Material nonlinearity and plastic hinges

By default, the FEM code in Ashes treats the structure as linear elastic: stresses are proportional to strains and the material never yields, no matter how large the load. For many wind-turbine load cases this is sufficient, because the design intent is precisely to keep the structure in its elastic range. To study the structural response beyond yield — overload events, accidental loads, pushover analyses, or the energy a member can dissipate during extreme cyclic loading — Ashes provides a material-nonlinear beam element based on concentrated plastic hinges.

The nonlinearity is introduced through a dedicated element type that combines an elastic  Euler-Bernoulli beam with a nonlinear moment–rotation law concentrated at the two element ends. The constitutive law is the Giuffré–Menegotto–Pinto (GMP) model with isotropic strain hardening, the same smooth elastoplastic law popularised by the OpenSees Steel02 material. Only the bending response is made nonlinear; the axial and torsional responses remain linear elastic.

The nonlinear element is an Euler-Bernoulli beam carrying a plastic hinge at each end and for each of the two bending axes — four hinges in total per element. A hinge is a zero-length rotational spring whose moment–rotation response follows the GMP law described in Section 3. While a hinge is elastic it is rigid (it adds no flexibility) and the element behaves as an ordinary elastic beam; once the end moment approaches the yield moment, the hinge softens and concentrates the inelastic rotation.

Each hinge yields when the end moment reaches the plastic yield moment of the cross section. For bending about a principal axis this is the material yield stress  times the plastic section modulus   of the section about that axis:

where the subscripts 1 and 2 denote the two principal bending axes of the cross section. Using the plastic (rather than elastic) section modulus means the yield moment corresponds to a fully plastified section, consistent with the lumped-hinge idealisation.

The GMP law is written in a generic stress–strain form (Section 3). The element reuses that same law for its moment–rotation hinge by mapping moment onto “stress” and a chord-rotation measure onto “strain”. With the element bending stiffness   and length  , a reference rotational stiffness is defined as

and the GMP material is evaluated with the substitutions

so that a call to the material law returns the hinge moment and its tangent rotational stiffness. This keeps a single, well-tested constitutive routine for both interpretations.

For each bending axis, the two end rotations  are related to the two end moments  through a Giberson one-component model: the elastic flexibility of the beam in series with a zero-length plastic hinge at each end. The total flexibility is

where the elastic part   is the standard Euler-Bernoulli end-rotation flexibility, and each hinge contributes

with   the current tangent stiffness of that hinge. The  clamp means that an elastic hinge ( ) adds no flexibility and the matrix collapses back to the pure elastic beam. The end-rotation stiffness is the inverse  . Because the two hinges are added independently, the element correctly represents the asymmetric case where only one end has yielded: that end softens while the other retains its elastic stiffness.

The hinge constitutive law is the smooth elastoplastic curve of Menegotto & Pinto (1973) with the isotropic strain-hardening extension of Filippou, Popov & Bertero (1983). Compared with a simple bilinear law, the GMP curve replaces the sharp corner at yield with a smooth, continuously curving transition between the elastic and post-yield branches, which both reproduces the real rounded response of structural steel and avoids the numerical kink of a bilinear model. The default parameters reproduce the OpenSees Steel02 material.

The curve is expressed in normalized coordinates measured from the last load-reversal point. Let  be the strain–stress state at the last reversal and  the intersection of the elastic and post-yield asymptotes for the current branch. The normalized strain and stress are

and the GMP law gives the normalized stress as a function of normalized strain:

Here   is the post-yield slope ratio (the ratio of the hardening tangent to the elastic modulus), and  is a curvature parameter controlling the sharpness of the transition. A large   produces a sharp, almost bilinear corner; in the limit   the curve collapses exactly to the bilinear law, while a small   gives a very rounded knee. The asymptote intersection is found by intersecting the elastic-slope line through the reversal point with the post-yield line of slope   through the (possibly hardened) yield point  :

with the sign chosen according to the loading direction of the new branch. The tangent modulus follows by differentiation:

A reversal is detected whenever the strain rate changes sign relative to the current branch. At a reversal the model stores the current point as the new anchor  , switches branch direction, and recomputes the asymptote intersection. This is what gives the model its Bauschinger-like hysteresis: unloading follows the elastic slope and reloading curves smoothly toward the opposite asymptote.

The curvature parameter is not constant. It softens with cumulative plastic deformation, so the knee of the curve becomes rounder as the material accumulates damage over many cycles. At each reversal   is updated from its initial value   as

where   is the cumulative bidirectional plastic ductility — the sum of the largest tensile and compressive plastic excursions, each normalized by the yield strain  . The constants   and   shape how quickly the transition rounds off with damage; the value of   is held fixed over each branch and is floored at a small positive value for numerical stability.

In addition to the kinematic-like response built into the reversal logic, the yield surface can expand with accumulated plastic strain (isotropic hardening). The effective yield stress grows with the largest plastic excursion seen in that direction:

The coefficient   scales the amount of isotropic hardening and   sets the plastic-ductility threshold (in multiples of the yield strain) below which no isotropic hardening occurs. The expansion is tracked independently in tension and compression, so after the first plastic half-cycle the yield radius can differ between the two directions. Setting   disables isotropic hardening entirely, leaving only the post-yield slope   as the source of hardening.

A nonlinear material is defined by the elastic properties shared by every Ashes material — Young's modulus  , Poisson's ratio  , density   and yield stress   — plus the six GMP parameters described above:

• Post-yield ratio 
  $$b$$
   — ratio of post-yield to elastic stiffness. Default
  $$b=0.015$$
  . Use 
  $$b=0$$
   for a perfectly plastic material; values approaching 1 suppress yielding.
• Initial curvature 
  $$R_0$$
   — sharpness of the elastic–plastic transition. Default
  $$R_0=20$$
  .
• Transition constants 
  $$c_{R1},\ c_{R2}$$
   — control how the curvature softens with cumulative damage. Defaults
  $$c_{R1}=0.925,\ c_{R2}=0.15$$
  .
• Isotropic-hardening parameters 
  $$a_1,\ a_2$$
   — amount and threshold of yield-surface expansion. Defaults
  $$a_1=0$$
   (no isotropic hardening) and
  $$a_2=1$$
  .

The default set   matches the recommended OpenSees Steel02 values, so a material defined with only   and   already behaves like a standard Steel02 curve.

In an Ashes structural input (text) file, a nonlinear material is declared with the BilinearGMP keyword in the materials block, following the pattern

name  E  nu  density  yieldStress  BilinearGMP  b  R0  cR1  cR2  a1  a2

for example a structural steel with  ,   and   reads as nlsteel 2.1e11 0.3 7850 2.5e8 BilinearGMP 0.015 18 0.9 0.15 0.0 1.0. Members assigned a BilinearGMP material are meshed with plastic-hinge beam elements automatically.

The element participates in the standard nonlinear time-domain solution. Within each Newton-Raphson iteration of a time step, every hinge runs the GMP update as a trial state computed from the committed (last converged) history: an elastic moment predictor is formed from the current end rotations, the GMP law returns the corrected hinge moment and tangent stiffness, and the internal forces and tangent stiffness matrix are updated accordingly. The committed history is only overwritten once the step converges; if the step is rejected the trial state is discarded, so the path-dependent history (reversal points, plastic excursions, accumulated damage) stays consistent.

To keep the iteration well-conditioned across the smooth GMP knee, the stiffness handed to the solver blends the local tangent with a secant measure and is bounded between the post-yield ratio   and the full elastic stiffness. This prevents stiffness from swinging too abruptly during monotonic loading while preserving the elastic stiffness on unloading branches.

Alongside the moments and stresses, each hinge reports diagnostics that are exposed through the beam-element sensors: whether the hinge has yielded, the accumulated plastic rotation, the effective (hardened) yield moment, the number of load reversals, and the dissipated hysteretic energy   accumulated over the simulation. The dissipated energy is the area of the hysteresis loops and is the natural measure of how much inelastic work a member has absorbed.