[1] Figure 1 demonstrates a retaining wall AC (breadth b and height h) rotating around the hinge at point A. The box is filled with dry soil having a unit weight y and a friction angle o, reaching a thickness t while the wall remains upright. Later, a strong cord CE attached to the midpoint of the top of the wall passes over pulley D and then runs vertically down to a dead weight given by a mass m. The wall is gently rotated either inward or outward. At the instant when the soil yields, an excessive movement of the wall occurs as dead weight reaches extreme values, limited by active and passive lateral earth pressure. If the results are unaffected by the friction between the soil, the wall, and the box, answer the following questions. Table 4.2 Bearing Capacity Factors h B THI A Soil F B= T if T≤BI√2, B'=B/√2 if T>B/√2, B" =√2B' $' No N₁ Ny $' No N₁ Ny 0 5.14 1.00 0.00 1 5.38 1.09 0.07 2 5.63 1.20 0.15 3 5.90 1.31 0.24 4 6.19 1.43 0.34 5 6.49 1.57 0.45 6 6.81 1.72 0.57 7 7.16 1.88 0.71 8 7.53 2.06 0.86 9 7.92 2.25 1.03 E 10 8.35 2.47 1.22 11 8.80 2.71 1.44 m 12 9.28 2.97 1.69 13 9.81 3.26 1.97 14 10.37 3.59 2.29 68922222222223 16 11.63 4.34 3.06 17 12.34 4.77 3.53 CN 1+0.24 18 13.10 5.26 4.07 B" L FS= 19 13.93 5.80 4.68 20 14.83 6.40 5.39 2+ H H 21 15.82 7.07 6.20 16.88 7.82 7.13 18.05 8.66 8.20 1 ,= 24 19.32 9.60 9.44 H Cav = [c₁H₁ + c₂H₂+...+cH] H 25 20.72 10.66 10.88 22.25 11.85 12.54 Mmax 27 23.94 13.20 14.47 (A)(s²) 8 M (C₁ +C₂)(s²) = max 8 25.80 14.72 16.72 27.86 16.44 19.34 K1-sin o' 30 30.14 18.40 22.40 K≈0.95-sin & Ko(overconsoldate) K o(normally consolidated) NOCR 15 10.98 3.94 2.65 31 32.67 20.63 25.99 (continued) Rankine's active earth pressure Coulomb's active earth pressure Pama Table 4.2 Bearing Capacity Factors (Continued) Active force Figure 1 $' No Na Ny $' N Na Ny 32 35.49 23.18 30.22 42 93.71 85.38 155.55 33 38.64 26.09 35.19 43 105.11 99.02 186.54 (1.1) To prevent the boundary effect on the wall, the length / must be sufficiently long. Using Coulomb's earth pressure theory, express the minimum required length 1, in terms of t and ø, ensuring that the rupture plane of any wedge failure in cohesionless soil does not reach side FG. (1.2) According to Rankine's earth pressure theory, draw the active and passive earth pressure diagrams for cohesionless soil behind the retaining wall AC. Also, indicate the expressions for the active and passive earth pressures at points A and B in each diagram. 34 42.16 29.44 41.06 44 118.37 115.31 224.64 35 46.12 33.30 48.03 45 133.88 134.88 271.76 σ Pa 36 50.59 37.75 56.31 46 152.10 158.51 330.35 H 37 55.63 42.92 66.19 47 173.64 187.21 403.67 38 61.35 48.93 78.03 48 199.26 222.31 496.01 39 67.87 55.96 92.25 49 229.93 265.51 613.16 H/3 40 75.31 64.20 109.41 50 266.89 319.07 762.89 41 83.86 73.90 130.22 (1.3) As a consequence, describe the expressions for the resultant force and its location from point A under the active and passive earth pressures, using 6, y, b, and t. (1.4) To balance the earth pressure and the cord's tension, the resultant forces and the dead weights must satisfy the equilibrium of moments. Derive the masses corresponding to active and passive states, respectively. (1.5) Experiments were conducted to determine the extreme values of mass m needed to maintain the wall's stability upon yielding. The height h of the wall was 0.30 m, the soil thickness t was 0.20 m, and the breadth b of the box perpendicular to the section was 0.25 m. The minimum mass m was 0.4 kg, while the maximum mass m was 5.0 kg. Determine the unknown y and ø, using the gravitational acceleration g as 9.8 m/s². 0.25 H * 0.25 H 0.5 H HAD σα 0.75 H 0.25 H K = cosa cosa v - √cos² α- cos² cosa+cos α-cos² Q' Q' Fed =1+0.4() d F =1 Equations and Tables: Bearing capacity equation: q=c'N FFF+q'N FFF +0.5yBN, FFF Shape factors by De Beer Depth factors by Hansen (1970) (1970) B. N F =1+(-)() LN F=1+() tan F =1-0.4() F₁ = F₁₁ = (1- Inclination factors by Meyerhof (1963) (a) σ = 0.657HK (b) the larger of σ =yH|1- 4c YH and Hanna and Meyerhof (1981) B° B -)² 90° D Fad 1+2 tan '(1-sin o')2. B Fx = (1-2)² and σ = 0.37H (c) σ = 0.2yH to 0.47H Wall movement away from C C₁ B (a) B-8 c'=0 w R sin² (B+') (b) K sin ẞsin(B-8) 1+ sin('+ 6) sin(' – α) sin(B-) sin(a + B)

[1] Figure 1 demonstrates a retaining wall AC (breadth b and height h) rotating around the hinge at point A. The box is filled with dry soil having a unit weight y and a friction angle o, reaching a thickness t while the wall remains upright. Later, a strong cord CE attached to the midpoint of the top of the wall passes over pulley D and then runs vertically down to a dead weight given by a mass m. The wall is gently rotated either inward or outward. At the instant when the soil yields, an excessive movement of the wall occurs as dead weight reaches extreme values, limited by active and passive lateral earth pressure. If the results are unaffected by the friction between the soil, the wall, and the box, answer the following questions. Table 4.2 Bearing Capacity Factors h B THI A Soil F B= T if T≤BI√2, B'=B/√2 if T>B/√2, B" =√2B' $' No N₁ Ny $' No N₁ Ny 0 5.14 1.00 0.00 1 5.38 1.09 0.07 2 5.63 1.20 0.15 3 5.90 1.31 0.24 4 6.19 1.43 0.34 5 6.49 1.57 0.45 6 6.81 1.72 0.57 7 7.16 1.88 0.71 8 7.53 2.06 0.86 9 7.92 2.25 1.03 E 10 8.35 2.47 1.22 11 8.80 2.71 1.44 m 12 9.28 2.97 1.69 13 9.81 3.26 1.97 14 10.37 3.59 2.29 68922222222223 16 11.63 4.34 3.06 17 12.34 4.77 3.53 CN 1+0.24 18 13.10 5.26 4.07 B" L FS= 19 13.93 5.80 4.68 20 14.83 6.40 5.39 2+ H H 21 15.82 7.07 6.20 16.88 7.82 7.13 18.05 8.66 8.20 1 ,= 24 19.32 9.60 9.44 H Cav = [c₁H₁ + c₂H₂+...+cH] H 25 20.72 10.66 10.88 22.25 11.85 12.54 Mmax 27 23.94 13.20 14.47 (A)(s²) 8 M (C₁ +C₂)(s²) = max 8 25.80 14.72 16.72 27.86 16.44 19.34 K1-sin o' 30 30.14 18.40 22.40 K≈0.95-sin & Ko(overconsoldate) K o(normally consolidated) NOCR 15 10.98 3.94 2.65 31 32.67 20.63 25.99 (continued) Rankine's active earth pressure Coulomb's active earth pressure Pama Table 4.2 Bearing Capacity Factors (Continued) Active force Figure 1 $' No Na Ny $' N Na Ny 32 35.49 23.18 30.22 42 93.71 85.38 155.55 33 38.64 26.09 35.19 43 105.11 99.02 186.54 (1.1) To prevent the boundary effect on the wall, the length / must be sufficiently long. Using Coulomb's earth pressure theory, express the minimum required length 1, in terms of t and ø, ensuring that the rupture plane of any wedge failure in cohesionless soil does not reach side FG. (1.2) According to Rankine's earth pressure theory, draw the active and passive earth pressure diagrams for cohesionless soil behind the retaining wall AC. Also, indicate the expressions for the active and passive earth pressures at points A and B in each diagram. 34 42.16 29.44 41.06 44 118.37 115.31 224.64 35 46.12 33.30 48.03 45 133.88 134.88 271.76 σ Pa 36 50.59 37.75 56.31 46 152.10 158.51 330.35 H 37 55.63 42.92 66.19 47 173.64 187.21 403.67 38 61.35 48.93 78.03 48 199.26 222.31 496.01 39 67.87 55.96 92.25 49 229.93 265.51 613.16 H/3 40 75.31 64.20 109.41 50 266.89 319.07 762.89 41 83.86 73.90 130.22 (1.3) As a consequence, describe the expressions for the resultant force and its location from point A under the active and passive earth pressures, using 6, y, b, and t. (1.4) To balance the earth pressure and the cord's tension, the resultant forces and the dead weights must satisfy the equilibrium of moments. Derive the masses corresponding to active and passive states, respectively. (1.5) Experiments were conducted to determine the extreme values of mass m needed to maintain the wall's stability upon yielding. The height h of the wall was 0.30 m, the soil thickness t was 0.20 m, and the breadth b of the box perpendicular to the section was 0.25 m. The minimum mass m was 0.4 kg, while the maximum mass m was 5.0 kg. Determine the unknown y and ø, using the gravitational acceleration g as 9.8 m/s². 0.25 H * 0.25 H 0.5 H HAD σα 0.75 H 0.25 H K = cosa cosa v - √cos² α- cos² cosa+cos α-cos² Q' Q' Fed =1+0.4() d F =1 Equations and Tables: Bearing capacity equation: q=c'N FFF+q'N FFF +0.5yBN, FFF Shape factors by De Beer Depth factors by Hansen (1970) (1970) B. N F =1+(-)() LN F=1+() tan F =1-0.4() F₁ = F₁₁ = (1- Inclination factors by Meyerhof (1963) (a) σ = 0.657HK (b) the larger of σ =yH|1- 4c YH and Hanna and Meyerhof (1981) B° B -)² 90° D Fad 1+2 tan '(1-sin o')2. B Fx = (1-2)² and σ = 0.37H (c) σ = 0.2yH to 0.47H Wall movement away from C C₁ B (a) B-8 c'=0 w R sin² (B+') (b) K sin ẞsin(B-8) 1+ sin('+ 6) sin(' – α) sin(B-) sin(a + B)

Fundamentals of Geotechnical Engineering (MindTap Course List)

5th Edition

ISBN:9781305635180

Author:Braja M. Das, Nagaratnam Sivakugan

Publisher:Braja M. Das, Nagaratnam Sivakugan

Chapter15: Retaining Walls, Braced Cuts, And Sheet Pile Walls

Section: Chapter Questions

Problem 15.3P

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