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Rainfall infiltration is a key factor that determines the active earth pressure on an earth-retaining structure. Previous investigations on the stability of earth retaining structures were mainly conducted under the assumption that the backfills was dry, saturated, or subjected to a steady unsaturated flow, ignoring the time-varying characteristics of the rainfall infiltration process. Based on the upper bound theorem of limit analysis, this study conducted a stability analysis of a three-dimensional (3D) earth retaining structure (ERS) subjected to various rainfall infiltration patterns. An analytical model that was, capable of calculating the pore water pressure distribution due to rainfall infiltration, was employed to calculate the time-dependent pore water pressure change of unsaturated soil. A horizontal slice method was employed to take into account the change of effective unit weight of soil. Afterward, an analytical expression of the active earth pressure coefficient is derived from the energy balance equation. The effects of the 3D characteristics of ERSs and rainfall patterns on the active earth pressure coefficient and the failure pattern of ERSs were investigated. It was found that compared with plain strain conditions, the 3D characteristics of ERSs lead to a lower value for the active earth pressure and better stability. The rainfall patterns not only directly determined the stability and active earth pressure of an ERS, but also played a dominant role in critical failure pattern of the ERS.

investigate the symmetric mixed isoperimetric deficit Δ2( K 0, K 1) of domains K 0 and K 1 in the Euclidean plane R 2 via the known kinematic formulas of Poincare and Blaschke in integral geometry. Then we obtain symmetric mixed isoperimetric inequality and some Bonnesen-style symmetric mixed inequalities. One of those inequalities strengthens Kotlyars inequality. Finally, we obtain some reverse Bonnesen-style symmetric mixed inequalities that generalize the known Bottemas result.

In this paper, we investigate the symmetric mixed isoperimetric deficit $\Delta_{2}(K_{0},K_{1})$ of domains $K_{0}$ and $K_{1}$ in the Euclidean plane $\mathbb{R}^{2}$ . Via the known kinematic formulae of Poincare and Blaschke in integral geometry, we obtain some Bonnesen-style symmetric mixed inequalities. These new Bonnesen-style symmetric mixed inequalities are known as Bonnesen-style inequalities if one of the domains is a disc. Some inequalities obtained in this paper strengthen the known Bonnesen-style inequalities.