Optimal Placement of Visco-Elastic Dampers and Supporting Members Under Variable Critical Excitations

Abstract

The concept of performance-based design has recently been introduced and is well accepted in the current structural design practice of buildings. In earthquake-prone countries, the philosophy of earthquake-resistant design to resist ground shaking with sufficient stiffness and strength of a building itself has also been accepted as a relevant structural design concept for many years. On the other hand, a new strategy based on the concept of active and passive structural control has been introduced rather recently in order to provide structural designers with powerful tools for performance-based design.

Appendices

Appendix 1: Equivalent Stiffness and Damping Coefficient of Damper Unit Including Supporting Member in N-Model (Eqs. (12.1) and (12.2))

Let δ _Fi , δ _1i , δ _2i denote the interstory drift, the internal nodal displacement in the Maxwell model in Fig. 12.2 relative to the (i−1)th floor and the displacement of the node between the damper unit and the supporting member in the i-th story, i.e. $\delta_{{{\text{F}}i}} = u_{{{\text{F}}i}} - u_{{{\text{F(}}i - 1)}}$, $\delta_{1i} = u_{1i} - u_{{{\text{F(}}i - 1)}}$ and $\delta_{2i} = u_{2i} - u_{{{\text{F(}}i - 1)}}$. The equations of dynamic equilibrium of the 3N model can be derived as

$$ k_{{{\text{b}}i}} \delta_{2i} = p_{i} \left(t \right) $$

(12.57)

$$ k_{{{\text{M}}i}} (\delta_{1i} - \delta_{2i}) + k_{{{\text{V}}i}} (\delta_{{{\text{F}}i}} - \delta_{2i}) + c_{{{\text{V}}i}} (\dot{\delta}_{{{\text{F}}i}} - \dot{\delta}_{2i}) - k_{{{\text{b}}i}} \delta_{2i} = 0 $$

(12.58)

$$ c_{{{\text{M}}i}} (\dot{\delta}_{{{\text{F}}i}} - \dot{\delta}_{1i}) - k_{{{\text{M}}i}} (\delta_{1i} - \delta_{2i}) = 0 $$

(12.59)

In Eq. (12.57) $p_{i} \left(t \right)$ denotes the internal force of the supporting member in the i-th story. Let $\Updelta_{{{\text{F}}i}} (\omega),\Updelta_{1i} (\omega),\Updelta_{2i} (\omega),P_{i} (\omega)$ denote the Fourier transforms of $\delta_{{{\text{F}}i}} (t)$, $\delta_{1i} (t)$, $\delta_{2i} (t)$and $p_{i} (t)$. From Eq. (12.57), $\Updelta_{2i} (\omega)$ can be described by $P_{i} (\omega)/k_{{{\text{b}}i}}$. By substituting this equation into Eq. (12.59) expressed in frequency domain, we can obtain $\Updelta_{1i} (\omega)$ as

$$ \Updelta_{1i} (\omega) = \frac{{{\text{i}}\omega k_{{{\text{b}}i}} c_{{{\text{M}}i}} \Updelta_{{{\text{F}}i}} (\omega) + k_{{{\text{M}}i}} P_{i} (\omega)}}{{k_{{{\text{b}}i}} \left({{\text{i}}\omega c_{{{\text{M}}i}} + k_{{{\text{M}}i}}} \right)}} $$

(12.60)

Substitution of these equations for $\Updelta_{1i} (\omega)$ and $\Updelta_{2i} (\omega)$ into Eq. (12.58) in frequency domain leads to the following relationship between $\Updelta_{{{\text{F}}i}} (\omega)$ and $P_{i} \left(\omega \right)$.

$$ \begin{gathered} \left({\frac{{{\text{i}}\omega k_{{{\text{b}}i}} k_{{{\text{M}}i}} c_{{{\text{M}}i}}}}{{k_{{{\text{M}}i}} + {\text{i}}\omega c_{{{\text{M}}i}}}} + k_{{{\text{b}}i}} k_{{{\text{V}}i}} + {\text{i}}\omega k_{{{\text{b}}i}} c_{{{\text{V}}i}}} \right)\Updelta_{{{\text{F}}i}} \left(\omega \right) \hfill \\ \quad \quad = \left\{{\left({k_{{{\text{b}}i}} + k_{{{\text{M}}i}} + k_{{{\text{V}}i}} + {\text{ i}}\omega c_{{{\text{V}}i}}} \right) - \frac{{k_{{{\text{M}}i}}^{2}}}{{k_{{{\text{M}}i}} + {\text{i}}\omega c_{{{\text{M}}i}}}}} \right\}P_{i} \left(\omega \right) \hfill \\ \end{gathered} $$

(12.61)

After some manipulations, Eq. (12.61) can be rewritten as

$$ \frac{{k_{{{\text{b}}i}} k_{{{\text{V}}i}} k_{{{\text{M}}i}}^{2} + \omega^{2} k_{{{\text{b}}i}} c_{{{\text{M}}i}}^{2} \left({k_{{{\text{V}}i}} + k_{{{\text{M}}i}}} \right) + {\text{i}}\omega k_{{{\text{b}}i}} \left\{{k_{{{\text{M}}i}}^{2} \left({c_{{{\text{V}}i}} + c_{{{\text{M}}i}}} \right) + \omega^{2} c_{{{\text{V}}i}} c_{{{\text{M}}i}}^{2}} \right\}}}{{\left({(k_{{{\text{b}}i}} + k_{{{\text{V}}i}} )k_{{{\text{M}}i}}^{2} + (k_{{{\text{b}}i}} + k_{{{\text{M}}i}} + k_{{{\text{V}}i}} )\omega^{2} c_{{{\text{M}}i}}^{2}} \right) + {\text{ i}}\omega \left\{{k_{{{\text{M}}i}}^{2} \left({c_{{{\text{V}}i}} + c_{{{\text{M}}i}}} \right) + \omega^{2} c_{{{\text{V}}i}} c_{{{\text{M}}i}}^{2}} \right\}}}\Updelta_{{{\text{F}}i}} \left(\omega \right) = P_{i} \left(\omega \right) $$

(12.62)

On the other hand, the force–displacement relation of the general Kelvin–Voigt model can be given by

$$ \left({K_{{{\text{E}}i}} + {\text{i}}\omega C_{{{\text{E}}i}}} \right)\Updelta_{{{\text{F}}i}} \left(\omega \right) = P_{i} \left(\omega \right) $$

(12.63)

where K _Ei and C _Ei are the equivalent stiffness and the damping coefficient of the frequency-dependent Kelvin–Voigt model in the i-th story defined by Eqs. (12.1) and (12.2). By comparing Eqs. (12.62) and (12.63), Eqs. (12.1) and (12.2) can be derived.

Appendix 2: Transformation Matrix from the Nodal Displacements to the Relative Displacements Between both Ends of Supporting Members

For evaluating the axial force of the supporting member, the relative displacements u _b between both ends of supporting members are expressed in terms of nodal displacements u _full by

$$ {\mathbf{u}}_{{\mathbf{b}}} = {\mathbf{T}}_{{\mathbf{b}}} {\mathbf{u}}_{\text{full}} $$

(12.64)

In Eq. (12.64) T _b denotes the transformation matrix. In the case of the 3-story building model, T _b can be given by

$$ {\mathbf{T}}_{{\mathbf{b}}} = \left[{\begin{array}{*{20}c} 0 & 0 & 0 & {} & {} & {} & {} & {} & {} \\ 0 & 1 & 0 & {} & {[{\mathbf{0}}]} & {} & {} & {[{\mathbf{0}}]} & {} \\ 0 & 0 & 0 & {} & {} & {} & {} & {} & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} \\ {- 1} & 0 & 0 & 0 & 1 & 0 & {} & {[{\mathbf{0}}]} & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & {} & {} & {} \\ {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 \\ {} & {[{\mathbf{0}}]} & {} & {- 1} & 0 & 0 & 0 & 1 & 0 \\ {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right] $$

(12.65)

Appendix 3: Second-Order Sensitivities of the Equivalent Stiffness and Damping Coefficient

The second-order sensitivities of the equivalent stiffness and damping coefficient for the N model can be derived as follows.

$$ \begin{aligned} K_{{\rm E}i,jk} &= - 2k_{{\rm b}i}^{2} \frac{{\left( {c_{1}^{2} c_{2} S_{{\rm d}i}^{3} + 3c_{1}^{2} c_{3} k_{{\rm b}i} S_{{\rm d}i}^{2} + 3c_{1} c_{2} c_{3} k_{{\rm b}i}^{2} S_{{\rm d}i}+ c_{3} (2c_{2}^{2} - c_{1} c_{3} )k_{{\rm b}i}^{3} } \right)}}{{(c_{1} S_{{\rm d}i}^{2} + 2c_{2} k_{{\rm b}i} S_{{\rm d}i} + c_{3} k_{{\rm b}i}^{2} )^{3} }} \\K_{\rm{E}i}^{,jk} &= \left( {S_{{\rm d}i}^{2}/k_{{\rm b}i}^{2} } \right)K_{{\rm E}i,jk} \\K_{{\rm E}i_{,j}}^{,k} &= 2k_{{\rm b}i} S_{{\rm d}i} \frac{{c_{1}^{2} c_{2} S_{{\rm d}i}^{3} + 3c_{1}^{2} c_{3} k_{{\rm b}i}S_{{\rm d}i}^{2} + 3c_{1} c_{2} c_{3} k_{{\rm b}i}^{2} S_{{\rm b}i} +\left( {2c_{2}^{2} c_{3} - c_{1} c_{2}^{2} }\right)k_{{\rm b}i}^{2} S_{{\rm d}i} }}{{(c_{1} S_{{\rm d}i}^{2} +2c_{2} k_{{\rm b}i} S_{{\rm d}i}+ c_{3} k_{{\rm b}i}^{2} )^{3} }} \\C_{{\rm E}i,jk} &= - 2k_{{\rm b}i}^{2} c_{5} \frac{{c_{1} c_{3}k_{{\rm b}i}^{2} S_{{\rm d}i} + 2c_{2} c_{3} k_{{\rm b}i}^{3} +c_{1}^{2} S_{{\rm d}i}^{3} }}{{(c_{1} S_{{\rm d}i}^{2} + 2c_{2}k_{{\rm b}i}S_{{\rm d}i} + c_{3} k_{{\rm b}i}^{2} )^{3} }} \\C_{{\rm E}i}^{,jk} &= - 2S_{{\rm d}i}^{2} c_{5} \frac{{3c_{1} c_{3}k_{{\rm b}i}^{2} S_{{\rm d}i} + 2c_{2} c_{3} k_{{\rm b}i}^{3} - c_{1}^{2} S_{{\rm d}i}^{3} }}{{(c_{1} S_{{\rm d}i}^{2} + 2c_{2}k_{{\rm b}i}S_{{\rm d}i} + c_{3} k_{{\rm b}i}^{2} )^{3} }} \\C_{{\rm E}i,j}^{,k} &= - 2k_{{\rm b}i} S_{{\rm d}i} c_{5}\frac{{2c_{2} c_{3} k_{{\rm b}i}^{3} + c_{1} c_{3} k_{{\rm b}i}^{2}S_{{\rm d}i} + c_{1}^{2} S_{{\rm d}i}^{3} }}{{(c_{1} S_{{\rm d}i}^{2}+ 2c_{2} k_{{\rm b}i} S_{{\rm d}i} + c_{3} k_{{\rm b}i}^{2} )^{3} }}\end{aligned} $$

(12.66a–f)

where c ₁, c ₂, c ₃ are given by Eq. (12.49c, d, e).