To determine the maximum shear stress (\(\tau_{max}\)) at the section where the internal shear is maximum in the T-beam, we need to follow these steps: --- ## Step 1: Identify the shear force \(V\) The shear force \(V\) at the section where the internal shear is maximum is generally equal to the total shear force acting on the beam. Given data: - \(w_1 = 28\, \text{kN/m}\) - \(w_2 = 42\, \text{kN/m}\) The total shear force \(V\) at the section is the sum of the distributed loads over the relevant length: \[ V = (w_1 + w_2) \times \text{length} \] Assuming the length of the loaded segment is 3 meters (from the figure), then: \[ V = (28 + 42) \times 3 = 70 \times 3 = 210\, \text{kN} \] --- ## Step 2: Determine the cross-sectional properties The maximum shear stress in a T-beam occurs at the junction of the flange and web. The shear flow \(q\) and shear stress \(\tau\) are related to the shear force \(V\) and the cross-sectional properties. --- ## Step 3: Calculate the web shear area The web shear area \(A_v\) at the section: \[ A_v = h \times t_w \] Where: - \(h\) is the web height - \(t_w\) is the web thickness From the figure: - The web thickness \(t_w = 20\, \text{mm} = .02\, \text{m}\) - The web height \(h = 200\, \text{mm} = .2\, \text{m}\) So, \[ A_v = .2 \times .02 = .004\, \text{m}^2 \] --- ## Step 4: Calculate shear stress \(\tau\) Shear stress in the web: \[ \tau = \frac{V}{A_v} \] \[ \tau = \frac{210\, \text{kN}}{.004\, \text{m}^2} = \frac{210,000\, \text{N}}{.004\, \text{m}^2} = 52.5\, \text{kPa} \] --- ## **Final answer:** \[ \boxed{ \tau_{max} \approx 52.5\, \text{kPa} } \] --- *Note:* This is an approximation based on the typical shear flow in the web; actual maximum shear stress might slightly vary depending on more detailed cross-sectional analysis.