## 1. Introduction

The comprehensive studies conducted by a number of researchers in the past few decades and investigations of the effects of past earthquakes have shown that in buildings with non-coincident the center of mass (CM) and the center of rigidity (CR), significant coupling may occur between the translational and the torsional displacements of the floor diaphragms even when the earthquake induces uniform rigid base translations (Kuo, 1974; Chandler & Hutchinson, 1986; Cruz & Chopra, 1986; Hejal & Chopra, 1989).

In investigating the seismic torsional response of structures to earthquakes, it is customary to assume that each point of the foundation of the structure is excited simultaneously. Under this assumption, if centers of mass and rigidity of the floor diaphragms lie along the same vertical axis, a horizontal component of ground shaking will induce only lateral or translational components of motion. On the other hand, if the centers of mass and rigidity do not coincide, a horizontal component of excitation will generally induce both lateral components of motion and a rotational component about a vertical axis. Structures for which the centers of mass and rigidity do not coincide will be referred to herein as eccentric structures. Torsional actions may also be induced in symmetric structures due to the fact that, even under a purely translational component of ground excitation, all points of the base of the structure are not excited simultaneously because of the finite speed of propagation of the ground excitation, (Kuo, 1974).

This seismic torsional response leads to increased displacement at the extremes of the torsionally asymmetric building systems and may cause suffering in the lateral load-resisting elements located at the edges, particularly in the systems that are torsionally flexible. More importantly, the seismic response of the systems, especially in the torsionally flexible structure is qualitatively different from that obtained in the case of static loading at the center of mass. To account for the possible amplification in torsion produced by seismic response and accidental torsion in the elastic range, the equivalent static eccentricities of seismic forces are usually defined by building codes with simple expressions of the static eccentricity. The equivalent static eccentricities of seismic forces are proposed by researchers, (Dempsey & Irvine, 1979, Tso & Dempsey, 1980 and De la Llera & Chopra, 1994). A clear and comprehensive study of the equivalent static eccentricities that are presented by Anastassiadis et al., (1998), included a set of formulas for a one-storey scheme, allow the evaluation of the exact additional eccentricities necessary to be obtained by means of static analysis the maximum displacements at both sides of the deck, or the maximum deck rotation, given by modal analysis. A procedure to extend the static torsional provisions of code to asymmetrical multi-storey buildings is presented by Moghadam and Tso, (2000). They have developed a refined method for determination of CM eccentricity and torsional radius for multi-storey buildings. However, the inelastic torsional response is less easily predictable, because the location of the center of rigidity on each floor cannot be determined readily and the equivalent static eccentricity varies storey by storey at each nonlinear static analysis step. The simultaneous presence of two orthogonal seismic components or the contemporary eccentricity in two orthogonal directions may have some importance, mainly in the inelastic range, (Fajfar et al., 2005). Consequently, the static analysis with the equivalent static eccentricities can be effective only if used in the elastic range. This can only be achieved, the location of the static eccentricity is necessary to change in each step of the nonlinear static procedure. It may be needed for the development of simplified nonlinear assessment methods based on pushover analysis.

However, the seismic torsional response of asymmetric buildings in the inelastic range is very complex. The inelastic response of eccentric systems only has been investigated in an exploratory manner, and, on the whole, it has not been possible to derive any general conclusions from the data that were obtained. No work appears to have been reported concerning the torsiona1 effects induced in symmetric structures deforming into the inelastic range (Tanabashi, 1960; Koh et al., 1969; Fajfar et al., 2005).

Torsional motion is produced by the eccentricity existing between the center of mass and the center of rigidity. Some of the situations that can give rise to this situation in the building plan are:

Positioning the stiff elements asymmetrically with respect to the center of gravity of the floor.

The placement of large masses asymmetrically with respect to stiffness.

A combination of the two situations described above.

Consequently, torsional-translational motion has been the cause of major damage to buildings vibrated by strong earthquakes, ranging from visible distortion of the structure to structural collapse (see Fig. 1). The purpose of this chapter is to investigate the torsional vibration of both symmetric and eccentric one-storey building systems subjected to the ground excitation.

## 2. Classification of vibration

Vibration can be classified in several ways. Some of the important classifications are as follows: Free and forced vibration: If a system, after an internal disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration. No external force acts on the system. The oscillation of the simple pendulum is an example of free vibration.

If a system is subjected to an external force (often, a dynamic force), the resulting vibration is known as forced vibration. The oscillation that arises in buildings such as earthquake is an example of forced vibration.

A building, for which the centers of mass and rigidity do not coincide, (eccentric building) will experience a coupled torsional-translational motion even when it is excited by a purely translational motion of the ground. The torsional component of response may contribute significantly to the overall response of the building, particularly when the uncoupled torsional and translational frequencies of the system are close to each other (see Fig. 2).

Failures of such structures as buildings and bridges have been associated with the torsional-translational motion.

### 2.1. Free vibration analysis

One of the most important parameters associated with engineering vibration is the natural frequency. Each structure has its own natural frequency for a series of different mode shapes such as translational and torsional modes which control its dynamic behaviour (see Fig. 3). This will cause the structures to be subjected to series structural vibrations, when they are located in environments where earthquakes or high winds exist. These vibrations may lead to serious structural damage and potential structural failure.

In buildings, both translational and torsional vibration modes arise, even if, little eccentricity in the transverse direction during earthquakes. The in-plane floor vibration mode such as arch-shaped floor vibration mode also arises during earthquakes. However these observational data are not enough at present. The causes of the torsional-translational vibration are thought as follows:

Input motion to the foundation has a possibility to contain the torsional component, which is the cause of the torsional vibration.

The torsional coupling, due to the eccentricity in both directions, is also a cause of the torsional vibration. It arises surely when the eccentricity in the transverse direction is large. However, even if the eccentricity is small, it is well-known that the strong torsional coupling also arises when the natural frequencies of the translational mode and the torsional mode approach closely to each other.

The eccentricity in the transverse direction is small in general, since sufficient attention is usually paid on the eccentricity to prevent the torsional vibration in the structural planning. On the other hand, the eccentricity in the longitudinal direction results often from necessity of architectural planning and/or from insufficiency of attention on the eccentricity in the structural planning, but it is also small as a necessity from the configuration of the floor plan.

#### 2.1.1. One-storey system with double eccentricities

The estimation of torsional-translational response of simplified procedure subjected to a strong ground motion, is a key issue for the rational seismic design of new buildings and the seismic evaluation of exacting buildings. This section is a vibration-based analysis of the simple one-storey model with double eccentricities, and it would be a promising candidate as long as buildings oscillate predominantly in the two lateral directions (Tabatabaei and Saffari, 2010).

#### 2.1.2. Basic parameters of the model

The one-storey system, considered in this section, may be modeled as shown in Fig. 4. The center of rigidity (CR) is the point in the plan of the rigid floor diaphragm through which a lateral force must be applied in order that it may cause translational displacement without torsional rotation. When a system is subjected to forces, which will cause pure rotation, the rotation takes place around the center of rigidity, which remains fixed. The location of the center of rigidity can be determined from elementary principles of mechanic.

The horizontal rigid floor diaphragm is constrained in the two lateral directions by resisting elements (columns). Let

Applying the geometric relationships between the centers of mass and rigidity, the equations of motion of undamped free vibration of the system may be written as follows

where

For free vibration analysis, the solution of Eqs. (1) may be taken in the form

where

Eqs. (3) have a nontrivial solution only if the determinate of the coefficients of

where,

and making use of the relation

where the values of

where

As a matter of fact, the numerical results have been evaluated over a wide range of the frequency ratio

In these Figures, the uncoupled natural frequencies of the systems are represented by the straight lines corresponding to

The curve for the lowest frequency always starts from the origin whereas the curve for the highest frequency starts from a value higher than the uncoupled translational frequencies of the system, depending on the value of the eccentricity. Both curves increase with the higher value of

It is interesting to note that the coupled dynamic properties depend only on the four dimension 1ess parameters

If

The ratio of

For the one-storey system with eccentricity considered in Fig. 4, the locus of the associated center of rotation is plotted in Fig. 9. It should be recalled that a value of

#### 2.1.3. One-storey system with single eccentricity

In the particularly case of

Numerical data have been evaluated over a wide range of the frequency ratio

As a result, Figs. 5 and 10, which refer to systems with double and single eccentricities, respectively, are very similar in form. This is due to the fact that, since

#### 2.1.4. Classification of torsionl behaviour

To obtain the knowledge on the torsional response of buildings, a key elastic parameter is the ratio of the two uncoupled frequencies,

where

If

### 2.2. Forced vibration analysis

Civil engineering structures are always designed to carry their own dead weight, superimposed loads and environmental loads such as wind or earthquake. These loads are usually treated as maximum loads not varying with time and hence as static loads. In some cases, the applied load involves not only static components, but also contains a component varying with time which is a dynamic load. In the past, the effects of dynamic loading have often been evaluated by use of an equivalent static load, or by an impact factor, or by a modification of the factor of safety.

Many developments have been carried out in order to try to quantify the effects produced by dynamic loading. Examples of structures, where it is particularly important to consider dynamic loading effects, are the construction of tall buildings, long bridges under wind-loading conditions, and buildings in earthquake zones, etc.

Typical situations, where it is necessary to consider more precisely, the response produced by dynamic loading are vibrations due to earthquakes. So it is very important to study the dynamic nature of structures.

Dynamic characteristics of damaged and undamaged buildings are, as a rule, different. This difference is caused by a change in stiffness and can be used for the detection of damage and for the determination of some damage parameters such as crack magnitude and location. In this connection, the use of vibration methods of damage diagnostics is promising. These methods are based on the relationships between the vibration characteristics (natural frequencies and mode shapes) or peculiarities of vibration system behaviour (for example, drift of building edges, the amplitudes of base shear, the resonance frequencies, etc.) and damage parameters.

Depending on the assumptions adopted, the type of analysis used the kind of the loading or excitation and the overall building characteristics, a variety of different approaches have been reported in the references to one-storey building systems.

#### 2.2.1. One-storey system with double eccentricities

In this section, the dynamic response of one-storey system to the horizontal components of ground motion is considered. At any instant of time, the ground motion is assumed to be the same at all points on the foundation. Also, the ground motion is considered to be plane shear waves propagating horizontally with a constant velocity and without change in shape.

Now, equations of motion are developed for system as shown in Fig. 11 subjected to a horizontally propagating ground excitation. Let the earthquake ground motion be defined by accelerations along the two axes. Therefore, a force applied along either of the two principal axes of rigidity will cause displacement in the same direction. The principal axes of rigidity are orthogonal and pass through the center of rigidity. For building plans of the type shown in Fig. 11, where principal axes of rigidity of individual column sections are all parallel to one another, the principal axes of rigidity of the complete building are parallel to those of the individual elements. Within the range of linear behaviour, the equations of motion of the system, written about the center of rigidity of the system, are as follows

where,

It is important to note that time is measured from the instant the ground motion reaches the first support on the left of the diagram shown in Fig. 11 and that the amplitude of the ground motion at any support is a function of both time and distance from the first, i.e.,

where

Since it is customary to specify the ground motion in terms of its acceleration, Eqs. (11) will be rewritten by making a proper coordinate transformation. Assuming that the damping coefficients are proportional to the corresponding stiffness coefficients, and letting

where

Substituting Eqs. (13) in Eqs. (11) and writing those equations, after introducing damping, in a convenient form yields

where

Eqs. (14) has been expressed in the most general form. It is applicable to both symmetric and eccentric systems subjected to a ground disturbance having a finite speed of propagation. It may be reduced to the equations derived from a conventional analysis in which the speed of propagation of the ground motion is assumed to be infinite.

#### 2.2.2. Time history response

The modal equations (14) can be solved numerically using the modal superposition method or by direct numerical integration. The modal superposition method, which involves the use of the characteristic values and functions of the system, uncouples the equations of motion so that each of the uncoupled equations may be integrated independently. Since it is based on the assumption that the structure behaves linearly, this method is applicable only to the elastic range of response. These issues have been further discussed in Ref., (Kuo, 1974).

The method of direct numerical integration, which integrates the equations of motion in their original form, may be applied to both the elastic and inelastic ranges of response. For the inelastic analysis, the material properties of the system are assumed to be linear within a small time interval, and they are modified at the end of each integration step when needed.

The modal superposition method is used to evaluate the elastic response of both symmetric and eccentric systems. Before solving Eqs. (14), it is desirable to rewrite it in matrix form. For consistency in units, the third equation is multiplied by

where the uncoupled circular natural frequencies

Symbolically, Eq. (15) may also be written as

Now, let

Substituting Eq. (18) into Eq. (17), multiplying the resulting equation by

where * n = 1,2,3*;

It should be recalled that the damping coefficients are assumed to be proportional to the stiffness coefficients. For specified

There are several different methods available for integrating numerically equations of the form of Eq. (19). One of the procedures is the linear acceleration method, which is simple but sufficiently accurate for all practical purposes if the integration increment is chosen suitably.

In this method, the acceleration is assumed to vary linearly with time,

where

with the acceleration

For specified initial values of

with

Once the values of

where

Note that the terms

#### 2.2.3. Equations for symmetric systems

The equations are applicable to both symmetric and eccentric systems. A symmetric system may experience torsional response even under the influence of a translational motion of the ground. This fact, which appears to have been first investigated by Newmark, can be seen clearly from Eqs. (14). In Newmark's approach, the rotational component of the response is first evaluated, and then it is combined with the translational component determined in the usual way. The method of combining the two components was not specified, (Newmark, 1969).

The method used herein consists of using Eqs. (14) with the

Note that these equations are independent of one another. Accordingly, each unknown coordinate may be evaluated independently by the method described in the preceding section and the column deformations computed by use of Eqs. (25).

If the speed of propagation of the ground motion were infinite, as is normally assumed, the right-hand members of the three equations would be, respectively, * 0*, and no torsional response could develop. It is also interesting to note that, if only a single component of ground shaking is considered, say

Now, assume that the system has the same number of columns in the x and y directions and that the total stiffness of the system in either direction is equally distributed among the columns. The Newmark approach may be determined by neglecting the damping term in the Eq. (27c). For a very small transit time,

#### 2.2.4. Spectral response

In the time history method, there have been two major disadvantages in the use of this approach. First, the method produces a large amount of output information that can require a significant amount of computational efforts to conduct all possible design checks, as a function of time. Second, the analysis must be repeated for several different earthquake motions in order to assure that all frequencies are excited, since a response spectrum for one earthquake in a specified direction is not a smooth function.

There are computational advantages in using the spectral response method of seismic analysis for prediction of displacements and member forces in structural systems. The method involves the calculation of only the maximum values of the displacements and member forces in each mode using smooth design spectrum that are the average of several ground motions.

If the ground motions in both x and y directions are characterized by the same response spectrum, then the maximum value of the modal response function (19) may be rewritten in the following form

In which

Finally, the most conservative method that is used to estimate a peak value of displacement or force within a structure is to use the sum of the absolute of the modal response values. This approach assumes that the maximum modal values, for all modes, occur at the same point in time. The relatively new method of modal combination is the Complete Quadratic Combination, CQC, method that was first published in 1981, (Wilson et al. 1981).

Now, the dynamic response of one-storey building model to the horizontal components of ground motion along x and y axes are investigated. For the objectives of this study it is considered the most appropriate to characterize ground motion by its response spectrum. The numerical results presented are for two idealized response spectrum (see Fig. 12), flat (or period independent) acceleration spectrum and hyperbolic acceleration spectrum (or flat velocity spectrum). The chosen spectrums have the advantage that the normalized response of the system does not depend on the periods of vibration, but only on their ratios.

For these two idealized spectrums, the normalized response does not depend on

Those for the case of hyperbolic spectrum are presented in Fig. 14. The maximum displacements of both flexible and stiff edges are calculated by modal superposition method, using complete CQC method. These values are then normalized by

Fig. 14 Normalized displacements of the stiff and flexible edges; hyperbolic spectrum.

The normalized flexible edge displacement

It is also of interest to note that resonance between uncoupled translational and torsional frequencies, i.e., when

Frequency resonance is not, therefore, a critical issue. Plots of normalized stiff edge displacement are shown in Fig. 13, again for different values of eccentricity

## 3. Torsional provisions in seismic codes as applied one-storey buildings

Most seismic building codes Formulate the design torsional moment at each storey as a product of the storey shear and a quantity termed design eccentricity. These provisions usually specify values of design eccentricities that are related to the static eccentricity between the center of rigidity and the center of mass. The earthquake-induced shears are applied through points located at the design eccentricities. A static analysis of the structure for such shears provides the design forces in the various elements of the structure. In some codes the design eccentricities include a multiplier on the static eccentricity to account for possible dynamic amplification of the torsion. The design eccentricities also include an allowance for accidental torsion. Such torsion is supposed to be induced by the rotational component of the ground motion and by possible deviation of the centers of rigidity and mass from their calculated positions. The design eccentricity formulae, given in building codes, can be written in two following parts:

The first part is expressed as some magnification factor times the structural eccentricity. This part deals with the complex nature of torsion and the effect of the simultaneous action of the two horizontal ground disturbances.

The second term is called accidental eccentricity to account for the possible additional torsion arising from variations in the estimates of the relative rigidities, uncertain estimates of dead and live loads at the floor levels, addition of wall panels and partitions; after completion of the building, variation of the stiffness with time and, Inelastic or plastic action. The effects of possible torsional motion of the ground are also considered to be included in this term. This terms in general a function of the plan dimension of the building in the direction of the computed eccentricity.

In Iranian code, in case of structures with rigid floors in their own plan, an additional accidental eccentricity is introduced through the effects generated by the uncertainties associate with the distribution of the mass level and/or the spatial variation of the ground seismic movement, (Iranian code 2800, 2005). This is considered for each design direction and for each level and also is related to the center of mass. The accidental eccentricity is computed with the relationship

where * i*from its nominal location, applied in the same direction at all levels;

where

## 4. Conclusion

A study of free vibration characteristics of an eccentric one-storey structural model is presented. It is shown in the previous sections that the significance of the coupling effect of an eccentric system depends on the magnitude of the eccentricity between the centers of mass and of rigidity and the relative values of the uncoupled torsional and translational frequencies of the same system without taking the eccentricity into account. The coupling effect for a given eccentricity is the greatest when the uncoupled torsional frequency,

In addition, it is shown that the locus of the associated center of rotation can be formulated corresponding for a given eccentricity. Note that, for all values of eccentricity, as the value of the uncoupled natural frequencies ratio increases the center of rotation shifts away from the center of rigidity for the first mode and approaches the center of mass for the higher mode. It is also shown that, the torsional behaviour of the model assembled, using our approach, can be classified based on the nature of the instantaneous center of rotation.

It is well known that asymmetric or torsionally unbalanced buildings are vulnerable to damage during an earthquake. Resisting elements in such buildings could experience large displacements and distress. With eccentricity defined for one-storey buildings, the torsional provisions or building codes can then be applied for a seismic design or such structures.

## Acknowledgments

The author gratefully acknowledges the financial support provided by the Office of Vice Chancellor for Research of Islamic Azad University, Kerman Branch.