Intro Doc
$\underline{Introduction}$ $\\$ Settlement of shallow foundations include both the elastic settlement, consolidation method, and secondary method. For sand and stiff and overconsolidated clay, the elastic settlement is the primary component and can be predicted using the elastic theory, Janbu's method, Mayne and Poulos's method, and Schmertmann's method. Consolidation settlement is the predominant settlement component in soft clays. Primary settlement needs to be considered for plastic soils. $\\$ $\\$ $\underline{Mayne\space and\space Poulos\space Method}$ $\\$ This method is more general and versatile for the elastic prediction of sand and stiff clay. \begin{equation}\label{eq5} \delta=I_G \cdot I_F \cdot I_E \cdot \frac{PB_e}{E_0}(1-\nu^2) \end{equation} where $I_G$, $I_F$, and $I_E$ are correction factors for the finite thickness and soil modulus, the footing rigidity, and the embedment depth, respectively; $B_e$ is the effective width of the footing, and $E_0$ the rigidity at the bottom of the footing. The correction factors can be calculated as follows \begin{equation}\label{eq6a} I_G=f \left( \frac{E_0}{k B_e} , \frac{H}{B_e} \right) \ \qquad \text{ see Fig. 7.6} \end{equation} \begin{equation}\label{eq6b} I_F=\frac{\mathrm{\pi}}{4} + \cfrac{1}{4.6+10 \left( \cfrac{E_\mathrm{f}}{E_0+B_e k/2} \right) \left( \cfrac{2t}{B_e} \right)^3} \ \qquad \text{ see Eq. 7.18 or Fig. 7.7} \end{equation} \begin{equation}\label{eq6c} I_E=1 - \frac{1}{3.5 \cdot \exp{(1.22 \nu-0.4)(\frac{B_e}{D}+1.6)}} \ \qquad \text{see Eq. 7.19 or Fig. 7.8} \end{equation} $\\$ where $E_\mathrm{f}$ is the modulus of footing. The effective width is calculated using the following equation: \begin{equation}\label{eq7} B_e=\sqrt{\cfrac{4 B L}{\mathrm{\pi}}},\text{for square/rectangular footing}\\ B_e=D, \text{for circular footing} \end{equation} $\\$ $\\$ Finally, $E=E_0+k z$.