Embedded anchor pull-out, Mohr-Coulomb interface (2D)

This benchmark verifies the stress-dependent axial capacity of *Embedded region, interaction, type=Mohr-Coulomb: a vertical anchor is embedded in a rigid host region carrying a geostatic stress field, so that the interface capacity varies with depth and the pull-out capacity is available in closed form.

Upcoming release

The *Embedded region, interaction feature will be available with the upcoming release

Model

A soil block of 1.0 m x 2.0 m is discretised with four stacked U4-SOLID elements. All soil nodes are fully fixed, rendering the host rigid; a geostatic initial stress field with \(\sigma_v(y) = -20 + 10\,y\) kPa (surface at \(y = 2.0\) m) and \(K_{01} = K_{02} = 0.5\) is prescribed and - since no soil node can move - remains unchanged throughout the analysis. A vertical truss (three U2-TRUSS elements, \(EA = 10^6\) kN) extends from \(y = 0.25\) m to \(y = 1.75\) m at \(x = 0.45\) m; the anchor nodes are located at the centroids of the host elements and coincide with no soil node. The interface is defined by

\[ k_t = k_n = 100 \; \mathrm{kN/m/m} , \qquad c = 5 \; \mathrm{kPa} , \qquad \varphi = 30^\circ , \qquad A_s = 0.4 \; \mathrm{m} . \]

For the vertical anchor axis, the mean lateral stress reduces to \(\sigma_n = \tfrac{1}{2}(\sigma_{xx} + \sigma_{zz}) = \tfrac{1}{2}(K_{01} + K_{02})\,\sigma_v = 0.5\,\sigma_v\), i.e. \(\sigma_n = -8.75, -6.25, -3.75, -1.25\) kPa at the four anchor nodes (compression increases the capacity, so the capacity is lowest at the top). The tension cutoff is not active anywhere.

Figure 1: Model setup (not to scale).

Input files

Download the input file here

Closed-form pull-out capacity

The local capacity \(t_{cap} = A_s (c - \sigma_n \tan\varphi)\) is linear along each guest element (the host stresses at the guest nodes are interpolated with the linear guest shape functions), so 2-point Gauss integration is exact and

\[ F_{max} = A_s \left( c\,L - \tan\varphi \int_\mathcal{L} \sigma_n \, \mathrm{d}s \right) = 0.4 \cdot 7.5 \cdot \left( 1 + \tan 30^\circ \right) = 3 + \sqrt{3} = 4.7320508 \; \mathrm{kN} , \]

using \(\int \sigma_n \, \mathrm{d}s = -7.5\) kPa·m (trapezoidal over the nodal values, \(L = 1.5\) m).

Step 1: elastic pull-out (load controlled)

An upward load of \(F = 3.0\) kN is applied to the anchor head. The elastic response is independent of the capacity law: as in the constant-capacity benchmark,

\[ u_2 = \frac{F}{k_t \, L} = \frac{3.0}{100 \cdot 1.5} = 0.02 \; \mathrm{m} . \]

The corresponding uniform traction \(k_t \, u_2 = 2.0\) kN/m stays below the smallest integration-point capacity (\(2.41\) kN/m, near the top of the anchor) - the interface is elastic everywhere and the step verifies that the Mohr-Coulomb law does not perturb the elastic regime. A comparison of the simulation results to the analytical solution is provided in Figure 2.

Figure 2: Simulation results vs. analytical solution.

Step 2: pull-out capacity (displacement controlled)

The anchor head displacement is increased to \(u_2 = 0.3\) m. All integration points reach their (depth-dependent) capacity and slide; the reaction force at the anchor head equals the closed-form pull-out capacity

\[ R_2 = 3 + \sqrt{3} = 4.7320508 \; \mathrm{kN} , \]

reproduced by numgeo to machine precision, since the linear capacity distribution is integrated exactly by the 2-point Gauss rule.