Wedge Failure Analysis Statement

A $20-$meter$-$wide road is proposed on a hillside where the slope of the hill is $\backslash (72^\{\backslash$circ$\}\backslash )$ with an estimated height of $38$ meters striking at an angle of $\backslash (178^\{\backslash$circ$\}\backslash ).$ The upper slope dips at a gentle angle of $\backslash (9^\{\backslash$circ$\}\backslash )$ and striking at an angle of $\backslash (183^\{\backslash$circ$\}\backslash ).$ This site is prone to slope instability due to failure involving a wedge formed by the intersection of two discontinuity planes. The measured data for both planes is given in the following table:

Tasks

$1.$ Fully Drained Condition:

Calculate the factor of safety against the sliding of the wedge block under fully drained conditions $($no pore water pressure$).$

$2.$ Fully Saturated Condition:

Recalculate the factor of safety for the wedge block considering full saturation of the tension crack $($i$.$e$.,$ pore water pressure due to the presence of water$).$

$3.$ Effect of Road Excavation:

If a $10-$meter$-$wide excavation is proposed for the road, determine the factor of safety considering the new slope face under both fully drained and fully saturated conditions.

$4.$ Effect of Slope Face Angle:

Establish a graphical relationship of the factor of safety with slope face angles ranging from $\backslash (40^\{\backslash$circ$\}\backslash )$ to $\backslash (90^\{\backslash$circ$\}\backslash )$ for both fully drained and fully saturated conditions.

$5.$ Effect of Dip Angle of Bedding Plane:

Investigate the influence of varying the dip angle of the bedding plane $($from $\backslash (30^\{\backslash$circ$\}\backslash )$ to $\backslash (50^\{\backslash$circ$\}\backslash ))$ on the factor of safety under fully drained and fully saturated conditions. Present the results graphically.

$6.$ Effect of Dip Angle of Joint Plane:

Analyze the effect of varying the dip angle of the joint plane $($from $\backslash (40^\{\backslash$circ$\}\backslash )$ to $\backslash (60^\{\backslash$circ$\}\backslash ))$ on the factor of safety under fully drained and fully saturated conditions. Present the results graphically.

$7.$ Combined Variation:

Establish a $3$D surface plot showing the factor of safety as a function of both the slope face angle $\backslash (\backslash$left$(40^\{\backslash$circ$\}\backslash$right$.\backslash )$ to $\backslash (\backslash$left$\mathrm{.}90^\{\backslash$circ$\}\backslash$right$)\backslash )$ and the dip angle of the bedding plane $\backslash (\backslash$left$(30^\{\backslash$circ$\}\backslash$right$.\backslash )$ to $\backslash (\backslash$left$\mathrm{.}50^\{\backslash$circ$\}\backslash$right$)\backslash ),$ assuming fully drained conditions.

$8.$ Effect of Rock Properties:

Explore how changes in rock cohesion $($from $50$ kPa to $100$ kPa $)$ and friction angle $($from $\backslash (20^\{\backslash$circ$\}\backslash )$ to $\backslash (30^\{\backslash$circ$\}\backslash ))$ impact the factor of safety under fully drained conditions. Present the relationship graphically.

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