Civil Engineering Reference
In-Depth Information
steps are taken, the method can become equivalent to a simple Euler “explicit” method.
In practice, the global stiffness matrix may be updated periodically and “body loads”
iterations employed to achieve convergence. In contrasting the two methods, the extra cost
of reforming and factorising the global stiffness matrix in the “variable stiffness” method
is offset by reduced numbers of iterations, especially as failure is approached.
A further possibility, introduced in later programs, is “implicit” integration of the rate
equations rather than the “explicit” methods just described. This helps to further reduce the
number of iterations to convergence.
Programs 6.1 to 6.4 employ the “constant stiffness” approach and explicit integration,
and are similar in structure to Program 4.5 described previously for plastic analysis of
frames. Both the viscoplastic method and the simple initial stress method are implemented.
Programs 6.5 and 6.6 introduce tangent stiffness algorithms, and Programs 6.7 and 6.8
describe procedures for embanking and excavation in which construction sequences can be
realistically modelled. Program 6.9 introduces a simple technique for modelling pore pres-
sures in a saturated soil, and Programs 6.10 and 6.11 complete the chapter with 3D elasto-
plastic analyses of slopes. Mesh-free solution strategies are described in Programs 6.2, 6.6,
and 6.11. Many of the examples are chosen because they have closed-form solutions for
comparison.
Before describing the programs, some discussion is necessary regarding the form of the
stress-strain laws that are to be adopted. In addition, two popular methods of generating
body loads for “constant stiffness” methods, namely “viscoplasticity” and “initial stress”
are described.
6.2 Stress-strain behaviour
Although non-linear elastic constitutive relations have been applied in finite element anal-
yses and especially soil mechanics applications (e.g. Duncan and Chang, 1970), the main
physical feature of non-linear material behaviour is usually the irrecoverability of strain.
A convenient mathematical framework for describing this phenomenon is to be found in
the theory of plasticity (e.g. Hill, 1950). The simplest stress-strain law of this type that
could be implemented in a finite element analysis involves elastic-perfectly plastic material
behaviour (Figure 6.3). Although a simple law of this type was described in Chapter 4
(Figure 4.28), it is convenient in solid mechanics to introduce a “yield” surface in principal
stress space which separates stress states that give rise to elastic and to plastic (irrecover-
able) strains. To take account of complicated processes like cyclic loading, the yield surface
may move in stress space “kinematically” (e.g. Molenkamp, 1987) but in this topic only
immovable surfaces are considered. An additional simplification introduced here is that the
yield and ultimate “failure” surfaces are identical.
Algebraically, the surfaces are expressed in terms of a yield or failure function
.
This function, which has units of stress, depends on the material strength and invariant
combinations of the stress components. The function is designed such that it is negative
within the yield or failure surface and zero on the yield or failure surface. Positive values
of
F
imply stresses lying outside the yield or failure surface which are illegal and which
must be redistributed via the iterative process described previously.
F
