Time History Analysis
The "Time History Analysis" system is mentioned to solve problems by the method of direct integration of the equations of motion in time. This enables the user to perform computer modelling of the response of a structure to a dynamic load both during its action and after its termination. Both material damping and external dampers (shock absorbers, seismic isolators) can be considered in an analysis.
The "Time History Analysis" system is used to solve both linear and nonlinear problems, providing an effective tool for more accurate and flexible modelling of dynamic processes, as well as checking the load-bearing capacity of structural elements (reinforced concrete, steel, masonry reinforcing structures).
Types of FEs for solving problems with time history analysis
Various types of finite elements are provided to solve problems with time history analysis. These types include all linear elements, one-way restraints (including elements for modelling damping of foundation base, seismic isolators considering unloading with initial stiffness, and considering friction), physically nonlinear elements (step and iterative), and geometrically nonlinear elements.
Iterative physically nonlinear finite elements (bar, plate, and solid) with account of unloading can be used in time history analysis; these finite elements realise the theory of elastic-plasticity. The iterative physically nonlinear finite elements are the most relevant in earthquake analysis and analysis on progressive collapse. An account of the plastic behaviour of the structures enables the user to significantly increase the economic efficiency of design solutions.
Two methods of unloading are implemented: method 1 - pico-oriented hysteresis model, method 2 - model with isotropic hardening. Plastic behaviour of materials is considered, unloading by initial modulus of elasticity.
Accounting for damping
An important aspect of the problems with time history analysis is the ability to account for damping.
Rayleigh damping takes into account the ability of buildings and structures to dissipate energy through material damping. Rayleigh damping coefficients can be automatically calculated from the specified frequencies and decrements of vibrations.
Rayleigh damping in LIRA-FEM program
In dynamic analysis by direct integration of the equations of motion, the behaviour of special damping elements can be considered using the finite element of viscous damping FE 62. The damping force in this element is proportional to velocity.
Element of viscous damping
External damping devices can be modelled with FE 66 and FE 65, in which the viscous damping coefficients are assigned in six directions. For these finite elements, it is assumed that viscous damping is realised, i.e., the resistance force is proportional to the corresponding velocity component. The viscous damping coefficients are defined independently for each nodal translation (rotation) and do not influence each other.
FE 65 and FE 66 (a 2-node, 1-node damper with six degrees of freedom) may also be used to model the damping of the foundation base when an elastic half-space is reduced to a set of springs and dampers.
Seismic isolators in the form of rubber-metal supports with a bilinear behaviour diagram are modelled with FE 255 and FE 256. These FEs consider the unloading with initial stiffness. Loading-unloading behaviour diagram: Strength-hardening non-degrading hysteretic model.
For FE 255 and FE 256, it is possible to specify behaviour graphs for vector sums of the following components: (X+Y) and (X+Y+Z). The graph is described by 3 values - 1st modulus of elasticity (R, t/m), 2nd modulus of elasticity (R2, t/m), graph break (N, t). You can define any set of graphs for individual components and combinations of component vector sums, but each component can only participate once. For example, if a behaviour graph is described separately for X, then X can no longer participate in any of the combinations of vector sums. The results for FE 255 and FE 256 are presented as forces along the corresponding directions of the local coordinate system Rx,Ry,Rz, Rux,Ruy,Ruz. As can be seen in the figure below, if the same parameters are defined for the X1 and Y1 axes, displacements and ultimate forces are controlled only along these directions. But if the load is applied at a different angle, the resultant (vector) sum of the forces can cause displacements that are greater than what a circular seismic isolator can handle. Hence, we can obtain the controlled values of parameters for the load at any angle in the plan by setting the parameters for the vector sum of the (X+Y) components.
Friction FE 263 and FE 264 with the activated option "unloading with initial stiffness" enable the user to describe, for example, the behaviour of a frictional seismic isolator, and when connected in parallel with the FE of an elastic spring - a frictional pendulum seismic isolator.
The option "Unloading with initial stiffness" allows the user to implement the hysteresis behaviour of FE 263 and FE 264 under cyclic loading: at the moment when the direction of movement is changed (when the velocity is equal to 0), it causes the friction force T=N*mf (mf is the friction coefficient defined in the stiffness parameters).
Dynamic loads available for time history analysis
To effectively solve the problems with time history analysis, it is possible to define the general diagram of dynamic load variation over time. The following types of dynamic loads at the nodes of the design model are available:
- Piecewise linear (polyline) graph of dynamic load with arbitrary step
- Sinusoidal graph of dynamic load
- Accelerogram in relative units
- Piecewise linear (polyline) graph of dynamic load with uniform step
- Seismograms
The diagram of dynamic load variation over time can be defined in different ways: manually, explicitly, or by reading from a file.
The Calculate Spectrum module is used to process earthquake accelerograms and obtain parameters of earthquake loads.
It is possible to convert the nodal static load into nodal dynamic load according to the specified conversion diagram (polyline with arbitrary step; sinusoidal; polyline with uniform step).
It is also possible to specify mass weights for nodes and elements or to collect masses automatically from the specified static load cases (mass can be accumulated either from static load cases (one or more) or from the material density specified when describing the stiffness parameters).
Design combinations of forces (DCF), Design combinations of loads (DCL)
In time history analysis, the DCF and DCL tables are generated automatically:
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For Eurocodes and building codes similar to the European standards for design in construction (EN 1990:2002+A1:2005, SP RK EN 1990:2002+A1/2011, etc.), there is an option to generate the characteristic forces DCL(c) based on the results of time history analysis.
Generating the DCL(c) for problems with time history analysis -
For the other building codes, the DCF table is generated automatically based on the results of the time history analysis.
Generating the DCF for problems with time history analysis -
With the help of the "Problem Integration" system (METEOR), the enveloping DCF can be generated for several problems with time history analysis.
Generating the enveloping DCF for problems with different scenarios of progressive collapse in dynamic statement
The load-bearing capacity of structural elements in reinforced concrete and steel structures is checked on the basis of automatically generated DCF or DCL.
Options to review, evaluate analysis results and prepare documentation
Nonlinear dynamic studies solve the problem of dynamic response as a function of time. The analysis results contain displacements, velocities, and accelerations of nodes; reactions in nodes; and forces and stresses in elements calculated for each moment of integration time (multiple of integration step).
Displacements (amplitudes), velocities, and accelerations can be displayed by means of mosaic plots for all nodes of the model for each moment of time. In addition, for each node of the model, it is possible to draw graphs of changes in amplitude, accelerations, and velocities in time. The spectrum graph is generated on the basis of the acceleration graph.
The forces and stresses calculated at each moment of integration time for all elements of the model can be presented as mosaic plots, diagrams, and contour plots. It is also possible to generate the graph of change in forces or stresses over time for each element of the design model.
For problems with Time History Analysis, it is possible to generate results for nodal reactions in elements. Nodal reactions are used in calculations of loads on a fragment, punching shear, bar analogues, loads in partitions. For structural analyses, dangerous combinations of forces/reactions are selected from all moments of time.
A convenient tool for evaluating and preparing documentation is provided for graphs of time history for displacements, forces, loads on a fragment, loads on a group of partitions, and graphs of kinetic energy. When you move the mouse pointer within the graph, the coordinates of appropriate values are displayed. Thus, it is possible to find out the value of the function at any point of the graph. Click any of the graphs to indicate some step as a reference time point. To delete the specified reference time point, just right-click the red line of this time point. It is also possible to use the mode in which, when the reference time points are added, they will be defined to the nearest local extremum.
After the analysis, we have the results of the stress-strain state for each time point. So, it enables the users to generate animations of the behaviour of structures under dynamic load.
For physically nonlinear time history analysis, the state of the section can be viewed at each integration time.
For physically nonlinear problems with iterative elements, there is a tool to view, evaluate and prepare documentation for the calculated stress-strain state parameters for standard, steel section types and plates (normal stress in the main material /reinforcement of plates and bars; relative strain in the main material /reinforcement of plates and bars; tangential stress τxy in the main material of plates; relative strain γxy in the main material of plates; maximum stress σmax in the main material of plates; relative strain εmax in the base material of the plates).
For physically nonlinear step-type elements of bars and plates, the calculated parameters of cracks can also be displayed.
To reduce the amount of the output data and time to save them, a technique is provided for iterative elements to output the results of the section state only at the user-defined integration steps. It is also possible to assign a list of iterative physically nonlinear elements in which section states should be computed in the analysis.
Thus, with the help of the "Time History Analysis" system it is possible to perform a complete nonlinear dynamic analysis of the model in the time interval for the design of active seismic isolation; to analyse the building structures on progressive collapse and at the same time to check the load-bearing capacity of structural elements (reinforced concrete, steel, masonry reinforcing).