We study the pressure of a natural medium with non-linear and non-associative failure criterion on a vertical wall. This is particularly useful for analysing rock media which are extensively fractured or weathered. The application of the Hoek and Brown (1980) failure criterion is extended to the case of walls retaining with heavily fractured rock masses or volcanic materials such as pyroclasts slightly cemented, as well as to rock fill or other materials with non-linear resistance laws. In this case, specific mention is made to materials with a Hoek and Brown (1980) failure criterion and a constant dilatancy angle of 0° and 4° respectively, according to Hoek and Brown (1997) recommendations.


The failure criterion of the material considered is a Coulomb type; as such, this is a criterion that only expresses the stress ratios over the fracture plane. A general parabolic law is used to be subsequently specified for the Hoek and Brown (1980) failure criterion. This criterion has historically been used to determine the stress conditions in tunnels, in the analysis of the stability of slopes and in the calculation of foundations bearing capacity.


The active pressure hypothesis exerted on a wall by the back material is analysed.

2.1 Adimensionalisation of variables
2.1.1 Adimensionalisation of stresses

In order to study pressure problems as generally as possible, adimensional variables are considered for all the physical magnitudes involved in the calculation process. To do so, the stresses are divided by an arbitrary pressure module β (MPa), to obtain the following result:

  • (Equation in full paper)


Figure 1 shows an example of the type of retaining walls considered in this work, some of the concepts and parameters used in the calculation;

(Figure in full paper)

4.1 Basic hypotheses

The different hypotheses taken into account for the calculation of the pressure exerted on the wall are the following:

  1. The fracture wedge is divided into small vertical slices of infinitesimal thickness (Figure 2).

  2. It is assumed that a shear stress (t) and a normal stress (n) (Figure 2) act on the lateral faces of each of the slices.

  3. The total exerted pressure acting on the wall is:

    • (Equation in full paper)

  4. The moment variable (m) is defined by m =Bi. n; where Bi is the height at which the pressure acts on the lateral faces of each slice, measured from the base of the slice.

  5. On the base of each slice, acting at rest, once the shearing resistance along the length of the possible fracture or displacement surface is mobilised, the unitary adimensional stresses σn and τm, are considered where σ n is the normal pressure and m t is the mobilised shear stress obtained from the expression τ m= τ R, where τ R = limit or ultimate shear stress of the rock mass and F = safety coefficient for wall stability.

(Figure in full paper)

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