IntroductionInundation events generated by tsunamis and storms are a major hazard in developed coastal regions. The Indian Ocean tsunami in 2004 (Athukorala and Resosudarmo 2005), the 2010 Chilean tsunami (Fritz et al. 2011; Khew et al. 2015), the Great East Japan earthquake and tsunami in 2011 (Suppasri et al. 2012; Davis et al. 2012); Hurricanes Katrina (Brunkard et al. 2008) and Sandy (Seil et al. 2016); and Typhoon Haiyan (Mikami et al. 2016) have all caused devastating loss of life and economic damage. Many postevent field surveys have shown that, in addition to failures from surge and wave loading (Lynett et al. 2003; Fritz et al. 2011; Fraser et al. 2013; Tomiczek et al. 2017; Izquierdo et al. 2018), significant damage has been associated with waterborne debris impacts. Waterborne debris includes a wide variety of objects carried inland by the flow, for example, structural fragments, such as concrete blocks and wooden poles (Hatzikyriakou et al. 2016), shipping containers (Mikami et al. 2016), boats and cars (Ghobarah et al. 2006), boulders (Fritz et al. 2011; Kennedy et al. 2017), and propane tanks (Stolle et al. 2020a). Residential structures frequently show damage from waterborne debris impacts (Reese et al. 2011; Yeh et al. 2013) but significant structural damage to bridges (Stearns and Padgett 2012; Ghobarah et al. 2006) and above-ground storage tanks (Bernier and Padgett 2020) has also been observed. An improved understanding of waterborne debris impacts and their loading is essential for increasing the resiliency of coastal communities after inundation events.Debris LoadingMost studies of debris impacts are based on a single debris object impact on a single structure using a single degree of freedom (SDOF) model (Nistor et al. 2017a). The SDOF model is a one-dimensional, simplified mechanical model that characterizes the dynamic behavior of a structural system by representing it as a mass–spring–damping model. The SDOF solves a displacement for a given applied load and is used in civil engineering to model structural dynamics. The SDOF model is commonly used in the context of debris impacts on coastal structures; it has been formulated using three equivalent methods: the impulse–momentum approach, the work–energy approach, and the effective contact–stiffness method (Haehnel and Daly 2004). Each of these three methods requires the debris velocity, the debris mass, and an additional parameter that depends on the method. The additional parameters in the impulse–momentum, work–energy, and effective contact–stiffness methods are the impact duration, the debris stopping distance, and the effective stiffness of the debris impact, respectively (Stolle et al. 2018). Other methods estimate the impact force based on empirical equations. For instance, Ikeno et al. (2016) proposed a model based on the Hertz equation and Matsutomi (2009) integrated results from experimental tests to derive an empirical formula for the impact force. Of all the methods, the contact–stiffness approach is most common (Nistor et al. 2017a). This approach estimates the maximum debris impact load F, assuming a rectangular pulse of duration td, as(1) where u = impact velocity; k = effective stiffness of the debris impact; and md = mass of the debris. The effective stiffness of the debris impact is defined as k=(ks−1+kd−1)−1, where ks = stiffness of the structure; and kd = stiffness of the debris (Stolle et al. 2018). Eq. (1) does not consider additional effects, such as the added mass of the displaced fluid, the load eccentricity, or the debris impact orientation. Haehnel and Daly (2004) studied these additional effects through reduced- and full-scale laboratory experiments of wood and steel impacts on an instrumented building on stationary water. These experimental results show that increases in load eccentricity and oblique impact orientations reduce the maximum impact force. However, when the longitudinal axis of the debris is aligned with the flow, the maximum load is well represented by Eq. (1), suggesting that Eq. (1) is an upper value and a conservative estimate for the maximum impact load F. Riggs et al. (2014) and Ko et al. (2015) also studied the effect of the added mass using 1:5 scaled tests of a shipping container and wood log impacts in both water and air. The maximum impact load was not significantly different between the in-water and the in-air tests, suggesting that an added mass coefficient is not necessary. A separate study of experimental full-scale in-air impacts of a wood pole, a steel tube, and a shipping container found that Eq. (1) agrees reasonably well with these experimental tests (Piran Aghl et al. 2014). Shafiei et al. (2016) found that in-air and in-water tests resulted in similar impact accelerations but the forces measured by loading cells attached to the structure were about 1.5 times larger for the in-water tests. This increase comes from the difference in the impacting mass. From the studies cited previously, it can be concluded that the effect of added mass is still to be determined.ASCE (2017, Chapter 6.11) provides design guidelines for tsunami-borne debris impacts using the contact–stiffness approach. For logs, poles, and shipping containers, the ASCE7-16 tsunami design instantaneous debris impact force (Fi) is estimated as(2) with(3) Fni=umaxmdkmdRmax=I0kmdRmaxwhere Fni = ASCE7-16 nominal maximum instantaneous debris impact force; Itsu is an importance factor, ranging from 1.0 to 1.25; CO is an orientation coefficient, equal to 0.65 for logs and poles and 0.75 for shipping containers; umax = maximum flow velocity; md = mass of the debris; k is the lesser value between the debris stiffness or the lateral stiffness of the impacted structural element; and Rmax is a dynamic response factor. The debris impulse, I0 = mdumax, is the fundamental momentum carried by the debris that can be imparted to the structure. The ASCE7-16 tsunami-borne debris impact load assumes a rectangular pulse of duration td, which is defined as(4) The load experienced by a structural mode will be modified by its dynamic properties. For a fundamental structural mode with period Tn and short duration loading td/Tn ≤ 0.2 (known as an impulsive loading condition, which is studied here), the dynamic response factor can be simplified as Rmax = 4td/Tn, which greatly simplifies the ASCE7-16 tsunami debris impact load experienced by the structure, to(5) Ftsu,impulsive=8TnI0ItsuCOwhere Ftsu,impulsive = ASCE7-16 impulsive tsunami debris impact force.Impact loads during flooding events are considered in ASCE (2017, Section C5.4.5), which estimates the maximum impact load using the impulse–momentum approach and assuming a half-sine loading pulse. For an impulsive loading condition, the ASCE7-16 flood debris impact load experienced by the structure can be expressed as(6) Fflood,impulsive=2πTnI0CICOCDCBwhere Fflood,impulsive = ASCE7-16 flood debris impact force for td/Tn ≤ 0.2; I0 = mdumax = debris impulse; Tn = structure or structural component fundamental period; CI is an importance coefficient, ranging from 0.6 to 1.3; CO is an orientation coefficient, equal to 0.8; CD is a depth coefficient that varies from 0 to 1 depending on the still water depth (SWD); and CB is a blockage coefficient that varies from 0 to 1 depending on the width of the flow path.Both ASCE7-16 tsunami [Eq. (5)] and ASCE7-16 flood [Eq. (6)] impulsive debris impact forces are fundamentally very similar. The only difference is the set of coefficients that multiply the quantity I0/Tn, which are 8 ItsuCO and 2πCICOCDCB for tsunami and flood loads, respectively.Debris TransportIn addition to predicting the debris impact loading, the occurrence probability of an impact is of particular interest in coastal engineering research. This has been addressed by studying the debris transport during inundation events. If the flow depth is large enough, debris are expected to spread in both the along-shore and the cross-shore directions. Based on a field survey after the 2011 Tohoku Tsunami, Naito et al. (2014) proposed and ASCE (2017) adopted a debris hazard region, defined as the area between two ±22.5∘ downstream headings, beginning from the debris source point. The ±22.5∘ headings are measured with respect to the wave propagation direction. Nistor et al. (2017b) found that the debris spreading angle follows an increasing function with respect to the number of debris elements, but is not larger than the spreading angle 22.5∘ proposed by Naito et al. (2014). Spreading angles obtained from the experiments of Park et al. (2021) and Goseberg et al. (2016) are also within this 22.5∘ cone. The debris spreading probability has also been represented as a Gaussian function, with the along-shore distance as the independent variable and with a variance that depends on the cross-shore direction (Matsutomi 2009; Stolle et al. 2020b). It is important to emphasize that, in all the studies mentioned, except for that of Naito et al. (2014), debris transport has been studied primarily on a flat surface or with a constant slope. Complex bathymetric profiles would evidently have a large influence on the distribution of debris.Debris Loading on Building ArraysDebris loading in current engineering standards is addressed assuming isolated structures. However, coastal cities are characterized by arrangements of many buildings. These arrangements, also referred to as structural arrays, will affect the flow conditions, influencing inundation levels and flow velocities, and consequently affecting waterborne debris loading. Evidence of flow disturbance through the presence of structural arrangements has been found in fieldwork (Reese et al. 2007; Leone et al. 2011; Hatzikyriakou et al. 2016), laboratory tests (Simamora et al. 2007; Rueben et al. 2011; Park et al. 2013; Goseberg 2013; Thomas et al. 2015; Nouri et al. 2010; Tomiczek et al. 2016; Moon et al. 2019; Moris et al. 2021), and numerical experiments (Nakamura et al. 2010; Ardianti et al. 2015; Yang et al. 2018; Sogut et al. 2019; Moris et al. 2021). The effect of structural arrays on wave loading generally results in a reduction in the maximum wave loading in the inland buildings, owing to a sheltering effect given by the front buildings (Simamora et al. 2007; Ardianti et al. 2015; Thomas et al. 2015; Tomiczek et al. 2016; Yang et al. 2018; Sogut et al. 2019; Moris et al. 2021). Although studies about the influence of obstacles on debris dispersion have considered arrays composed of one (Kihara and Kaida 2020; Park et al. 2021) or two rows of obstacles (Goseberg et al. 2016), the effect of structural arrays on debris loading is not known. Field survey data show that sheltering and debris are explanatory variables for damage from inundation events (Reese et al. 2011); however, debris loading predictions in sheltered buildings, up to this date, do not, so far as we know, exist. This is an issue because current estimates for impact debris loads [e.g., Eqs. (5) and (6)] could result in overly conservative load predictions for sheltered structures.Limited research has been done regarding the effect of building arrays on debris loading under flooding conditions. Hence, we attempt to address that knowledge gap by investigating the effect of a building array on debris loading under flooding conditions by:
•obtaining the structural response of waterborne debris impacts on buildings within a building array from laboratory experimental data,•applying a methodology based on a SDOF structural system to estimate the applied debris impulse from the collisions,•quantifying the debris collision probability and the collision impulse magnitude and its dependence on the number of sheltering rows within the building array,•developing a predictive equation for maximum structural loading based on the debris momentum, and•comparing the predictive equation from the aforementioned point with current design estimates.ResultsDebris Impact ProbabilityCollisions 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(19) where Nrow 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 ImpulseFor each collision event, the impulse applied by the debris in the cross-shore direction (I0,x¯) was estimated in all instrumented structures and the impulse applied by the debris in the along-shore direction (|I0,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 I0,x¯ or |I0,y¯| in a particular event was considered representative; for the debris events with no collision (q = 0), a value of I0,x¯=0 or |I0,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 Pi, estimated as(20) where Ri = rank number of the impulse I0,xi¯ or I0,yi¯; and NDE = number of debris events in the area of the instrumented structure. The empirical exceedance probabilities of I0,x¯ and |I0,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 Pi, the magnitude of the debris impulse decreases with the number of sheltering rows. Moreover, for a given Pi, 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 (I0,x(2nd)¯) and the along-shore (|I0,y(2nd)¯|) directions in all debris events. If the collision event has only one collision (q = 1), then I0,x(2nd)¯=0 and |I0,y(2nd)¯|=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 I0,x¯, I0,x(2nd)¯, |I0,y¯|, and |I0,y(2nd)¯| were also obtained including only collision events, i.e., neglecting cases where q = 0, as(21) where NCE = 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 Pi (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 I0,x,max¯≈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 I0,x,max¯≈0.8 for the wave–current condition [Fig. 11(a)]. Regarding the dimensionless along-shore impulses, a dimensionless upper limit of I0,y,max¯≈0.6 might be applicable [Fig. 11(c)].Maximum Structural LoadUltimately, 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 Fx(t) = I0,xδ(t) and Fy(t) = I0,yδ(t), the maximum loading perceived by the structural system corresponds to the maximum structural loading response Fr,x,max and Fr,y,max, respectively. This maximum loading response is obtained analytically as(22) (23) where(24) λ(ξ)=exp⁡(−ξcos−1⁡ξ1−ξ2)Together, these predict the maximum structural load as a function of the impulse applied by the debris, the structural fundamental period Tn, and the damping ratio ξ. The fundamental period was estimated as Tn=2(t2−t1)1−ξ2, 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 Fr,x,max and Fr,y,max with the experimental maximum loading response Fr,x,max,exp and Fr,y,max,exp, respectively, for each collision event, where Fr,x,max and Fr,y,max are computed using estimated values of I0,x [Eq. (13)], I0,y [Eq. (15)], Tn, and λ(ξ) (Table 1), whereas Fr,x,max,exp and Fr,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 Fr,x,max and Fr,x,max,exp and Fr,y,max and Fr,y,max,exp are presented in Fig. 12, along with the best-fit lines passing through the coordinate system origin.DiscussionDebris Collision ProbabilityThe distribution of the collision events in the building array [Figs. 8(a and b)] shows that the buildings in the central portion of the first row are most likely to be impacted by the debris. Similar debris spreading distributions have been identified both in the laboratory (Nistor et al. 2017b; Park et al. 2021; Goseberg et al. 2016; Stolle et al. 2020b) and in the field (Naito et al. 2014), but these distributions are defined for debris spreading and not necessarily for collisions. However, debris dispersion is not a major focus of this paper: a complete analysis for this wave–current flow will be given by Cinar et al. (G. E. Cinar, A. Keen, and P. Lynett, “Motion of a debris line-source under currents and waves: An experimental study,” submitted, J. Waterway, Port, Coastal, Ocean Eng., ASCE, Reston, Virginia).The decrease in both CR [Fig. 8(c)] and q (Fig. 9) with increasing row number means that not only does the probability of a collision event decrease with sheltering, but the probability of repeated collisions decreases as well. It was found that CR(n) becomes very low as more rows provide shelter: the wave–current condition CR(n) decreases from CR(n = 1) = 0.42 to CR(n = 10) = 0.015, which means that a collision in the first row is CR(n = 1)/CR(n = 10) = 28 times more likely than one in the tenth row. This can be explained by the debris being channeled through the “streets” in between buildings. This observation has also been identified in a tsunami-like flow through a smaller building array (Goseberg et al. 2016). In the first three rows, the CR for the wave–current condition is higher than for the current-only condition, which can be explained by the enhanced effect of the wave orbital velocities transporting the debris onto the front face of the structures. However, once the debris passed the third row, there was not a significant difference in CR with respect to the hydrodynamic conditions, suggesting that once a debris element passes the first rows of buildings, the collision probability does not depend on the flow type (wave–current or current-only).Cases with a number of collisions in the same debris event (q > 1) could be explained by the debris trajectory. In some cases, after the debris first impacts the structure [e.g., Fig. 3(d)], the debris stays on the front face for some time, experiencing repeated collisions [e.g., Figs. 3(e–h)] until it is carried inland by the flow [e.g., Fig. 3(i)]. Similar results were found for tsunami flow over an initially dry test section (Derschum et al. 2018), where the repeated collisions were explained by the debris being trapped in the surface roller of the tsunami bore. Although our results were obtained for only two hydrodynamic conditions, one debris material and dimension, and one building distribution, the potential impact of generalizing these results to more conditions could result in the quantification of collision damage risk and the improvement of fragility curves, which could significantly increase the resilience of coastal communities to inundation events.Debris Impulse in Unobstructed Row 1The empirical exceedance probability of I0,x¯ for unobstructed Row 1 [Figs. 10(a) and 11(a)] shows that it seems to be an upper limit for the maximum value for I0,x¯ that does not exceed 1.0 for unobstructed Row 1. This means that I0,x = 1.0 · pn might be a reasonably conservative upper limit for the dimensional impulse applied by a debris element in the average flow direction. The empirical exceedance probability of |I0,y¯| for unobstructed Row 1 [Figs. 10(c) and 11(c)], shows that although the mean flow direction was predominantly in the cross-shore direction, the along-shore flow variations cannot be neglected. It seems that there is an upper value for |I0,y¯| that does not exceed 0.6, however this upper limit is not as clear as the upper limit for I0,x¯.Collision impulse statistics from the current-only condition could only be analyzed for structure M-LC-1, where 11 collision events were registered [Fig. 8(b)] from the four current-only trials. The empirical exceedance probability regarding the current-only condition for both I0,x¯ and |I0,y¯| shows a similar trend with respect to the wave–current condition.The empirical exceedance probabilities of I0,x(2nd)¯ and I0,y(2nd)¯ for the unobstructed Row 1, with wave–current forcing, show upper limits of about 0.6 and 0.25, respectively [Figs. 10(b–d) and 11(b–d)]. This means that secondary collisions, even though they are not as high as the largest collision, should not be neglected. The structural damage from repeated impacts can be substantial, as the structural elements can deteriorate because of the number of applied impacts (Derschum et al. 2018). For the current-only condition, the second largest impulse was found to be much weaker than the wave–current condition. This could be explained by the fact that a flow without waves somewhat decreases the probability of debris to significantly impact a building after the first impact has taken place. However, a general statement about the second largest impulse for the current-only condition cannot be made because we registered only 11 current-only collision events, of which only 5 were collisions with q ≥ 2.Collision Magnitude and Its Distribution Within the Building ArrayThe empirical exceedance probability functions of I0,x¯ and I0,x(2nd)¯ show that the impulse applied by debris decreases as sheltering increases [Figs. 10(a and b) and 11(a and b)]. Similar results have been found for maximum force reduction (Simamora et al. 2007; Nakamura et al. 2010; Ardianti et al. 2015; Yang et al. 2018; Sogut et al. 2019; Moon et al. 2019; Moris et al. 2021), maximum pressure reduction (Tomiczek et al. 2016), and observed damage reduction (Tomiczek et al. 2017; Hatzikyriakou et al. 2016; Dall’Osso et al. 2010, 2016; Izquierdo et al. 2018; Reese et al. 2011) in building arrays. Our results present, for the first time, to our knowledge, the decay of the applied debris loading in a building array for a coastal inundation event, showing that there is a compounded sheltering effect in the building array, owing to (1) a lower CR; (2) a lower number of multiple collisions q; and (3) lower I0,x¯ and I0,x(2nd)¯ as the row number increases. The most significant difference is found between Row 1 (unobstructed) and Rows 2 to 5 (sheltered), suggesting that once the debris passes the first row, the collision physics is similar for Rows 2 to 5. It was noted from the video recordings that, for some debris collisions in Row 1, the debris stopped after the impact, transferring most of its momentum to the instrumented structures, whereas most of the debris in the inner rows, Rows 2 to 5, did not come to a stop after the collisions, which means that only a fraction of the momentum was transferred.The experimental flow had two hydrodynamic components: the steady current and the waves. For the steady current, there is a nonnegligible increase in the steady current between Row 1 and Row 2. This is due to the reduction in the cross-sectional area, where the flow enters the building array area. The steady velocity between Rows 1 and 2 increases from 0.26 to 0.41 m/s, giving an important increase of 58%. Regarding the waves, the maximum significant linear orbital velocity (Hs,ngdn/2dn) gradually decreases with the row number, which indirectly shows how the wave energy is being dissipated, owing to the breaking of the waves. Therefore, these two hydrodynamic components somewhat offset each other (see pn, Table 2). Although pn considers both a current and a wave component, the debris velocity might be lower than the sum of both components, owing to the inertia of the debris. It is also more likely that the debris could move with the steady current, but not necessarily with the orbital velocity, which would correspond to an upper value for the wave component velocity of the debris. Similar findings were obtained in experiments of tsunami-borne downscaled shipping container collisions (Derschum et al. 2018), where the measured debris velocity was lower than the flow velocity in the front of a tsunami-like bore. This issue suggests that a probabilistic approach to estimating I0,x¯ is more appropriate, given the uncertainty of the actual debris velocity at the impact time, which is addressed by the empirical exceedance probability curves presented in Figs. 10 and 11.Maximum Loading ResponseThe maximum loading responses Fr,x,max and Fr,y,max were predicted using Eqs. (22) and (23) and compared with the respective experimental maximum loading response (Fig. 12). Extremely high correlations are seen between the predicted and the experimental maximum loading response, with correlation coefficients r2 = 0.92 − 0.98. These demonstrate the impulsive nature of the debris loading, where the debris impulse translates directly to loads on the structural system. However, the best-fit lines are not 1:1, with slopes over the range [1.11, 1.34]; thus the maximum measured load is somewhat larger than that predicted by Eqs. (22) and (23). Although quantitative details of the difference will need to wait for a future paper, the differences are almost certainly related to the negligence of higher-order structural modes in our analysis. When obtaining the fundamental structural frequency and damping ratios, clear secondary modes with higher frequencies were also observed. These modes also appear to have been excited by debris impacts: as can be seen in Eqs. (22) and (23), a secondary mode with a shorter period (higher frequency) will increase the maximum load experienced by the structure, as seen in Fig. 12. A better representation for impacts with more than one mode is the two degree of freedom system proposed by Stolle et al. (2019); however, its implementation is beyond the scope of this paper, but Stolle et al. stress that it should be considered in future research. However, the excitation of higher modes does not appear to be a problem with a simple analytic solution, for which a safety-factor approach may be a relatively straightforward workaround, as will be discussed later.ASCE7-16 Impulsive Flood Debris Impact ForceASCE (2017, Section C.5.4.5) estimates the impulsive flood debris impact force using Eq. (6). It uses as impulse I0 = umaxmd, which only explicitly considers steady currents. By replacing I0 in Eq. (6) with the impulse pn defined in Eq. (18), the ASCE7-16 impulsive flood debris impact force can be extended to wave–current conditions:(25) Fflood,impulsive=2πTnpnCICOCDCB,Δt/Tn≤0.2which is evaluated using coefficients CI = CD = CB = 1.0, and setting the orientation coefficient as CO = 0.8, following ASCE7-16 recommendations. These loads are compared with the 2% and 10% experimental exceedance probability loads for a given collision event, denoted EP2CE and EP10CE, respectively (Table 3).Table 3. Comparison between 2% and 10% empirical exceedance loads, ASCE7-16 impulsive debris flood impact force, and proposed impulsive impact forceTable 3. Comparison between 2% and 10% empirical exceedance loads, ASCE7-16 impulsive debris flood impact force, and proposed impulsive impact forceI-LC-1I-LC-2I-LC-3I-LC-4I-LC-5Hydrodynamic conditionImpact force descriptionFx (N)Fy (N)Fx (N)Fx (N)Fx (N)Fx (N)Fx (N)Current-onlyEP10CE24.26.6N/A*N/A*N/A*N/A*N/A*ASCE7-16 impulsive debris flood impact force, Eq. (25)44.7N/A**25.638.738.236.140.8Proposed impulsive impact force, Eqs. (26) and (27)–currentEP2CE69.332.992.375.185.6N/A*N/A*EP10CE58.423.571.559.761.553.454.1ASCE7-16 impulsive debris flood impact force, Eq. (25)119.7N/A**68.776.379.273.465.3Proposed impulsive impact force, Eqs. (26) and (27)95.257.1125.395.7101.393.082.0Results show that the ASCE load overestimates both the EP2CE and EP10CE loads on M-LC-1, mainly because ASCE7-16 does not account for the structural damping ratio in its formulation. The response of highly damped structural elements will result in a reduction of the load according to the factor λ(ξ) [Eq. (24)]. The ASCE load slightly underestimates the I-LC-1 EP10CE load and slightly overestimates the EP10CE loads in the sheltered Rows 2 to 5. This shows the lack of consideration of a sheltered condition in the ASCE formulation.An alternative and simple approach to address these issues is the proposed maximum impulsive debris impact load (Fx,max,proposed) given by(26) Fx,max,proposed=2πTnI0,x,max¯pnλ(ξ)CLwhere CL is a load correction coefficient that accounts for the possible excitation of different modes and departure from the SDOF model. A value of CL = 1.3 is recommended, given that the slopes of the best-fit lines presented in Fig. 12 are in the range [1.13, 1.34]. Figures 11(a and b) show a plausible upper limit for the cross-shore dimensionless impulse of I0,x,max¯≈1.0 for the unobstructed Row 1, and I0,x,max¯≈0.8 for the sheltered Rows 2 to 5; thus, it is recommended to use in Eq. (26) a value of I0,x,max¯=1.0 for exposed structures, and a value of I0,x,max¯=0.8 for sheltered structures. Here, a sheltered structure is defined as having at least one structure in front of it that covers its entire along-shore width for a given flow direction.For the along-shore load, an expression similar to Eq. (26) is proposed:(27) Fy,max,proposed=2πTnI0,y,max¯pnλ(ξ)CLwhere, in this case, a plausible upper limit for the along-shore dimensionless impulse of I0,y,max¯≈0.6 is suggested; thus, it is recommended to use a value of I0,y,max¯=0.6 in Eq. (27). Owing to the lack of along-shore load data in the sheltered rows, it is not possible to suggest any potential inland reduction.Both Eqs. (26) and (27) are evaluated with respect to the experimental hydrodynamic conditions for both the current-only and the wave–current conditions (Table 3). The loads from the proposed equations have the advantage of (1) considering the damping ratio; (2) accounting for sheltering through Eq. (26); and (3) providing conservative predictions with respect to the observed EP2CE and EP10CE loads.Loading Duration Effects on the Maximum Loading ResponseThe assumption that the impulsive debris loading can be represented by Fx(t) = I0,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/Tn) is significant, the impulsive debris loading must be represented with a finite impact duration. In this section, we analyze how the maximum loading response Fr,x,max varies depending on the ratio Δt/Tn. 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 I0,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 Tn, the loading response Fr,x(t) is obtained by solving Duhamel’s integral [Eq. (8)] with a finite applied force, defined as(28) Fx(t)={0t<0I0,xπ4Δtsin⁡πt2Δt0≤t≤2Δt0t>2ΔtBy substituting Eq. (28) into Eq. (8), the maximum structural loading response Fr,x,max can be estimated as(29) Fr,x,max=2πTnI0,xγ(Δt/Tn,ξ)where γ(Δt/Tn, ξ) is a coefficient that only depends on the ratio Δt/Tn 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/Tn, ξ) takes values between 0 and 1. The longer the I0,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/Tn → 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 Fr,x,max and Fr,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/Tn, 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/Tn, ξ = 0), for an applied triangular loading history. If the ratio of impact duration to the fundamental period of the structure, (Δt/Tn), 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/Tn) without any damping, γ takes the upper limit of γ = 1, which yields the maximum loading response of Fr,x,max = 2πI0,x/Tn.Application to Moment-Resisting FramesThis section details the effects of finite loading duration and damping applied to generic steel and reinforced concrete (RC) moment-resisting frames. The fundamental period of moment-resisting frames is estimated as Tn=0.0724hn0.8 and Tn=0.0466hn0.9 for steel and RC, respectively, with hn as the structural height (ASCE 2017, Section The damping ratios in these two cases are estimated as ξ = 0.013/Tn (steel) and ξ = 0.014/Tn (RC) (Satake et al. 2003). A story height of 3.6 m is assumed; thus, the structural height is expressed as hn = 3.6Ns, with Ns equal to the number of stories of the moment-resisting frame. ASCE (2017) recommends an impact duration of Δt = 0.03 s for full-scale flood debris impacts. This yields different γASCE,flood values (Table 4 and Fig. 13) for steel and RC frames of three different heights. The different γASCE,flood values represent the reductions in maximum loading with respect to the fully impulsive and undamped responses.Table 4. Fundamental period (Tn), damping ratio (ξ), γASCE,flood, λ, and γASCE,flood/λ for moment-resisting steel and RC frames of different heightsTable 4. Fundamental period (Tn), damping ratio (ξ), γASCE,flood, λ, and γASCE,flood/λ for moment-resisting steel and RC frames of different heightsStructure typeNshn (m)Tn (s)ξγASCE,floodλγASCE,flood/λSteel moment-resisting frame13.60.200.0650.830.910.9227.20.350.0370.920.940.97310.80.480.0270.950.960.99RC moment-resisting frame13.60.150.0940.750.870.8627.20.280.0500.890.930.96310.80.400.0350.930.950.98The most important result here is that all cases presented in Table 4 and Fig. 13 are within the impulsive region (Δt/Tn < 0.25). The loading duration effect is assessed by taking the ratio γASCE,flood/λ (Table 4): although this effect is somewhat relevant for single-story structures (Ns = 1), it is negligible for two- and three-story frames (Ns ≥ 2). Consideration of the relative impact duration and structural damping will yield a more accurate maximum structural response; however, a more conservative alternative for structural design involves assuming a fully impulsive load with zero damping (γ = 1.0), which is particularly recommended when detailed information about the structural system or structural component is not available.Considerations and Scope of the ResultsThe building array presented in this paper has unobstructed transverse and longitudinal streets, which allows the development of a clear cross-shore flow on which debris elements are transported. Different results would have been obtained if the building array were composed of staggered rows. This would result in a variation of the flow, resulting in different debris loads. Also, only one building array geometry was used in the tests, with a building array blockage ratio of 0.4. Different ratios will give different collision probabilities and different flow focusing effects. Thus, the probabilities and loads presented in this paper should only be applied to an obstruction ratio similar to the one used in this document.Only one type of debris was tested, which consisted of a unique debris mass, density, material, and dimensions; and only one wave–current condition and one current-only condition over a constant SWD were considered. Although this paper presents results in terms of dimensionless quantities, different debris and flow characteristics should be tested in the future to ensure the applicability of our results to a wider range of conditions.Appendix. Maximum Loading Response for an Impact Duration Greater than Zero (Δt > 0)Let the applied loading be a half-sine pulse, with a duration Δt from when the pulse starts until the pulse is at a maximum. The following dimensionless quantities are defined (* denotes dimensionless):(30) (31) (32) (33) which allows us to express the dimensional finite applied debris force [Eq. (28)] in terms of the dimensionless finite applied debris loading Fx∗(t), as(34) Fx∗(t∗)={0t∗<0π4sin⁡πt∗20≤t∗≤20t∗>2Substituting the dimensionless quantities of Eqs. (30)–(33) into Duhamel’s integral [Eq. (9)], the structural loading response can be written as(35) Fr,x(t)=2πTnI0,x11−ξ2∫−∞t∗F∗(τ∗)e−ξ2πΔt/Tn)(t∗−τ∗)sin⁡2πΔtTn11−ξ2(t∗−τ∗)dτ∗⏟Γ(Δt/Tn,ξ,t∗)Thus, the structural loading response can be expressed as(36) Fr,x(t)=2πTnI0,xΓ(ΔtTn,ξ,t∗)with(37) Γ(ΔtTn,ξ,t∗)=11−ξ2∫−∞t∗Fx∗(τ∗)e−ξ2π(Δt/Tn)(t∗−τ∗)sin⁡2πΔtTn11−ξ2(t∗−τ∗)dτ∗The structural response is zero at the impact time Fr,x(t = 0) = 0 and, since it has a decaying oscillatory response, the response will decay to 0 after some time; therefore, there must exist a time, tFr,x=Fr,x,max∗, that gives a maximum value of Fr,x,max. We define, for given values of Δt/Tn and ξ, the maximum value of the structural response as(38) Fr,x,max=2πTnI0,xγ(Δt/Tn,ξ)where γ is the maximum value of Γ for a given Δt/Tn and ξ:(39) γ(ΔtTn,ξ)=Γ(ΔtTn,ξ,t∗=tFr,x=Fr,max∗)References Ardianti, A., H. Mutsuda, and Y. Doi. 2015. “2015A-GS19-3 interactions between run-up tsunami and structures using particle based method.” In Vol. 21 of Conf. Proc. 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