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MODULE 1. Fundamentals of Soil Mechanics

MODULE 2. Stress and Strength

MODULE 3. Compaction, Seepage and Consolidation of...

MODULE 4. Earth pressure, Slope Stability and Soil...

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## LESSON 27. Rankine’s Theory of Earth Pressure and Numerical Exercise

**27.1 Earth Pressure on Retaining Wall **

**27.1.1. c-ø**

**backfill**

In case of *c***-***f* backfill, negative active earth pressure is developed upto the *Z*_{0 }depth from the ground level as shown in Figure 27.1. It is clear that the net active pressure is zero upto a depth of 2*Z*_{0}. Thus, in case of cohesive soil, vertical cut can be made with out any lateral supported upto the depth equal to 2*Z*_{0}. The depth 2*Z*_{0 }is called as critical depth of vertical cut *H*_{c} in a cohesive soil and can be expressed as:

\[{H_c}=2Z{}_0={{4c} \over {\gamma \sqrt {{K_A}} }}\] (27.1)

where *c* is the cohesion of the soil, \[\gamma\]* *is the effective unit weight of the soil and *K _{A}* is the coefficient of active earth pressure and can be expressed as:

\[{K_A}={{1 - \sin \phi } \over {1 + \sin \phi }}\] (27.2)

During the calculation of total active force (*P*_{A}) in case of *c -*ø backfill, negative zone is neglected and only the active earth pressure due to the positive zone is considered. Thus,

\[{P_A}={1\over 2}{K_A}\gamma\,{\left({H-{Z_0}}\right)^2}={1\over 2}{K_A}\gamma\,{H^2}-2cH\sqrt{{K_A}}+{{2{c^2}}\over\gamma}\] (27.3)

*P _{A}* acts at the height of (

*H-Z*

_{0})/3 from the base of the wall.

**Fig. 27.1. Active earth pressure for c-**ø

**backfill.**

**Fig. 27.2. Passive earth pressure for c-**ø

**backfill.**

**Figure 27.2 shows the passive earth pressure distribution for c-ø backfill. The passive force P_{P}_{1} and P_{P}_{2 }can be determined as:**

The passive force *P _{P}*

_{1}and

*P*

_{P}_{2 }can be determined as:

\[{P_{P1}}={1 \over 2}\gamma\,{K_P}{H^2}\] acts at a height of *H*/3 from the base (27.4)

\[{P_{P2}}=2cH\sqrt{{K_P}}\] acts at a height of *H*/2 from the base (27.5)

Thus the total force *P*_{P} can be determined as:

\[{P_P}={1\over 2}\gamma {K_P}{H^2} + 2cH\sqrt{{K_P}}\] (27.6)

where \[{K_P}={{1 + \sin \phi}\over{1-\sin\phi}}\]

**Problem**

Determine the active earth pressure distribution on the retaining as shown in Figure 27.3. Also determine the total active force and point of application of the force.

**Solution:**

*K*_{A1 }for layer I = \[{K_A}={{1-\sin\phi}\over{1+\sin\phi}}\] = \[{{1-\sin 28^\circ}\over{1+\sin 28^\circ}}\] = 0.36

*K*_{A2 }for layer II = \[{K_A}={{1-\sin\phi}\over{1+\sin\phi}}\] = \[{{1-\sin 32^\circ}\over{1+\sin 32^\circ}}\ = 0.31

Active *P*ressure distribution for layer I

At *Z* = 0m, *P _{A }*= 0 kN/m

^{2}

At *Z* = 1.5m, *P _{A}*

_{ }= 0.36 x 18 x 1.5 = 9.72 kN/m

^{2}

At *Z* = 6 m, *P _{A}*

_{ }(due to soil) = 9.72 + 0.36 x (20 – 10) x 4.5 = 9.72 + 16.2 =25.92 kN/m

^{2}(take unit weight of the water is 10 kN/m

^{3}).

At *Z* = 6 m, *P*_{A }(due to water) = 4.5 x 10 = 45 kN/m^{2}

Active *P*ressure distribution for layer II.

At *Z* = 6 m, *P*_{A }(due to surcharge of the layer I) = 0.31 (1.5 x 18 + 4.5 x 10) = 22.32 kN/m^{2}

At *Z* = 6 m, *P*_{A }(due to the water *P*ressure above this level) = 45 kN/m^{2}

At *Z* = 11 m, *P*_{A }(due to soil) = 67.32 + 0.31 x (20 – 10) x 5 = 67.32 + 15.5 =82.82 kN/m^{2}

At *Z* = 6 m, *P*_{A }(due to water) = 5 x 10 = 50 kN/m^{2}

**Fig. 27.3. Active earth Pressure distribution of problem 1.**

The active force of the various levels is calculated as:

*P*_{A1 }= 0.5 x 9.72 x 1.5 = 7.29 kN/m acts at a height of 10 (=1.5/3+4.5+5) m from the base

*P*_{A2 }= 9.72 x 4.5 = 43.74 kN/m acts at a height of 7.25 (=4.5/2+5) m from the base

*P*_{A3 }= 0.5 x (16.2 + 45) x 4.5 = 137.7 kN/m acts at a height of 6.5 (=4.5/3+5) m from the base

*P*_{A4 }= 67.32 x 5 = 336.6 kN/m acts at a height of 2.5 (=5/2) m from the base

*P*_{A5 }= 0.5 x (15.5 + 50) x 5 = 163.75 kN/m acts at a height of 1.67 (=5/3) m from the base

*P*_{A} = *P*_{A1}+ *P*_{A2 }+ *P*_{A3}+ *P*_{A4}+ *P*_{A5} = 7.29 + 43.74 + 137.7 + 336.6 + 163.75 = 689.08 kN/m

*P*oint of a*PP*lication of the resultant force *P*_{A} is

_{ }

** **

**References**

Ranjan, G. and Rao, A.S.R. (2000). Basic and Applied Soil Mechanics. New Age International Publisher, New Delhi, India.

** **

**Suggested Readings**

Ranjan, G. and Rao, A.S.R. (2000) Basic and Applied Soil Mechanics. New Age International Publisher, New Delhi, India.

Arora, K.R. (2003) Soil Mechanics and Foundation Engineering. Standard Publishers Distributors, New Delhi, India.

Murthy V.N.S (1996) A Text Book of Soil Mechanics and Foundation Engineering, UBS Publishers’ Distributors Ltd. New Delhi, India.