外文翻譯--由隧道排水引起的地面反應(yīng)曲線(文件)
【正文】 有彈性變形和塑性變形，并表示為 Eqn。 當(dāng)使用流動(dòng)規(guī)則，塑性發(fā)展趨勢(shì)函數(shù) Q，如下： 其中： Ψ為擴(kuò)展角參數(shù)； 塑性部分的徑向圓周應(yīng)變可以表示如下 ： Eqn. 結(jié)合 (20)~(23)可有下面的微分方程： Eqn。 however, mathematical solutions of ground reaction curves influenced by seepage forces have not been suggested. In this study, based on these previous studies, the theoretical solutions of the ground reaction curve considering seepage forces due to groundwater flow under steadystate flow were derived. THEORETICAL SOLUTION OF GROUND REACTION CURVE WITH CONSIDERATION OF SEEPAGE FORCES Theoretical solution for stress It is assumed that a soilmass behaves as an isotropic, homogeneous and permeable medium. Also, an elastoplastic model based on a linear MohrCoulomb yield criterion is adopted in this study, as indicated in Figure 1. σ1′ = kσ3′ + (k ?1)a (1) Here σ1′ indicates the major principal,σ3′ is the minor principal stress, k=tan2(45 + ?2),a= ctan?, where k and a are the MohrCoulomb constants, c is the cohesion, and ? is the friction angle. Figure 2 shows a circular opening of radius r0 with k0 soilmass subject to a hydrostatic in situ stress,σ0′ . The opening inner surface is subject to the outward radial pressure to the tunnel surface,pi(k0 means the ratio of effective vertical stress and horizontal stress). Considering all the stresses on an infinitesimal element abcd of unit thickness during excavation of a circular tunnel in Figure 3, when ?θ is small, the equilibrium of radial forces with respect to r and can be expressed as follows: If the tunnel is excavated under the groundwater table, then it acts as a drain. The body force is the seepage stress, as illustrated in Figure 3. In this state, ir and iθ are the hydraulic gradients in the r and θ directions,respectively, and γw is the unit weight of the groundwater. Therefore, (2) and (3) can be rewritten as follows: If the stress distribution is symmetrical with respect to the axis O in Figure 3, then the stress ponents do not vary with angular orientation, θ , and therefore, they are functions of the radial distance r only. Accordingly, (6) reduces to the single equation of equilibrium as follows: For the plastic region, (1) can be modified as follows: Where kr=tan2(45 + ?r2 ),ar = crtan?r, where ke and ar are the MohrCoulomb constants, c r is the cohesion, and ? ris the friction angle in the plastic region. Substituting (9) into (8) and solving it with the boundary conditions σr′ = pi at r=r0 Then, the radial and circumferential effective stresses in the plastic region are as follows (Shin et al., 2020): where,r0 is the distance from ground to the center of tunnel In this equation, pi is all the support pressure developed by in situ stress and seepage. Subscripts rp and θp are the radial and tangential effective stresses in the plastic region, respectively. In order to estimate the effective stress in the elastic region,the superposition concept is used. As shown in Figure 4, the effective stress considering the seepage force can be assumed as a bination of the solution of the equilibrium equation for the dry condition and the effective stress only considering seepage. The Kirsch solutions are applied to solve the effective stress
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