4 DOF Timoshenko element shape functions

Plotting the shape function of 4 DOF Timoshenko elements using the github example code.

The track model implemented here is based on the work of Gimenez et al [Gimenez2018]. The finite element implementations feature two choices of Timoshenko beam elements:

The conventional 4 DOF Timoshenko (TIM4) element.
A 4+1 DOF Timoshenko (TIM4eb) element which is supported by an elastic foundation distributed along the whole length of the element.

The modified 4+1 DOF Timoshenko element (TIM4eb) has a 5th DOF which corresponds to the contribution from the vertical displacement of the sleeper.

The nodal displacement vector for a TIM 4 element composed of a beam with node 1 and 2 is:

\[u^{(e)}=\left\{ w_1,\theta_1,w_2,\theta_2\right\}\]

where \(w\) is the vertical displacement and \(\theta\) is the nodal bending rotation.

The following figure illustrates the interpolation functions for the response quantities of a standard TIM4 element:

The interpolation functions for the response quantities of a TIM4 element on the elastic ballast is illustrated as:

_images/shapeFunctions_Timoshenko4_elasticbedding.png

The difference between the TIM4 and TIM4eb at changing stiffness can be illustrated as:

_images/shapeFunctions_comparison_Timoshenko4_elasticbedding.png

Influence of the sleeper stiffness on the interpolation function for the sleeper displacement:

_images/shapeFunctions_sleeper_stiffness_sleeperdisplacement.png

Track Frequency Response Evaluation

This example evaluates the frequency response of a track composed of TIM4 or TIM4eb (eb: elastic bedding) elements using the github example code.

Frequency response of the track for point excitation at mid-span or over the sleeper for changing ballast pameters:

_images/FrequencyResponse_ChangingBallastParams.png

Frequency response of the track for point excitation at mid-span or over the sleeper for changing pad pameters:

_images/FrequencyResponse_ChangingPadParams.png

Frequency response observed at different DOFs of the track for point excitation at mid-span or over the sleeper compared for PT (TIM4) and EB (TIM4eb):

_images/FrequencyResponse_point_distributed_support.png

Frequency response observed at different DOFs of the track for point excitation at mid-span or over the sleeper compared for PT (TIM4) and EB (TIM4eb):

Modal response amplitude and shapes for mid-span excitation at peak response frequencies (peak picking on the FRF):

Modal response amplitude and shapes for excitation over the sleeper at peak response frequencies (peak picking on the FRF):

_images/mode_shapes_load_oversleeper.png

Time integration for impulse

This example evaluates the response of the axle during the crossing of a vehicle with the parameters of an EW4 Panorama waggon using the github example code.

Time series of the axle response for the crossing of a defect after 0.25 seconds:

_images/time_integration_timeseries_axle.png

Time series of the bogie response for the crossing of a defect after 0.25 seconds:

_images/time_integration_timeseries_bogie.png

Time series of the body response for the crossing of a defect after 0.25 seconds:

_images/time_integration_timeseries_body.png

Frequency response observed at different DOFs of the track for point excitation at mid-span or over the sleeper compared for PT (TIM4) and EB (TIM4eb):

_images/time_integration_timeseries_railSleeper.png

Time integration with varying track parameters

This example evaluates the response of the system during the crossing of a vehicle with the parameters of an EW4 Panorama waggon using the github example code.

Time series of the axle response for the crossing of a defect after 0.25 seconds at varying speeds: