Examples and Applications

A cantilever bridge abutment retains a soil backfill. The total vertical load acting on the base of the abutment (including the weight of the concrete, soil on the heel, and superstructure dead load) is $V = 1500\text{ kN}$. The resultant of this vertical force acts at a distance of $x_v = 2.0\text{ m}$ from the toe of the abutment.

The total horizontal active earth pressure from the backfill is $H = 400\text{ kN}$. The resultant of this horizontal force acts at a height of $y_h = 2.5\text{ m}$ above the base of the footing.

Calculate the Factor of Safety (FS) against overturning. The minimum required FS against overturning is typically $2.0$ for service loads.

Using the same abutment from Problem 1, verify its stability against sliding.

The abutment rests on a cohesionless soil (sand) with an angle of internal friction $\phi = 30^\circ$. The coefficient of friction between the concrete base and the soil is $\mu = \tan(30^\circ) \approx 0.577$. Ignore any passive earth pressure at the toe for a conservative estimate.

Calculate the Factor of Safety (FS) against sliding. The minimum required FS is typically $1.5$ for service loads.

A rectangular bridge pier footing has a width of $B = 4\text{ m}$ and a length of $L = 6\text{ m}$. The total vertical load acting at the centroid of the footing base is $P = 6000\text{ kN}$. A longitudinal wind load on the pier creates an overturning moment at the base of $M_L = 1200\text{ kN}\cdot\text{m}$ (acting parallel to the $L$ dimension).

Calculate the maximum and minimum soil bearing pressures ($q_{max}$ and $q_{min}$) under the footing.