Factors affecting bearing capacity; water table

Bearing capacity of a footing depends on the soil strength and on the geometry of the footing, and it is reduced when the soil is submerged. Terzaghi's equation for a strip footing is $q_u = cN_c + \gamma D_f N_q + 0.5\gamma B N_\gamma$.

Factors affecting bearing capacity

• Size (width B): in cohesionless soil the third term $0.5\gamma BN_\gamma$ grows with $B$, so a wider footing has a larger ultimate capacity. In clay ($\phi = 0$, $N_\gamma = 0$) the capacity does not depend on $B$. A wider footing, however, stresses a deeper zone and settles more, so settlement often governs the design.
• Shape: a square or circular footing mobilises three-dimensional resistance. Terzaghi uses shape factors: strip $1.0 / 0.5$, square $1.3 / 0.4$, circular $1.3 / 0.3$ (for the $cN_c$ and $\gamma BN_\gamma$ terms). A rectangle lies between a strip and a square.
• Depth of foundation: a deeper base increases the surcharge $q = \gamma D_f$ and so the $N_q$ term.
• Soil properties: $c$, $\phi$ (which fix $N_c, N_q, N_\gamma$) and unit weight $\gamma$. Loose or soft soil fails in local shear and gives a lower capacity.
• Water table: it reduces the effective unit weight of the soil (see below).
• Eccentricity and inclination of load: these reduce the effective width and the capacity.

Effect of water table position

Submergence reduces the effective stress, so the shear strength of the soil falls. This can be allowed for in either of two ways (use one, not both):

• (a) use the submerged unit weight $\gamma'$ for the soil below the water table, or
• (b) use the bulk unit weight $\gamma$ in Terzaghi's equation with the water table correction factors

$$q_u = cN_c + \gamma D_f N_q R_{w1} + 0.5\gamma B N_\gamma R_{w2}$$

with $R_{w1} = 0.5\left(1 + \dfrac{Z_{w1}}{D_f}\right)$ and $R_{w2} = 0.5\left(1 + \dfrac{Z_{w2}}{B}\right)$. Here $Z_{w1}$ is the depth of the water table below the ground surface (taken as $D_f$ if the water table is at or below the base), and $Z_{w2}$ is the depth of the water table below the base (zero if it is at or above the base, and not more than $B$).

 Case 1: WT above base           Case 2: WT below base
 GL ______________________       GL ______________________
 WT ~~~~~~~~~~~~~  (Zw1)            |  gamma (moist)
    |  gamma' (submerged)      Base ======================
 Base ====================      WT ~~~~~~~~~~~~~~  (Zw2 below base, < B)
    |  gamma' (submerged)           |  gamma' (submerged)

1. WT above the base ($Z_{w1} < D_f$, $Z_{w2} = 0$): the soil beside the footing (below the water table) and the soil below the footing are submerged. With method (a), $\gamma'$ is used in the $N_\gamma$ term and the surcharge becomes $q = \gamma Z_{w1} + \gamma'(D_f - Z_{w1})$. With method (b), $R_{w2} = 0.5$ and $R_{w1}$ lies between $0.5$ and $1$. If the water table is at the ground surface, both terms are about halved, so the capacity of a cohesionless soil falls by about half.
2. WT at base level: $R_{w1} = 1$ and $R_{w2} = 0.5$, so only the $N_\gamma$ term is halved.
3. WT below the base (at depth $Z_{w2} < B$ below it): $R_{w1} = 1$ and $R_{w2}$ lies between $0.5$ and $1$. The effect is only on the $\gamma B N_\gamma$ term, because the failure wedge below the base is partly submerged.
4. WT deeper than $B$ below the base: $Z_{w2} \ge B$, so $R_{w2} = 1$ and the water table has no effect.

In clay with $\phi = 0$, the water table has little effect on $q_u$ because $N_q = 1$ and $N_\gamma = 0$, except through the change in surcharge.