1. **State the problem:** We have a simply supported beam of length 12 m with a uniform load of 10 kN/m and two equal point loads of 150 kN each at the third points (4 m and 8 m). We need to check if the steel section W30 x 90 is adequate given Fy = 345 MPa. 2. **Calculate reactions:** Total uniform load = $10 \times 12 = 120$ kN Point loads = $150 + 150 = 300$ kN Total load = $120 + 300 = 420$ kN 3. **Calculate maximum moment:** For uniform load, max moment $M_u = \frac{wL^2}{8} = \frac{10 \times 12^2}{8} = 180$ kNm For point loads at 4 m and 8 m, max moment occurs at center: Moment due to point loads at center: $$M_p = 150 \times 4 + 150 \times 4 = 1200 \text{ kNm}$$ Total max moment $M_{max} = 180 + 1200 = 1380$ kNm 4. **Convert moment to Nmm:** $$M_{max} = 1380 \times 10^6 \text{ Nmm}$$ 5. **Calculate required section modulus $S_{required}$:** Using formula $$M = F_y \times S$$ Rearranged: $$S = \frac{M}{F_y}$$ $$S_{required} = \frac{1380 \times 10^6}{345} = 4.0 \times 10^6 \text{ mm}^3$$ 6. **Compare with given section modulus:** Given $S_x = 4015 \times 10^3 = 4.015 \times 10^6$ mm$^3$ 7. **Conclusion:** Since $S_x = 4.015 \times 10^6 > S_{required} = 4.0 \times 10^6$, the steel section is adequate. **Final answer:** The W30 x 90 steel section is adequate for the given loading conditions.