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Problem $3.$

A load that can be modeled as a point load is applied on the soil profile below. Point C is $1\mathrm{.}5$ m directly below the applied load.

a$)$ Calculate the change in total stress at point $"C",$ due to the point load.

$P=50kN$

$z=1\mathrm{.}5$

$r=1\mathrm{.}5$

Stress caused by a pt$.$ lad: ${}_z=\frac{3p{z}^3}{2(r^2+z^2{)}^{\frac{5}{2}}}$

${}_z=\frac{3(50kN)(1\mathrm{.}5(m){)}^3}{(2ubrace(1\mathrm{.}5\text{u}\text{b}\text{r}\text{a}\text{c}\text{e}{)}_{1\mathrm{.}5}+{1\mathrm{.}5}^2{)}^{\frac{5}{2}}}$

b$)$ Calculate the total stress at point $"$C$"$ after the point load was applied.

${}_B=(0\mathrm{.}5m)(15\frac{kN}{m^3})=7\mathrm{.}5\frac{kN}{m^2}$

${}_C=7\mathrm{.}5\frac{kN}{m^2}+(1m)(16\frac{kN}{m^3})=23\mathrm{.}5\frac{kN}{m^2}$

${}_c=(1m)(9\mathrm{.}81\frac{kN}{m^3})-9\mathrm{.}81\frac{kN}{m^2}$

${}_{c,\text{i}\text{n}\text{t}\text{t}\text{i}\text{a}\text{l}}=23\mathrm{.}5\frac{kN}{m^2}$

${}_{c,\text{f}\text{i}\text{n}\text{a}\text{l}}=73\mathrm{.}5\frac{k{N}^2}{m^2},={}_0+J$

$4$

$-5$