Intro Doc
$\underline{Introduction}$ $\\$ The axial bearing capacity of driven piles is composed of point resistance $Q_p$ and side resistance $Q_s$. For $Q_p$, Mayerhof's method and Vesic's method will be introduced; while for $Q_s$, $\alpha$ method and $\beta$ method will be introduced. The bearing capacity here is represented in terms of forces instead of stress. \begin{equation}\label{eq0} q_u=c' \cdot N^*_c + q \cdot N^*_q + \gamma \cdot D \cdot N^*_\gamma \end{equation} where the third term on the right-hand side of the equation is negligible because $D$ is small. $\\$ $\\$ $\underline{Point\space Resistance, Q_p}$ $\\$ $\underline{Mayerhof's\space Method}$ $\\$ A piecewise equation is used for the calculation of $Q_p$.$\\$ $\bullet{Sand}$ $\\$ $\circ L \lt; L_{cr}, Q_p=A \cdot q_p=A \cdot q' \cdot N^*_q + c'*N^*_c \le A \cdot q_l,$ where $N^*_q$ is a function of $\phi '$ and can be obtained in Table 9.5 in Das' book $\\$ $\circ L \ge L_{cr}, Q_p=A \cdot q_l=A \cdot \cfrac{1}{2} \cdot p_a \cdot N^*_q \cdot \tan \phi '$, where p_a is the standard air pressure, 1 atm = 100 kPa. $\\$ The tip resistance increases from 0 at the ground surface to $q_l$ at the depth of $L_{cr} \approx (16\sim18) D$. $\\$ $\bullet{Clay}$ $\\$ $ Q_p=A \cdot C_u \cdot N^*_c \approx 9 \cdot Cu \cdot A$. $\\$ $\underline{Vesic's\space Method}$ $\\$ $\bullet {Sand}$ $\\$ For sand, the following equation is used: \begin{equation} Q_p=A \cdot q_p=A \cdot (q' N^*_q+B \gamma ' N^*_\gamma) \end{equation} $\\$ where, $\\$ $N^*_a=\cfrac{(1+2K) N_\sigma}{3}\\$ $K=(1-\sin \phi ') OCR^{\sin \phi '}\\$ $N_\sigma=\cfrac{3}{3-\sin \phi '} e ^ {\frac{(\frac{\mathrm{\pi}}{2}-\phi ') \mathrm{\pi}}{180}} \cdot \tan ^2 (\cfrac {\mathrm{\pi}}{4} + \cfrac{\phi '}{2}) \cdot I_r^ {\frac{4 \sin \phi '}{3 (1+ \sin \phi ')}}\\$ $I_r=\cfrac{E}{2(1+\nu) q' \tan \phi '},\\ \text {in which $I_r$ is the soil rigidity.}$ $\\$ $\\$ $\bullet{Clay}$ $\\$ For clay, the following equation is used: $Q_p=A \cdot C_u \cdot N^*_c$ $\\$ where, $\\$ $N^*_c=\cfrac{4}{3} (\ln I_{rr}+1) + \cfrac{\pi}{2} + 1$ $\\$ $I_{rr}=\cfrac{I_r}{1+I_r \cdot \Delta}$ $\\$ $\Delta = 0.005 (1-\cfrac{\phi ' -25*\pi/180}{20*\pi/180}) \cfrac{q'}{p_a}$ $\\$ $\underline{Side\space Resistance, Q_s}$ $\\$ $\underline{\beta\space Method}$ $\\$ $\text {The equation for the } \beta \text { method is}$ $Q_s=\sum (f \cdot P \cdot \Delta L) \text { where } f=\beta \cdot \sigma '_{z0}$ $\\$ $\bullet{Sand}$ $\\$ $\beta=0.18+0.65 D_r$ $\\$ $\bullet{Clay}$ $\\$ $\beta=(1-\sin \phi ') \tan \phi ' \cdot \sqrt{OCR}$ $\\$ $\underline{\alpha\space Method}$ $\\$ The $\alpha$ method applies to clays only. \begin{equation} \alpha= \begin{cases} 0.5(\cfrac{C_u}{\sigma ' _{z0}})^{-0.5}, \text{if } \cfrac{C_u}{\sigma ' _{z0}} \le 1\\ 0.5(\cfrac{C_u}{\sigma ' _{z0}})^{-0.25}, \text{if } \cfrac{C_u}{\sigma ' _{z0}} \gt 1 \end{cases} \end{equation} $\underline{General\space Bearing\space Capacity\space Theory}$ $\\$ $\\$ The general bearing capcity theory requires the following input: $B$, $D$, $c'$, $\phi'$, $\gamma$, $\gamma_{sat}$, $D_w$ and foundation type. Different from Terzaghi's bearing capacity theory, the general theory do not use different equations for different types of foundations. Instead, the general theory use one shape factor to consider the influence of foundation types on the bearing capacity. In addition, two more factors are added to consider the influences of the foundation depth and load inclination. The equation for the bearing capacity prediction in the general theory is as follows: $\\$ \begin{equation}\label{eq1} q_u=c' \cdot N_c \cdot F_\mathrm{cs} \cdot F_\mathrm{cd} \cdot F_\mathrm{ci}+q \cdot N_q \cdot F_\mathrm{qs} \cdot F_\mathrm{qd} \cdot F_\mathrm{qi}+\frac{1}{2}\cdot \gamma \cdot B \cdot N_{\gamma} \cdot F_\mathrm{\gamma s} \cdot F_\mathrm{\gamma d} \cdot F_\mathrm{\gamma i} \end{equation} where the bearing capacity factors can be obtained using the following equations (Eqs. 4.27-4.29 or Table 4.2 in Das' book). One major difference is that we now use $\frac{\mathrm{\pi}}{4}+\frac{\phi^\prime}{2}$ rather than $\phi^\prime$, leading to the following equations: \begin{equation}\label{e2a} N_c=\frac{N_q-1}{\tan(\phi^\prime)} \end{equation} \begin{equation}\label{e2b} N_q= \tan^2 \left( \frac{\mathrm{\pi}}{4} + \frac{\phi^\prime}{2} \right) e^{\mathrm{\pi} \tan{\phi^\prime}} \end{equation} \begin{equation}\label{e2c} N_{\gamma} = 2(N_q+1)\tan(\phi^\prime) \end{equation} $\\$ $\underline{Shape\space factors (Debeer,1970)}$ $$F_\mathrm{cs}=1+\left(\cfrac{B}{L}\right) \left(\cfrac{N_q}{N_c} \right)$$ $\\$ $$F_{qs}=1+ \left(\cfrac{B}{L} \right)\tan{\phi^\prime}$$ $\\$ $$F_{\gamma s}=1-0.4 \left(\cfrac{B}{L} \right)$$ $\\$ $\underline {Depth\space factors (Hansen,1970)}$ $\\$ $\text If\space \cfrac{D}{B} \le 1\space \text and\space \phi=0$ $\\$ $$F_{cd}=1+0.4 \left(\cfrac{D}{B} \right)$$ $\\$ $$F_{qd}=1$$ $\\$ $$F_{\gamma d}=1$$ $\\$ $\text If \space\cfrac{D}{B} \le 1\space\text and\space \phi \gt 0$ $\\$ $$F_{cd}=F_{qd}-\cfrac{1-F_{qd}}{N_c \tan{\phi^\prime}}$$ $\\$ $$F_{qd}=1+2 \tan{\phi^\prime} (1-\sin{\phi^\prime})^2 \left( \cfrac{D}{B} \right)$$ $\\$ $$F_{\gamma d}=1$$ $\\$ $\text If\space \cfrac{D}{B} \gt 1\space \text and \space\phi=0$ $\\$ $$F_{cd}=1+0.4 \tan ^{-1} \left(\cfrac{D}{B} \right)$$ $\\$ $$F_{qd}=1$$ $\\$ $$F_{\gamma d}=1$$ $\\$ $\text If\space \cfrac{D}{B} \gt 1\space \text and\space \phi \gt 0$ $\\$ $$F_{cd}=F_{qd}-\cfrac{1-F_{qd}}{N_c \tan{\phi^\prime}}$$ $\\$ $$F_{qd}=1+2 \tan{\phi^\prime} (1-\sin{\phi^\prime})^2 \tan ^ {-1} \left( \cfrac{D}{B} \right)$$ $\\$ $$F_{\gamma d}=1$$ $\\$ $\underline{Inclination\space Factors (Meyerhof,1963;\\ Hanna\space and \space Meyerhof,1981)}$ $\\$ $$F_{ci}=\left( 1- \cfrac{2\beta}{\mathrm{\mathrm{\pi}}} \right)^2$$ $\\$ $$F_{q i}=\left( 1- \cfrac{2\beta}{\mathrm{\mathrm{\pi}}} \right)^2$$ $\\$ $$F_{\gamma i}= \left( 1- \cfrac{\beta}{\phi^\prime} \right)^2$$ $\\$ $\underline{Bearing\space Capacity\space of\space Eccentrically\space \\Loaded\space Foundations}$ $\\$ ${One-way\space eccentric\space loading}$ $\\$ To ensure equilibrium, the following equations need to be satisfied \begin{equation}\label{e3a} \sum F=F-F^\prime \end{equation} \begin{equation}\label{e3b} \sum M=M-F^\prime \cdot e \end{equation} These equations lead to \begin{equation}\label{eq4} e=\frac{M}{F} \end{equation} If eccentricity occurs along the width ($B$) direction, we have \begin{equation}\label{e5a} q_{\mathrm{max}}=q_{\mathrm{F}}+q_{\mathrm{M}}=\frac{F}{BL}+\frac{6M}{B^2L}=\frac{F}{BL} (1+\frac{6e}{B}) \end{equation} \begin{equation}\label{e5b} q_{\mathrm{min}}=q_{\mathrm{F}}-q_{\mathrm{M}}=\frac{F}{BL}-\frac{6M}{B^2L}=\frac{F}{BL} (1-\frac{6e}{B}) \end{equation} Before starting evaluating the bearing capacity of one-way eccentrically loaded footings, we need to check the following condition: \begin{equation}\label{eq6} \text{If } 1-\frac{6e}{B} \gt 0, \text{then } B=B^\prime \\ \text{If } 1-\frac{6e}{L}\gt 0, \text{then } L=L^\prime \end{equation} If the above equation(s) is not valid, then excessive tilting will happen; and accordingly, tensile stresses will appear at the bottom of the foundation. In that case, we do not need to move to the following section to evaluate the bearing capacity. $\\$ $\\$ $\underline{Effective\space Area\space Method}$ $\\$ The following procedure can be taken to tell whether the bearing capacity of the eccentrically-loaded foundation is adequate for design. $\\$ 1. Calculate $B^\prime$, $L^\prime$, and $A^\prime$. $\\$ $ \space \space \space \bullet$ $B^\prime=B-2e$ if eccentricity occurs along the width direction; $\\$ $\space \space \space \bullet$ $L^\prime=L-2e$ if eccentricity occurs along the length direction. $\\$ 2. Calculate the bearing capacity using the general theory.$\\$ $\space \space \space \bullet$ Use $B^\prime$ and $L^\prime$ to calculate $F_\mathrm{cs}$, $F_\mathrm{qs}$, and $F_\mathrm{\gamma s}$; $\\$ $\space \space \space \bullet$ Use $B$ and $L$ to calculate $F_\mathrm{cd}$, $F_\mathrm{qd}$, and $F_\mathrm{\gamma d}$; $\\$ $\space \space \space \bullet$ The others are not dependent on $B$ and $L$. $\\$ 3. Calculate the load $p=\frac{F}{A^\prime}=\frac{F}{B^\prime L^\prime}$.$\\$ 4.Check if $\frac{q_u}{p} \ge SF_\text{design}$.