Soil Mechanics Test 7

Concept:

The total active earth pressure is given by,

P = \(\sqrt {P_H^2 + \;P_V^2} \)

P_H = \(\frac{1}{2}{K_H}{H^2}\) and P_V = \(\frac{1}{2}{K_V}{H^2}\)

Calculation:

Given that: K_H = 760 kg/m²/m and K_V = 100 kg/m²/m, H = 6 m

\({P_H} = \frac{1}{2} \times 760 \times {6^2}\) = 136.8 kN/m

\({P_V} = \frac{1}{2} \times 100 \times {6^2}\) = 18 kN/m

\(P = \sqrt {{{136.8}^2} + \;{{18}^2}} \)

Hence, P = 138 kN/m

Concept:

Stability of Finite slopes:

Stability analysis is generally done either using total stress analysis or effective stress analysis.

Total stress analysis is used for immediately after construction stability check while Effective stress analysis is used for long term after construction stability check.

Various methods based on Total stress analysis:

(1) Swedish circle Method

(2) Friction circle Method

(3) ϕ_U = 0 Analysis.

In this case we use Unconsolidated undrained test results.

Various methods based on Effective stress analysis are:

(1) Taylor’s method

(2) Bishop’s method.

In this case we use CU or CD test analysis results.

Concept:

Critical depth of unsupported cut is given by:

\({{\rm{H}}_{\rm{c}}} = \frac{{4{\rm{c'}}}}{{{\rm{\gamma }}\sqrt {{{\rm{k}}_{\rm{a}}}} }}\)

C’ = effective cohesion

γ = unit weight of soil

\({{\rm{k}}_{\rm{a}}} = {\rm{active\;earth\;pressure\;coefficient}} = \frac{{1 - {\rm{sin\;}}\phi {\rm{'}}}}{{1 + {\rm{sin\;}}\phi {\rm{'}}}}\)

Calculation:

Case: 1 when c’ = 25 kN/m², ϕ’ = 20°

\({{\rm{H}}_{{{\rm{c}}_1}}} = \frac{{4 \times 25}}{{18\sqrt {{{\rm{k}}_{{{\rm{a}}_{\rm{'}}}}}} }}\)

\({{\rm{k}}_{{{\rm{a}}_1}}} = \frac{{1 - \sin 20^\circ }}{{1 + \sin 20^\circ }} = 0.490\)

\({{\rm{H}}_{{{\rm{c}}_1}}} = \frac{{4 \times 25}}{{18\sqrt {0.490} }} = 7.936{\rm{m}}\)

Case: 2 \({\rm{c'}} = 25{\rm{kN}}/{{\rm{m}}^3},\phi {\rm{'}} = \phi _2^1,{\rm{\;}}{{\rm{H}}_{{{\rm{c}}_2}}} = 1.1{{\rm{H}}_{\rm{c}}}\)

\({{\rm{H}}_{{{\rm{c}}_2}}} = 1.1 \times 7.936 = 8.729{\rm{m}}\)

\(8.729 = \frac{{4 \times 25}}{{18\sqrt {{{\rm{k}}_{{{\rm{a}}_2}}}} }}\)

\({{\rm{k}}_{{{\rm{a}}_2}}} = 0.405\)

\(\frac{{1 - {\rm{sin\;}}\phi _2^{\rm{'}}}}{{1 + {\rm{sin\;}}\phi _2^{\rm{'}}}} = 0.405\)

\(1 - {\rm{sin\;}}\phi _2^{\rm{'}} = 0.405 + 0.405{\rm{\;sin\;}}\phi _2^{\rm{'}}\)

\(0.595 = 1.405{\rm{\;sin\;}}\phi _2^{\rm{'}}\)

Concept:

Factor of safety with respect to cohesion \(= \frac{{Cohesive\;strength}}{{Mobilised\;cohesion}}\)

Taylor’s stability Number is a dimensionless parameter and for purely cohesive soil, it is given as

\({S_n} = \frac{c}{{{F_c}{\gamma _t}H}}\)

Where, c = cohesion, Fc = Factor of safety with respect to cohesion, and H = Depth of excavation.

Also, S_n = cos β × sin β

Calculation:

c = 30 kN/m², β = 20°, γ_t = 16 kN/m³

Mobilised cohesion = 0.75 C

Factor of safety = \(\frac{{Cohesive\;strength}}{{Mobilised\;cohesion}}\) \(\Rightarrow \frac{C}{{{F_C}}} = {c_m} \Rightarrow \frac{C}{{{F_C}}} = 0.75\;c = 0.75 \times 30 = 22.5\)

Also, S_n = cos β × sin β = cos 20° × sin 20° = 0.321 

\({S_n} = \frac{C}{{{F_C}\; \times \;{\gamma _{t\;}}\; \times \;H}} \Rightarrow H = \frac{C}{{{F_C}\; \times \;{\gamma _t} \;\times\; {S_n}}} = \frac{{22.5}}{{16\; \times \;0.321}}\) = 4.38 m

Important Point:

Taylor’s stability number (sn) = cos β × sin β

Where, β = slope angle

Theoretical maximum value of stability number is 0.5 and practically its value is in the range of 0.25 - 0.30.