Ultimate Bearing Capacity of Low-Density Volcanic Pyroclasts: Application to Shallow Foundations

Abstract

The present research focuses on the calculation of the bearing capacity of low-density volcanic Pyroclasts. First, the theoretical basis that define an adequate failure criterion for collapsible rocks based on the parameters that characterize them are developed here. Second, a mathematical characteristic lines method is proposed to resolve the ultimate load of shallow foundation rocks with collapsible failure criterion. This method leads to an analytical solution that differentiates two possible rupture mechanisms depending on the rock parameters and external confining load: (a) plastic failure wedge; (b) failure due to destructuring. The analytical solution is also represented in design abacuses to make it easy and quick. Finally, the proposed formulation is validated using numerical models by implementing the collapse criterion as a model defined by the user in a finite difference code. The result is a satisfactory comparison of bearing capacity values and the analytical rupture mechanisms.

Appendix: Theoretical bases of the failure criterion

The change in volumetric plastic deformation ($\dot{v}^{\rm p}$) and angular distortion ($\dot{\gamma }^{\rm p}$) can be expressed as a function of the major principal strain ($\varepsilon_{1}$) and minor principal strain ($\varepsilon_{3}$):

$$\dot{v}^{\rm p} = \dot{\varepsilon }_{1}^{\rm p} + 2\dot{\varepsilon }_{3}^{\rm p}$$

$$\dot{\gamma }^{\rm p} = \frac{2}{3}\left( {\dot{\varepsilon }_{1}^{\rm p} - \dot{\varepsilon }_{3}^{\rm p} } \right)$$

From the obliquity ($\eta = q_{\rm KR} /p_{\rm KR}$) and the dilatancy ($\chi = \dot{v}^{\rm p} /\dot{\gamma }^{\rm p}$) an energy consumption function (F) can be defined as the sum of both: $F = \chi + \eta$, and the consumed physical work in the plastic failure ($\dot{W}^{\rm p}$) can be expressed by:

$$\dot{W}^{\rm p} = p_{\rm KR} \dot{v}^{\rm p} + q_{\rm KR} \dot{\gamma }^{\rm p} = \left( {\chi + \eta } \right)p_{\rm KR} \dot{\gamma }^{\rm p} = F p_{\rm KR} \dot{\gamma }^{\rm p}$$

And therefore, with R being the consumption rate: $R = \dot{W}^{\rm p} /\dot{\gamma }^{p}$, results $R = F p_{\rm KR}$.

Considering Drucker's stability postulate (1964):

$$\dot{p}_{\rm KR} \dot{v}^{\rm p} + \dot{q}_{\rm KR} \dot{\gamma }^{\rm p} \ge 0 \to \left( {\chi + \frac{{\dot{q}_{\rm KR} }}{{\dot{p}_{\rm KR} }}} \right) \ge 0$$

In a material with associated dilatancy the condition must be strictly verified, that is:

$$\chi + \frac{{\dot{q}_{\rm KR} }}{{\dot{p}_{\rm KR} }} = 0$$

(A1)

Since $q_{\rm KR} = \eta p_{\rm KR}$, differentiating it will obtain:

$$\dot{q}_{\rm KR} = \eta \dot{p}_{\rm KR} + p_{\rm KR} \dot{\eta } \to \frac{{\dot{q}_{\rm KR} }}{{\dot{p}_{\rm KR} }} = \eta + p_{\rm KR} \frac{{\dot{\eta }}}{{\dot{p}_{\rm KR} }} \mathop \to \limits^{{{\text{Using}}~\left( 3 \right)}} \chi + \eta + p_{\rm KR} \frac{{\dot{\eta }}}{{\dot{p}_{\rm KR} }} = 0$$

Therefore:

$$F + p_{\rm KR} \frac{{\dot{\eta }}}{{\dot{p}_{\rm KR} }} = 0$$

(A2)

An energy consumption law is proposed, which is an extension of the Granta Gravel model by Roscoe et al. (1963) to cohesive materials. Following these ideas, the energy consumption at failure has two addends, one ($R_{\rm f} = M p_{\rm KR}^{2}$) produced by friction (where M is a frictional parameter) and the second ($R_{\rm c} = q_{i}$) due to cohesion. In this way, and since it was deduced that $R = F p_{\rm KR}$, it is possible to obtain:

$$R = R_{\rm f} + R_{\rm c} = M p_{\rm KR}^{2} + q_{i} = F p_{\rm KR} \to F = M p_{\rm KR} + \frac{{q_{i} }}{{p_{\rm KR} }}$$

(A3)

From (A2) and (A3), the following differential equation is obtained:

$$p_{\rm KR} \frac{{\rm d}\eta }{{{\rm d}p_{\rm KR} }} + M p_{\rm KR} + \frac{{q_{i} }}{{p_{\rm KR} }} = 0 \to \frac{{\rm d}\eta }{{{\rm d}p_{\rm KR} }} = - \left( {M + \frac{{q_{i} }}{{ p_{\rm KR}^{2} }}} \right)$$

The solution of this differential equation is immediate, and the solution can now be a function of the constant $C_{1}$, resulting in:

$$\eta = - M p_{\rm KR} + \frac{{q_{i} }}{{p_{\rm KR} }} + M C_{1} \to q_{\rm KR} = - M p_{\rm KR}^{2} + C_{1} M p_{\rm KR} + q_{i}$$

(A4)

Equation (A4) can be expressed in the previously defined physical canonical variables ($p_{\rm KR} = p_{\rm K} - \zeta$, $q_{\rm KR} = q_{\rm K}$), so that:

$$q_{\rm K} = - M p_{\rm K}^{2} + M \left( {C_{1} + 2\zeta } \right)p_{\rm K} + \left( {q_{i} - M \zeta^{2} - M C_{1} \zeta } \right)$$

(A5)

Applying the known boundary condition ($p_{\rm K} = 0$, $q_{\rm K} = 0$) in Eq. (A5), results into:

$$q_{i} = M\zeta \left( { \zeta + C_{1} } \right)$$

(A6)

And considering in (A5) the other boundary condition ($p_{\rm K} = 1$, $q_{\rm K} = 0$), obtains:

$$- M + M\left( {C_{1} + 2\zeta } \right) + \left( {q_{i} - M \zeta^{2} - M C_{1} \zeta } \right) = 0\begin{array}{c}{{{\text{Using}} \left( 8 \right)}}\\ \rightarrow\end{array}\ C_{1} = 1 - 2\zeta$$

(A7)

Finally, substituting (A6) and (A7) in (A5) the collapse equation that complies with what is indicated in (2) for the most common case of λ = 1 is obtained:

$$q_{\rm K} = Mp_{\rm K} \left( { 1 - p_{\rm K} } \right)$$