) until midspan, drops to \(-P/2\) after; moment is triangular with max at midspan. - **4-Point Bending:** Shear is \(+P/2\) until first load, zero between loads, \(-P/2\) after; moment is constant between loads, max in that region. ### **B. Flexural Strength Calculations** - **3-Point Bending:** \[ \sigma_{f,3pt} = \frac{3PL}{2 b h^2} \] - **4-Point Bending:** \[ \sigma_{f,4pt} = \frac{3P a}{2 b h^2} \] where \( a \) is the distance from the support to the load. --- **If you need the diagrams drawn, please specify which format (ASCII, hand sketch, etc.). Let me know if you want a numerical example using a specific load!**