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Interaction surface: exact fixed-host pull-out (A0)

This benchmark verifies the shaft quadrature, the axial beam transfer, the constant-capacity return mapping and the committed plastic slip of *Embedded region, interaction-surface against a closed-form solution, before host-stress recovery and soil nonlinearity are introduced.

Model

A straight vertical pile (nine u2-beam-3D elements between \(z=10\) m and \(z=5\) m) is embedded in a block of u8-solid-3d elements (720 elements, 2.83 m x 2.83 m in plan with an O-grid around the pile, 10 m high) whose nodes are all fixed, rendering the host rigid. Only the shaft interaction surface is active (no base disk, no ghost zone); the pile head is pulled upwards by a prescribed displacement of \(w=3\) mm and the toe is axially free. The interaction surface uses four coupling points per ring (np=4).

Quantity Value
Diameter of the interaction surface \(D\) \(0.20\,\mathrm m\)
Embedded length \(L\) \(5.0\,\mathrm m\)
Axial rigidity of the beam \(EA\) \(9.42478\times10^{5}\,\mathrm{kN}\) (\(A=0.031416\,\mathrm{m^2}\), \(E=3\times10^{7}\,\mathrm{kPa}\))
Tangential stiffness \(k_t\) \(1.0\times10^{5}\,\mathrm{kN/m^3}\)
Normal stiffness \(k_n\) \(0\) (no normal coupling)
Constant shaft capacity \(t_{\mathrm{ult}}\) \(100\,\mathrm{kPa}\)
Base coupling, ghost zone inactive 
*Embedded region, interaction-surface, type=constant, np=4, ghost=no
embedded-pile-1-shaft, host, none, 0.2, 100000, 0, 0, 100, 0

*Step, name=step 1, inc=1e4, maxiter=32, miniter=1
*Static
0.01, 1, 0.001, 0.1
*Boundary, amplitude=amplitude 1
embedded-pile-1-head, u3, 0.003
*Boundary
host, u1, 0
host, u2, 0
host, u3, 0

Input files

Download the input files (mesh, calculation, material definitions) and the evaluation script here. Run with numgeo inp=embedded-beam-benchmark-A0-calculation out=A0.

The circumference is \(P=\pi D\). With fixed host displacements, the elastic transfer parameter is

\[ \lambda=\sqrt{\frac{Pk_t}{EA}}=0.258199\,\mathrm{m^{-1}} . \]

The same \(\lambda\), \(Q_y\) and \(Q_{\mathrm{ult}}\) are obtained for any combination of \(D\), \(k_t\) and \(t_{\mathrm{ult}}\) with equal \(Pk_t\) and \(Pt_{\mathrm{ult}}\); the manuscript version of this benchmark uses \(D=1\) m, \(k_t=2\times10^4\) kN/m³ and \(t_{\mathrm{ult}}=20\) kPa.

Closed-form solution

For prescribed head displacement \(w\) and axially free toe, the elastic solution (distance \(x\) from the pile head) is

\[ u(x)=w\frac{\cosh[\lambda(L-x)]}{\cosh(\lambda L)}, \qquad \tau(x)=k_tu(x), \qquad N(x)=EA\lambda w\frac{\sinh[\lambda(L-x)]}{\cosh(\lambda L)}, \qquad Q(w)=EA\lambda w\tanh(\lambda L). \]

First yield occurs at the pile head at

\[ w_y=\frac{t_{\mathrm{ult}}}{k_t}=1.0\,\mathrm{mm}, \qquad Q_y=209.129\,\mathrm{kN}. \]

If \(a\) denotes the yielded length measured from the pile head and \(u_y=t_{\mathrm{ult}}/k_t\), the partially yielded solution reads

\[ u_e(x)=u_y\frac{\cosh[\lambda(L-x)]}{\cosh[\lambda(L-a)]}\quad(x\ge a), \qquad w(a)=u_y+a\lambda u_y\tanh[\lambda(L-a)] +\frac{Pt_{\mathrm{ult}}a^2}{2EA}, \qquad Q(a)=Pt_{\mathrm{ult}}a +EA\lambda u_y\tanh[\lambda(L-a)]. \]

Full yield gives

\[ Q_{\mathrm{ult}}=Pt_{\mathrm{ult}}L=314.159\,\mathrm{kN}, \qquad w_{\mathrm{full}}=1.83333\,\mathrm{mm}. \]

Results


Figure 1: pile-head response (left) and shaft traction distribution at three stages of loading (right) compared with the closed-form solution. The dashed lines mark \(w_y\) and \(w_{\mathrm{full}}\).

The numerical curve coincides with the closed-form solution through the elastic, partially yielded and fully yielded ranges. In the elastic range the pile-head load is 0.12 % above the closed-form value - the discretisation error of nine two-node beam elements with two integration stations each - and the fully mobilised capacity \(Q_{\mathrm{ult}}=314.159\) kN is reproduced exactly (the tributary areas of the 72 coupling points sum to \(\pi DL\)). The shaft tractions read from the diagnostic file embedded-surface-interactions.res follow the analytical distribution, including the yielded zone that propagates from the pile head at \(w=1.19\) mm. Every increment converges in one equilibrium iteration (two in the first increment).

\(w\) (mm) \(Q\) numgeo (kN) \(Q\) closed form (kN) deviation
0.492 103.10 102.98 0.12 %
0.904 189.36 189.14 0.12 %
1.191 244.66 244.51 0.06 %
1.548 292.88 292.64 0.08 %
1.995 314.16 314.16 0.00 %
3.000 314.16 314.16 0.00 %

This verifies:

  • the integration of the shaft area \(\pi D L\);
  • the equal-and-opposite beam-host force transfer;
  • the axial beam stiffness and the sign convention;
  • the radial return to the constant shaft capacity;
  • the storage and commitment of the plastic shaft slip;
  • the algorithmic tangential stiffness (quadratic convergence).


Figure 2: reconstructed shaft surface (VTK output variable embedded) during progressive yielding.

The displayed quadrilaterals are post-processing cells, not interaction finite elements. Every cell represents one coupling-point tributary area. The small axial spaces separate cells centred at adjacent beam integration stations and have no mechanical significance.