The active lateral earth pressure and displacement are calculated without considering the influence of pile side friction. It is assumed that the vertical stress component, Rli, remains constant in the vertical direction, while the horizontal stress, R3ai, decreases (i.e., unloading occurs). For each excavation step, the following equations are applied: Rli = Czi + q, where q represents the surcharge, zi is the soil thickness at node i, which remains unchanged during excavation, and C is the unit weight of the soil. Rli denotes the vertical stress at node i. R3ai = R0i - qai, where R3ai is the horizontal stress at node i. The initial value of R0i is k0Czi, with k0 = 1 - sinφ, where φ is the effective internal friction angle of the soil. After each excavation step, R0i - qai is updated and used as the initial value for the next step. Qai = Fai / bid, where d is the diameter of the slope protection pile, bi is the length of the pile segment, and Fai is the increment of active earth pressure at the interface of node i, representing the tensile force caused by the extension of the active soil spring. This results in an increase in the pull force. Before foundation pit excavation, the consolidation ratio of the soil is defined as k0 = R3/R1, and R2 = R3 = k0R1, with Rm being the initial mean consolidation pressure, given by Rm = (R1 + R2 + R3)/3. Considering the unloading modulus calculation based on the stress path, the stress-strain relationship of soft soil is determined through unloading stress path tests, which can be described by a hyperbolic model as follows: (R1 - R3) - 3(1 - k0)/(1 + 2k0) * Rm = Eaa + bEa. Here, (R1 - R3) is the deviatoric stress, and Ea is the axial strain. The parameters a and b are defined as a = 1/Eui, where Eui is the initial unloading modulus, and b = 1 / [(R1 - R3)ult - 3(1 - k0)/(1 + 2k0) * Rm]. The denominator in equation (8) represents the asymptotic value 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]. When (R1 - R3)f is the deviatoric stress at failure, b can be expressed as b = Rf[(R1 - R3)f - 3(1 - k0)/(1 + 2k0) * Rm]. Thus, equation (9) can be rewritten as (R1 - R3) = Ea / Eui + Ea * Rf[(R1 - R3)f - 3(1 - k0)/(1 + 2k0) * Rm] + 3(1 - k0)/(1 + 2k0) * Rm. Equation (10) provides a clear framework for analyzing stress increments. For this equation, the tangent slope of the stress-strain curve (R1 - R3) vs. Ea at any point is given by tanθ = ∂(R1 - R3)/∂Ea. Deriving the horizontal resistance coefficient involves understanding its critical role in finite element analysis of bar systems. The horizontal resistance coefficient, ks, is a key parameter and highly sensitive to input conditions. Common methods for estimating ks include the Zhang Youling method, the c method, the m method, and JEBowles’ approach. Jin Yabing made a reasonable adjustment to the JEBowles law. However, the Zhang Youling, c, and m methods do not account for the size effect of the loaded area. JEBowles is also an empirical method. In reality, ks depends not only on soil properties but also on the loaded area (size effect) and the unloading mode of soft soil, which is closely related to the stress path.
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