Answer :
In Rankine's earth pressure theory, the coefficient of lateral pressure [tex]K[/tex] for active and passive scenarios is determined by the angle of internal friction [tex]\phi[/tex] of the soil. For a horizontal backfill scenario, the coefficient of active earth pressure [tex]K_a[/tex] and passive earth pressure [tex]K_p[/tex] can be calculated using:
- [tex]K_a = \frac{1 - \sin\phi}{1 + \sin\phi}[/tex]
- [tex]K_p = \frac{1 + \sin\phi}{1 - \sin\phi}[/tex]
The problem states that the coefficient of lateral pressure equals 4, which is a scenario typically relating to [tex]K_p[/tex], the coefficient of passive earth pressure. Thus:
[tex]4 = \frac{1 + \sin\phi}{1 - \sin\phi}[/tex]
By rearranging and solving for [tex]\sin\phi[/tex]:
[tex]4(1 - \sin\phi) = 1 + \sin\phi[/tex]
[tex]4 - 4\sin\phi = 1 + \sin\phi[/tex]
[tex]3 = 5\sin\phi[/tex]
[tex]\sin\phi = \frac{3}{5}[/tex]
Now, to find the angle [tex]\theta[/tex] since [tex]\phi = \theta[/tex] in this context, we calculate the arcsin:
[tex]\theta = \arcsin\left(\frac{3}{5}\right)[/tex]
[tex]\theta \approx 36.9^\circ[/tex]
Thus, the angle [tex]\theta[/tex] that results in a coefficient of lateral pressure equal to 4 is approximately 36.9 degrees.
Therefore, the correct option is:
- 36.9
