Vibrations of Beams for Seismic Response Estimation

Introduction

The last decades have seen a growing interest for design and construction of tall buildings. In order to accurately estimate the dynamic response to environmental loads of such complex constructions, structural models are usually characterized by a large number of degrees of freedom; as a consequence, structural optimization algorithms, used to reduce the cost and the energy required to construct such buildings while maintaining appropriate safety levels, might be time consuming and even not feasible given the large dimension of the problem. As far as the global response is concerned, it is of interest to develop reduced models that are capable of giving approximate optimal solutions by significantly reducing the computation time. Therefore, a reduced model based on a Timoshenko beam will be presented in this paper which can be used to estimate the response of tall buildings subjected to environmental loads, such those induced by earthquakes.

Flexural Beam Models

In the present section, the main beam models are recalled, focusing mainly on the Timoshenko model, which is able to describe the flexural behavior of the beam taking into account, besides the flexural stiffness and inertia due to deflections, also the shear deformability and the rotational inertia. Nevertheless, before introducing the Timoshenko beam model, the two models that can considered to be the ones on which it is based, the Euler-Bernoulli and the simple shear models, will be briefly recalled.

The Cartesian reference system consists of the principal axes of the section, x and y, and the axis of the beam (the line where centroids of the sections lie), z. The beam is loaded with a distributed load q(z), acting in the direction y, and a distributed moment m(z) whose axis is in x direction (see Fig. 1). The analysis is limited to the plane y-z, and the only displacements of interest are those in y direction, the deflection v(z). In the following, since the dependence on z is obvious, it will be dropped.

Vibrations of Beams for Seismic Response Estimation, Fig. 1

Reference system of the beam model

In what follows, the balance equations, which relate the loads q and m to the bending moment M and shear force V, will be used:

$$ \frac{\mathrm{dM}}{\mathrm{dz}}=\mathrm{V}-\mathrm{m} $$

(1)

$$ \frac{\mathrm{dV}}{\mathrm{dz}}=-\mathrm{q} $$

(2)

The Euler-Bernoulli Beam

It is assumed that the deflection of the beam is due to the moment only, and therefore this behavior is known as “pure bending.” The constitutive equation which relate the bending moment M to the slope of deflection curve is

$$ \begin{array}{lll}\mathrm{M}=\mathrm{E}\mathrm{I}\frac{\mathrm{d}\uptheta}{\mathrm{d}\mathrm{z}}\hfill & \mathrm{with}\hfill & \uptheta =-\frac{\mathrm{d}v}{\mathrm{d}\mathrm{z}}\hfill \end{array} $$

(3)

where EI is the bending stiffness, the product of the Elastic (Young’s) modulus, E, and the second moment of area (or moment of inertia) for rotation around x, I. The previous relation takes into account the fact that the sections of the beam remain plane in the deformed configuration and normal to the axis.

Using Eqs. 3 and 1, the following equation is obtained:

$$ \mathrm{E}\mathrm{I}\frac{{\mathrm{d}}^2\uptheta}{{\mathrm{d}\mathrm{z}}^2}=\mathrm{V}-\mathrm{m}\Rightarrow -\mathrm{E}\mathrm{I}\frac{{\mathrm{d}}^3v}{{\mathrm{d}\mathrm{z}}^3}=\mathrm{V}-\mathrm{m} $$

Assuming m = 0, differentiating again, and using Eq. 2, the Euler-Bernoulli equation for the beam is obtained:

$$ \frac{{\mathrm{d}}^4v}{{\mathrm{d}\mathrm{z}}^4}=\frac{\mathrm{q}}{\mathrm{EI}} $$

Moreover, the following are also valid:

$$ \begin{array}{l}\mathrm{M}=-\mathrm{E}\mathrm{I}\frac{{\mathrm{d}}^2v}{{\mathrm{d}\mathrm{z}}^2}\hfill \\ {}\mathrm{V}=-\mathrm{E}\mathrm{I}\frac{{\mathrm{d}}^3v}{\mathrm{d}{z}^3}\hfill \end{array} $$

The Euler-Bernoulli model can be extended to the dynamic case using the D’Alembert principle and adding to the external load q the load equivalent to inertia forces associated with deflections, q _i :

$$ {\mathrm{q}}_i=-\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2} $$

where ρ is the density of mass and A is the area of the section. Assuming ρ and A constant along the axis, the equation that rules the problem is

$$ \mathrm{E}\mathrm{I}\frac{\partial^4v}{\partial {\mathrm{z}}^4}+\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=\mathrm{q} $$

The free motion equation is

$$ \frac{\partial^4v}{\partial {\mathrm{z}}^4}+\frac{\uprho \mathrm{A}}{\mathrm{EI}}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=0 $$

with solution that can be written through separation of variables as

$$ v\left(\mathrm{z},\mathrm{t}\right)=\tilde{v}\left(\mathrm{z}\right) \cos \left(\upomega \mathrm{t}+\upphi \right) $$

(4)

where

$$ \tilde{v}\left(\mathrm{z}\right)={\mathrm{D}}_1 \cos \left(\mathrm{k}\mathrm{z}\right)+{\mathrm{D}}_2 \sin \left(\mathrm{k}\mathrm{z}\right)+{\mathrm{D}}_3 \cosh \left(\mathrm{k}\mathrm{z}\right)+{\mathrm{D}}_4 \sinh \left(\mathrm{k}\mathrm{z}\right) $$

and

$$ \mathrm{k}=\sqrt[4]{\frac{\upomega^2\uprho \mathrm{A}}{\mathrm{EI}}} $$

The parameters D₁, D₂, D₃, and D₄ can be obtained using the boundary conditions and determining ω to avoid the trivial solution. For example, in the case of cantilever beam clamped at z = 0 and free at z = L, the following boundary conditions hold

$$ \begin{array}{llll}v\left(0,\mathrm{t}\right)=0,\hfill & \uptheta \left(0,\mathrm{t}\right)=0,\hfill & \mathrm{M}\left(\mathrm{L},\mathrm{t}\right)=0,\hfill & \mathrm{V}\left(\mathrm{L},\mathrm{t}\right)=0\hfill \end{array} $$

and the values of ω are given by the following

$$ \cos \sqrt[4]{\frac{\upomega^2\uprho \mathrm{A}}{\mathrm{EI}}}\mathrm{L}\ \cosh \sqrt[4]{\frac{\upomega^2\uprho \mathrm{A}}{\mathrm{EI}}}\mathrm{L}+1=0 $$

which has solutions

$$ {\upomega}_{\mathrm{j}}={\left({\mathrm{a}}_{\mathrm{j}}\mathrm{L}\right)}^2\frac{1}{{\mathrm{L}}^2}\sqrt{\frac{\mathrm{EI}}{\uprho \mathrm{A}}} $$

with

$$ {\mathrm{a}}_1\mathrm{L}=1.875, {\mathrm{a}}_2\mathrm{L}=4.694, {\mathrm{a}}_3\mathrm{L}=7.855,{\mathrm{a}}_{\mathrm{j}}\mathrm{L}\approx {\mathrm{a}}_3\mathrm{L}+\left(\mathrm{j}-3\right)2\uppi \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=4,5,\dots $$

The Simple Shear Beam

The simple shear beam model is based on the assumption that the beam undergoes only shear deformation. In this case, the constitutive equation which relates V to the slope of deflection curve is

$$ \begin{array}{lll}\mathrm{V}=\mathrm{G}\mathrm{K}\upgamma \hfill & \mathrm{with}\hfill & \upgamma =\frac{\mathrm{d}v}{\mathrm{d}z}\hfill \end{array} $$

(5)

where GK is the shear stiffness, the product of the tangential modulus, G, and shear area K. Moreover, the shear area is related to A by means of a shear factor κ, K = κA, which depends on the shape of the section and is determined by means of energetic equivalence.

Using the preceding and Eq. 2, the following equation is obtained:

$$ \frac{{\mathrm{d}}^2v}{{\mathrm{d}\mathrm{z}}^2}=-\frac{\mathrm{q}}{\mathrm{GK}} $$

In dynamics, by means of D’Alembert principle, the following equation of motion is obtained:

$$ \mathrm{G}\mathrm{K}\frac{\partial^2v}{\partial {\mathrm{z}}^2}-\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=-\mathrm{q} $$

and in free motion

$$ \frac{\partial^2v}{\partial {\mathrm{z}}^2}-\frac{\uprho}{\mathrm{G}\upkappa}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=0 $$

Adopting the separation of variables Eq. 4, the following expression for deflection is obtained:

$$ \tilde{v}\left(\mathrm{z}\right)={\mathrm{D}}_1 \cos \left(\mathrm{k}\mathrm{z}\right)+{\mathrm{D}}_2 \sin \left(\mathrm{k}\mathrm{z}\right) $$

and

$$ \mathrm{k}=\sqrt{\frac{\upomega^2\uprho}{\mathrm{G}\upkappa}} $$

The parameters D₁, D₂ can be obtained using the boundary conditions and determining ω to avoid the trivial solution. For example, in the case of cantilever beam clamped at z = 0 and free at z = L, the boundary conditions are

$$ \begin{array}{ll}v\left(0,\mathrm{t}\right)=0,\hfill & \mathrm{V}\left(\mathrm{L},\mathrm{t}\right)=0\hfill \end{array} $$

and therefore

$$ \begin{array}{ll}{\upomega}_{\mathrm{j}}=\left(2\mathrm{j}-1\right)\frac{\uppi}{2\mathrm{L}}\sqrt{\frac{\mathrm{G}\upkappa}{\uprho}}\hfill & \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=1,2,\dots \hfill \end{array} $$

The Timoshenko Beam Model

In the Timoshenko beam model (Timoshenko 1921, 1922), it is assumed that the slope of the deflection curve is the sum of those due to bending moment and those given by shear deformation, each one assumed to act alone

$$ \frac{\mathrm{d}v}{\mathrm{d}\mathrm{z}}=\upgamma -\uptheta $$

(6)

The first part of both Eqs. 3 and 5 is unchanged, while the second parts are modified according to Eq. 6, and therefore

$$ \begin{array}{ll}\mathrm{V}=\mathrm{G}\mathrm{K}\upgamma =\mathrm{G}\mathrm{K}\left(\frac{\mathrm{d}v}{\mathrm{d}\mathrm{z}}+\uptheta \right),\hfill & \mathrm{M}=\mathrm{E}\mathrm{I}\frac{\mathrm{d}\uptheta}{\mathrm{d}\mathrm{z}}\hfill \end{array} $$

Assuming EI and GK constant along the axis, the following equations are obtained:

$$ \left\{\begin{array}{l}\mathrm{G}\mathrm{K}\left(\frac{{\mathrm{d}}^2v}{{\mathrm{d}\mathrm{z}}^2}+\frac{\mathrm{d}\uptheta}{\mathrm{d}\mathrm{z}}\right)=-\mathrm{q}\hfill \\ {}\mathrm{E}\mathrm{I}\frac{{\mathrm{d}}^2\uptheta}{{\mathrm{d}\mathrm{z}}^2}-\mathrm{G}\mathrm{K}\left(\frac{\mathrm{d}v}{\mathrm{d}\mathrm{z}}+\uptheta \right)=-\mathrm{m}\hfill \end{array}\right. $$

In dynamics the D’Alembert principle is used again. Moreover, in the Timoshenko beam model, the inertial forces associated to rotations θ are also taken into account

$$ {\mathrm{m}}_{\mathrm{i}}=-\uprho \mathrm{I}\frac{\partial^2\uptheta}{\partial {\mathrm{t}}^2} $$

and the equations of motion are therefore

$$ \left\{\begin{array}{l}\mathrm{G}\mathrm{K}\left(\frac{\partial^2v}{\partial {\mathrm{z}}^2}+\frac{\partial \uptheta}{\partial \mathrm{z}}\right)-\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=-\mathrm{q}\hfill \\ {}\mathrm{E}\mathrm{I}\frac{\partial^2\uptheta}{\partial {\mathrm{z}}^2}-\mathrm{G}\mathrm{K}\left(\frac{\partial v}{\partial \mathrm{z}}+\uptheta \right)-\uprho \mathrm{I}\frac{\partial^2\uptheta}{\partial {\mathrm{t}}^2}=-\mathrm{m}\hfill \end{array}\right. $$

(7)

In free motion the following equations hold:

$$ \left\{\begin{array}{l}\mathrm{G}\mathrm{K}\left(\frac{\partial^2v}{\partial {\mathrm{z}}^2}+\frac{\partial \uptheta}{\partial \mathrm{z}}\right)-\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2}=0\hfill \\ {}\mathrm{E}\mathrm{I}\frac{\partial^2\uptheta}{\partial {\mathrm{z}}^2}-\mathrm{G}\mathrm{K}\left(\frac{\partial v}{\partial \mathrm{z}}+\uptheta \right)-\uprho \mathrm{I}\frac{\partial^2\uptheta}{\partial {\mathrm{t}}^2}=0\hfill \end{array}\right. $$

and considering that

$$ \uptheta =\upgamma -\frac{\partial v}{\partial \mathrm{z}}\Rightarrow \frac{\partial \uptheta}{\partial \mathrm{z}}=\frac{\partial \upgamma}{\partial \mathrm{z}}-\frac{\partial^2v}{\partial {\mathrm{z}}^2}=\frac{1}{\mathrm{G}\mathrm{K}}\frac{\partial \mathrm{V}}{\partial \mathrm{z}}-\frac{\partial^2v}{\partial {\mathrm{z}}^2}=-\frac{\uprho}{\mathrm{G}\upkappa}\frac{\partial^2v}{\partial {\mathrm{t}}^2}-\frac{\partial^2v}{\partial {\mathrm{z}}^2} $$

the following equation can be obtained:

$$ \mathrm{E}\mathrm{I}\frac{\partial^4v}{\partial {\mathrm{z}}^4}+\uprho \mathrm{A}\frac{\partial^2v}{\partial {\mathrm{t}}^2}-\uprho \mathrm{I}\left(1+\frac{\mathrm{E}}{\mathrm{G}\upkappa}\right)\frac{\partial^4v}{\partial {\mathrm{z}}^2\partial {\mathrm{t}}^2}+\frac{\uprho^2\mathrm{I}}{\mathrm{G}\upkappa}\frac{\partial^4v}{\partial {\mathrm{t}}^4}=0 $$

The solution is given by

$$ v\left(\mathrm{z},\mathrm{t}\right)=\tilde{v}\left(\mathrm{z}\right) \cos \left(\upomega \mathrm{t}+\upphi \right) $$

where

$$ \tilde{v}\left(\mathrm{z}\right)=\left\{\begin{array}{ll}{\mathrm{D}}_1 \cos \left({\mathrm{k}}_1\mathrm{z}\right)+{\mathrm{D}}_2 \sin \left({\mathrm{k}}_1\mathrm{z}\right)+{\mathrm{D}}_3 \cos\;\mathrm{h}\left({\mathrm{k}}_2\mathrm{z}\right)+{\mathrm{D}}_4 \sin\;\mathrm{h}\left({\mathrm{k}}_2\mathrm{z}\right)\hfill & \upomega <{\upomega}_{\mathrm{c}}\hfill \\ {}{\mathrm{D}}_1 \cos \left({\mathrm{k}}_1\mathrm{z}\right)+{\mathrm{D}}_2 \sin \left({\mathrm{k}}_1\mathrm{z}\right)+{\mathrm{D}}_3 \cos\;\left({\mathrm{k}}_2\mathrm{z}\right)+{\mathrm{D}}_4 \sin\;\left({\mathrm{k}}_2\mathrm{z}\right)\hfill & \upomega\;>{\upomega}_{\mathrm{c}}\hfill \end{array}\right. $$

(8)

where ω_c is a cut-off frequency given by

$$ {\upomega}_{\mathrm{c}}=\frac{\mathrm{G}\upkappa \mathrm{A}}{\uprho \mathrm{I}} $$

As usual, ω is found by imposing the suitable boundary conditions.

It can be interesting to plot the values of ω in terms of variation of slenderness λ = L/r where $\mathrm{r}=\sqrt{\mathrm{I}/\mathrm{A}}$, as in Fig. 2 (the values have been obtained assuming κ = 0.85 and E/G = 2.6). It can be noted that as λ increases, the values ω of the Euler-Bernoulli model are achieved

$$ \frac{\upomega_{\mathrm{j}}\mathrm{L}}{\sqrt{\frac{\mathrm{E}}{{\uprho \uplambda}^2}}}={\left({\mathrm{a}}_{\mathrm{j}}\mathrm{L}\right)}^2 $$

while as λ decreases the values of the simple shear model are recovered

$$ \frac{\upomega_{\mathrm{j}}\mathrm{L}}{\sqrt{\frac{\mathrm{E}}{{\uprho \uplambda}^2}}}=\left(2\mathrm{j}-1\right)\frac{\uppi}{2}\sqrt{\frac{\mathrm{G}}{\mathrm{E}}\upkappa}\cdot \uplambda $$

It is worth noting that the convergence to the Euler-Bernoulli or simple shear beam tends to slow down as mode number increases. For intermediate values of slenderness, the natural frequencies ω depend on both the bending stiffness EI and the shear stiffness GK.

Vibrations of Beams for Seismic Response Estimation, Fig. 2

Pulsations of the first (blue), second (red), and third (green) mode of cantilever beam with κ = 0.85 and E/G = 2.6: (—) Timoshenko beam model, (— —) Euler-Bernoulli beam model, (— · —) simple shear model

Example

The case of a cantilever beam with the following characteristics:

$$ \begin{array}{lll}\mathrm{A}=0.120 {\mathrm{m}\mathrm{m}}^2,\hfill & \mathrm{I}=1.60\cdot {10}^{-3} {\mathrm{m}\mathrm{m}}^4\hfill & \mathrm{K}=0.102\;{\mathrm{m}\mathrm{m}}^2\hfill \\ {}\mathrm{E}=30,000 \mathrm{N}/{\mathrm{m}\mathrm{m}}^2,\hfill & \mathrm{G}=11,538 \mathrm{N}/{\mathrm{m}\mathrm{m}}^2\hfill & \hfill \\ {}\uprho =2,500\;\mathrm{kg}/{\mathrm{m}}^3\hfill & \mathrm{L}=2.0\;\mathrm{m}\hfill & {\upomega}_{\mathrm{c}}=542.43 \mathrm{rad}/\mathrm{s}\hfill \end{array} $$

is considered in what follows.

The first 12 modal shapes are shown in Fig. 3, and the parameters of Eq. 8 are shown in Tables 1 and 2. In the tables also the phase velocity, defined as the ratio between the angular frequency, ω, and the wavenumber, k₁, is reported: as well known, the phase velocity is unbounded for the Euler-Bernoulli beam, while in the case of Timoshenko beam the two different families of waves present when ω &gt; ω_c, with wavenumbers k₁ and k₂, tend to the following values (Hagedorn and DasGupta 2007):

$$ \begin{array}{l}\underset{{\scriptscriptstyle \mathrm{j}\to +\infty }}{ \lim}\frac{\upomega_{\mathrm{j}}}{{\mathrm{k}}_1}=\sqrt{\frac{\mathrm{G}\upkappa}{\uprho}}=62.63 \mathrm{rad}/\mathrm{s}\hfill \\ {}\underset{{\scriptscriptstyle \mathrm{j}\to +\infty }}{ \lim}\frac{\upomega_{\mathrm{j}}}{{\mathrm{k}}_2}=\sqrt{\frac{\mathrm{E}}{\uprho}}=109.54 \mathrm{rad}/\mathrm{s}\hfill \end{array} $$

where j is the mode number.

Vibrations of Beams for Seismic Response Estimation, Fig. 3

First 12 modal shapes of the Euler-Bernoulli beam (dashed red) and Timoshenko beam (solid blue)

Vibrations of Beams for Seismic Response Estimation, Table 1 Parameters of first 12 modes of Euler-Bernoulli beam

Vibrations of Beams for Seismic Response Estimation, Table 2 Parameters of first 12 modes of Timoshenko beam

Deterministic Excitation-Response Relations

The unit impulse applied at the section z = z_o at the instant t = t_o is considered:

$$ \mathrm{f}\left(\mathrm{z},\mathrm{t}\right)=\updelta \left(\mathrm{z}-{\mathrm{z}}_{\mathrm{o}}\right)\updelta \left(\mathrm{t}-{\mathrm{t}}_{\mathrm{o}}\right) $$

Assuming that prior of the unit impulse application the beam is at rest with zero displacement, the response can be expressed by (Robson et al. 1971; Crandal 1979; Lin 1967; Elishakoff 1988)

$$ \mathrm{v}\left(\mathrm{z},\mathrm{t}\right)=\mathrm{h}\left(\mathrm{z},\mathrm{t}-{\mathrm{t}}_{\mathrm{o}};{\mathrm{z}}_{\mathrm{o}}\right) $$

where h(z, t − t_o; z_o) is the impulse response function or Green’s function. A f(z, t) arbitrary space-time action can be regarded as a superposition of unit impulses at the time τ in generic section $0\le \upxi \le \mathrm{L}$

$$ \mathrm{f}\left(\mathrm{z},\mathrm{t}\right)={\displaystyle {\int}_{-\infty}^{+\infty}\mathrm{d}\uptau}{\displaystyle {\int}_0^{\mathrm{L}}\mathrm{f}\left(\upxi, \mathrm{t}-\uptau \right)\updelta \left(\mathrm{z}-\upxi \right)\updelta \left(\uptau \right)\mathrm{d}\upxi} $$

If the impulse response function is known, then the response at the time t in the generic section $0\le \upxi \le \mathrm{L}$ can be obtained by the following convolution integral or superposition integral:

$$ \mathrm{v}\left(\mathrm{z},\mathrm{t}\right)={\displaystyle {\int}_{-\infty}^{+\infty}\mathrm{d}\uptau}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{f}\left(\upxi, \mathrm{t}-\uptau \right)\mathrm{h}\left(\mathrm{z},\uptau; \upxi \right)\mathrm{d}\upxi} $$

The unit complex sinusoidal action with the fixed frequency ω applied at the section z = z_o is

$$ \mathrm{f}\left(\mathrm{z},\mathrm{t}\right)=\updelta \left(\mathrm{z}-{\mathrm{z}}_{\mathrm{o}}\right){\mathrm{e}}^{\mathrm{i}\upomega \mathrm{t}} $$

Given the complex frequency response function H (z, ω; z_o), the response is given by

$$ \mathrm{v}\left(\mathrm{z},\mathrm{t}\right)=\mathrm{H}\left(\mathrm{z},\upomega; {\mathrm{z}}_{\mathrm{o}}\right){\mathrm{e}}^{\mathrm{i}\upomega \mathrm{t}} $$

Let F (z, ω) be the Fourier transform of the action f (z, t) which is assumed to be an integrable function, so that

$$ \mathrm{F}\left(\mathrm{z},\upomega \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{f}\left(\mathrm{z},\mathrm{t}\right)}{\mathrm{e}}^{-\mathrm{i}\upomega \mathrm{t}}\mathrm{d}\mathrm{t} $$

and

$$ \mathrm{f}\left(\mathrm{z},\mathrm{t}\right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{F}\left(\mathrm{z},\upomega \right)}{\mathrm{e}}^{+\mathrm{i}\upomega \mathrm{t}}\mathrm{d}\upomega $$

The complex frequency response function is the Fourier transform of the impulse response function

$$ \begin{array}{l}\mathrm{H}\left(\mathrm{z},\upomega; {\mathrm{z}}_{\mathrm{o}}\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{h}}\left(\mathrm{z},\uptau; {\mathrm{z}}_{\mathrm{o}}\right){\mathrm{e}}^{-\mathrm{i}\upomega \tau}\mathrm{d}\uptau \hfill \\ {}\mathrm{h}\left(\mathrm{z},\uptau; {\mathrm{z}}_{\mathrm{o}}\right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{H}\left(\mathrm{z},\upomega, {\mathrm{z}}_{\mathrm{o}}\right){\mathrm{e}}^{\mathrm{i}\upomega \tau}\mathrm{d}\upomega}\hfill \end{array} $$

Introduced the Fourier transform of the response

$$ \mathrm{V}\left(\mathrm{z},\upomega \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{v}\left(\mathrm{z},\mathrm{t}\right){\mathrm{e}}^{-\mathrm{i}\upomega \mathrm{t}}\mathrm{d}\mathrm{t}} $$

and

$$ \mathrm{v}\left(\mathrm{z},\mathrm{t}\right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{V}\left(\mathrm{z},\upomega \right){\mathrm{e}}^{+\mathrm{i}\upomega \mathrm{t}}\mathrm{d}\upomega} $$

the excitation-response relation for Fourier transform is

$$ \mathrm{V}\left(\mathrm{z},\upomega \right)={\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{H}\left(\mathrm{z},\upomega; \upxi \right)\mathrm{F}\left(\upxi, \upomega \right)\mathrm{d}\upxi} $$

so that

$$ v\left(\mathrm{z},\mathrm{t}\right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{e}}^{\mathrm{i}\upomega \mathrm{t}}\mathrm{d}\upomega}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{H}\left(\mathrm{z},\upomega; \upxi \right)\mathrm{d}\upxi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{f}\left(\upxi, \uptau \right){\mathrm{e}}^{-\mathrm{i}\upomega \tau}\mathrm{d}\uptau} $$

and

$$ \mathrm{V}\left(\mathrm{z},\upomega \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{e}}^{-\mathrm{i}\upomega \tau}\mathrm{d}\upomega}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{h}\left(\mathrm{z},\uptau; \upxi \right)\mathrm{d}\upxi}\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{F}\left(\upxi, \upomega \right){\mathrm{e}}^{+\mathrm{i}\upomega \tau}\mathrm{d}\upomega} $$

Statistical Characteristics of the Excitation

Let f(z, t) be a random space-time processes. In order to describe this processes, it is introduced the ensemble average or expected value

$$ \mathrm{E}\left[\mathrm{f}\left(\mathrm{z},\mathrm{t}\right)\right]={\mathrm{m}}_{\mathrm{f}}\left(\mathrm{z},\mathrm{t}\right) $$

the space-time correlation

$$ \mathrm{E}\left[\mathrm{f}\left({\mathrm{z}}_1,{\mathrm{t}}_1\right)\mathrm{f}\left({\mathrm{z}}_2,{\mathrm{t}}_2\right)\right]={\mathrm{R}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,{\mathrm{t}}_1,{\mathrm{t}}_2\right) $$

and the space-time covariance

$$ \mathrm{E}\left[\left(\mathrm{f}\left({\mathrm{z}}_1,{\mathrm{t}}_1\right)-{\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{t}}_1\right)\right) \left(\mathrm{f}\left({\mathrm{z}}_2,{\mathrm{t}}_2\right)-{\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_2,{\mathrm{t}}_2\right)\right)\right]={\Gamma}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,{\mathrm{t}}_1,{\mathrm{t}}_2\right) $$

It is assumed that the previous characteristics are sufficient to describe the excitation.

Considering the case of weak stationary process, the mean is independent on t, and the space-time correlation depends on τ = t₂ − t₁ so that

$$ \begin{array}{c}\hfill \mathrm{E}\left[\mathrm{f}\left(\mathrm{z},\mathrm{t}\right)\right]={\mathrm{m}}_{\mathrm{f}}\left(\mathrm{z}\right)\hfill \\ {}\hfill \mathrm{E}\left[\mathrm{f}\left({\mathrm{z}}_1,\mathrm{t}\right)\mathrm{f}\left({\mathrm{z}}_2,\mathrm{t}+\uptau \right)\right]={\mathrm{R}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right)\hfill \\ {}\hfill \mathrm{E}\left[\left(\mathrm{f}\left({\mathrm{z}}_1,\mathrm{t}\right)-{\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_1\right)\right)\left(\mathrm{f}\left({\mathrm{z}}_2,\mathrm{t}+\uptau \right)-{\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_2\right)\right)\right]=\hfill \\ {}\hfill ={\Gamma}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\ \uptau \right)={\mathrm{R}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right)-{\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_1\right){\mathrm{m}}_{\mathrm{f}}\left({\mathrm{z}}_2\right)\hfill \end{array} $$

If z₁ = z₂ = z, the autocovariance function is obtained

$$ {\Gamma}_{\mathrm{f}}\left(\mathrm{z},\uptau \right)={\mathrm{R}}_{\mathrm{f}}\left(\mathrm{z},\uptau \right)-{\mathrm{m}}_{\mathrm{f}}^2\left(\mathrm{z}\right) $$

that for τ = 0 reduces to the variance

$$ {\upsigma}_{\mathrm{f}}^2(x)=\mathrm{E}\left[{\mathrm{f}}^2\left(\mathrm{z}\right)\right]-{\mathrm{m}}_{\mathrm{f}}^2\left(\mathrm{z}\right) $$

where ${\left.{\mathrm{R}}_{\mathrm{f}}\left(\mathrm{z},\uptau \right)\right|}_{\tau =0}=\mathrm{E}\left[{\mathrm{f}}^2\left(\mathrm{z}\right)\right]$ is the mean-square value.

Let S_f(z₁, z₂, ω) be the space-time cross-spectral density function; this function is connected to the cross-correlation function by the Wiener-Khintchine relations:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\upomega \right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{R}}_{\mathrm{f}}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right){\mathrm{e}}^{-\mathrm{i}\upomega \uptau}\mathrm{d}\uptau \hfill \\ {}{\mathrm{R}}_{\mathrm{f}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{S}}_{\mathrm{f}}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\upomega \right){\mathrm{e}}^{\mathrm{i}\upomega \uptau}\mathrm{d}\upomega \hfill \end{array} $$

If z₁ = z₂ = z and τ = 0, the mean-square value can be expressed by the cross-spectral density function

$$ \mathrm{E}\left[{\mathrm{f}}^2\left(\mathrm{z}\right)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{S}}_{\mathrm{f}}\left(\mathrm{z}, \mathrm{z}, \upomega \right)\mathrm{d}\upomega} $$

and for excitation with zero mean

$$ {\upsigma}_{\mathrm{f}}^2\left(\mathrm{x}\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{S}}_{\mathrm{f}}\left(\mathrm{z},\mathrm{z},\upomega \right)\mathrm{d}\upomega} $$

It is worth noting that earthquake-induced base accelerations are samples of non-stationary random process. Anyway, there is an interval, called strong motion phase, where the base accelerations can be considered samples of stationary random process. The length of strong motion phase is T_S. It is commonly assumed that, if T_S is much greater than the period of the elementary oscillator, both the base accelerations and the response can be considered samples of stationary random process with zero mean.

Statistical Characteristics of the Response

The statistical characteristics of the response can be obtained from corresponding characteristics of the excitation by taking the averages of the previously introduced relations. The expected value of the response can be obtained by equation

$$ \mathrm{E}\left[\mathrm{v}\left(\mathrm{z},\mathrm{t}\right)\right]=\mathrm{E}\left[{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}\uptau}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{f}\left(\upxi, \mathrm{t}-\uptau \right)\mathrm{h}\left(\mathrm{z},\uptau, \upxi \right)\mathrm{d}\upxi}\right] $$

and interchanging the integral operators and mean operator, the following equation is obtained:

$$ \mathrm{E}\left[\mathrm{v}\left(\mathrm{z},\mathrm{t}\right)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}\uptau}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{E}\left[\mathrm{f}\left(\upxi, \mathrm{t}-\uptau \right)\right]\mathrm{h}\left(\mathrm{z},\uptau; \upxi \right)\mathrm{d}\upxi} $$

where E[f(z, t)] is the mean of the excitation. If the excitation is a stationary processes, its mean value is constant, E[f(z, t)] = m_f(z), and also the mean value of the response is constant, E[v(z, t)] = m_v(z), where

$$ {\mathrm{m}}_{\mathrm{v}}\left(\mathrm{z}\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{d}\uptau}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{m}}_{\mathrm{f}}\left(\upxi \right)\mathrm{h}\left(\mathrm{z},\uptau; \upxi \right)\mathrm{d}\upxi} $$

If in addition m_f(z) = 0, as for the seismic excitation, then m_v(z) = 0.

The space-time correlation function of the response is obtained by average of the product of v(z₁, t) and v(z₂, t + τ), which are given by the impulse response function

$$ \mathrm{E}\left[\mathrm{v}\left({\mathrm{z}}_1,\mathrm{t}\right),\mathrm{v}\left({\mathrm{z}}_2,\mathrm{t}+\uptau \right)\right]=\mathrm{E}\left[{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_1}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{f}\left({\upxi}_1,\mathrm{t}-{\uptheta}_1\right)\mathrm{h}\left({\mathrm{z}}_1,{\uptheta}_1;{\upxi}_1\right){\mathrm{d}\upxi}_1}\right.\times \left.{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_2}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{f}\left({\upxi}_2,\mathrm{t}+\uptau -{\uptheta}_2\right)\mathrm{h}\left({\mathrm{z}}_2,{\uptheta}_2;{\upxi}_2\right){\mathrm{d}\upxi}_2}\right] $$

and interchanging the integral operators and mean operator

$$ \mathrm{E}\left[\mathrm{v}\left({\mathrm{z}}_1,\mathrm{t}\right)\mathrm{v}\left({\mathrm{z}}_2,\mathrm{t}+\uptau \right)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_1}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_2}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{d}\upxi}_1}\times {\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{E}\left[\mathrm{f}\left({\upxi}_1,\mathrm{t}-{\uptheta}_1\right)\mathrm{f}\left({\upxi}_2,\mathrm{t}+\uptau -{\uptheta}_2\right)\right]\mathrm{h}\left({\mathrm{z}}_1,{\uptheta}_1;{\upxi}_1\right)\mathrm{h}\left({\mathrm{z}}_2,{\uptheta}_2;{\upxi}_2\right){\mathrm{d}\upxi}_2} $$

If the excitation is stationary, then its space-time correlation is independent of t and so is the response space-time correlation

$$ {\mathrm{R}}_{\mathrm{v}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_1}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{d}\uptheta}_2}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{d}\upxi}_1}\times {\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{R}}_{\mathrm{f}}\left({\upxi}_1,{\upxi}_2,\uptau +{\uptheta}_1-{\uptheta}_2\right)\mathrm{h}\left({\mathrm{z}}_1,{\uptheta}_1;{\upxi}_1\right)\mathrm{h}\left({\mathrm{z}}_2,{\uptheta}_2;{\upxi}_2\right){\mathrm{d}\upxi}_2} $$

Applying the Wiener-Khintchine relations, the space-time cross-spectral density of the response can be obtained by the following:

$$ {\mathrm{S}}_{\mathrm{v}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\upomega \right)={\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{d}\upxi}_1}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{d}\upxi}_2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{h}\left({\mathrm{z}}_1,{\uptheta}_1;{\upxi}_1\right){\mathrm{d}\uptheta}_1}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{h}\left({\mathrm{z}}_2,{\uptheta}_2;{\upxi}_2\right){\mathrm{d}\uptheta}_2}\times \frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{R}}_{\mathrm{f}}\left[{\upxi}_1,{\upxi}_2,\uptau +{\uptheta}_1-{\uptheta}_2\right]{\mathrm{e}}^{-\mathrm{i}\upomega \uptau}\mathrm{d}\uptau} $$

Introducing in the previous integrals the unity products ${\mathrm{e}}^{-{\mathrm{i}\upomega \uptheta}_1}{\mathrm{e}}^{{\mathrm{i}\upomega \uptheta}_1}$ and ${\mathrm{e}}^{-{\mathrm{i}\upomega \uptheta}_2}{\mathrm{e}}^{{\mathrm{i}\upomega \uptheta}_2}$, by the frequency response function

$$ \begin{array}{l}\mathrm{H}\left({\mathrm{z}}_1,-\upomega; {\upxi}_1\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{h}\left({\mathrm{z}}_1,{\uptheta}_1;{\upxi}_1\right){\mathrm{e}}^{{\mathrm{i}\upomega \uptheta}_1}{\mathrm{d}\uptheta}_1}\hfill \\ {}\mathrm{H}\left({\mathrm{z}}_2,\upomega; {\upxi}_2\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\mathrm{h}\left({\mathrm{z}}_2,{\uptheta}_1;{\upxi}_2\right){\mathrm{e}}^{-{\mathrm{i}\upomega \uptheta}_2}{\mathrm{d}\uptheta}_2}\hfill \end{array} $$

and introducing the space-time cross-spectral density of the excitation,

$$ {\mathrm{S}}_{\mathrm{f}}\left({\upxi}_1,{\upxi}_2,\upomega \right)=\frac{1}{2\uppi}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{R}}_{\mathrm{f}}\left[{\upxi}_1,{\upxi}_2,\uptau +{\uptheta}_1-{\uptheta}_2\right]{\mathrm{e}}^{-\mathrm{i}\upomega \left(\uptau +{\uptheta}_1-{\uptheta}_2\right)}\mathrm{d}\left(\uptau +{\uptheta}_1-{\uptheta}_2\right)} $$

the following expression to determine S_v(z₁, z₂, ω) is obtained:

$$ {\mathrm{S}}_{\mathrm{v}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\upomega \right)={\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}\mathrm{H}\left({\mathrm{z}}_1,- \upomega, {\upxi}_1\right){\mathrm{S}}_{\mathrm{f}}\left[{\upxi}_1,{\upxi}_2,\upomega \right]\mathrm{H}\left({\mathrm{z}}_2,\upomega, {\upxi}_2\right){\mathrm{d}\upxi}_1{\mathrm{d}\upxi}_2}} $$

Natural Modes and Frequencies

The dynamic proprieties of the systems can be described in terms of natural modes, ṽ, and corresponding natural frequencies, ω. In what follows, the natural modes will be expressed in terms of both deflection, ${v}_{\mathrm{j}}\left(\mathrm{z}\right)=\tilde{v}\left(\mathrm{z}\right)$, and rotation, θ_j(z), where j is the mode number. Once the natural modes have been found, the equations of motion Eq. 7 can be reduced to

$$ {\mathrm{m}}_{\mathrm{j}}\frac{{\mathrm{d}}^2{\mathrm{g}}_{\mathrm{j}}}{{\mathrm{d}\mathrm{t}}^2}+{\mathrm{k}}_{\mathrm{j}}{\mathrm{g}}_{\mathrm{j}}={\mathrm{f}}_{\mathrm{j}} $$

where m_j, k_j, and f_j are, respectively, the generalized mass, stiffness, and force corresponding to mode j, given by

$$ \left\{\begin{array}{l}{\mathrm{m}}_{\mathrm{j}}={\displaystyle {\int}_0^{\mathrm{L}}\left({\uprho \mathrm{A}\mathrm{v}}_{\mathrm{j}}^2+{\uprho \mathrm{I}\uptheta}_{\mathrm{j}}^2\right)} \mathrm{d}\mathrm{z}\hfill \\ {}\begin{array}{ll}{\mathrm{k}}_{\mathrm{j}}={\displaystyle {\int}_0^{\mathrm{L}}\left(-\mathrm{E}\mathrm{I}{\left(\frac{{\mathrm{d}\uptheta}_{\mathrm{j}}}{\mathrm{d}\mathrm{z}}\right)}^2+\mathrm{G}\mathrm{K}{\left(\frac{{\mathrm{d}\upgamma}_{\mathrm{j}}}{\mathrm{d}\mathrm{z}}\right)}^2\right)}\mathrm{d}\mathrm{z},\hfill & \mathrm{with} {\mathrm{k}}_{\mathrm{j}}={\upomega}_{\mathrm{j}}^2{\mathrm{m}}_{\mathrm{j}}\hfill \end{array}\hfill \\ {}{\mathrm{f}}_{\mathrm{j}}={\displaystyle {\int}_0^{\mathrm{L}}\left({\mathrm{qv}}_{\mathrm{j}}+{\mathrm{m}\uptheta}_{\mathrm{j}}\right)}\mathrm{d}\mathrm{z}\hfill \end{array}\right. $$

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and g_j is the jth principal coordinate. In the preceding, the following orthogonality properties of the modes have been used (Bishop and Price 1977):

$$ \left\{\begin{array}{l}{\displaystyle {\int}_0^{\mathrm{L}}\left({\uprho \mathrm{A}\mathrm{v}}_{\mathrm{j}}{\mathrm{v}}_{\mathrm{k}}+{\uprho \mathrm{I}\uptheta}_{\mathrm{j}}{\uptheta}_{\mathrm{k}}\right)}\mathrm{d}\mathrm{z}={\mathrm{a}}_{\mathrm{k}\mathrm{j}}{\updelta}_{\mathrm{k}\mathrm{j}}\hfill \\ {}\begin{array}{lll}{\displaystyle {\int}_0^{\mathrm{L}}\left(-\mathrm{E}\mathrm{I}\frac{{\mathrm{d}\uptheta}_{\mathrm{j}}}{\mathrm{d}\mathrm{z}}\frac{{\mathrm{d}\uptheta}_{\mathrm{k}}}{\mathrm{d}\mathrm{z}}+\mathrm{G}\mathrm{K}\frac{{\mathrm{d}\upgamma}_{\mathrm{j}}}{\mathrm{d}\mathrm{z}}\frac{{\mathrm{d}\upgamma}_{\mathrm{k}}}{\mathrm{d}\mathrm{z}}\right)}\mathrm{d}\mathrm{z}={\mathrm{a}}_{\mathrm{k}\mathrm{j}}{\upomega}_{\mathrm{k}}^2{\updelta}_{\mathrm{k}\mathrm{j}},\hfill & \mathrm{with}\hfill & {\updelta}_{\mathrm{k}\mathrm{j}}=\left\{\begin{array}{ll}1\hfill & \mathrm{if} \mathrm{k}=\mathrm{j}\hfill \\ {}0\hfill & \mathrm{if} \mathrm{k}\ne \mathrm{j}\hfill \end{array}\right.\hfill \end{array}\hfill \end{array}\right. $$

Once the g_j are known, the values of displacements are given by

$$ \begin{array}{ll}\mathrm{v}\left(\mathrm{z},\mathrm{t}\right)={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\mathrm{v}}_{\mathrm{j}}\left(\mathrm{z}\right){\mathrm{g}}_{\mathrm{j}}\left(\mathrm{t}\right),}\hfill & \uptheta \left(\mathrm{z},\mathrm{t}\right)={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\uptheta}_{\mathrm{j}}\left(\mathrm{z}\right){\mathrm{g}}_{\mathrm{j}}\left(\mathrm{t}\right)}\hfill \end{array} $$

If a modal damping v_j is assumed, the equations of motion are

$$ {\mathrm{m}}_{\mathrm{j}}\frac{{\mathrm{d}}^2{\mathrm{g}}_{\mathrm{j}}}{{\mathrm{d}\mathrm{t}}^2}+2{\upnu}_{\mathrm{j}}\sqrt{{\mathrm{k}}_{\mathrm{j}}\;{\mathrm{m}}_{\mathrm{j}}}\frac{{\mathrm{d}\mathrm{g}}_{\mathrm{j}}}{\mathrm{d}\mathrm{t}}+{\mathrm{k}}_{\mathrm{j}}{\mathrm{g}}_{\mathrm{j}}={\mathrm{f}}_{\mathrm{j}} $$

that can also be written as

$$ \frac{{\mathrm{d}}^2{\mathrm{g}}_{\mathrm{j}}}{{\mathrm{d}\mathrm{t}}^2}+2{\upnu}_{\mathrm{j}}{\upomega}_{\mathrm{j}}\frac{{\mathrm{d}\mathrm{g}}_{\mathrm{j}}}{\mathrm{d}\mathrm{t}}+{\upomega}_{\mathrm{j}}^2{\mathrm{g}}_{\mathrm{j}}=\frac{1}{{\mathrm{m}}_{\mathrm{j}}}{\mathrm{f}}_{\mathrm{j}} $$

Let h_j(t) be the single degree of freedom unit impulse response function. With initial conditions g_i = 0 and dg_j/dt = 0, and assuming distributed load q = δ (z – z₀) δ(t) and distributed moments m = 0, the solution is

$$ {\mathrm{g}}_{\mathrm{j}}\left(\mathrm{t};{\mathrm{z}}_0\right)={\mathrm{h}}_{\mathrm{j}}\left(\mathrm{t}\right){\mathrm{v}}_{\mathrm{j}}\left({\mathrm{z}}_0\right) $$

where

$$ {\mathrm{h}}_{\mathrm{j}}\left(\mathrm{t}\right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \mathrm{t}\le 0\hfill \\ {}\hfill \frac{1}{{\mathrm{m}\upomega}_{\mathrm{j}}\sqrt{1-{\upnu}_{\mathrm{j}}^2}}{\mathrm{e}}^{-{\upnu}_{\mathrm{j}}{\upomega}_{\mathrm{j}}\mathrm{t}} \sin \left({\upomega}_{\mathrm{j}}\sqrt{1-{\upnu}_{\mathrm{j}}^2} \mathrm{t}\right)\hfill & \hfill \mathrm{t}>0\hfill \end{array}\right. $$

so that

$$ \mathrm{h}\left(\mathrm{z},\mathrm{t};{\mathrm{z}}_0\right)={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\mathrm{v}}_{\mathrm{j}}\left(\mathrm{z}\right){\mathrm{v}}_{\mathrm{j}}\left({\mathrm{z}}_0\right){\mathrm{h}}_{\mathrm{j}}\left(\mathrm{t}\right)} $$

and using the Fourier transform relation, the complex frequency response is

$$ \mathrm{H}\left(\mathrm{z},\upomega; {\mathrm{z}}_0\right)={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\mathrm{v}}_{\mathrm{j}}\left(\mathrm{z}\right){\mathrm{v}}_{\mathrm{j}}\left({\mathrm{z}}_0\right){\mathrm{H}}_{\mathrm{j}}\left(\upomega \right)} $$

where the single degree of freedom unit complex frequency response function is given by

$$ {\mathrm{H}}_{\mathrm{j}}\left(\upomega \right)=\frac{1}{\mathrm{m}\left({\upomega}_{\mathrm{j}}^2-{\upomega}^2+2{\upnu}_{\mathrm{j}}{\upomega}_{\mathrm{j}}\upomega \mathrm{i}\right)} $$

Inserting in the previous equations, the following expressions for space-time correlation and space-time cross-spectral density are obtained:

$$ {\mathrm{R}}_{\mathrm{v}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\uptau \right)={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\displaystyle \sum_{\mathrm{k}=1}^{\infty }{\mathrm{v}}_{\mathrm{j}}\left({\mathrm{z}}_1\right){\mathrm{v}}_{\mathrm{k}}\left({\mathrm{z}}_2\right)}}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{h}}_{\mathrm{j}}\left({\uptheta}_1\right){\mathrm{h}}_{\mathrm{k}}\left({\uptheta}_2\right){\mathrm{d}\uptheta}_1{\mathrm{d}\uptheta}_2}}\times {\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{v}}_{\mathrm{j}}\left({\upxi}_1\right){\mathrm{v}}_{\mathrm{k}}\left({\upxi}_2\right){\mathrm{R}}_{\mathrm{f}}\left[{\upxi}_1,{\upxi}_2,\uptau +{\uptheta}_1-{\uptheta}_2\right]{\mathrm{d}\upxi}_1{\mathrm{d}\upxi}_2}} $$

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$$ \begin{array}{ll}{\mathrm{S}}_{\mathrm{v}}\left({\mathrm{z}}_1,{\mathrm{z}}_2,\upomega \right)\hfill & ={\displaystyle \sum_{\mathrm{j}=1}^{\infty }{\displaystyle \sum_{\mathrm{k}=1}^{\infty }{\mathrm{v}}_{\mathrm{j}}\left({\mathrm{z}}_1\right){\mathrm{v}}_{\mathrm{k}}\left({\mathrm{z}}_2\right){\mathrm{H}}_{\mathrm{j}}\left(-\upomega \right){\mathrm{H}}_{\mathrm{k}}\left(+\upomega \right)}}\times {\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\displaystyle \underset{0}{\overset{\mathrm{L}}{\int }}{\mathrm{v}}_{\mathrm{j}}\left({\upxi}_1\right){\mathrm{v}}_{\mathrm{k}}\left({\upxi}_2\right){\mathrm{S}}_{\mathrm{f}}\left[{\upxi}_1,{\upxi}_2,\upomega \right]{\mathrm{d}\upxi}_1{\mathrm{d}\upxi}_2}}\hfill \end{array} $$

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Application to the Beam Under Seismic Load

The concepts illustrated in the previous sections are applied to the case of the cantilever beam seen previously in the example, subjected to a seismic load. The seismic load is modeled by means of an elastic response spectrum according to the European standard for seismic actions (Eurocode 8 2004). It has been assumed a type 1 elastic response spectrum with a ground type C, with parameters:

$$ \begin{array}{lll}{\mathrm{a}}_{\mathrm{g}}=0.355\;\mathrm{g}\;\left(\mathrm{Peak} \mathrm{g}\mathrm{round} \mathrm{acceleration}\right),\hfill & \mathrm{S}=1.15 \left(\mathrm{Soil} \mathrm{factor}\right)\hfill & \hfill \\ {}{\mathrm{T}}_{\mathrm{B}}=0.20\;\mathrm{s},\hfill & {\mathrm{T}}_{\mathrm{C}}=0.60\;s,\hfill & {\mathrm{T}}_{\mathrm{D}}=2.00\;\mathrm{s}\hfill \end{array} $$

A 5 % damping, ν = 0.05, was adopted. The response spectrum in terms of pseudo-accelerations (RSA) is shown in Fig. 4a.

Vibrations of Beams for Seismic Response Estimation, Fig. 4

Elastic response spectrum according to Eurocode 8 and corresponding power spectral density function

In order to apply what presented in previous sections, the power spectral density function, ${\mathrm{G}}_{{\ddot{\mathrm{u}}}_{\mathrm{g}}}$, corresponding to the assumed response spectrum is needed. The method proposed in (Cacciola et al. 2004) is used. The time observing window (coincident with the strong motion phase) T_S has been set to 10s, while the cut-off frequency is ω_u = 150 rad/s and the frequency step is Δω = 2π/T_S = 0.63 rad/s. An iterative scheme has been used as suggested in (Cacciola 2010), and the power spectral density function obtained is shown in Fig. 4b.

The generation of a time history of base acceleration is achieved by means of superposition of N_a harmonics with random phases Φ_i as follows:

$$ {\ddot{\mathrm{u}}}_{\mathrm{g}}\left(\mathrm{t}\right)=\upvarphi \left(\mathrm{t}\right){\displaystyle \sum_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{a}}}\sqrt{2{\mathrm{G}}_{{\ddot{\mathrm{u}}}_{\mathrm{g}}}\left(\mathrm{i}\Delta \upomega \right)\Delta \upomega}} \cos \left(\mathrm{i}\Delta \upomega +{\Phi}_{\mathrm{i}}\right) $$

where φ(t) is the modulating function (which changes the total energy but not its distribution among the frequencies; see Fig. 4b) (Cacciola 2010):

$$ \upvarphi \left(\mathrm{t}\right)=\left\{\begin{array}{ll}{\left(\frac{\mathrm{t}}{{\mathrm{t}}_1}\right)}^2\hfill & \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\le {\mathrm{t}}_1\hfill \\ {}1\hfill & \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{t}}_1<\mathrm{t}\le {\mathrm{t}}_2\hfill \\ {}{\mathrm{e}}^{-\upbeta \left(\mathrm{t}-{\mathrm{t}}_2\right)}\hfill & \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}>{\mathrm{t}}_2\hfill \end{array}\right. $$

with t₂ = t₁ + T_S. In the present case, it was assumed t₁ = 10s and β = 0.3 s⁻¹. An example of the generated ü_g(t) is shown in Fig. 5.

Vibrations of Beams for Seismic Response Estimation, Fig. 5

Example of a time history of acceleration compatible to power spectral density function ${\mathrm{G}}_{{\ddot{\mathrm{u}}}_{\mathrm{g}}}$

The problem of the beam subjected to the base acceleration given by ü_g(t) can be solved by means of modal analysis, as formulated in Eq. 9.

In the case of seismic action represented by base accelerations ü_g(t), the external loads are

$$ \left\{\begin{array}{l}\mathrm{q}=-\uprho \mathrm{A}{\ddot{\mathrm{u}}}_{\mathrm{g}}\left(\mathrm{t}\right)\hfill \\ {}\mathrm{m}=0\hfill \end{array}\right. $$

and therefore:

$$ {\mathrm{f}}_{\mathrm{j}}=-{\displaystyle {\int}_0^{\mathrm{L}}\uprho}\mathrm{A}{\ddot{\mathrm{u}}}_{\mathrm{g}}\left(\mathrm{t}\right){\mathrm{v}}_{\mathrm{j}}\mathrm{d}\mathrm{z}=-{\Gamma}_{\mathrm{j}}{\ddot{\mathrm{u}}}_{\mathrm{g}}\left(\mathrm{t}\right),\mathrm{with} {\Gamma}_{\mathrm{j}}={\displaystyle {\int}_0^{\mathrm{L}}\uprho}{\mathrm{Av}}_{\mathrm{j}}\mathrm{d}\mathrm{z} $$

Once the time histories of g_j have been determinated, the solution is given by

$$ \begin{array}{ll}\mathrm{v}\left(\mathrm{z},\mathrm{t}\right)={\displaystyle \sum_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{p}}}{\mathrm{v}}_{\mathrm{j}}}\left(\mathrm{z}\right){\mathrm{g}}_{\mathrm{j}}\left(\mathrm{t}\right),\hfill & \uptheta \left(\mathrm{z},\mathrm{t}\right)={\displaystyle \sum_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{p}}}{\uptheta}_{\mathrm{j}}}\left(\mathrm{z}\right){\mathrm{g}}_{\mathrm{j}}\left(\mathrm{t}\right)\hfill \end{array} $$

where N_p is the number of modes considered in the solution. For example, the displacement at the free end, ν(L), for the base acceleration shown in Fig. 5 and assuming N_p = 10 (the modal contribution tends to be negligible quite rapidly) is shown in Fig. 6.

Vibrations of Beams for Seismic Response Estimation, Fig. 6

Displacement at free end for base accelerations of Fig. 5

The power spectral density function of the displacement at free end, S_ν(L, L, ω) is shown in Fig. 7, compared to that evaluated by mean of Eq. 11. As can be appreciated, the same results are obtained using both procedures.

Vibrations of Beams for Seismic Response Estimation, Fig. 7

Power spectral density function of free end displacement by means of modal analysis with generated spectrum compatible accelerograms and by means of stochastic analysis

Cross-References

• Random Process as Earthquake Motions

• Stochastic Analysis of Linear Systems

• Stochastic Ground Motion Simulation

• Time History Seismic Analysis