Question

For cohesion-less soil, the ratio of major to minor principal stress is __________

(a) \(\frac{σ_1}{σ_3} = cos^2 (45°+\frac{φ}{2})\)

(b) \(\frac{σ_1}{σ_3} = sin^2 (45°+\frac{φ}{2})\)

(c) \(\frac{σ_1}{σ_3} = tan^2 (45°+\frac{φ}{2})\)

(d) \(\frac{σ_1}{σ_3} = tan^2 (45°-\frac{φ}{2})\)

I had been asked this question in examination.

My doubt is from Earth Pressure Introduction in chapter Failure Envelopes & Earth Pressure of Geotechnical Engineering II

Answer 1

The correct answer is (c) \(\frac{σ_1}{σ_3} = tan^2 (45°+\frac{φ}{2})\)

To explain I would say: The stress condition during plastic equilibrium is given by,

\(σ_1=2c \,tan⁡(45°+\frac{φ}{2})+σ_3 tan^2 (45°+\frac{φ}{2}),\)  and for cohesion-less soil, c=0,

∴ \(\frac{σ_1}{{σ_3}} = tan^2 (45°+\frac{φ}{2}).\)