Relation of shear strength to bearing strength of soil by formula

Question

From formulas:

• Shear Strength: $τ = c+ σ'\cdot sin\phi$
• Bearing strength such as: $1.3 cN_c+ qN_q + 0.4B_γN_γ$

Now I know that bearing capacity decreases as the shear strength of soil decreases. But from these two formulas this relation is not visible as they seem unrelated. How are shear strength and bearing strength related by formula? And no, don't say because of c, cohesion, because for sands it is zero so you can eliminate the first terms with c in both formulas, but still for sands also bearing strength decreases with shear strength decrease.

Edit: think in terms of liquefaction for example. During liquefaction the shear stress decreases due to decrease of effective stress per the formula above. But how does the bearing capacity decrease (and as a result structures settle)? I do not see an effective stress term in the bearing capacity formula above for instance. Or what must be the bearing capacity formula in such case?

Answer 1

The bearing capacity of soil increases with the increase of the effective friction angle as indicated in Terzaghi’s bearing capacity equation:

$q_u = c’N_c + \gamma D_f N_q + 0.5 \gamma B N_{\gamma}$

$N_c, N_q, N_{\gamma}$ are the Terzaghi’s bearing capacity factors. These dimensionless factors are dependents of the angle of internal shearing resistance $\phi$. Equations to find the bearing capacity factors are:

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