饱和土非饱和的初始地应力-有效应力公式推导

问题：非饱和土的一维固结分析（abaqus 在土木工程中的应用 page158）9 m0 I9 L5 z2 O4 b$ P/ Z7 r 一个 2m 高的土体，水位线在土体的中央，即 1m 处，所以土体下半部分饱和，上半部分非饱 和。土体的重度=18KN/M3，水的重度为 10kN/M3,土的初始孔隙比 e0=1，顶部施加一均布荷 载为 10kpa则其初始地应力如下：& s+ U9 E' w* T *initial conditions, type=stress, geostatic“ w6 K# K( t: J, c7 v: g nup,-9.789，2，-22.97，1，0.5 ndown,-22.97，1，-35.97，0，0.5 y=2 时的地应力为-9.789 是如何得出的？根据单向固结理论，有效应力=总应力-孔隙压应力。在非饱和问题中，ABAQUS 采用的是：有效应力=总应力-饱和度*孔隙水压力。又由于 ABAQUS 对孔压正负号的规定和土力学中一致，而对应力正负号的规定和土力学中刚好相反，故上面公式中的“-”号要改为“+”号。手册中是分饱和，半饱和，以及干燥三部分说的，但是其给出的公式是两个，第一个是把饱和和非饱和结合到一块的，第二个是单存干燥的。手册里地应力平衡和饱和度的计算为：VerticalVertical equilibriumequilibrium inin a a porousporous mediummediumMost geotechnical problems begin from a geostatic state, which is a steady-state equilibrium configuration of the undisturbed soil or rock body under geostatic loading. The equilibrium state usually includes both horizontal and vertical stress components. It is important to establish these initial conditions correctly so that the problem begins from an equilibrium state. Since such problems often involve fully or partially saturated flow, the initial void ratio of the porous medium, , the initial pore pressure, , and the initial effective stress must all be defined.If the magnitude and direction of the gravitational loading are defined by using the gravity distributed load type, a total, rather than excess, pore pressure solution is used (see “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1). This discussion is based on the total pore pressure formulation.The z-axis points vertically in this discussion, and atmospheric pressure is neglected. We assume that the pore fluid is in hydrostatic equilibrium during the geostatic procedure so that where is the user-defined specific weight of the pore fluid (see “Permeability,” Section 22.7.2). (The pore fluid is not in hydrostatic equilibrium if there is significant steady-state flow of pore fluid through the porous medium: in that case a steady-state coupled pore fluid diffusion/stress analysis must be performed to establish the initial conditions for any subsequent transient calculationssee “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1.) If we also take to be independent of z (which is usually the case, since the fluid is almost incompressible), this equation can be integrated to define where is the height of the phreatic surface, at which and above which and the pore fluid is only partially saturated. We usually assume that there are no significant shear stresses , . Then, equilibrium in the vertical direction is where is the dry density of the porous solid material (the dry mass per unit volume), g is the gravitational acceleration, is the initial porosity of the material, and s is the saturation, (see “Permeability,” Section 22.7.2). Since porosity is the ratio of pore volume to total volume and the void ratio is the ratio of pore volume to solids volume, is defined from the initial void ratio by Abaqus/Standard requires that the initial value of the effective stress, , be given as an initial condition (“Initial conditions,” Section 29.2.1). Effective stress is defined from the total stress, , by where is a unit matrix. Combining this definition with the equilibrium statement in the z-direction and hydrostatic equilibrium in the pore fluid gives again using the assumption that is independent of z. is the position of the surface that separates the dry soil from the partially saturated soil. The soil is assumed to be dry () for , and it is assumed to be partially saturated for and fully saturated for . In many cases s is constant. For example, in fully saturated flow everywhere below the phreatic surface. If we further assume that the initial porosity, , and the dry density of the porous medium, , are also constant, the above equation is readily integrated to give where is the position of the surface of the porous medium, . In more complicated cases where s, , and/or vary with height, the equation must be integrated in the vertical direction to define the initial values of .HorizontalHorizontal equilibriumequilibrium inin a a porousporous mediummediumIn many geotechnical applications there is also horizontal stress, typically caused by tectonic action. If the pore fluid is under hydrostatic equilibrium and , equilibrium in the horizontal directions requires that the horizontal components of effective stress do not vary with horizontal position: only, where is any horizontal component of effective stress.而如果要把饱和非饱和分开来则可按abaqus 在土木工程中的应用所写公式，对该公式 的推导如下：where is a unit matrix. （0）（1）（2）（3）（4）上边这些公式为手册中的公式，这里推导过程中为了跟岩土书中一致，故将 z 换成y,z0 换成 y0在水面以上，即非饱和时，地表面设为 y0,将公式（1）， （2）进行积分即：（5）)(000yysngdydwyyyy yy（6）)(00yydyduuwyyw w将（5）（6）式代入（0）式可以得到：（7） )(1 ()()(00000yynsgyysyysngwww 七式正是书中表达的非饱和土，即水面以上地表以下，但是不包括地表，因为地表不需要总应力积分，地表如果也为非饱的话，可以直接按公式：来表示wsu而对于饱和土来说，也同样是从地表往下任一深度 y 积分，只是其中 s=1，并且为了与书中统一，将 000 1e en则：（8）)(1e(0 000yye