Q$1.($i$)$ Identify the following $3$ modes of failure from the failure surfaces. Draw the load$-$settlement curves for these failure modes. Which is the most catastrophic failure among these three and why? $(2$M$)$

$($a$)$

$($b$)$

$($c$)$

$($ii$)$ Define the terms $($a$)$ Gross bearing capacity $($b$)$ ultimate bearing capacity $(2$M$)$

$($iii$)$ Discuss the nature of variation of lateral earth pressure with the magnitude of wall movement. $(2$M$)$

$($iv$)$ Explain in brief about negative skin friction in pile groups. $(2$M$)$

$($v$)$ What are the different factors of safety used in the stability of slopes? $(2$M$)$

Question $2-4($ANSWER ANY TWO$*)*-$ If you answer $3,$ then best of $2$ will be considered

Q$2.$ A rectangular footing is founded at the ground surface in a cohesionless soil having the following properties: $e=0\mathrm{.}67,G_s=2\mathrm{.}67,c=0k\frac{N}{m^2},=30.$ The water table is at $2$ m below the ground level. If the width of the footing is $3$ m $,$ what is the length required to carry a gross allowable load of $6500$ kN with a factor of safety $=3?$ Assume local shear failure. Use Terzaghi's bearing capacity equation. $(5$M$)$

Q$3.$ A friction pile $300$ mm in diameter is proposed to be driven in a layer of uniform cohesive soil. The pile tip is assumed to carry $20\%$ of the load. The skin friction between the pile surface and the soil is assumed to be $50k\frac{N}{m^2}.$ Determine the length of the pile required to carry a safe load of $200$ kN with a FS of $4\mathrm{.}0.(5$M$)$

Q$4.$ The $15$ m long pile is loaded as shown in figure below. Take $EI$ of the pile is $3\mathrm{.}7{10}^{7}N{m}^2$ and the modulus of horizontal subgrade reaction as $10{10}^6\frac{N}{m^3}.$

$($a$)$ Show that the pile is classified as a long pile. $(1$M$)$

$($b$)$ If the deflection of the free head of the pile at ground level is $7$ mm $,$ determine the moment $(M_g)$ at the pile head. $(4$M$)$