SeismoStruct: The effect of dampers

Athors: Barbier, Delobelle, Verhulst

This project is made by Jolien Barbier, Manon Delobelle and Yaron Verhulst.

In this wiki-page we will analyze the effect of a damper in structures subjected to earthquakes by the following sections:

Introduction

The goal of this project is to analyze the effect of dampers on buildings which are subjected to earthquakes. These earthquakes are vibrations that are the result of a huge release of energy in the earth's crust. Those vibrations cause excitations of buildings. This can cause damage to structures that can't resist those shocks. We can use dampers to reduce damage of buildings caused by earthquakes. Dampers are passive control systems. Those devices are smoothing out the vibrations by absorbing the energy. Because the dampers dissipate this energy, the building can resist more vibrations than before. Therefore the building will be less damaged after an earthquake.

This is a major advantage as the costs to repair the building after an earthquake will be reduced a lot. In most cases, only the dampers will need to be replaced after an earthquake. The dampers are an easy method to improve the resistance and safety of structures. This is very useful in existing buildings, so no huge renovations of foundations or connections are needed to improve them. 

A time history analysis is used to investigate the effect of dampers on the evolution of buildingresponse. The finite element model 'SeismoStruct' will be used to simulate the results.

There are different types of dampers: hysteretic, viscous, tuned mass and self-centering dampers. The hysteretic damper will be used in further calculations.

A lot of dampers are already used in several structures: buildings, bridges, ... In the two picture below, you can see applications of those dampers in practice:

Background of a damped seismic system

In this project a two-floor building will be used which can be simplified to a 3D-framework. A model of the structure is described in section 'SeismoStruct models'. To explain the basic principles, this MDOF system can be simplified to a SDOF framework with only one floor. This one floor-framework can be represented by a mass connected with an equivalent spring and damper which are connected to the ground. This SDOF system can be shown in following figure:

The floors are the main structural components which contribute to the mass of the system. The columns and the beams have certain stifness and damping parameters. The bending stiffness of the columns can be approximated with following formula:

k = 12*EI/L³ 

E represents the Young's Modulus of the column material, I the moment of inertia of the cross section and L the length of a column. Rigid connections between the floors and the columns are considered here. For this simplification the following equation of motion can be constructed:

In this equation u_g (t) is the ground displacement and u(t) the displacement of the floor relative to the ground. To find the displacement, this differential equation can be solved analytically by using the Duhamel integral. This results in:

In this equation ω_D is the damped eigenfrequency of the system. ζ is the damping ratio of the system which is related to the critical damping of the system. For the standard model without extra damping systems, this time history analysis can be done to find the response of the system. 

To improve the structural behaviour of the structure, vibration control systems can be introduced, such as external dampers. Dampers increase the energy dissipation in the building which results in a better behaviour. When external damping systems are added to the structure, a different approach can be used to determine the response of the structure. The equation of motion can be modified a bit to take into account these external dampers:

f(x,v) is the force generated in the beams and columns of the structure. In the vibration control system (external damper), a force F_VC is initiated. p(t) are the external forces acting on the structure caused by the earthquake. These forces can be related to a ground motion. The dampers used in this project can be approximated by hysteretic dampers. In this kind of damping, some of the energy involved in the repetitive internal deformation and restoration to original shape is dissipated in the form of random vibrations and random kinetic energy. This behaviour can be shown in following figure:

This non-linear behaviour can be simplified to an equivalent linearization. This can be done by determining an equivalent stiffness and damping in the equation of motion above:

This linearization can be visualized in the following figure:

The hysteretic behaviour represented by the blue striped area can be simplified by a linear behaviour represented by the red line. The effective stiffness can be determined by using the maximal forces and displacements in the damper, while the effective damping constant also uses the energy dissipated in the damper. This energy is equal to the area of the blue striped curve.

With all these aspects, the structural behaviour of a building can be analyzed using SeismoStruct.  

SeismoStruct models

First of all we need to construct steelstructures in SeismoStruct, before we can analyze any results. We created different models to investigate several effects of dampers on a structure. A model without damper is used as referencemodel to compare the results to models with damper. This analysis is done with a structure that is not fully symmetric, this allows us to analyze the effect of different orientations/positions of a damper in a building.

We created three models to investigate the effect of the location of the dampers:

The following parameters are assumed in the model:

There are three loads applied on the modelled steelstructure. First of all we have the selfweight of the elements which causes a gravityforce with an acceleration of 9.81 m/s². Additionally, some permanent loads are acting on the building due to for example furniture inside. We have choosen to take a general value of 50 kN. Finally there is the earthquake which causes nodal loads on the nodes at the base. Data of the earthquake TABAS which took place in Iran in 1978, is used in further calculations. 

Results

We will investigate different parameters: relative displacements, accelerations, support forces, support moments and damping forces. Each time we want to analyze a parameter, we start with analyzing the referencemodel. This way, we can search for the biggest problems due to the earthquake. Based on this, we will know in which direction we need to analyze the parameters in the other models with dampers. Additionally, if we look to the orientations of the damper in the models, we will know which direction will give the most significant differences: The damper is orientated in Model 1 in the xz-plane, so we will investigate those directions. In models 2 and 3, the dampers are orientated in the yz-plane. We have choosen to analyze the nodes close to the dampers, to detect the most significant differences between the several models.

Relative displacements

Firstly, the relative displacements of the referencemodel are analyzed. The two nodes above a base node will be compared to eachother, this will be done in all directions. We can see that the relative displacements in the x- and y-direction on both floors are much larger than the one in the z-direction: (n221 = node at the roof, n121 = node at the first floor) 

Secondly, the relative displacements of the different models will be compared to the relative displacements of the referencemodel.

The ratio of the relative displacement in the several models compared to reference model is given in the table between brackets (x%). Based on the results we can observe that a decrease of the relative displacement occurs almost everywhere. It seems that Model 2, where the brace is in the y-direction, is more efficient than the first model. Model 3 shows some negative results. The relative displacement of the first floor is increased compared to the reference model. Only the relative displacement of the roof is decreased, but the decrease is smaller than in the other 2 models. This is what we expect. If a displacement of a floor is decreased, the floors above will also have an effect of this decrease in displacement. The lower floors don't have this effect. If the damper is placed at the first level, the bottom part of the structure doesn't experience a large influence of the damper above which results in a lower decrease. In this case it's even an increase of the displacement. 

Acceleration

The acceleration will be investigated in the x- and y-direction, because the differences of accelerations in z-direction are quite small. The increase/decrease of the absolute value of the acceleration from the several models compared to the referencemodel can be founded between brackets in the table.

Based on these results some conclusions can be made. When adding a damper, the acceleration in the x-direction increases and in the y-direction it decreases. An earthquake has a certain response in time. With a Fast Fourier Transformation this time response can be transformed to an equivalent response in the frequency domain. Some frequencies can be more 'represented' than others, so if the eigenfrequency of the structure is close to these frequencies, a larger response is expected. Adding a damper with braces increases the damping, but also the stifness of the building. The eigenfrequency of a structure can be determined with following formula:

ω_n = sqrt(k/m)

If the stiffness k increases, the eigenfrequency increases. If this eigenfrequency gets closer to the dominated frequency of the earthquake, this can cause an increase in acceleration. But if the eigenfrequency gets further away from the dominated frequency of the earthquake, a decrease of acceleration is possible. 

In our case, the acceleration increases in the x-direction and decreases in Model 1 and 3 in the y-direction. In Model 2 the acceleration in y-direction increases, probably because of the fact that the eigenfrequency is different compared to the other models. The earthquake may have a different response in the frequency domain for the x- and the y-direction. So other frequencies may dominate between both directions which causes different behaviour in other directions.  

Total support forces and moments

Generally, the analysis shows us that the total support forces in the direction of the damper will increase by approximately 10%. This means that when de damper is the x-direction (Model 1) , the forces in the x-direction will increase. Analogous to this, when the damper is in the y-direction (Model 2 and 3), the forces in y-direction will increase.

As for the moments, the values will approximately stay the same.

Global damping forces

The damping forces in the z-direction will not be taken into account, since the damper mainly damps the horizontal forces. The structure originally has a large damping from itself in z-direction, but the external dampers don't have a lot of effect on the damping forces in this direction.

Additionally, we will again only consider the direction of the damper itself, just like we did in the section of 'Relative displacements'. We will do this, because the dampers don't have a large effect in the other direction.

We can see an increase of 55.9% on the global damping force for Model 2. Because of this, we can conclude that model 2 damps the most. Consequently, this model will absorb way more energy than the other two models. This is what we expected, based on the results of 'Relative displacements' and 'Accellerations'. 

Conclusion

From previous results, we can conclude that Model 2 is probably the best model. This model has one damper on the groundlevel, on the long side of the structure. Graphs show us that the relative displacements decrease a lot by using this damper. This is because of the fact that this damper has a high increase of damping forces, which is discussed in previous paragraph. Likewise, the large increase in accelerations in x- and y-direction shows us that this damper damps the most compared to other models.