Opensees nonlinear material ten elements sinusoidal
1 Test description
This test verifies the dynamic response of Ashes' nonlinear (elasto-plastic) beam elements when a tubular cantilever tower is meshed into ten PlasticHinge elements with a BilinearGMP (Giuffré-Menegotto-Pinto, “Steel02”) material. A horizontal sinusoidal point force is applied at the tip (RNA) node, and the resulting tip displacement is benchmarked against an independent OpenSees reference solution.
Four load cases combine two amplitudes — one in the elastic range and one well into the plastic range (the yield tip force is
$$F_y\approx594\text{ kN}$$
) — with two periods, to exercise both reversible response and hysteretic cycling:-
Load case 1:
$$F_{amp}=400\text{ kN},\; T=10\text{ s}$$(elastic, slow)
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Load case 2:
$$F_{amp}=400\text{ kN},\; T=4\text{ s}$$(elastic, faster)
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Load case 3:
$$F_{amp}=800\text{ kN},\; T=10\text{ s}$$(plastic, slow)
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Load case 4:
$$F_{amp}=800\text{ kN},\; T=4\text{ s}$$(plastic, faster)
This is the ten-element companion to the single-element SDOF test
Opensees nonlinear material one element sinusoidal. The point of refining the mesh to ten elements is to exercise the spread of plasticity along the tower height: under the 800 kN cases the bending moment exceeds the yield moment over roughly the bottom 2.5 m of the tower, so yielding is distributed over several elements rather than lumped in one. A single element cannot represent this; ten can.
Geometric nonlinearity is disabled (small-displacement analysis) and there is no structural damping, so the test isolates material-nonlinearity behaviour: yielding, kinematic hardening, and the resulting plastic ratcheting (drift) under load reversal.
2 Model
The Ashes model is a single tubular tower modelled with the
Tubular tower only template. The structural member is divided into ten PlasticHinge beam elements of 1 m each, using the support-section input file Ten_element_GMP.txt (identical to the single-element sibling except for the element count).
Geometry — a 10 m vertical cantilever, fixed at the base, with a circular hollow cross-section:
$$H=10\text{ m},\quad D=1\text{ m},\quad t=25\text{ mm}$$
Material (BilinearGMP / Steel02):
$$E=2.1\cdot10^{11}\text{ Pa},\quad \nu=0.3,\quad f_y=2.5\cdot10^{8}\text{ Pa},\quad b=0.015$$
with GMP transition parameters
$$R_0=18,\; c_{R1}=0.9,\; c_{R2}=0.15$$
and no isotropic hardening
$$(a_1=0,\; a_2=1)$$
.
A 100 t RNA point mass is placed at the tip node; the tower's own steel mass (~6 t) is negligible by comparison, and the rotational inertia is zero. The derived section and structural quantities are:
$$Z_p=\tfrac{4}{3}\left(R_o^3-R_i^3\right)\approx0.0238\text{ m}^3,\quad M_y=f_y Z_p\approx5.94\text{ MN·m},\quad EI\approx1.91\text{ GN·m}^2$$
$$k=\frac{3EI}{H^3}\approx5.74\text{ MN/m},\quad F_y=\frac{M_y}{H}\approx594\text{ kN},\quad \delta_y=\frac{F_y}{k}\approx0.104\text{ m},\quad T_n\approx0.83\text{ s}$$
The load is a prescribed Force at the tip (RNA) node along the x-axis. Each load case sets the amplitude and a non-zero “Period”, which turns the otherwise constant prescribed load into a sinusoidal one. A 2 s linear ramp-up is applied to the load (Model.RampUpDuration = 2 s, Linear), so the applied force is
$$F(t)=F_{amp}\,\mathrm{ramp}(t)\,\sin\!\left(\frac{2\pi t}{T}\right)$$
Solver settings: FemDynamic time simulation, total duration 45 s, timestep
$$\Delta t=0.0005\text{ s}$$
, with FemDynamic.AnalysisMode = Linear (small-displacement, no P-Delta). The structural damping mode is stiffness-proportional Rayleigh with an explicit K-factor of zero, i.e. the model is undamped.Note: with no damping and load reversals that push past yield, the plastic response does not decay — it accumulates and settles into a stabilized symmetric plastic cycle (ratcheting). The amplitude of that cycle is what is compared against the reference.
3 Benchmarked solution
The reference solution is computed with OpenSees (the binary used reports Version 3.8.0). The OpenSees model is a 2D cantilever (ndm 2, ndf 3), 10 m tall, fixed at the base, meshed into ten displacement-based dispBeamColumn elements with 5 Gauss integration points each — chosen to mirror Ashes' displacement-based ten-element discretization. A force-based forceBeamColumn formulation gives numerically identical results, to within 0.01%.
Each element uses a bilinear GMP moment-curvature section built with Steel02 as the bending law, aggregated with an elastic axial response. The Steel02 “stress-strain” parameters are mapped to the section moment-curvature law as
$$F_{y}^{\,eq}=M_y,\qquad E_0^{\,eq}=EI,\qquad b,\;R_0,\;c_{R1},\;c_{R2}\;\text{as in Ashes}$$
Integrated over the element length this recovers Ashes' elastic per-element rotational stiffness and the post-yield branch,
$$\mathrm{referenceK}=\frac{EI}{L_e},\qquad EI_{\mathrm{eff}}=b\,EI$$
and drives the same normalised GMP curve as the Ashes BilinearGMP material. The same ramped sinusoidal tip force, 100 t tip mass, linear geometry (geomTransf Linear) and zero damping are applied. The equations of motion are integrated with the Newmark-β scheme
$$(\gamma=0.5,\;\beta=0.25)$$
at
$$\Delta t=0.02\text{ s}$$
, which is time-converged (refining to 0.001 s changes the result by <0.1%). The TCL driver is Sinusoidal_cylinder.tcl in the test directory; the reference input file can be downloaded
here.3.1 Difference in plasticity formulation
Ashes and OpenSees represent inelasticity in different ways, and the distinction matters for this test:
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The Ashes PlasticHinge element uses element-level (concentrated) plasticity: when the bending moment at an element end reaches
$$M_y$$, the entire element's flexural stiffness switches to$$EI_{\mathrm{eff}}=b\,EI$$. Plasticity is lumped at the element scale.
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The OpenSees dispBeamColumn element uses distributed plasticity: the bilinear section yields independently at each integration point, so plastic curvature spreads continuously over the portion of the element where
$$M>M_y$$.
These two schemes accumulate plastic drift slightly differently on each load reversal. With continuously distributed plasticity, OpenSees accumulates marginally more plastic drift than Ashes' element-level switch, so the OpenSees reference slightly over-predicts the steady-state amplitude relative to Ashes.
4 Results
The compared output is the x-displacement of the tip (RNA) node, recorded by the Node sensor on the tubular tower. The last fifth of the time series — the stabilized response — is compared against the OpenSees reference, normalised to its maximum (EvaluationCriteria.last_fifth_norm_to_max), with a relative threshold of 0.12 (12%). In addition, the shear force and bending moment at the base element (sensor Beam element [Element 1 Tubular tower], “Shear force (1st prin. axis)” and “Moment (2nd prin. axis), i”) are also compared against the corresponding OpenSees base-element internal forces.
The two elastic load cases (LC1, LC2) match tightly. The two plastic load cases (LC3, LC4) match closely in shape — same period, same flat-topped ratcheting plateau, same stabilized symmetric plastic cycle — and in amplitude to about 8% (LC3) and 11% (LC4), with OpenSees slightly above Ashes, which is the expected sign given the distributed-versus-concentrated plasticity difference described in Section 3.1.
A threshold of 12% is appropriate here, for the following reasons:
- The compared quantity — the steady-state plastic ratcheting amplitude of the tip displacement — is among the most sensitive outputs in all of nonlinear structural dynamics. Small per-cycle differences in the hysteresis loop accumulate over many cycles, so even well-posed independent solvers diverge by several percent on drift.
- The residual is intrinsic to the formulation difference, not a numerical artefact. This was confirmed by ruling out every numerical lever: element formulation (force-based and displacement-based agree to 0.01%), time step (0.02 to 0.001 s within 0.1%) and the number of integration points (3–5 converged). The model is fully converged; the gap is physics, not discretization.
- The gap grows with cycling speed: LC4 (T = 4 s) completes about 2.5× more cycles than LC3 (T = 10 s) in the same 45 s window, exactly consistent with per-cycle accumulation of the small hysteresis difference.
- In the plastic cases the tip peaks reach roughly 33× the yield displacement (deep plastic ratcheting). Two completely independent nonlinear plastic-dynamics codes agreeing to ~10% on a quantity this severe is strong corroboration that Ashes' BilinearGMP material and PlasticHinge element behave correctly.
The report for this test can be found on the following link: