Correct Answer - Option 4 :
\(\frac{3}{4}\)
Concept:
Shear Strength:
- The capability of a soil to support loading from a structure, or to support its own overburden, or to sustain a slope in equilibrium is governed by its shear strength.
- The shear strength of a soil is of prime importance for foundation design, earth, and rockfill dam design, highway and airfield design, stability of slopes and cuts, and lateral earth pressure problems.
- Shear strength of the soil is given by,
\(\tau = {\sigma _n} + 2C\tan \varphi \)
Where, \({\sigma _n}\) = Effective normal stress at failure, C = Effective cohesive strength of soil and φ = Angle of shearing resistance.
- Methods to determine shearing strength of soil
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Laboratory Methods
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Field Methods
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Direct shear test
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Vane shear test
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Triaxial shear test
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Unconfined compression test
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Ring shear test
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Triaxial Shear Test:
It is a versatile method that can be used for any type of soil and test condition.
As per triaxial test, effective major principal stress at the failure of the specimen is given by,
\(\sigma {'_1} = \sigma {'_2}{\tan ^2}\left( {45 + \frac{{\varphi '}}{2}} \right) + 2c'\tan \left( {45 + \frac{{\varphi '}}{2}} \right)\)
Where, \(\sigma {'_2}\) = effective minor principal stress, φ’ = effective angle of shearing resistance and c’ = effective cohesive strength of the soil.
Note:
1. For Clays, \(c = \frac{1}{2}\left( {unconfined\;compressive\;strength} \right)\)
2. For sands, c = 0
Calculation:
\({\sigma _1} = 90kPa,\)
\({\sigma _2} = 30kPa\)
\(c' = 0\;\left( {sandy\;soil} \right)\)
\(and\;u = 10kPa\)
\({\sigma '_1} = \;{\sigma _1} - u = 90 - 10 = 80kPa,\)
\({\sigma '_2} = \;{\sigma _2} - u = 30 - 10 = 20kPa\)
\({\sigma '_1} = {\sigma '_2}{\tan ^2}\left( {45 + \frac{{\varphi '}}{2}} \right) + 2{c^{'\tan \left( {45 + \frac{{\varphi '}}{2}} \right)}}\)
\( \Rightarrow \;80 = 20{\tan ^2}\left( {45 + \frac{{\varphi '}}{2}} \right)\)
\(\Rightarrow \varphi ' = 0.60\)
Tangent of angle of shearing resistance, \(\tan \varphi ' = 0.75 \approx \frac{3}{4}\)
Alternatively formula for angle of shearing resistance:-
\(\sin \varphi ' = \frac{{{{\sigma '}_1} - {{\sigma '}_3}}}{{{{\sigma '}_1} + {{\sigma '}_3}}}\)
