Intro Doc
$\underline{Understanding\space of\space Schmertmann's\space Method}$ $\\$ The method was developed on the elastic theory with calibration from field and lab tests. The method uses the strain factor instead of the stress influence factors to account for the finite size of the footing ("stress bulb" - strain influence factor distribution). Besides, the method also uses the equivalent soil modulus, $E_s$ instead of Young's modulus. $\\$ $\\$ $\underline{Strain\space Influence\space Factor}$ $\\$ Different types of foundations have different distributions of the strain influence factors. The strain influence factor $I_{\epsilon}$ linearly increases from $I_{\epsilon 0}$ to $I_{\epsilon p}$ at $z=z_1$, which $z=0$ is at the bottom of the foundation. After the peak, $I_{\epsilon}$ then decreases to 0 at $z=z_2$. \begin{equation}\label{eq1} I_{\epsilon p}=0.5+0.1 \sqrt{\cfrac{p-\sigma^\prime_{zD}}{\sigma^\prime_{zp}}} \end{equation} $\\$ $\\$ $\underline{Square/Circular\space Footing (L/B=1)}$ $\\$ For square/circular foundations, $z_1=B/2$ and $z_2=2B$. The strain influence factor varies in the following way: \begin{equation} I_{\epsilon p}=0.1+{\cfrac{z}{B}}*(2I_{\epsilon p}-0.2)\space\text when\space 0\lt z\lt B/2\\ I_{\epsilon p}=0.667 I_{\epsilon p} (2-z/B)\space\text when\space B/2 \lt z \lt 2B \end{equation} $\\$ In addition, $E_s=3.5 q_c$ $\\$ $\\$ $\underline{Continuous\space Footing\space (L/B \ge 1)}$ $\\$ For continuous footings, $z_1=B$ and $z_2=4B$. The strain influence factor varies in the following way: \begin{equation}\label{eq2} I_{\epsilon p}=0.2+{\cfrac{z}{B}}*(I_{\epsilon p} - 0.2)\space\text when\space 0 \lt z \lt B\\ I_{\epsilon p}=0.333 I_{\epsilon p} (4-z/B)\space\text when\space B \lt z \lt 4B \end{equation} $\\$ In addition, $E_s=3.5 q_c$ $\\$ $\\$ $\underline{Rectangular\space Footing (1\le L/B \le 10)}$ $\\$ For rectangular foundations, $z_1=0.5 B +0.0555(L-B)$ and $z_2=2B+0.222(L-B)$. The strain influence factor varies in the following way: \begin{equation}\label{eq3} I_{\epsilon p}=I_{\epsilon s}+0.111(I_{\epsilon c}-I_{\epsilon s}) \left( \cfrac{L}{B} -1 \right) \end{equation} $\\$ In addition, $E_s=2.5 q_c + (L/B -1)/9$. $\\$ $\\$ $\underline{Settlement\space Prediction}$ $\\$ ${Prediction Equation}$ \begin{equation}\label{eq4} \delta=C_1 \cdot C_2 \cdot C_3 \cdot (p-\sigma^\prime_{zD}) \sum{\left( \cfrac{I_\epsilon}{E_s} \Delta z \right)} \end{equation} $\underline{Procedure}$ $\\$ $\bullet$ Calculate $C_1$, $C_2$, $C_3$, $p$, and $\sigma^\prime_{zD}$.$\\$ $\bullet$ Provide the compressible layers within the depth of influence into layers, usually 5-10 layers, for hand calculations and the layer boundaries are determined by the natural soil layers.$\\$ $\bullet$ Calculate $E_s$ and $I_\epsilon$ for each layer (mid-point) to obtain $\sum{\left( \cfrac{I_\epsilon}{E_s} \Delta z \right)}$$\\$ $\bullet$ Calculate $\delta=C_1 \cdot C_2 \cdot C_3 \cdot (p-\sigma^\prime_{zD}) \sum{\left( \cfrac{I_\epsilon}{E_s} \Delta z \right)}$.