Opensees nonlinear material one element dynamic
1
Test description
This test analyses the dynamic response of a one-element cantilever beam featuring a nonlinear steel material (Giuffré-Menegotto-Pinto / Steel02) subjected to a transverse tip force that is ramped up linearly over 2 s and then held constant for a total simulation duration of 60 s. Results produced by
OpenSees 3.7.1
are used to benchmark Ashes.
The purpose of the test is to validate the
BilinearGMP material model in Ashes under dynamic loading conditions, complementing the static sister test
Opensees nonlinear material one element.
The benchmark covers the elastic range, the elastic-to-plastic transition, and the post-yield hardening regime by sweeping the peak tip force from 0 to 1 MN in 50 kN increments (21 load cases). Because stiffness-proportional Rayleigh damping is applied (see Section 2), the dynamic transient settles to the same static bilinear F-u curve at every load level, so the test simultaneously serves as a consistency check against the static sister test.
2
Model
The model is identical to the one used in the static sister test: a vertical cantilever beam built with the
Tubular tower only template, 10 m tall, fixed at the base and free at the tip, discretized with a single
PlasticHinge beam element. A point mass of 100 t is placed at the tip node (RNA mass).
The cross section is circular hollow with diameter
$$D=1.0\text{ m}$$
and thickness
$$t=0.025\text{ m}$$
.
The material is a
BilinearGMP nonlinear steel matching the parameters of the OpenSees
Steel02 uniaxial material:
$$E=2.1\cdot10^{11}\text{ Pa},\quad \nu=0.3,\quad f_y=2.5\cdot10^{8}\text{ Pa}$$
$$b=0.015,\quad R_0=18,\quad cR_1=0.9,\quad cR_2=0.15$$
where
$$b$$
is the strain-hardening ratio and
$$R_0,\, cR_1,\, cR_2$$
control the curvature of the elastic-to-plastic transition.
Gravity, aerodynamic and wave loads are disabled. The Ashes geometric formulation is linear; the only source of nonlinearity is the material. The tip force is applied in the global x-direction at the free node. The simulation runs for
$$T=60\text{ s}$$
with a timestep of
$$\Delta t=0.02\text{ s}$$
.
The 21 load cases are distinguished by the peak tip force:
$$F_i = (i-1)\cdot 50\text{ kN},\quad i=1,\dots,21$$
i.e. from 0 N (Load case 1) up to 1 MN (Load case 21).
2.1
Load ramp-up
Each load case applies its peak force via a linear ramp over the first
$$T_\text{ramp}=2\text{ s}$$
,
after which the force remains constant for the remainder of the 60 s simulation. The ramp is configured using the Ashes built-in
Duration / Linear ramp-up scheme.
2.2
Damping setup
Ashes applies stiffness-proportional Rayleigh damping with coefficient
$$\beta_K = 0.05\text{ s/rad}$$
This level of damping is sufficient to suppress the dynamic transient so that the tip displacement settles to the static bilinear F-u value at every load level (e.g. 4.82 m at F = 1 MN, 2.49 m at 800 kN, 1.33 m at 700 kN). Consequently, the dynamic test doubles as an independent consistency check against the static sister test: if the static and dynamic results agree at every load level, both the static solver and the dynamic integrator are exercising the same material curve.
OpenSees uses mass-proportional Rayleigh damping instead:
$$\alpha_M = 2.87\text{ rad/s}, \quad \beta_K^\text{OS} = 0$$
This is a deliberate workaround for an OpenSees limitation: OpenSees silently ignores stiffness-proportional Rayleigh damping on
zeroLength + uniaxialMaterial elements. Any value of
$$\beta_K^\text{OS}$$
produces the same undamped output.
For a single-DOF system the damping force
$$C\dot{u}$$
is identical whether
$$C$$
is built as
$$\alpha_M \cdot m$$
or as
$$\beta_K \cdot k$$
, provided the two coefficients satisfy
$$\alpha_M = \beta_K \cdot \frac{k_\text{init}}{m}$$
With the initial elastic stiffness
$$k_\text{init} = 3EI/H^3 = 5.737\cdot10^6\text{ N/m}$$
and the tip mass
$$m = 10^5\text{ kg}$$
:
$$\alpha_M = 0.05 \times \frac{5.737\cdot10^6}{10^5} = 2.87\text{ rad/s}$$
This value reproduces Ashes' damped settled response exactly at every load level in the elastic regime.
Equivalence caveat: the
$$\alpha_M \leftrightarrow \beta_K \cdot k$$
equivalence is exact only while the system remains elastic. After yielding, Ashes' damping coefficient
$$\beta_K \cdot k_\text{tangent}$$
varies with the tangent stiffness, whereas the OpenSees mass-proportional coefficient
$$\alpha_M \cdot m$$
stays constant. For deeply plastic load cases the transient shape can therefore differ slightly between Ashes and the reference. The
settled value (evaluated over the last fifth of the time series) still matches because the same total kinematic hardening state is reached — which is why the acceptance criterion is
last_fifth_time_series rather than full time series.
3
Benchmarked solution
The reference solution is computed with OpenSees 3.7.1 using a SDOF formulation that collapses the bending response of the cantilever onto a single uniaxial Steel02 spring at the tip, identical in construction to the static sister test. The flexural stiffness and yield force of the equivalent spring are derived from the cross section and beam geometry:
$$k_\text{init} = \frac{3EI}{H^3},\qquad F_y = \frac{f_y \cdot Z_p}{H}$$
where
$$I$$
is the second moment of area of the hollow circular section,
$$Z_p$$
is its plastic section modulus and
$$H=10\text{ m}$$
is the cantilever length.
The OpenSees model (
Uniform_cylinder_dynamic.tcl) is a one-node system with a
zeroLength spring using a
Steel02 uniaxial material. A 100 t point mass is attached to the free node. The tip force is applied as a dynamic time history: a linear ramp over the first 2 s followed by a constant force for the remaining 58 s of the 60 s simulation. The same
$$\Delta t = 0.02\text{ s}$$
timestep is used. Mass-proportional Rayleigh damping is applied as described in Section 2.2. The tip displacement time series is recorded and interpolated onto the Ashes time grid for comparison.
The static convergence of the OpenSees reference to the same F-u curve as the static pull test (verified at F = 700 kN, 800 kN and 1 MN) provides robust evidence that the mass-proportional workaround does not introduce a systematic bias in the settled values.
4
Results
For each load case, the global x-displacement of the tip node (sensor
Node RNA_node Tubular tower, displacement DOF x) is compared between Ashes and OpenSees. The comparison is performed over the last fifth of the 60 s time series (i.e. from t = 48 s to t = 60 s), by which point the ramp-up transient has fully settled. The test is considered passed if the relative error between the Ashes tip displacement and the OpenSees reference displacement stays below
2% in that window.
The same acceptance criterion is used for all 21 load cases, which together cover the purely elastic response, the elastic-to-plastic transition around
$$F_y$$
, and the post-yield hardening branch. The agreement between the settled dynamic tip displacement in Ashes and the OpenSees reference — at each load level matching the static F-u curve — is the key result validated by this test.
The report for this test can be found on the following link: