﻿ Formulation of the element matrices and vectors for elasticity problem > Consistent element nodal loads > Pore pressure loads

The effect of free surface over soil massive is taken into account in two ways; the first consists in computation of the hydrostatic spectra computing the water pore pressures either through a rigorous steady state flow analysis and through an external loading due to reservoir water that act over the massive.

The pore pressures are computed at all submerged (Gauss integration samples) points and subtracted from the total normal stresses computed at the same locations following the application of surface and gravity loads. The resulting effective stresses are then used in the remaining parts of the algorithm relating to the assessment of yield functions and elasto-plastic stress redistribution.

If we denoted p the water pore pressures, meaning a potential of the volume forces, then over an infinitesimal element acting in both directions x-y of the plane the following forces:

 (69)

Let’s assume an finite element e with nodes i,j,k and the potential of the volume forces at those nodes are pi, pj, pk, then the forces potential for the considered element can be written as (stores the nodal values of the prescribed pore pressures) :

 (70)

Because for this potential we can chose the same shape functions as for the displacement field of the finite element, we can write analogous:

 (71)

The effect of water pore pressure produce normal stresses , for plane strain analysis given by:

 (72)

where for plane strain cases:

 (73)

The total stress vector then assumes the following form:

 (74)

Consequently, the general equilibrium relationship, given by the equation (21)  of the finite element can be generalized, introducing the following term in the element equivalent nodal loads vector taking into account in this way the effect of the water pore pressures:

 (75)