Opensees nonlinear material one element sinusoidal
1 Test description
This test exercises the nonlinear (Giuffré-Menegotto-Pinto, “Steel02”) material model under cyclic loading in a transient analysis. A single-element tubular tower is driven by a sinusoidal tip force, and the resulting tip displacement is benchmarked against an equivalent single-degree-of-freedom (SDOF) OpenSees model that uses the same material law and Rayleigh damping.
Four load cases cover two amplitudes — one in the elastic range and one well into the plastic range — combined with two periods, to exercise both reversible response and hysteretic cycling:
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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)
The companion static-pull test is
Opensees nonlinear material one element; the present test extends that comparison to dynamic cyclic loading.
2 Model
The Ashes model is a single tubular tower modelled with one beam element. Geometry and material are identical to the static-pull sibling and use the same support-section input file (One_element_GMP.txt).
Geometry:
$$H=10\text{ m},\quad D=1\text{ m},\quad t=25\text{ mm}$$
Material (Steel02 / GMP):
$$E=2.1\cdot10^{11}\text{ Pa},\quad f_y=2.5\cdot10^{8}\text{ Pa},\quad b=0.015$$
Solver: FemDynamic, linear analysis mode, timestep
$$\Delta t=0.02\text{ s}$$
, total duration 30 s, with a 2 s linear ramp-up applied to the load.
Damping is stiffness-proportional Rayleigh with explicit K-factor:
$$\beta_K=0.05\text{ s/rad}$$
Loading is a prescribed Force at the tip (RNA) node along the x-axis. The amplitude and the “Period” parameter of the prescribed load are set per load case (see Section 1); a non-zero Period turns the otherwise constant prescribed load into a sinusoidal one, so that the applied force is
$$F(t)=F_{amp}\,\sin\!\left(\frac{2\pi t}{T}\right)$$
3 Benchmarked solution
The reference solution is computed with OpenSees 3.7.1 using an SDOF formulation: the cantilever flexural response is collapsed onto a uniaxial Steel02 spring at the tip via a zeroLength element, with the same material parameters as Ashes. A lumped mass equal to half the steel mass of the tower is placed at the tip node:
$$m_{tip}=\tfrac{1}{2}\,\rho\,A\,H$$
with steel density
$$\rho=7850\text{ kg/m}^3$$
and the annular cross-section area
$$A=\pi\,(R_o^2-R_i^2)$$
.
Stiffness-proportional Rayleigh damping is applied with the same K-factor
$$\beta_K=0.05\text{ s/rad}$$
as the Ashes run. The sinusoidal load is produced via OpenSees' Trig timeSeries with matching amplitude and period, and the equations of motion are integrated with the Newmark-β scheme at
$$\Delta t=0.02\text{ s}$$
.
The TCL driver is Sinusoidal_cylinder.tcl in the test directory; it expects the command-line arguments
$$(F_{amp}\text{ [N]},\;T\text{ [s]},\;t_{end}\text{ [s]},\;\Delta t\text{ [s]})$$
.4 Results
The compared output is the x-displacement of the tip (RNA) node, recorded by the Node sensor on the tubular tower. The full time series is compared point-by-point against the OpenSees reference (EvaluationCriteria.full_time_series) with a relative threshold of 0.03 (3%).
The report for this test can be found on the following link: