# Relation of shear strength to bearing strength of soil by formula

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?

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:

• Yes, both shear strength and bearing capacity increase as the angle of internal friction increases. That is fine, and I know that part (still thanks for the effort). But what I am asking is, how is shear strength related to bearing strength. For example during liquefaction, the shear strength of sand decreases due to decrease in effective stress (per the formula I wrote). I understand that too. But how / why bearing strength also decreases during liquefaction and structures settle? The angle of internal friction is still the same. Do those N factors also depend on sigma then? or what / how? Commented Nov 1, 2021 at 14:08
• Then you should post another question for stresses of "liquified" soil, which is dynamic in nature, that wasn't considered in the early soil theories.
– r13
Commented Nov 1, 2021 at 14:19
• But two things for you to think about - what is the shear strength of saturated sand and what is the effect of shaking on loose sand?
– r13
Commented Nov 1, 2021 at 14:24
• I made an edit. and asked it in terms of considering liquefaction. Commented Nov 1, 2021 at 14:38
• As I mentioned above, you are comparing the apple with an orange. The equation was developed without consideration of the effect of dynamic loadings that cause the change in water content and rearrangement of soil particles. The simple answer is the friction angle changes during the dynamic event, depending on the type and state of the soil, liquefication may or may not occur.
– r13
Commented Nov 1, 2021 at 14:52