# Time History Analysis

Available in: 3D Structure

FEM-Design gives the possibility to run so-called linear time-history analysis (LTHA) on structures based on 3 different methods.

The dynamic load and response of the structure can be calculated from forced vibration (e.g.: time dependent excitation force) or from ground accelerations (e.g: seismic effect). The results of the structural responses primarily the dynamic amplification factor (DAF), the dynamic displacements and accelerations at different points of the structure.

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Contents:

- Structural response analysis
- Level acceleration response spectra calculation
- Forced vibration with arbitrary functions

## Structural response analysis

### Input

In FEM-Design, we can define so-called ground acceleration functions (also known as "accelerogram") in time and in horizontal/vertical direction with the *Ground acceleration *command (*Loads *tab).

These functions (at *Diagrams*) can be independent or dependent from each other. If we copy them from an earthquake database, the database may contain coherent accelerograms, which were recorded in the same time at the same place in global X, Y and Z directions. But the given accelerogram in one direction can be independent from the others.

After the insertions of the accelerograms we can calculate the acceleration response spectras (*Response spectra* button) by setting different calculation parameters such as calculation integration scheme (Newmark β or Wilson θ), damping factor, time steps, time period and behavior factor. Based on the settings, *Calculate *generates the horizontal, vertical and average response spectras.

With *Set spectrum* the selected spectra can be exported as an input unique spectra of the Modal analysis/Static linear shape/Static mode shape seismic calculation.

We can also combine the ground acceleration diagrams (defined at *Diagrams*) by time steps and by directions with given multiplication factors (*Combinations*).

### Calculation request and settings

Similarly to other FEM-Design calculation requests, Time history analysis can be set in *Calculation *dialog.

Just activate “Time history, ground acceleration” and setup additional parameters under *Setup:*

*Time history calculation*should be selected for structural response analysis.*n result*– during the analysis (direct integration method) we should calculate the displacements and accelerations of the nodes in every time step to get proper solution, but to save calculation time and hard drive space at the end of the calculation as results only every*n-*th time step result will be saved and available.*t end*[s] – this time moment will be the last during the calculation. It is useful because it could happen that one of the accelerogram function in the specific combination shorter in one direction (X, Y or Z) but the calculation will performed further than this time moment. If one function ends earlier in any direction than the others, the shorter function will be neglected and obviously only the remaining ones will be involved into the last time interval of the calculation.

- Choose an
*Integration scheme*method:*Newmark*(default) or*Wilson-Theta*direct integration method. - At
*Rayleigh damping matrix*, we should set the damping parameters (*Alpha*and*Beta*) that determine the damping behavior of our 3D structure, and based on the relevant eigenfrequencies of the structure and its critical damping ratios. - (
*Damping factor*is not considered in structural response analysis.)

### Results

- Dynamic relative to the ground displacements (translations or rotations) in the selected time step and their envelope absolute maximum.
- Detailed result diagram of the selected nodal point about the nodal displacements (translations or rotations) as a function of time.
- Absolute accelerations in the selected time step and their envelope absolute maximum.
- Detailed result diagram of the selected nodal point about the nodal accelerations as a function of time.
- The dynamic reactions in the selected time step and their envelope absolute maximum, positive maximum and negative maximum.
- The internal forces in the structural elements in the selected time step and their envelope absolute maximum, positive maximum and negative maximum.

## Level acceleration response spectra calculation

### Input

In case of the ground acceleration (earthquake) under the structure we can calculate the response of the main structure (e.g.: nodal displacements, nodal accelerations). During the analysis if we have some machine or equipment on one of the structural floor the base foundation (supporting structure) of the equipment between the floor (level) and the equipment is usually not part of the structural analysis, but in reality it has a stiffness property which affects the response of the equipment.

The level spectra calculation will show the maximum acceleration response of the equipment during the ground acceleration (earthquake) as a function of time period of the equipment (and its supporting structure) which depends on *m *mass, *k *stiffness and *ξ *damping.

To calculate the level spectra it is necessary to define at least one *Storey *(or more) in the FEM-Design model. During the calculation first of all a time-history structural response analysis will be run in the background with the defined settings and ground accelerations (see *Structural response analysis*). One of the results of this calculation is the nodal (X,Y and Z directional) absolute accelerations. According to the defined storey(s) we will get the average values of this nodal acceleration vectors storey by storey. It means that we collect all the nodal acceleration vectors on one storey and calculate an average acceleration vector function storey by storey.

During the level spectra calculation these average storey absolute acceleration functions will be the basics of the level spectra calculation. Basically we will perform the same calculation as written at *Structural response analysis*, but instead of using the original ground acceleration functions we will use these average level acceleration response functions as excitation on the single degree of freedom system.

As by the original response spectra calculation, the level spectra calculation also has X, Y and Z component, because the level acceleration vectors have three different component.

The same input is required as with the method *Structural response analysis*, because a complete structural response analysis (only displacements and accelerations) is necessary to get the level responses. Therefore ground accelerations and ground acceleration combinations what were introduced above are necessary to define.

### Calculation request and settings

Similarly to *Structural response analysis*, just activate “Time history, ground acceleration” and setup additional parameters under *Setup:*

Unlike the *Structural response analysis*, select the *Level acceleration spectra* option, fill its parameters, and set *Damping factor*:

### Results

- Level pseudo acceleration response spectras to the storeys based on the adjusted ground accelerograms.
- A level spectra result contains three different diagrams, the global X, Y and Z directional results respectively.

## Forced vibration with arbitrary functions

### Input

In FEM-Design, we can define specific force multiplier functions in time by using the *Excitation force* command (*Loads *tab), and by defining excitation force diagrams.

These functions (at *Diagrams*) can have unique name and arbitrary time steps. If we have some table data about our multiplier functions we may export/import or copy them

We can also combine the multiplier functions by time steps and by load cases involved in the dynamic combination with given multiplication factors (*Combinations*). Every excitation force combinations may contain arbitrary number of load cases with the selected multiplier functions. During the dynamic calculation FEM-Design will apply in the specific time step the dynamic load according to the multiplication of the defined multiplication factors, the multiplier function values (at the actual time step) and the intensity of the selected loads in the adjusted load cases. One excitation force combination may contain different load cases with different multiplier functions which provides a wide range of opportunities to analyze several dynamic problems.

### Calculation request and settings

Similarly to other FEM-Design calculation requests, excitation force-based Time history analysis can be set in *Calculation *dialog.

Just activate “Time history, excitation force” and setup additional parameters under *Setup:*

*Calculation parameters*:*n result*– during the analysis (direct integration method) we should calculate the displacements and accelerations of the nodes in every time step to get proper solution, but to save calculation time and hard drive space at the end of the calculation as results only every*n-*th time step result will be saved and available.*t end*[s] – this time moment will be the last during the calculation. It is useful because it could happen that one of the involved dynamic force on the structure stops and others keep continuing. Thus if we applied a shorter multiplier function compared to others and the end time moment is longer than the shorter multiplier functions on the remaining time interval of the calculation that specific load case will be involved with zero multiplier and only the relevant ones will be considered.

- Choose an
*Integration scheme*method:*Newmark*(default) or*Wilson-Theta*direct integration method. - At
*Rayleigh damping matrix*, we should set the damping parameters (*Alpha*and*Beta*) that determine the damping behavior of our 3D structure, and based on the relevant eigenfrequencies of the structure and its critical damping ratios. - (
*Damping factor*is not considered in excitation force-based calculations.)

### Results

- Static displacements (translations or rotations) in the selected time step considering the static loads from the selected excitation force function combination in the selected time step. The envelope absolute maximum also available.
- Dynamic displacements (translations or rotations) in the selected time step considering the dynamic effects from the selected excitation force function combination. The envelope absolute maximum also available.
- Detailed result diagram of the selected nodal point about the nodal displacements (translations or rotations) as a function of time. Both static and dynamic displacements.
- Accelerations in the selected time step in the selected time step considering the static loads from the selected excitation force function combination in the selected time step. The envelope absolute maximum also available.
- Detailed result diagram of the selected nodal point about the nodal accelerations as a function of time.
- The dynamic reactions in the selected time step and their envelope absolute maximum, positive maximum and negative maximum.
- The internal forces in the structural elements in the selected time step and their envelope absolute maximum, positive maximum and negative maximum.
- Dynamic amplification factors in the selected time step and their envelope absolute maximum.
- Detailed result diagram of the selected nodal point about the nodal dynamic amplification factors as a function of time.
- Normalized dynamic amplification factors in the selected time step and their envelope absolute maximum.
- Detailed result diagram of the selected nodal point about the nodal normalized dynamic amplification factors as a function of time.