Unsafe Error in Conventional Shape Factor for Shallow Circular Foundations in Normally Consolidated Clays

In practice, a factor of 1.2 is generally accepted as the shape factor for undrained bearing capacity of surface foundations in homogeneous clays between axisymmetric (AS) and plane strain (PS) conditions. This empirical shape factor first proposed by Skempton has been widely adopted in design guidelines for shallow foundations to determine general undrained bearing capacity of circular foundations from standard solutions for strip foundations. However, this factor is unsafe and should not be used for shallow circular foundations in normally consolidated (NC) clay with shear strength increasing linearly with depth from zero at the ground surface.

Fig. 1 shows the shape factors ($S_c$) between circular and strip foundations, which were numerically derived using finite element limit analysis (FELA) in OptumG2 software (Krabbenhoft et al. 2015). Examples of predicted failure mechanisms are illustrated in Fig. 2. Numerical models of shallow foundations correspond to weightless rigid piers with embedded depth ($L$) at full size ($D$) (i.e., diameter for circular foundations and width for strip foundations) in a weightless clay layer with strength zero at the ground surface and increasing linearly with depth by linear strength gradient ($\rho$). The undrained strength of clay obeys the Tresca yield criterion with the associated flow rule. The foundation base is modelled as rough and smooth interfaces and its side is smooth to directly obtain the ultimate bearing capacity from FELA. For all cases, computed plasticity solutions are accurate to within 1%. Thus, the derived shape factor $S_c$ is defined as the ratio of the ultimate pressure ($q_{\text{ult}}$) of circular foundations to that of strip foundations, where these pressures correspond to the average of upper bound and lower bound solutions.

As the embedded depth ratio ($L/D$) decreases, $S_c$ decreases nonlinearly from 1.2 at $L/D=1.2$ and converges to 0.67 (i.e., $4/6$) when $L/D$ tends toward zero. This limiting condition is theoretically validated by the exact solutions of Salençon and Martar (1982) and Davis and Booker (1973) for surface circular and strip foundations, respectively, on such strength profiles, where $q_{\text{ult}}=\rho D/6$ (AS) and $\rho D/4$ (PS). The error in shape factor increases as the depth of circular foundations decreases. For example, it is 20% for $L/D=0.2$ but can be as much as 20–80% for very shallow circular foundations with $L/D=0.2–0$.