theory) (Solution (C) is independent of π2).
Source: Reproduced from Zienkiewicz et al. (1980) by permission of the Institution of Civil Engineers.
(a) π2 ≤ 10−3. (b) π2 = 10−2. (c) π2 = 10−1. (d) π2 = 100. (e) π2 = 101. (f) π1 = 10−1 π2 = 102. Reproduced from Zienkiewicz (1980) by permission of the Institution of Civil Engineers.
In the study, the following values were assumed:
and
Figure 2.2 Zones of sufficient accuracy for various approximations: Zone 1, B = Z = C, slow phenomena (
Figure 2.2 summarizes the conclusions by indicating three zones in which various approximations are sufficiently accurate.
We note that, for instance, fully undrained behavior is applicable when Π1 < 10−2 and when Π1 > 102, the drainage is so free that fully drained condition can be safely assumed.
To apply this table in practical cases, some numerical values are necessary. Consider, for instance, the problem of the earthquake response of a dam in which the typical length is characterized by the height L = 50 m, subject to an earthquake in which the important frequencies lie in the range
Thus, with the wave speed taken as
we have
the parameter Π2 is, therefore, in the range 3.9 × 10−3 < Π2 < 39
and Π1 is dependent on the permeability k with the range defined by
According to Figure 2.2, we can, with reasonable confidence:
1 assume fully undrained behavior when Π1 = 97k′ < 10−2 or the permeability k′ < 10−4 m/s.
2 We can assume u–p approximation as being valid when k′ < 10−3 m/s to reproduce the complete frequency range. However, when k′ < 10−1 m/s, periods of less than 0.5 seconds are still well modeled.
We shall, therefore, typically use the u–p formulation appropriately in what follows reserving the full form for explicit transients where shocks and very high frequency are involved.
2.3 Partially Saturated Behavior with Air Pressure Neglected (pa = 0)
2.3.1 Why Is Inclusion of Partial Saturation Required in Practical Analysis?
In the previous, fully saturated, analysis, we have considered both the water pore pressure and the solid displacement as problem variables. In the general case of nonlinear nature, which is characteristic of the problems of soil mechanics, both the effective stresses and pressures will have to be determined incrementally as the solution process (or computation) progresses step by step. In many soils, we shall encounter a process of “densification” implied in the constitutive soil behavior. This means that the history of straining (associated generally with shear strain) induces the solid matrix to contract (or the material to densify). Such densification usually will cause the pore pressure to increase, leading finally to a decrease of contact stresses in the soil particles to near‐zero values when complete liquefaction occurs. Indeed, generally, failure will occur prior to the liquefaction limit. However, the reverse may occur where the soil “dilation” during the deformation history is imposed. This will imply the development of negative pressures which may reach substantial magnitudes. Such negative pressures cannot exist in reality without the presence of separation surfaces in the fluid which is contained in the pores, and consequent capillary effects. Voids will therefore open up during the process in the fluid which is essentially incapable of sustaining tension. This opening of voids will probably occur
