1. **Problem Statement:** Determine the design moment of a rectangular beam with width $b=400$ mm, total height $h=800$ mm, concrete compressive strength $f_c'=28$ MPa, and steel yield strength $f_y=420$ MPa. The beam has 5 bottom reinforcement bars of 32 mm diameter. 2. **Relevant Formulas and Concepts:** - The design moment capacity $M_u$ of a reinforced concrete beam is calculated using the formula: $$M_u = A_s f_y (d - a/2)$$ where: - $A_s$ is the area of tensile steel reinforcement, - $d$ is the effective depth (distance from compression face to centroid of tensile steel), - $a$ is the depth of the equivalent rectangular stress block, given by: $$a = \beta_1 c$$ - $c$ is the neutral axis depth, found by equilibrium: $$A_s f_y = 0.85 f_c' \beta_1 c b$$ - $\beta_1$ depends on $f_c'$; for $f_c' = 28$ MPa, $\beta_1 \approx 0.85$. 3. **Calculate Area of Steel $A_s$:** - Diameter $d_b = 32$ mm, number of bars $n=5$. - Area of one bar: $$A_{bar} = \pi \frac{d_b^2}{4} = \pi \frac{32^2}{4} = \pi \times 256 = 804.25 \text{ mm}^2$$ - Total steel area: $$A_s = 5 \times 804.25 = 4021.25 \text{ mm}^2$$ 4. **Determine Effective Depth $d$:** - Total height $h=800$ mm. - Assume concrete cover plus half bar diameter for centroid of steel: $$d = 800 - (75 + \frac{32}{2}) = 800 - (75 + 16) = 709 \text{ mm}$$ 5. **Find Neutral Axis Depth $c$:** - Using equilibrium: $$A_s f_y = 0.85 f_c' \beta_1 c b$$ $$4021.25 \times 420 = 0.85 \times 28 \times 0.85 \times c \times 400$$ Calculate left side: $$4021.25 \times 420 = 1,688,925 \text{ N}$$ Calculate right side constants: $$0.85 \times 28 \times 0.85 \times 400 = 8084$$ Solve for $c$: $$c = \frac{1,688,925}{8084} = 208.95 \text{ mm}$$ 6. **Calculate $a$:** $$a = \beta_1 c = 0.85 \times 208.95 = 177.61 \text{ mm}$$ 7. **Calculate Design Moment $M_u$:** $$M_u = A_s f_y (d - \frac{a}{2})$$ $$= 4021.25 \times 420 \times (709 - \frac{177.61}{2})$$ Calculate inside parentheses: $$709 - 88.805 = 620.195 \text{ mm}$$ Calculate moment in N.mm: $$M_u = 4021.25 \times 420 \times 620.195 = 1,047,000,000 \text{ N.mm}$$ Convert to kN.m: $$M_u = \frac{1,047,000,000}{1,000,000} = 1047 \text{ kN.m}$$ 8. **Compare with given options:** Closest option is 986.77 kN.m, but our calculation yields approximately 1047 kN.m. **Note:** Differences may arise due to assumptions in effective depth or $\beta_1$ value. **Final answer:** Approximately **1047 kN.m** (not listed among options).