Question

A retaining wall with vertical back retains a cohesionless dry backfill at an inclination of $\beta$ with the horizontal. The backfill has an angle of internal friction $\varphi ,$ unit weigth $\gamma$ and height of the wall is $H.$ The passive earth pressure on the wall is given by (where $P_p$= Total passive earth pressure)

• A. $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta -\sqrt{{\cos }^2\beta -{\cos }^2\beta }}{\cos \beta +\sqrt{{\cos }^2\beta -{\cos }^2\beta }}\right]$
• B. $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta -\sqrt{{\cos }^2\beta +{\cos }^2\beta }}{\cos \beta +\sqrt{{\cos }^2\beta +{\cos }^2\beta }}\right]$
• C. $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta +\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}{\cos \beta -\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}\right]$
• D. $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta +\sqrt{{\cos }^2\beta +{\cos }^2\varphi }}{\cos \beta -\sqrt{{\cos }^2\beta +{\cos }^2\varphi }}\right]$

Answer 1

The correct option is C $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta +\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}{\cos \beta -\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}\right]$

(c) $P_p=\frac{1}{2}\gamma H^2\cos \beta \left[\frac{\cos \beta +\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}{\cos \beta -\sqrt{{\cos }^2\beta -{\cos }^2\varphi }}\right]$