Seismic Nonlinear Time History Analysis#
Introduction#
Welcome to the Seismic Nonlinear Time History Analysis (NTLHA) of Bridge Piers tutorial.
In this tutorial, we assess the structural performance of a two-column concrete bridge pier subjected to seismic excitation in the form of base acceleration. The structure is modelled in SOFiPLUS, where we assign the material’s nonlinear behaviour using concentrated plasticity models—plastic hinges—that can accommodate cyclic loading and exhibit hysteretic behaviour. The workflow is implemented in SOFiSTiK Structural Desktop using a new nonlinear dynamics solver, resulting in response time histories. These are assessed using strain- and rotation-based criteria, thereby evaluating the structural performance.
We divide the Seismic NLTHA workflow into three distinct steps:
Modelling Plastic Hinge for Cyclic Loading
Conducting Nonlinear Time History Analysis
Assessing Structural Performance using Strain and Deformation Criteria
By the end of this tutorial, you should be able to:
Define plastic hinges with specified loading and unloading laws, and assign them to a model in SOFiPLUS.
Set up and perform a Nonlinear Time History Analysis with physical and geometrical nonlinearity to determine the time-dependent structural behaviour.
Perform checks of the structural performance by evaluating the performance levels of plastic hinges and the internal forces of structural elements.
Project Description#
We conduct this tutorial on the same two-column reinforced concrete pier with a header beam used in the Pushover Tutorial, as a continuation of the seismic analysis tutorials. The structure is 13 metres high, with an 8-metre spacing between the columns. The circular columns have a radius of 1.6 metres, and the rectangular header beam is 1.8 metres deep and 1.6 metres wide. The structure is loaded with three concentrated forces of 3000 kN to simulate the permanent loading of the superstructure, which is transferred through the main girders.
Unlike in the Pushover Tutorial, the columns in this analysis are considered confined. The serviceability stress-strain relationship of the core material is modified based on confined C35/45 concrete. The concrete cover of the columns and the header beam is standard C35/45 concrete. B500 is considered as the material for the reinforcing steel.
![Nonlinear Time History Analysis: Structure and additional dead load [kN].](../../_images/DescriptionExample.png)
Nonlinear Time History Analysis: Structure and additional dead load [kN]. (click on image to enlarge)#
Nonlinear Time History Analysis: Cross section of header beam (left) and confined column (right) (click on image to enlarge).#
Plastic Hinge for Cyclic Loading#
A plastic hinge suitable for dynamic time history analysis consists of two components: an idealised moment-rotation backbone curve that defines the sectional capacity, and a loading-unloading material model capable of capturing hysteretic behaviour. In this example, the backbone curve is obtained using cross-sectional analysis for a specified normal force level and is subsequently idealised using the Caltrans method. The Modified Takeda material model is used to represent the hysteretic behaviour during cyclic loading, making the plastic hinge well suited for reinforced concrete structures.
We perform this analysis using the Plastic Hinge Task.
Additionally, we define strain-based Performance Categories, as explained in the Pushover Analysis - Bridge Piers: Performance Categories and Criteria Tutorial. In this way, we enable performance assessment based on the behaviour of the concrete cover, the confined concrete core, and the reinforcement.
Note
The theoretical background for the cross-sectional analysis and the Modified Takeda model is provided in the CSA and FEABENCH manuals.
Nonlinear Time History Analysis: Modified Takeda Model for load reversal with large (left) and small (right) amplitudes. The backbone curve is depicted in blue (click on image to enlarge).#
Nonlinear Time History Analysis#
We determine the dynamic response of the structure by performing a Nonlinear Time History Analysis. The El Centro record is used as a prescribed base acceleration, and the response is determined by time integration using the Newmark-Beta algorithm. The integration begins from a system state under permanent loads, which serves as the initial loading condition.
The time step size is set equal to the sampling interval of the earthquake ground acceleration record. A smaller time step does not necessarily improve accuracy and may even introduce spurious frequency content due to linear interpolation between sample points.
Nonlinearity is considered in both material and geometry. Material nonlinearity is concentrated in the plastic hinges at both ends of the columns. Degradation of the material along the height of the columns is taken into account by reducing their elastic bending stiffness using an effective stiffness factor obtained from cross-sectional analysis. We apply Second Order Theory to account for geometric nonlinearity, considering additional moments resulting from large concentrated forces of the superstructure and structural deformation.
An initial performance evaluation can also be carried out by plotting specific time histories of the structural response, such as displacements, accelerations, or internal forces.
All of this is integrated into the Nonlinear Dynamic Solver from the FEABENCH module, which significantly improves computational performance.
Tip
A suitable time step size larger than the sampling interval of the particular ground acceleration record can be found and verified by a time step convergence study using the input command DTC. In this procedure, the dynamic response is efficiently computed several times with different time step sizes. To make sure that the ground acceleration is always evaluated only at the sample points, the DTC command can be specified within the FEABENCH input as follows:
let#sampling_intv 0.02
let#fac0 0.05
dtc item ux fmax #fac0/(2.0*#sampling_intv) fac0 #fac0 facr 0.5 csmx 2 lts on
where the sampling interval of the ground acceleration is 0.02 sec and the monitored result is the displacement of all nodes in X-direction. The value of fmax is selected such that the reference response time history is computed using the sampling interval.
Note
Rayleigh damping in the analysis is based on natural frequencies determined from a dynamic eigenmode analysis. The Rayleigh coefficients remain constant throughout the analysis.
Tip
Nonlinear Time History Analysis is an advanced task that requires computational algorithms such as nonlinear iteration solvers and time integration methods with adaptive time stepping. These algorithms rely on parameters that may vary depending on the task at hand. While default values are provided, advanced users have the option to customise these settings. Further information on these parameters and their theoretical background can be found in the FEABENCH manual.
Performance Assessment#
Having determined the dynamic response of the structure due to the El Centro earthquake, we assess the structural performance by evaluating the rotation- and strain-based criteria in the plastic hinges. We use Graphic and Result Viewer as standard postprocessnig tools.
To gain an understanding of the strain energy dissipated in the plastic hinges, it is best to look at the moment-rotation hysteresis. With the new capabilities in the Result Viewer from SOFiSTiK 2026-02, accessing the data and plotting the hysteresis is straighforward.
Nonlinear Time History Analysis: Moment-rotation hysteresis of a plastic hinge (click on image to enlarge).#
Additional Resources#
Related information can be found in: