overbar With ampersand c period dotab semicolon Subscript n Baseline normal upper Delta t plus beta overbar Subscript 1 Baseline normal upper Delta ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon Subscript n Baseline normal upper Delta t EndLayout"/>
where
The optimal choice of these values is a matter of computational convenience, the discussion of which can be found in literature. In practice, if the higher order accurate “trapezoidal” scheme is chosen with β2 = β1 = 1/2 and
or
Dewoolkar (1996), using the computer program SWANDYNE II in the modelling of a free‐standing retaining wall, reported that the first set of parameters led to excessive algorithmic damping as compared to the physical centrifuge results. Therefore, the second set was used and gave very good comparisons. However, in cases involving soil, the physical damping (viscous or hysteretic) is much more significant than the algorithmic damping introduced by the time‐stepping parameters and the use of either sets of parameters leads to similar results.
Inserting the relationships (3.43) into Equations (3.41) and (3.42) yields a general nonlinear equation set in which only
This set can be written as
(3.44a)
(3.44b)
where
(3.45)
In this,
The equation will generally need to be solved by a convergent, iterative process using some form of Newton–Raphson procedure typically written as
(3.46a)
