The active lateral earth pressure and displacement are calculated without considering the influence of pile side friction. It is assumed that the vertical stress, Rli, remains constant in the vertical direction, while the horizontal stress, R3ai, decreases (unloading) during excavation. For each excavation step, the following equations are applied: Rli = Czi + q, (1) where q represents the surcharge, zi is the thickness of soil at node i, which remains unchanged during excavation, and C is the unit weight of the soil. Rli represents the vertical stress at node i. R3ai = R0i - qai, (2) where R3ai is the horizontal stress at node i. Initially, R0i is set as k0Czi, with k0 = 1 - sinφ, where φ is the effective internal friction angle of the soil. After each excavation step, R0i is updated for the next stage. Qai = Fai / bid, where d is the diameter of the retaining pile, bi is the length of the pile segment, and Fai is the increment of active earth pressure at the interface of the i-th node, representing the tensile force acting on the soil spring. This leads to an increase in the pulling force. Before foundation pit excavation, the consolidation ratio of the soil is defined as k0 = R3/R1, with R2 = R3 = k0R1, and Rm = (R1 + R2 + R3)/3, where Rm represents the initial consolidation pressure before excavation. By considering the unloading modulus calculation formula along the stress path, the stress-strain relationship of soft soil can be determined through unloading tests, typically described by a hyperbolic model. (R1 - R3) - 3(1 - k0)/(1 + 2k0) * Rm = Eaa + bEa. (7) Here, (R1 - R3) is the deviatoric stress, and Ea is the axial strain. The parameters a and b are defined as follows: a = 1/Eui (where Eui is the initial unloading modulus), and b = 1 / [(R1 - R3)ult - 3(1 - k0)/(1 + 2k0) * Rm]. (8) The denominator in equation (8) represents the asymptote of the hyperbola in equation (7). The failure ratio is defined as Rf = [(R1 - R3)f - 3(1 - k0)/(1 + 2k0) * Rm] / [(R1 - R3)ult - 3(1 - k0)/(1 + 2k0) * Rm]. (9) When (R1 - R3)f is the deviatoric stress at failure, b can be rewritten as b = Rf[(R1 - R3)f - 3(1 - k0)/(1 + 2k0) * Rm]. Thus, equation (9) becomes: (R1 - R3) = Ea[1/Eui + Rf(R1 - R3)f - 3(1 - k0)/(1 + 2k0) * Rm + 3(1 - k0)/(1 + 2k0) * Rm]. (10) Equation (10) provides a practical way to analyze stress increments. The slope of the stress-strain curve (R1 - R3) vs. Ea at any point is given by tanθ = ∂(R1 - R3)/∂Ea. Derivation of the horizontal resistance coefficient: In the finite element analysis of pile systems, the horizontal resistance coefficient ks is a crucial and sensitive parameter. Commonly used methods to estimate ks include the Zhang Youling, c, m, and JEBowles approaches. Jin Yabing made a reasonable modification to the JEBowles method. However, the Zhang Youling, c, and m methods do not account for the size effect of the loaded area. Similarly, JEBowles is an empirical approach. In reality, ks depends not only on soil properties but also on the size of the loaded area and the unloading behavior of soft soils, which is closely related to the stress path.
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