Concept:
Shear Strength:
\(\tau = {\sigma _n} + 2C\tan \varphi \)
Where, \({\sigma _n}\) = Effective normal stress at failure, C = Effective cohesive strength of soil and φ = Angle of shearing resistance.
Laboratory Methods 
Field Methods 
Direct shear test 
Vane shear test 
Triaxial shear test 

Unconfined compression test 

Ring shear test 

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:
\(\sin \varphi ' = \frac{{{{\sigma '}_1}  {{\sigma '}_3}}}{{{{\sigma '}_1} + {{\sigma '}_3}}}\)