An approach for minimum rotational response of medium-rise asymmetric structures under seismic excitations

Abstract

Studied in this article is the elastic and inelastic rotational behavior of asymmetric multi-story buildings under unidirectional and bidirectional excitations in relation to the location of the modal center of rigidity. The concept of this point in mono-symmetric uniform buildings has been outlined by the author in earlier papers, and its significance lies on the property that when its location is within a close distance from the mass axis the torsional response is mitigated. At present, a generalized definition of this point is given for buildings with double eccentricity, using (1) the approach of the continuum medium and (2) the stiffness matrix methodology. To determine the location of modal center of rigidity in practical terms, a few modal quantities are required (element frequencies and effective modal masses), which may be easily obtained by commercial engineering software. To verify the aforesaid property of modal center of rigidity under unidirectional and bidirectional excitations, the rotational response of 10-story wall-frame structures is examined under the ground motions of Erzincan 1992 and Kobe 1995.

Keywords

asymmetric buildings earthquake engineering inelastic structures modal center of rigidity

Introduction

The issue of predicting the seismic rotational response of asymmetric buildings has been the objective of many research projects. Recognizing that structural asymmetry may be the major reason for poor performance after a severe ground excitation, researchers have paid much effort of how to control and, even more, minimize this response. Starting from the response of multi-story linear systems, it has been shown that for a special class of asymmetric buildings having resisting elements with stiffness matrices which are proportional to each other (proportionate buildings) and in which the centers of mass (CMs) of all floors lie on the same vertical line, the seismic behavior can be accurately assessed by analyzing the total responses of two simpler systems: (1) the response of the corresponding uncoupled multi-story structure and (2) for each mode of vibration of the latter structure, by analyzing an associated torsionally coupled single-story system (Athanatopoulou et al., 2006; Hejal and Chopra, 1989; Kan and Chopra, 1977a). The same analysis can also be applied in shear-type buildings with different static eccentricities at the various floor levels (Kan and Chopra, 1977b). Recently, this analysis was extended by the author to non-proportionate buildings (structural systems composed by different types of bents: shear walls, moment resisting frames, coupled walls, etc.), by introducing the concept of the modal stiffness of the bents which provide the lateral stiffness of a given structure. It is shown (Georgoussis, 2010, 2012) that the peak elastic response of medium-height buildings can be derived by analyzing two equivalent single-story modal systems, each of which has a mass equal to the k-mode effective mass, $M_{k}^{*}$ (k = 1, 2), of the uncoupled multi-story structure, and is supported by elements with stiffnesses equal to the product of $M_{k}^{*}$ with the first mode (when k = 1) or second mode (when k = 2) squared frequencies of the corresponding real bents (element frequencies) of the assumed multi-story structure. These element frequencies are simply evaluated from the corresponding bents when they are assumed to carry, as planar frames, the mass of the complete structure. The formulation presented in the aforementioned author’s papers applies to mono-symmetric buildings and it was shown that in structures, in which the fundamental period of the uncoupled structure is in the acceleration sensitive region, the first mode (k = 1) equivalent single-story system is sufficient to provide reasonable estimates of the peak values of base resultant forces. The stiffness center of this modal system constitutes the first modal center of rigidity (m-CR) and when this point lies close to the axis passing through the centers of floor masses, the rotational response sustained by an elastic asymmetric building system is minimum (Georgoussis, 2009). In other words, the closeness of this point with the mass axis implies a reduced rotational response and the vertical axis which passes through m-CR may be regarded as a reference axis; in the sense that, when this axis coincides with the mass axis, an almost translational response is obtained in the event of a ground motion.

In the majority of tall buildings, the lateral and torsional displacements are coupled and in strict terms there is not always an “elastic axis” with properties similar to those of the stiffness center of single-story systems. This has led to many investigations about the issue of establishing a set of points located at the floor levels of a multi-story building with properties similar to those of the elastic axis. Since the oscillatory response of single-story systems is due to the distance between CM and CR, usually referred to as static eccentricity, early studies on elastic multi-story systems (e.g. Humar, 1984; Jiang et al., 1993; Poole, 1977; Smith and Vezina, 1985) have led to different definitions about the magnitude of this eccentricity at the various floor levels. Cheung and Tso (1986) proposed the “rigidity centers (CRs)” as a reference axis for structural applications. These are the points that when a given distribution of lateral loading passes through them, only translational movement of the floors will occur. However, apart from the proportionate structures, these points are load-dependent and their space distribution is very irregular, even in uniform structures composed of different types of bents. More recently, Makarios and Anastassiadis (1998a, 1998b) introduced the “axis of optimum torsion” as a reference axis with promising results (Makarios, 2005, 2008; Makarios et al., 2006). This axis can be determined by means of an indirect static analysis by applying a set of floor torques equal in magnitude to the lateral forces at the same floors. An alternative mathematical procedure was proposed by Marino and Rossi (2004).

Recently, several researches have focused on the rotational response of multi-story asymmetric structures composed by inelastic bents (qualitative reviews have been presented by De Stefano and Pintucchi (2008) and Anagnostopoulos et al. (2013)). An alternative strategy for controlling this response in multi-story structures designed to withstand ground motions into the inelastic region is presented by Aziminejad et al. (2008) and Aziminejad and Moghadam (2009). In their studies, the problem of element strength distribution on the rotational response of the structure is studied using a proper configuration of the centers of mass, strength, and stiffness according to the findings obtained from single-story systems with elements having strength-dependent stiffness (Myslimaj and Tso, 2002, 2004). Interesting results are also highlighted by Lucchini et al. (2008) concerning the response of shear-type three-story inelastic buildings under strong ground motions and, also, by Stathopoulos and Anagnostopoulos (2005).

The first objective of this study is to provide a generalized formulation of the aforementioned author’s methodology (Georgoussis, 2009, 2010, 2012) applicable to linear multi-story systems with double eccentricity. The backbone of this formulation is outlined in section “Basic mathematical model,” by means of the approximate continuum method, because of the advantage of this method to provide the response of building structural systems in a parametric form (Georgoussis, 2008). Besides, as the aim of the methodology is to demonstrate the conditions under which a multi-story building may respond in a practically translational mode, the location of the center of stiffness of the equivalent single-story system (m-CR point) is also given in terms of data that can be obtained from commercial engineering software.

The second objective is to demonstrate that the location of m-CR can also be derived using the discrete element approach (stiffness method). The procedure proposed in the past (Cheung and Tso, 1986; Goel and Chopra, 1993) for the determination of the CRs at the floor levels of a multi-story building is used for this purpose. This procedure, in section “An alternative method to define the location of the first m-CR by using the discrete element approach,” is focused on the first modal response, which is the dominant mode of vibration in low- or medium-height building structures.

For structural applications, the equation of motion of the first mode (k = 1) equivalent single-story system, which provides the basic dynamic properties (frequencies, resultant base shears, and torques) of low- or medium-height buildings, is presented in brief in section “The equation of motion of the equivalent single-story modal system—modifications for systems composed by dissimilar bents,” in terms again of modal data (first mode effective modal mass of the uncoupled structure and element frequencies) that may be easily obtained by standard commercial engineering software. For systems composed by very dissimilar bents, a minor modification of the aforesaid formulation is also presented in this section. In such cases, the use of the effective element frequencies is recommended (Georgoussis, 2014; Georgoussis et al., 2013a), instead of the element frequencies. This is demonstrated in the multi-story elastic models examined in section “Systems analyzed under unidirectional and bidirectional excitations,” which are composed by dissimilar types of resisting bents. In other words, the structural configuration of minimum torsion is best predicted when the m-CR is defined on the grounds of the effective element frequencies rather than on the element frequencies.

The third objective of the article is to show that the response of systems composed by inelastic bents is virtually translational in the post-elastic phase when (1) the bent strength assignment has been derived from a planar static analysis under a code lateral loading and (2) the mass axis passes through (or in a close distance from) m-CR. This behavior has already been demonstrated in the special case of mono-symmetric multi-story systems with inelastic bents under unidirectional excitations (Georgoussis, 2014; Georgoussis et al., 2013b), and it is explained as follows: when the elastic behavior is practically translational, the effective seismic forces developed on a medium- or low-height structure are basically proportional to the first translational mode of vibration. Therefore, a strength assignment obtained from a planar static analysis under a set of lateral loads simulating the aforesaid mode of vibration represents a system in which all potential plastic hinges at the critical sections (at the ends of the beams and the foot of the ground floor columns and walls, according to the strong column-weak beam concept) are formed at the same instant. In other words, the almost concurrent yielding of all the lateral load–resisting bents preserves the translational response, attained at the end of the elastic phase, to the post-elastic one. At present, this behavior is examined in a broader field, that is, in similar structural systems under strong bidirectional excitations. Here, it should be mentioned that to apply the aforesaid methodology in real structures is not always an easy task. This is because the locations and the size of the lateral load–resisting elements are sometimes predetermined by architectural or aesthetic considerations and, also, because other aspects may have an impact on the overall design, as for example the presence and the magnitude of the gravity loads, which add some complexity to the problem. Note here that the concept of this approach (that the coincidence of m-CR with the mass axis implies a response of minimum rotational behavior) originates from the response of eccentric single-story systems. Such systems, with coincident the centers of mass and rigidity and elastoplastic elements having a strength distribution proportional to the stiffness distribution (derived, for example, by a planar static analysis), present a purely translational inelastic response under strong ground excitations. For this reason, they are used as “reference” models in relevant studies (e.g. Chandler et al., 1996; Correnza et al., 1994; Wong and Tso, 1994). This behavior is attained because yielding is initiated at the same instant in all elements and the element force balance about CM is preserved into the inelastic phase, leading to a translational response throughout the ground shaking.

The model examined is a common dual building structure with simple eccentricity, composed by two shear walls and a moment resisting frame in the direction of asymmetry. Various configurations of this model building have been analyzed: the two walls were assumed at fixed locations, while the frame was assumed to take any possible position along the axis of symmetry. The element strength distribution was derived by a simple planar static analysis under a code lateral loading. The initial stiffness of the various elements was considered independent of the assigned strength, as recommended by the current building codes. Undoubtedly, in reinforced concrete walls, the element stiffness is strength-dependent (Paulay, 2001; Priestley et al., 2007), but this issue is beyond the scope of this work. Note here that modern building codes (e.g. Eurocode EC8-2004, clause 4.3.1(5)) recommend such a structural detailing when the regularity in plan criteria is satisfied. Although all configurations thus constructed satisfy the regularity criteria of the aforesaid code, their response under strong excitations is quite different. The analyses were performed by the accurate SAP2000 computer program and the assumed excitations are the ground motions of Erzincan 1992 (components NS and EW) and Kobe 1995 (components KJM/000 and 090), selected from the strong ground motion database of the Pacific Earthquake Engineering Research (PEER) Center. For comparison reasons and also in order to clarify the significance of having the location of m-CR as close as possible to the mass axis, the response of the aforesaid structural configurations is presented in the elastic phase and also when the inelastic systems are subjected to just one component of the assumed ground motions.

Basic mathematical model

To study the coupled translational–rotational response of a uniform multi-story building with a plan-asymmetric configuration, consider the structure shown in Figure 1(a) and (b). For simplicity, the resisting elements are aligned in two orthogonal directions and the lateral resistance, in both directions, is provided by different types of bents: shear walls, rigid frames, assemblies of wall and frames, and, also, core elements. The floors of the building are assumed to act as rigid diaphragms in their own planes (flexible out of plane, but infinitely rigid in plane so that each floor undergoes a rigid body movement in plan) and all bents have a constant cross section over the height of the building, so that the structure can be treated as a continuous medium and the governing equations of motion can be expressed by differential equations, as presented below.

Figure 1.

(a) Vertical section of a uniform multi-story building, (b) asymmetric floor plan configuration, and (c) typical symmetric counterpart plan configuration.

In planar bents of low or medium height, in which the axial deformations of the vertical members can be neglected and have no effect on the their stiffnesses, the equation governing the static equilibrium of any i-bent, aligned in the x-direction, at the height, z, is given by

w_{xi} = E I_{xi} u_{i}^{iv} - G A_{xi} u_{i}^{ii}

(1)

where w_xi and u_xi are the intensity of the distributed lateral loading and the deflection of the bent at height z, respectively, EI_xi is the sum of flexural rigidities of columns or walls, and GA_xi is the racking shear rigidity of the bent as developed by the horizontal or diagonal members connecting the vertical components of the bent (Smith et al., 1981). Note here that bents, in which the axial deformations in columns or walls are significant and should be taken into account (e.g. tall frames, coupled walls), can be treated with equivalent shear rigidity (Georgoussis, 2006). Similar is the equation of equilibrium of the j-bent, aligned in they-direction

w_{yj} = E I_{yj} v_{j}^{iv} - G A_{yj} v_{j}^{ii}

(2)

Similarly, for a c-core element, the equation of equilibrium in a matrix form is

\begin{matrix} {\begin{matrix} w_{x} \\ w_{y} \\ t \end{matrix}}_{c} = {[\begin{matrix} E I_{x} & 0 & 0 \\ 0 & E I_{y} & 0 \\ 0 & 0 & E I_{w} \end{matrix}]}_{c} {\begin{matrix} u \\ v \\ θ \end{matrix}}_{c}^{iv} \\ - {[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & G J_{d} \end{matrix}]}_{c} {\begin{matrix} u \\ v \\ θ \end{matrix}}_{c}^{ii} \end{matrix}

(3)

where EI_wc and GJ_dc are the warping and St Venant torsional stiffnesses, respectively (Heidebrecht and Stafford Smith, 1973a); $W_{c} = 〈 w_{x} w_{y} t 〉_{c}^{T}$ and $U_{c} = 〈 u v θ 〉_{c}^{T}$ are, respectively, the load and displacement vectors in the principal local coordination system O_cx_cy_cz_c of the particular element. Equation (3) is a general expression and both the differential equations (1) and (2) can be set in the form of equation (3). For example, equation (1) reflects the first row of the set of equations (3) when the first terms of the main diagonals of the stiffness matrices above are replaced by EI_xi and GA_xi, respectively. Assuming that the global coordination system Oxyz has the vertical axis, z, passing through the CMs of the various floors, the undamped equation of motion of free vibration of the complete structure (Heidebrecht, 1975; Ng and Kuang, 2000) takes the form

\begin{matrix} m [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ρ^{2} \end{matrix}] {\begin{matrix} \overset{\cdot\cdot}{u} \\ \overset{\cdot\cdot}{v} \\ \overset{\cdot\cdot}{θ} \end{matrix}} + [\begin{matrix} E I_{x} & 0 & - y_{m} E I_{x} \\ 0 & E I_{y} & x_{m} E I_{y} \\ - y_{m} E I_{x} & x_{m} E I_{y} & E I_{w} \end{matrix}] \\ {\begin{matrix} u \\ v \\ θ \end{matrix}}^{iv} - [\begin{matrix} G A_{x} & 0 & - y_{s} G A_{x} \\ 0 & G A_{y} & x_{s} G A_{y} \\ - y_{s} G A_{x} & x_{s} G A_{y} & G J_{w} \end{matrix}] {\begin{matrix} u \\ v \\ θ \end{matrix}}^{ii} = 0 \end{matrix}

(4)

where $U = 〈 u v θ 〉^{T}$ is the displacement vector in the global coordination system (Oxyz); m and ρ are the mass per unit height of the structure and its radius of gyration about CM, respectively, and

\begin{matrix} E I_{x} = Σ E I_{xi} + Σ E I_{xc}, E I_{y} = Σ E I_{yj} + Σ E I_{yc} \\ E I_{w} = Σ y_{i}^{2} E I_{xi} + Σ x_{j}^{2} E I_{yj} + Σ y_{c}^{2} E I_{xc} + Σ x_{c}^{2} E I_{yc} + Σ E I_{wc} \\ y_{m} = \frac{Σ y_{i} E I_{xi} + Σ y_{c} E I_{xc}}{E I_{x}}, x_{m} = \frac{Σ x_{j} E I_{yj} + Σ x_{c} E I_{yc}}{E I_{y}} \end{matrix}

(5)

\begin{matrix} G A_{x} = Σ G A_{xi}, G A_{y} = Σ G A_{yj} \\ y_{s} = \frac{Σ y_{i} G A_{xi}}{G A_{x}}, x_{s} = \frac{Σ x_{j} G A_{yj}}{G A_{y}} \\ G J_{w} = Σ y_{i}^{2} G A_{xi} + Σ x_{j}^{2} G A_{yj} + Σ G J_{dc} \end{matrix}

It is interesting to mention here that the pair of coordinates x_m, y_m specifies the location of the center of the flexural rigidities (CF), while the pair x_s, y_s specifies the center of shear rigidities (CS). A more convenient form of equation (4) for the evaluation of the coupled frequencies is

\begin{matrix} \frac{m}{E I_{x}} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {\bar{ρ}}^{2} \end{matrix}] {\begin{matrix} \overset{\cdot\cdot}{u} \\ \overset{\cdot\cdot}{v} \\ r \overset{\cdot\cdot}{θ} \end{matrix}} \\ + [\begin{matrix} 1 & 0 & - {\bar{y}}_{m} \\ 0 & c_{m} & {\bar{x}}_{m} c_{m} \\ - {\bar{y}}_{m} & {\bar{x}}_{m} c_{m} & 1 \end{matrix}] {\begin{matrix} u \\ v \\ r θ \end{matrix}}^{iv} \\ - α_{x}^{2} [\begin{matrix} 1 & 0 & - {\bar{y}}_{s} \\ 0 & c_{s} & {\bar{x}}_{s} c_{s} \\ - {\bar{y}}_{s} & {\bar{x}}_{s} c_{s} & {\bar{r}}_{g}^{2} \end{matrix}] {\begin{matrix} u \\ v \\ r θ \end{matrix}}^{ii} = 0 \end{matrix}

(6)

where

$c_{m} = E I_{y} / E I_{x}$ , $c_{s} = G A_{y} / G A_{x}$ , $r^{2} = E I_{w} / E I_{x}$ , $r_{g}^{2} = G J_{w} / G A_{x}$ , $α_{x}^{2} = G A_{x} / E I_{x}$ , ${\bar{y}}_{m} = y_{m} / r$ , ${\bar{x}}_{m} = x_{m} / r$ , ${\bar{r}}_{g} = r_{g} / r$ , and $\bar{ρ} = ρ / r$ .

The exact solution of equation (6) can be obtained by a separation of variables, that is

U = {〈 \begin{matrix} u & v & r θ \end{matrix} 〉}^{T} = Σ Ψ_{n} (z) Y_{n} (t)

(7)

where the shape vector Ψ_n(z) is assumed to be of the form

Ψ_{n} (z) = {〈 \begin{matrix} Ψ_{x} & Ψ_{y} & Ψ_{θ} \end{matrix} 〉}_{n}^{T} = {〈 \begin{matrix} c_{x} & c_{y} & c_{θ} \end{matrix} 〉}_{n}^{T} e^{λ_{n} z}

(8)

Introducing equations (7) and (8) into equation (6), the following matrix equation is obtained

[\begin{matrix} λ_{n}^{4} - α_{x}^{2} λ_{n}^{2} - m ω^{2} / E I_{x} & 0 & - λ_{n}^{2} (λ_{n}^{2} {\bar{y}}_{m} - α_{x}^{2} {\bar{y}}_{s}) \\ 0 & c_{m} (λ_{n}^{4} - α_{y}^{2} λ_{n}^{2} - m ω^{2} / E I_{y}) & λ_{n}^{2} c_{m} (λ_{n}^{2} {\bar{x}}_{m} - α_{y}^{2} {\bar{x}}_{s}) \\ - λ_{n}^{2} (λ_{n}^{2} {\bar{y}}_{m} - α_{x}^{2} {\bar{y}}_{s}) & λ_{n}^{2} c_{m} (λ_{n}^{2} {\bar{x}}_{m} - α_{y}^{2} {\bar{x}}_{s}) & λ_{n}^{4} - α_{w}^{2} λ_{n}^{2} - m ρ^{2} ω^{2} / E I_{w} \end{matrix}] {\begin{matrix} c_{x} \\ c_{y} \\ c_{θ} \end{matrix}}_{n} = 0

(9)

where $α_{y}^{2} = G A_{y} / E I_{y}$ and $α_{w}^{2} = G J_{w} / E I_{w}$ .

The coupling form of the equations above vanishes when the centers CF and CS coincide with CM. In such a case (y_m = x_m = x_s = y_s = 0), the aforementioned set of equations takes the form of three uncoupled equations as follows

λ_{n}^{4} - α_{p}^{2} λ_{n}^{2} - \frac{m_{p} ω^{2}}{E I_{p}} = 0

(10)

where p = x, y, w and m_p = m when p = x, y;m_p = mρ² when p = w. Equation (10) represents the undamped equations of free vibration of a symmetrical counterpart structure, similar to that of Figure 1(c). The frequencies of these uncoupled systems can be simply evaluated from the formula

ω_{p}^{2} = (λ_{1} H)^{4} [1 - \frac{{(α_{p} H)}^{2}}{{(λ_{1} H)}^{2}}] \frac{E I_{p}}{m_{p} H^{4}}

(11)

The roots λ₁H, for the first three modes of vibration and for values of the stiffness ratio a_pH into the practical range from 0 to 7, are approximately equal to those of the slender flexural cantilever (Heidebrecht, 1975; Heidebrecht and Stafford Smith, 1973b). This means that the frequencies of equation (11) may be computed by setting

λ_{1} H = \pm 1.875 i, \pm 4.694 i, \pm 7.855 i

(12)

Envisaging equation (9), it is evident that in asymmetric building systems of low or medium height in which the first mode of vibration plays the dominant role, their response under translational ground excitations will be practically translational when the value of (λ₁H)² = −1.875² makes the off-diagonal terms of the matrix of equation (9) equal to 0. That is, when the coordinates of the centers CF and CS are related by the equations

\frac{x_{m}}{x_{s}} = - \frac{{(α_{y} H)}^{2}}{{1.875}^{2}}, \frac{y_{m}}{y_{s}} = - \frac{{(α_{x} H)}^{2}}{{1.875}^{2}}

(13)

In other words, when the first mode root λ₁H produces zero off-diagonal terms in equation (9), then the first three frequencies of the assumed system are uncoupled. Higher frequencies may be coupled, but their contribution on common medium-height buildings is of minor importance. Similarly, when the second mode root λ₁H produces zero off-diagonal terms in equation (9), the first three frequencies are coupled, but the next three higher frequencies are uncoupled.

On the grounds of this consideration, there are a few points that need a special attention:

1. Simple mathematic calculations show that the translational frequencies of the uncoupled structure (Figure 1(c) and equation (11), p = x, y) for the first three modes of vibration (n = 1, 2, 3) may be obtained from the following formulae

ω_{xn}^{2} = Σ ω_{in}^{2} + Σ ω_{xcn}^{2}, ω_{yn}^{2} = Σ ω_{jn}^{2} + Σ ω_{ycn}^{2}

(14)

where ω_in (or ω_xcn) denotes the n-mode frequency of a planar subsystem which has the same mass as the actual structure, but its lateral stiffness along the x-direction depends entirely on the i-bent (or the c-core element). This subsystem’s frequency can be calculated through equation (11) by replacing the overall stiffness ratio α_pH with the element ratio $α_{i} H = \sqrt{G A_{i} / E I_{i}} H$ , provided that the latter is less than 7. In the same way, the frequencies in the second of equation (14) can be evaluated by a similar formula. Therefore

\begin{matrix} ω_{in}^{2} = (λ_{1} H)^{4} [1 - \frac{{(α_{i} H)}^{2}}{{(λ_{1} H)}^{2}}] \frac{E I_{xi}}{m H^{4}}, \\ ω_{jn}^{2} = (λ_{1} H)^{4} [1 - \frac{{(α_{j} H)}^{2}}{{(λ_{1} H)}^{2}}] \frac{E I_{yj}}{m H^{4}} \end{matrix}

(15)

Note here that in the case of the c-core element, to calculate the lateral frequencies ω_xcn and ω_ycn, the corresponding element stiffness ratios α_xcH and α_ycH are 0, but to calculate the rotational frequencies ω_wcn, the corresponding element torsional stiffness ratio is equal to $α_{c} H = \sqrt{G J_{dc} / E I_{wc}} H$ .

2. In the earthquake analysis of elastic structures, the n-mode response of a given building can be derived through its corresponding modal system. This system is single-degree-of-freedom (SDOF) equivalent system with frequency and mass equal to those of the n-mode of vibration of the real structure. These dynamic data are adequate to provide the peak modal base shear through a design acceleration spectrum. For example, the n-mode response of the uncoupled structure of Figure 1(c), in the case of a ground excitation along the x-direction, can be obtained from the response of an equivalent SDOF system which has a mass equal to $M_{xn}^{*}$ (the effective n-mode mass of the uncoupled multi-story structure) and stiffness equal to

k_{xn}^{*} = ω_{xn}^{2} M_{xn}^{*}

(16a)

This is shown in Figure 2(a) and (b) for the first mode of vibration. As the frequency ω_xn can be given from the first of equation (14), it can be concluded that the contribution of the individual i-bent (or c-core) to the total n-mode stiffness of the complete structure, $k_{xn}^{*}$ , is proportional to its square frequency ω_in (or ω_xcn) as defined above. That is, the SDOF modal oscillator of Figure 2(b) is supported by elements (Figure 2(c)) with stiffnesses

k_{in}^{*} = ω_{in}^{2} M_{xn}^{*} and k_{xcn}^{*} = ω_{xcn}^{2} M_{xn}^{*}

(16b)

and its modal stiffness is equal to

k_{xn}^{*} = Σ k_{in}^{*} + Σ k_{x c n}^{*}

(16c)

Figure 2.

(a) Typical first mode deformation profile of a multi-story building, (b) the first mode single-story system with the modal stiffness, and (c) the contribution of each bent to the modal stiffness.

Similarly, the n-mode response of the assumed uncoupled structure in the y-direction is given by an equivalent SDOF system, having a mass equal to $M_{yn}^{*}$ and supported by elements with stiffnesses equal to

k_{jn}^{*} = ω_{jn}^{2} M_{yn}^{*} and k_{ycn}^{*} = ω_{ycn}^{2} M_{yn}^{*}

(17a)

which provide its total modal stiffness as

k_{yn}^{*} = Σ k_{jn}^{*} + Σ k_{y c n}^{*}

(17b)

3. The question which arises in the case of a building with an asymmetric plan configuration (Figure 1(a) and (b)) is whether it would be possible to evaluate basic dynamic quantities (frequencies, base shear, and torque) of its n-mode response through an equivalent single-story asymmetric system with a mass equal to $M_{xn}^{*}$ (or $M_{yn}^{*}$ , depending on the axis closer to the direction of the ground motion), radius of gyration equal to the floor radius of the real structure, and supported by elements located at the positions of the real bents (Figure 1(b)) with stiffnesses equal to the product of their squared modal frequency with the effective mass mostly excited ( $M_{xn}^{*}$ or $M_{yn}^{*}$ ). This approach is investigated further below in comparison with the results obtained by the accurate SAP2000 software, but at first it should be examined whether the special case, in which the real bents are at locations that satisfy equation (13), reflects a particular configuration of the equivalent first mode single-story system, where the center of rigidity (m-CR) is coincident with CM. This is a requirement, since in the special case in which equations (13) are satisfied the first three frequencies of the assumed structure are uncoupled, as implies a coincident location of CM and m-CR in the equivalent first mode single-story system.

4. To prove the special case above, let us assume that the stiffnesses of the elements that provide the lateral resistance of the aforesaid first mode single-story system are given by the products of their squared frequency (ω_i₁, ω_xc₁ for the elements in the x-direction and ω_j₁, ω_yc₁ for the elements in the y-direction, as shown in equation (15) for n = 1) with $M_{y 1}^{*}$ (for simplicity, let us assume that the major ground excitation is along the global y-axis). The response of this system is presented in detail in section “The equation of motion of the equivalent single-story modal system—modifications for systems composed by dissimilar bents,” but in the general case of an arbitrary coordinate reference system, the coordinates of the CR (m-CR) of this equivalent single-story system are equal to

\begin{matrix} e_{x 1} = \frac{Σ ω_{j 1}^{2} x_{j} + Σ ω_{yc 1}^{2} x_{c}}{Σ ω_{j 1}^{2} + Σ ω_{yc 1}^{2}}, e_{y 1} = \frac{Σ ω_{i 1}^{2} y_{i} + Σ ω_{xc 1}^{2} y_{c}}{Σ ω_{i 1}^{2} + Σ ω_{xc 1}^{2}} \end{matrix}

(18a)

Using equation (14) for n = 1, the expressions above may be written in the general form

e_{x 1} = \frac{Σ ω_{j 1}^{2} x_{j}}{ω_{y 1}^{2}}, e_{y 1} = \frac{Σ ω_{i 1}^{2} y_{i}}{Σ ω_{x 1}^{2}}

(18b)

Introducing the expressions of equation (15) into equation (18a), it can easily be shown that when e_x₁ = 0, then the first of equation (13) is satisfied, and when e_y₁ = 0, the second of equation (13) is also valid.

An alternative method to define the location of the first m-CR using the discrete element approach

The procedure presented in the past (Cheung and Tso, 1986; Goel and Chopra, 1993) for the determination of the CRs at the floor levels of a multi-story building can be used for this purpose. The equilibrium equation between forces and displacements of any N-story building with rigid floor diaphragms is as follows

[\begin{matrix} K_{xx} & 0 & K_{xz} \\ 0 & K_{yy} & K_{yz} \\ K_{zx} & K_{zy} & K_{zz} \end{matrix}] {\begin{matrix} Δ_{x} \\ Δ_{y} \\ Δ_{θ} \end{matrix}} = {\begin{matrix} P_{x} \\ P_{y} \\ T \end{matrix}}

(19)

The equation above displays a set of 3N equations for the displacement vectors Δ_x and Δ_y, and the rotation vector Δ_θ (all of order N) in an arbitrary coordinate system Oxyz. For buildings with the lateral load–resisting elements in two orthogonal directions, the submatrices of the above stiffness matrix are given as

\begin{matrix} K_{xx} = Σ K_{i} \\ K_{yy} = Σ K_{j} \\ K_{xz} = K_{zx}^{T} = - Σ y_{i} K_{i} \\ K_{yz} = K_{zy}^{T} = Σ x_{j} K_{j} \\ K_{zz} = Σ x_{j}^{2} K_{j} + Σ y_{i}^{2} K_{i} \end{matrix}

(20)

where the element submatrices K_i and K_j above (of order N × N) represent the stiffness matrices of the i-bent (along the x-direction) and j-bent (along the y-direction), respectively. In general, the location of CRs can be determined by assuming that in the case of zero external floor torques (T = 0) when the lateral load vectors P_x and P_y (of order N) are acting through these centers, the structure will have no rotational deformation, that is, Δ_θ = 0. In such cases, the x- and y-coordinates of CRs, represented as the elements of the diagonal matrices X_L and Y_L (of order N × N), can be found by the following matrix equations (Cheung and Tso, 1986)

K_{zx} K_{xx}^{- 1} P_{x} = - Y_{L} P_{x}, K_{zy} K_{yy}^{- 1} P_{y} = X_{L} P_{y}

(21)

When the building, under a ground motion along the y-axis, is deformed along the same axis with no rotation, its displacements may be evaluated as a superposition of the modal contributions of the uncoupled system, that is

v = Σ Φ_{yr} Y_{r}

(22)

where Y_r (r = 1, 2, …) are the scalar multiplies (modal coordinates) and Φ_yr are the corresponding shape vectors of the uncoupled structure. In the case of a low- or medium-height building, the shape of the first mode vector Φ_y1, which is the dominate mode, can be simply obtained by assuming that at centers of rigidities are acting the corresponding modal forces, which are proportional to the matrix product MΦ_y1. Using the basic relationship between natural frequencies and mode shapes of the uncoupled structure (Chopra, 2008)

P_{y} = K_{yy} Φ_{y 1} = ω_{y 1}^{2} M Φ_{y 1}

(23)

where M is the diagonal mass matrix and ω_y₁ is the first mode frequency of the uncoupled system. Inserting equation (23) into the second of equation (21), it follows

K_{zy} K_{yy}^{- 1} M Φ_{y 1} = X_{L} M Φ_{y 1}

(24a)

\frac{1}{ω_{y 1}^{2}} Σ x_{j} K_{j} Φ_{y 1} = X_{L} M Φ_{y 1}

(24b)

The matrix equation (24b) represents a set of N equations, each of which reflects the equilibrium of forces in a particular floor. The matrix products K_jΦ_y1 represent the elastic force vector developed in the j-bent which is at a distance equal to x_j from the origin of the assumed coordinate system, and each term of the vector MΦ_y1 represents the floor load applied through the corresponding CR (e.g. at the nth floor, with a mass equal to m_n, the load m_nΦ_n,y1 is applied at a distance equal to x_Ln). As the only unknown in each of these equations is the x-coordinate of CR in the corresponding floor level (say x_Ln, n = 1, …, N), its determination is routine procedure. The coordinates of CRs present a significant scattering along the height of the building and for this reason cannot be used as reference points, in the same sense, as the center of stiffness is used in single-story buildings. As an alternative means, the axis of optimum torsion has been recently proposed (Makarios, 2005, 2008; Makarios et al., 2006; Marino and Rossi, 2004). The concept of this approach is to define a vertical line, through which a set of in-plane floor forces causes a translational response with minimum torsion. An alternative procedure to define this axis is to assume a virtual displacement proportional to Φ_y1 and compute the virtual work done by the forces shown in equation (24b). This is equivalent of pre-multiplying both parts of this equation by $Φ_{y 1}^{T}$ , that is

\frac{1}{ω_{y 1}^{2}} Σ x_{j} Φ_{y 1}^{T} K_{j} Φ_{y 1} = Φ_{y 1}^{T} X_{L} M Φ_{y 1}

(25)

The equation above contains scalar quantities and it is derived on the grounds that the system undergoes a translational deformation. Assuming now a set of in-plane forces, given by the vector $P_{y} = ω_{y 1}^{2} M Φ_{y 1}$ , which produces, through the aforesaid virtual displacement, a work equal to $ω_{y 1}^{2} Φ_{y 1}^{T} M Φ_{y 1}$ , it can be seen that by the aforementioned procedure (equating external and internal work), the distance of the plane of the induced forces from the origin of the coordinate system should be given by the formula

e_{x 1} = \frac{1}{ω_{y 1}^{2}} Σ x_{j} \frac{Φ_{y 1}^{T} K_{j} Φ_{y 1}}{Φ_{y 1}^{T} M Φ_{y 1}}

(26a)

The in-plane forces acting in such a location will cause a coupled translational–rotational response in the assumed building, but the rotational effect will be smaller than in any other case (different locations of the in-plane forces), since it is compatible with the assumed translational displacement.

Note here that the first frequency of any planar subsystem (say the j-bent along the y-direction), which has the same mass per floor as the actual structure, is given by the formula

ω_{j 1}^{2} = \frac{Φ_{j 1}^{T} K_{j} Φ_{j 1}}{Φ_{j 1}^{T} M Φ_{j 1}}

(26b)

where Φ_j1 is the first mode vector of the particular j-bent. A close estimate of this frequency, however, may be obtained using a similar shape vector, as for example the first mode shape of the uncoupled structure Φ_y1. The ratio thus produced represents a Rayleigh quotient, which provides an upper bound of the first mode frequency. For cantilever structures, however, this quotient is a very close estimate (Georgoussis, 2009) of the fundamental frequency of the particular planar subsystem (j-bent). Therefore, a good estimate of the e_x₁ coordinate of the optimum torsion axis, which in the case of a uniform building may be seen as the x-coordinate of the mass axis, can be obtained as

e_{x 1} = \frac{Σ ω_{j 1}^{2} x_{j}}{ω_{y 1}^{2}}

(27)

This expression is identical to the first of equation (18b), and the second of them can be obtained with a similar procedure. It is evident that when the coordinates of the mass axis, in any multi-story structural configuration, are given by the aforementioned equations, the seismic rotational response will be significantly reduced, as the first mode effective inertia forces produce minimal rotational distortion.

The equation of motion of the equivalent single-story modal system—modifications for systems composed by dissimilar bents

Assume that the asymmetric building of Figure 1(a) and (b) is subjected to a ground motion, with components ${\overset{\cdot\cdot}{u}}_{gx} (t)$ and ${\overset{\cdot\cdot}{u}}_{gy} (t)$ along the global axes. Assuming further that the y-component is the major excitement, the evaluation of the basic dynamic data (frequencies, base shear, and torque), for the first three modes of vibration, is implemented through an equivalent single-story system, which has the same floor plan as the typical story of the real structural and a mass equal to the first mode effective mass of the uncoupled system, $M_{y 1}^{*}$ . Following the analysis presented in section “Basic mathematical model,” concerning the stiffnesses of the lateral load–resisting elements, the undamped equation of motion of the aforesaid equivalent system, in the global coordination system Oxy, is as follows

M_{e}^{*} {\overset{\cdot\cdot}{U}}_{e} + K_{e}^{*} U_{e} = - M_{e}^{*} {\overset{\cdot\cdot}{u}}_{g}

(28)

where $M_{e}^{*} = M_{y 1}^{*} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ρ^{2} \end{matrix}]$ is the mass matrix; $U_{e} = {〈 \begin{matrix} u_{e} & v_{e} & θ_{e} \end{matrix} 〉}^{T}$ is the displacement vector at CM; and $K_{e}^{*} = M_{y 1}^{*} [\begin{matrix} k_{xx}^{*} & 0 & k_{xz}^{*} \\ 0 & k_{yy}^{*} & k_{yz}^{*} \\ k_{zx}^{*} & k_{zy}^{*} & k_{zz}^{*} \end{matrix}]$ is the stiffness matrix

{\overset{\cdot\cdot}{u}}^{T} = {〈 \begin{matrix} {\overset{\cdot\cdot}{u}}_{gx} & {\overset{\cdot\cdot}{u}}_{gy} & 0 \end{matrix} 〉}^{T}

k_{xx}^{*} = Σ ω_{i 1}^{2} + Σ ω_{xc 1}^{2} = ω_{x 1}^{2}, k_{yy}^{*} = Σ ω_{j 1}^{2} + Σ ω_{yc 1}^{2} = ω_{y 1}^{2}

\begin{matrix} k_{zz}^{*} = Σ x_{j}^{2} ω_{j 1}^{2} + Σ x_{c}^{2} ω_{yc 1}^{2} + Σ y_{i}^{2} ω_{i 1}^{2} \\ + Σ y_{c}^{2} ω_{xc 1}^{2} + Σ ω_{wc 1}^{2} = ω_{θ 1}^{2} \end{matrix}

k_{xz}^{*} = k_{zx}^{*} = - (Σ ω_{i 1}^{2} y_{i} + Σ ω_{xc 1}^{2} y_{c}) = - e_{y 1} k_{xx}^{*}

k_{yz}^{*} = k_{zy}^{*} = (Σ ω_{j 1}^{2} x_{j} + Σ ω_{yc 1}^{2} x_{c}) = e_{x 1} k_{yy}^{*}

(29)

Equation (28) represents the undamped equation of motion of the first mode (k = 1) equivalent single-story system. It is based on the first mode effective mass of the uncoupled real structure, $M_{y 1}^{*}$ , and the first mode element frequencies of the individual bent-subsystems. That is, the translational element frequencies ω_i₁ and ω_xc₁ of the bents aligned in the x-direction, the translational element frequencies ω_j₁ and ω_yc₁ of the bents in the y-direction, and also the rotational frequencies ω_wc₁ of the cores. All these modal data may be easily obtained by standard engineering software applicable to discrete member models, and therefore the formulation of equation (28) is a routine procedure. This equation provides the first three frequencies of the coupled system by means of the following eigenvalue equation

[K_{e}^{*} - ω^{2} M_{e}] Φ_{e} = 0

(30a)

which in an analytical form is as follows

[\begin{matrix} ω_{x 1}^{2} - ω^{2} & 0 & - e_{y 1} ω_{x 1}^{2} / ρ \\ 0 & ω_{y 1}^{2} - ω^{2} & e_{x 1} ω_{y 1}^{2} / ρ \\ - e_{y 1} ω_{x 1}^{2} / ρ & e_{x 1} ω_{y 1}^{2} / ρ & ω_{θ 1}^{2} - ω^{2} \end{matrix}] {\begin{matrix} α_{x 1} \\ α_{y 1} \\ α_{θ 1} \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}}

(30b)

Note here that in an earlier article (Georgoussis, 2012), the same eigenvalue equation was derived by making use of the potential of Rayleigh’s quotients and also the same equation is given by Kan and Chopra (1977a) for the special class of proportionate buildings.

In a similar manner, the undamped equation of motion of the second mode (k = 2) equivalent single-story system will be based on the second mode data of the uncoupled system, $M_{y 2}^{*}$ , and the individual element frequencies (ω_i₂, ω_xc₂, ω_j₂, ω_yc₂, and ω_wc₂). The second mode equivalent single-story system provides the frequencies and base resultant forces of the next three higher modes of vibration of the real structure. Full description of the modal quantities that can be derived by the proposed analysis, in the simpler case of mono-symmetric systems, is given in Georgoussis (2009).

It should be noticed here that in the case of systems composed by dissimilar bents, the aforesaid analysis attains a higher accuracy when the effective element frequencies (denoted, for example, as ${\bar{ω}}_{i 1}$ , ${\bar{ω}}_{xc 1}$ , ${\bar{ω}}_{j 1}$ , ${\bar{ω}}_{yc 1}$ for the first mode of vibration) are used instead of the element frequencies ω_i₁, ω_xc₁, ω_j₁, and ω_yc₁ (Georgoussis, 2014; Georgoussis et al., 2013a). An effective element frequency (say that one corresponding to j-bent aligned in the y-direction) is defined as

{\bar{ω}}_{j 1} = ω_{j 1} \sqrt{\frac{M_{j 1}^{*}}{M_{y 1}^{*}}}

(31)

where $M_{j 1}^{*}$ is first mode effective mass of the j-bent when it is assumed to carry, as a planar system, the mass of the complete structure. It is evident that when similar types of bents provide the lateral resistance of a given structure, the ratio of modal masses under the square root in the equation above is approximately equal to unity and the effective element frequency $({\bar{ω}}_{j 1})$ is practically equal to the element frequency $(ω_{j 1})$ . In systems, however, composed by dissimilar bents, each of which deflects in a different shape (e.g. in the first mode of vibration, shear walls deflect in a flexural mode in the lower part (concavity downwind), while rigid frames deflect in a shear mode in the upper part (concavity upwind)), the use of the effective element frequencies, particularly in the equation of motion (equation (29)) of the first (k = 1) mode equivalent single-story system, improves the accuracy of the proposed approximate method and this is demonstrated in the numerical results presented below.

Systems analyzed under unidirectional and bidirectional excitations

To demonstrate the accuracy of the proposed method, a 10-story mono-symmetric model building, with an orthogonal floor plan of 20 × 10 m (Figure 3(a)), was analyzed. The total mass per floor is m = 160 kNs²/m, uniformly distributed over the floor slab, the radius of gyration about CM is ρ = 6.455 m, the story height is 3.5 m, and the modulus of elasticity is assumed equal to 26 × 10⁶ kNs²/m, typical for concrete structures. The lateral resistance along the y-direction is provided with three resisting elements, two of which are flexural shear walls (Wa, Wb) of a cross section 30 cm × 600 cm the first and 30 cm × 400 cm the second and, also, by a moment resisting frame (FR) with two columns 80 cm × 80 cm, 5 m apart, which are connected by floor beams 35 cm × 70 cm. The lateral resistance along the x-axis is provided by a pair of flexural shear walls (Wx) of a cross section 30 cm × 500 cm, located symmetrically to the axis of symmetry at distances $\pm 4 m$ as shown in Figure 3(a). The analyzed building represents a typical dual system in the y-direction and a wall system in the x-direction. To investigate the accuracy of the proposed method to a broader range of building structures, different structural configurations of the example structure are examined as follows: walls Wa and Wb are assumed to be located at a fixed positions, the first on the left of CM in a distance equal to 4 m and the second on the right of CM at a distance of 7 m, while the frame FR is taking all the possible locations along the x-axis.

Figure 3.

(a) Floor plan of the 10-story example structure and (b) the EC8-2004 recommended spectrum.

The elastic response of the assumed models is at first examined. The accuracy of the proposed methodology to predict periods of vibrations and base resultant forces, in the case of a dynamic excitation (along the y-direction) characterized by the EC8-2004 recommended response spectrum (Figure 3(b)), is investigated by comparison with the results derived by means of the computer program SAP2000-V11. As the lateral load–resisting bents along the y-direction are of dissimilar type (shear walls and moment resisting frame), the results of the approximate method are derived first using the element frequencies and second using the effective element frequencies. In practical terms, the difference between the two approximate procedures is based on the grounds that in the first case the formulation of equation (28) is based on the element frequencies $ω_{jk}$ and $ω_{ik}$ , while in the second case this formulation is based on the effective element frequencies: ${\bar{ω}}_{jk}$ and ${\bar{ω}}_{ik}$ (k = 1, 2).

To apply the proposed method, the first pair of element frequencies of the various bent-subsystems is required and, also, their effective modal masses. Denoting with M the total mass of the structure (M = 10 m = 1600 kNs^2/m), these quantities, found by the SAP2000 program are as follows:

For wall Wa: ω_wa₁ = 4.505/s, ω_wa₂ = 26.146/s and ${\bar{M}}_{wa 1}^{*} = M_{wa 1}^{*} / M = 0.651$ , ${\bar{M}}_{wa 2}^{*} = 0.209$ .

For wall Wb: ω_wb₁ = 2.472/s, ω_wb₂ = 14.983/s and ${\bar{M}}_{wb 1}^{*} = 0.648$ , ${\bar{M}}_{wb 2}^{*} = 0.203 .$

For frame FR: ω_f₁ = 2.476/s, ω_f₂ = 8.163/s and ${\bar{M}}_{wb 1}^{*} = 0.765$ , ${\bar{M}}_{wb 2}^{*} = 0.123$ .

For wall Wx: ω_wx₁ = 3.442/s, ω_wx₂ = 20.447/s and ${\bar{M}}_{wx 1}^{*} = 0.649$ , ${\bar{M}}_{wx 2}^{*} = 0.206 .$

The first two effective modal masses of the uncoupled structure, in the y-direction, normalized with respect to the total mass, are, respectively, equal to ${\bar{M}}_{y 1}^{*} = M_{y 1}^{*} / M = 0.659$ and ${\bar{M}}_{y 2}^{*} = 0.200$ , while the corresponding quantities of the same structure along the x-direction are the same as those of wall Wx. From these data, the effective element frequencies of all the bents shown in Figure 3(a) can be determined. For wall Wa: ${\bar{ω}}_{wa 1} = 4.477 / s$ , ${\bar{ω}}_{wa 2} = 26.728 / s$ ; for wall Wb: ${\bar{ω}}_{wb 1} = 2.451 / s$ , ${\bar{ω}}_{wb 2} = 15.095 / s$ ; and for frame FR: ${\bar{ω}}_{f 1} = 2.668 / s$ , ${\bar{ω}}_{f 2} = 6.402 / s$ . Similarly for the walls Wx, along the x-direction: ${\bar{ω}}_{wx 1} = 3.416 / s$ , ${\bar{ω}}_{wx 2} = 20.751 / s$ .

The inelastic response of the assumed model structures was investigated under two characteristic ground motions of Erzincan 1992 (components NS and EW) and Kobe 1995 (components KJM090 and 000), as shown in Figure 4, selected from the strong ground motion database of the PEER Center. All resisting elements (bents) are assumed to have only in-plane stiffness and their strength assignment is based on a planar static analysis under an external lateral loading with floor forces having the shape of the “inverted triangle” and summing to a base (design) shear equal to V_d = 3200 kN (approximately equal to one-fifth of the total weight of the structure). More specifically, allowing for plastic hinges at the bases of walls Wa and Wb and detailing frame FR according to the strong column-weak beam philosophy (i.e. allowing plastic hinges at the ends of the beams and at the foot of the ground floor columns), this static analysis leads to the following data: (1) the bending (yield) capacity at the plastic hinge at the base of wall Wa equals 48,377 kN m and for wall Wb is equal to 15,680 kN m and (2) the bending (yield) capacity of the plastic hinges at the ends of the beams of FR (from the top downward) is equal to 680, 814, 814, 814, 801, 766, 701, 601, 458, and 276 kN m, respectively, while the corresponding capacity at the plastic hinges at the base of the ground columns of FR equals 452 kN m. The bending capacity of walls Wx (at the plastic hinges allowed to be formed at their base sections) is equal to 39,200 kN m, which corresponds again to a lateral base shear equal to 3200 kN.

Figure 4.

Ground motions considered.

The aforesaid structural detailing is based on the recommendations of Eurocode EC8-2004, clause 4.3.1(5)), which suggests a planar static analysis when the regularity in plan criteria of the code is satisfied. These criteria are actually satisfied for all possible locations of frame FR along the axis of symmetry, when the walls Wa and Wb are located at the fixed positions shown in Figure 3(a). The various structural configurations, depending on the particular location of frame FR, provide different locations of m-CR, but according to the analysis described in section “Basic mathematical model,” minimum elastic rotational response is expected when m-CR lies on (or is at a close distance from) the vertical mass axis. Because of the structural symmetry of the assumed models along the x-direction, m-CR is positioned on this axis, but to have its location coincident to CM, its eccentricity from CM (as given from the first of equation (18b)) must be equal to 0. Using the first mode element frequencies of the bents aligned in the y-direction, the optimum location of frame FR was found at x = 6.26 m ( $\bar{x} = x / ρ = 0.97$ ), while the use of their effective frequencies provides an optimum location FR at x = 5.36 m ( $\bar{x} = x / ρ = 0.83$ ).

The response of the assumed structural configurations is first examined under purely translational excitations along the y-direction. The assumed unidirectional motions are the components NS of Erzincan and KJM090 of Kobe. These responses are compared with those when the transverse excitations of the mentioned earthquakes (components EW of Erzincan and KJM000 of Kobe) are also concurrently acting along the x-direction. In the elastic state, the bidirectional response is simply derived as the sum of two independent responses along the principal x- and y-directions, but in the inelastic phase the overall behavior is coupled with a distinct difference from that of a unidirectional excitation along the y-direction. In the latter case, because of the structural symmetry in the x-direction, the corresponding resisting elements (walls Wx) are always under reverse bending, but when the system is under a bidirectional excitation this behavior is completely altered, especially in the post-elastic phase. During such an excitation, the x-direction resisting elements experience different loading paths, yielding and unloading at different instants. What is important to investigate now is whether this behavior affects the overall rotational response when the mass axis passes through the point m-CR.

All analyses were performed by means of the program SAP2000-V11, using inelastic link elements at the assumed locations of plastic hinges. The moment–rotation relationships of these elements were assumed bilinear with a post-yielding stiffness ratio of the generalized load–deformation curve, equal to 4%. The nonlinear response history analyses were performed using the numerical implicit Wilson-θ time integration method, with the parameter θ taken equal to 1.4. The damping matrix is proportional to the mass matrix and the instantaneous stiffness matrix, and the target damping is 5% in the first two modes.

Discussion of results

The elastic response of the assumed models is shown in Figures 5 and 6. The first four periods of vibration of the example structural configurations (Figure 3(a)) are shown in Figure 5. These are the periods obtained by the coupled lateral (along the y-direction) torsional response of the system. The aforesaid periods, computed by the proposed approximate method on the grounds of (1) the element frequencies (red lines) and (2) the effective element frequencies (green lines), for different locations of the frame FR (indicated by the normalized coordinates $\bar{x} = x / ρ$ ), are shown in Figure 5, together with the accurate, in black lines, computer values (along the uncoupled x-direction the periods for the first four translational modes of vibration are, respectively, equal to 1.290, 0.217, 0.084, and 0.048 s for any location of frame FR).

Figure 5.

Periods of vibration of the assumed structural configurations.

Figure 6.

Peak-normalized (a) base shears and (b) torques for the case of EC8-2004 spectrum.

For the first mode of vibration, both the approximate procedures present data which are very close to the accurate computer ones. For the second mode of vibration, the proposed procedure when it is based on the effective element frequencies (green dashed line) presents values closer to the accurate ones (black dashed line). For the next two higher modes of vibration (third and fourth), both the approximate procedures present period values which are almost identical to the accurate ones. It is worth noting here that the first pair of the approximate periods (first and second) is derived from the first mode (k = 1) equivalent single-story system, while the second pair of the approximate periods (third and fourth) is obtained from the second mode (k = 2) equivalent single-story system.

Elastic base shears (in the y-direction) and torques, for the case of the spectrum of Figure 3(b), are shown in Figure 6. Normalized base shears and torques ${\bar{V}}_{m}$ and ${\bar{T}}_{m}$ (green lines) represent the approximate peak results obtained by the proposed procedure on the grounds of the effective element frequencies. The aforesaid base shears are normalized with respect to the total shear, V_o, of the uncoupled structure and the mentioned torques are normalized with respect to the product: ρV_o. Note here that the first mode (k = 1) equivalent single-story system provides the first two peak modal data, which, when combined by the complete quadratic combination (CQC) rule, produce the normalized forces ${\bar{V}}_{m, 1 - 2}$ and ${\bar{T}}_{m, 1 - 2}$ (blue solid lines). From the second mode (k = 2) equivalent single-story system, the normalized forces ${\bar{V}}_{m, 3 - 4}$ and ${\bar{T}}_{m, 3 - 4}$ (blue dotted lines) are evaluated in a similar way. The expected total maximum ${\bar{V}}_{m}$ (or ${\bar{T}}_{m}$ ) is finally calculated by a combination of ${\bar{V}}_{m, 1 - 2}$ and ${\bar{V}}_{m, 3 - 4}$ (or ${\bar{T}}_{m, 1 - 2}$ and ${\bar{T}}_{m, 3 - 4}$ ) through the square-root-of-sum-of-squares (SRSS) rule. In a similar manner, the total maxima ${\bar{V}}_{ea}$ and ${\bar{T}}_{ea}$ (red lines) are derived on the grounds of the element frequencies. In the same figure are also shown the accurate data ${\bar{V}}_{com}$ and ${\bar{T}}_{com}$ (black lines) given by the computer program SAP2000-V11 on the basis of the first 12 peak modal values combined according to the CQC rule (the damping ratio in each mode of vibration was taken equal to 5%). Envisaging Figure 6, the first rough conclusion is that when the approximate procedure is based on the effective element frequencies, then data of superior accuracy are obtained (the green lines are closer to the black lines, which are the results of the computer analysis, than the red lines which are based on element frequencies). Besides, the location of FR, which produces minimum base torque according to SAP2000 results, is closer to $\bar{x} = x / ρ = 0.82$ , which is given by the first mode equivalent system (zero ${\bar{T}}_{m, 1 - 2}$ ) when it is based on the effective element frequencies (the same system when based on the element frequencies gives $\bar{x} = x / ρ = 0.97$ ). In other words, the structural configuration of minimum torsion is best predicted when the m-CR is defined on the grounds of the effective element frequencies rather than on the element frequencies.

The torsional response of the model structural systems under the assumed excitations of Erzincan and Kobe is shown in Figures 7 and 8. Three response parameters are shown for both the elastic and inelastic systems in the case of the unidirectional excitations along the y-direction (Figures 7(a) and 8(a)): top rotations, θ, normalized base shears, and normalized base torques. The red lines represent the peak elastic response (top rotations: θ_e, are shown by dashed lines, normalized base shears along the y-direction: ${\bar{V}}_{ey} = V_{ey} / V_{d}$ by solid lines, and normalized base torques: ${\bar{T}}_{e} = T_{e} / ρ V_{d}$ by dotted lines) and the corresponding black lines represent the peak response of the inelastic systems (θ_in, ${\bar{V}}_{iny} = V_{iny} / V_{d}$ , ${\bar{T}}_{in} = T_{in} / ρ V_{d}$ ). For both the assumed ground excitations, minimum rotational response (θ_e and ${\bar{T}}_{e}$ ) of the elastic systems appears when the frame FR approaches the coordinate $\bar{x} = 0.76$ , which is closer to the value $\bar{x} = 0.82$ , predicted by the proposed approximate procedure when the effective element frequencies are taken into account. The response of the inelastic systems is smoother and the overall rotational behavior is smaller than that obtained by the elastic behavior. This finding confirms observations on single-story systems that after yielding asymmetric systems have the tendency to deform further in a translational mode (e.g. Ghersi and Rossi, 2001; Kan and Chopra, 1981). Besides, minimum torsional response, in terms of θ_in and ${\bar{T}}_{in}$ , is attained when FR takes locations, $\bar{x}$ , into the interval 0.80–0.85. These findings are in agreement with those observed in an author’s earlier article (Georgoussis et al., 2013b).

Figure 7.

Top rotations (×10⁻², rads) and normalized base shears and torques of assumed models under the Erzincan 1992 ground motion: (a) unidirectional excitation and (b) bidirectional excitation.

Figure 8.

Top rotations (×10⁻², rads) and normalized base shears and torques of assumed models under the Kobe 1995 ground motion: (a) unidirectional excitation and (b) bidirectional excitation.

The response of the model structures under the bidirectional excitations is shown in Figures 7(b) and 8(b). In these figures, the peak base shears along the x-direction are also shown (the elastic normalized base shears, ${\bar{V}}_{ex} = V_{ex} / V_{d}$ , are shown by dashed-dotted red lines and the corresponding shears of the inelastic systems, ${\bar{V}}_{inx} = V_{inx} / V_{d}$ , are shown by black lines). As can be seen, the response of the elastic systems, in terms of θ_e, ${\bar{V}}_{ey}$ , and ${\bar{T}}_{e}$ , is not affected by the presence of the concurrent excitation along the x-direction, because of the stiffness symmetry along this direction (uncoupled direction of motion). The red curves of the aforementioned quantities are the same in Figure 7(a) and (b) and also in Figure 8(a) and (b). For the same reason, the value of ${\bar{V}}_{ex}$ in Figures 7(b) and 8(b) remains constant for any location of FR. The torsional response of the inelastic systems (θ_in, ${\bar{T}}_{in}$ ) under bidirectional excitation is generally of the same order as that one under unidirectional excitation. This response confirms observations on single-story systems that the inelastic response is affected to minor way by the presence of the orthogonal component of the seismic action (Ghersi and Rossi, 2001). The important finding here is that minimum values of these quantities are attained when FR takes locations into the interval 0.80–0.85, as in the case of the unidirectional excitations. Note that at such locations of FR, the mass axis is practically passing through the m-CR, implying that the elastic response of the system along the y-direction is virtually translational. As the strength distribution has been determined by a planar static analysis, this response results in an almost in-phase yielding of the bents aligned in the y-direction, leading to a minimum rotational response. It is worth reminding here a statement by Lucchini et al. (2009) concerning the behavior of single-story buildings: their nonlinear response depends on how the building enters the nonlinear range, which, in turn, depends on its elastic properties (i.e. the stiffness and mass distributions) and on the capacities of its resisting elements (i.e. the strength distribution).

Another important feature of Figures 7(b) and 8(b) is that the peak base shear, ${\bar{V}}_{inx}$ , along the x-direction is almost constant for any location of FR. This response may be explained by a finding of Lucchini et al. (2008) that deep into the nonlinear range, the maximum displacements of the different lateral load–resisting bents tend to be reached by the same deformed configuration of the system, which corresponds to story shears that are located at the centers of strength of each floor. This finding implies that the peak base shear, which causes this maximum displacement profile, represents a set of lateral loads passing through the centers of floor masses, because of the symmetry of the assumed structural configurations along the x-axis. As all lateral load–resisting bents along this axis are well displaced in the nonlinear range in the same direction, it is not surprising that the peak base shear receives an almost constant value for any location of FR.

Here, it should be mentioned that the findings concerning the inelastic behavior of multi-story building are based on a limited number of numerical examples and further research is required to investigate the impact of other ground motions, with different frequency content and seismic intensity, on such systems. The problem of earthquake-induced torsion in buildings which are stressed beyond the elastic limits is an open problem and further studies are required to check the validity of the present results.

Conclusion

Frequencies and basic earthquake response (resultant base shears and torques) of asymmetric, medium-height uniform buildings, can be estimated by means of an approximate method, by analyzing two equivalent, single-story, modal systems. The dimensions of the deck of each of these systems are the same of a typical floor plan of the real structure and their masses are, respectively, equal to the first and second mode effective masses of the uncoupled multi-story structure. The lateral resistance of each system is provided by elements, at the locations of the real bents, with stiffnesses which are equal to the aforesaid masses multiplied, respectively, by the first and second mode squared element frequencies. These frequencies are determined from the corresponding individual bents when they are assumed to carry, as planar frames, the mass of the complete structure. In buildings composed by very dissimilar bents, the use of the effective element frequencies provides an increased accuracy to the proposed method, particularly in the assessment of the location of the CR of the first of the equivalent, single-story, modal systems. When this point (m-CR) lies close to the axis passing through the centers of floor masses, the rotational response sustained by an elastic asymmetric building system is minimum and this response is preserved into the post-elastic phase when the lateral load–resisting bents are detailed as planar structures under a code load. This is demonstrated in a limited number of 10-story, wall-frame asymmetric building configurations under two characteristic strong unidirectional and bidirectional ground motions. The procedure to check whether the m-CR lies within a close distance to the mass axis (equation (27)) is quite simple and the practicing engineer, at the stage of the preliminary structural design, particularly when he is allowed to make decisions on the location and the dimensions of one or more lateral load–resisting bents, may have an overview of the overall response under seismic excitations.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been co-financed by the European Union (European Social Fund (ESF)) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: ARCHIMEDES III: Investing in knowledge society through the European Social Fund.

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