Rankine's theory of earth pressure is a widely accepted theory used to analyze the lateral pressure exerted by soil on retaining walls. It assumes certain conditions about the back of the wall, which helps in simplifying the calculations and analysis. According to Rankine's theory, the back of the wall is assumed to be vertical and smooth.

Explanation:

Rankine's theory of earth pressure:
Rankine's theory is based on the assumption that the soil mass in front of the retaining wall is in a state of plastic equilibrium. It considers the soil as a cohesive-frictional material and assumes that failure occurs along a certain plane called the Rankine active wedge. This theory is widely used in the design of retaining walls and other structures that interact with soil.

Assumption of vertical back:
The assumption of a vertical back of the wall means that the back face of the retaining wall is assumed to be vertical, i.e., perpendicular to the base of the wall. This assumption simplifies the analysis by considering only horizontal forces acting on the wall. It allows for the calculation of the active and passive earth pressures acting on the wall.

Assumption of smooth back:
The assumption of a smooth back of the wall implies that the back face of the retaining wall is assumed to be free from any irregularities or roughness. This assumption simplifies the analysis by assuming that there is no frictional resistance between the wall and the soil. It allows for the calculation of the active and passive earth pressures based solely on the cohesive and frictional properties of the soil.

Advantages of assuming a vertical and smooth back:
1. Simplified analysis: Assuming a vertical and smooth back allows for the use of simple equations and calculations to determine the active and passive earth pressures on the wall.
2. Conservative design: The assumption of a smooth back tends to provide a more conservative estimate of the earth pressures, which ensures the safety of the retaining wall.
3. Applicability: The assumption of a vertical and smooth back is applicable to a wide range of soil conditions and retaining wall geometries.

In conclusion, Rankine's theory of earth pressure assumes that the back of the wall is vertical and smooth. This assumption simplifies the analysis and allows for the calculation of the active and passive earth pressures acting on the retaining wall.