Stress Resultants 841D7F

1. **State the problem:** We have a rectangular beam with width $b = 300$ mm and height $h = 500$ mm. It carries a concentric prestress $P = 900$ kN and an external bending moment $M = 120$ kN\cdot m. We need to find the stresses at the top and bottom fibers of the beam. 2. **Relevant formulas:** The total stress at a fiber due to axial load and bending moment is given by: $$\sigma = \frac{P}{A} \pm \frac{M y}{I}$$ where: - $A = b \times h$ is the cross-sectional area, - $I = \frac{b h^3}{12}$ is the moment of inertia about the neutral axis, - $y$ is the distance from the neutral axis to the fiber (top or bottom), - The $+$ sign is for the fiber where bending causes tension, - The $-$ sign is for the fiber where bending causes compression. 3. **Calculate cross-sectional area $A$:** $$A = 300 \times 500 = 150000 \text{ mm}^2$$ 4. **Calculate moment of inertia $I$:** $$I = \frac{300 \times 500^3}{12} = \frac{300 \times 125000000}{12} = \frac{37500000000}{12} = 3125000000 \text{ mm}^4$$ 5. **Convert units for $P$ and $M$ to N and N\cdot mm:** $$P = 900 \text{ kN} = 900000 \text{ N}$$ $$M = 120 \text{ kN}\cdot m = 120000 \text{ N}\cdot m = 120000 \times 1000 = 120000000 \text{ N}\cdot mm$$ 6. **Calculate axial stress $\sigma_a$:** $$\sigma_a = \frac{P}{A} = \frac{900000}{150000} = 6 \text{ N/mm}^2$$ 7. **Calculate bending stress $\sigma_b$ at top and bottom fibers:** Distance from neutral axis to top fiber: $$y_{top} = \frac{h}{2} = \frac{500}{2} = 250 \text{ mm}$$ Distance from neutral axis to bottom fiber: $$y_{bottom} = -250 \text{ mm}$$ Calculate bending stress magnitude: $$\sigma_b = \frac{M y}{I}$$ For top fiber: $$\sigma_{b,top} = \frac{120000000 \times 250}{3125000000} = \frac{30000000000}{3125000000} = 9.6 \text{ N/mm}^2$$ For bottom fiber: $$\sigma_{b,bottom} = \frac{120000000 \times (-250)}{3125000000} = -9.6 \text{ N/mm}^2$$ 8. **Calculate total stresses:** Top fiber stress (compression due to bending): $$\sigma_{top} = \sigma_a - \sigma_{b,top} = 6 - 9.6 = -3.6 \text{ N/mm}^2$$ Bottom fiber stress (tension due to bending): $$\sigma_{bottom} = \sigma_a - \sigma_{b,bottom} = 6 - (-9.6) = 6 + 9.6 = 15.6 \text{ N/mm}^2$$ 9. **Interpretation:** - Negative stress means compression. - Positive stress means tension. **Final answer:** - Top fiber stress = $-3.6$ N/mm$^2$ (compression) - Bottom fiber stress = $15.6$ N/mm$^2$ (tension)