It is planned to construct a building on a silty sand deposit. Assume the site conditions: Silty sand deposit has a level ground surface, a total unit weight $\backslash$gamma t above the groundwater table of $18\mathrm{.}9$ kN$/$m$3,$ and a submerged unit weight $\backslash$gamma b of $9\mathrm{.}84$ kN$/$m$3.$ The groundwater table is located $1\mathrm{.}5$m below ground surface. Standard penetration tests were performed in a $10-$cm$-$diameter borehole. At a depth of $3$ m below ground surface, a standard penetration test was performed using a doughnut hammer with a blow count of $3$ blows for the first $15$ cm$,4$ blows for the second $15$ cm$,$ and $5$ blows for the third $15$ cm of diving penetration. Calculate the factor of safety against liquefaction for the soil located at a depth of $3$m below ground surface.

a$.$ assume that sand contains $15\%$ nonplastic fines, amax$/$g $=0\mathrm{.}1$ and the earthquake magnitude M $=7\mathrm{.}5.$

b$.$ assume that sand contains no fines, amax$/$g $=0\mathrm{.}2$ and the earthquake magnitude M$=7\mathrm{.}5.$

c$.$ assume that sand contains no fines, amax$/$g $=0\mathrm{.}4,$ the earthquake magnitude M $=7\mathrm{.}5$ and qc$=3\mathrm{.}9$ Mpa

d$.$ assume that sand contains no fines, amax$/$g $=0\mathrm{.}4,$ the earthquake magnitude M $=7\mathrm{.}5$ and shear wave velocity Vs$=150$m$/$s$.$

e$.$ Assume that the liquefied soil layer is $1\mathrm{.}0$ m thick. For question $$b$$ estimate the soil settlement, using two methods: Method by Ishihara and Yoshimine & method by Tokimatsu and Seed.