Seismic Modal Contribution Factors

Abstract

Over the years, the belief that the first mode of vibration governs the seismic response of shear-type frame structures has been widely accepted and proved to be effective for preliminary structural design. Indeed, most of the actual seismic design procedures are based on drift profiles which are typically an approximation of the shape of the fundamental mode of vibration. In this paper, an analytical study on the dynamic properties of multi-storey shear-type frames is carried out with the purpose of precisely identifying the contribution of the modes of vibration to the seismic response of such structures, both in terms of maximum inter-storey displacement profiles (which govern the beams and columns maximum actions) and maximum inter-storey velocity profiles (which govern the viscous dampers maximum forces, of fundamental importance for building structures equipped with additional viscous dampers). A new parameter, referred to as Seismic Modal Contribution Factor, which represents the contribution of the generic mode to the seismic response of the structure, is introduced. With respect to the well-known Modal Contribution Factor, grounded on the concept of modal static response, the Seismic Modal Contribution Factor explicitly takes into account also the dynamic nature of the response due to earthquake excitation. The Seismic Modal Contribution Factor could be a meaningful parameter to be implemented in a professional structural design software and used in conjunction with the common modal participating mass ratios to identify the number of modes to be included in the analyses.

Appendix

Let us consider an N-storey shear-type (ST) structure with uniform floor mass m and uniform lateral storey stiffness k. Let us assume to schematize the ST structure as the N-elements discrete mass-spring (DMS) chain depicted in Fig. 11, with m and k indicating the element mass and the stiffness of the single spring, respectively. It is assumed that all masses are equally spaced at distance a, thus the position of the jth mass with the respect to the origin of the x- axis and the length of the N-elements DMS system are:

$$ x_{j} = ja $$

(31)

$$ L = Na $$

(32)

Fig. 11

The N-elements discrete mass-spring (DMS) chain and its equivalent homogenous continuous shear-beam (CSB) model

The dynamic equilibrium equation of the jth mass can be written as follows:

$$ m\textit{\"{u}}_{j} + 2ku_{j} - k(u_{j + 1} + u_{j - 1} ) = 0 $$

(33)

where $\textit{\"{u}}_{j}$ and u _j are the acceleration and displacement of the jth mass along the y-direction.

Figure 11 displays the N-elements discrete mass-spring chain system together with its equivalent homogenous continuous shear-beam (CSB) model. If G, A, ρ denote the shear modulus, the cross section area and the density of the equivalent continuous shear-beam, respectively, the following relationships hold between the DMS system and its equivalent CSB model:

$$ k = GA/a $$

(34)

$$ m = \rho Aa $$

(35)

$$ a = L/N $$

(36)

A solution of the dynamic equilibrium equation for theDMS system characterised by infinite number of elements (periodic structure) is sought in the form of a harmonic plane (Hussein et al. 2014):

$$ u(x_{j} ,t) = u_{j} (t) = \tilde{u}e^{{i(\pi \kappa x_{j} - \omega t)}} = \tilde{u}e^{i(\pi \kappa ja - \omega t)} $$

(37)

where κ is the wave number, ω is the frequency of the harmonic motion, $\tilde{u}$ is the amplitude of the wave motion.

By imposing the boundary conditions (e.g. cantilever system), the nth modal shape of the infinite-elements DMS system is characterized by a wave length equal to (see Fig. 12):

$$ \lambda_{n} = \frac{4aN}{2n - 1} $$

(38)

and the corresponding wave number is given by:

$$ \kappa_{n} = \frac{2n - 1}{4aN} $$

(39)

Fig. 12

The first three mode shapes of the DMS system

Substituting Eq. (39) in Eq. (37) leads to:

$$ u_{j} (t) = \hat{u}_{j} e^{ - i\omega t} = e^{{\frac{i\pi j(2n - 1)}{2N}}} e^{ - i\omega t} $$

(40)

Similarly:

$$ u_{j + 1} (t) = \hat{u}_{j + 1} e^{ - i\omega t} = e^{{\frac{i\pi (j + 1)(2n - 1)}{2N}}} e^{ - i\omega t} $$

(41)

$$ u_{j - 1} (t) = \hat{u}_{j - 1} e^{ - i\omega t} = e^{{\frac{i\pi (j - 1)(2n - 1)}{2N}}} e^{ - i\omega t} $$

(42)

where $\hat{u}_{j}$, $\hat{u}_{j}$, $\hat{u}_{j}$ indicates the spatial part (i.e. the shapes) of the solution.

The second time derivative of $u_{j} (t)$ is equal to:

$$ u_{j} (t) = - \omega^{2} u_{j} (t) $$

(43)

Substituting Eqs. (40)–(43) in the equilibrium equation (Eq. 33) yields to:

$$ \tilde{u}\left[ { - \omega^{2} m + 2k - k\left( {e^{{\frac{{i\pi \left( {2n - 1} \right)}}{2N}}} + e^{{\frac{{ - i\pi \left( {2n - 1} \right)}}{2N}}} } \right)} \right]e^{{ - i\left( {\frac{\pi j}{2N} - \omega t} \right)}} = 0 $$

(44)

Nontrivial ($\tilde{u} \ne 0$) solutions are obtained by solving the eigenvalues problem:

$$ - \omega^{2} m + 2k - k\left( {e^{{\frac{{i\pi \left( {2n - 1} \right)}}{2N}}} + e^{{\frac{{ - i\pi \left( {2n - 1} \right)}}{2N}}} } \right) = - \omega^{2} m + 2k - \left( {1 - \cos \left( {\frac{{\pi \left( {2n - 1} \right)}}{2N}} \right)} \right) = 0 $$

(45)

leading to the following expression of the natural frequencies:

$$ \omega_{DMS} = \sqrt {\frac{2k}{m}\left( {1 - \cos \left( {\frac{{\pi \left( {2n - 1} \right)}}{2N}} \right)} \right)} = 2\sqrt {\frac{k}{m}} \sin \left( {\frac{{\pi \left( {2n - 1} \right)}}{4N}} \right) $$

(46)

From structural dynamics (Den Hartog 1985), it is well known that the natural frequencies of a CSB system are equal to:

$$ \omega_{CSB} = \left( {n - \frac{1}{2}} \right)\pi \frac{{\sqrt {G/\rho } }}{L} $$

(47)

Substituting Eqs. (33)–(35) into Eq. (47) leads to:

$$ \omega_{CSB} = \left( {n - \frac{1}{2}} \right)\pi \frac{{\sqrt {G/\rho } }}{L} = \left( {n - \frac{1}{2}} \right)\pi \frac{{\sqrt {k/m} }}{N} $$

(48)

It can be noted that the natural frequencies of the CSB model are coincident with the first term of the Taylor expansion of Eq. (46) (i.e. the natural frequencies of the DMS system).

Figure 13 compares the natural frequencies of a low rise (5-storey) and high rise (50-storey) shear type systems, characterized by uniform mass and stiffness distributions, with the ones of the equivalent infinite-elements DMS system and CSB models as given by Eqs. (46) and (48), respectively (assuming unit values of m and k). Furthermore, the relative differences between the natural frequencies of the ST structures and the equivalent DMS and CSB systems are graphically compared in Fig. 14. It can be noted that the natural frequencies of the DMS system rapidly approach to those of the ST structure, as the number of degrees of freedom increases (for the 50-storey structure the relative differences are <1 % for all modes). The largest discrepancy between the ST and the DMS frequencies is observed for the first frequency. On the contrary, as the number of degrees of freedom increases, the frequencies of the ST structure diverge from those of the equivalent CSB system (for the 50-storey structure the relative differences in the higher frequencies are larger than 50 %).

Fig. 13

Natural frequencies of a uniform ST structure together with the corresponding DMS system and CSB model. a 5-storey structure; b 50-storey structure

Fig. 14

Relative differences between the natural frequencies of a uniform ST structure together with the corresponding DMS system and CSB model. a 5-storey structure; b 50-storey structure