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Embedded beam with implicit interaction surface


*Embedded region, interaction-surface extends the embedded beam approach from a line coupling to a coupling on an implicit interaction surface: a virtual cylinder of radius \(R = D/2\) around the beam axis and, optionally, a base disk at the pile toe. The formulation follows Truty (2023) and Granitzer et al. (2024) and is intended for pile-type structures where the finite diameter matters (single piles, pile groups, energy piles), while the line coupling remains the method of choice for slender inclusions (anchors, nails).

Upcoming release

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

Coupling points on the shaft

At each line integration station \(\xi_g\) of a guest element (\(n_{gp} = 2\) for linear, \(3\) for quadratic interpolation), \(n_p\) coupling points are distributed over the circumference of the interaction surface,

\[ \tilde{\mathbf{x}}_{gj} = \mathbf{x}_{axis}(\xi_g) + \mathbf{r}_{cs,j}, \qquad \mathbf{r}_{cs,j} = R \left( \cos\varphi_j \, \mathbf{e}_1 + \sin\varphi_j \, \mathbf{e}_2 \right), \qquad \varphi_j = \frac{2\pi (j - \tfrac{1}{2})}{n_p}, \]

with a deterministic orthonormal frame \((\mathbf{e}_1, \mathbf{e}_2)\) perpendicular to the axis direction \(\mathbf{a}(\xi_g)\). Each point carries the analytically assigned area

\[ A_{gj} = \frac{2 \pi R}{n_p} \, J_s(\xi_g) \, w_g , \]

which sums exactly to the shaft area \(\pi D L\) of the guest element. For each coupling point, the host element containing it is determined and the host shape functions \(N^h_k(\tilde{\mathbf{x}})\) are frozen at the reference configuration (internal points, cf. the line coupling). Points outside the host region are skipped and their area is dropped.

Kinematics: beam-side mapping with rotations


The relative displacement at a coupling point compares the motion of the beam cross-section with the host displacement field,

\[ \Delta \mathbf{u}(\tilde{\mathbf{x}}) \;=\; \underbrace{\sum_i N_i(\xi_g) \left( \mathbf{u}_i + \boldsymbol{\theta}_i \times \mathbf{r}_{cs} \right)}_{\text{beam cross-section}} \;-\; \underbrace{\sum_k N^h_k(\tilde{\mathbf{x}}) \, \mathbf{u}^h_k}_{\text{host}} . \]

The lever-arm term \(\boldsymbol{\theta}_i \times \mathbf{r}_{cs}\) (interpolated with the same 1D shape functions, i.e. shear-flexible Timoshenko kinematics) is essential: it makes the coupling exact for arbitrary rigid-body motions including rotations (patch test) and transfers interaction moments — skin friction acting at the radial offset \(R\) produces torsional and bending moments on the beam. In matrix form, the participation of beam node \(i\) is

\[ \mathbf{W}_i = N_i \left[ \; \mathbf{I} \;\; -\mathbf{S}(\mathbf{r}_{cs}) \; \right] \in \mathbb{R}^{3 \times 6}, \qquad \mathbf{S}(\mathbf{r})\,\mathbf{v} = \mathbf{r} \times \mathbf{v}, \]

and of host node \(k\): \(\mathbf{W}^h_k = -N^h_k \, \mathbf{I}\). With the interface traction \(\mathbf{t}\) and its consistent modulus \(\mathbf{D} = \partial \mathbf{t} / \partial \Delta \mathbf{u}\), the nodal forces and the stiffness contributions follow as

\[ \mathbf{f}_p = -\mathbf{W}_p^T \, \mathbf{t} \, A, \qquad \mathbf{K}_{pq} = \mathbf{W}_p^T \, \mathbf{D} \, \mathbf{W}_q \, A , \]

i.e. beam nodes receive forces and moments (\(\mathbf{W}_i^T \mathbf{t} = N_i \, [\, \mathbf{t} ;\; \mathbf{r}_{cs} \times \mathbf{t} \,]\)), host nodes receive forces. The resulting stiffness is symmetric (for symmetric \(\mathbf{D}\)) and the total interface force and moment vanish identically (action = reaction).

Interface law on the shaft

In the local triad \((\mathbf{n}, \mathbf{e}_{c}, \mathbf{a})\) (outward radial normal, circumferential, axial):

  • Normal direction: linear elastic, \(t_n = k_n \, \Delta u_n\). No gapping (tension cutoff of the normal contact) is modelled, consistent with Truty (2023) and Granitzer et al. (2024).
  • Tangent plane: elastic — perfectly plastic Coulomb-circle slider with a single tangential stiffness \(k_t\) (Granitzer et al. 2024 use identical axial and circumferential stiffnesses, Eq. 23),
\[ \mathbf{t}^{tr} = k_t \left( \Delta \mathbf{u}_{tan} - \boldsymbol{\gamma}^{n} \right), \qquad \| \mathbf{t}_{tan} \| \leq t_{cap} . \]

Radial return for \(\|\mathbf{t}^{tr}\| > t_{cap}\): \(\mathbf{t}_{tan} = t_{cap} \, \mathbf{m}\) with \(\mathbf{m} = \mathbf{t}^{tr}/\|\mathbf{t}^{tr}\|\), slip update \(\boldsymbol{\gamma}^{n+1} = \Delta \mathbf{u}_{tan} - \mathbf{t}_{tan}/k_t\), and the consistent (algorithmic) tangent

\[ \mathbf{D}_{tan} = \frac{t_{cap}}{\|\mathbf{t}^{tr}\|} \, k_t \left( \mathbf{I}_2 - \mathbf{m} \otimes \mathbf{m} \right), \]

which is symmetric positive semi-definite. The plastic slip is stored per coupling point and committed upon convergence of the increment.

  • Capacity:
    • type=constant: \(t_{cap} = t_{ult}\) (input); \(t_{ult} \leq 0\) deactivates the slider (elastic shaft).
    • type=Mohr-Coulomb: \(t_{cap}(\tilde{\mathbf{x}}) = \max\left( c - \sigma_n(\tilde{\mathbf{x}}) \tan\varphi, \; 0 \right)\) with the local host normal stress \(\sigma_n = \mathbf{n} \cdot \boldsymbol{\sigma} \cdot \mathbf{n}\) (tension-positive convention; the tension cutoff clamps the capacity at zero).

Stress recovery on the interaction surface

For the Mohr-Coulomb law, \(\boldsymbol{\sigma}\) is recovered per coupling point from the converged host stresses (effective stresses for multi-phase hosts) by inverse-distance weighting over the integration points of all host elements sharing a node with the host element of the point,

\[ \boldsymbol{\sigma}(\tilde{\mathbf{x}}) = \frac{\sum_m w_m \, \boldsymbol{\sigma}^{0}_m}{\sum_m w_m}, \qquad w_m = \frac{1}{\| \mathbf{x}^{ip}_m - \tilde{\mathbf{x}} \|} , \]

whereby integration points inside the virtual pile volume (distance to a guest chord \(\leq R\), or inside the half sphere of radius \(R\) below a toe) are excluded: they carry the (ghost) stress state of the structural inside, not the soil state governing the skin friction (Granitzer et al. 2024). If a recovery patch contains no admissible integration point, the nearest outside integration point of the host region is used (warning). The recovery mapping is precomputed at the reference configuration. Since converged stresses enter, \(t_{cap}\) is constant within the increment (capacity lags one increment) and the consistent linearisation of the slider is preserved.

Base disk

Each toe node given in the base nset carries \(n_b\) coupling points: one central point and \(n_b - 1\) ring points at \(0.75R\), each with area \(\pi R^2 / n_b\) (Granitzer et al. 2024, Eq. 14). The relative displacement uses the toe cross-section motion \(\mathbf{u}_{toe} + \boldsymbol{\theta}_{toe} \times \mathbf{r}_{cs}\), so the base disk also transfers moments. The interface response is an isotropic spring \(k_b\); the axial (outward) traction carries an elastic — perfectly plastic one-sided cap at the compressive base capacity \(\sigma_{lim}\) (state variable per base point; algorithmic stiffness zero when capped). The base capacity is a direct input rather than recovered from the host stresses, since the stress recovery at the tip is unreliable (singular stress concentration; Granitzer et al. 2024). Base tension is not limited (no gapping).

Ghost (elastic) zone

The soil elements inside the virtual pile volume would spuriously plastify under the interface loading transferred through the coupling. With ghost=yes (default), all host integration points inside the pile volume (+ half sphere below each toe) are switched to a linear elastic response with user-defined \(E_{ghost}\), \(\nu_{ghost}\) ("similar to the soil stiffness", Granitzer et al. 2024) and frozen state variables:

\[ \Delta \boldsymbol{\sigma} = \lambda \, \mathrm{tr}(\Delta \boldsymbol{\varepsilon}) \, \mathbf{I} + 2 \mu \, \Delta \boldsymbol{\varepsilon} . \]

The override is implemented as a guarded per-integration-point branch in the constitutive dispatch of the elements.

Verification (closed forms)

For a rigid host and a fully embedded guest element of length \(L\) (verified to machine precision by the automated checks):

Case Closed form
Rigid-body motion (translation + rotation) \(\Delta \mathbf{u} = \mathbf{0}\), zero interface forces
Lateral translation \(\delta\) \(F = \pi R (k_n + k_t) L \, \delta\)
Rotation \(\theta\) about a lateral axis through the centre \(M = \theta \left[ k_t \pi R^3 L + (k_t + k_n) \pi R L^3 / 12 \right]\)
Axial pull-out \(F_{max} = \pi D L \; t_{cap}\)
Base push-in \(F_{max,b} = \sigma_{lim} \, \pi R^2\)

The first term of the rotation stiffness stems from the axial skin tractions acting at the radial offset \(R\) (\(\mathrm{d}A = R \, \mathrm{d}\varphi \, \mathrm{d}s\)) — this moment transfer is the key mechanical difference to the line coupling. The circumferential sums are exact for the offset ring points (\(\sum \cos^2 \varphi_j = \sum \sin^2 \varphi_j = n_p/2\) for \(n_p \geq 3\)).

Limitations

  • 3D models and 3D beam guests only.
  • No gapping (neither shaft nor base); base tension is unbounded elastic.
  • Circular cross-sections only (rectangular sections require an orientation input and a perimeter segmentation; planned).
  • Constant user-defined interface stiffnesses (the automatic \(G_{el}\)-based stiffness of Granitzer et al. 2024, Eqs. 22-24, is a planned extension).
  • Geometrically linear coupling (reference configuration).
  • No explicit dynamics.

References

  • Truty, A. (2023): Improved formulation of embedded beams with interaction surface. Studia Geotechnica et Mechanica, 45(2).
  • Granitzer, A.-N., Tschuchnigg, F., et al. (2024): Construction and verification of an improved embedded beam formulation with an explicit interaction surface. Computers and Geotechnics.