1. **Problem Statement:** Calculate the allowable load on a simply supported precast concrete beam with given dimensions and material properties, and determine the cracking moments for positive and negative bending moments. 2. **Given Data:** - Span length, $L = 3$ m - Concrete compressive strength, $f'_{c} = 17$ MPa - Concrete unit weight, $\gamma = 17$ kN/m$^3$ - Dimensions: $a = 250$ mm, $d = 200$ mm, $b = 350$ mm, $e = 600$ mm, $c = 100$ mm - Cracking moment, $M_{cr} = 27$ kN-m 3. **Formulas and Concepts:** - The cracking moment $M_{cr}$ is related to the allowable bending moment before cracks appear. - For a simply supported beam with uniform load $w$, the maximum moment is $M_{max} = \frac{wL^2}{8}$. - To find allowable load $w$, rearrange: $w = \frac{8M_{cr}}{L^2}$. - Cracking moment depends on the modulus of rupture $f_r$ and section modulus $S$. - Modulus of rupture $f_r$ for concrete can be approximated by $f_r = 0.62 \sqrt{f'_{c}}$ MPa. - Section modulus $S = \frac{I}{y}$, where $I$ is moment of inertia and $y$ is distance from neutral axis to extreme fiber. 4. **Step 1: Calculate allowable load $w$ using given $M_{cr}$** $$ w = \frac{8M_{cr}}{L^2} = \frac{8 \times 27}{3^2} = \frac{216}{9} = 24 \text{ kN/m} $$ 5. **Step 2: Calculate modulus of rupture $f_r$** $$ f_r = 0.62 \sqrt{17} = 0.62 \times 4.123 = 2.56 \text{ MPa} $$ 6. **Step 3: Calculate moment of inertia $I$ and neutral axis $y$ for T-section** - Convert all dimensions to meters: - $a = 0.25$ m, $b = 0.35$ m, $d = 0.20$ m, $e = 0.60$ m, $c = 0.10$ m - The T-section consists of flange and web: - Flange area $A_f = e \times a = 0.60 \times 0.25 = 0.15$ m$^2$ - Web area $A_w = b \times d = 0.35 \times 0.20 = 0.07$ m$^2$ - Calculate centroid $\bar{y}$ from bottom: - Flange centroid at $a/2 = 0.125$ m above flange bottom, flange bottom is at $d + a = 0.20 + 0.25 = 0.45$ m from bottom, so flange centroid from bottom: $$y_f = 0.45 + 0.125 = 0.575 \text{ m}$$ - Web centroid at $d/2 = 0.10$ m from bottom - Total area $A = A_f + A_w = 0.15 + 0.07 = 0.22$ m$^2$ - Neutral axis $\bar{y}$: $$ \bar{y} = \frac{A_f y_f + A_w y_w}{A} = \frac{0.15 \times 0.575 + 0.07 \times 0.10}{0.22} = \frac{0.08625 + 0.007}{0.22} = \frac{0.09325}{0.22} = 0.424 \text{ m} $$ 7. **Step 4: Calculate moment of inertia $I$ about neutral axis** - Flange moment of inertia about its own centroid: $$ I_f = \frac{e a^3}{12} = \frac{0.60 \times 0.25^3}{12} = \frac{0.60 \times 0.015625}{12} = 0.00078125 \text{ m}^4 $$ - Distance from flange centroid to neutral axis: $$ d_f = |0.575 - 0.424| = 0.151 \text{ m} $$ - Using parallel axis theorem: $$ I_{f,total} = I_f + A_f d_f^2 = 0.00078125 + 0.15 \times 0.151^2 = 0.00078125 + 0.15 \times 0.0228 = 0.00078125 + 0.00342 = 0.0042 \text{ m}^4 $$ - Web moment of inertia about its own centroid: $$ I_w = \frac{b d^3}{12} = \frac{0.35 \times 0.20^3}{12} = \frac{0.35 \times 0.008}{12} = 0.0002333 \text{ m}^4 $$ - Distance from web centroid to neutral axis: $$ d_w = |0.424 - 0.10| = 0.324 \text{ m} $$ - Using parallel axis theorem: $$ I_{w,total} = I_w + A_w d_w^2 = 0.0002333 + 0.07 \times 0.324^2 = 0.0002333 + 0.07 \times 0.105 = 0.0002333 + 0.00735 = 0.00758 \text{ m}^4 $$ - Total moment of inertia: $$ I = I_{f,total} + I_{w,total} = 0.0042 + 0.00758 = 0.01178 \text{ m}^4 $$ 8. **Step 5: Calculate section modulus $S$ for positive bending (tension at bottom fiber)** - Distance from neutral axis to bottom fiber: $$ y_{bottom} = 0.424 \text{ m} $$ - Section modulus: $$ S = \frac{I}{y_{bottom}} = \frac{0.01178}{0.424} = 0.0278 \text{ m}^3 $$ 9. **Step 6: Calculate cracking moment for positive bending** $$ M_{cr,pos} = f_r \times S = 2.56 \times 0.0278 = 0.0712 \text{ MN-m} = 71.2 \text{ kN-m} $$ 10. **Step 7: Calculate section modulus $S$ for negative bending (tension at top fiber)** - Distance from neutral axis to top fiber: $$ y_{top} = e - \bar{y} = 0.60 - 0.424 = 0.176 \text{ m} $$ - Section modulus: $$ S_{neg} = \frac{I}{y_{top}} = \frac{0.01178}{0.176} = 0.0669 \text{ m}^3 $$ 11. **Step 8: Calculate cracking moment for negative bending** $$ M_{cr,neg} = f_r \times S_{neg} = 2.56 \times 0.0669 = 0.171 \text{ MN-m} = 171 \text{ kN-m} $$ **Final answers:** - a) Allowable load on the beam: $\boxed{24 \text{ kN/m}}$ - b) Cracking moment for positive bending: $\boxed{71.2 \text{ kN-m}}$ - b) Cracking moment for negative bending: $\boxed{171 \text{ kN-m}}$