Numerical implementation of MCC and GCC models

An important step, ensuring a reliable application of MCC and GCC both models, is the determination of the initial preconsolodation pressure p_cⁱⁿ and the corresponding bulk modulus Kⁱⁿ. These two parameters, however, are not directly specified by the user. Instead, they are derived by the program based on the assumed distribution of intial geostatic stress. Recall three basic options to derive the intial geostatic stress:

1. Using Ko procedure

The use of K_o procedure yields the following value of the initial mean stress:

where:	Ko	-	coefficient of earth pressure at rest
	γ	-	bulk weight of soil
	h	-	current depth below terrain

Assuming normal consolidation the value of p_cⁱⁿ is determined such that the stress derived using the K_o procedure fullfils the yield condition:

where:	Mcs	-	slope of critical state line
	J	-	equivalent deviatoric stress
	σm	-	mean stress

Values J and σ_m of are defined by the following expressions:

where:	Ed	-	equivalent deviatoric stress
	eij	-	deviatoric strain tensor
	εij	-	overall strain tensor
	εv	-	volumetric strain
	σij	-	stress tensor
	sij	-	deviatoric stress tensor
	δij	-	Kronecker’s delta
	Dijkl	-	elastic stiffness tensor
	G	-	elastic shear modulus
	K	-	elastic bulk modulus
	E	-	Young’s modulus
	ν	-	Poisson’s number

In the case of triaxial compression or extension it is possible to determine the slope of critical state line M_cs from the following expressions:

In the case of overconsolidated soils the initial value of p_cⁱⁿ is modified as:

The initial value of the bulk modulus follows from:

where the current void ratio e is written as:

For small stresses we get:

,

2. Standard (elastic) analysis

Recall that the program allows for replacing the material model between stages of construction. Providing the K_o procedure cannot be used it is possible to carry out the analysis assuming elastic response of the clayey soil. The resulting stresses are used to derive the initial values of p_cⁱⁿ and Kⁱⁿ employing the previously defined expressions.  In the next stages of construction the original elastic material model is replaced by the required MCC or GCC models.

3. Standard (plastic) analysis

This option allows the soil to be consolidated under the assumption of nonlinear behavior when generating the geostatic stress. This results in the evolution of plastic strains already in the first stage of construction. As in the K_o procedure we consider a normally consolidated soil which, during the course of deformation, moves down the normal consolidation line with the initial values of p_cⁱⁿ and Kⁱⁿ given by:

,

Before the next analysis step the resulting plastic strains are set equal to zero. In some cases such an analysis may fail to converge.