The fiber method¶

GenSec analyses cross-sections using the fiber method, a numerical approach that discretizes the section into small elements (fibers) and computes internal forces by integrating stresses over the fiber assembly.

This page describes the mathematical formulation implemented in FiberSolver.

Hypotheses¶

The fiber method rests on three fundamental assumptions:

Plane sections remain plane (Bernoulli–Navier hypothesis): the strain distribution over the section is linear.
Perfect bond between all constituents: each fiber experiences the strain dictated by its position in the strain plane.
Uniaxial stress state: each fiber carries only axial stress \(\sigma(\varepsilon)\), governed by its own constitutive law. Shear stress and confinement effects are neglected at the section level.

Strain plane¶

Given three scalar parameters \((\varepsilon_0,\,\chi_x,\,\chi_y)\), the strain at fiber \(i\) located at \((x_i,\,y_i)\) is:

\[\varepsilon_i = \varepsilon_0 + \chi_x \, (y_i - y_{\text{ref}}) + \chi_y \, (x_i - x_{\text{ref}})\]

where \((x_{\text{ref}},\,y_{\text{ref}})\) is the reference point (the geometric centroid by default).

\(\varepsilon_0\) — axial strain at the reference point.
\(\chi_x\) — curvature about the \(x\)-axis (bending in the \(y\)-direction).
\(\chi_y\) — curvature about the \(y\)-axis (bending in the \(x\)-direction).

Uniaxial bending is the special case \(\chi_y = 0\).

Internal forces¶

Once every fiber stress \(\sigma_i\) is computed through the material constitutive law, the resultant internal forces are obtained by summation over all \(n_f\) bulk fibers and \(n_r\) rebar fibers:

\[\begin{split}N &= \sum_{i=1}^{n_f} \sigma_i^{(b)} \, A_i^{(b)} + \sum_{j=1}^{n_r} F_{\text{net},j} \\[6pt] M_x &= \sum_{i=1}^{n_f} \sigma_i^{(b)} \, A_i^{(b)} \, (y_i - y_{\text{ref}}) + \sum_{j=1}^{n_r} F_{\text{net},j} \, (y_j - y_{\text{ref}}) \\[6pt] M_y &= \sum_{i=1}^{n_f} \sigma_i^{(b)} \, A_i^{(b)} \, (x_i - x_{\text{ref}}) + \sum_{j=1}^{n_r} F_{\text{net},j} \, (x_j - x_{\text{ref}})\end{split}\]

where \(F_{\text{net},j}\) accounts for embedded bar correction (see below).

Embedded bar correction¶

For rebars lying inside the bulk material, the concrete area that the bar displaces is already counted in the bulk fiber sum. To avoid double-counting, the net rebar contribution is:

\[F_{\text{net},j} = \bigl[\sigma_{\text{rebar},j}(\varepsilon_j) - \sigma_{\text{bulk}}(\varepsilon_j)\bigr] \, A_{s,j}\]

For external elements (embedded = false):

\[F_{\text{net},j} = \sigma_{\text{rebar},j}(\varepsilon_j) \, A_{s,j}\]

Direct and inverse problems¶

Direct problem — given \((\varepsilon_0,\,\chi_x,\,\chi_y)\), compute \((N,\,M_x,\,M_y)\). This is a straightforward evaluation of the equations above.

Inverse problem — given a target \((N_{\text{target}},\, M_{x,\text{target}})\) (uniaxial) or \((N_{\text{target}},\, M_{x,\text{target}},\,M_{y,\text{target}})\) (biaxial), find the strain plane parameters that produce the specified internal forces.

GenSec solves the inverse problem with a Newton–Raphson iteration using the analytical tangent stiffness matrix. A backtracking line search ensures global convergence even for strongly nonlinear constitutive laws (e.g. softening concrete at ultimate).

Analytical tangent stiffness¶

The 3×3 tangent stiffness matrix relates infinitesimal changes in the strain-plane parameters to changes in internal forces:

\[\mathbf{K} = \frac{\partial (N,\,M_x,\,M_y)} {\partial (\varepsilon_0,\,\chi_x,\,\chi_y)} = \sum_i E_{t,i} \, A_i \; \boldsymbol{\varphi}_i \, \boldsymbol{\varphi}_i^T\]

where the shape-function vector for fiber \(i\) is

\[\boldsymbol{\varphi}_i = \bigl[1,\; (y_i - y_{\text{ref}}),\; -(x_i - x_{\text{ref}})\bigr]^T\]

and \(E_{t,i} = d\sigma_i / d\varepsilon_i\) is the tangent modulus at the current strain, computed by the material’s tangent_array method.

This analytical approach replaces the former finite-difference Jacobian (which required 3–4 extra integrate() calls per Newton iteration), halving the cost of each iteration.

Batch integration¶

The integrate_batch() method evaluates internal forces for many strain configurations at once. All inputs are 1-D arrays of length \(n\); the computation builds a 2-D strain matrix of shape \((n, n_{\text{fibers}})\), passes it through the constitutive law in a single vectorized call, and sums forces and moments along the fiber axis.

This is used by the capacity generators (generate(), generate_biaxial(), generate_mx_my()) to eliminate the Python-loop overhead of calling integrate() once per configuration.

Multi-material bulk¶

When the section contains multiple bulk material zones (e.g. a composite section with regions of different concrete grades), the integrator groups fibers by material index and evaluates each constitutive law on its own subset. The summation then runs over all groups:

\[N = \sum_{k=1}^{n_{\text{mat}}} \sum_{i \in G_k} \sigma^{(k)}(\varepsilon_i) \, A_i + \sum_{j=1}^{n_r} F_{\text{net},j}\]

This is handled transparently by the solver when the section exposes a mat_indices array.