Certainly! Let's work through the problem step-by-step to find the maximum shear stress in the cross-section of the steel beam. --- ## **Problem Restatement** - **Beam Type:** Rectangular steel beam - **Width (b):** 60 mm - **Depth (h):** 100 mm - **Shear Force (V):** 12 kN (which is 12,000 N) - **Material:** Uniform, no stress concentration - **Objective:** Determine the maximum shear stress (\(\tau_{max}\)) in the cross-section --- ## **Step 1: Understand the shear stress formula** In a rectangular cross-section, the shear stress at any point is given by the formula: \[ \tau = \frac{V \times Q}{I \times t} \] However, for maximum shear stress in a rectangular section, there's a simplified relation derived from shear stress distribution: \[ \tau_{max} = \frac{3}{2} \times \frac{V}{A} \] where: - \(A\) is the cross-sectional area, - \(V\) is the shear force. This relation is valid for maximum shear stress at the neutral axis in a rectangular section. --- ## **Step 2: Calculate the cross-sectional area \(A\)** \[ A = b \times h \] \[ A = 60\, \text{mm} \times 100\, \text{mm} = 600\, \text{mm}^2 \] --- ## **Step 3: Convert units** Since the shear force is in Newtons and the area in \(\text{mm}^2\), convert the area to \(\text{m}^2\) for consistency, or keep units consistent in mm: - Shear force \(V = 12,000\, \text{N}\) - Area \(A = 600\, \text{mm}^2\) To keep units consistent, note that: \[ 1\, \text{mm}^2 = 1 \times 10^{-6}\, \text{m}^2 \] So, \[ A = 600 \times 10^{-6} = 6 \times 10^{-3}\, \text{m}^2 \] --- ## **Step 4: Calculate the shear stress \(\tau_{max}\)** Using the simplified maximum shear stress formula: \[ \tau_{max} = \frac{3}{2} \times \frac{V}{A} \] Substitute the known values: \[ \tau_{max} = \frac{3}{2} \times \frac{12,000\, \text{N}}{6 \times 10^{-3}\, \text{m}^2} \] Calculate the shear force per unit area: \[ \frac{12,000}{6 \times 10^{-3}} = 12,000 \div .006 = 2,000,000\, \text{Pa} \quad (\text{or N/m}^2) \] Now multiply by \(\frac{3}{2}\): \[ \tau_{max} = \frac{3}{2} \times 2,000,000 = 1.5 \times 2,000,000 = 3,000,000\, \text{Pa} \] which is: \[ \boxed{\tau_{max} = 3\, \text{MPa}} \] --- ## **Final Answer:** **The maximum shear stress in the cross-section of the steel beam is \(\boxed{3\, \text{MPa}}\).** --- ## **Summary:** - Calculated cross-sectional area. - Used the simplified maximum shear stress formula for a rectangular section. - Substituted values and performed calculations in consistent units. - Final shear stress is 3 MPa, which indicates the maximum shear stress experienced at the neutral axis. Let me know if you'd like more details on shear stress distribution or other related topics!