Assuming a log-spiral sliding surface passing through the toe of retaining wall

based on energy method

the upper bound of active earth pressure for rough walls with inclined backfill is formulated. The formula can be applied to sandy soil and cohesive soil. Extracting the screw-in angle ? 0 and screw-out angle ? h through the slope as variables

hybrid particle swarm optimization is used to search the most dangerous sliding surface globally. Hence

the optimal solution of active earth pressure is obtained. For sandy soil

the earth pressure coefficients in this work agree with the classical upper bound solutions of limit analysis when the inclination of wall is small

but obviously bigger than the classical solutions while the wall angle is larger than 30°. Compared to other well-known solutions obtained by limit equilibrium based on optimum principle

the solution herein is more approximate than that of the classical upper bound solution. For cohesive soil

employing the formula developed by this work to calculate the active earth pressure in an engineering practice

it is found that the relative error between calculation results and measured data is 5.4%.