1. **Problem Statement:** Determine the dimensions of a counterfort retaining wall retaining earth with a horizontal top 5.5 m above ground level. Given data: - Height of retained earth, $H = 5.5$ m - Density of earth, $\gamma = 16$ kN/m$^3$ - Angle of internal friction, $\phi = 30^\circ$ - Safe bearing capacity (SBC) of soil = 200 kN/m$^2$ - Coefficient of friction, $\mu = 0.6$ - Concrete grade = M20 - Steel grade = Fe 415 2. **Calculate active earth pressure:** Using Rankine's theory, active earth pressure coefficient: $$K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right) = \tan^2(45^\circ - 15^\circ) = \tan^2(30^\circ) = (0.577)^2 = 0.333$$ 3. **Calculate active earth pressure at the base:** $$p_a = K_a \times \gamma \times H = 0.333 \times 16 \times 5.5 = 29.33\text{ kN/m}^2$$ 4. **Calculate total active force on the wall:** $$P_a = \frac{1}{2} p_a H = \frac{1}{2} \times 29.33 \times 5.5 = 80.66\text{ kN/m}$$ 5. **Determine base width (B) for stability:** Assuming the base width $B$ is to be found such that the bearing pressure does not exceed SBC. Total vertical load includes: - Weight of the wall (assumed initially) - Weight of soil on the base - Active earth pressure 6. **Assume initial dimensions:** - Thickness of stem $t_s = 0.3$ m - Thickness of base slab $t_b = 0.5$ m - Thickness of heel and toe each $0.5$ m 7. **Calculate weight of concrete:** Density of concrete $\approx 24$ kN/m$^3$ Volume of stem per meter length: $$V_s = t_s \times H = 0.3 \times 5.5 = 1.65\text{ m}^3$$ Weight of stem: $$W_s = 1.65 \times 24 = 39.6\text{ kN}$$ Volume of base slab per meter length: $$V_b = B \times t_b = B \times 0.5$$ Weight of base slab: $$W_b = 24 \times 0.5 B = 12 B\text{ kN}$$ 8. **Calculate soil weight on heel:** Assuming heel length $L_h = 0.6 B$ Volume of soil on heel: $$V_{soil} = L_h \times 1 \times H = 0.6 B \times 5.5 = 3.3 B\text{ m}^3$$ Weight of soil on heel: $$W_{soil} = 3.3 B \times 16 = 52.8 B\text{ kN}$$ 9. **Calculate total vertical load:** $$W = W_s + W_b + W_{soil} = 39.6 + 12 B + 52.8 B = 39.6 + 64.8 B$$ 10. **Calculate resultant horizontal force:** $$P_a = 80.66\text{ kN}$$ 11. **Check for overturning stability:** Moment due to vertical loads about toe: Assuming centroid of stem at $H/2 = 2.75$ m from base Moment of stem: $$M_s = W_s \times \frac{B}{2}$$ Moment of base slab: $$M_b = W_b \times \frac{B}{2}$$ Moment of soil: $$M_{soil} = W_{soil} \times \frac{L_h}{2} = 52.8 B \times 0.3 B = 15.84 B^2$$ Moment of active force about toe: $$M_a = P_a \times \frac{H}{3} = 80.66 \times 1.83 = 147.6\text{ kNm}$$ 12. **Calculate factor of safety against overturning:** Sum of resisting moments: $$M_r = W_s \times \frac{B}{2} + W_b \times \frac{B}{2} + M_{soil} = 39.6 \times \frac{B}{2} + 12 B \times \frac{B}{2} + 15.84 B^2 = 19.8 B + 6 B^2 + 15.84 B^2 = 19.8 B + 21.84 B^2$$ Factor of safety: $$FS = \frac{M_r}{M_a} = \frac{19.8 B + 21.84 B^2}{147.6}$$ Set $FS \geq 1.5$ for safety: $$\frac{19.8 B + 21.84 B^2}{147.6} \geq 1.5$$ $$19.8 B + 21.84 B^2 \geq 221.4$$ Solve quadratic: $$21.84 B^2 + 19.8 B - 221.4 = 0$$ Using quadratic formula: $$B = \frac{-19.8 \pm \sqrt{19.8^2 + 4 \times 21.84 \times 221.4}}{2 \times 21.84}$$ Calculate discriminant: $$\Delta = 392.04 + 19332.7 = 19724.74$$ $$\sqrt{\Delta} = 140.44$$ Positive root: $$B = \frac{-19.8 + 140.44}{43.68} = \frac{120.64}{43.68} = 2.76\text{ m}$$ 13. **Check bearing pressure:** Total vertical load: $$W = 39.6 + 64.8 \times 2.76 = 39.6 + 178.85 = 218.45\text{ kN}$$ Bearing pressure: $$q = \frac{W}{B} = \frac{218.45}{2.76} = 79.15\text{ kN/m}^2$$ Since $79.15 < 200$, SBC is safe. 14. **Design stem and counterfort:** - Use M20 concrete and Fe415 steel. - Calculate bending moments and shear forces from earth pressure. - Design reinforcement accordingly (detailed structural design beyond scope here). **Final dimensions:** - Base width $B = 2.76$ m - Stem thickness $t_s = 0.3$ m (adjust as per design) - Base slab thickness $t_b = 0.5$ m - Heel length $L_h = 0.6 B = 1.66$ m These dimensions ensure stability against overturning and bearing capacity requirements.