Wave–Current Impulsive Debris Loading on a Coastal Building Array

Introduction

Experimental Data

Debris Test Characteristics

Thin rectangular wooden debris elements with dimensions 15 × 15 × 1.9 cm were painted and then submerged in water over the course of 12 days. The soaked debris elements were weighed daily until the mass did not increase any more. This ensured that the center of gravity remained the geometric center and all debris elements weighed about the same, reaching a mass of m_d = 359 g ( ${\sigma }_{m_d}$ = 9.4 g). The debris was released from a box (the Houdini box) with a horizontal trapdoor where 39 individual debris elements were placed in a line with a separation of 0.6 cm. The debris was marked so that each test had the debris placed in the same location. The debris was released by manually pulling a latch, which quickly opened the horizontal trapdoor, resulting in the dropping of the debris elements [Fig. 2(a)]. The Houdini box was located above the sloped section in the center of the concrete beach [Figs. 1(a and c)]. For the wave–current condition, 30 trials were conducted. In each trial, 39 individual debris elements were released for a total of N = 39 × 30 = 1,170 individual debris elements. For the current-only condition, four trials were conducted, totaling N = 39 × 4 = 156 individual debris elements.

Methods

Debris Collision Probability

The combined field of view of the four video cameras gave visual evidence of debris events and collision events for each of the 100 structures, whether the structure was instrumented or not. This was done manually by watching all the experimental video recordings and by registering the individual collisions. An example of a collision event is given in Fig. 3. In addition, and only for instrumented structures, the number of times a debris element collided with an instrumented structure during a collision event was obtained by analyzing the structural response time series, from which repeated collisions from the same piece of debris were identified. Confirmation that the observed multiple collisions were caused by the same piece of debris came from manual observation of the video. Figure 4 shows an example of a collision event for M-LC-1, with four impacts by the same piece of debris.

Debris Structural Impulse and Load

The load cells do not provide a direct record of the debris load–time history on the structure; they instead provide the structural response to this loading in the form of cross-shore base shear (I-LC-n), or cross-shore and long-shore base shear (M-LC-1). The difference between these measured quantities and the raw debris load arises because the structure has a finite mass, stiffness, and damping; thus, impulsive debris loads on the front of the structure are modified by the structure to generate the measured response. The analysis presented here will use the measured loading records at the base of the structure to estimate the applied debris impulse on the front face. We note that the overturning moment is not considered here because its response was very resonant even without debris, preventing the separation into flow and debris loading components. The torsional moment is also not included in this work because its response was more complex and not possible to analyze using the SDOF model described next.

Applied Debris Load and Structural Response in Cross-Shore Direction

The structural loading response to a debris load in the cross-shore direction on the instrumented structures is estimated using a model based on the one proposed by Haehnel and Daly (2004), which is a standard SDOF system:

$$m\ddot{x}(t)+c\dot{x}(t)+kx(t)=F_x(t)$$

(7)

where m = structural mass; c is the damping coefficient; k = effective stiffness of the structural system; t = time; x = displacement in the x-direction; and F_x(t) = applied debris force in the x-direction. The response of the SDOF system defined in Eq. (7) can be obtained using Duhamel’s integral:

$$x(t)=\frac{1}{m{\omega }_d}{\int }_{-\mathrm{\infty }}^t{F}_x(\tau )e^{-\xi {\omega }_n(t-\tau )}\sin {\omega }_d(t-\tau )d\tau$$

(8)

where ω_n = 2π/T_n = fundamental angular frequency; ξ = c/(2mω_n) = damping ratio; and ${\omega }_d={\omega }_n\sqrt{1-{\xi }^2}=$ damped angular frequency. Using the relation ${\omega }_n^2=k/m$, the structural loading response in the x-direction, F_r,x(t) = kx(t), is evaluated as

$$F_{r,x}(t)=\frac{{\omega }_n^2}{{\omega }_d}{\int }_{-\mathrm{\infty }}^t{F}_x(\tau )e^{-\xi {\omega }_n(t-\tau )}\sin {\omega }_d(t-\tau )d\tau$$

(9)

To analyze the applied debris loading, an assumption regarding the applied force type must be considered, as load cells measure the structural response F_r,x and not the directly applied debris force F_x. Categories of applied force are typically given as quasi-static, dynamic, or impulsive (Chen et al. 2019). This classification depends on the ratio between the loading duration to the fundamental period of the structure (t_d/T_n). Combining Eq. (4) and $T_n=2\pi \sqrt{m/k}$, this ratio can be estimated for the tests considered here as $t_d/T_n={\pi }^{-1}\sqrt{m_d/m}=0.042$; thus, all results are categorized as impulsive loading, defined as t_d/T_n < 0.25 (Chen et al. 2019).

The impulsive loading exerted by the debris can be accurately approximated by an instantaneously applied load using a Dirac delta function as F_x(t) = I_0,xδ(t), with I_0,x = impulse applied by the debris in the x-direction. This is defined as $I_{0,x}\equiv {\int }_{t_a}^{t_b}{F}_{d,x}(t)dt$, where F_d,x(t) = actual debris loading on the front of the structure; t_a = beginning time of a collision; and t_b = ending time of a collision. The difference between F_x(t) and F_d,x(t) is that the former is a mathematically convenient applied debris force used in the SDOF and the latter, F_d,x(t), is the true applied debris force. However, when integrated over time, F_x(t) and F_d,x(t) yield the same impulse, I_0,x. This impulse will be the applied loading to be estimated in all of our analyses and, as will be shown, is a more fundamental quantity than the structural response. Considering this, Eq. (9) becomes

$$F_{r,x}(t)=I_{0,x}\frac{{\omega }_n^2}{{\omega }_d}{e}^{-\xi {\omega }_{n}t}\sin {\omega }_{d}t$$

(10)

To estimate I_0,x, the area below the first peak was obtained from the structural response F_r,x(t) recorded by the load cells. This was evaluated using the integral of F_r,x(t) evaluated between the impact time (t₁ = 0) and the time when the structural response reached zero again (t₂ = π/ω_d). This integral is defined as the effective impulse of the structural response I_r,x (Fig. 5), which is evaluated as

$$I_{r,x}={\int }_{t_1}^{t_2}{F}_{r,x}(t)dt$$

(11)

$$I_{r,x}=I_{0,x}{\int }_0^{\pi /{\omega }_d}\frac{{\omega }_n^2}{{\omega }_d}{e}^{-\xi {\omega }_{n}t}\sin {\omega }_{d}tdt$$

(12)

Solving Eq. (11) for I_0,x yields

$$I_{0,x}=C(\xi )I_{r,x}$$

(13)

where C(ξ) is a factor that can be analytically expressed as

$$C(\xi )={(1+e^{-\pi \xi /\sqrt{1-{\xi }^2}})}^{-1}$$

(14)

The factor C(ξ) is a strictly increasing function that only depends on the damping ratio ξ, defined for 0 ≤ ξ < 1, and bounded between C(ξ = 0) = 0.5 and C(ξ → 1) → 1.

Eq. (13) is used in this paper to estimate the impulse applied by the debris (I_0,x), using the effective impulse of the structural response recorded by the load cells (I_r,x). It assumes an initially unloaded structure (Fig. 5); however, during the experimental tests, the debris did not impact a completely unloaded structure. The debris impacted the structures in an already loaded state, dominated by the flow field consisting of a steady current and, if applicable, random waves. Thus, the experimental I_r,x associated with the debris impact was obtained by following these steps. Step 1: the impact time (t₁) was selected as the point when a sudden spike rose in the time series. Step 2: confirmation that this rise in the time series corresponded to a debris collision was obtained from the video camera recordings. Step 3: a horizontal line passing through F_r,x(t) = F_r,x(t = t₁) was generated. Step 4: the intersection between this horizontal line and the structural response after the first peak was calculated using a linear interpolation scheme between the two adjacent F_r,x data points. Step 5: the area below the first peak and above the horizontal line F_r,x(t) = F_r,x(t = t₁) was calculated. This area was assigned the value of I_r,x. The theoretical and experimental I_r,x are denoted in Figs. 5 and 6, respectively, as a gray shaded region.

In this paper, we use the applied impulse I_0,x instead of the maximum applied loading F_x,max because the applied impulse is a more stable and fundamental quantity than the applied maximum force (Bullock et al. 2007; Chen et al. 2019), particularly in cases like those here, where the impulsive loading has a much shorter duration than the fundamental period of the structure. An advantage of working with the impulses of Eq. (13) is that the relation between I_0,x and I_r,x only depends on one structural property, the damping ratio ξ.

The damping ratio was obtained from the experimental structural response of each debris impact on the instrumented structures. The damping ratio ξ of each of I-LC-n was estimated using the half-power bandwidth method (Wu 2015). This method was applied to the structural response for each of the debris collisions for I-LC-n between the impact time and the time when the oscillating response completely decayed. The mean of the individual damping ratio estimates from each collision event for I-LC-n is considered as the representative value for ξ. The structural response recorded on M-LC-1 was highly damped; therefore, the damping ratio of M-LC-1 could not be estimated using the half-power bandwidth method because this method is only valid when ξ < 0.383 (Wu 2015). The M-LC-1 damping ratio was estimated by fitting the M-LC-1 structural response to a theoretical SDOF structural response to a F_x(t) = I_0,xδ(t) load. To fit the theoretical loading response, the M-LC-1 structural response was zeroed as follows. A low-pass window-based far-infrared (FIR) filter with a cutoff frequency of 20 Hz was applied to the structural response. This filtered response was subtracted from the original response to obtain the zeroed signal (Fig. 7). The theoretical structural response was generated by varying (1) the damping ratio ξ; (2) the fundamental period T_n; (3) the impulse applied by the debris I_0,x; (4) and a time offset between the time when the measured response began and the time when the theoretical structural response started. Figure 7 shows an example of a best fit using this procedure for a M-LC-1 collision event. This procedure was applied to all the M-LC-1 collision events from which a representative value of ξ was obtained, allowing for calculation of the mean of the individual values of the damping ratio of each collision event. The ξ estimates for M-LC-1 and I-LC-n, along with their respective standard deviations, are given in Table 1.

Load cell	Measurement	ξ	σξ	Tn (s)	${\sigma }_{T_n}$ (s)	k (MN/m)	NCE	C (ξ)	λ (ξ)
M-LC-1	Fr,x	0.64	0.0982	0.0105	0.0017	7.09	182	0.929	0.482
	Fr,y	0.64	0.0982	0.0105	0.0017	7.09	182	0.929	0.482
I-LC-1	Fr,x	0.021	0.0101	0.0183	0.0024	2.41	87	0.516	0.968
I-LC-2	Fr,x	0.033	0.0099	0.0191	0.0035	2.21	56	0.526	0.951
I-LC-3	Fr,x	0.017	0.0051	0.0184	0.0022	2.38	37	0.513	0.974
I-LC-4	Fr,x	0.018	0.0051	0.0185	0.0017	2.36	22	0.514	0.972
I-LC-5	Fr,x	0.020	0.0082	0.0177	0.0015	2.57	22	0.515	0.970

Applied Load and Structural Response in Along-Shore Direction

The debris impulse in the y-direction (along-shore) applied to M-LC-1 (I_0,y) was estimated identically to the cross-shore procedure:

$$I_{0,y}=C(\xi )I_{r,y}$$

(15)

where I_r,y is the effective impulse of the structural response in the y-direction, and is obtained using the same method as for I_r,x.

Dimensionless Impulse

The impulses applied by the debris in the cross-shore direction (I_0,x) and along-shore direction (I_0,y) are made dimensionless using the hydrodynamics and debris properties in the following manner:

$$\overline{I_{0,x}}=\frac{I_{0,x}}{p_n}$$

(16)

$$\overline{I_{0,y}}=\frac{I_{0,y}}{p_n}$$

(17)

The normalizing impulse p_n is defined by the debris mass and the local hydrodynamic conditions in Row n using the following relationship:

$$p_n=\{\begin{array}{cc}{m}_d{u}_n& \mathrm{for}\mathrm{the}\mathrm{current}\text{-}\mathrm{only}\mathrm{condition}\\ m_d{u}_n+m_d\frac{H_{s,n}}{2d_n}\sqrt{g{d}_n}& \mathrm{for}\mathrm{the}\mathrm{wave}\text{-}\mathrm{current}\mathrm{condition}\end{array}$$

(18)

where u_n is the cross-shore component of the steady current; $(H_{s,n}\sqrt{g{d}_n}/2d_n)=$ maximum orbital velocity from the local significant wave height, H_s,n, calculated using linear theory and shallow water conditions; d_n = local SWD; and g = gravitational acceleration. The linear maximum orbital velocity is taken as the representative value for the wave component in the wave–current condition, but this does not imply that the wave is linear; most of the waves broke close to the structures, beginning around the end of the 1:20 slope. All hydrodynamic variables are evaluated at the center of the unobstructed cross-shore channel, 0.1 m seaward of the corresponding row. For instance, for an impact on I-LC-2, located at the second row, p_n is evaluated at point number 3 according to Fig. 1(d). Values associated with the hydrodynamic conditions in the test section are given in Table 2.

Row	Location [Fig. 1(d)]	md (kg)	un (m/s)	$(H_{s,n}/2d_n)\sqrt{g{d}_n}$ (m/s)	pn,wave-current (N · s)	pn,current-only (N · s)
1	1	0.359	0.26	0.45	0.25	0.093
2	3	0.359	0.41	0.41	0.29	0.147
3	5	0.359	0.39	0.43	0.29	0.140
4	7	0.359	0.37	0.39	0.27	0.133
5	9	0.359	0.40	0.24	0.23	0.144

Results

Debris Impact Probability

Collisions are more frequent in the first row and decrease in rows further inland for both the wave–current and the current-only conditions [Figs. 8(a and b)]. A collision ratio (CR) is defined as

$$\mathrm{CR}(n)=N_{\mathrm{row}n}/N$$

(19)

where N_{row n} is the sum of the collision events in all structures that occurred in Row n and N is the sum of the debris events in all structures that occurred in Row n. Values of CR(n) are given in Fig. 8(c), indicating that the collision probability strongly decreases with row number.

The number of times the same individual debris element collided with an instrumented building during a collision event was denoted q. For the wave–current condition, the fraction of debris events with q or more collisions is presented in Fig. 9. Two features are immediately evident. First, it is not at all uncommon for a debris element to have more than one collision with the same structure: up to seven individual collisions from one piece of debris on a single structure were noted in the instrument record. A number of collisions could occur when a debris element was trapped for a short time in front of a structure, or when different edges of the debris element impacted sequentially. An example of this situation is presented in Fig. 4, with q = 4. The second observation is that the fraction of debris events with q or more collisions strongly decreases with the row number, indicating that the more shelter is given, the fewer times the same piece of debris collides with the structure. In Row 5, which contained the most sheltered structure with a load cell, very few multiple collisions were recorded.

Debris Impulse

For each collision event, the impulse applied by the debris in the cross-shore direction $(\overline{I_{0,x}})$ was estimated in all instrumented structures and the impulse applied by the debris in the along-shore direction $(|\overline{I_{0,y}}|)$ was estimated in M-LC-1 using Eqs. (16) and (17), respectively. It must be noted that absolute values were considered in the along-shore direction because there could be either positive or negative values depending on the angle of impact. For debris events with q > 1, the collision with the largest value of $\overline{I_{0,x}}$ or $|\overline{I_{0,y}}|$ in a particular event was considered representative; for the debris events with no collision (q = 0), a value of $\overline{I_{0,x}}=0$ or $|\overline{I_{0,y}}|=0$ was recorded. For each instrumented structure and for each direction, the dimensionless impulses were then sorted in ascending order. This allowed the calculation of an empirical exceedance probability P_i, estimated as

$$P_i=1-\frac{R_i}{N_{\mathrm{DE}}+1}$$

(20)

where R_i = rank number of the impulse $\overline{I_{0,x_i}}$ or $\overline{I_{0,y_i}}$; and N_DE = number of debris events in the area of the instrumented structure. The empirical exceedance probabilities of $\overline{I_{0,x}}$ and $|\overline{I_{0,y}}|$ are presented in Figs. 10(a and c). The same procedure was applied to the current-only condition; however, here M-LC-1 was the only instrument with enough data points to generate meaningful probabilities (orange symbols, Fig. 10). Results show how, for a given P_i, the magnitude of the debris impulse decreases with the number of sheltering rows. Moreover, for a given P_i, the impulses in the along-shore direction are lower than the impulses in the cross-shore direction, but higher than might be expected with an average flow in the cross-shore direction. To assess the relevance of repeated impacts by the same debris element, the second largest impulse applied by the debris was evaluated for both the cross-shore $(\overline{I_{0,x}^{(2\mathrm{nd})}})$ and the along-shore $(|\overline{I_{0,y}^{(2\mathrm{nd})}}|)$ directions in all debris events. If the collision event has only one collision (q = 1), then $\overline{I_{0,x}^{(2\mathrm{nd})}}=0$ and $|\overline{I_{0,y}^{(2\mathrm{nd})}}|=0$. These empirical exceedance probabilities are presented in Figs. 10(b and d), which show that the impulse from the second largest collision from the same debris element can still exert an important structural load, especially for the first row.

The exceedance probabilities of $\overline{I_{0,x}}$, $\overline{I_{0,x}^{(2\mathrm{nd})}}$, $|\overline{I_{0,y}}|$, and $|\overline{I_{0,y}^{(2\mathrm{nd})}}|$ were also obtained including only collision events, i.e., neglecting cases where q = 0, as

$$P_i=1-\frac{R_i}{N_{\mathrm{CE}}+1}$$

(21)

where N_CE = number of collision events in the instrumented structure. These exceedance probabilities, under the assumption that a collision event will occur, also show that there is a reduction in the probabilistic magnitude of debris impacts with increasing row number for a given P_i (Fig. 11).

Considering only the first and exposed row, where the loads are largest, a plausible upper limit for the cross-shore dimensionless impulse might be $\overline{I_{0,x,max}}\approx 1.0$ for both wave–current and current-only conditions; and for the sheltered condition (Rows 2 to 5) a plausible upper limit for the cross-shore dimensionless impulse might be $\overline{I_{0,x,max}}\approx 0.8$ for the wave–current condition [Fig. 11(a)]. Regarding the dimensionless along-shore impulses, a dimensionless upper limit of $\overline{I_{0,y,max}}\approx 0.6$ might be applicable [Fig. 11(c)].

Maximum Structural Load

Ultimately, an estimate for the debris collision design load is needed for any structural system or component that is being considered. One of the most important aspects of this work is the emphasis that the load experienced by a particular structural system is a function not only of the debris impulse, but also of the structural properties. Under the assumption of the impulsive loading functions F_x(t) = I_0,xδ(t) and F_y(t) = I_0,yδ(t), the maximum loading perceived by the structural system corresponds to the maximum structural loading response F_r,x,max and F_r,y,max, respectively. This maximum loading response is obtained analytically as

$$F_{r,x,max}=\frac{2\pi }{T_n}{I}_{0,x}\lambda \left(\xi \right)$$

(22)

$$F_{r,y,max}=\frac{2\pi }{T_n}{I}_{0,y}\lambda \left(\xi \right)$$

(23)

where

$$\lambda \left(\xi \right)=\exp \left(\frac{-\xi {\cos }^{-1}\xi }{\sqrt{1-{\xi }^2}}\right)$$

(24)

Together, these predict the maximum structural load as a function of the impulse applied by the debris, the structural fundamental period T_n, and the damping ratio ξ. The fundamental period was estimated as $T_n=2(t_2-t_1)\sqrt{1-{\xi }^2}$, where (t₂ − t₁) was directly obtained from the structural response measurements (Table 1). The predictability of Eqs. (22) and (23) is assessed by comparing the predicted maximum loading response F_r,x,max and F_r,y,max with the experimental maximum loading response F_r,x,max,exp and F_r,y,max,exp, respectively, for each collision event, where F_r,x,max and F_r,y,max are computed using estimated values of I_0,x [Eq. (13)], I_0,y [Eq. (15)], T_n, and λ(ξ) (Table 1), whereas F_r,x,max,exp and F_r,y,max,exp are obtained from the load cell recordings by subtracting the experimental debris loading response at the time of the debris impact from the maximum experimental debris loading response (Fig. 6). The comparisons between F_r,x,max and F_r,x,max,exp and F_r,y,max and F_r,y,max,exp are presented in Fig. 12, along with the best-fit lines passing through the coordinate system origin.

Discussion

Loading Duration Effects on the Maximum Loading Response

The assumption that the impulsive debris loading can be represented by F_x(t) = I_0,xδ(t) is only valid when the impact duration is very short with respect to the fundamental period of the structure. If the ratio of the impact duration to the fundamental period of the structure (Δt/T_n) is significant, the impulsive debris loading must be represented with a finite impact duration. In this section, we analyze how the maximum loading response F_r,x,max varies depending on the ratio Δt/T_n. A half-sine pulse has been found to be representative for the applied debris impact load (Piran Aghl et al. 2014); therefore, a half-sine pulse of magnitude I_0,x and a duration of Δt, measured between the start of the pulse and when the pulse reaches its maximum value, is considered for the analysis. Assuming a SDOF system with a damping ratio ξ and a fundamental period T_n, the loading response F_r,x(t) is obtained by solving Duhamel’s integral [Eq. (8)] with a finite applied force, defined as

$$F_x(t)=\{\begin{array}{cc}0& t<0\\ {\displaystyle \frac{I_{0,x}\pi }{4\mathrm{\Delta }t}\sin \frac{\pi t}{2\mathrm{\Delta }t}}& 0\le t\le 2\mathrm{\Delta }t\\ 0& t>2\mathrm{\Delta }t\end{array}$$

(28)

By substituting Eq. (28) into Eq. (8), the maximum structural loading response F_r,x,max can be estimated as

$$F_{r,x,\max }=\frac{2\pi }{T_n}{I}_{0,x}\gamma (\mathrm{\Delta }t/T_n,\xi )$$

(29)

where γ(Δt/T_n, ξ) is a coefficient that only depends on the ratio Δt/T_n and the damping ratio ξ, the values of which are presented in Fig. 13. The derivation of Eq. (29) can be found in the Appendix.

The function γ(Δt/T_n, ξ) takes values between 0 and 1. The longer the I_0,x pulse with respect to the fundamental period and the larger the damping, the smaller the maximum response (Fig. 13). The particular case of a fully impulsive impact (γ(Δt/T_n → 0, ξ) → λ(ξ)) corresponds to the case represented by Eq. (22), which was used in this study to estimate the fully impulsive maximum responses F_r,x,max and F_r,y,max presented in Fig. 12. Another particular case corresponds to the case when damping is neglected (ξ = 0) but an impact duration greater than 0 is considered (γ(Δt/T_n, ξ = 0)). This case has been presented by Chen et al. (2019), who proposed a modified dynamic response factor, similar to the factor γ(Δt/T_n, ξ = 0), for an applied triangular loading history. If the ratio of impact duration to the fundamental period of the structure, (Δt/T_n), is significant, impulsive debris loading must be represented with a finite impact duration. In the case where an impulsive impact is considered with a small (Δt/T_n) without any damping, γ takes the upper limit of γ = 1, which yields the maximum loading response of F_r,x,max = 2πI_0,x/T_n.

Conclusions

Appendix. Maximum Loading Response for an Impact Duration Greater than Zero (Δt > 0)

Data Availability Statement

Acknowledgments

Notation

C

factor that relates I_r,x and I_0,x;

C_B

blockage coefficient;

C_D

depth coefficient;

C_I

importance coefficient;

C_L

load correction coefficient;

C_O

ASCE7-16 orientation coefficient;

CR(n)

collision ratio in Row n;

c

damping coefficient;

d_n

steady current mean water depth at location of Row n;

EP2CE

2% experimental exceedance probability load, given a collision event;

EP10CE

10% experimental exceedance probability load, given a collision event;

F

contact–stiffness maximum debris impact load;

F_d,x

actual debris loading on front of structure;

F_{flood,impulsive}

ASCE7-16 impulsive flood debris impact force;

F_i

ASCE7-16 tsunami design instantaneous debris impact force;

F_ni

ASCE7-16 nominal maximum instantaneous debris impact force;

F_r,x

structural loading response in x-direction;

F_r,x,max

predicted maximum loading response in x-direction;

F_r,x,max,exp

experimental maximum loading response in x-direction;

F_r,y,max

predicted maximum loading response in y-direction;

F_r,y,max,exp

experimental maximum loading response in y-direction;

F_{tsu,impulsive}

ASCE7-16 impulsive tsunami debris impact force;

F_x

applied force in x-direction;

F_{x,max,proposed}

proposed maximum impulsive debris impact load in x-direction;

F_{y,max,proposed}

proposed maximum impulsive debris impact load in y-direction;

g

gravitational acceleration;

H_s,n

significant wave height at location of Row n;

h_n

structural height of moment-resisting frame;

I-LC-n

structure located in Row n, instrumented with inline load cell;

I_r,x

effective impulse of structural response in x-direction;

I_r,y

effective impulse of structural response in y-direction;

I_tsu

ASCE7-16 tsunami importance factor;

I₀

debris impulse;

I_0,x

impulse applied by debris in x-direction;

$\overline{I_{0,x}}$

largest dimensionless impulse applied by debris in x-direction;

$\overline{I_{0,x,max}}$

upper limit for cross-shore dimensionless impulse;

$\overline{I_{0,x}^{(2\mathrm{nd})}}$

second largest dimensionless impulse applied by debris in x-direction;

I_0,y

impulse applied by debris in y-direction;

$\overline{I_{0,y}}$

largest dimensionless impulse applied by debris in y-direction;

$\overline{I_{0,y,max}}$

upper limit for along-shore dimensionless impulse;

$\overline{I_{0,y}^{(2\mathrm{nd})}}$

second largest dimensionless impulse applied by debris in y-direction;

k

effective stiffness;

k_d

debris stiffness;

k_s

structural stiffness;

M-LC-1

structure located in Row 1, instrumented with multiaxis load cell;

m

mass of instrumented structures;

m_d

debris mass;

N

sum of debris events in all structures that occurred in Row n;

N_CE

number of collision events in corresponding instrumented structure;

N_DE

number of debris events in area of corresponding instrumented structure;

N_{row n}

sum of collision events in all structures that occurred in Row n;

N_s

number of stories of moment-resisting frame;

P_i

empirical exceedance probability;

p_n

normalizing impulse determined by debris mass and hydrodynamic conditions;

Q

approximate flow of experimental tests;

q

number of times same individual debris piece collided with instrumented building during collision event;

R_i

rank number;

R_max

dynamic response factor;

T_n

fundamental structural period;

T_p

peak period;

t

time;

t_a

beginning time of collision;

t_b

end time of collision;

t_d

rectangular pulse duration;

t₁

debris impact time;

t₂

time after debris impact when structural response reaches zero again;

u

impact velocity;

u_max

maximum impact velocity;

u_n

cross-shore velocity component of steady current in Row n;

γ

gamma factor;

Δt

duration of half-sine pulse from pulse start to when it reaches its maximum value;

δ(t)

Dirac delta function;

λ

lambda factor;

ξ

damping ratio;

${\sigma }_{m_d}$

debris mass standard deviation;

ω_d

damped angular frequency; and

ω_n

fundamental angular frequency.

References