Geotechnical Engineering Part 1

$e = \dfrac{V_v}{V_s}= \dfrac{n}{1 - n}

HYDRAULIC CONDUCTIVITY
same as
COEFFICIENT OF PERMEABILITY
(k)

$n = \dfrac{V_v}{V_T}=\dfrac{e}{1 + e}$

$S = \dfrac{V_w}{V_v}\;\;\;\;\;\;\;\;\;\;\; Se=wG_s

total volume = volume of soilds + volume of voids
$V = V_s + V_v$

volume of voids = volume of water + volume of air
$V_v = V_w + V_a$

total weight = weight of solids + weight of water
$W = W_s + W_w$

$\omega = \dfrac{W_w}{W_s} \times 100\%$

$\gamma_m = \dfrac{W}{V} $
$\gamma_m= \dfrac{(G_s + Se)\gamma_w}{1 + e} $
$\gamma_m= \dfrac{G_s\gamma_w( 1 + w)}{1 + e}$

$\gamma_d = \dfrac{W_s}{V} = \dfrac{G_s\gamma_w}{1 + e}= \dfrac{\gamma_m}{1 + w}

$\gamma_{sat} =\dfrac{W_{sat}}{V_{t}}= \dfrac{(G_s + e)\gamma_w}{1 + e}

$\gamma_b = \gamma_{sat} - \gamma_w $
$\gamma_b= \dfrac{(G_s - 1)\gamma_w}{1 + e}

$G_s = \dfrac{\gamma_{s}}{\gamma_w}$

$i_{cr} = \dfrac{\gamma_b}{\gamma_w} $
$i_{cr}= \dfrac{G_s - 1}{1 + e}

$D_R = \dfrac{e_{max} - e}{e_{max} - e_{min}}$
$D_R=\frac{{{(\gamma}_d)}_{max}}{\gamma_d}\left(\frac{\gamma_{d\ }-{\ (\gamma_d)}_{min}}{{\ (\gamma_d)}_{max}-{\ (\gamma_d)}_{min}}\right)$
$D_R = \dfrac{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{\gamma_d}}{\dfrac{1}{(\gamma_d)_{min}} - \dfrac{1}{(\gamma_d)_{max}}}

$e_{min}=\frac{G_s-\frac{\gamma_{max}}{\gamma_w}}{\frac{\gamma_{max}}{\gamma_s}-1}

$e_{max}=\frac{G_s-\frac{\gamma_{min}}{\gamma_w}}{\frac{\gamma_{min}}{\gamma_w}-1}

$q=\frac{\pi\,k\left({h_1}^2-{h_2}^2\right)}{ln\left(\frac{r_1}{r_2}\right)}

$q=\frac{2\pi\,kH\left(h_1-h_2\right)}{ln\left(\frac{r_1}{r_2}\right)}

$PI = LL − PL

$LI = \dfrac{w - PL}{PI}

$TI=\frac{PI}{FI}

$SI=PL−SL

$A_c = \dfrac{PI}{\mu}$

$V_v = \dfrac{eV}{1 + e}

$V_s = \dfrac{V}{1 + e}

$V_w = \dfrac{SeV}{1 + e}

$W_w = \dfrac{SeV \gamma_w}{1 + e}

$W = \dfrac{V(G_s + Se)\gamma_w}{1 + e}

$FI = \frac{w_1-w_2}{log\left(\frac{N_2}{N_1}\right)}

$SL = \left(\dfrac{m_i - m_d}{m_d} - \dfrac{(V_i - V_d)\rho_w}{m_d}\right)x\ 100\ $
$SL=\left(\frac{W_w}{W_d}-\frac{1}{G_s}\right)x\ 100\

$SR = \dfrac{m_d}{V_d \rho_w}

$G_s = \dfrac{1}{\dfrac{1}{SR} - \dfrac{SL}{100}}

$Air\ Void\ Ratio =(1-S)$

$Air\ Content\ =n(1-S)$
$Air\ Content\ =\frac{e-wG_s}{1+e}

$\gamma_{zav}=\frac{G_s\gamma_w}{1+wG_s}

$V_s=\frac{V_i-V_d}{V_d}x100

$D_S=\frac{V_i-V_d}{V_i}x100

$L_S=\left(1-\sqrt[3]{\frac{100}{V_s+100}\ }\right)x100

$\gamma_m = \dfrac{W}{V_{(m)}}\;\;\;\;\;\;\;\;;V_{(m)}=\dfrac{1}{30}ft^3

$R(\%)=\left(\frac{\gamma_{d(field)}}{\gamma_{d(max-lab)}}\right)x100

$i=\frac{h}{L}

$V=ki

$q=kiA

Velocity thru voids
$V_s=\dfrac{V}{n}

$Le=\dfrac{K'}{b'}

$a=\dfrac{K}{\dfrac{K'}{b'}}

$B=\sqrt{\frac{Kb}{\frac{K\prime}{b\prime}}}

$\bar{K}=\frac{K\ \eta}{\gamma_w}

$T=\bar{K}\;b

$D_{10}

$C_u=\dfrac{D_{60}}{D_{10}}

$C_C=\frac{{(D_{30})}^2}{D_{60}D_{10}}

$S_o=\sqrt{\frac{D_{75}}{D_{25}}}

$D_{50}

$S_n=1.7\sqrt{\frac{3}{{(D_{50})}^2}+\frac{1}{{(D_{20})}^2}+\frac{1}{{(D_{10})}^2}}

$GI=(F_{200}-35)(0.005LL)$
$\ \ \ \ \ \;\;\;\;\;\;\;\; +0.01(F_{200}-15)(PI-10)

$\sigma=\frac{3P}{2\pi\ z^2\left(1+\left(\frac{r}{z}\right)^2\right)^\frac{5}{2}}

$\sigma=\frac{P}{\pi\ z^2\left(1+{2\left(\frac{r}{z}\right)}^2\right)^\frac{3}{2}}

$\sigma_{Zlinear}=\frac{2qz^3}{\pi{(x^2+z^2)}^2}

$K=\dfrac{QL}{hAt}

$K=\frac{aL}{At}\;ln\left(\frac{h_1}{h_2}\right)

$R_T=2.42-0.475\ ln(T)

$K_{H(eq)}=\frac{k_1H_1+k_2H_2+k_3H_3}{H_T}+...

$K_{V\left(eq\right)}=\frac{H_T}{\frac{H_1}{k_1}+\frac{H_2}{k_2}+\frac{H_3}{k_3}+...}

$\ K_{20\degree\,C}=R_TK

$T=\frac{q\,ln\left(\frac{r_1}{r_2}\right)}{2 \pi(h_1-h_2)}=kH=\frac{q\,ln\left(\frac{r_1}{r_2}\right)}{2\pi(z_2-z_1)}

$\frac{\Delta H}{H_0}=\frac{\Delta e}{1+e_0}=\frac{e_0-e_f}{1+e_0}

$\Delta H=\ \frac{H}{1+\ e_0}\ C_c\ log\ \left(\frac{P_0+\ \Delta P}{P_0}\right)$

$P_0+\ ∆P <P_c$
$\Delta H\ =\ \frac{H}{1+e_0}\ C_s\ log\ \left(\frac{P_0+\ \Delta P}{P_0}\right)$

$P_0+\ ∆P >P_c$
$\Delta H\ =\ \frac{H}{1+e_0}\ \left[ \ C_s\ log\ \left(\frac{P_c}{P_0}\right) + C_c\ log\ \left(\frac{P_0+\ \Delta P}{P_c}\right)\ \right] $

$∆_t = ∆_i+∆_p +∆_s$

$C_c=\ \frac{-\left(\Delta e\right)}{log\ \left(\frac{\Delta P+\ P_0}{P_0}\right)}$

$C_c=0.007\left(LL-10\right)

$\Delta_s\ =\ \frac{C_\alpha\ H}{1+\ e_f}\ log\ \left(\frac{t_2}{t_1}\right)$

MINIMUM PRINCIPAL NORMAL STRESS
in MOHR's CIRCLE

a.k.a
Chamber Confining Pressure
All-Around Pressure
Lateral Pressure
Cell Pressure

ADDITIONAL STRESS @ FAILURE
in MOHR's CIRCLE

a.k.a
Deviator Stress
Plunger Stress

$C_C=\frac{{(D_{30})}^2}{D_{60}D_{10}}

COEFFICIENT of CURVATURE

$C_s$ or $C_r $$=\frac{C_c}{5}$ to $\frac{C_c}{10}$

$C_c=0.009\left(LL-10\right)

$OCR=\frac{P_c}{P_0}$

$a_v=\ \frac{e_0-e_f}{P_f-P_0}$

$m_{av}=\ \frac{a_v}{1+e_{ave}\ }$

$C_v=\ \frac{\left(H_{dr}\right)^2\ T_v}{t\ }$

$k=C_v{\ m}_v\ \gamma_w$

TOTAL AXIAL STRESS @FAILURE
is same as
MAXIMUM NORMAL STRESS
in MOHR's CIRCLE
$\left( \sigma_1\right)$

$U(\%)=\left(\dfrac{h_f-h_i}{h_f}\right)(100)