Experimental Modeling of Horizontal and Vertical Wave Forces on an Elevated Coastal Structure

Abstract

A large-scale physical model was created in Oregon State University’s Large Wave Flume to collect an extensive dataset measuring wave-induced horizontal and vertical forces on an idealized coastal structure. Water depth was held constant while wave conditions included regular, irregular, and transient (tsunami-like) waves with different significant wave heights and peak periods for each test. The elevation of the base of the test specimen with respect to the stillwater depth (air gap) was also varied from at-grade to 0.28 m above the stillwater level to better understand the effects of raising or lowering a nearshore structure on increasing or decreasing the horizontal and vertical wave forces. Results indicate that while both horizontal and vertical forces tend to increase with increasing significant wave height, the maximum and top 0.4% of forces increased disproportionally to other characteristic values such as the mean or top 10%. As expected, the horizontal force increased as the test specimen was more deeply submerged and decreased as the structure was elevated to larger air gaps above the stillwater level. However, this trend was not true for the vertical force, which was maximized when the elevation of the base of the structure was equal to the elevation of the stillwater depth. Small wave heights were characterized by low horizontal to vertical force ratios, highlighting the importance of considering vertical wave forces in addition to horizontal wave forces in the design of coastal structures. The findings and data presented here may be used by city planners, engineers, and numerical modelers, for future analyses, informed coastal design, and numerical benchmarking to work toward enabling more resilient nearcoast structures.

1. Introduction

1.1. Background and Literature Review

With the population densities in coastal regions being almost three times that of the global average (Nicholls and Small, 2002), coastal communities provide important economic, transport, and recreational services to large numbers of people. However, these coastal communities are vulnerable to damage by extreme events such as tropical cyclones or tsunamis, and studies indicate that the potential destructiveness of tropical cyclones, based on the storm’s intensity and total lifetime, has increased over the past 30 years (Emanuel, 2005). The potential damage of these storms is exacerbated in regions characterized by aging infrastructure and structures built according to outdated design standards. For example, recent storms in the United States such as Hurricane Ike (2008) and Hurricane Sandy (2012) generated strong waves and storm surge that caused extensive damage to aging infrastructure in the affected communities. As shown in Fig. 1, waves and surge cause extensive damage to elevated structures through a combination of horizontal and vertical wave and surge-induced forces. While age and deterioration play an important role, structural elevation has been shown to be a critical variable affecting damage and loss. Hurricane Ike caused significant damage to structures located on the Bolivar Peninsula, a narrow region along the Gulf of Mexico located east of Galveston, Texas, USA, in 2008. After the storm, reconnaissance surveys by Kennedy et al. (2011a) identified a sharp distinction between houses that were destroyed or survived with minimal structural damage based on the elevation of the lowest horizontal structural member of a residence with respect to the local storm surge elevation above ground. Likewise, Tomiczek et al. (2014) found much greater destruction rates on the peninsula for pile elevated residences built before the establishment of Flood Insurance Rate Maps (FIRMs) than for newer homes built according to updated guidelines. Similarly, many residences that were severely damaged by Hurricane Sandy had been built before the National Flood Insurance Program (NFIP) established base flood elevations (BFEs) for vulnerable areas, while others had been built using construction guidelines that had been published over 25 years prior to the storm (Frantz, 2012).

Fig. 1:

Examples of wave and surge-induced damage to elevated coastal structures located in Toms River, New Jersey occurring after Hurricane Sandy (2012). Photographs by Tori Tomiczek.

Recent efforts have been made to retrofit structures or improve coastal protection and damage mitigation plans in coastal communities in order to increase community resilience, defined as the ability to recover quickly after a disaster (NRC, 2012). However, in order to effectively retrofit old structures or design new structures to resist damage due to hurricanes, engineers require an accurate estimation of both the wave climate and the resulting loads. Therefore, the estimation of wave-induced loads on structures has been the subject of many theoretical, experimental, and numerical studies. Bagnold (1939) studied impulsive wave pressures induced by waves on a vertical wall, and noted the importance of the wave’s breaking condition (breaking or nonbreaking) as well as the effects of trapped air on the recorded impact pressure. Since this work, the shape of the incoming wave has been shown to affect the type of breaking and the maximum pressure on a structure (Cooker and Peregrine, 1995, Peregrine, 2003). Lundgren (1969) characterized wave impact phenomena into ventilated, compression, and hammer shock loads and noted that entrained air may cause overestimation when scaling laboratory pressures to prototype scales according to the Froude Number. Kirkgöz (1995) also characterized the breaking condition as early breaking (with or without air entrainment), late breaking, and perfect breaking, concluding that perfect breaking waves produced the largest impact pressures on a vertical face. Due to the extremely short duration of the force caused by wave impact, additional studies have considered the quasi-hydrostatic forces caused by waves affecting coastal structures. Morison et al. (1950) published a pioneering model to estimate wave forces on small piles as the combination of drag and inertial forces. Based on large-scale experimental measurements of wave impacts on a vertical seawall, Cuomo et al. (2010) developed prediction formulae for impulsive and quasi-hydrostatic wave loads and overturning moments on a vertical seawall based on the incident wavelength, height, and the normalized difference between the water depth at the structure and the water depth at breaking. Seiffert et al. (2014) and Hayatdavoodi et al. (2015) experimentally measured horizontal and vertical forces caused by Cnoidal waves on a flat plate elevated at varied distances above or below the free surface on a 1:35 scale. Additional studies have further investigated horizontal breaking wave-induced pressures and loads on coastal structures of varying geometries (e.g. El-Ghamry et al., 1965, Cuomo et al., 2011, Schoeman et al., 2012, Wiebe et al., 2014, Tomiczek et al., 2016a).

Many studies have also given attention to the horizontal and vertical wave-induced pressures and forces on bridge superstructures (e.g. French, 1969, Wang et al., 1970, Broughton and Horn, 1987, Shih and Anastasiou, 1992, Bricker and Nakayama, 2014, Azadbakht and Yim, 2015, Hayatdavoodi and Ertekin, 2015, Wei and Dalrymple, 2016). A review article by Hayatdavoodi and Ertekin (2016) provides a comprehensive review of the state of the art in understanding bridge failure mechanisms and experimental, theoretical, and numerical advances in understanding wave loads on coastal bridge decks. Early experimental work by French (1969) considered the rapidly-varying and slowly-varying wave-induced horizontal and vertical pressures acting on a horizontal platform elevated above the stillwater level and found that the slowly varying positive force was dependent on deck clearance and wave height. Broughton and Horn (1987) performed experiments on a 1:50 scale to measure the horizontal and vertical wave loads on elevated platforms. Based on these experiments they proposed a method for evaluating the impulsive force based on the change in momentum of the wave crest at the moment of impact as a function of the deck geometry, height and length of the wave crest, celerity, and the wave’s water particle velocity components. After the damage caused by Hurricane Katrina (2005) to bridges in the Gulf Coast of Mexico, wave loads and effects on bridges became a subject of significant concern (e.g. O’Conner, 2005, FEMA, 2006). Bradner et al. (2011) performed a series of experiments on a 1:5 scale bridge superstructure and found a relationship between force and wave height while noting that the sharp pressure peaks induced by wave impact had a small effect on the horizontal and vertical forces on bridge superstructure, as suggested by Lomonaco et al., 2016.

Based on experimental results, empirical methods for wave force estimation have been developed. For example, Suchithra and Koola (1995) calculated slamming coefficients for the vertical wave-induced force on a horizontal slab and found that the vertical force was dependent on the relative elevation of the slab with respect to the water depth, the incoming wave length and frequency, and area over which the wave impacted the slab. Using the results of 1:25 scale experiments, Cuomo et al. (2007) derived equations for the horizontal and vertical impulsive and quasi-static wave loads on the deck and beams of exposed jetties. These equations were later extended to account for the role of air in wave impacts at prototype scales (Cuomo et al., 2009). Other models have similarly been developed to relate the incident wave climate with the forces on coastal structures such as piers, jetties and bridge superstructures (e.g. Kaplan, 1992, McConnell et al., 2004, Hayatdavoodi and Ertekin, 2014a, Azadbakht and Yim, 2015). Wiebe et al. (2014) modified the equations for wave pressures on caissons presented by Goda (1974, 2010) to estimate wave pressures on elevated structures with geometries similar to a coastal houses elevated on piles or piers. Methodologies for relating wave conditions to forces on structures are also presented in design standards (FEMA, 2011, ASCE, 2016).

In recent years, progress has been made in the subject of computational fluid dynamics (CFD). Numerical models have been developed and refined to model wave interaction with the built environment (e.g. Cox et al., 2009, Xiao and Huang, 2008, Park et al., 2013, Carratelli et al., 2016). Jin and Meng (2011) used the CFD software Flow 3-D to compute wave loads on submerged bridges. Open source software such as OpenFOAM has also been used to compute wave forces on coastal bridge decks (Hayatdavoodi et al., 2014b, Seiffert et al., 2015, Wu et al., 2016). Wei and Dalrymple (2016) applied GPUSPH, a weakly compressible smooth particle hydrodynamics model to determine tsunami-induced forces on abridge superstructure. They compared results to lab experiments and investigated methods for mitigating the tsunami force, finding that the presence of a service road bridge or offshore breakwater ban reduce the tsunami force on the bridge assuming the structures do not fail under the tsunami loading. Recently, Do et al. (2016) performed a numerical study of waves interacting with an elevated coastal structure and proposed a performance-based design approach using fragility functions.

Although previous studies have resulted in a better understanding of wave forces on structures, several questions remain when considering potential damage caused by wave and surge-induced loads on coastal residences. Many residential structures are wood-framed constructions and elevated on piles; the geometries and response of these structures differ from those of coastal bridges and jetties, limiting the application of many of the models discussed above when predicting wave forces on coastal residences. Additionally, the measurement of horizontal and vertical wave forces from small-scale physical models will be subject to Reynolds number and scaling issues when considering air entrainment in impulsive waves (Liu et al., 2008). Moreover, most of previous experiments have been performed with limited wave types such as regular or transient (tsunami-like) waves for simplicity. However, these waves have different scaling effects and can cause different structural responses than the random (irregular) waves associated with tropical cyclones. Finally, equations given in design standards are often simplified by using depth-limited wave heights and may need further refinement to accurately predict wave forces on structures (Tomiczek, et al. 2016b). The floor elevation of a structure with respect to the water surface (air gap) has been shown to affect the degree of damage (Tomiczek et al., 2014), but the explicit effects of an increasingly positive or negative air gap on amplifying or reducing the horizontal or vertical forces on a coastal house have remained largely unexplored. This study uses a large scale experimental dataset and specifically analyzes the effects of irregular wave loads on the unique geometry of an elevated coastal structure. Based on these experimental results, relationships between significant wave height and force are presented that give useful information for coastal engineers and city planners.

The objective of this experiment was to understand the relationship between a structure’s air gap and the resulting horizontal and vertical forces generated by a range of wave conditions. Large scale experiments in Oregon State University’s Large Wave Flume (LWF) collected benchmark data to measure wave heights, pressures, and loads for regular, irregular, and transient waves. This paper presents the methodology of the experimental regime and significant relationships between wave height, period, air gap, and load for irregular wave conditions. Section 2 introduces the experimental set up, instrumentation, and test conditions, which were based on a 1:10 length scale idealization of Hurricane Ike’s impact on the Bolivar Peninsula, Texas. Section 3 presents experimental data obtained from wave gauges, pressure transducers, and load cells installed in the flume and on the experimental specimen, while Section 4 discusses the impact of varying the incident wave conditions or structural air gaps on changing the characteristic horizontal and vertical forces recorded by the structure. Finally, Section 5 discusses the major conclusions of this work while laying a framework for future analytical and computational analyses that will be performed using this dataset.

2. Experimental Setup

2.1. Characteristic experimental conditions from Hurricane Ike

Experimental conditions were selected based on an idealization of Hurricane Ike affecting elevated and at-grade structures located on the Bolivar Peninsula. On 13 September 2008, Hurricane Ike made landfall on Galveston Island as a Category 2 storm with winds up to 49 m/s (110 mph). The storm was characterized by a very large wind field; these winds combined with the broad continental slope of the Gulf of Mexico to generate a large forerunner storm surge and similarly large wave heights that inundated the low-lying Bolivar Peninsula (Berg, 2009, Kennedy et al., 2011a). Temporary wave gauges deployed by Kennedy et al. (2011b) recorded nearshore maximum surge levels and wave heights between 3.3–4.6 m and 4.8–5.6 m, respectively (North America Vertical Datum, NAVD88). This combination of waves and surge was the primary cause of damage to the building stock on the Bolivar Peninsula, which comprised predominantly of wood-framed, pile-elevated coastal residences. The distinction between survival and failure was shown to be dependent on the elevation of the structure with respect to the wave conditions at the structure’s location (Kennedy et al., 2011a, Tomiczek et al., 2014), making the scenario of Hurricane Ike an ideal test bed for experiments.

2.2. Physical Model Setup in Large Wave Flume

Figure 2 shows a profile and plan view of the Large Wave Flume at Oregon State University’s Hinsdale Wave Research Laboratory (HWRL). The flume measures 104 m long, 3.66 m wide, and 4.6 m deep, with adjustable bathymetry. The piston-type wave maker assembly is designated in Fig. 2 and specified as the origin for wave generation (x=0.00 m). The hydraulic actuator assembly of the wave maker is capable of generating regular, irregular, and tsunami-type waves and is equipped with an active wave absorption system for large reflected waves. Upon generation, waves propagated across a flat offshore section for 14.07 m before reaching the first bathymetric concrete slab, a horizontal, 3.64 m long slab elevated 0.15 m above the base of the flume. The sloping bathymetry of the flume, selected based on flume dimensions and wave conditions, was composed of two sections: a 10.98 m long, ~1:11 slope followed by a 14.64 m long, ~1:24 slope, at which point (x=43.33 m) waves propagated across a flat section representing the Bolivar Peninsula, elevated 1.75 m above the base of the flume. This slope is relatively steep with respect to the broad continental shelf in the Gulf of Mexico, which may affect the location of wave breaking; however, nonbreaking, breaking, and broken wave conditions were observed at the location of the experimental specimen. Thus, the wave-induced forces recorded here are applicable to a range of field conditions. The elevated flat section of the flume measured 36.60 m before a 7.50 m, 1:12 slope extended to the end of the flume. For numerical modeling, the wave maker may be modelled using a moving boundary with inputs from the wave maker displacement time series, while the end of the flume may be modelled using either a radiation or sponge layer boundary condition or a fixed beach profile. Detailed bathymetric slab positions are summarized in the Appendix Table A.1.

Fig. 2.

Large wave flume: profile (top) and plan (bottom) view.

Based on flume dimensions, experiments were scaled using a 1:10 length scale under Froude Number similarity. These large-scale experiments allow confident comparisons between model and prototype scales; however, the authors note that at full scale air compressibility can affect the dynamics of high aeration impacts (Bullock et al., 2007). Cuomo et al. (2010) determined a correction coefficient for experimentally measured pressures to account for scaling effects in breaking-wave impacts based on the Bagnold number. Therefore, while the forces presented here for breaking waves are qualitatively applicable to prototype scenarios, care must be taken when making direct comparisons with prototype breaking-wave-induced forces.

Table 1 lists combinations of test conditions for each experimental trial. Three types of waves were considered: 1) irregular waves (TMA), for comparison with the scenario of Hurricane Ike on the Bolivar Peninsula, 2) regular waves (REG), to check instrumentation and generalize results, and 3) transient waves (TRAN), for future benchmarking tests and analyses. Transient waves were generated using an error function displacement time series for the paddle stroke, which maximized the tsunami inundation duration, thus minimizing errors between model and prototype time scales that have been reported for solitary waves (Thomas and Cox, 2012). These waves were characterized by their amplitude, $A$, and representative period, $T_R$, which was defined as the time for which the water surface elevation exceeded 1% of its amplitude. Table 1 also lists the duration of the paddle stroke, $t_s$, for these trials, which was incrementally increased from 10 to 45 s to keep the full 4-m stroke (Thomas and Cox, 2012). For these trials, the offshore stillwater depth, $h$, was set to the maximum allowed for the wave maker to generate these wave conditions ($h=2.00\mathrm{m}$).

Table 1.

Experimental wave conditions.

Exp.	TMA				REG				TRAN
	H1/3 (m)	TP (s)	h (m)	Dur. (min)	$\overline{H}\left(\text{m}\right)$	$\overline{T}\left(\text{s}\right)$	h (m)	Dur. (min)	A (m)	TR (s)	h (m)	ts (s)
X1	0.10	3.72	2.15	40.0	0.10	4.10	2.15	4.00	0.51	36.4	2.00	10.0
X2	0.19	3.86	2.15	40.0	0.21	4.10	2.15	4.00	0.34	51.0	2.00	15.0
X3	0.29	4.10	2.15	40.0	0.29	4.10	2.15	4.00	0.28	87.2	2.00	20.0
X4	0.40	4.10	2.15	40.0	0.40	4.10	2.15	4.00	0.21	109	2.00	25.0
X5	0.50	3.86	2.15	40.0	0.50	4.10	2.15	4.00	0.18	117	2.00	30.0
X6	0.16	2.52	2.15	25.0	0.16	2.52	2.15	2.50	0.16	120	2.00	35.0
X7	0.21	2.98	2.15	30.0	0.23	2.98	2.15	3.00	0.14	154	2.00	40.0
X8	0.25	3.28	2.15	35.0	0.26	3.64	2.15	3.50	0.13	162	2.00	45.0
X9	0.34	4.68	2.15	45.0	0.35	4.68	2.15	4.50
X10	0.39	5.04	2.15	50.0	0.42	5.04	2.15	5.00

For irregular and regular waves, trials were characterized by the significant or mean wave height, H_1/3 and $\bar{H}$, and peak or mean period, $T_p$ or $\bar{T}$, respectively. A range of wave conditions, with significant wave heights (measured at wg1, located x=14.17 m from the wave maker) between 0.10–0.50 m and periods between 2.5–5.0 s, were generated to assess the effects of increasing wave height or period on increasing the maximum horizontal or vertical force recorded by the structure. Offshore water depths at the wave maker were set to 2.15 m, giving local stillwater depths of 0.40 m at the location of the experimental specimen (x=44.51 m). Irregular waves were generated using the TMA (Texel-MARSEN-ARSLOE) spectrum for shallow water waves with γ=3.3. The duration of each spectral case was determined by requiring at least 600 waves for each condition; based on the expected wave peak period, experimental durations ranged from 25–50 minutes. The data acquisition system recorded data for four minutes longer than the experimental durations reported in Table 1 in order to capture a complete record of each trial as the water surface transitioned from calm, still water, to wave ramp up, full propagation, and ramp down, before returning to a calm surface after sufficient time with no wave maker motion. Processes associated with wave ramp up and ramp down were not considered in data analyses. Based on the 1:10 length scale, the water depths and wave heights tested here for TMA (hurricane) waves correspond to prototype surge levels and wave heights of 4.0 m and 1.0–5.0 m, respectively, which are within the ranges measured on the Bolivar Peninsula during Hurricane Ike. Likewise, the prototype-scale wave period ranged from 7.9–15.8 s. Monochromatic wave trials generated 20 waves in order to obtain a representative average.

Water surface elevations were measured using three wire resistance wave gauges (wg) along the flume, and six ultrasonic wave gauges (uswg, TS-30S1-IV, Senix) were installed around the specimen; wave gauge locations are shown in Fig. 2, and specific x, y, and z coordinates are listed in Table 2. All test results were recorded through the HD integrated camera (HDC-SD80, Panasonic) at Bay 9 with a 60 fps sample speed to analyze the detailed wave-structure interaction.

Table 2.

Instrument locations (wave gages, pressure gages, and load cells).

Instrument description	Instrument	x (m)	y (m)	z (m)
Wave maker displacement	Wmdisp	-	0.00	-
Wave maker wave gage	Wmwg	-	0.00	-
Resistive wave gage	wg1	14.17	−1.39	-
Resistive wave gage	wg2	32.44	−1.38	-
Resistive wave gage	wg3	39.57	−1.38	-
Ultrasonic wave gage	uswg1	39.70	−1.38	3.65
Ultrasonic wave gage	uswg2	43.18	−1.30	3.59
Ultrasonic wave gage	uswg3	43.22	−0.11	3.55
Ultrasonic wave gage	uswg4	45.68	−0.01	3.56
Ultrasonic wave gage	uswg5	47.09	−1.28	3.59
Ultrasonic wave gage	uswg6	44.37	−1.28	3.54
Pressure gage (front)	press1	43.84	−0.01	2.16
Pressure gage (front)	press2	43.84	−0.01	2.20
Pressure gage (front)	press3	43.84	−0.01	2.23
Pressure gage (front)	press4	43.84	−0.01	2.27
Pressure gage (front)	press5	43.84	−0.01	2.31
Pressure gage (front)	press6	43.84	−0.01	2.35
Pressure gage (front)	press7	43.84	−0.01	2.39
Pressure gage (front)	press8	43.84	−0.01	2.43
Pressure gage (front)	press9	43.84	−0.01	2.46
Pressure gage (rear)	press10	44.86	−0.02	2.16
Pressure gage (bottom)	press11	43.97	−0.02	2.14
Pressure gage (bottom)	press12	44.22	−0.02	2.14
Pressure gage (bottom)	press13	44.47	−0.02	2.14
Pressure gage (bottom)	press14	44.72	−0.02	2.14
Load cell (vertical)	load1	44.92	−0.43	2.77
Load cell (vertical)	load2	44.92	0.43	2.77
Load cell (vertical)	load3	45.77	−0.43	2.77
Load cell (vertical)	load4	45.77	0.43	2.77
Load cell (horizontal)	load5	44.86	−0.02	2.57

Figure 3 shows a detailed view of the experimental specimen and locations of pressure transducers and load cells on the elevated structure, while Fig. 4 presents photographs of the front and rear side of the specimen and the instrumentation (ultrasonic and wire resistance wave gauges, pressure transducers, and load cells) used for experiments. The structure was constructed of steel, with dimensions 1.02 m × 1.02 m × 0.61 m high; for the experiments, the specimen was positioned 0.51 m inland from the onset of the flat beach (x=43.84 m). The structure was mounted on a frame that could be raised or lowered to increase or decrease the air gap, a, between the base of the structure and the elevation of the stillwater. Note here that the z coordinates of pressure and load cell transducers in Table 2 correspond to a=0.00 m. Table 3 presents the range of air gaps tested: trials considered varying structural elevations ranging from at-grade to 0.68 m above the flume bathymetry (a=−0.40 – 0.28 m (TMA & REG); a=−0.25–0.13 m (TRAN)). The dimensions of the structure scale to a house or building with base dimensions 10.2 m × 10.2 m.

Fig. 3.

Sketch of specimen and instruments: (a) side view of specimen-frame system elevated at air gap a above the stillwater level; (b) clockwise from top left: front, rear, top, and bottom views of the specimen. In the bottom and top views, the wave propagates from left to right.

Fig. 4.

Images of specimen and instruments. (a) Front view of the specimen before filling the tank, (b) Rear view of the specimen, (c) Wire-resistance wave gage and ultrasonic wave gage, (d) 9 pressure gages installed on the front face of the specimen, (e) Load cell measuring horizontal force.

Table 3.

Air gap conditions.

	a (m)
Air Gap cases	REG TMA	TRAN
a0	−0.40	−0.25
a1	−0.30	−0.15
a2	−0.20	−0.05
a3	−0.10	0.05
a4	−0.05	0.10
a5	0.00	0.15
a6	0.05	0.20
a7	0.10	0.25
a8	0.20	0.35
a9	0.28	0.43

As seen in Fig. 3b and listed in Table 2, pressure gauges were installed along the centerline of the structure on its front, rear, and bottom faces to measure the dynamic pressure associated with horizontal and vertical wave impacts. The nine frontal pressures gauges were spaced 0.04 m apart, with the lowest gauge elevated 0.02 m above the base of the structure. The close spacing of the frontal pressure gauges allowed pressures to be integrated across their effective areas and compared with the total load measured by the horizontal load cell. A rear pressure gauge was installed 0.02 m above the base of the structure to determine any wave-induced pressures on the back of the structural element, and four pressure gauges were installed on the centerline of the structure’s base. These bottom-mounted pressure gauges were spaced 0.25 m apart, with the front and rear gauges positioned 0.13 m from the front and rear edges of the experimental specimen. Pressures measured by the bottom sensors were also integrated across the base area of the structure to obtain estimations of the vertical loads caused by a wave. The horizontal load cell was mounted to the rear of the specimen, and four vertical load cells were installed on the top corners of the elevated specimen as shown in Fig. 3. The total weight of the specimen, including the frame and instrumentation, was 680.4 kg; the combined weight of the specimen plus instrumentation (without the frame) was 197.8 kg; the weight of only the specimen was 173.8 kg.

Pressure, force, and water surface elevation measurements were synchronized and recorded at 500 Hz; pressure and force data were processed to remove noise using a low pass filter for frequencies above 50 Hz (Oliveras et al., 2012). Cooker (2002) noted that the duration of a pressure impact can last ~100 ms; therefore, when considering peak pressures, sampling rates between 400–10000 Hz have been reported (e.g. Wood et al., 2000, Bullock et al., 2007). Tomiczek et al. (2016b) performed sensitivity tests at 1000, 500, and 200 Hz and found that a 200 Hz sampling rate was sufficient to capture peak pressures in tsunami impact experiments. Therefore, the sampling rate used in these experiments was sufficient to accurately characterize forces caused by nonbreaking, impulsive breaking, and broken waves.

In addition, repeatability trials were completed for five iterations of test TRAN_X2, with the structure positioned at the base of the flume (a₀), and the water depth set to h=1.75 m (no storm surge level). This condition was selected as the repeatability metric for future numerical benchmarking and validations. Table A.2 in the Appendix shows the peak mean absolute error (MAE) of pressures, loads, and water surface elevations recorded by the instrumentation for each of the five repeated trials. As expected, the most variability is observed for pressure measurements; peak MAE is typically about 10%, and the time-series showing the rise and fall of the water surface or pressures indicated good agreement for repeated trials as shown in the Appendix Fig. A.1. Results of the repeatability trials indicate that wave gauge, force, and pressure measurements were repeatable. Additionally, peak values of raw and filtered data were compared for each of the pressure gauge and load cell time series recorded during the repeatability trials. Percent differences between the filtered and unfiltered values may be seen in Table A.3 of the Appendix and were on average less than 3%.

The experimental regime consisted of three test phases, listed in Table 4. The first diagnostic phase consisted of all wave trials described in Table 1, performed over a bare earth bathymetry without the structure. This diagnostic test held three purposes: (1) to check wave breaking conditions and identify the ideal position for measuring wave-induced forces and pressures; (2) to provide inputs for numerical models; and (3) to evaluate the increase in wave reflection caused by submerging the specimen. Sample time series of water surface elevations at wg2 (x=32.44 m) are provided in the Appendix Fig. A.2 and data from all trials is available for download at the NHERI DesignSafe Website. The ideal x-position of the structure was chosen such that a range of nonbreaking, breaking, and broken wave conditions impacted the specimen. To evaluate wave reflection induced during negative air gap conditions, the water surface recordings with the structure submerged to a₂=−0.20 m and to the base of the flume (a₀=−0.40 m) were compared with those recorded without the structure. For both cases, the change in peak period was less than 1%, and the change in the significant wave height was on average 6.3% at wg2. With a blocking ratio, defined as the ratio of the box width to the flume width, equal to 0.25, the slight reflection caused by the obstruction is expected. As suggested by Nouri et al. (2010) and experimental comparisons presented here, a blocking ratio of less than 0.4 may not significantly affect downstream flow. Therefore, wave reflection did not significantly change experimental measurements.

Table 4.

Summary of Trials.

Phase	Wave Conditions	Air Gap Conditions	Description
TEST1	TMA, REG, TRAN	n/a	Without specimen
TEST2	TMA, REG, TRAN	a0 – a9	With two vertical load cells (load1 & load4)
TEST3	TMA, REG, TRAN	a5	With all new vertical load cells
	TMA_X3, REG_X2	a1, a2, a3, a7, a8, a9

The second phase of experiments involved installing the structure in the wave flume to record pressure and force data for all wave trials. Post-processing revealed that two vertical load cells (load2 and load3) malfunctioned during experiments. While time constraints prevented the repetition of all experimental trials, new vertical load cells were installed and selected trials were repeated in a final phase (Test 3) to check previously measured pressure and horizontal force data as well as to include confident vertical force measurements in analyses. In Test 3, all TMA, REG, and TRAN wave conditions were generated with the structure elevated 0.40 m above the base of the flume bathymetry (a₅=0 m, TMA/REG and a₅=0.15 m, TRAN), and the conditions of TMA_X3 and REG_X2 were generated for air gap conditions ranging from a₁=−0.3 to a₉=0.28 m.

3. Qualitative Wave Observations

3.1. Wave characterization and experimental time series

When waves approach a beach, the horizontal location of breaking is affected by the wave height (H), wave length (L), and local water depth (h). Ratios of H/h and h/L give useful information regarding the application of well-known wave theories (e.g. Stokes, Airy) based on wave nonlinearity and shallowness; these ratios are often compared using the Ursell Number, Ur= (H/h)/(h/L)². Waves in deep water can be characterized by a small Ursell Number and are often well-represented using Stokes theory, while a large Ur indicates long waves with large heights, better characterized using cnoidal or solitary wave theory (Fenton, 1990). Initial wave nonlinearity and shallowness characterizations are shown in Fig. 5. It plots characteristic wave conditions measured at wg1 (filled), and wg3 (hollow), normalized by the local water depth, against the ratio of water depth to wave length for all experimental trials for a constant air gap condition (a₅). Characteristic wave heights were defined using the measured significant wave height for TMA trials (circles), the mean wave height for REG trials (triangles), and the maximum amplitude for TRAN trials (squares). As mentioned above, the wavelengths of TRAN trials were defined as the length for which a positive wave train exceeded 1% of the maximum amplitude and used to calculate the representative tsunami wave period, T_R, as described in previous tsunami run-up studies (e.g. Park et al., 2015). At wg1, all TMA and REG wave conditions are in intermediate water depths and fall within the limits of Airy Wave Theory. TMA and REG_X1−X5 trials generated varying wave heights with a constant period; thus, wave characterizations show increasing values of H/h while maintaining a constant h/L ratio. On the other hand, TMA and REG_X6−X10 trials varied both wave height and period while retaining a constant wave slope (H/L) and therefore show a linearly decreasing trend with logarithmically increasing values of h/L. TRAN_X1−X8 trials are near the limit of Solitary Wave Theory and increase linearly as the stroke displacement time increases from 10–45 s. At wg3, due to wave shoaling and wave breaking processes, most of the TMA and REG trials fall within the limits of Cnoidal Wave Theory, and they approach the wave breaking limit. At this wave gauge, TRAN trials also show an increase of H/h and approach this depth-limited breaking criterion. Generally, trials involving larger wave heights approached the breaking wave limit as waves propagated along the flume bathymetry. Wave shoaling and breaking were commonly observed on the slope and at the location of the elevated structure for TMA and REG trials X3−5 and X8−10.

Fig. 5.

Experimental wave classification (H/h vs. h/L) at wg1 and wg3. Each circle, triangle, and rectangle presents the characterized measured wave condition of TMA_X1−X10, REG_X1−X10, and TRAN_X1−X8 at wg1 (filled) and at wg3 (hollow) with the structure positioned at a₅ (zero-air gap).

The photographic images (Fig. 6), extracted from high speed videos of experiments, show broken (REG_X3) and breaking (REG_X8) waves impacting the elevated structure. While few broken waves were observed for irregular wave trials with lower significant wave heights (H_1/3<0.21 m), most waves of trials TMA_X1−2 and X6−7 propagated unbroken past the elevated structure. Significant wave heights of between 0.25 and 0.29 m (TMA_X3 and X8) produced the greatest number of waves that broke directly on the elevated structure, while experiments with larger significant wave heights typically broke along the sloping beach and impacted the structure as fully broken, turbulent bores. Thus, the varying trials induced a range of nonbreaking, breaking, and broken wave conditions at the location of the instrumented specimen and can be used to generalize relationships between wave height, period, and horizontal or vertical force.

Fig. 6.

Images of experiment. (a) A broken wave just before impacting the structure, (b) Lateral view of a wave about to break directly on the structure.

Incoming wave conditions are useful in characterizing offshore wave climates and are necessary inputs for numerical models of wave-structure interaction. It is also important to compare time series of wave gauges with pressures or forces measured on the instrumented specimen. As a wave interacts with a coastal structure, the horizontal forces induced on the front face of the structure will have a phase difference with the vertical forces induced on the floor assembly. Thus, the maximum horizontal force will occur at a different time than the maximum vertical force. In this experiment, all pressure gauge, load cell, and wave gauge time series were synchronized to allow for accurate relation between the pressure and force data and to better understand this time shift between horizontal and vertical wave forces. An example of the wave, load and pressure time series over a 20 second snapshot from one of the irregular wave trials, with the base of the structure positioned at the water level (TMA_X3_a₅), is shown in Fig. 7. This trial, with a 0.29 m incident significant wave height, produced a range of breaking conditions: Fig. 7 shows a breaking wave impacting the structure and inducing a sharp pressure peak on the frontal pressure gauges (from top to bottom: press8, press6, press4, and press2) at t=48 s. This wave was identified from the video of the experiment to break almost directly on the specimen; the breaking wave impact is similarly recorded by the horizontal load cell (load5); note that load5 further shows the damped oscillations induced by waves impacting the frame-structure system.

Fig. 7.

Sample time series of wave gauges, pressure sensors and load cells from TEST3_TMA_X3_a₅. Subplots show free surface elevations at (a) wg2 and (b) uswg2; vertical forces at (c) load1 and (d) load4; bottom pressures at (e) press11 and (f) press13; horizontal force at (g) load5; front pressures at (k) press8, (j) press6, (i) press4, and (h) press2; and rear pressure at (l) press10.

A pluck test was conducted to determine the frequency of the oscillations and the rate of decay of the system for varying structural elevations. The frame-structure’s natural response frequency was between 3.8−9.2 Hz, with increasing frequencies at higher elevations when the structure was not submerged. While the geometry tested here differs from a caisson breakwater, these response frequencies are in agreement with typical periods of vibration reported by Cuomo et al. (2011) in a probabilistic assessment of wave loads on caisson breakwaters. The rate of decay of the amplitude of oscillation ranged from 0.26–0.79 [1/s], with higher rates of decay associated with higher elevations; however, for the same structural elevation, adding water to partially submerge the structure likewise increased the rate of decay, as expected. Table A.4 in the Appendix provides details about the natural response frequencies and rates of decay tested at three different elevations. Design engineers may consider the effects of a wave-induced force on causing structural vibrations based on a structure’s response, which will be affected by building material and elevation above grade.

Comparison of the horizontal and vertical force time series in Fig. 7 indicates that although the breaking wave induced large horizontal pressures and force, the vertical load cells and pressure transducers did not record a similar spike due to the wave impact at t=48 s. Rather, the vertical sensors registered a more prolonged, constant vertical load. Note that nonbreaking waves that followed (impacting the specimen at t=58 s) generated vertical forces and pressures of similar magnitudes to the first wave; this phenomenon contrasts with the significantly smaller nonbreaking horizontal force and pressures recorded compared with those caused by the breaking wave. Therefore, while breaking wave conditions generate horizontal forces that are significantly larger than those induced by nonbreaking waves, significant vertical forces may still be felt by an elevated structure under nonbreaking wave conditions. These forces will also be dependent on the elevation of the structure above or below the stillwater depth (air gap). Based on these observations, the relationship between wave height, wave length, structural air gap, and characteristic wave-induced forces were more fully explored and are discussed in the following sections.

3.2. Pressure integrations and load cells

As discussed above, two vertical load cells malfunctioned during the experimental sequence Test 2. While time constraints prevented data recollection for all air gap-wave trial combinations upon the installation and verification of new load cells (Test 3), useful relationships between vertical force and wave height may be obtained from integrating the bottom pressure sensors over the base area of the specimen, provided that pressure integrations accurately represented the total load. Note that all pressure measurements considered the dynamic pressure due to wave loading. Thus, load cell and pressure data from Test 3 were compared to those from Test 2 and used to verify pressure integrations as accurate horizontal and vertical force estimations.

Figure 8 shows a detailed comparison of pressure integrations with the horizontal and vertical forces over the duration of the breaking wave shown in Fig. 7 for structural air gaps −0.10 m, 0.00 m, and 0.10 m. Horizontal pressure integrations were calculated assuming the measured pressure extended across the entire base of the structure over a vertical distance extending from 0.02 m below to 0.02 m above the gauge (the halfway point between two pressure gauges). The distance to the point of zero pressure from the top pressure gauge was calculated assuming a hydrostatic distribution above the highest pressure gauge. While this assumption can cause overestimation of the horizontal force for conditions in which a wave breaks directly on the highest pressure gauge (press9), in most cases the maximum pressure occurred at a pressure gauge elevated below press9, and the pressure measured by this topmost gauge was very near zero. Thus the integrated pressures gave reasonable force approximations. Vertical forces were estimated by assuming that the pressure gauge recordings were constant over an area extending across the alongshore width of the structure (B) and half the x- distance to the neighboring pressure gauge.

Fig. 8.

Detailed time series comparing pressure integrations (black dotted line) and load cell measurements (red solid line) of horizontal (right) and vertical (left) forces for TEST3_TMA_X3_a₃ (top), a₅ (middle), and a₇ (bottom).

The pressure integrations agree well for both horizontal and vertical load measurements. In particular, peak loads recorded by the vertical load cells closely match pressure-integrations; this phenomenon was tested and confirmed for all of the wave conditions generated in Test 3. Thus, although the vertical load cell data from Test 2 was deemed invalid, integrations of the base pressures may be confidently used to relate wave conditions to the uplift force. Note also in Fig. 8 that the horizontal wave impact induced a sharp pressure peak recorded by the front-mounted pressure gauges, resulting in an integrated horizontal force that exceeds the force recorded by the horizontal load cell at the moment of wave impact. Bradner et al. (2009) found similar results in that large inertia structures may not record a wave impact force due to the extremely short duration of these impact pressures. However, the forces associated with wave impact are important when considering localized damage. Additionally, the horizontal pressure measurements were not affected by oscillations of the frame-structure system. Therefore, all remaining figures use integrated pressures for both vertical and horizontal force relationships.

4. Experimental Results

4.1. Wave height and characteristic forces

Hurricane waves are often characterized by the significant wave height, H_1/3; however, other definitions of wave height, such as the maximum wave height, H_Max, or the average of the highest specified percentage of the waves, e.g. H_1/10, may be used. Similarly, the wave-induced forces impacting the structure may be analyzed using different definitions for a characteristic force. Several characteristic forces have been considered previously; for example, Cuomo et al. (2007) considered a characteristic F_1/250, the average of the highest 0.4% of all forces. The maximum force, F_Max, may also be important for resisting critical loads, and other definitions may be considered based on a structure’s importance or use. Therefore, F_mean, F_1/3, F_1/10, F_1/50, F_1/100, F_1/250, and F_Max, representing the mean, significant, average of the highest 10%, 2%, 1%, 0.4%, and maximum force, respectively, were calculated for each of the TMA wave trials (X1−X10) with the structure elevated at the stillwater depth (a₅=0.00 m) in the horizontal and vertical directions. These characteristic forces are plotted in Fig. 9 against the significant wave height measured at wg3, averaged over all air gaps for a given trial. The significant wave height for each trial and air gap are presented in the Appendix Table A.5, along with the ensemble average and standard deviation. The good agreement of wave heights across all trials further indicates that wave reflection did not significantly affect experimental results. Wave gauge 3 was chosen to define wave characteristics because this gauge was located closest to the distance from the structure where wave height is defined in common design equations (Goda, 1974, 2010). To clarify the specific effect of wave heights on wave-induced forces, only TMA trials X1–5 are plotted in Fig. 9, because these trials hold wave period constant while varying wave height. The figure shows that both the horizontal and vertical forces tend to increase with increasing H_1/3 until the breaking wave limit. After the onset of wave breaking, waves with a larger input H_1/3 tended to break earlier along the flume’s bathymetry, causing wave energy to dissipate before the wave impacted the specimen. As a result, the horizontal force tended to decrease while the vertical force remained relatively constant. In general, the characteristic force chosen for analysis impacts the magnitude of the force but not the general trend of the force-wave height relationship, with the notable exception that the horizontal F_Max and F_1/250 increase disproportionately for the largest wave heights tested here. This disproportionate increase is due to the random nature of the wave spectra and the variability inherent in forces generated by breaking waves due to turbulence and impacts. With 600 waves generated for irregular wave trials, any outliers well over the mean were identified in this extreme force analysis. This result indicates that although extreme forces can be significantly larger than other characteristic force definitions, depending on the importance of a structure, it may not be economically feasible to design the structure to resist the maximum expected force.

Fig. 9.

Characterized F_H and F_V versus H_1/3: F_mean (blue diamond), F_1/3 (magenta circle), F_1/10 (black square), F_1/50 (yellow inverse triangle), F_1/100 (red upright triangle), F_1/250 (hollow diamond) and F_Max (hollow circle).

4.2. Effects of air gap on horizontal and vertical forces

While Fig. 9 shows that the magnitudes of horizontal and vertical forces are dependent on the incoming wave climate, loads on a structure will also change depending on the structure’s elevation above or below the stillwater depth. Therefore, a representative force, F_1/10 was normalized by the incident wave energy over the width of the structure, ρgH_1/3²B, where ρ is the density of water, g is the gravitational acceleration, H_1/3 is the ensemble average significant wave height for each trial taken at wg3, and B is the alongshore width of the structure equal to 1.02 m. Figure 10 depicts these normalized forces plotted against each air gap, normalized by the local water depth (a/h), for all constant-period wave trials (X1-X5). Although the normalized F_1/10 is presented in Fig. 10, other characteristic forces (e.g. F_1/3) yielded similar results. The normalized horizontal force shows a decreasing trend with increasing air gap, with the largest normalized forces for all wave conditions occurring at the deepest level of submersion. It is interesting to note that the nonbreaking wave cases, X1 and X2, resulted in the largest normalized horizontal forces for the more deeply submerged cases. Likely this phenomenon indicates less energy dissipation associated with nonbreaking waves and the corresponding small wave heights used to normalize the characteristic forces. Additionally, more deeply submerged conditions cause a larger frontal surface area to be impacted by a nonbreaking wave; it was noted that large impulsive pressures caused by breaking and broken wave impacts were distributed over a relatively narrow vertical section of the specimen localized above the water surface for all air gap conditions. Thus, increasing the air gap reduced the characteristic horizontal force on the structure more dramatically for nonbreaking waves, which did not generate impulsive slamming forces. For trials with low nominal wave heights, increasing the air gap will decrease the number of waves that impact the elevated specimen until the specimen is sufficiently elevated such that all waves pass beneath the structure unimpeded. Therefore, the number of impacts for each trial was checked for each air gap and may be seen in Fig. A.3 in the Appendix.

Fig. 10.

Normalized (a) horizontal and (b) vertical forces, F_1/10/(ρgH_1/3²B) vs. a/h for constant-period TMA wave trials: X1 (green diamond), X2 (yellow inverse triangle), X3 (blue square), X4 (red upright triangle), and X5 (black circle).

The reduction in horizontal force with increasing air gap is expected; however, the largest normalized vertical force does not correspond with the most deeply submerged air gap condition but rather the condition with the structure elevated at the water level (a/h=0). This phenomenon was especially true for nonbreaking waves, where the zero air gap condition likely allowed wave impact forces to affect the base of the structure. These findings are in agreement with the work of Hayatdavoodi et al. (2014) and Seiffert et al. (2015), who experimentally measured and numerically modelled horizontal and vertical forces induced by Cnoidal waves on a flat plate at varied elevations above or below the free surface. Hayatdavoodi et al. (2014) and Seiffert et al. (2015) noted that the horizontal and vertical forces generally increased with increasing wave height and that the maximum vertical force occurred when the plate was elevated to the stillwater level. These results are significant for engineering design: even though elevating a structure will reduce the expected horizontal load during a wave and surge event, the vertical load may not necessarily be reduced depending on the initial surge level. The reduction of vertical loading with increasingly negative air gap is due to the vertical velocities of water particles below a wave, which decrease with depth. Therefore, a submerged structure will experience smaller vertical wave velocities, dynamic pressures, and forces than one elevated closer to the stillwater level. However, the authors also note that if a structure is elevated to a height sufficient to prevent any wave impact, both the horizontal and vertical wave forces on the structure (neglecting any forces on piles) will be zero.

4.3. Horizontal to vertical force ratio

Although the normalized forces plotted in Fig. 10 show that the forces caused by nonbreaking waves are large with respect to the incident wave energy, qualitative wave observations from Fig. 7 showed that the structure recorded a larger horizontal force when subjected to breaking waves than when impacted by smaller, nonbreaking waves. In contrast, the specimen recorded vertical wave forces on similar orders of magnitude for all wave conditions. This phenomenon is more fully developed in Fig. 11, which depicts the relationship between the ratio of the horizontal force to the vertical force, F_H/F_V, and the significant wave height, H_1/3. This figure shows the horizontal to vertical force ratio for all characteristic force definitions described above for the condition with the structure located at the water line (a₅), which was shown in Fig. 10 to be the condition in which the structure recorded the maximum vertical force. A similar trend to Fig. 9 can be seen in that the maximum force ratio occurs when taking the maximum force as the characteristic force, while the ratio of the mean horizontal and vertical forces is a minimum. This trend indicates that for all incident wave conditions, the random waves generated during TMA trials produced a wider distribution of horizontal forces and allowed for extreme forces, while vertical forces were more closely distributed around the mean. However, all characteristic force ratios increased with wave height. This trend is important to note because Fig. 11 indicates that as wave height increases, the horizontal force becomes larger with respect to the vertical force. The horizontal force has been noted in design equations to increase nonlinearly with increasing wave height (e.g. Cuomo et al., 2010, FEMA, 2011, Wiebe et al., 2014). Thus, increasing wave height is expected to increase the magnitude of the horizontal force. Additionally, horizontal wave slamming forces may be significantly greater than the vertical wave impact forces for a breaking wave. Thus, the increase in significance of the horizontal force for larger wave heights makes intuitive sense. Therefore, for small wave heights, the vertical force may become a significant design parameter, and for large wave heights, the horizontal force may become more important.

Fig. 11.

Characterized F_H/F_V versus H_1/3. The ratio of horizontal to vertical force considering F_mean (blue diamond), F_1/3 (magenta circle), F_1/10 (black square), F_1/50 (yellow inverse triangle), F_1/100 (red upright triangle), F_1/250(hollow diamond), and F_Max (hollow circle).

Figure 12 shows the horizontal to vertical force ratio for the characteristic force F_1/10 plotted against increasing air gap conditions for each of the constant peak period irregular wave trials (X1-X5). The horizontal force is generally larger than the vertical force for submerged conditions with a/h<−0.2, with the smallest wave heights of TMA_X1 always generating a larger vertical force than horizontal. The vertical force is largest with respect to the horizontal when the base of the structure is elevated to the stillwater level (a/h=0.00), which may be expected based on the conclusions of Fig. 10, which showed that the vertical force was a maximum for the zero air gap condition. Further elevating the structure above a/h=0.00 increases the F_H/F_V ratio, but the vertical force is still larger than the horizontal force for all wave conditions. Therefore, depending on a structure’s air gap condition, the vertical force may be a critical failure-inducing load that must be accounted for in design. This result is important not only when considering structures elevated to different heights above grade but also for determining possible effects of an increased or decreased initial storm surge elevation, as both of these conditions affect a structure’s air gap.

Fig. 12.

F_H(1/10)/F_V(1/10) vs. a/h for each of the constant peak period TMA Trials (X1-X5): X1 (green diamond), X2 (yellow inverse triangle), X3 (blue square), X4 (red upright triangle), and X5 (black circle).

Thus far, focus has been on the relationship between wave height, air gap, and vertical and horizontal forces. However, other factors that could influence the magnitude of the load on a structure subject to hurricane waves include the wave period or wavelength as well as the base dimensions of the structure. Figure 13 shows characteristic horizontal and vertical forces (F_H and F_V), plotted against the representative wavelength, L, for the zero air gap experiment. Forces are again normalized by ρgH_1/3²B, which may be thought of as a representation of the energy of the incoming wave applied over the width of the structure in the alongshore, (y-), direction. The wavelength is normalized by the base dimension of the structure, W, measured in the x- direction, going with the wave. Figure 13 shows that normalized horizontal forces are generally insensitive to wavelength while vertical forces show a linear, decreasing trend as wavelength increases. A slightly undulating behavior may be seen for normalized horizontal characteristic forces larger than F_1/10 as L/W increases. This increase and decrease of the normalized force corresponds with waves in these trials transforming from nonbreaking to impulsively breaking to fully broken bores as they impact the structure. Impulsively breaking waves such as those observed in X8 (L/W=7.3) are associated with impact pressures and forces larger and more variable than forces corresponding with other wave conditions (Peregrine, 2003, Tomiczek et al., 2016a). Thus, trials in which many waves impacted the box as breaking waves resulted in larger characteristic normalized forces than trials in which the majority of the waves propagated past the structure as either nonbreaking waves or turbulent bores that had broken along the slope. Note that the base dimensions of the structure (1.02 × 1.02 m) were kept constant during this experiment. Increasing the area of the base of the structure is expected to likewise increase the total uplift force; future work may examine the effects of changing the base area of a structure on the resulting vertical force due to wave impact. Also note that for this experiment trials X6-X10 held wave steepness constant while varying wave height and wavelength; therefore, the larger normalized vertical forces observed for smaller wavelengths also indicate the smaller wave heights associated with those conditions. Increasing the wave height as wavelength increased resulted in a decreased normalized vertical force. Future work may consider increasing wavelength while holding wave height constant.

Fig. 13.

Normalized characteristic forces (a) F_H and (b) F_V vs. L/W for TMA trials X6, X7, X3, X8, X9, and X10 for a₅=0.00. Characteristic forces F_mean (blue diamond), F_1/3 (magenta circle), F_1/10 (black square), F_1/50 (yellow inverse triangle), F_1/100 (red upright triangle), F_1/250 (hollow diamond), and F_Max (hollow circle).

Figure 14 shows the normalized characteristic 1/10^th horizontal and vertical forces plotted against L/W for varying air gap conditions. Generally, the horizontal force does not seem to be impacted by the length scale of the wave and the structure. However, the most deeply submerged condition (a₁=−0.30) is associated with the largest normalized horizontal forces for all wavelength conditions. The vertical force shows a generally decreasing trend with increasing L/W for all air gap conditions; the largest forces are observed when the structure is elevated near the water level (a=−0.05, 0.00, or 0.05 m). In general, when the structure was elevated above the stillwater level, it recorded lower vertical and horizontal forces compared to when it was subjected to wave impacts under negative air gap conditions. This result is not unexpected, because raising a structure is expected to reduce the area over which a wave will impact the structure, thus reducing the overall force. Nonetheless, this phenomenon indicates the importance of elevating coastal structures well above the expected flood elevation.

Fig. 14.

Normalized forces (a) F_H and (b) F_V vs. L/W for various air gap conditions. The spectrum of red symbols represents the negative air gap conditions, with lighter shades indicating increasing air gaps, a₁=0.30 (diamond), a₂=−0.20 (inverse triangle), a₃=−0.10 (square), and a₄=−0.05 (upright triangle). Zero air gap trials (a₅=0.00) are represented with hollow circles, and increasingly positive air gap conditions are shown in increasingly dark shades of blue: a₆=0.05 (upright triangle), a₇=0.10 (square), a₈=0.20 (inverse triangle), and a₉=0.28 (diamond).

5. Summary and Conclusions

A comprehensive dataset of irregular, regular, and transient wave-induced pressures and loads on an elevated structure has shown that wave height, pressure, and air gap significantly impact the magnitude of the horizontal and vertical loads on a structure. This work has made manifest several important relationships between wave conditions, structural elevation, and characteristic force:

1. The choice of characteristic force (i.e. F_mean, F_1/10, F_1/250) has important implications for design. In general, the characteristic forces show similar increasing trends with increasing significant wave heights; however, considering the average of the highest 0.4% of all forces or the maximum force indicates that F_1/250 and F_Max increase disproportionally for larger values of H_1/3. An engineer must select the design force based on expected wave conditions, structural importance, and economic constraints. While designing a structure to resist F_Max will certainly result in a more robust structure, such a design may only be economically realistic for hospitals or other structures of vital community importance.

2. The vertical or horizontal force caused by a given set of wave conditions may increase or decrease depending on the structure’s elevation above the water level (air gap). As expected, the horizontal force generally decreased with increasingly positive air gap conditions and increased for more deeply submerged trials. In contrast, the maximum vertical force was measured when the structure was positioned at the water level (a₅=0.00 m). Thus, elevating a structure may be hypothesized to reduce horizontal forces but may not necessarily guarantee mitigated vertical forces depending on the specific wave-surge conditions.

3. The horizontal to vertical force ratio was low for small wave heights, indicating that for such conditions, the vertical force is larger with respect to the horizontal force and thus may be the limiting force causing structural failure. As the ratio of F_H/F_V increased with increasing H_1/3, the horizontal force became larger with respect to the vertical uplift force. In design, both horizontal and vertical forces must be carefully considered in order to prevent structural failure.

Note that the measured data and analyzed results are based on idealized conditions. However, during a real hurricane event, structures will be subject to not only wave impacts but also flow currents, scour, and debris impact, which can cause other structural failure mechanisms (i.e. foundation settlement). Likewise, a structure’s distance from shore will have an effect on the horizontal and vertical forces measured for similar wave inputs. Wave forces are expected to increase or decrease depending on the horizontal location of wave breaking; likewise, the directions of wave impact may generate significantly different pressures on the same structures. Future work may consider varying the x- location of the structure and angle of incidence of incoming waves. Current research is considering the effects of shielding elements in diverting flow and creating asymmetric forces on the structure as well as the loads generated by various debris configurations.

The wave-induced force exerted on a structure will vary significantly depending on the incoming wave characteristics: nonbreaking, impulsively breaking, or fully broken. As reported in previous experiments (e.g. Peregrine, 2003, Tomiczek et al., 2016a, 2016b), waves that broke directly on the specimen generated large impulsive pressures that corresponded with extreme forces recorded by the idealized structure. Due to the spectral nature of the irregular waves analyzed here, generally each of the three breaking types was observed during each trial, as might be expected during a real hurricane event. Therefore, this analysis does not distinguish between nonbreaking, breaking, or broken wave forces but rather identifies characteristic forces, which consider all of the wave forces for a given trial. Furthermore, breaking waves have been reported to induce highly variable pressures and forces even for the same nominal wave condition (Witte, 1988, Shih and Anastasiou, 1992). Current work is focused on quantifying the lateral variation and uncertainty associated with repeated wave impacts. This methodology can give realistic information for coastal design as each of the three breaking types may impact a coastal structure during a hurricane event; however the accurate calculation of the magnitude of a wave-induced force for a given wave height and breaking classification is an important task for design engineers.

The size and geometry of the building is also important, especially when considering the uplift pressure. The ratio of the horizontal force to the vertical force may change significantly depending on the ratio of the areas of the front and bottom faces of a building. Another variable that must be considered in future experiments is the area of impact in both the horizontal and vertical directions. Ongoing research is testing wave impacts on the specimen with an attached subassembly to characterize the effects of the subassembly on the vertical force. Regarding the horizontal force, a decreased impact area due to an opening (for example, doors, windows, or breakaway walls) will decrease the overall force. However, distributions of pressure may not be uniform depending on the spatial location and size of opening. It is also noted that while a window or door breaking may decrease the horizontal wave force on a structure, water intrusion and flooding will cause substantial interior damage.

Another significant contribution of this work is a complete dataset that can be used for benchmarking tests. Further analysis of this dataset is underway to determine force exceedance probabilities and wave-induced pressure distributions under varied wave-air gap conditions. Work is ongoing to model this data numerically and compare experimentally measured pressure distributions with those expected based on well-known load-estimation equations.

Nomenclature

Symbol	Descriptions	Unit
a	Air gap	L
FH	Horizontal force	MLT−2
FV	Vertical force	MLT−2
g	Gravitational acceleration	LT−2
H	Wave height	L
H1/3	Significant wave height	L
A	Transient wave amplitude	L
<H1/3>	Ensemble average of H1/3 over all air gaps for one trial	L
L	Wave length	L
P	Pressure	ML−1T−2
T	Wave period	T
TP	Peak wave period	T
TR	Representative tsunami wave period	T
t	Time	T
tS	Stroke time of wave-maker for a transient wave	T
Ur	Ursell number	-
W	Wave direction width of the structure	L
B	Alongshore direction width of structure	L
ρ	Density of water	ML−3
(–)	Mean of ()	-

Appendix

Table A.1. Bathymetric slab positions.

Slab Loc.	x (m)	z (m)
Bay 2	14.07	0.15
Bay 3	17.71	0.15
Bay 6	28.69	1.05
Bay 10	43.33	1.75
Bay 20	79.93	1.75
Bay 22	87.43	2.37

Table A.2. Repeatability Test (TRAN_X2; structure positioned on floor, h=1.75 m).

Gauge	MAE (%)
wg1	0.37
wg2	0.48
uswg1	0.82
uswg2	0.45
press1	11.11
press2	12.91
press3	10.54
press4	7.28
press5	6.25
press6	4.70
press7	3.15
press8	2.43
press9	4.21
press10	1.72

Fig. A.1.

Repeatability Test (TRAN_X2; structure positioned on floor, h=1.75 m)

The repeatability test was performed using five trials of TRAN_X2 with the specimen positioned at the base of the flume and an offshore water depth of $h=1.75\mathrm{m}$. The mean absolute errors (MAE) of peak wave height and pressure were calculated using the following equation:

$$MAE=\frac{1}{n}\sum _{i=1}^n\left|e_i\right|$$ (1).

Here, $n$ is the total number of repeatability tests, and $e_i$ is the percent error of each gage, calculated by:

$$e_i(\mathrm{\%})=\frac{p_i-\bar{p}}{\bar{p}}\times 100$$ (2),

where, $p_i$ is the peak value of i-th trial, and $\bar{p}$ is the averaged of peak value.

Table A.3. Percent difference between unfiltered and filtered pressure and load data, for each of the five trials used for the repeatability test (TRAN_X2; structure positioned on floor, h=1.75 m).

	Percent Difference (%)
Instrument	Trial 1	Trial 2	Trial 3	Trial 4	Trial 5	AVERAGE
load5	1.23	0.06	0.07	0.75	1.89	0.80
press1	3.12	7.48	4.96	0.00	3.78	3.87
press2	0.72	1.81	4.75	1.69	2.64	2.32
press3	5.26	1.37	1.45	0.60	3.52	2.44
press4	−1.91	3.30	5.67	4.29	2.12	2.70
press5	1.91	−0.91	5.39	1.83	4.21	2.49
press6	3.48	31.60	−1.27	4.65	−0.58	7.58
press7	−0.11	0.53	0.13	5.51	2.79	1.77
press8	5.12	0.13	0.63	6.39	2.87	3.03
press9	5.96	1.95	3.77	1.27	−1.67	2.26
press10	1.41	2.25	1.13	0.22	8.51	2.70

Table A.4. Pluck Test Results.

Number of iterations for each test, water depth, air gap, natural response frequency of structure-frame system, and rate of decay of oscillations. Note that the structure was elevated above the water level for Tests 1, 3, and 4, and partially submerged for Test 2.

Test	Iterations	h (m)	a (m)	fn (Hz)	α (1/s)
1	6	1.75	−0.3	3.8	−0.26
2	5	2.15	−0.3	3.4	−0.39
3	5	2.15	0	5.7	−0.61
4	5	2.15	0.28	9.2	−0.79

Table A.5. Significant wave heights at wg3, used to normalize characteristic forces.

<H> and STD indicate the ensemble average and standard deviation of significant wave heights measured from all air gaps for each trial. The “–” symbol indicates conditions in which no data was collected because no waves impacted the box in the previous trial.

	a1 (m)	a2 (m)	a3 (m)	a4 (m)	a5 (m)	a6 (m)	a7 (m)	a8 (m)	a9 (m)	<H> (m)	STD (m)
X1	0.107	0.108	0.101	0.095	0.095	0.090	0.100	0.101	–	0.100	0.0058
X2	0.230	0.235	0.226	0.205	0.205	0.199	0.220	0.223	0.226	0.219	0.0118
X3	0.347	0.348	0.341	0.317	0.341	0.316	0.349	0.348	0.350	0.340	0.0128
X4	0.406	0.423	0.394	0.382	0.385	0.368	0.408	0.398	0.398	0.396	0.0152
X5	0.416	0.430	0.409	0.388	0.399	0.378	0.412	0.405	0.403	0.404	0.0143
X6	0.151	0.156	0.145	0.141	0.148	0.141	0.152	0.154	–	0.148	0.0053
X7	0.224	0.231	0.224	0.207	0.216	0.204	0.220	0.219	0.215	0.218	0.0079
X8	0.300	0.303	0.291	0.272	0.280	0.264	0.292	0.290	0.287	0.287	0.0120
X9	0.364	0.376	0.360	0.337	0.353	0.337	0.377	0.372	0.381	0.362	0.0157
X10	0.382	0.396	0.378	0.356	0.379	0.368	0.406	0.406	0.421	0.388	0.0196

Fig. A.2.

Water surface elevation time series for regular wave trials of TEST 1 (without specimen): (a) X1, (b) X2, (c) X3, (d) X4, (e) X5, (f) X6, (g) X7, (h) X8, (i) X9, and (j) X10.

Fig. A.3.

Number of measured horizontal (left) and vertical (right) forces at the specimen for all TEST2 cases.