Feasibility of Kinetic Umbrellas as Deployable Flood Barriers during Landfalling Hurricanes

All dam-break experiments were conducted within a channel constructed using transparent acrylic panels 25.4 mm thick elevated 0.63 m above ground level via an aluminum frame [Figs. 2(a) and 3(a)]. The inner dimensions of the channel measured $3\times 0.56\times 0.305\mathrm{m}$ ($\mathrm{length}\times \mathrm{width}\times \text{height}$). An aluminum gate at a distance of $L_0=0.616\mathrm{m}$ from the upstream wall provided containment for water within the reservoir [Fig. 2(b)]. This gate was slotted into aluminum channels embedded into each side wall of the flume, and petroleum jelly (white petrolatum) was applied to ensure a watertight seal. A 4.76-mm-diameter steel wire rope was connected to the gate, which fed into a pulley system and attached at the opposite end to a 25-kg drop weight suspended approximately 2 m above ground level [Fig. 2(a)]. The initiation of each dam-break test involved the release of this weight using a trigger mechanism built into the aluminum pulley beam. Once triggered, the weight experienced free fall over a vertical distance of 1 m before placing the wire rope into tension, thus raising the gate. This effectively increased its initial velocity, reducing the time required to clear the reservoir to satisfy the sudden dam-break condition (Lauber and Hager 1998). Within this study, three impoundment depths of water in the reservoir ($d_0$), 100, 150, and 200 mm, were implemented to vary the force of hydrodynamic impact on the umbrellas.

Scale-model specimens of kinetic umbrellas were created for dam-break testing based on the geometry of their full-scale counterparts presented by Wang et al. (2021c). The full-scale prototypes and their models are summarized in Table 1. At full scale, it was envisioned that each square hypar umbrella would cover a projected area of $A=64{\mathrm{m}}^2$, corresponding to each side measuring 8 m and with a diagonal distance between two corners $S=11.3\mathrm{m}$. The extent of geometrical shell warping therefore can be controlled via modification of the rise ($r$), defined as the distance between the hypar vertex and the point bisecting $S$ [Fig. 1(a)]. This study considered three values of $r$ at full scale, resulting in rise to area ($r/A$) ratios of 0.015, 0.030, and $0.045{\mathrm{m}}^{-1}$. Because scaling must be dimensionless, the scaling was proportional to the ratio $r/S$. Hence, the $r/A$ values corresponded to $r/S$ ratios of 0.085, 0.170, and 0.255, respectively (Table 1). This resulted in a scaling factor of approximately $1:57$. All models were fabricated via stereolithography (SLA) 3D printing using Accura Xtreme resin (3D Systems, Rock Hill, South Carolina) with solid density (${\rho }_s$) and Young’s modulus ($E_s$) of $\mathrm{1,200}{\mathrm{kgm}}^{-3}$ and 1,885 MPa, respectively. A constant projected thickness of 4.5 mm was maintained across all three geometries. The material properties are such that the umbrellas would remain essentially rigid under testing. Although the properties of the resin do not reflect those of the real structure, this would not influence any conclusions pertaining to the verification of the SPH-FEM technique.

Because individual kinetic umbrellas were designed to operate as part of a continuous impermeable barrier, two additional hypars with the same $r/S$ were placed adjacent to the central hypar to enforce this boundary condition [Figs. 2(c and d)]. Only the central umbrella was monitored in terms of the hydrodynamic force history. A 3D printed column was attached to the central umbrella via a stainless-steel binding barrel and screw slotted through a hole near the top of the column. This screw also passed through two cleats attached to the umbrella via holes offset 8 mm from the vertex on the rear face of the panel so that the column was free to rotate [Fig. 3(b)]. The base of the central column consisted of a V-shaped wedge 15 mm in length with the tip entering a 9.53-mm-diameter hole drilled through the channel floor. This tip rested within a slot cut into the end of a 9.5-mm-diameter aluminum rod sitting snugly within the hole, enabling free motion along the vertical axis [Fig. 3(b)]. The opposite end of this rod contacted a load cell attached to the underside of the channel via a steel U-bracket [Fig. 2(c)]. Hence, a pin–pin boundary condition effectively was imposed upon the ends of the column, at which the transducer captured only the vertical reaction at the umbrella vertex (vibrational responses were not considered). The total length of the central column was controlled such that umbrellas across each $r/S$ ratio maintained a constant angle of inclination to the horizontal ($\theta$) of 70° [Fig. 3(b)]. Table 1 summarizes the vertical distance $h$ between the column hinge and floor level for each $r/S$.

To integrate the umbrella specimens into the dam-break channel, a 3-mm-diameter hole travelling through the lower edge was incorporated into each 3D-printed hypar [Fig. 3(b)]. This enabled a single 2.38-mm aluminum rod to feed though the base of both the boundary and the central umbrellas while passing through four aluminum cleats [Fig. 2(c)]. The cleats were attached to an aluminum baseplate 4.76 mm thick [Fig. 3(a)] recessed into the tank with the top surface flush with the channel floor. To maintain $\theta =70°$, the cleats were positioned such that the distance between the base hinge and the front edge of the baseplate $d^{\prime }$ (Fig. 3) adhered to that in Table 1 for a given geometry defined by $r/S$. For each boundary umbrella, a separate column was printed containing a flat end that could be fixed to the baseplate via machine screws [Fig. 2(c)]. Similar to the central column, its overall length was based on that required to maintain $\theta =70°$ for each $r/S$. To control seepage between each boundary umbrella and the central umbrella, a 1-mm-thick aluminum strip was attached to the side edges of each panel with cyanoacrylate. The small gap remaining between each umbrella subsequently was filled with petroleum jelly [Fig. 2(d)].

To capture the force history constituting the reaction of the central column, a Loadstar Sensors (Fremont, California) RSB2 resistive load cell recording at 50 Hz was adopted [Fig. 3(b)]. This sampling frequency is consistent with that in other dam-break studies (Wei et al. 2015), and effectively filters out extremely short duration impulsive pressures because they do not affect the structural performance significantly (Goda 2000; Reeve et al. 2018). As previously stated, the purpose of this transducer was to capture the vertical column reaction at the umbrella vertex ($R_v^{\exp }$). The recorded column reaction effectively acted to balance the torque generated about the pin at the umbrella base by dam-break impact to maintain static equilibrium. Evidently, the magnitude of this reaction at any given time is governed by a dynamic fluid pressure field acting on the hypar. Hence, comparisons with its numerical counterpart enabled an indirect assessment of the numerical technique’s ability to reproduce accurately the spatial and temporal hydrodynamic force distribution on a range of hypar geometries. In addition, two Basler acA1440-220um (Exton, Pennsylvania) monochrome high-speed cameras were synchronized for flow visualization. The first camera (C1) was placed at the gate location to capture the exact instant of dam-break initiation, whereas the second (C2) was directed toward the specimen. A third camera (C3) was directed toward the upstream end of the channel to capture potential overtopping and flow around the specimen.

The decoupled SPH-FEM scheme was described in detail by Wang et al. (2020a) and is discussed only briefly herein. In this work, the open-source CUDA-accelerated SPH solver DualSPHysics version 4.2 (Crespo et al. 2015) was implemented to resolve the governing equations as detailed in the Appendix. The dimensions of the dam-break channel [Fig. 4(a)] defined the SPH computational domain. Based on previous sensitivity studies of particle resolution (Wang et al. 2020a), fluid particles with an initial interparticle spacing ($d_p$) of 2 mm constituted the impounded reservoir, whereas boundary particles with the same $d_p$ comprised the umbrella specimens and walls of the channel. For the three impoundment depths of 100, 150, and 200 mm, this corresponded to a total of 4,413,948, 6,577,648, and 8,741,348 fluid particles, respectively. Because the sudden dam-break condition was satisfied, the gate was not included explicitly within the simulation (St-Germain et al. 2014). The front surface of the central umbrella in contact with the initial bore, referred to as the shield [Fig. 4(b)], was defined according to the geometric formulations presented by Wang et al. (2020a) based on the parameters in Table 1. This surface was discretized into 36 SPH cells arranged in a $6\times 6$ grid for each quadrant, thus producing 144 cells in total as optimal for square hypars [Fig. 4(c)]. Each cell represented a collection of boundary particles in which the Cartesian components of the hydrodynamic force history were computed as a function of time [Fig. 4(c), ${\mathit{f}}_s^w$]. The two boundary umbrellas flanking the central hypar were modeled in SketchUp Pro 2020 and imported into the computational domain via discretization into a triangular lattice by the DualSPHysics preprocessor before being converted into boundary particles. Each dam-break simulation spanned 5 s of physical time and was performed utilizing an NVIDIA GeForce RTX 2080Ti GPU (Santa Clara, California) with 4,352 CUDA cores. The average SPH simulation time pertaining to $d_0=100$, 150, and 200 mm was 21.7, 42.2, and 65.8 h, respectively.

An overview of the SPH-FEM modeling framework is presented in Fig. 5. After the spatial and temporal distribution of hydrodynamic force vectors, ${\mathit{f}}_s^w(t)$, were computed using SPH as previously described, they were mapped to an identical finite-element representation of the central kinetic umbrella based on the same geometry as the SPH model [Fig. 4(c)] to compute its structural behavior assuming small deformations. This assumption was shown to hold under both hydrostatic conditions (Wang et al. 2020a) and for the dam-break experiments conveyed in the section “Validation against Dam-Break Results at Model Scale.” It was necessary to ascertain that an accurate description of the dynamic structural response can be obtained when following the procedure outlined in Fig. 5. Therefore, the open-source finite element solver OpenSees version 2.5.0. was implemented to validate the SPH-FEM technique against the aforementioned dam-break experiments. Following each dam-break simulation, forces on the umbrella surface as output from DualSPHysics were input into OpenSees (Fig. 5). In OpenSees, the umbrella was modeled using eight-node hexahedral stabilized single-point integration brick elements (SSPbrick) utilizing the ElasticIsotropic material (McGann et al. 2011) adopting material properties consistent with the experiment. Each quadrant composed of 36 SPH cells was discretized into 144 elements [Fig. 4(c)], and ${\mathit{f}}_s^w(t)$ was assigned to the central node of each four-element cluster mapped to each cell. To model the column, nine uniaxialMaterial links of arbitrarily large stiffness were adopted to connect the nine nodes [Fig. 4(d)] adjacent to the vertex on the rear surface (referred as the soffit) to a node offset 8 mm from the vertex. This node effectively acted as the column hinge [Fig. 3(b)] and was fixed in translation along the $y$- and $z$-axes while being free to rotate. Likewise, all nodes along the base were fully pinned (fixed in translation but free to rotate about the $y$-axis) to simulate the base hinge depicted in Fig. 3(b). The vertical reaction at the column hinge obtained via the FEM model $R_v^{\mathrm{SPH}}(t)$ therefore was the numerical counterpart to the experimental force time history $R_v^{\exp }(t)$ measured by the load cell. Hence, by evaluating the synchronicity between $R_v^{\exp }$ and $R_v^{\mathrm{SPH}}$ as a function of time, the structural model could be validated against experimental results under the hydrodynamic regime. Although any FEM software can be used for validation purposes, OpenSees was selected in this instance due to its support for the assignment of temporal force data and the ease with which the reaction time history can be obtained.

Fig. 6 summarizes the comparison between experimental ($R_v^{\exp }$) and numerical SPH-FEM ($R_v^{\mathrm{SPH}}$) hydrodynamic force histories associated with the central column reaction across $r/S$ ratios of 0.085, 0.170, and 0.255 at each reservoir impoundment depth of 100, 150, and 200 mm. The initial flat segment of loading reflects only the structural self-weight (attributed to the hypar, hinge, and column) prior to dam-break impact. The self-weight was compared with its numerical counterpart to confirm that friction associated with the sealing between the umbrellas did not significantly affect the measured force. For each experimental configuration, three tests were conducted to ensure data repeatability (Fig. 6, shaded lines). The solid black line in each subplot is the averaged force history across all three trials. A total of 27 individual dam-break experiments across the three $r/S$ ratios and impoundment depths were conducted.

At an impoundment depth of 100 mm, the decoupled SPH-FEM scheme proved capable of capturing the hydrodynamic behavior of the central column [Figs. 6(a–c)]. The dual-peaked nature of the experimental force history was reproduced successfully, and the peak reaction was well described by the numerical technique (Table 2). However, at $d_0=150\mathrm{mm}$ [Figs. 6(d–f)], numerical reactions across all $r/S$ ratios underestimated the first spike associated with the initial bore impact, and overestimated the final peak immediately preceding the gradual transition into the hydrostatic regime. The corresponding peak column reaction also was underestimated by the SPH-FEM method (Table 2). At the highest impoundment depth, 200 mm [Figs. 6(g–i)], simulated column reactions for $r/S$ ratios of 0.170 and 0.255 compared well with their experimental counterparts across the entire monitored interval. However, at the lowest level of hypar warping at $r/S=0.085$, the peak force was overestimated.

To explain the observations in Fig. 6, experimental and numerical flow profiles at various time intervals were examined. This comparison was facilitated by horizontal and vertical gridlines (spaced at 50 mm) applied to the inner surface of each side wall of the channel. Fig. 7 presents snapshots of the flow profile for $r/S=0.170$ across all impoundment depths at the times marked in Figs. 6(b, e, and h). These times were selected to capture the approximate peak force for both $R_v^{\exp }$ and $R_v^{\mathrm{SPH}}$. Interactions between the dam-break bore and kinetic umbrella models resulted in the formation of a plunging-type breaker in the opposing direction. As the bore contacted the specimen, the geometry of each inclined hypar shell initially directed the flow upward following its curvature profile. After an upper limit was reached, the crest curled toward the upstream direction before plunging back into the flow, resulting in highly turbulent mixing with significant air entrainment. The collapse of this breaker thus corresponded to the first peak of the force history measured on the central column (Fig. 6). After an initial reduction, a second force peak was reached from pressure accumulation on the shield as the reservoir continued to drain.

At $d_0=100\mathrm{mm}$, experimental flow characteristics were well captured by the SPH model [Figs. 7(a and b)]. However, the initial runup induced by hypar geometry was underestimated for $d_0=150\mathrm{mm}$ [Fig. 7(c)]. Although the plunging crest had yet to contact the upstream flow in the experiment, simulations appeared to depict the flow at a later stage of breaking. Experimental results at $d_0=150\mathrm{mm}$ also reflected higher levels of overtopping that were not captured numerically [Figs. 7(c and d)]. The lower peak force produced by the SPH-FEM scheme at this impoundment depth therefore can be linked to its underestimation of the maximum level of initial runup, hence underpredicting the maximum moment exerted about the base hinge of the central umbrella. The subsequent overestimation of the final peak [Figs. 6(d–f)] followed from a larger quantity of water retained on the upstream side of the specimen supplementing the hydrodynamic pressures induced by the remaining incoming flow. At the highest impoundment depth, $d_0=200\mathrm{mm}$, continuous overtopping was observed [Figs. 7(e and f)], and SPH results also predicted an earlier formation of the plunging breaker (similar to $d_0=150\mathrm{mm}$). However, the synchronicity between experimental and numerical column force histories for $r/S=0.170$ and 0.255 [Figs. 6(h and i), respectively] demonstrated the capability of the SPH-FEM technique to capture the main features of hydrodynamic FSI for flows in which overtopping can be represented reasonably. Evidently, the observed discrepancies may result from the exclusion of the gaseous phase so that the effects of air entrainment on the flow cannot be captured. Multiphase SPH simulations (Grenier et al. 2013; Mokos et al. 2015) therefore would be required to advance further the study of violent hydrodynamic interactions with kinetic umbrellas. However, the corresponding increase in computational cost may not prove feasible from a design perspective (De Padova et al. 2020). For purposes of predicting the structural response to inform the design of kinetic umbrellas in engineering practice, the single-phase implementation of SPH as described herein likely would prove to be sufficient based on the correlations observed in Fig. 6.

The previous section validated the SPH-FEM scheme against experimental data using model-scale and hypar surfaces. This section validates the SPH-FEM scheme at full scale on flat surfaces. To achieve full-scale validation against the forces of wave impact, Goda’s formulations were used as applicable to conventional sloped barriers without hypar warping (Tanimoto and Kimura 1985). Because there are no existing formulations dealing with wave impact on hypar surfaces, the use of a barrier without warping was necessary. An $8\times 8\text{-}\mathrm{m}$ flat panel (with $r=0$) inclined at 65° was selected to maintain consistency with the geometric parameters previously adopted by Wang et al. (2021c), and placed at the top of a sloping beach 10 m long and 2 m high [Fig. 8(a)]. Three water depths ($d$), 3.8, 4.7, and 5.6 m, were considered, which translated to levels of inundation ($d_w$) matching $0.25h^{*}$, $0.37h^{*}$, and $0.5h^{*}$, where $h^{*}=7.25\mathrm{m}$ is the total deployed height of the barrier. For each water depth, regular waves with wave height $H_w$ propagating at frequency $f$ (with period $T=1/f$) and wavelength $L_w$ were selected based on the maximum $H_w$ that can be produced via Stokes second-order wave theory defined by the Le Mehaute abacus (Le Mehaute 1976) [Fig. 8(b)] (summarized in Table 3). The maximum Goda wave pressures $p_1$, $p_2$, and $p_3$ [Fig. 8(a)] were determined as follows (Goda 2000):where ${\rho }_w$ = water density; $g$ = gravitational acceleration; andwhere $h_b$ = water depth at a distance $5H_w$ from the barrier toe; $h_c=h^{*}-d_w$ [Fig. 8(a)]; and ${\eta }^{*}=1.5H_w$. The wavelength $L_w=2\pi /k$ is determined from the wave period or frequency through the dispersion relationwhere $\omega =2\pi f=2\pi /T$; and $k$ = wavenumber, which equals the number of wave cycles per unit length (simply defined as $2\pi /L_w$). Eq. (1) combined with the corresponding linear hydrostatic pressure distribution therefore enables determination of the maximum base shear ($R_b$) and vertex reaction ($R_v$) [Fig. 8(a)] for the inclined panel via principles of static equilibrium. This was compared against their numerical counterparts from SPH-FEM modeling to validate the numerical scheme. An offset of 0.3 m between the vertex hinge and panel centroid was imposed to mirror the geometry in Wang et al. (2021c).

The domain illustrated in Fig. 8(a) subsequently was simulated via SPH across a total width of 16 m. This effectively accommodated one full panel modeled using 144 SPH cells ($36\mathrm{cells}/\mathrm{quadrant}$) flanked by two half-panels acting as the boundary condition [Fig. 8(c)]. An interparticle spacing of $d_p=0.12\mathrm{m}$ was implemented to maintain the same resolution as that of the dam-break simulation described in the section “SPH-FEM Modeling.” An open boundary was adopted at the upstream end of the domain at which buffer particles were used to create an inlet layer where the horizontal components of the fluid velocity, surface elevation, and pressure were imposed externally (Verbrugghe et al. 2019). The total length of the flat upstream section of the numerical flume was adjusted for each water depth to ensure that it was longer than 1 wavelength $L_w$ [Fig. 8(a)]. To generate the waves, the horizontal velocity time history $V_b(t)$ of all fluid particles constituting the open boundary were prescribed. This was achieved through Stokes second-order wave theory as defined by Madsen (1971) to suppress the generation of spurious secondary waves (Altomare et al. 2017)which is applicable only when $U_N<8{\pi }^2/3\approx 26$, where $U_N=H_w{L}_w^2/d^3$ is the Ursell number. Furthermore, $\xi$ denotes the Biesel transfer function assuming irrotational and incompressible fluid behavior with constant pressure at the free surface

Finally, to ensure numerical stability, $r_t(t)$ represents an artificial ramping function (Ning et al. 2017)

Wave forces acting on each SPH cell for a given loading scenario subsequently were extracted and applied to its FEM counterpart (following the same modeling approach defined in the section “SPH-FEM Modeling” and Fig. 4) to obtain the maximum base shear and vertex reaction. Across all three water depths, absolute errors pertaining to $R_v$ and $R_b$ between Goda’s relations and the SPH-FEM scheme were less than 5% (Table 3). As a supplement to the experimental results from the section “Validation against Dam-Break Results at Model Scale,” the modeling technique presented herein may therefore be adopted with confidence for the analysis of full-scale kinetic umbrellas subject to wave attack. Because direct comparisons were not made of the hydrodynamic response between model and full-scale simulations, scaling effects (such as Froude similarities) do not apply.

The validated SPH-FEM scheme was implemented to assess the performance of kinetic umbrellas subjected to a realistic storm event to ascertain their structural feasibility in an engineering context. Hurricane Sandy, the costliest cyclone on record to impact the New York metropolitan area, was selected for this case study. Making landfall near Brigantine, New Jersey at $23:30$ UTC on October 29, 2012, Sandy induced record levels of storm surge across the Mid-Atlantic states, with the highest inundation, 2.7 m, observed at Monmouth, New Jersey (Blake et al. 2013). Therefore, the Atlantic shoreline of Monmouth Beach located at 40°20′04″ N 73°58′28″ W was adopted as the study site [Fig. 9(a)], and its corresponding beach profile during fall 2012 [Fig. 9(b)] was extracted from the New Jersey Beach Profile Network (NJBPN) database compiled by Coastal Research Center (2021). An existing seawall approximately 3.5 m high prevents visibility and access to the shoreline [Fig. 9(c)]. Figs. 9(d and e) illustrate the proposed use of kinetic umbrellas as an adaptable alternative to conventional armoring at this site, the structural behavior of which was evaluated in response to combined surge/wave loading consistent with Hurricane Sandy. The hinge in Figs. 9(d and e) is shown only for conceptual purposes. A possible design of the hinge mechanism was explored by Wang et al. (2021a).

The idealized beach profile depicted in Fig. 9(b) was implemented for the SPH simulation [Fig. 10(a)]. A total water depth ($d$) of 9.1 m ensured that an inundation ($d_w$) of 2.7 m was achieved to reflect the maximum witnessed during Hurricane Sandy at Monmouth. The depth-limited wave spectrum [Fig. 10(b)] was estimated based on the maximum sustained surface wind speed of $21.6{\mathrm{ms}}^{-1}$ (42 kn) recorded at Monmouth Beach approximately 15 min after landfall (Blake et al. 2013). This spectrum was generated via a simplified parametric swell and wind-sea model utilizing the Donelan et al. (1985) spectral form modified for depth-limited regimes (Wang et al. 2021b). The model produced a significant wave height ($H_{m0}$) of 3.64 m, with peak period ($T_p$), frequency ($f_p$), and wavelength ($L_p$) of 8.33 s, $0.12{\mathrm{s}}^{-1}$, and 71 m, respectively. Unlike Joint North Sea Wave Project (JONSWAP), in the Donelan spectrum the high-frequency face of the spectral energy ($S_D$) is proportional to $f^{-4}$ as follows (Young 2003):whereand

The adopted $L_p$ of 71 m constitutes a rough approximation of the sea state, because using Stokes fifth-order yields $L_p=75.9\mathrm{m}$, and in stream function theory, $L_p=76.03\mathrm{m}$.

Prior to the 3D simulation of kinetic umbrellas, it was necessary to obtain the wave conditions yielding the maximum hydrodynamic force associated with Hurricane Sandy at the proposed umbrella location. For this purpose, irregular waves (which provide a more realistic representation of waves found in nature) were generated via the deterministic spectral simulation method. To ascertain the desired forcing conditions for structural analysis, a two-dimensional (2D) SPH simulation of the domain [Fig. 10(a)] was implemented. This enabled the force history on an arbitrary obstacle at the proposed umbrella location to be captured over a total storm duration of 5,000 s (equivalent to approximately 660 waves propagating at the mean period $T_m\approx 0.9T_p$), with the inlet velocity [Fig. 10(c)] defined as follows:where $A_i$ = amplitude associated with each discrete frequency $f_i\in \{0.05,0.30\}{\mathrm{s}}^{-1}$ of wave spectrum comprising $N=100$ frequency bandswhere $\mathrm{\Delta }f$ = constant width of each frequency band; ${\omega }_i=2\pi f_i$; andwhere $k_i$ is calculated using the dispersion relation [Eq. (3)]. To capture the stochastic nature of realistic sea states, ${\tilde{\varphi }}_i\sim U(-\pi ,\pi )$ imposes an arbitrary phase shift randomly sampled from a uniform distribution between $-\pi$ and $\pi$. A 60-s excerpt of the inlet velocity [Fig. 10(d)] responsible for producing the maximum wave force on the obstacle [Fig. 10(c)] subsequently was identified and adopted as the forcing conditions for the 3D structural analysis of kinetic umbrellas. Fig. 10(e) shows the force on the obstacle when utilizing the inlet velocity sequence in Fig. 10(d).

Full-scale kinetic umbrellas measuring $8\times 8\mathrm{m}$ ($S=11.3\mathrm{m}$ and $A=64{\mathrm{m}}^2$) [Fig. 1(a)] with $r/S$ ratios 0, 0.085, 0.170, and 0.255, all inclined at $\theta =65°$, were adopted for the case study. This mirrored the geometries previously examined by Wang et al. (2021c) in their parametrization of the hydrostatic response. To examine the influence of waves approaching at oblique angles, umbrellas with $r/S=0.170$ rotated counterclockwise about the vertical axis at the base at angles ${\theta }_w=15°$, 30°, and 45° also were implemented. A SPH domain width of 16 m was adopted for the beach profile presented in Fig. 10(a), of which the central 8 m wide umbrella was modeled using 144 SPH cells and which was flanked by two half-umbrellas, each 4 m wide, acting as the boundary condition. The umbrellas were positioned 50 m shoreward from the top of the sloping beach, and experienced 2.7 m of still water inundation. Fig. 11 illustrates the full SPH computational domain and showcases all umbrella geometries and rotations considered. For each configuration, over 14.7 million fluid particles were simulated for a total physical time of 60 s, per the inlet velocity history in Fig. 10(d), to cpature the peak hydrodynamic force occuring at approximately $t=46.5\mathrm{s}$ [Fig. 10(e)]. Due to computational limitations, it would be infeasible to simulate the entire 5,000-s storm; hence only the 60-s duration yielding the maximum force was considered. Each simulation took approximately 63 h to complete.

Only the peak hydrodynamic SPH force measurements generated from DualSPHysics were used for structural analysis (as opposed to temporal force data, which were used in the validation studies); thus it was simpler and more effective and efficient to use the commercial finite-element software SAP2000 v20 for the structural analysis of kinetic umbrellas, rather than OpenSees. Furthermore, the use of SAP2000 facilitated a straightforward approach for the determination of internal shell stresses. FEM models and structural detailing followed those described by Wang et al. (2021c), for which only a brief overview is presented here. All panels were 100 mm thick (modeled using 576 Mindlin–Reissner shell elements) and incorporated a standard steel 125-mm parallel flange channel framing the perimeter for additional stiffening [Fig. 12(b)]. High-strength concrete with compressive capacity ($f_c^{\prime }$) and Young’s modulus ($E_c$) of 65 and 37,900 MPa, respectively, were used as the shell material. A mesh of 16-mm glass fiber–reinforced polymer (GFRP) bars with Young’s modulus $E_f=66\mathrm{GPa}$ and tensile strength $f_u=\mathrm{1,135}\mathrm{MPa}$ acted as reinforcement (Abdelkarim et al. 2019). GFRP was used to mitigate the issue of reinforcement corrosion associated with traditional steel-reinforced concrete exposed to marine environments (Medeiros and Helene 2009). In contrast to the material properties adopted for model-scale experiments (reflecting the 3D printed resin relevant only for validating the SPH-FEM modeling technique), the properties defined herein pertain to umbrellas built for real-world applications. Nonlinear compression-only gap elements were utilized along the base of each umbrella such that displacements were permitted only along the positive $z$-ordinate, whereas translation along the $x$- and $y$-axes was restrained [Fig. 12(a)]. This reflected the panel base nested within a groove recessed into the ground allowing only uplift to occur. To simulate the behavior of the supporting column, lateral springs with stiffness $k_v=3E_c{I}_c/h_c^3$ along the $x$- and $y$-ordinates were assigned to the hinge (offset 0.3 m from the vertex), where $I_c$ is the second moment of area for a 1-m-square concrete column [Fig. 12(c)], and $h_c$ denotes the column height. Moment–curvature relations for the umbrella shell and column computed using the RC sectional analysis program Response-2000 version 1.0.5 (Bentz and Collins 2001) are shown in Figs. 12(d and e), respectively. A detailed description of the FEM model and material characterisics were given by Wang et al. (2021c).

Application of the spatial forces corresponding to the peak total hydrodynamic force (computed via SPH) to the FEM structural model (constructed using SAP2000) enabled the commencement of structural analysis. Fig. 13 summarizes the maximum out-of-plane bending ($M^{*}$), shear ($V^{*}$), and in-plane normal ($N^{*}$) forces within the 100-mm-thick hypar umbrella (averaged over a 1-m width) across the various $r/S$ ratios (with ${\theta }_w=0$) and ${\theta }_w$ angles (with $r/S=0.170$) considered at the instant of maximum wave impact occurring at approximately $t=45\mathrm{s}$. At ${\theta }_w=0$, the moment and shear demands [Fig. 13(a)] dramatically decreased with the introduction of hypar geometry (i.e., increasing $r/S$ beyond 0), which is consistent with the hydrostatic behavior reported by Wang et al. (2021c). The reduction in $M^{*}$ and $V^{*}$ was attributed to the marked increase in compressive (C) and tensile (T) normal stresses [Fig. 13(b)] facilitated by the double curvature of hypar geometry dramatically increasing the shell’s resistance against out-of-plane deformation (Wang et al. 2020b). This effect is observed clearly via the reduction in maximum shell deflections in Fig. 13(b), which confirms that the assumption of small deformations was valid for umbrellas exhibiting hypar warping when considering the geometries and hydrodynamic loads examined in this study. Evidently, this behavior is highly desirable because the structural design of thin plates and shells is limited primarily by out-of-plane bending and shear effects. From a design perspective, the nominal moment capacity for the 100-mm-thick shell was approximately $41\mathrm{kNm}/\mathrm{m}$ [Fig. 12(d)]. This is significantly lower than the $M^{*}=137\mathrm{kNm}/\mathrm{m}$ hydrodynamic moment computed for the flat plate ($r/S=0$) [Figs. 13(a) and 14(a)], signifying flexural failure of the shell. However, as hypar geometry was introduced commencing with $r/S=0.085$, the moment demand decreased sharply to $28.9\mathrm{kNm}/\mathrm{m}$, only 70% of the flexural capacity. As the extent of hypar warping ($r/S$) further increased, critical shell demands across $M^{*}$, $V^{*}$, and $N^{*}$ continued to decrease. Hence, it was demonstrated that umbrellas adopting hypar geometry are not susceptible to bending failure, unlike a flat plate with zero warping. This also applies to shear demands, as is explained in the following section. When considering oblique angles of wave attack on umbrellas with $r/S=0.170$, bending moment decreased modestly, accompanied by an increase in shear as ${\theta }_w$ increased [Fig. 13(d)]. Compressive normal stresses also increased initially at ${\theta }_w=15°$ but decreased at larger angles of incidence [Fig. 13(e)]. A corresponding decrease in maximum shell deflections also occurred. The 1-m-square concrete column supporting the vertex exhibited decreased moment ($M_c^{*}$), and shear ($V_c^{*}$) with larger ${\theta }_w$ [Fig. 13(f)]. However, tensile normal stresses in the shell and axial compression in the column ($N_c^{*}$) were not affected significantly by ${\theta }_w$.

Fig. 14 illustrates the distribution of shell moment, shear, and normal demands along the $u$- and $v$-ordinates for panels with $r/S=0$ and 0.170 at ${\theta }_w=0.$ In shell theory, $M_u$ and $M_v$ respectively denote bending in the of $u$- and $v$-directions, with positive moments signifying tension on the shield. Locations at which critical demands manifested (Fig. 14, boxes) migrated from the hinge zone [Figs. 14(a–c)] to the lower panel [Figs. 14(d–f)] with the introduction of hypar geometry, which reflected the behavior reported by Wang et al. (2021c) for purely hydrostatic scenarios. Due to the relatively large compression forces present in the hypar, a simplified yet conservative buckling assessment was undertaken. Fig. 14(f) reveals critical compression forces along the transverse spine and lower longitudinal edge with buckling lengths ($L_b$) of 3.7 and 3 m, respectively. Euler’s critical buckling load ($P_{cr}$) was determined for the transverse spine via the approach proposed by Wang et al. (2020b) for square hypars, and $P_{cr}$ for the longitudinal edge was obtained from the buckling capacity of the 125-mm parallel flange channel framing the perimeter edge. Comparisons with the critical force ($N_{cr}^{*}$) revealed that the umbrella sustained only 16% and 29% of the ultimate buckling capacity at the spine and edge, respectively, which clearly affirmed Candela’s belief in the superior buckling performance of four-sided hypar umbrellas (Candela 1955). The influence of oblique waves relating to ${\theta }_w=45°$ on hypar stresses for $r/S=0.170$ is shown in Fig. 15. Comparing these results with those for (${\theta }_w=0°$) [Figs. 14(d–f)] showed that differing angles of wave incidence did not significantly affect the spatial manifestation of critical demands. The decrease in critical moment with ${\theta }_w$ [Fig. 13(d)] stemmed from a reduction in transverse bending about the longitudinal spine on the lower panel [Fig. 15(a)]. Conversely, elevated shear demands at larger ${\theta }_w$ resulted from larger shear stresses generated along the transverse base on the right side [Fig. 15(b)] as the direction of wave propagation became more orthogonal with the right half of the umbrella (Fig. 11). Likewise, an increase in the buckling force ($N_{cr}^{*}$) along the left longitudinal edge also occurred, but was only 38% of the ultimate buckling capacity ($P_{cr}$).

Wang et al. (2021c) demonstrated that a kinetic umbrella with $r/S=0.170$ designed in accordance with Figs. 12(b and c) successfully can resist hydrostatic inundation matching the entire deployed height (i.e., $d_w=h^{*}=7.25\mathrm{m}$). To ascertain its overall structural feasibility against surge and wave attack generated by Hurricane Sandy, all critical hydrodynamic demands pertaining to the umbrella shell and supporting column were compared with those of its hydrostatic counterpart at a given level of static inundation. The hydrodynamic demand ratio (HDR) is introducedwhere $D_{\mathrm{dyn}}^{*}$ and $D_{\%}^{*}$ = critical demand of any component (bending, shear, and normal forces) associated with maximum wave impact and hydrostatic inundation $d_w$ [Fig. 10(a)], respectively. Note that the percent symbol % in ${\mathrm{HDR}}_{\%}$ and $D_{\%}^{*}$ represents the percentage of the total deployed height subject to inundation (e.g., $D_{100}^{*}$ = critical hydrostatic demand for complete inundation at 100% of the deployed height). A HDR value less than 1 thus implies that the structure experiences lower forces from the peak dynamic load case than the corresponding hydrostatic scenario. Fig. 16 reports the HDR across all relevant components of the demand for $r/S=0.170$ compared with the hydrostatic results in Wang et al. (2021c) for inundation equal to 100% of $h^{*}$, representing the most critical loading condition. All demands resulting from Hurricane Sandy were smaller than their static equivalents at the maximum inundation level (${\mathrm{HDR}}_{100}$). Therefore, it is demonstrated that the 100-mm-thick kinetic umbrella reported in this study can be designed successfully to serve as an adaptable flood barrier against combined surge and wave loading attributed to Hurricane Sandy at Monmouth Beach from a structural engineering perspective. Evidently, if larger loads from potentially more-severe storms must be resisted, the shell thickness can be increased accordingly. As an example, increasing the thickness to 150 mm and adopting two layers of 16-mm GFRP bars increased flexural capacity 173%, from 41 to $112\mathrm{kNm}/\mathrm{m}$.