Hydrostatic Response of Deployable Hyperbolic-Paraboloid Umbrellas as Coastal Armor

Prior to the commencement of structural analysis, it is important to characterize the geometry of a tilted hypar shell via a mathematically consistent approach. The Cartesian coordinates $\{\mathbf{x},\mathbf{y},\mathbf{z}\}$, with corresponding orthonormal ordinates $\{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}$, of any given four-sided singly symmetric hypar panel in its deployed configuration (${\mathbf{\Omega }}_p$) are parameterized via a set constituting seven fundamental geometric parameters, $S_i=\{b_1,b_2,c,r,h,t_{pv},t_{pe}\}$ illustrated in Fig. 3, where

Fig. 3 illustrates the geometry of an arbitrary tapered hypar characterized by $S_i$ with such parameters accentuated within boxes. Furthermore, the local basis $\{\mathbf{u},\mathbf{v},\mathbf{w}\}$ defines the planar projection of the surface of inclination ${\mathbf{\Pi }}_{\mathbf{u}\mathbf{v}}\in [-c,c]\times [0,b_1+b_2]$ as shown in Figs. 3(a and d). The angle between ${\mathbf{\Pi }}_{\mathbf{u}\mathbf{v}}$ and the horizontal plane is referred to as the angle of inclination and is denoted as ${\theta }_1^{*}$ as shown in Fig. 3(c). Various special terminologies are also introduced which are referenced throughout this paper. The concave (front in contact with fluid) and convex (rear) surfaces are referred to as the shield and soffit, respectively. Coordinates constituting such surfaces are henceforth denoted, respectively, by ${\mathbf{\Pi }}_k$ and ${\mathbf{\Pi }}_m$, as indicated in Fig. 3(c). In addition, the deployed panel is divided into lower and upper sections separated by the transverse spine [Fig. 3(b)]. The term $\beta \in \{1,2\}$ is thus adopted to distinguish between the two sections as highlighted in Figs. 3(a and c). In addition, ${\mathbf{\Omega }}_p$ is mapped via the orthogonal ratios ${\lambda }_1,{\lambda }_2,{\lambda }_3\in [0,1]$, increasing along $\pm \mathbf{u}$, $\mathbf{v}$, and $–\mathbf{w}$, respectively, as shown in Figs. 3(b and d). This ensures that the full geometry can be easily represented via a 3D grid of structured cells which streamlines its application within numerical models as implemented within this paper. The reader is referred to Appendix I for a complete derivation pertaining to the geometric parameterization of ${\mathbf{\Omega }}_p$ by any given set $S_i$ via $\lambda$ discretization.

Relations are derived for net reactions at the vertex and base hinges for a deployed panel with geometry delineated via $S_i$ under self-weight and hydrostatic loading on ${\mathbf{\Pi }}_k$. The structural system considered in the analysis is presented in Fig. 4. To consider the highest loading scenario, the relations do not consider potential fluid accumulation on ${\mathbf{\Pi }}_m$. To simplify the procedure, the contribution of normal force to deformation is neglected as commonly accepted in structural analysis (Hibbeler 2008). The vertex hinge is idealized as unrestrained in $\mathbf{x}$ to simulate a flexible support offering minimal lateral resistance, while the base is continuously pinned along $\mathbf{y}$. This effectively reflects a panel supported at the vertex by a vertical column exhibiting axial stiffness much greater than that in the lateral direction. Additionally, the approach ensures that flexural failure at the column base would not result in global failure of the deployed system. The validity of a flexible vertex design philosophy is to be examined in the context of uplift suppression at the soffit base. The effective height of water submersion measured from the base, $h_w\ge 0$ [Fig. 3(c)], is introduced as an additional parameter for the quantification of hydrostatic forces. The net reactions at the base (${\mathit{f}}_b$) and vertex (${\mathit{f}}_v$) hinge under combined hydrostatic and self-weight loading adhering to the flexible support condition are analytically derived via principles of static equilibrium. Appendixes II and III detail the derivation pertaining to the hydrostatic (${\mathit{f}}_v^w,{\mathit{f}}_b^w$) and self-weight (${\mathit{f}}_v^g,{\mathit{f}}_b^g$) components, respectively. The total net reactions are subsequently encapsulated via

Figs. 4(a and b) depict an upright and deployed hypar, respectively, with their idealized boundary conditions shown in Figs. 4(c and d). Supplementary to Appendixes II and III, the concept of panel uplift at the base is illustrated by applying graphic statics to resolve the force polygon associated with the total hydrostatic and self-weight applied force resultant (${\mathit{f}}^w+{\mathit{f}}^g$), and reactions at the base (${\mathit{f}}_b$) and vertex hinges (${\mathit{f}}_v$). If the applied force resultant vector intersects the vertical reaction vector below $\mathbf{z}=0$, uplift is suppressed [Fig. 4(c)]. A line must be drawn between that point of intersection and the base support to maintain equilibrium. This line represents the direction of the reaction vector at the base where the vertical component ${\mathit{f}}_b·\hat{\mathbf{z}}>0$. Conversely, should the intersection of the resultant force be above $\mathbf{z}=0$ as shown in Fig. 4(d), ${\mathit{f}}_b·\hat{\mathbf{z}}<0$ occurs and the presence of vertical restraints are necessary along the panel base.

It is evident that if ${\mathit{f}}_b·\hat{\mathbf{z}}<0$, uplift is expected to occur at the panel base, assuming no vertical restraints are present. Hence, the relationship between panel geometry and net vertical reactions at the soffit base as a function of inundation height ($h_w$) is subsequently explored. This effectively enables exploration into the conditions governing panel uplift at ground level, allowing for the optimization of geometric forms such that uplift can be minimally suppressed. A parametric study was implemented, the parameters of which are summarized in Table 1 and include the ratio between the projected lower-to-upper longitudinal dimension of the panel ($b_1/b_2$), along with changes to the rise-to-area ratio ($r/A$), with $A$ denoting the total area of the inclined surface ${\mathbf{\Pi }}_{\mathbf{u}\mathbf{v}}$. In this study, the total projected area $A$ of the hypar was held constant at $64{\mathrm{m}}^2$, reflecting a square with sides of 8 m. In addition, $b_1/b_2$ was varied from 1 to 4, corresponding to the lower projected length $b_1$ increasing from 4 to 6.4 m and $b_2$ decreasing from 4 to 1.6 m in response. Values of $r^{*}/A$ were selected between 0 and $0.047{\mathrm{m}}^{-1}$, inferring a total vertex rise $r^{*}$ from 0 to 3 m, respectively. Note that Candela recommends a rise/area ratio greater than $0.015{\mathrm{m}}^{-1}$ for the minimization of short- and long-term deflections of upright concrete hypar umbrellas (Garay 1994). Furthermore, three angles of inclination (${\theta }_1^{*}$) constituting 55°, 65°, and 75° were adopted in addition to three inundation heights $h_w$ for each level of tilt. $h_w$ was designated as a proportion of the maximum effective panel height ($h^{*}$) with 0.5 $h^{*}$, 0.75 $h^{*}$, and $h^{*}$ considered. For simplicity, the panel thickness was neglected with $t_{pv}=t_{pe}=0$. Hence, only hydrostatic demands were examined via the parametric investigation while ignoring the influence of self-weight and shell-thickness gradients. This effectively leads to a conservative assessment of inundation-induced panel uplift. Table 1 summarizes the parameters and the range of their respective values implemented for the parametric investigation.

The net vertical reaction at the panel base ${\mathit{f}}_b·\hat{\mathbf{z}}$ was computed via Eq. (1) for the geometries and inundation heights considered within the parametric study. Fig. 5 depicts critical uplift lines reflecting the geometric configuration in terms of $r/A$ and $b_1/b_2$ for a given submersion height and angle of inclination such that the net vertical base reaction is zero. This implies that the hydrostatic resultant from Fig. 4 intersects the $\mathbf{z}$-axis at $\mathbf{z}=0$. If one begins with a design constraint of $h_w$ and ${\theta }_1^{*}$, combinations of $r/A$ and $b_1/b_2$ to the right of any given line thus encompass the entire range of hypar geometries to which a laterally unrestrained vertex hinge would not result in panel uplift. If one begins with a design geometry of $r/A$ and $b_1/b_2$, then critical uplift lines to the left (holding $b_1/b_2$ constant) represent $h_w$ and ${\theta }_1^{*}$ values that would not result in uplift. As an example, if Candela’s limit of $r/A=0.015{\mathrm{m}}^{-1}$ were adopted, for $b_1/b_2\ge 1.25$, uplift is avoided for ${\theta }_1^{*}=55°$ up to $h_w=0.75h^{*}$ or ${\theta }_1^{*}=65°$ up to $h_w=0.5h^{*}$. Furthermore, it is observed that the critical uplift lines exhibit a rightward shift with increasing inundation heights and angles of inclination, constraining the range of geometric combinations available to prevent uplift. For a constant $b_1/b_2$, increasing the $r/A$ ratio appears to reduce the uplift potential of the panel for any given ${\theta }_1^{*}$ and $h_w$. This is attributed to greater concavity of the shield maximizing the gravity component of the hydrostatic demand acting against the uplift tendency of the deployed hypar. Likewise, an increase in the $b_1/b_2$ ratio while maintaining a constant $r/A$ also aids in the suppression of panel uplift due to the greater volume of water acting on the lower shield ($\beta =1$). Fig. 5 therefore informs the selection of geometric parameters for a squarely projected hypar pertinent to the parametric study such that hydrostatic uplift is avoided. If such conditions are violated, additional structural modifications may include the provision of lateral rigidity at the vertex (via a diagonal strut or otherwise) and/or vertical restraints along the base.

Smoothed particle hydrodynamics (SPH) is a mesh-free Lagrangian particle method commonly adopted for the simulation of FSI involving free surface fluid behavior (Liu and Liu 2010). While mesh-based techniques such as FEM have been utilized to model FSI problems in isolation (Zhu and Scott 2014), expensive considerations such as mesh rezoning are often required to accommodate excessive deformations, potentially introducing additional errors (Liu and Gu 2005). With SPH, all geometries are nonconstrained irrespective of complexity and the extent of their evolution beyond initial conditions (Crespo et al. 2008), which reinforces its applicability toward coastal engineering applications (Altomare et al. 2015).

In this study, SPH was implemented via DualSPHysics to model hydrostatic fluid accumulation on the concave surface of the deployed hypar. The fluid continuum was represented as a set of discrete particles, each exhibiting physical properties including position, velocity, density, and pressure. Such quantities for any given particle are computed via the integral interpolant of adjacent particles within its support domain. The extent to which each nearby particle influences a given property is dependent upon the interparticle distance, to which a kernel function ($W$) was adopted to represent this contribution. The kernel function is governed by a smoothing length ($h_s$) which delineates its zone of influence and should be greater than the initial particle separation (Barreiro et al. 2013). The weighted interpolant approximating any quantity field $B(\mathit{r})$ is thus expressed aswhere $\mathit{r}$ = position vector; and ${\mathit{r}}^{\prime }$ denotes the position of all remaining particles within the support domain $\mathrm{\Omega }$. Eq. (2) may subsequently be expressed in discrete form to compute any quantity $B$ for particle $i$ aswhere $m_j$ and ${\rho }_j$ = mass and density of particle $j$, respectively; and $W_{ij}=W({\mathit{r}}_i-{\mathit{r}}_j,h_s)$ is the weighting kernel. $W_{ij}$ governs the interaction between adjacent particles and must be selected in strict accordance with a prescribed set of conditions (Liu et al. 2003; Liu and Liu 2010). In this work, the quintic kernel by Wendland (1995) was adopted:where $q$ = ratio between the interparticle distance and smoothing length; and ${\alpha }_D=21/(16\pi h_s^3)$ for analyses in ${\mathbb{R}}^3$. Particle clumping is avoided via use of the tensile instability correction developed by Monaghan (1999). The determination of fluid forces results from changes in the velocity field stipulated by the conservation of linear momentum. This was implemented via the popular artificial viscosity scheme developed by Monaghan (1992):where ${\mathit{v}}_i$ = particle velocity; $P_{i,j}$ = pressure; and $\mathit{g}$ = gravitational acceleration vector. ${\mathrm{\Gamma }}_{ij}$ is the viscous term representing an artificial viscosity model (Monaghan 1994):with${\mathit{v}}_{ij}={\mathit{v}}_i-{\mathit{v}}_j$ and ${\mathit{r}}_{ij}={\mathit{r}}_i-{\mathit{r}}_j$, respectively denote the relative velocity and position of the particle; ${\zeta }^2=0.01h_s^2$; and ${\overline{c}}_{ij}=0.5(c_i+c_j)$ is the average speed of sound. For CFD applications regarding water, it is suggested that $\alpha$ falls between 0.01 and 0.1 (Dalrymple and Rogers 2006; Monaghan 1994). The choice of $\alpha$ is designed to prevent numerical instability (Crespo et al. 2011), to which Altomare et al. (2015) and Barreiro et al. (2013) confirm that 0.01 is the most suitable value for the consideration of fluid interaction with coastal structures. In addition, changes in the fluid density were determined via the equation governing the conservation of mass:

As such, the weakly compressible assumption for fluids was adopted such that an equation of state may be implemented to determine the pressure-density relationship. The following equation of state for water enabled the determination of fluid pressure via its density (Batchelor 2000):where ${\rho }_w=\mathrm{1,000}\mathrm{kg}/{\mathrm{m}}^{-3}$ is the fluid reference density; $\kappa =7$ (Monaghan 1994); and $c_0$ is an artificial speed of sound to suppress density fluctuations within 1% of ${\rho }_w$ (Monaghan 2012). This ensures that the fluid behavior will not deviate significantly from the incompressible approach. Through the resolution of Eq. (5), the spatial distribution of fluid forces imposed upon boundary particles constituting the shield of the deployed panel at any given timestep was determined viawhere $m_b$ = mass of a boundary particle; and $K$ = total number of particles forming the impacted surface over which ${\mathit{f}}_s^w$ acts. In this work, dynamic boundary conditions proposed by Crespo et al. (2007) were adopted where boundary particles exert a repulsive force on approaching fluid particles. All boundary particles exhibit the same equations of continuity and state as their fluid counterparts but are fixed in position. The repulsive force is generated by an increase in pressure as the distance between boundary and fluid particles fall within the kernel threshold ($2h_s$), effectively preventing fluid penetration.

The SPH computational domain in this study consists of a box 6 m long in $\mathbf{x}$, 8 m wide in $\mathbf{y}$, and 10 m tall in $\mathbf{z}$. The concave side of the tilted panel ${\mathbf{\Pi }}_k$ in contact with the fluid was discretized into a collection of quadrilaterals with corner coordinates derived per Eq. (23) from Appendix I and positioned 0.5 m from the front wall of the domain with the origin ${\mathbf{x}}_0=\{0.5,0,0\}$ as shown in Fig. 6. The height of fluid inundation ($h_w$) was 7.4 m, matching the total effective height of the deployed panel ($h^{*}$). An initial interparticle spacing ($d_p$) of 0.1 m was implemented along with an artificial viscosity constant $\alpha =0.01$. A total 15 s of physical time was simulated in order to provide sufficient time for initial particle oscillations to dampen (Fig. 6). The $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ components of the total hydrostatic force ${\mathit{f}}_s^w$ acting on each quadrilateral $s$ constituting ${\mathbf{\Pi }}_k$ were subsequently determined via the time averaging of fluid forces over the final 7.5 s of the simulation as demonstrated for a typical cell in Fig. 6. Time-stepping was conducted via the computationally efficient Verlet algorithm (Verlet 1967), with each time step determined according to the Courant-Friedrichs-Lewy condition (Monaghan and Kos 1999). The simulation was performed within DualSPHysics utilizing a NVIDIA GeForce GTX 1060 GPU with 1280 CUDA cores (Santa Clara, California).

The computation of hydrostatic demands via SPH subsequently enables the determination of reaction forces at the vertex and base supports via finite element analysis. OpenSees was implemented for evaluation of the structural response in which the hypar continuum was modeled via eight-node hexahedral stabilized single-point integration brick elements (SSPbrick), eliminating the potential for volumetric or shear locking (McGann et al. 2011). Each SPH cell was represented using four brick elements [Figs. 7(a and b)] where the eight corner nodes of each element occupied coordinates determined via Eq. (25) in Appendix I. Hence, each element exhibits four nodes on both ${\mathbf{\Pi }}_k$ and ${\mathbf{\Pi }}_m$. The individual Cartesian components of the hydrostatic force on each SPH cell ${\mathit{f}}_s^w$ were subsequently applied to the center node on ${\mathbf{\Pi }}_k$ of each four-element group constituting a given cell [Fig. 7(c)]. Due to symmetry of the panel about the $\mathbf{x}\mathbf{z}$ plane, only half of the hypar extending in the positive $\mathbf{y}$ ordinate was modeled. As such, nodes forming the longitudinal spine were translationally constrained in the $\mathbf{y}$ direction, effectively restricting rotation about $\mathbf{z}$. The implementation of this boundary constraint was shown to be applicable for the modeling of symmetrical hypar shells as demonstrated by Levine (2018). Furthermore, the vertex node was free to translate only in $\mathbf{x}$ while all nodes at the base were fully translationally constrained, permitting only rotation about $\mathbf{y}$.

For the determination of panel self-weight, each brick element was discretized into eight sectors, with the volume of each sector approximated via computation of the minimum volume encapsulating the interpolated coordinates of its eight corner positions [Fig. 7(d)] using the convex hull algorithm. The total computed volume of the panel arising from the summation of all sector volumes was then compared against the true theoretical hypar volume ($V_p$) from Eq. (46) in Appendix III. This allows for the determination of a global volume correction factor ${\gamma }_c$:where $c_i$ = volume of sector $i$ derived via the convex hull; and $C$ = total number of element sectors constituting the entire half-hypar, equal to eight times the total number of brick elements. The self-weight force vector of a given sector is therefore ${\gamma }_c{c}_i{\rho }_s\mathit{g}$ and was assigned to the corner node of the brick element pertaining to sector $c_i$, as shown in Fig. 7(d). The mechanical properties of the panel material matching that of typical structural concrete were adopted, with density ${\rho }_s=\mathrm{2,400}\mathrm{kg}/{\mathrm{m}}^3$, Poisson’s ratio $\nu =0.15$, and Young’s modulus $E=\mathrm{30,000}\mathrm{MPa}$. Implementation of the ElasticIsotropic OpenSees material thus results in panel behavior constrained entirely within the elastic regime. Note that while concrete properties were adopted for the validation study, the most ideal material for such structures remains under investigation.

The decoupled SPH-FEM scheme was validated via comparing the net reactions at the vertex and base to their analytically derived counterparts per Eq. (1). The total hydrostatic demand imposed along the positive $\mathbf{y}$ ordinate was also evaluated against the theoretical result given by Eq. (37) in Appendix II. The validation was carried out by means of a convergence study in which the total number of SPH cells constituting a panel quadrant was increased from 1 to 49, corresponding to an increase in the number of FEM elements from 4 to 196. In addition, the deflection at five locations on the upper panel was evaluated as a function of mesh resolution to obtain the minimum cell count necessary for adequate convergence.

Fig. 8(a) summarizes the relative error between reaction forces produced via the numerical scheme and solutions to the analytical expressions derived in the appendices as a function of modeling resolution. Particular forces constituted the total horizontal base reaction ${\mathit{f}}_b·\hat{\mathbf{x}}$ (Base-X), total vertical base reaction ${\mathit{f}}_b·\hat{\mathbf{z}}$ (Base-Z), and the vertex hinge vertical reaction ${\mathit{f}}_v·\hat{\mathbf{z}}$ (Vertex-Z) as shown in Fig. 4. The total transverse hydrostatic force acting on each half of the deployed hypar ${\mathit{f}}_{half}^w·\hat{\mathbf{y}}$ (Half Panel-Y) per Eq. (37) in Appendix II was also included. As expected, Fig. 8(a) reveals that increasing the resolution of the numerical model generally enhances the overall convergence between the analytical and numerical results. When adopting 36 SPH cells per quadrant (corresponding to 144 brick elements), the forces computed via the SPH-FEM scheme were Base-X = $-\mathrm{2,120}\mathrm{kN}$, Base-Z = $-857\mathrm{kN}$, Vertex-$\mathrm{Z}=\mathrm{2,143}\mathrm{kN}$, and Half Panel-Y = 310 kN. This compares well to the analytically derived values of $-\mathrm{2,149}$, $-832$, 2,127, and 296 kN, respectively, exhibiting differences of $-1.35\%$, 3.00%, 0.75%, and 4.73%. The assumption of small deformations pertaining to the deformed structural configuration is therefore satisfied. In addition, total OpenSees runtimes were extracted for each modeling resolution [Fig. 8(b)]. The relative increase in FEM runtimes (all less than 1 s) were seen to be negligible compared to the 30 min required to process SPH simulations across all cell discretizations. Note that increasing the number of SPH cells while maintaining a fixed initial interparticle spacing of $d_p=0.1\mathrm{m}$ does not alter the total number of particles, thus resulting in a constant processing time. As a further point of comparison, the delta-SPH formulation (Molteni and Colagrossi 2009) introducing a diffusive term into the continuity equation [Eq. (8)] was also investigated in addition to modifications to the artificial viscosity constant at the fluid-boundary interface. The recommended delta-SPH coefficient (${\delta }_{\mathrm{\Phi }}$) of 0.1 was considered alongside a viscosity scaling factor ($f_{vb}$) of 10 to compare against the default implementation of ${\delta }_{\mathrm{\Phi }}=0$ and $f_{vb}=1$ for 36 SPH cells per quadrant. Fig. 9(a) reveals that under hydrostatic conditions, the incorporation of ${\delta }_{\mathrm{\Phi }}$ and/or changes to $f_{vb}$ does not have a significant effect on the synchronicity between numerical and analytical results. However, an overall reduction in relative error is produced by reducing the initial interparticle spacing as illustrated via Fig. 9(b). By adopting $d_p=0.06\mathrm{m}$, the maximum error in absolute terms was only 0.89%, compared to 4.73% for $d_p=0.1\mathrm{m}$. Nevertheless, this level of detail results in a runtime over seven times that of $d_p=0.1\mathrm{m}$ at 214 min due to the increase in overall particle count from 185,025 to 786,133.

The implementation of 36 SPH cells per quadrant of the squarely projected hypar is also observed to result in the convergence of nodal deflections taken at five locations on the panel computed via OpenSees (Fig. 10). Because analytical or experimental results pertaining to panel deflections under hydrostatic loading are not available, the convergence study provides insight into the optimal cell resolution required to achieve an acceptable deformation profile for square hypars. Thus, the decoupled SPH-FEM numerical scheme proved capable in assessing the behavior of deployed hypar panels under complete fluid inundation in the hydrostatic regime. This validation effectively reinforces the applicability of such models for future FSI studies involving hydrostatic and hydrodynamic actions on hypar surfaces.

The structural geometry modeled via the SPH-FEM scheme within the previous section was selected as characteristic of a typical deployed hypar. For this exercise, however, the projected thickness $t_{pv}$ and $t_{pe}$ was increased to 116 mm to obtain a true thickness of $116\cos {\alpha }^{*}=105\mathrm{mm}$ along the spine [where ${\alpha }^{*}={\alpha }_{\beta }{|}_{{\lambda }_1=0}=25.6°$ per Eq. (14) in Appendix I]. This modification ensures that a minimum clear cover of 1.5″ (38 mm) can be achieved when considering a single mesh composed of #4 reinforcement within the panel as stipulated by ACI 318 (ACI 2014) for exposed structures. The height of hydrostatic inundation $h_w$ was selected based on observations of historical storm surge and storm tide inundation events impacting the east coast of the United States. Fig. 11 summarizes 70 observations of the maximum storm surge or storm tide inundation heights associated with individual hurricane events from 1899 to 2012 (Needham et al. 2015). The region captured encompassed coastal zones extending from Massachusetts to Florida. Note that all recorded inundations were below the maximum deployed panel height of 7.4 m, with mean and median heights of 2.5 and 2.3 m, respectively. A conservative fluid inundation height of $0.75h^{*}$ equating to 5.6 m was henceforth adopted for the application of hydrostatic loading on the tilted hypar. This was attributed to Fig. 11(b) revealing only a single event (associated with Hurricane Wilma in 2005) exceeding this threshold over the entire observational period [Fig. 11(b)]. While ongoing climate change may result in future surges of greater severity (Needham et al. 2015), the assumption of data stationarity is adopted for the proof of concept. A submersion height of 5.6 m may therefore conservatively capture the extent of coastal inundation expected from storm surges and storm tides impacting the eastern United States.

A 500-mm square reinforced concrete column was adopted to support the hypar considered for the proof of concept. The dimensions are such that slenderness effects may be neglected per ACI 318 (ACI 2014). Structural concrete with a compressive strength ($f_c^{\prime }$) of 40 MPa was implemented for the column and panel, incorporating A706 Grade 80 reinforcing steel. The yield and ultimate strength of G80 reinforcement were taken as 552 and 689 MPa (80 and 100 ksi), while the strain at the onset of strain hardening and the ultimate tensile strain were 0.0074 and 0.0954, respectively (Overby et al. 2015). Longitudinal reinforcement within the column consisted of eight #7 bars evenly positioned around the perimeter confined with #4 stirrups at 200 mm spacing. The resulting 54 mm of clear cover to transverse reinforcement thus complies with durability requirements per ACI 318. The hypar shell was reinforced with a mesh of #4 bars spaced at approximately 170 mm in the longitudinal and transverse direction located about the neutral axis of the cross section.

The panel capacity was assessed in terms of the bending moment demand associated with hydrostatic loading acting on the shield. A 3D finite element model was created within OpenSees based upon the implementation of 36 SPH cells per panel quadrant. For the computation of shell-bending moments, 144 element sets were used for each quadrant, where each set was represented by 10 layers of SSPbrick elements, effectively translating to 10 integration points through the thickness of the panel [Fig. 12(b)]. This was achieved by discretizing ${\lambda }_3$ into 10 uniformly spaced increments (Appendix I), giving a total of $144\times 10=\mathrm{1,440}$ elements per quadrant. The bending moment per unit width along the $\mathbf{u}$ and $\mathbf{v}$ ordinates, respectively denoted $M_u$ and $M_v$, were determined for each element stack as follows:where $t_s$ = projected thickness along the integration points; $w$ = distance along the positive $\mathbf{w}$ ordinate measured from the neutral axis; and ${\sigma }_u$ and ${\sigma }_v$ = components of stress resolved along $\mathbf{u}$ and $\mathbf{v}$, respectively. The integral in Eq. (12) was discretized within OpenSees, with $n=10$ being the number of SSPbrick integration points through the thickness, $w_i$ denoting the $\mathbf{w}$ component of a vector from the neutral axis to integration point $i$, andin which ${\mathbf{T}}_i$ = Cauchy stress tensor evaluated at integration point $i$. Positive values of $M_u$ and $M_v$ were selected to reflect tension on the shield, hence $i$ increases along the negative $\mathbf{w}$ ordinate as shown in Fig. 12(b).

As only the left half of the deployed hypar (extending along the negative $\mathbf{u}$ ordinate) was modeled, the same boundary conditions as those implemented for the validation case in the previous section were adopted. To account for the column, a linear spring with stiffness $k_v^{\prime }=3E{I}_c^{\prime }/h^3$ was implemented at the vertex hinge, to which $I_c^{\prime }$ represents half the second moment of area of the square column cross section [Fig. 12(a)]. This allows for the determination of axial, shear, and moment demands associated with the column due to hydrostatic inundation. Fig. 12(c) illustrates the maximum enveloped bending moment distribution of the panel along both the transverse ($\mathbf{u}$) and longitudinal ($\mathbf{v}$) ordinates determined per Eq. (12). Note that the spatial moment distribution computed via OpenSees was independently verified using the commercial finite element software SAP2000 via Mindlin-Reissner shell approximations. Moments associated with the area immediately adjacent to the vertex node [enclosed by dashed lines in Fig. 12(c)] were ignored because the boundary condition does not account for the tributary area associated with the vertex hinge and will be highly overestimated at this location. The design and detailing of the hinging mechanism will be investigated as part of future research and is beyond the scope of this paper. The maximum bending moment averaged over 1 m (${\overline{M}}_p^{*}$) was subsequently determined to be $14\mathrm{kN}·\mathrm{m}/\mathrm{m}$ from Fig. 12(c). In addition, the axial ($N_c^{*}={\mathit{f}}_v·\hat{\mathbf{z}}$), shear ($V_c^{*}=-{\mathit{f}}_v·\hat{\mathbf{x}}$), and moment ($M_c^{*}=V_c^{*}h$) demands attributed to the column were 1,013 kN, 47 kN, and 141 kN·m, respectively. The small shear relative to the axial demand thus affirms the assumption of a laterally unrestrained vertex hinge for purposes of structural analysis of the hypar shell per Fig. 4.

The nonlinear sectional analysis program for reinforced concrete, Response-2000 (Bentz and Collins 2001), was utilized to determine the theoretical capacities of the hypar shell and supporting column. The layout of reinforcement within the 500-mm square concrete column and a 1-m segment of the 105-mm-thick panel as previously described are illustrated in Figs. 13(a and b), respectively. The trilinear moment-curvature relationship characteristic of reinforced concrete is also shown, which was produced via sectional analyses implemented within Response-2000, including considerations of axial-shear interaction for the column. The yield moment ($M_n$) of the concrete column and hypar shell were subsequently computed to be 494 kN · m and $18.6\mathrm{kN}·\mathrm{m}/\mathrm{m}$, respectively, exceeding their respective moment demands of 141 kN · m and $14\mathrm{kN}·\mathrm{m}/\mathrm{m}$. Hence, the detailing of structural members adopted for the proof-of-concept study proved adequate for the hydrostatic forces exerted on the deployed hypar. The conceptual feasibility of such structures for use as coastal armor against storm surge inundation in terms of a simplified hydrostatic situation is therefore demonstrated. Future studies will involve the geometric optimization of hypar forms for pertinent FSI scenarios as well as hydrodynamic loads.