The Town Effect: Dynamic Interaction between a Group of Structures and Waves in the Ground

Abstract

In a conventional approach, the mechanical behaviour of a structure subjected to seismic or blast waves is treated separately from its surroundings, and in many cases, the dynamic coupling effect between multiple structures and the waves propagating in the ground is disregarded. However, if many structures are built densely in a developed urban area, this dynamic interaction may not become negligible. The first purpose of this contribution is to briefly show the effect of multiple interactions between waves and surface buildings in a town. The analysis is based on a recently developed, fully coupled, rigorous mathematical study, and for simplicity, each building in the town is represented by a rigid foundation, a mass at the top and an elastic spring that connects the foundation and mass. The buildings stand at regular spatial intervals on a linear elastic half-space and are subjected to two-dimensional anti-plane vibrations. It is found that the buildings in this model significantly interact with each other through the elastic ground, and the resonant (eigen) frequencies of the collective system (buildings or town) become lower than that of a single building with the same rigid foundation. This phenomenon may be called the “town effect” or “city effect.” Then, second, it is shown that the actual, unique structural damage pattern caused by the 1976 Friuli, Italy, earthquake may better be explained by this “town effect,” rather than by investigating the seismic performance of each damaged building individually. The results suggest that it may also be possible to evaluate the physical characteristics of incident seismic/blast waves “inversely” from the damage patterns induced to structures by the waves.

Appendix: Mathematical Background

In this Appendix, the mathematical treatment of the elastodynamic problem in Sect. 2 is briefly described. Utilising the theory of the single layer potential and taking the effect of the edges of foundations into account, the harmonic solution that satisfies Eq. 1 and the outgoing Sommerfeld radiation condition at infinity may be expressed in a general form as (Ghergu and Ionescu 2009)

$$ w(x,y,t) = \text {Re} \left[ {\frac{i}{4}e^{i\omega t} \sum\limits_{j = 1}^{N} {\int_{{a_{j} }}^{{b_{j} }} {H_{0}^{(1)} \left( {\omega \sqrt {(x - s)^{2} + y^{2} } /c_{\text{S}} } \right){\frac{\phi (s)}{{\sqrt {(s - a_{j} )(b_{j} - s)} }}}{\text{d}}s} } } \right], $$

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where H ⁽¹⁾₀ (x) is the Hankel function of the first kind of order zero, and ϕ(x) is a continuous function on a _j  < x < b _j , to be determined by the boundary conditions. Upon modifications and corrections of the method (and results) presented in (Ghergu and Ionescu 2009), the boundary conditions posed by the rigid foundations of the buildings (2) and the mass–spring–foundation system (3) read

$$ \left[ {\xi^{2} - \left( {{\frac{{(c_{\text{S}} )_{\text{b}} }}{{c_{\text{S}} }}}} \right)^{2} \left( {{\frac{{l_{\text{b}} }}{h}}} \right)^{2} } \right]\tau_{k} (\xi ) = 2\xi^{2} {\frac{{\rho_{\text{b}} }}{\rho }}{\frac{h}{{l_{\text{b}} }}}\left\{ {\xi^{2} - \left( {1 + {\frac{{m_{0} }}{{m_{1} }}}} \right)\left[ {\xi^{2} - \left( {{\frac{{(c_{\text{S}} )_{\text{b}} }}{{c_{\text{S}} }}}} \right)^{2} \left( {{\frac{{l_{\text{b}} }}{h}}} \right)^{2} } \right]} \right\}, $$

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to be satisfied for the kth vibration mode of the town (1 ≤ k ≤ N) at ξ = ξ _k (>0), with τ _k (ξ) (τ₁ ≤ τ₂ ≤···≤ τ _N ) being the eigenvalues of the N × N matrix T that is associated with the definite integral in Eq. 3 and expressed as

$$ T_{j,k} = \text{Re} \left[ {\frac{1}{2}\sum\limits_{p = 1}^{2M} {\sum\limits_{q = 1}^{2M} {\left[ {\arcsin (\bar{x}_{p} + 1/(2M)) - \arcsin (\bar{x}_{p - 1} + 1/(2M))} \right]M_{p + 2M(j - 1),q + 2M(k - 1)}^{ - 1} } } } \right],\quad 1 \le j \le N,\,1 \le k \le N. $$

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Here, $\bar{x}_{p} = (2p - 1)/(2M) - 1$ for 0 ≤ p ≤ 2M, and the 2MN × 2MN matrix M ⁻¹_{p+2M(j−1),q+2M(k−1)} is the inverse of M _{p+2M(j−1),q+2M(k−1)} that is given by

$$ \begin{aligned} M_{p + 2M(j - 1),q + 2M(k - 1)} = & - {\frac{{J_{0} (\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|)}}{{2\pi \sqrt {1 - \bar{x}_{q} } }}}\int_{{\bar{x}_{q - 1} + 1/(2M)}}^{{\bar{x}_{q} + 1/(2M)}} {{\frac{{\ln \left| {g_{j} (\bar{x}_{p} ) - g_{k} (u)} \right|}}{{\sqrt {1 + u} }}}{\text{d}}u} \\ + & \left[ {A_{0} (\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|) - {\frac{{J_{0} (\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|)}}{2\pi }}\ln {\frac{\xi }{2}}} \right] \\ & \times \left[ {\arcsin (\bar{x}_{q} + 1/(2M)) - \arcsin (\bar{x}_{q - 1} + 1/(2M))} \right],\quad {\text{for}}\,1 \le q \le M \\ \end{aligned} $$

$$ \begin{aligned} M_{p + 2M(j - 1),q + 2M(k - 1)} = & - {\frac{{J_{0} (\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|)}}{{2\pi \sqrt {1 + \bar{x}_{q} } }}}\int_{{\bar{x}_{q - 1} + 1/(2M)}}^{{\bar{x}_{q} + 1/(2M)}} {{\frac{{\ln \left| {g_{j} (\bar{x}_{p} ) - g_{k} (u)} \right|}}{{\sqrt {1 - u} }}}{\text{d}}u} \\ + & \left[ {A_{0} \left( {\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|} \right) - {\frac{{J_{0} \left( {\xi \left| {g_{j} (\bar{x}_{p} ) - g_{k} (\bar{x}_{q} )} \right|} \right)}}{2\pi }}\ln {\frac{\xi }{2}}} \right] \\ & \times \left[ {\arcsin (\bar{x}_{q} + 1/(2M)) - \arcsin (\bar{x}_{q - 1} + 1/(2M))} \right],\quad {\text{for }}\;M + 1 \le q \le 2M \\ \end{aligned} $$

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for 1 ≤ j ≤ N, 1 ≤ k ≤ N and 1 ≤ p ≤ 2M, with g _j (u) = [a _j  + b _j  − (a _j  − b _j )u]/(2l _b), J ₀(x) being the Bessel function of the first kind of order zero, A ₀(x) = iH ⁽¹⁾₀ (x)/4 + J ₀(x) ln(x/2)/(2π) if x ≠ 0 and (iπ − 2γ)/(4π) if x = 0, and finally, γ is the Euler-Mascheroni constant (γ = 0.57721566…). In obtaining Eq. 8, J ₀, A ₀ and ϕ are approximated as constant functions on each interval $l_{\text{b}} g_{j} (\bar{x}_{p - 1} + 1/(2M)) < x < l_{\text{b}} g_{j} (\bar{x}_{p} + 1/(2M))$ along the rigid foundation of the jth building (1 ≤ j ≤ N, 1 ≤ p ≤ 2M). This way of discretisation gives fast convergence of the calculation, and with relatively smaller M precise results may be obtained (Ghergu and Ionescu 2009). In the calculations in this study, M = 100 is used.

From Eq. 6, the normalised eigenfrequency ξ _k [or f _k  = ξ _k c _S/(2πl _b) in a dimensional form] associated with the kth vibration mode of the town is obtained through the eigenvalue τ _k (ξ _k ). Note that ξ _k is controlled not only by m ₁/m ₀, ρ_b/ρ and (c _S)_b/c _S but also implicitly by d/l _b, h/l _b and l _t/l _b (or N) through the function g _j , i.e., a _j  = −l _t + (2l _b + d) (j − 1) and l _t = b _N  = a _N  + 2l _b. The normalised eigenvectors α ^k _j associated with the matrix T _j,k(ξ = ξ _k ) and its eigenvalue τ _k (ξ _k ) give the normalised displacement amplitudes of the kth vibration mode of the foundation j as w ^m0 _j (t) = α ^k _j e ^iωt (and w ^m1 _j (t) = α ^k _j /{1 − ξ ² _k (h/l _b)²[c _S/(c _S)_b]²} e ^iωt for the mass at the top), and then the function ϕ for that vibration mode is obtained in a discretised form as $\phi (x) = \text {Re} \left[ {\sum\nolimits_{l = 1}^{N} {\sum\nolimits_{q = 1}^{2M} {M_{p + 2M(j - 1),q + 2M(l - 1)}^{ - 1} \alpha_{l}^{k} } } } \right]$ for $l_{\text{b}} g_{j} (\bar{x}_{p - 1} + 1/(2M)) < x < l_{\text{b}} g_{j} (\bar{x}_{p} + 1/(2M))$ (again, 1 ≤ j ≤ N, 1 ≤ p ≤ 2M), and the displacement in the homogeneous, isotropic linear elastic half-space w(x, y, t) may be calculated using Eq. 5 for the kth vibration mode and normalised eigenvectors α ^k _j .

Equation 6 implies that $\xi = [(c_{\text{S}} )_{\text{b}} /c_{\text{S}} ](l_{\text{b}} /h)\sqrt {1 + m_{1} /m_{0} }$ when there is no structure–wave–structure interaction [i.e., τ _k (ξ) = 0 and hence the eigenfrequency for a single m ₁–k–m ₀ (mass–spring–foundation) system] and $\xi = [(c_{\text{S}} )_{\text{b}} /c_{\text{S}} ](l_{\text{b}} /h)$ when 1/τ _k (ξ) → 0 [normalised eigenfrequency for a single (or equivalently an N–) m ₁–k (mass–spring) system on top of a rigid half-space where no displacement is allowed, and the structure–wave–structure interaction is “infinite”], i.e., the eigenfrequencies ξ _k of the N building system lie in the range

$$ {\frac{{(c_{\text{S}} )_{\text{b}} }}{{c_{\text{S}} }}}\;{\frac{{l_{\text{b}} }}{h}}( \equiv \xi_{\infty } ) < \xi_{N} \le \xi_{N - 1} \le \cdots \le \xi_{1} < {\frac{{(c_{\text{S}} )_{\text{b}} }}{{c_{\text{S}} }}}\;{\frac{{l_{\text{b}} }}{h}}\sqrt {1 + {\frac{{m_{1} }}{{m_{0} }}}} ( \equiv \xi_{0} ). $$

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It should be noted that the first vibration mode corresponding to the smallest eigenvalue τ₁(ξ₁) gives the highest eigenfrequency ξ₁ (f ₁), and vice versa. In Figs 3 and 4, a lower eigenfrequency is related to more complex “out-of-phase” eigenvectors of the town. The alternate feature mentioned in Sect. 3.2 (“reverse” or “normal” order of vibration modes) is not recognised in Ghergu and Ionescu (2009), where the eigenfrequencies are calculated for the cases of N = 1, 3 and 21 buildings in a town with the same geometrical and mechanical properties as in this study, but the results are inversely presented, i.e., their eigenfrequency for the kth vibration mode is actually that for the (N + 1 − k)th mode (1 ≤ k ≤ N).