Influence of flexible boundary on damage quantification in shear buildings by FRF-based approach

Abstract

Vibration-based damage detection techniques have gained popularity in structural health monitoring due to their non-destructive nature. Most of such damage detection techniques on buildings have developed considering fixed base foundation, that is without considering effect of soil underneath. The objective of the present study is to develop a closed-form expression for determining damage severity in a shear building considering flexible boundary condition that is soil-structure interaction (SSI) using frequency response function (FRF)-based approach. The main concern is to understand the influence of SSI on structural damage quantification during post-seismic mitigation through numerical as well as experimental study. A numerical simulation has been performed on a 14-storey shear building with various soil conditions, namely fixed, dense, medium and soft soil. In the experimental investigation, the dimensions of the soil mass have been considered in such a way that free-field response can be replicated. By similitude laws, a geometric scale factor has been applied to develop a small-scale model and an equivalent shear beam (ESB) container. Damage severity has been determined for both numerical and experimental studies. The effectiveness of the proposed approach has been further studied for a real structure. The novelty of the study lies in the mathematical development involving minimum number of sensors as well as in modelling the effect of semi-infinite soil layer under a scaled-down model. The proposed approach is effective in identifying intermediate and ground storey damage. However, further investigation is required for quantifying complex damage patterns.

Keywords

natural frequency modeshape frequency response function closed-form expression soil-structure interaction

Introduction

On the aftermath of a disastrous event like earthquake, cyclone, flood, etc. It is necessary to monitor the structural health of all important civil structures. Modal parameter-based approaches of vibration-based structural health monitoring techniques have gained immense popularity. These modal properties are generally calculated by considering a structure as fixed base. This assumption suits well when the soil below the structure is stiff enough. However in case of soft soil, the modal properties vary resulting to misleading damage identification. Therefore the influence of SSI should be considered on modal properties sensitive to damage. To understand its effect, SSI can be described mainly in two parts: kinematic and inertial interaction. The kinematic interaction deals with the alteration of the free-field response under the base of the structure whereas, inertial interaction deals with the structural deformation induced soil dynamic responses (Hokmabadi et al. (2015)). Based on the history of past earthquakes around the world, it has become necessary to consider the properties of soil like natural frequency, site amplification characteristics, etc. In the extraction of modal properties of a structure. Seed et al. (1988) discussed that there may be significant difference in the shaking intensity of an earthquake in different regions due to varying soil properties. Again, Esper and Tachibana (1998) concluded from the Kobe earthquake (1995) that SSI plays significant role not only for heavy structures but also for light structures depending on the soil type. Lo and Wang (2012) studied about six recent earthquakes and concluded that the secondary effects due to earthquakes, namely tsunami, landslides, liquefaction, etc. Were the main cause of casualties. Therefore, SSI should be considered in damage determination to mitigate structural damage.SSI influences the modal frequency of a structure as can be seen from various literature survey (Luco et al. (1988); Kohler et al. (2005); Todorovska and Trifunac (2007)). Trifunac et al. (2001) showed that the fundamental frequency of a structure varies with change in earthquake magnitude due to the nonlinearity of the soil below foundation. Various field tests have been performed with real-time data to determine the influence of SSI in structural modal parameters (Luco et al. (1988); Celebi and Şafak (1992); Todorovska and Trifunac (2007)). However, with the development of computational technology, simulation studies to model SSI for designing important structures gained popularity with the help of finite element method (Dutta and Roy (2002)), boundary element method (Hall and Oliveto (2003); Fu et al. (2018)), hybrid methods (Guarracino et al. (1992)), etc. Now to validate these simulation studies, experimental investigations especially by shaking table tests (Wang et al. (2017); Nardin et al. (2022)) and centrifuge model tests (Park et al. (2017)) became very important. However, it is very difficult to imitate the dynamic behaviour of soil in a small-scale experimental setup. Various propositions such as rigid container (Cheney et al. (1990); Adalier and Elgamal (2002); Ha et al. (2011)), laminar container (Hushmand et al. (1988); Jafarzadeh (2004); Prasad et al. (2004); Turan et al. (2009); Wang et al. (2019)) and equivalent shear beam (ESB) container (Zeng and Schofield (1996); Teymur and Madabhushi (2003); Carvalho et al. (2010); Lee et al. (2012)) have been recommended to perform laboratory tests for SSI investigations. Now based on Zeng and Schofield (1996), ESB container has been adopted in this study such that shear beam phenomena and minimal effect of compressible waves can be maintained along with proper stress distribution.Lately, researchers have started to explore the influence of SSI instead of considering a structure as fixed base. Roy et al. (2012) performed damage detection on a shear building considering fixed base, elastic base and nonlinear base where the nonlinear base has been modelled with beams-on-nonlinear-Winkler foundation. It has been observed that in case of nonlinear base, the mode shape curvature (Shokrani et al. (2018)) changes not only at the damage location but also between ground and first storey due to increased flexibility. Chen et al. (2019) also localized damage in shear building including SSI. Motlagh et al. (2021) detected damage in a wind turbine tower involving SSI using heat waves. Won and Shin (2021) developed machine learning-based approach for seismic damage prediction with SSI effect. It has been observed that neither any closed-form expression for quantifying damage has been proposed to date nor any small-scale experimental investigation associating SSI in damage identification has been performed so far. Now, for structural damage quantification, modal frequency and mode shape-based approaches (Huang et al. (2018); Liu et al. (2019); Masciotta and Pellegrini (2022); Jahangiri et al. (2019); Hassani et al. (2022)) are widely used. However, mode shape extraction may become computationally expensive along with error accumulation for complex systems. These limitations can be minimized if damage is quantified by FRF, bypassing the procedure of mode shape extraction. It has been observed that FRF-based approaches mainly involved model updating, pattern recognition (Santos et al. (2017)) and decision making technologies (Min et al. (2012); Entezami and Shariatmadar (2018); Lee et al. (2021); Hou and Xia (2021)). Again, FRF-based methods have some drawbacks (Das and Roy (2022)) like data incompleteness, large dataset management, long computation time, etc. Therefore, the motivation of this study is to develop an FRF-based closed-form expression for quantifying structural damage considering SSI.

In this study, an FRF-based closed-form expression for damage severity of an axially vibrating prismatic bar considering one partially restrained and another free-end has been derived. Emphasis has been provided to obtain an efficient formulation at estimated fundamental frequency of the structure, since it is difficult to capture structural response at higher frequency range in reality. The dynamic behaviour of the bar has been correlated to a shear building mounted on a rigid surface foundation along with the influence of SSI. Following the mathematical development and illustration on a shear building, a numerical study on a 14-storey shear building has been performed. The efficiency of the proposed formulation in determining intermediate and ground storey damage in presence of different soil types has been demonstrated. Finally, a detailed experimental investigation has been carried out on a miniature model to determine the performance of the proposed approach with real data. An ESB container has been developed to imitate the free-field motion of the field soil in a scaled-down model. To highlight the robustness of the proposed expression, various damage scenarios of single and multi-damage severity have been estimated in numerical, small-scale experimental study and also on real structure.

Mathematical development for damage severity

The homogenous elastic halfspace of soil below the massless rigid surface foundation of a shear building is shown in Figure 1(a). K_t and K_r represent translational and rotational stiffness of the compliant system, respectively. C_t and C_r represent translational and rotational damping coefficients of the compliant system, respectively. Veletsos and Meek (1974) and Renzi et al. (2013) described that the soil-structure system can be idealised as a stick model as shown in Figure 1(b). To maintain uniform units across all stiffness terms, K_r is represented by a vertical translational spring positioned at the foundation edge. Both the stiffness and damping characteristics of soil-foundation system can be condensed into two terms $\tilde{K}$ and $\tilde{C}$ , respectively as shown in Figure 1(c). The vibrational response of a shear building is analogous to an axially vibrating bar (Das and Roy (2023)). Therefore, the soil-structure system considered in this study can be represented in the form of Figure 1(d). Let E, ρ, L and A₀ be the modulous of elasticity, mass per unit length, bar length and cross-sectional area of the bar, respectively. The SSI analysis considers both rotational and translational properties of the soil.

Figure 1.

(a) Soil-structure model (Ahmadi et al. (2015)), (b) soil-structure idealised by stick model, (c) condensed model (Veletsos and Meek (1974); Renzi et al. (2013)) and (d) analogous axially vibrating prismatic bar with flexible boundary.

Let, k_eq = A₀E/L be the equivalent stiffness of the uniformly discritized bar and $\tilde{K}$ be the soil stiffness, such that the interaction factor $β = k_{e q} / \tilde{K}$ . Based on practical scenario, the soil translational stiffness is usually greater than the structural stiffness that is the critical condition may occur when $k_{e q} = \tilde{K}$ . Therefore, the range of β can be given as 0 ≤ β ≤ 1. When β = 0, it represents fixed base condition $(\tilde{K} \to \infty)$ that is without SSI influence or very stiff soil. The expression of free vibration ϕ(x) at any location x of an axially vibrating prismatic bar (Clough and Penzien (2003)) can be given as,

ϕ (x) = C_{1} cos (\frac{ω x}{c}) + C_{2} \sin (\frac{ω x}{c})

(1)

where,

c = \sqrt{E / ρ}

is phase velocity, ω is circular frequency and C₁, C₂ are the arbitrary constants depending on the boundary conditions. Now due to partially restrained end,

\begin{gathered} E A_{0} \frac{\partial ϕ (0)}{\partial x} = \tilde{K} ϕ (0) \\ \Rightarrow C_{2} = C_{1} (\frac{\tilde{K} c}{E A_{0} ω}) \end{gathered}

(2)

and at the free end,

\begin{gathered} ϕ^{'} (L) = 0 \\ \Rightarrow C_{2} = C_{1} tan (\frac{ω L}{c}) \end{gathered}

(3)

Now, equating equations (2) and (3),

tan (\frac{ω L}{c}) = \frac{\tilde{K} c}{E A_{0} ω}

(4)

Consider ωL/c = z, therefore equation (4) can be expressed as,

z \sin (z) - \frac{cos (z)}{β} = 0

(5)

The above equation is transcendental in nature that is the closed-form solution is difficult to obtain. Thus, to determine the approximate solution for z, Newton-Raphson method (Grewal (1996)) can be implemented. From literature survey Kohler et al. (2005), Todorovska and Trifunac (2007), it has been observed that the frequency of vibration of any structure diminishes due to influence of SSI in comparison to the fixed base condition. Therefore, the initial value (z₀) for the numerical method has been considered as the expression obtained for a cantilever bar at any modal frequency (Clough and Penzien (2003))

((2 n - 1) π c / 2 L)

that is z₀ = (2n − 1)π/2. Now if f(z) = z sin(z) − cos(z)/β, then by Newton-Raphson method the expression obtained after single iteration is

z_{1} = z_{0} - \frac{f (z_{0})}{f^{'} (z_{0})} = (\frac{2 n - 1}{1 + β}) \frac{π}{2}

(6)

Now, as the Newton-Raphson method follows quadratic convergence thus the expression obtained from first two iterations that is z₁ and z₂ are alike (refer Appendix). Therefore, the expression of natural frequency of the undamaged system obtained from first iteration can be given as

ω_{n} = (\frac{2 n - 1}{1 + β}) \frac{π c}{2 L}

(7)

Substituting equations (3) and (7) in equation (1), the expression of any n^th mode shape can be given as,

ϕ^{(n)} (x) = b cos [(\frac{2 n - 1}{1 + β}) \frac{π}{2 L} (L - x)]

(8)

where, b is a constant governed by structural and soil characteristics. Now, the system has been uniformly discritized into N elements of length ϵ each (Figure 1(d)) such that ϵ ≪ L. Let, damage is present at p^th element and the stiffness of the element is reduced by δk_p. Here, δk_p = η_pA₀E/ϵ, where η_p stands for damage severity at the p^th element and EA₀ is the rigidity modulous of the system. The change in any m^th damaged and undamaged orthonormal mode shape of the bar that is

{\hat{ϕ}}^{(m)}

and ϕ^(m) respectively can be obtained from a combination of undamaged orthonormal mode shapes ϕ⁽ⁿ⁾ and first-order sensitivity coefficient τ_n (Zhu et al. (2011)) as given in equation (9). Note that

\hat{(\cdot)}

has been used to indicate damaged system throughout this study.

{\hat{ϕ}}^{(m)} (x) - ϕ^{(m)} (x) = \frac{\partial ϕ^{(m)}}{\partial k_{p}} δ k_{p} = (\sum_{n = 1}^{N} τ_{n} ϕ^{(n)}) δ k_{p}

(9)

where,

τ_{n} = \{\begin{cases} - \frac{(ϕ^{(n)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(n)} (x_{p} - \frac{ϵ}{2})) (ϕ^{(m)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(m)} (x_{p} - \frac{ϵ}{2}))}{ω_{n}^{2} - ω_{m}^{2}}, n \neq m \\ 0, n = m \end{cases}

From equation (7), the expression of

(ω_{n}^{2} - ω_{m}^{2})

can be obtained as,

ω_{n}^{2} - ω_{m}^{2} = {(\frac{π c}{2 L (1 + β)})}^{2} [{(2 n - 1)}^{2} - {(2 m - 1)}^{2}]

(10)

Considering ϵ to be too small and using equation (8) , $(ϕ^{(m)} (x_{p} + ϵ / 2) - ϕ^{(m)} (x_{p} - ϵ / 2))$ can be derived as,

\begin{aligned} ϕ^{(m)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(m)} (x_{p} - \frac{ϵ}{2}) = b [cos (\frac{(2 m - 1) π (L - (x_{p} + \frac{ϵ}{2}))}{(1 + β) 2 L}) \\ - cos (\frac{(2 m - 1) π (L - (x_{p} - \frac{ϵ}{2}))}{(1 + β) 2 L})] \\ = \frac{(2 m - 1) b π ϵ}{(1 + β) 2 L} \sin (\frac{(2 m - 1) π (L - x_{p})}{(1 + β) 2 L}) \end{aligned}

(11)

Similarly,

\begin{align} \begin{aligned} ϕ^{(n)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(n)} (x_{p} - \frac{ϵ}{2}) \\ = & \frac{(2 n - 1) b π ϵ}{(1 + β) 2 L} \sin (\frac{(2 n - 1) π (L - x_{p})}{(1 + β) 2 L}) \end{aligned} \end{align}

(12)

Substituting equations (10)–(12) in equation (9), the difference between damaged and undamaged mode shapes at any location x can be obtained as,

\begin{aligned} \frac{\partial ϕ^{(m)}}{\partial k_{p}} δ k_{p} = \sum_{\begin{array}{c} n = 1 \\ n \neq m \end{array}}^{N} \frac{b^{3} ϵ^{2} (2 m - 1) \sin (\frac{(2 n - 1) π (L - x_{p})}{(1 + β) 2 L})}{c^{2} (2 n - 1) [1 - {(\frac{2 m - 1}{2 n - 1})}^{2}]} \\ \sin (\frac{π (2 m - 1) (L - x_{p})}{(1 + β) 2 L}) cos (\frac{π (2 n - 1) (L - x)}{(1 + β) 2 L}) δ k_{p} \end{aligned}

(13)

Now, the term ((2m − 1)/(2n − 1))² decreases with increase in n for the fundamental mode (m = 1). Therefore, neglecting the terms of ((2m − 1)/(2n − 1))² for n ≥ 2, the expression of equation (13) for the fundamental mode can be written as,

\begin{aligned} δ ϕ^{(1)} (x) & = {\hat{ϕ}}^{(1)} (x) - ϕ^{(1)} (x) \\ = \frac{b^{3} ϵ^{2} \sin (\frac{π (L - x_{p})}{(1 + β) 2 L})}{2 c^{2}} (\sum_{n = 1}^{N} \frac{\sin (\frac{(2 n - 1) π (2 L - x - x_{p})}{(1 + β) 2 L}) + \sin (\frac{(2 n - 1) π (x - x_{p})}{(1 + β) 2 L})}{2 n - 1} - 2 \sin (\frac{π (L - x_{p})}{(1 + β) 2 L}) cos (\frac{π (L - x)}{(1 + β) 2 L})) δ k_{p} \end{aligned}

(14)

Now for simplification, consider

2 \sin (\frac{π (L - x_{p})}{(1 + β) 2 L}) cos (\frac{π (L - x)}{(1 + β) 2 L}) = f_{p} (x)

(15)

The value of $b^{3} ϵ^{2} / 2 c^{2} \sin (π (L - x_{p}) / (1 + β) 2 L) δ k_{p}$ remains constant based on damage location and severity. For further simplification of equation (14), series convergence (Zwillinger (2018)) has been applied such that

\lim_{N \to \infty} \sum_{n = 1}^{N} \frac{\sin (\frac{(2 n - 1) π (x - x_{p})}{(1 + β) 2 L})}{2 n - 1} = \{\begin{cases} - \frac{π}{4}, for x < x_{p} \\ 0, for x = x_{p} \\ \frac{π}{4}, for x > x_{p} \end{cases}

(16)

Again, the minimum and maximum values of x and x_p can be 0 and L, respectively. Therefore, the above expression converges to π/4 because for n = 1,

\begin{gathered} 0 \leq \frac{π (2 L - x - x_{p})}{(1 + β) 2 L} \leq \frac{π}{(1 + β)} \\ \Rightarrow 0 \leq (L - \frac{x + x_{p}}{2}) \leq L \end{gathered}

(17)

Therefore,

\lim_{N \to \infty} \sum_{n = 1}^{N} \frac{\sin (\frac{(2 n - 1) π (2 L - x - x_{p})}{(1 + β) 2 L})}{2 n - 1} = \frac{π}{4}

(18)

Now, substituting equations (15)–(18) in equation (14), the fundamental mode shape difference can be expressed as,

δ ϕ^{(1)} (x) = \frac{b^{3} ϵ^{2}}{2 c^{2}} \sin (\frac{π (L - x_{p})}{(1 + β) 2 L}) δ k_{p} \times {\begin{cases} [\frac{π}{4} - f_{p} (x)], x = x_{p} \\ [- f_{p} (x)], x < x_{p} \\ [\frac{π}{2} - f_{p} (x)], x > x_{p} \end{cases}

(19)

The change in mode shape difference at immediate left and right of the damage location x_p can be given as,

\begin{aligned} δ ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - δ ϕ^{(1)} (x_{p} - \frac{ϵ}{2}) \\ = \frac{b^{3} ϵ^{2}}{2 c^{2}} \sin (\frac{π (L - x_{p})}{(1 + β) 2 L}) δ k_{p} [(\frac{π}{2} - f_{p} (x_{p})) - (- f_{p} (x_{p}))] \\ = \frac{b^{3} ϵ^{2}}{2 c^{2}} \sin (\frac{π (L - x_{p})}{(1 + β) 2 L}) δ k_{p} \frac{π}{2} \end{aligned}

(20)

The difference in undamaged fundamental mode shape at the immediate left and right of x_p can be given as,

ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(1)} (x_{p} - \frac{ϵ}{2}) = \frac{b π ϵ}{(1 + β) 2 L} \sin (\frac{π (L - x_{p})}{(1 + β) 2 L})

(21)

Now, dividing equation (20) by (21), the expression can be simplified into,

\frac{δ ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - δ ϕ^{(1)} (x_{p} - \frac{ϵ}{2})}{ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(1)} (x_{p} - \frac{ϵ}{2})} = \frac{b^{2} ρ A_{0} L (1 + β) η_{p}}{2}

(22)

Here, b can be obtained by mass normalization of the fundamental mode shape $(ϕ^{(1)})$ as given in equation (23).

b = \sqrt{\frac{2}{ρ A_{0} L (1 + \frac{\sin (\frac{π}{1 + β})}{\frac{π}{1 + β}})}}

(23)

Again, the fundamental modal displacement ϕ⁽¹⁾ (x) of an axially vibrating bar can be defined as the FRF magnitude H (x, ω₁) at any degree of freedom (DoF) x normalized by the FRF magnitude recorded at the free-end H (L, ω₁) of the system that is $ϕ^{(1)} (x) = H (x, ω_{1}) / H (L, ω_{1}) = \frac{R (x, ω_{1})}{R (0, ω_{1})} / \frac{R (L, ω_{1})}{R (0, ω_{1})}$ , as described in Figure 2. R (x, ω₁) and R (L, ω₁) represent the amplitude of time-history response data in the frequency domain at x and free-end, respectively. R (0, ω₁) indicates the input excitation in the frequency domain.

\begin{aligned} η_{p} & = (\frac{1}{1 + β} + \frac{\sin (\frac{π β}{1 + β})}{π}) (\frac{δ ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - δ ϕ^{(1)} (x_{p} - \frac{ϵ}{2})}{ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - ϕ^{(1)} (x_{p} - \frac{ϵ}{2})}) \\ = χ_{s} (\frac{{\hat{ϕ}}^{(1)} (x_{p} + \frac{ϵ}{2}) - {\hat{ϕ}}^{(1)} (x_{p} - \frac{ϵ}{2})}{ϕ^{(1)} (x_{p} + \frac{ϵ}{2}) - ϕ (x_{p} - \frac{ϵ}{2})} - 1) \\ = χ_{s} (\frac{\frac{\hat{H} (x_{p} + \frac{ϵ}{2}, {\hat{ω}}_{1}) - \hat{H} (x_{p} - \frac{ϵ}{2}, {\hat{ω}}_{1})}{\hat{H} (L, {\hat{ω}}_{1})}}{\frac{H (x_{p} + \frac{ϵ}{2}, ω_{1}) - H (x_{p} - \frac{ϵ}{2}, ω_{1})}{H (L, ω_{1})}} - 1) \\ = χ_{s} (\frac{\frac{\frac{\hat{R} (x_{p} + \frac{ϵ}{2}, {\hat{ω}}_{1})}{\hat{R} (0, {\hat{ω}}_{1})} - \frac{\hat{R} (x_{p} - \frac{ϵ}{2}, {\hat{ω}}_{1})}{\hat{R} (0, {\hat{ω}}_{1})}}{\frac{\hat{R} (L, {\hat{ω}}_{1})}{\hat{R} (0, {\hat{ω}}_{1})}}}{\frac{\frac{R (x_{p} + \frac{ϵ}{2}, ω_{1})}{R (0, {\hat{ω}}_{1})} - \frac{R (x_{p} - \frac{ϵ}{2}, ω_{1})}{R (0, {\hat{ω}}_{1})}}{\frac{R (L, ω_{1})}{R (0, {\hat{ω}}_{1})}}} - 1) \\ = χ_{s} (\frac{\frac{\hat{R} (x_{p} + \frac{ϵ}{2}, {\hat{ω}}_{1}) - \hat{R} (x_{p} - \frac{ϵ}{2}, {\hat{ω}}_{1})}{\hat{R} (L, {\hat{ω}}_{1})}}{\frac{R (x_{p} + \frac{ϵ}{2}, ω_{1}) - R (x_{p} - \frac{ϵ}{2}, ω_{1})}{R (L, ω_{1})}} - 1) \end{aligned}

(24)

Figure 2.

Relationship between output response, frequency response function (FRF) and mode shape at fundamental frequency.

FRF can be defined as the ratio of output response to input excitation in frequency domain. Considering all the output responses being recorded simultaneously, the influence of input excitation gets normalized. Therefore, the output response in frequency domain can be directly considered as desired FRF amplitude. The FRF at any DoF can be obtained from the recorded time-history response data at that location by applying fast Fourier transform (FFT). Here, the magnitude obtained at the fundamental frequency has been considered because they remain in same phase irrespective of any location of the bar. Now, based on Figure 2 and substituting equation (23) in (22), the expression of structural damage severity (η_p) at any location x_p can be expressed as shown in equation (24).

The term $χ_{s} = (1 / 1 + β + \sin (\frac{π β}{1 + β}) / π)$ is the factor corresponding to soil stiffness properties influencing the damage severity of the structure. When fixed base condition or very stiff soil is considered then β = 0 such that χ_s = 1. Thus, the structural damage severity in presence of soil can be determined from the FRF magnitude obtained at the fundamental frequency recorded at the free-end and at the adjacent locations of the damage. The computational procedure to be followed to estimate the damage severity at the desired locations has been shown in Figure 3. The FRF amplitudes at the estimated fundamental frequency have been obtained by well-known ‘peak-picking’ technique (Brincker and Ventura (2015)).

Figure 3.

Flowchart to determine damage severity at desired locations.

Illustration on shear building

The well-known procedure to calculate the modal properties of a shear building mathematically is to solve an eigenvalue problem comprising lumped mass at each floor level and storey stiffness (Figure 4) such that the eigenvalues and eigen vectors represent modal frequencies (ω_i) and corresponding mode shapes (ϕ⁽ⁱ⁾), respectively. In case of fixed base condition, the eigen value problem (Clough and Penzien (2003)) can be given as,

K Φ = Λ M Φ

(25)

where, M, K, Λ and Φ indicate lumped mass matrix, stiffness matrix, eigenvalue matrix comprising square of the modal frequencies at the diagonal elements and mass-normalized mode shape matrix of the total structure, respectively. The expressions of M and K can be given as,

M = [\begin{matrix} m_{1} & 0 & \dots & \dots & 0 \\ 0 & m_{2} & 0 & ⋮ \\ ⋮ & 0 & ⋱ & ⋱ & ⋱ \\ ⋮ & ⋱ & m_{i} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & \dots & 0 & m_{N} \end{matrix}]

(26)

K = [\begin{matrix} k_{1} + k_{2} & - k_{2} & \dots & \dots & \dots & 0 \\ - k_{2} & k_{2} + k_{3} & - k_{3} & ⋮ \\ ⋮ & - k_{3} & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & k_{i - 1} + k_{i} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & - k_{N} \\ 0 & \dots & \dots & - k_{N} & k_{N} \end{matrix}]

(27)

Figure 4.

Lumped mass and stiffness representation of an N-storey of shear building considering SSI.

Now, while considering the effect of soil in the dynamic behaviour of the structure as presented in Figure 4, the mass and stiffness matrix of the eigen value problem get modified with additional submatrices of soil and structure coupling effect. The eigen value problem for the fundamental mode gets modified into,

[\begin{matrix} K_{S} & 0 \\ 0 & K \end{matrix}] Φ_{S}^{(1)} = Λ_{S}^{(1)} [\begin{matrix} M_{S} & M_{SSI} \\ M_{SSI}^{T} & M \end{matrix}] Φ_{S}^{(1)}

(28)

where,

M_{S} = [\begin{matrix} I_{0} + \sum_{i = 1}^{N} (I_{i} + m_{i} h_{i}^{2}) & \sum_{i = 1}^{N} m_{i} h_{i} \\ \sum_{i = 1}^{N} m_{i} h_{i} & m_{0} + \sum_{i = 1}^{N} m_{i} \end{matrix}]

(29)

M_{SSI} = [\begin{matrix} m_{1} h_{1} & m_{2} h_{2} & \dots & m_{i} h_{i} & \dots & m_{N} h_{N} \\ m_{1} & m_{2} & \dots & m_{i} & \dots & m_{N} \end{matrix}]

(30)

\begin{array}{l} Φ_{S}^{(1)} = {[\begin{matrix} ϕ_{θ}^{(1)} & ϕ_{h}^{(1)} & ϕ_{h_{1}}^{(1)} & \dots & ϕ_{h_{i}}^{(1)} & \dots & ϕ_{h_{N}}^{(1)} \end{matrix}]}^{T} \end{array}

\begin{align} = {[H_{θ} (ω_{1}) H_{h} (ω_{1}) H (h_{1}, ω_{1}) \dots H (h_{i}, ω_{1}) \dots H (h_{N}, ω_{1})]}^{T} \end{align}

(31)

$Φ_{S}^{(1)}$ represents modified mass-normalized fundamental mode shape matrix including soil effect. In Equation (29), I₀ and m₀ indicate moment of inertia and mass of the rigid foundation, respectively. m_i, h_i and I_i denote mass, height from soil surface and moment of inertia of the i^th floor of the shear building, respectively. $ϕ_{θ}^{(1)}$ , $ϕ_{h}^{(1)}$ and $ϕ_{h_{i}}^{(1)}$ denote fundamental modal deformations corresponding to soil rotation, soil translation and lateral displacement of i^th floor of the shear building. As the proposed formulation equation (24) has been derived for first resonating frequency of the structure thus, equation (28) has been expressed for fundamental mode. Now, as the modal displacements of all DoFs remain in same phase at the fundamental mode, it can be directly related to FRF amplitudes H (h_i, ω₁). The influence of soil properties on the surface foundation can be described in form of horizontal (k_h) and rocking (k_θ) stiffness. The expressions of the soil stiffness can be obtained from Wolf (1994) and is given as,

\begin{aligned} K_{S} & = [\begin{matrix} k_{θ} & 0 \\ 0 & k_{h} \end{matrix}] \\ = [\begin{matrix} \frac{2.9 G_{s} I_{0}^{0.75}}{1 - μ} {(\frac{L_{f}}{B_{f}})}^{0.15} & 0 \\ 0 & \frac{2 G_{s} L_{f}}{2 - μ} [2 + 2.5 {(\frac{L_{f} \times B_{f}}{4 L_{f}^{2}})}^{0.85}] \end{matrix}] \end{aligned}

(32)

The soil stiffness terms depend on soil properties that is shear modulous (G_s) and Poisson’s ratio (μ) and also on the foundation dimensions (length, L_f and breadth, B_f). This eigenvalue representation of the shear building with SSI effect has been adopted in the numerical study to simulate various damage conditions at any storey of the shear building resting on various soil types.

Numerical study

Modelling

A simulation study has been performed on a 14-storey shear building of uniform mass and stiffness. The effectiveness of the proposed formulation has been studied for various soil conditions comprising fixed base (without SSI), dense, medium and soft soil. The damage has been simulated by reducing storey stiffness. The structural properties of the simulated undamaged model has been adapted from Mendes et al. (2019) with modifications in the data corresponding to number of storeys, storey stiffness and foundation dimensions. The details have been enlisted in Table 1.

Table 1.

Details of model for numerical simulation.

Parameters	Values	Parameters	Values
No. of storeys (N)	14	Moment of inertia of each storey (I_i)	1.31 × 10⁸ kg m²
Each storey height (ϵ)	4 m	Rigid circular foundation length (L_f)	40 m
Mass of each storey (m_i)	9.8 × 10⁵ kg	Mass of foundation (m₀)	1.96 × 10⁶ kg
Stiffness of each storey (k_i)	1.6 × 10⁹ N/m	Moment of inertia of foundation (I₀)	2.6 × 10⁸ kg m²
Equivalent stiffness of structure (k_eq)	1.14 × 10⁸ N/m

The shear modulous (G_s) and Poisson’s ratio (μ) are the two important soil properties required for determining the soil characteristics experienced by the desired surface foundation system. In practical scenario, μ can be obtained from the standard tables (Bowles (1988)) for different soil classifications. The value of G_s = E_s/2 (1 + μ) (Mendes et al. (2019)) can be obtained from the relation of μ and the elastic modulous (E_s) of soil which can be determined from static-triaxial test in laboratory or from the standard penetration test data of any site. The soil properties (G_s, μ, ρ_s and V_s) have been adopted from Liu et al. (2008). The horizontal and rocking stiffness as well as damping of the soil have been implemented on the simulation model following Liu et al. (2008) as given in Table 2.The first three modal frequencies of the undamaged structure for different base conditions have been listed in Table 3. It has been observed that the natural frequencies of the structure diminishes with the influence of SSI. In other words, denser the soil, more is the modal frequencies of the desired system.

Table 2.

Characteristics of Various soil Types (Liu et al. (2008)).

Soil type	G_s (N/m²) × 10⁸	μ	ρ_s (kg/m³)	V_s (m/s)	$k_{θ} = \frac{8 G_{s} B_{f}^{3}}{3 (1 - μ)}$ (N-m) × 10¹³	$k_{h} = \frac{8 G_{s} B_{f}}{2 - μ}$ (N/m) × 10¹⁰	β	$c_{θ} = \frac{0.4 ρ_{s} V_{s} B_{f}^{4}}{1 - μ}$ (N-s-m) × 10¹¹	$c_{h} = \frac{4.6 ρ_{s} V_{s} B_{f}^{2}}{2 - μ}$ (N-s/m) × 10⁹
Dense	6	0.33	2400	500	1.91	5.75	0.0018	1.15	1.32
Medium	1.71	0.48	1900	300	0.702	1.80	0.006	0.702	0.69
Soft	0.18	0.49	1800	100	0.075	0.191	0.05	0.226	0.22

Table 3.

Modal frequencies of the undamaged structure considering different soil types.

Frquency (Hz)	Fixed	Dense	Medium	Soft
1^st mode	0.69	0.69	0.69	0.64
2^nd mode	2.08	2.05	1.99	1.50
3^rd mode	3.44	3.41	3.36	3.02

Both single and multi-damage conditions have been simulated in the study for intermediate storey and ground storey damages considering various damage intensities. To determine the effectiveness of the formulation, three cases have been generated for each scenario as given in Table 4. It has been observed that in past few decades, damage detection and localization techniques have developed extensively due to their non-destructive approaches. Some of them include frequency change, modal analysis, mode shape curvature and FRF curvature method, etc. Now, structural damage identification includes damage detection, its location, quantification and also estimates prognosis of the remaining service life of a structure (Rytter (1993); Sohn and Farrar (2001)). Therefore, as various methodologies regarding damage detection and localization have already developed, the recent progression is concentrating on developing the damage quantification techniques. Therefore in this study, emphasis has been provided to develop a one-shot FRF-based closed-form expression for determining damage severity considering SSI.

Table 4.

Summary of damage Cases Considered in Numerical Study.

Cases	At intermediate storeys		Involving ground storey
	Single-damage	Multi-damage	Single-damage	Multi-damage
	@ D₈	@ D₅ & D₁₀	@ D₁	@ D₁ & D₈
1	10%	20% & 20%	10%	20% & 20%
2	15%	10% & 20%	15%	10% & 20%
3	20%	20% & 10%	20%	20% & 10%

Damage at intermediate storeys

Results corresponding to single and multi-damage at intermediate storeys have been shown in Figures 5 and 7, respectively. Damage severity can be estimated with efficacy for both single as well as multiple damage scenario. In case of single damage scenario (Figure 5), the proposed formulation overestimates damage severity and its extent increases with increase in the applied damage intensity. The estimation using FRF amplitudes are shown in Figure 6. However in multi-damage scenario (Figure 7), the proposed formulation overestimates for higher damage severity and underestimates for lower damage intensity. Again, the errors at the higher storey damage are comparatively higher than the lower storey damage irrespective of the soil condition. Absolute error less than 3.3% has been observed in each case.

Figure 5.

Comparison of estimated damage severity for different soil conditions due to damage applied at D₈ (- - -): (a) Case 1, (b) Case 2 and (c) Case 3.

Figure 6.

Intermediate storey damage estimation using FRF amplitudes for different soil conditions: (a) fixed base, (b) dense, (c) medium and (d) soft soil.

Figure 7.

Comparison of estimated damage severity for different soil conditions due to damage applied at D₅ (- - -) and D₁₀ (– ⋅ –): (a) Case 1, (b) Case 2 and (c) Case 3.

Damage involving ground storey

The effect of SSI should be maximum at the ground storey level due to rocking and translatory motion of the soil underneath the surface foundation. Results corresponding to single and multi-damage have been shown in Figures 8 and 9, respectively. It has been observed that the proposed formulation is efficient in estimating structural damage intensity for fixed base, dense and medium soil type with a maximum absolute error of 4%. The estimation using FRF amplitudes are shown in Figure 10. The variation in χ_s with respect to β among fixed base, dense and medium soil is negligible. However, its trend drops drastically for soft soil conditions, as shown in Figure 11. Throughout the study, the damage estimation has been considered as a post-disaster mitigation event. In this case, the structural response is affected by both stiffness and damping of soil whereas the proposed formulation involved only the soil stiffness properties. The soil damping has not been incorporated into the formulation due to the inherent challenges associated with its modelling due to the nonlinear and hysteretic nature of soil behaviour, as well as its variability across different soil types. The primary objective was to develop a simple, closed-form expression for damage estimation, which would become significantly more complex if soil damping were included in the model. The proposed approach needs to be further modified to enhance its efficiency in estimating ground storey damage resting on soft soil and also for a situation where degradation of soil properties and structural damage coexist. Here, it has been assumed that the soil-types remained unaltered before and after damage of the structure. The approach may be further extended considering the change in soil-type along with structural damage.

Figure 8.

Comparison of estimated damage severity for different soil conditions due to damage applied at D₁ (- - -): (a) Case 1, (b) Case 2 and (c) Case 3.

Figure 9.

Comparison of estimated damage severity for different soil conditions due to damage applied at D₁ (- - -) and D₈ (– ⋅ –): (a) Case 1, (b) Case 2 and (c) Case 3.

Figure 10.

Ground storey damage estimation using FRF amplitudes for different soil conditions: (a) fixed base, (b) dense, (c) medium and (d) soft soil.

Figure 11.

Variation of χ_s with respect to β for different soil types.

Effectiveness in damage localization

The proposed formulation involving the FRF amplitudes only at three sensors is a one-shot solution to determine the damage severity with a priori knowledge of damage location. If responses of all storeys are known, the proposed expression can identify undamaged and damaged storeys along with damage intensity. Results corresponding to two cases that is (10% & 20%) damage at (5^th & 10^th) storeys and again in (1^st & 8^th) storeys respectively are shown in Figure 12(a) and (c). It has been observed that at the undamaged storeys the proposed expression of damage severity returns negligible values whereas, at the damaged locations the results return the estimated damage intensity. Therefore, undamaged and damaged locations can be identified with estimated damage severity successfully by the proposed approach. Additionally, the proposed approach can estimate low damage intensity (5% and 8% damage at 5^th and 10^th storeys, respectively) as observed in Figure 12(b).

Figure 12.

Damage localization considering different soil conditions for (a) 10% (- - -) & 20% (– ⋅ –) damage at 5^th & 10^th storeys, (b) low intensity damage and (c) 10% (- - -) & 20% (– ⋅ –) damage at 1^st & 8^th storeys, respectively.

Comparison with existing techniques

The FRF-based damage severity formulation has been applied to estimate both single as well as multi-damage scenarios in the structures described in various literature. Hosseinzadeh et al. (2016) estimated 15% stiffness degradation in 5^th storey of a 7-storey shear building with varying mass and stiffness by modal flexibility matrix. Anjneya and Roy (2021) proposed a response surface-based approach to estimate 15% damage at 4^th storey of a 6-storey shear building. Chaudhary et al. (2021) quantified 15% damage at 10^th storey of a 14-storey shear building with uniform mass and stiffness at each floor and storey respectively by mode shape-based approach. All the structures have been simulated in MATLAB R2018a (The MathWorks Inc, 2018) and the single damage quantification obtained for each case has been described in Table 5. For estimating the multi-damage conditions, the proposed approach has been implemented in the structures described in Roy (2023) and Niu (2021). A 16-storey shear building model has been considered in Roy (2023). 15% damage has been introduced in 5^th and 13^th storeys. The FRF amplitudes at the fundamental frequencies estimated at the adjacent floors of the damaged storeys and the top storey have been noted.

Table 5.

Comparison of single damage estimation.

Hosseinzadeh et al. (2016)			Anjneya and Roy (2021)			Chaudhary et al. (2021)
Parameters	Intact	15% @ 5^th storey	Parameters	Intact	15% @ 4^th storey	Parameters	Intact	15% @ 10^th storey
H ₇	0.0569	0.2421	H ₆	0.000234	0.000683	H ₁₄	0.0222	0.0522
H ₅	0.0491	0.2097	H ₄	0.000223	0.000652	H ₁₀	0.0196	0.0462
H ₄	0.0373	0.1525	H ₃	0.000213	0.000617	H ₉	0.0184	0.0428
$\frac{H_{5} - H_{4}}{H_{7}}$	0.2066	0.2362	$\frac{H_{4} - H_{3}}{H_{6}}$	0.0429	0.0503	$\frac{H_{10} - H_{9}}{H_{14}}$	0.0559	0.0653
severity (%)			severity (%)			severity (%)
by study	-	14.298	by study	-	17.156	by study	-	16.889
by literature	-	15	by literature	-	15.5	by literature	-	14.42

Finally, damage severity at the respective storeys have been determined by the proposed formulation. Niu (2021) described application of Neumann series expansion in FRF-based damage detecting approach by quantifying damages of 10% and 20% at 2^nd and 9^th storeys of a 12-storey shear building, respectively. Damage quantification of the same structure with non-uniform mass and stiffness has also been performed in this study. The results obtained from the multi-damage scenario have been shown in Table 6. In each case, the proposed formulation predicted damage severity with efficacy. Thus, the proposed formulation produces results that are on par with, or even surpass, those of existing techniques.

Table 6.

Comparison of multi-damage estimation.

Roy (2023)			Niu (2021)
Parameters	Intact	15% @ 5^th & 15% @ 13^th storey	Parameters	Intact	10% @ 2^nd & 20% @ 9^th storey
H ₁₆	0.0977	0.02468	H ₁₂	0.00108	0.00149
H ₁₃	0.0924	0.0233	H ₉	0.00109	0.00151
H ₁₂	0.0889	0.0224	H ₈	0.00106	0.00146
H ₅	0.0448	0.0114	H ₂	0.000393	0.000563
H ₄	0.0364	0.0089	H ₁	0.000198	0.000268
$\frac{H_{13} - H_{12}}{H_{16}}$	0.03545	0.04084	$\frac{H_{9} - H_{8}}{H_{12}}$	0.0269	0.0316
$\frac{H_{5} - H_{4}}{H_{16}}$	0.0866	0.0999	$\frac{H_{2} - H_{1}}{H_{12}}$	0.1804	0.1971
Severity (%)			Severity (%)
D₁₃ by study	-	15.339	D₉ by study	-	17.566
D₁₃ by literature	-	13.28	D₉ by literature	-	20.12
D₅ by study	-	15.297	D₂ by study	-	9.209
D₅ by literature	-	13.26	D₂ by literature	-	9.96

Experimental study

Prototype characteristics

A 9-storey steel moment resisting frame consisting of single-bay in each direction has been considered in this study. The sectional properties of the structure have been given in Table 7.

Table 7.

Characteristics of prototype model.

Beam	0.51 m × 0.51 m	Span in Y direction	5.7 m
Column	0.55 m × 0.6 m	Span in X direction	5.7 m
Floor thickness	200 mm	Storey height	5.7 m

Similitude relationships

For modelling any structure along with the soil-foundation system for small-scale shaker tests, some scaling laws have to be followed. These scaling laws are the rules through which geometry, material properties, boundary conditions, loading conditions, etc. Of the model are decided according to the prototype such that the response of the model and prototype can be related. Selecting an appropriate geometric scaling factor (λ) is one of the most important steps in the design of a small-scale model for the shaker test. Here, the maximum value of λ has been governed by the shaker plate dimensions (354 mm × 354 mm) and payload capacity (45 kg). Acceleration due to gravity has been considered same for the prototype as well as the model. If V_s is the shear wave velocity, $λ = {({(V_{s})}_{prototype} / {(V_{s})}_{model})}^{2}$ satisfies Cauchey’s condition as described in Turan et al. (2009) to ensure adequate model for the experimental study. In this investigation, λ has been considered as 60 and the similitude relationships with other variables like length (λ) and frequency $(1 / \sqrt{λ})$ have been adopted from Meymand (1998).

Components of experimental setup

Small-scale building model

A 9-storey miniature model has been developed after employing the geometric scaling factor of 60. Same material has been considered for prototype and model. The model building has been analyzed in SAP2000(CSI) to decide the section dimensions. The height and plan dimensions have been decided according to scaling relationships. However, the floor thickness, beam and column dimensions have been finalized after several trials to match the natural frequency of the model building. Different section dimensions have been shown in Table 8.

Table 8.

Dimension details of the miniature model.

Structural element	Dimension
Storey height	95 mm
Undamaged column	1.6 mm × 10 mm
Span in X-direction	95 mm
Damaged column	1.2 mm (average) × 10 mm
Span in Y-direction	95 mm
Beam dimension	1 mm × 25 mm
Floor thickness	6 mm

A schematic diagram of the intact experimental model has been shown in Figure 13. To perform the experiment on damaged structure the original columns of 1.6 mm thickness have been replaced by columns of reduced thickness (1.2 mm average) such that the stiffness of the storey degrades. Four cases have been generated for single damage scenario at the 5^th (D₅) storey. Again, two cases of multi-damage scenario have been generated at 3^rd (D₃) and 6^th (D₆) storeys. The details of the damage scenarios has been given in Tables 9 and 10. The fundamental frequency of the undamaged model in fixed base condition is 4.92 Hz.

Figure 13.

Schematic diagram of (a) prototype and (b) model considered for experimental study, (c) 3D view of equivalent shear beam container and (d) FFT result of ESB container for sinusoidal motion of 50 Hz as base excitation.

Table 9.

Details of single damage at 5^th storey.

Case	Applied	Damaged	Width of columns (mm)
Case	Damage (%)	Columns	Col. 1	Col. 2	Col. 3	Col. 4
1	14.45	1	1.2	1.6	1.6	1.6
2	28.9	2	1.2	1.6	1.2	1.6
3	45.78	3	1.2	1.2	1.1	1.6
4	57.37	4	1.2	1.2	1.1	1.3

Table 10.

Details of multi-damage at 3^rd and 6^th storeys.

Case	Damage at 3^rd storey					Damage at 6^th storey
	Applied	Width of columns (mm)				Applied	Width of columns (mm)
	Damage (%)	Col. 1	Col. 2	Col. 3	Col. 4	Damage (%)	Col. 1	Col. 2	Col. 3	Col. 4
1	45.4	1.2	1.2	1.4	1.4	28.5	1.3	1.6	1.1	1.6
2	45.4	1.2	1.2	1.4	1.4	37.3	1.1	1.5	1.4	1.5

Foundation and soil system

Similar to the model building, the foundation also needs to be designed as per the scaling relationships. Thus, to imitate a rigid surface foundation, a base plate of 150 mm × 150 mm × 5 mm has been provided. As the dimensions are very small, grooves have been made in the bottom of the base plate to provide appropriate interaction between soil and base plate during shaking. For the experiment, the soil of the IIT Patna campus has been considered. The density of the soil has been considered equal to the prototype. The site soil has been identified as medium sand with very less percentage of coarse sand particles in it. The site soil properties, namely density, shear wave velocity and Poisson’s ratio have been measured as 1500 kg/m³, 211 m/s and 0.25, respectively. Now, Kramer (1996) has proposed an expression to determine the frequency (Hz) of the soil mass as f = V_s/(4H) where, V_s is the shear-wave velocity of the soil mass at bedrock depth H. The frequency of the model soil imposing geometric scaling factor has been calculated as 40.06 Hz.

Equivalent shear beam (ESB) container

In reality, the soil surrounding the structure has an infinite domain. However in laboratory tests, the model has a finite size such that it replicates the field conditions with minimum possible error. If the shaking in the soil mass is due to vertically propagating waves, the motion varies with depth and also at constant depth the motion in the horizontal direction does not vary (Wolf (1994)). The soil mass, can therefore, be considered as a shear beam. The design principles of ESB container include:

• The end walls should maintain minimum soil interaction and should also behave as a shear beam with the adjacent soil resulting to negligible generation of body wave.

• Must maintain same stress distribution as may be generated in prototype due to base shaking by ensuring similar friction between all end walls and adjacent soil.

• Frictionless sidewalls to ensure no induced shear stress generation.

Zeng and Schofield (1996) proposed an ESB container consisting alternate layers of aluminium and rubber to imitate the soil vibration in a small-scale setup. To ensure true vibration of the soil, the natural frequency of the container should match with the soil frequency such that the container walls also behave as an equivalent shear beam as the soil. Here, natural rubber has been used along with aluminium. To design an ESB container, the modulous of elasticity (E_r) of rubber need to be determined. Three different rubber samples of size 30 mm × 30 mm × 30 mm cube has been tested in universal testing machine to obtain E_r. The average value has been obtained as 0.617 MPa. Employing λ, the length, width and height of the container has been calculated as 480 mm, 290 mm, and 170 mm, respectively. For finding the dimensions of the aluminium and rubber layers, a 3D model of the container has been modelled in SAP2000 (CSI SAP2000, 2016). Aluminium hollow sections and rubber layers have been modelled with 1D frame elements and 2D shell elements, respectively. An ESB container with 7 aluminium rings and 6 rubber layers has been proposed as shown in Figure 13(c). Again to calculate the outer dimensions, different sections of rubber and aluminium have been used to match the natural frequency of the soil mass and ESB container. The final dimensions of the section have been given in Table 11. A timber base plate with 8 mm diameter holes has been attached to the box to mount on the shaker. Coarse-grained soil has been glued to end walls and base to maintain appropriate friction such that required shear stresses can be achieved. The natural frequency of the ESB container has been determined by mounting the ESB container on shaker and a sinusoidal motion of frequency 50 Hz has been applied as base excitation. An accelerometer has been attached at the top edge of the box to record the horizontal acceleration response. The natural frequency of the empty container turned out to be 39.14 Hz (Figure 13(d)). This ensures that as the natural frequency of container and soil are almost equal, therefore under same shaking the two will behave as an equivalent shear beam assembly.

Table 11.

Dimensions of Proposed ESB Container.

Aluminium in X-direction	480 mm × 30 mm × 11 mm
Aluminium in Y-direction	290 mm × 20 mm × 11 mm
Rubber in X-direction	480 mm × 30 mm × 15.5 mm
Rubber in Y-direction	290 mm × 20 mm × 15.5 mm

Instruments involved in experiment

The experiment has been performed on the miniature structure mounted on APS400 long-stroke shaker. Base excitation has been applied to the shaker through keysight 33500B series trueform waveform generator. A power amplifier has been connected to the waveform generator to amplify the induced input signal before exciting the shaker. Type 4507-B-001 DeltaTron piezoelectric accelerometers have been attached at floor levels to capture the horizontal acceleration response at each storey of the shear building model. Brüel & Kjær data acquisition system (DAQ) has been connected to the accelerometers by SubMiniature version B cables to collect the recorded responses. Brüel & Kjær Pulse Lab. shop and Pulse reflex software have been used to record and digitize the data.

Procedure

Now for the experiment, the ESB container has been mounted on the shaker. Based on the volume of the ESB container and density of the campus soil, the total weight of the soil required to fill the shear box has been calculated beforehand. The complete filling of the ESB box with the total calculated quantity of soil is very essential, otherwise the semi-infinite soil medium below the structure will not be restored. After the assembly of the ESB container and soil on the shaker, the small-scale model of the structure has been rested on the soil surface such that the base plate behaves as a rigid surface foundation on the soil. The complete setup for performing the experiment has been shown in Figure 14. The desired approach to evaluate structural health in the post-seismic scenario is by determining modal parameters in ambient vibration condition. However in this study, emphasis has been provided on the influence of soil dynamics on the structure. Thus to recreate an ambient environment in the best possible way, low-level base excitation (0.1 g peak ground acceleration) has been applied. Again, it has been considered that the structure is already damaged due to earthquake. Therefore, experiments have been performed with Gaussian white signal as the base excitation restricting further deterioration of the structure. The horizontal acceleration response has been recorded for 30s with sampling frequency of 4096 Hz. For this study, both single and multiple damages have been considered. To simulate the damage condition all the columns have been replaced with thinner columns at desired storeys. The reduced column thickness imitates the stiffness degradation similar to a damaged system. The same experimental procedure has been followed for both the undamaged and damaged conditions.

Figure 14.

Experimental setup for flexible base condition.

Results and discussion

The FRF magnitude at the fundamental frequency of the intact and different damaged structures have been obtained from the horizontal acceleration response. Four single damage cases at the 5^th storey have been shown in Figure 15. Two multi-damage cases at the 3^rd and 6^th storeys have been shown in Figure 16. It has been observed that the proposed formulation can estimate the damage severity with efficacy in presence of soil underneath the foundation. A maximum absolute error of 3.2% has been observed. This shows an agreement with the numerical study for damage at the intermediate storeys of the structure. The error between the applied and estimated damage intensity may have generated due to various factors influencing the vibrational response of the structure. Therefore, the proposed methodology is capable enough to quantify damage in spite of circumstances involved with real responses.

Figure 15.

Comparison of actual and estimated damage severity for single damage scenario at 5^th storey for: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4.

Figure 16.

Comparison of actual and estimated damage severity for multiple damage scenario at 3^rd and 6^th storeys respectively:(a) Case 1 and (b) Case 2.

Application on Van Nuys hotel in California

The proposed formulation has been applied to estimate the damage severity between the 2^nd and 3^rd floors of one of the mostly studied civil structures, the 7-storey Van Nuys hotel (Figure 17(c)) in California as shown in Figure 17(b). The total height of the building is 18.9 m. The floor height of 1st storey is 4.11 m whereas all the other storeys are approximately 2.64 m height each. Dynamic responses of the building have been recorded by sensors placed at 1^st, 2^nd, 3^rd, 6^th floors and roof. Insignificant structural and mainly nonstructural damage have been observed before 1994 Northridge earthquake (M_w 6.7) Ni and Zhang (2019) . Thus, the responses recorded during 1992 Big Bear earthquake (M_w 6.5) have been considered as the responses corresponding to undamaged structure. Again, the data obtained during the 1994 Northridge earthquake have been considered corresponding to damaged structure. Figure 17(b) indicated various damages that have been observed from the south view of the building. The diagonal crack in the column between 2nd and 3rd floor contributed the most in the stiffness degradation of that storey. The other damages included damages at beams and joints which may not be evaluated by the proposed method as the formulation has been developed especially for storey stiffness degradation.Now, as the proposed FRF-based approach is directly related to modal displacement at the fundamental frequency, the modal data here has been adopted from Ni and Zhang (2019) (Figure 17(a)). Trifunac et al. (1999) reported damage severity of 33% between 2^nd and 3^rd floors. Damage intensity of 36.68% (Figure 17(d)) has been estimated by applying the proposed damage severity expression on modal displacements corresponding to 2^nd, 3^rd and top floors (Figure 17(c)) with an absolute error of 3.68% as given in Table 12. It is not possible to determine the stiffness degradation at other storeys by this approach. This is because the response data corresponding to adjacent floors of all damaged storeys are not available. Therefore, the estimated damage intensity determined between 2^nd and 3^rd floors indicated that the proposed method is capable of quantifying damage at intermediate storeys of real structures effectively.

Figure 17.

(a) Damaged and undamaged mode shape corresponding to Northridge earthquake and Big Bear earthquake respectively, (b) storey damage between 2^nd and 3^rd floor indicated in the south view of Van Nuys hotel citeTrifunac1999 along with sensor locations, (c) labelled representation of real structure to indicate damage location and (d) predicted damage severity by proposed approach.

Table 12.

Real data of Van Nuys hotel digitized from Ni and Zhang (2019).

Big bear earthquake (1992)				Northridge earthquake (1994)				Damage severity
H _roof	H _flr 3	H _flr 2	$\frac{H_{flr 3} - H_{flr 2}}{H_{roof}}$	${\hat{H}}_{roof}$	${\hat{H}}_{flr 3}$	${\hat{H}}_{flr 2}$	$\frac{{\hat{H}}_{flr 3} - {\hat{H}}_{flr 2}}{{\hat{H}}_{roof}}$	η_est (%)	η_report (%)	Err
−0.679	−0.294	−0.182	0.165	−0.719	−0.214	−0.052	0.226	36.68	33	3.68

Conclusion

In this study, an FRF-based closed-form expression for structural damage severity considering SSI has been developed involving minimum number of sensors. An axially vibrating prismatic bar has been correlated to a shear building to determine damage intensity at any of its storeys, since their dynamic responses are analogous. The stiffness and damping of the soil-foundation system have been portrayed as a partially restrained end of a bar-type structure, free at the other end. The efficiency of the proposed expression has been evaluated for various damage scenarios along with different soil conditions through numerical simulation. The modal parameters of any structure alters due to presence of damage and it further varies due to consideration of SSI. It has been observed that the proposed formulation is effective in determining damage intensity with good accuracy in case of damages at intermediate storeys for dense, medium and soft soil conditions. However, damage severity is overestimated for damages at ground storey in presence of soft soil condition. For better understanding of the soil-structure behaviour in reality, an experimental investigation of damage quantification with the SSI effect has been demonstrated on a miniature model. The free-field motion in the soil has been imitated with an ESB container such that the container and soil medium vibrates as an assembly. The experimental study has been performed to estimate damage severity at the intermediate storeys for both single and multi-damage scenarios considering SSI effect. Results showed that structural damage severity with desired accuracy can be determined by the proposed formulation even with real measurements. Error less than 3.5% has been observed in every case. The experimental outcomes validate the numerical study described for the structural stiffness degradation at the intermediate storeys. The efficiency of the proposed approach has been also studied for a real structure. However, the effectiveness needs to be further validated for ground storey damage and also with other soil-types. The experimental study could be more realistic with undisturbed soil samples. The study can be further extended to understand the influence of nonlinear and hysteretic behaviour of soil in identifying more complex damage patterns. The formulation may be further developed for damage quantification using higher modes of the structure.

Footnotes

Acknowledgement

The authors would like to thank Mr Nishant Nilay (former graduate of the Department of Civil and Environment Engineering, Indian Institute of Technology (IIT), Patna) for providing data obtained from multi-channel analysis of surface waves (MASW) conducted within the IIT Patna campus.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would also like to extend gratitude towards ministry of education (MoE), Government of India for financial support.

ORCID iDs

Saranika Das

Koushik Roy

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