Evaluation of the Dynamic Parameters Under Seismic Conditions for a Maxwell Rheological Base Isolation System

Abstract

The connections of seismic isolation devices for mitigating seismic shocks in the fundamental excitation mode are designed and implemented based on the serial combination of elastomeric isolators, which are primarily elastic, with fluid-based isolators, which are primarily viscous. The energy dissipated in the fluidic isolators represents a significant parameter for ensuring the attenuation degree of the amplitude of the displacement of the system as well as for its energy dissipation capacity as a direct effect on deformability and speed of the heat transfer. For bridges, viaducts, and buildings, families of elastomeric and fluid isolators connected in series are used to enable both analytical and experimental evaluations of the system’s dynamic isolation and energy dissipation capacities. Based on the results obtained from specialized isolation devices from Italy and the numerical and experimental evaluations carried out by ICECON S.A. Bucharest, Romania, this article will address the aforementioned topic.

1. Introduction

The linear viscoelastic connections employed are based on the serial coupling of the elastomeric devices with the equivalent stiffness k with the fluid-based devices having the equivalent viscous damping constant c [1,2,3].

For large displacements permitted during seismic motions [4], which do not compromise the structural functionality and safety of the structure, base isolation solutions using the Maxwell model are adopted [5]. These solutions involve bearing groups with horizontal connections between the superstructure of the building and the supporting base embedded in the ground subjected to seismic motion [5,6].

Each bearing group is constructed as a unit, achieved by serially connecting a linear elastomeric isolator with a linear fluidic device [7,8,9], with the primary function of providing dynamic isolation in the horizontal direction [10,11,12,13].

The computational model that is presented in this paper forms the basis for the design and development of highly efficient base isolation systems [14,15].

The computational relations are derived from the dynamic model of a single degree of freedom system, applicable to a rigid-type structure with translational motions along the three principal orthogonal axes.

Essentially, the approach focuses on a single direction of instantaneous translational motion, with the determination of the displacement parameters, dynamic isolation, and energy dissipation in the fluidic damper [16,17]. For different dissipation devices with distinct values of viscous damping c_i, i = 1, …, n, then the compound damping established in [18] can be used.

The novelty of the concept and the results obtained in this study lies in the fact that, based on favorable connections such as total rigidity and compound damping, the designer can choose from the category of CE-marked certified products the most suitable standard dimensions that can ensure a base isolation system with seismic energy isolation and dissipation performance at the values established on efficiency criteria.

2. System Amplitudes

The model of the dynamic system with a Maxwell rheological connection [5], which transmits the seismic kinematic excitation from point O to the receiver point A along the fundamental component of the seismic spectrum, is shown in Figure 1. The viscoelastic Maxwell model, having an elastic component connected in series with a viscous component (E–V), has equivalent viscosity coefficients c [19] and equivalent stiffness coefficients k.

The equivalent dynamic scheme in Figure 1 represents the physical model relative to the reference axis O₁X, which is considered fixed. The moving reference point O signifies the point of seismic excitation of the fundamental spectral excitation component (ω, X₀) having the pulsation ω = 2π/T₀, where T₀ represents the fundamental period of the excitation spectral line, and the amplitude X₀ corresponding to the acceleration a₀ = ω²X₀ of the fundamental excitation spectral line (ω, a₀) [20,21]. In this case, the seismic excitation following the fundamental, unidirectional spectral component can be expressed by the equation $x_0=x_0\left(t\right)=X_0\sin \omega t$ or, in the complex domain, by ${\tilde{x}}_0=X_0{e}^{\sin \omega t}$ where $x_0\left(t\right)=\mathrm{I}\mathrm{m}\left({\tilde{x}}_0\right)$.

Depending on the order of the rheological parameters c and k, the deformations in the viscous and elastic media will differ [3,4,6]. This paper focuses on the analysis of the Maxwell dynamic system—CKM.

Based on the dynamic analysis of the system, the parameters for the instantaneous viscous and elastic deformations, as well as the corresponding amplitude values, are determined. In the end, the calculation formulas for the seismic excitation transmissibility and dissipated energy will be presented.

In the moving coordinate system, with the origin in point O, the instantaneous relative displacements concerning the reference point O are x = x(t) and y = y(t) and the instantaneous deformation of the viscous damper with equivalent constant c is h = h(t) = y − x₀ [22]. For the entire viscoelastic system (E–V), the instantaneous deformation is d = d(t) = x − x₀.

The calculation parameters for the case study are the following: $k=4800\mathrm{ kN}/\mathrm{m};$ m $=3\times {10}^6\mathrm{ Kg};$ $X_0=0.3\mathrm{ m};$ $\omega =3.15\mathrm{ rad}/\mathrm{s};$ ${\omega }_n=1.25\mathrm{ rad}/\mathrm{s};$ $a_0=3\mathrm{ m}/{\mathrm{s}}^2.$

The differential equations of relative motion can be written as follows:

$$\left\{\begin{array}{c}m\ddot{x}=-k(x-y)\\ c\left(\dot{y}-{\dot{x}}_0\right)=k(x-y)\end{array}\right.$$

(1)

or, in the complex formulation, the differential equations of motion can be expressed as follows:

$$\left\{\begin{array}{c}m\ddot{\tilde{x}}=-k(\tilde{x}-\tilde{y})\\ c\left(\dot{\tilde{y}}-{\dot{\tilde{x}}}_0\right)=k(\tilde{x}-\tilde{y})\end{array}\right.$$

(2)

in which

$$\begin{array}{cc}{\tilde{x}}_0=X_0{e}^{j\omega t};& {\dot{\tilde{x}}}_0={j\omega e}^{j\omega t}\\ \tilde{x}=\tilde{A}{e}^{j\omega t};& \tilde{A}=A{e}^{j{\phi }_2}\\ \tilde{y}=\tilde{B}{e}^{j\omega t};& \tilde{B}=B{e}^{j{\phi }_1}\end{array}.$$

(3)

Introducing $\tilde{x}$, $\ddot{\tilde{x}}$, $\tilde{y}$, $\dot{\tilde{y}}$ and ${\dot{\tilde{x}}}_0$ in Equation (2), the following system is obtained using $\tilde{A}$ and $\tilde{B}$:

$$\left\{\begin{array}{c}\left(k-m{\omega }^2\right)\tilde{A}-k\tilde{B}=0\\ -k\tilde{A}+\left(k+jc\omega \right)\tilde{B}=jc\omega X_0\end{array}\right..$$

(4)

The determinant of the unknowns $\tilde{A}$ and $\tilde{B}$ results in

$$\tilde{D}=-km{\omega }^2+jc\omega (k-m{\omega }^2).$$

(5)

$\tilde{A}$ and $\tilde{B}$ from Equation (4) will result in

$$\left\{\begin{array}{c}\tilde{A}=-X_0{\displaystyle \frac{jc\omega k}{\tilde{D}}}\\ \tilde{B}=X_0{\displaystyle \frac{jc\omega (k-m{\omega }^2)}{\tilde{D}}}\end{array}\right.,$$

(6)

or in the form of complex functions as follows:

$$\left\{\begin{array}{c}\tilde{A}=-X_0{\displaystyle \frac{1}{D^2}}\left[c^2{\omega }^{2}k\left(k-m{\omega }^2\right)-j{k}^{2}mc{\omega }^3\right]\\ \tilde{B}=X_0{\displaystyle \frac{1}{D^2}}\left[c^2{\omega }^2{\left(k-m{\omega }^2\right)}^2-jkmc{\omega }^3(k-m{\omega }^2)\right]\end{array}\right.,$$

(7)

in which

$${\left|\tilde{D}\right|}^2=D^2=k^2{m}^2{\omega }^4+c^2{\omega }^2{(k-m{\omega }^2)}^2.$$

(8)

From Equation (7), the following relation can be obtained:

$${\left|\tilde{A}\right|}^2=A^2=X_0^2{\displaystyle \frac{k^2{c}^2{\omega }^2}{D^2}},$$

(9)

from which

$$A\left(c,\omega \right)=X_0{\displaystyle \frac{kc\omega }{\sqrt{k^2{m}^2{\omega }^4+c^2{\omega }^2{\left(k-m{\omega }^2\right)}^2}}}.$$

(10)

Alternatively, the equations can be expressed in terms of the relative parameters Ω, the dimensionless frequency ratio, and $\zeta$, the damping ratio [17,19], as follows:

$$A\left(\zeta ,\Omega \right)=X_0{\displaystyle \frac{2\zeta \Omega }{\sqrt{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}},$$

(11)

in which $A\left(c,\omega \right)$ or $A\left(\zeta ,\Omega \right)$ is the amplitude of the instantaneous relative displacement of the mass m relative to the moving reference point O.

Also, the amplitude of the instantaneous relative displacement of point B relative to the mobile reference point O can be determined from relation (7), resulting in the following:

$${\left|\tilde{B}\right|}^2=B^2=X_0^2{\displaystyle \frac{c^2{\omega }^2(k-m{\omega }^2)}{D^2}},$$

(12)

From relation (12), B results in

$$B\left(c,\omega \right)=X_0{\displaystyle \frac{c\omega (k-m{\omega }^2)}{\sqrt{k^2{m}^2{\omega }^4+c^2{\omega }^2{\left(k-m{\omega }^2\right)}^2}}},$$

(13)

and in relative parameters as follows:

$$B\left(\zeta ,\Omega \right)=X_0{\displaystyle \frac{2\zeta \Omega (1-{\Omega }^2)}{\sqrt{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}}.$$

(14)

The connection between $A\left(\zeta ,\Omega \right)$ and $B\left(\mathsf{\zeta },\mathsf{\Omega }\right)$ results using relations (11) and (14) and has the following form:

$$B\left(\zeta ,\Omega \right)=(1-{\Omega }^2)A\left(\zeta ,\Omega \right).$$

(15)

Figure 2 shows the variation curves of the relative amplitude $A\left(\zeta ,\Omega \right)$. The family of curves $B\left(\zeta ,\Omega \right)$ is shown in Figure 3.

3. The Instantaneous Deformation of the Viscous Equivalent Damper

The instantaneous deformation h(t) = $H_{0v}\sin (\omega t-\theta )$ [6,19] of the viscous equivalent damper c results as

$$h=h\left(t\right)=y-x_0$$

(16)

and in the complex domain as

$$\tilde{h}=\tilde{y}-{\tilde{x}}_0$$

(17)

resulting in

$${\tilde{H}}_v=\tilde{B}-X_0.$$

(18)

Introducing Equation (18) in the relation for $\tilde{B}$ from Equation (7) it results in

$${\tilde{H}}_v=X_0[{\displaystyle \frac{c^2{\omega }^2{\left(k-m{\omega }^2\right)}^2}{k^2{m}^2{\omega }^4+c^2{\omega }^2{\left(k-m{\omega }^2\right)}^2}}-1]+j{\displaystyle \frac{X_0}{D^2}}kmc{\omega }^2(k-m{\omega }^2)$$

(19)

in which D has the relationship from Equation (8).

Thus, ${\tilde{H}}_v$ becomes

$${\tilde{H}}_v={\displaystyle \frac{X_0}{D^2}}[{-k}^2{m}^2{\omega }^4+jc\omega km{\omega }^2{(k-m{\omega }^2)}^2]$$

(20)

from which

$${\left|{\tilde{H}}_v\right|}^2=H_{ov}^2={\displaystyle \frac{k^2{m}^2{\omega }^4{D}^2}{D^4}}{X}_0^2={\displaystyle \frac{k^2{m}^2{\omega }^4}{D^2}}{X}_0^2$$

(21)

Finally, it results in

$$H_{ov}(c,\omega )={\displaystyle \frac{km{\omega }^2{X}_0}{\sqrt{k^2{m}^2{\omega }^4+c^2{\omega }^2{(k-m{\omega }^2)}^2}}}$$

(22)

From the expression above, Equation (22) can be rewritten in relative parameters as follows:

$$H_{ov}(\zeta ,\Omega )=X_0{\displaystyle \frac{{\Omega }^2}{\sqrt{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}}.$$

(23)

4. The Maximum Transmitted Force

The maximum transmitted force [1,14,23] is the amplitude $Q_0$ of the instantaneous force $Q\left(t\right)=Q_0\sin (\omega t-\phi )$, with a phase shift $\phi$ between the displacement x₀(t) and the force $Q\left(t\right)$. In this case, the maximum transmitted force $Q_0=Q_0^v$ is given by the maximum velocity of the viscous deformation, meaning cH₀. Thus, it results in the following:

$$Q_0=c\omega H_{0v}$$

(24)

or

$$Q_0^v=Q_0=X_0{\displaystyle \frac{ckm{\omega }^3}{\sqrt{k^2{m}^2{\omega }^4+c^2{\omega }^2{(k-m{\omega }^2)}^2}}}.$$

(25)

5. The Kinematic Transmissibility

For the Maxwell model, the kinematic transmissibility T_c is given by the relation below:

$$T_c={\displaystyle \frac{A}{X_0}}$$

(26)

such that

$$T_c=T(c,\omega )={\displaystyle \frac{kc\omega }{\sqrt{k^2{m}^2{\omega }^4+c^2{\omega }^2{(k-m{\omega }^2)}^2}}}$$

(27)

or in relative parameters,

$$T(\zeta ,\Omega )={\displaystyle \frac{2\zeta \Omega }{\sqrt{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}}.$$

(28)

6. The Dissipated Energy

The energy dissipated on the equivalent viscous damper [5,24] c is of the form

$$W_d=\pi c\omega H_{0v}^2$$

(29)

in which $H_{ov}(c,\omega )$ and $H_o(\zeta ,\Omega )$ are introduced, thus resulting in the following:

$$W_d\left(c,\omega \right)=\pi X_0^2{\displaystyle \frac{ckm{\omega }^5}{k^2{m}^2{\omega }^4+c^2{\omega }^2{(k-m{\omega }^2)}^2}}.$$

(30)

For the function of $\Omega$ and $\zeta$, the energy dissipated is

$$W_d\left(\zeta ,\Omega \right)=\pi X_0^2\left(2\zeta \Omega \right)k{\displaystyle \frac{{\Omega }^4}{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}$$

(31)

or

$$W_d\left(\zeta ,\Omega \right)=\pi X_0^2{\displaystyle \frac{2\zeta k{\Omega }^5}{{\Omega }^4+{\left(2\zeta \Omega \right)}^2{(1-{\Omega }^2)}^2}}$$

(32)

Figure 4 shows the variation in dissipated energy $W_d\left(\zeta ,\Omega \right)$.

7. Conclusions

Based on the calculation relations established in the case of the linear Maxwell viscoelastic connections, the dynamic isolation system assembly can be evaluated as follows:

a.

The amplitude of the total deformation of the fluidic antiseismic systems $H_{ov}$ can be calculated with relation (22) or (23). This is dependent on the amplitude $X_0$ of the first seismic excitation mode, the first order pulsation of the seismic signal spectrum ɷ, the structural sizes of the base isolation system k, c, and the mass m of the building.

b.

The maximum force transmitted to the superstructure expressed by $Q_0^v$ is given by relation (25) and is proportional to the amplitude $X_0$ and the dynamic stiffness of the viscoelastic system $ckm{\omega }^3/D$.

c.

The kinematic transmissibility denoted by $T_c$ is directly influenced by the damping c or the fraction of the critical damping $\zeta$ as it results from relations (27) and (28).

d.

The dissipated energy can be calculated with relation (32), which essentially depends on $X_0$, $\zeta$, $\Omega$, and k, namely on the excitation factors ($X_0$, $\Omega$) and the structure of the insolation system ($\zeta$, k). It is also added that the dissipated energy must be evaluated at the maximum value because the temperature corresponding to the heat transport to the outside can lead to sudden expansions of the piston in the cylinder with mechanical blockages.

Drawing from the computational relationships established in the article, a base isolation system can be specified, comprising elastomeric isolation groups connected in series with fluid dampers, thereby allowing the Maxwell model to be fully defined [25,26,27,28].

The variation curves of parameters such as stiffness, damping, amplitude, transmitted force, and dissipated energy can be analyzed to determine the optimal configuration solution for the base isolation system of buildings subjected to seismic action.