Bearing Capacity Factors of Ring Footings

Abstract

In the present paper, bearing capacity factors were determined for ring footings. The footing roughness is considered in the analyses including smooth and a perfectly rough footing base. The computations were achieved by numerical simulation of ring footings using the finite difference method for different inner to outer ring radii ratio. Based on the results, it is shown that the value of all three bearing capacity factors, i.e., $N_{\gamma }^{*}$, $N_{q}^{*}$, and $N_{c}^{*}$ reduces with the increase in inner to outer ring radii ratio. The reduction effect is more sensible for $N_{\gamma }^{*}$ factor. Furthermore, it is seen that the footing roughness augments the value of the three bearing capacity factors. The results of the present work for $N_{\gamma }^{*}$ were compared with those of the previous published data including experimental and numerical studies, which reveals acceptable agreements. The obtained values of factors were compared for the case of circular footings as a benchmark in the analyses, which are in accordance with the previously published data. In the end, by applying some examples, it was shown that using the proposed bearing capacity equation, which is based on the superposition of different plastic conditions, is conservative and safe with respect to complete plastic analysis.

Appendix 1

In this appendix, an attempt is made to estimate the peak internal friction of the soil, which is mobilized under the footings corresponding to the ultimate bearing capacity. This is achieved by assuming that a triaxial stress condition under the footings exists and the ultimate bearing capacity has the role of the major principal stress induced in the soil.

Several triaxial compression tests were conducted on silica sand with a wide range of confining pressure $p^{\prime}$ = 25–2500 kPa. The results reported by Clark (Clark 1998) indicate that there is a linear relationship between the peak friction angle (ϕ _peak) and logarithm of stress level. The following expression can be derived:

$$ \phi_{\text{peak}} = 49 - 5\log \left( {\frac{{p^{\prime}}}{25}} \right) $$

(6)

The mean confining pressure ($p^{\prime}$) in the soil under the footing base for the triaxial condition is regarded as:

$$ p^{\prime} = \frac{{\sigma_{v}^{'} + 2\sigma_{h}^{'} }}{3} $$

(7)

where $\sigma_{v}^{'}$ and $\sigma_{h}^{'}$ are considered as the mean vertical and horizontal stress components in the soil below the ring footing. $\sigma_{v}^{'}$ can be estimated as the applied vertical pressure over the footing and $\sigma_{h}^{'}$ can be obtained by the lateral coefficient in the active condition (K _a) in the following way:

$$ \frac{{\sigma_{h}^{'} }}{{\sigma_{v}^{'} }} = K_{a} = \tan^{2} \left( {45 - \frac{\phi }{2}} \right) $$

(8)

Since the bearing capacity (q _ult) of ring footings was reported as the maximum applied vertical pressure over the footings, it is assumed that ϕ = ϕ _peak and $\sigma_{v}^{'} = q_{\text{ult}}$. The mean confining pressure can be rewritten in the following form:

$$ p' = \left( {\frac{{1 + 2K_{a} }}{3}} \right)q_{\text{ult}} $$

(9)

By considering Eqs. (6) and (9) and following several successive calculations for the bearing capacity of every ring footing, the following results, according to Table 5, are obtained.

Table 5 Results from back-calculation of bearing capacity of ring footings