upper T Baseline ModifyingAbove bold-italic epsilon With ampersand c period dotab semicolon plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper Q EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0"/>
This reduced equation system is precisely the same as that used conventionally in the study of consolidation if the dynamic terms are omitted or even of static problems if the steady state is reached and all the time derivatives are reduced to zero. Thus, the formulation conveniently merges with procedures used for such analyses. However, some loss of accuracy will be evident for problems in which high‐frequency oscillations are important. As we shall show in the next section, these are of little importance for earthquake analyses.
In eliminating the variable wi(w), we have neglected several terms but have achieved an elimination of two or three variable sets depending on whether the two‐ or three‐dimensional problem is considered. However, another possibility exists for obtaining a reduced equation set without neglecting any terms provided that the fluid (i.e. water in this case) is compressible.
With such compressibility assumed, Equation (2.16) can be integrated in time, provided that we introduce the water displacement
(2.22a)
or
(2.22b)
where the division by the porosity n is introduced to approximate the true rather than the averaged fluid displacement. We now can rewrite (2.16) after integration with respect to time as
(2.23a)
or
(2.23b)
and thus we can eliminate p from (2.11) and (2.13).
The resulting system which is fully discussed in Zienkiewicz and Shiomi (1984) is not written down here as we shall derive this alternative form in Chapter 3 using the total displacement of water U = UR + u as the variable. It presents a very convenient basis for using a fully explicit temporal scheme of integration (see Chan et al. 1991) but it is not applicable for long‐term studies leading to steady‐state conditions, as the water displacement U then increases indefinitely.
It is fortunate that the inaccuracies of the u–p version are pronounced only in high‐frequency, short‐duration, phenomena, since, for such problems, we can conveniently use explicit temporal integration. Here a very small time increment can be used for the short time period considered (see Chapter 3).
Table 2.1 summarizes various forms of governing equations used.
2.2.3 Limits of Validity of the Various Approximations
It is, of course, important to know the degree of approximation involved in various differential equation systems. Thus, it is of interest to know under what circumstances undrained conditions can be assumed, to define the behavior of the material and when the simplified equation system discussed in the previous section is applicable, without introducing serious error. An attempt to answer these problems was made by Zienkiewicz et al. (1980). The basis was the consideration of a one‐dimensional set of linearized equations of the full systems (2.11), (2.13), and (2.16) and of the approximations (2.20) and (2.21). The limiting case in which wi,i = 0 (representing undrained conditions) was also considered.
Table 2.1 Comparative sets of coupled equations governing deformation and flow.
| u − w − p equations (exact) [(2.11), (2.13), and (2.16)] |
|
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| u − p approximation for dynamics of lower frequencies. Exact for consolidation [(2.20), (2.21)] |
|
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| u − U, only convective terms neglected (3.72) |
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