Let's solve the problem step by step: --- # Given Data - **Beam span, \( L \) = 4. m** - **Central point load, \( P \) = 30 kN = 30,000 N** - **Section arrangement:** - **Top Flange:** 100 mm × 25 mm - **Web:** 200 mm × 20 mm - **Bottom Flange:** 100 mm × 25 mm - **Material Properties:** - **Young's modulus, \( E \) = 12 GPa** - **Allowable shear stress, \( \tau_{allow} \) = 2. MPa** --- ## (a) Determine the Maximum Shear Force, \( V_{max} \) For a simply supported beam with a central point load: \[ V_{max} = \frac{P}{2} \] \[ V_{max} = \frac{30,000~N}{2} = \boxed{15,000~N} \] --- ## (b) Shear Flow at Neutral Axis ### **Step 1: Section Geometry** #### **Overall Height, \( h \):** \[ h = 25~\text{mm}~(\text{top flange}) + 200~\text{mm}~(\text{web}) + 25~\text{mm}~(\text{bottom flange}) = 250~\text{mm} \] #### **Find the Neutral Axis (NA) Location:** Let’s set the bottom of the section as the reference ( \( y = \) ). - **Bottom Flange:** Area, \(A_b = 100 \times 25 = 250~\text{mm}^2\) Centroid, \(y_b = 12.5~\text{mm}\) above bottom - **Web:** Area, \(A_w = 200 \times 20 = 400~\text{mm}^2\) Centroid, \(y_w = 25 + 100 = 125~\text{mm}\) above bottom - **Top Flange:** Area, \(A_t = 100 \times 25 = 250~\text{mm}^2\) Centroid, \(y_t = 225 + 12.5 = 237.5~\text{mm}\) above bottom Weighted centroid (neutral axis): \[ y_{NA} = \frac{A_b y_b + A_w y_w + A_t y_t}{A_b + A_w + A_t} \] \[ y_{NA} = \frac{(250 \times 12.5) + (400 \times 125) + (250 \times 237.5)}{250 + 400 + 250} \] \[ = \frac{31,250 + 500,000 + 593,750}{900} \] \[ = \frac{1,125,000}{900} = 125~\text{mm} \] So, the neutral axis is 125 mm from the bottom. --- ### **Step 2: Second Moment of Area (\(I\))** Use the parallel axis theorem for each part. #### **Bottom Flange:** \[ I_b = \frac{1}{12}bh^3 + A_b d^2 \] Here, \(d\) is the distance from centroid of the part to NA. \[ d_b = y_{NA} - y_b = 125 - 12.5 = 112.5~\text{mm} \] \[ I_b = \frac{1}{12}(100)(25^3) + 250 \times (112.5^2) \] \[ = \frac{1}{12}(100)(15,625) + 250 \times 12,656.25 \] \[ = 130,208.33 + 31,640,625 = 31,770,833.33~\text{mm}^4 \] #### **Web:** \[ d_w = |y_{NA} - y_w| = |125 - 125| = \] \[ I_w = \frac{1}{12}(20)(200^3) = \frac{1}{12}(20)(8,000,000) \] \[ = \frac{1}{12}(160,000,000) = 13,333,333.33~\text{mm}^4 \] #### **Top Flange:** \[ d_t = y_t - y_{NA} = 237.5 - 125 = 112.5~\text{mm} \] \[ I_t = \frac{1}{12}(100)(25^3) + 250 \times (112.5^2) \] (Same as bottom flange) \[ = 130,208.33 + 31,640,625 = 31,770,833.33~\text{mm}^4 \] **Total \(I\):** \[ I = I_b + I_w + I_t = 31,770,833.33 + 13,333,333.33 + 31,770,833.33 \] \[ I = 76,875,000~\text{mm}^4 \] --- ### **Step 3: Shear Flow at Neutral Axis** Shear flow \(q\) at neutral axis (in the web) is: \[ q = \frac{V Q}{I} \] Where: - \(V\) = Shear force at the section (\(15,000~N\)) - \(Q\) = First moment of area above (or below) the point where shear flow is being calculated (about the NA) - \(I\) = Second moment of area #### **First Moment of Area (\(Q\)) at NA:** Take area above NA: - **Top flange:** - Area = 250 mm² - Distance from NA to its centroid = 237.5 - 125 = 112.5 mm - \(Q_{tf} = 250 \times 112.5 = 281,250~\text{mm}^3\) - **Upper part of web (from NA to top flange):** - Height = 100 mm (from NA at 125 mm to top at 225 mm) - Area = 20 mm × 100 mm = 200 mm² - Centroid distance from NA = 125 + 50 = 175 mm, but **relative to NA**, centroid is at 50 mm above NA. - \(Q_{web,upper} = 200 \times 50 = 100,000~\text{mm}^3\) Total \(Q\): \[ Q = Q_{tf} + Q_{web,upper} = 281,250 + 100,000 = 381,250~\text{mm}^3 \] #### **Calculate \(q_{NA}\):** \[ q_{NA} = \frac{15,000 \times 381,250}{76,875,000} \] \[ = \frac{5,718,750,000}{76,875,000} \] \[ = 74.4~\text{N/mm} \] --- ## (c) Shear Flow at Top Flange-Web Interface At the interface, **Q** is just the area of the top flange times the distance from NA: \[ Q_{interface} = Q_{tf} = 281,250~\text{mm}^3 \] So, \[ q_{interface} = \frac{15,000 \times 281,250}{76,875,000} \] \[ = \frac{4,218,750,000}{76,875,000} \] \[ = 54.88~\text{N/mm} \] --- # **Final Answers** ### **(a) Maximum shear force \( V_{max} \):** \[ \boxed{15,000~\text{N}} \] ### **(b) Shear flow at the neutral axis:** \[ \boxed{74.4~\text{N/mm}} \] ### **(c) Shear flow at the top flange-web interface:** \[ \boxed{54.88~\text{N/mm}} \] --- **Let me know if you need further clarification or a diagram!**