Hyperbolic Paraboloidal Shells Research Articles

Parametric studies on the structural response of hyperbolic paraboloidal shell footing with variation in geometrical configurations

Parametric studies on the structural response of hyperbolic paraboloidal shell footing with variation in geometrical configurations

An interior penalty coupling strategy for isogeometric non-conformal Kirchhoff–Love shell patches

AbstractThis work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the $$C^1$$ C 1 requirement of Kirchhoff–Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationship accounts for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the $$L^2$$ L 2 , $$H^1$$ H 1 , and $$H^2$$ H 2 norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method’s potential for real-world industrial applications.

The process for designing and constructing the Sunpu Kaikan

AbstractThis study aims to reveal the difficulty of realizing large hyperbolic paraboloid (HP) shell structures comprising reinforced concrete in earthquake‐prone Japan in the 1950s. The analysis focused on the design and construction of Sunpu Kaikan, focusing on the roles of Kenzo Tange and Yoshikatsu Tsuboi, alongside their architectural laboratory staff. It revealed how the experience gained in the Sunpu Kaikan project contributed to two architectural projects in the 1960s: the Niigata City Gymnasium, with its steel‐frame concrete HP shell, and the Yoyogi National Stadium, with its suspension roof structure.

Quadrilateral scaled boundary spectral shell elements with functionally graded piezoelectric materials

Accurate, efficient and versatile computational methods for functionally graded piezoelectric (FGP) shells are still restricted. This contribution proposes new FGP shell finite elements with superior comprehensive performance. In the formulation, a shell element is treated as a three-dimensional continuum and its middle surface is represented with a quadrilateral spectral element. Along the thickness, the shell geometry is described by scaling the middle surface while the displacements and electric potential are approximated via a consistent quadratic Lagrange interpolation. The electric potential is scaled by a factor λ to avoid numerical instability and the assumed natural strain method is applied to eliminate numerical locking. The piezoelectric material properties are assumed to vary continuously through the shell thickness obeying a simple power-law function. Binomial theorem is utilized to perform analytical integral along the thickness in evaluating the stiffness and mass matrices and nodal load vector. The validity of the developed approach is demonstrated by solving piezoelectric or FGP plate problems with reference solutions. A simply supported FGP square plate, a fully clamped FGP cylindrical shell and a partly clamped FGP hyperbolic paraboloid shell are systematically analyzed. The influence of the power-law index and span-to-thickness ratio on the static and free vibration behaviors of the FGP structures is investigated. The optimal value of λ for general FGP shells is also determined.

The Influence of Curved Profiles on the Vibro-Acoustic Characteristics of Light-Weighted Sandwich Shell: An Analytical Solution

The overall aircraft noise is significantly minimized using curved light weighted sandwich shells as an airframe element. However, there is a lack of studies on how the curvature can influence vibro-acoustic characteristics. To the best of our knowledge, this paper presents an analytical solution that predicts the vibro-acoustic characteristics of simply supported light-weighted sandwich shells with various curved profiles for the first time. Natural frequencies, displacement and velocity responses, sound pressure level (SPL), sound transmission loss (STL), and radiation efficiency are the critical parameters when analyzing vibro-acoustic characteristics of any structure and are presented in this paper. The generalized kinematic equations are established for the sandwich shells, and the governing differential equations are derived using Hamilton’s principle. The solutions are obtained using Navier’s method, and the vibration response is determined by solving the first five unknown mode coefficients. The Rayleigh Integral is used to extract the acoustic responses and is validated. In addition, the effect of face sheet to core thickness ratio [Formula: see text] of curved sandwich shells on vibro-acoustic characteristics is critically examined and presented. It is found that introducing the positive curvature to the sandwich shell reduces the bending moment, thus improves improving the vibro-acoustic characteristics in the lower frequency range. An increase in the [Formula: see text] enhances the shell’s vibro-acoustic characteristics until a threshold value [Formula: see text], beyond which the mass contribution deteriorates the characteristics. For a hyperbolic paraboloid shell [Formula: see text], this effect was observed at a lower-frequency value of 425[Formula: see text]Hz.

Structural analysis of Félix Candela’s hexagonal hyperbolic paraboloidal umbrellas

Félix Candela was a celebrated builder of thin concrete hyperbolic paraboloidal (hypar) shells with a lasting legacy on the rationalist architectural movement of the 20th century. The iconic 4-edged square hypar umbrella became one of Candela most defining works due to its structural efficiency, economical constructability, and striking elegance. Yet, Candela’s propensity for geometric experimentation led to the development of hexagonal 6-edged umbrellas capable of supporting parabolic discontinuities bisecting each tympan into two separate leaves for projects such as the Villahermosa Cathedral. Despite their artistic appearance, the structural performance of such topologies has never been subjected to rigorous analysis. In response, this paper presents a parametric investigation into the behaviour of 6-edged umbrellas, with and without parabolic bisectors, based on the proposed Villahermosa Cathedral. Finite element modelling revealed that hexagonal umbrellas exhibit markedly reduced deflections and stresses relative to equivalent square and triangular geometries, which are inversely related to the rise-to-area ratio. Unlike square umbrellas, critical bending moments are not sensitive to the edge rise for hexagonal 6-leaved variants with a constant projected area. However, 12-leaved hexagonal umbrellas adopting parabolic discontinuities suffer from disruptions in the flow of internal forces which ultimately yield a less efficient design.

A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries

The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells.

Influence of the concrete strength and the type of supports on the stress-strain state of a hyperbolic paraboloid shell footbridge structure

Relevance. This work is the first in a series of publications on the selection of a suitable analytical surface for implementation as a self-supporting structure for a thin shell footbridge. The study on the influence of concrete strength, live load position and support types on the stress-strain state of a hyperbolic paraboloid (hypar) shell is presented. Objective - to define the initial design parameters such as the appropriate concrete strength and the support type that generates the best structural behaviour to perform the subsequent structural design of a thin shell footbridge. Methods. The static finite element analysis was performed for 4 compressive strengths of concrete (28, 40, 80, 120 MPa) which correspond normal, high and ultra-high resistance concrete, 5 different live load arrangements and 3 different support conditions. Results. The shell model with pinned (two-hinged) supports shows the same vertical displacements as the model with fixed supports (hingeless). For the studied shell thickness, in terms of stress behaviour, the model with pinned ends is more efficient. The combination of two-hinged supports with 80 MPa concrete strength shows a better structural performance.

Accurate Results for Free Vibration of Doubly Curved Shallow Shells of Rectangular Planform (Part.2 Thickness effect)

This paper presents a follow-up study of a previous work that deals with the free vibration of moderately thin isotropic shallow shells under general edge conditions. The same semi-analytical method is used in this study for identical shape and degree of curvature in doubly curved geometry, and accurate natural frequencies are tabulated for a wide range of the shell edge conditions. Emphasis is made, however, to present the frequency parameters for the shallow shells with very thin thickness (representative length/shell thickness=100). In numerical experiments, convergence test is made against series terms in the case of very thin shallow shells. Twenty-one sets of frequency parameters are tabulated for three shell shapes (spherical, cylindrical and hyperbolic paraboloidal shells) and two curvature ratios. These two papers (Part.1 and 2) will constitute the accurate standard in the area of shallow shell vibration of rectangular planform and serve for future comparison and practical design purpose.

Nonlinear vibration of multilayer shell-type structural elements with double curvature consisting of CNT patterned layers within different theories

In this article, the nonlinear vibration of moderately thick multilayer shell-type structural elements with double curvature consisting of carbon nanotube (CNT) patterned layers is investigated within different shell theories. The first order shear deformation theory has been generalized on the motion for moderately thick multilayer shell-type structural elements with double curvature consisting of CNT patterned layers for the first time. Then, by applying Galerkin and semi-inverse perturbation methods to motion equations, and the frequency-amplitude relationship is obtained. From these formulas, the expressions for nonlinear frequencies of multilayer spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panels patterned by CNTs within shear deformation and classical shell theories are obtained in special cases. The reliability of obtained results is verified by comparison with other results reported in the literature. The effects of transverse shear strains, volume fraction, sequence and number of nanocomposite layers on nonlinear frequency are discussed in detail.

THE PROCESS FOR DESIGNING AND CONSTRUCTING THE SUNPU KAIKAN

The purpose of this study was to reveal how difficult it was to realize large structures of reinforced concrete hyperbolic paraboloidal (H.P.) Shells in the earthquake-prone Japan of the 1950s. The analysis examined the design and construction of the Sunpu Kaikan, focusing on the roles of Kenzo Tange, Yoshikatsu Tsuboi, and the staff of their architectural laboratories. This study brings out the following four points.

Influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures

In this study, the influences of material gradient and nonlinearity on the forced vibration of orthotropic shell structures under external excitations are investigated for first time. The mathematical model of inhomogeneous orthotropic double-curved shallow shells is built using the Hamilton principle and von Karman-type nonlinearity. The basic equations are reduced to nonlinear ordinary differential equations using the Galerkin procedure. Using the multiscale method, the frequency-amplitude relations of double-curved shallow shells and the nonlinear frequency response of forced vibrations are obtained for first time. The reliability of the obtained expressions is checked by comparison with the literature data. In numerical analysis, the influence of inhomogeneity, orthotropy, nonlinearity, and the external excitation parameter on the frequency of forced vibrations is investigated in detail by performing unique numerical calculations taking into account various profiles of inhomogeneous orthotropic shallow spherical and hyperbolic paraboloidal (or hypar) shells.

An approach to the solution of nonlinear forced vibration problem of structural systems reinforced with advanced materials in the presence of viscous damping

In this study, the nonlinear forced vibration of composite structural systems such as plates, panels and shells reinforced with advanced materials in the presence of linear viscous damping is investigated. Hamilton principle and von Kármán-type nonlinear theory are used to obtain the theoretical model of double-curved shells reinforced by carbon nanotubes (CNTs). The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method. By using the multiscale method, the frequency-amplitude relation and nonlinear forced vibration frequency of structural systems are obtained for the first time. Since double-curved shells can be transformed into other structural systems such as spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panel in special cases, the expressions for nonlinear frequencies can also be used for them. In additional, the backbone curve and the nonlinear frequency/linear frequency ratio are determined as a function of the amplitude in primary resonance for the first time. The results are verified by comparing the reliability and accuracy of the proposed formulation with those in the literature. Finally, a systematic study is aimed at controlling the influence of nonlinearity and types of distribution of CNTs on the frequencies and their quantitative and qualitative variation in the presence of external excitation and viscous damping.

Static and free vibration analysis of doubly-curved functionally graded material shells

Static and free vibration analysis of doubly-curved (cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid) functionally graded material (FGM) shells is presented here using various equivalent single-layer shell theories considering the shear deformation and rotary inertia effects. The kinematics of the various shell theories are presented using a generalized shell theory in which the displacement field is independent of the selection of transverse shear strain function which results in a theoretical unification of various equivalent single-layer shell theories. The present generalized formulation yields traction free boundary conditions on the top and bottom surfaces of the shell using constitutive relations. Equations of motion are derived by using Hamilton’s principle which are further solved analytically using the Navier’s technique of assuming unknown variables in the double trigonometric series. Displacements, stresses, and frequencies are presented for functionally graded cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid shells. The present results are compared with previously published results wherever possible to verify the accuracy and efficiency of the present generalized shell theory. Also, this study contributes many new numerical results which can be useful for future researchers.

Nonlinear Dynamic Behavior of Hyperbolic Paraboloidal Shells Reinforced by Carbon Nanotubes with Various Distributions

The analytical solution of the nonlinear dynamic behavior of CNT-based hyperbolic paraboloidal shallow shells (HYPARSSs) with various distribution shapes is presented. A theoretical model was created for HYPARSSs reinforced with CNTs using the von Karman -type nonlinearity. Then the nonlinear basic equations are reduced to ordinary nonlinear differential equations using Galerkin methods and the correlation for frequency-amplitude relationship is obtained using the Grigolyuk method. In addition, the nonlinear frequency/linear frequency (NL/L) ratio is determined as a function of amplitude. Comparisons with reliable results in the literature were made to test the accuracy of the formulas. Finally, a systematic investigate is performed to control the influences of CNTs in the matrix, CNT distribution types and nonlinearity on the vibration frequency-amplitude relationship.

A third-order polynomial for the free vibration response of 3D braided curved panels using various boundary conditions

3D braided composite is gaining popularity over the traditional composite because it has a higher impact resistance, fatigue strength, stiffness in the thickness direction, anti-delamination capability, etc. Based on bridging models, the fibers are considered as transversely isotropic and matrix isotropic. The equivalent material properties of the composites are computed using a volume averaging method (VAM). The four steps 1 × 1 braided technique are introduced to manufacture the 3D braided singly and doubly curved shell panels. The bridging model claimed to be more accurate in predicting mechanical properties and shear modulus in comparison with the other method. In the present study, the free vibration response of 3D braided curved panels is obtained using finite element methods (FEM). A third-order shear deformation (TSDT) with twelve degrees of freedom per node is for the first time employed in 3D braided panels using eight nodded isoparametric formulation. In this theory, the transverse displacement is the function of the thickness coordinate. In reality, the normal stress of the thick shells in the thickness direction can't be neglected. This theory having the capability of predicts the normal stress in the thickness direction. Hence, to predict the accurate results of the 3D braided shell, it has more capability compared to the other shell theory. The free vibration results of the shell panels' viz. cylindrical shells (CYS), elliptical paraboloid shells (EPS), and hyperbolic paraboloid shells (HPS) are computed at a different braided angle, braided volume fraction, aspect ratio, thickness ratio, curvature ratio, and boundary conditions, etc. HIGHLIGHTS The equivalent material properties of the 3D braided composites are calculated using bridging models based on a volume averaging method (VAM). The bridging model claimed to be more accurate in predicting mechanical properties and shear modulus in comparison with the other method. A third-order shear deformation (TSDT) with twelve degrees of freedom per node is for the first time employed in 3D braided panels using eight nodded isoparametric formulation. In this theory, the transverse displacement is the function of the thickness coordinate. To predict the 3D braided shell's accurate results, it has more capability than the other shell theory. The free vibration results of the shell panels' viz. cylindrical shells (CYS), elliptical paraboloid shells (EPS), and hyperbolic paraboloid shells (HPS) are computed at a different braided angle, braided volume fraction, aspect ratio, thickness ratio, curvature ratio, and boundary conditions, etc.

Parametric Modeling of Depth-Limited Wave Spectra under Hurricane Conditions with Applications to Kinetic Umbrellas against Storm Surge Inundation

This paper presents a simplified parametric model for the estimation of depth-limited hurricane wave spectra, accounting for swell and wind-sea components, for coastal engineering applications. The model was evaluated against observations obtained from three shallow water sites in Florida during Hurricane David in September 1979. It was revealed that the parametric approach increases in accuracy with decreasing distance to the storm center and generally provides a conservative representation of the significant wave height, albeit overestimating the peak wave frequency. The model was subsequently adopted to evaluate the performance of tilted hyperbolic paraboloidal (hypar) shells (referred to as “kinetic umbrellas”) as an adaptable alternative to conventional floodwalls via smoothed particle hydrodynamics (SPH). The introduction of hypar geometry proved superior to conventional sloped barriers in reducing overtopping waves but decreases in effectiveness at levels of inundation greater than two-thirds the deployed height. Furthermore, umbrellas exhibiting larger geometrical warping were more capable at suppressing overtopping but must sustain larger base shear forces when subjected to irregular waves consistent with landfalling hurricanes.

Navier type exact analytical solutions for vibrations of thin-walled shallow shells with rectangular planform

Exact analytical solutions for the free vibrations of Thin-Walled shallow shells are presented in this work. The shells considered have rectangular planform and two principal constant curvatures. Past treatment of this problem included only one possible case of boundary conditions, and was limited to symmetric modes only. The new solutions address six different cases of in-plane and out-of-plane boundary conditions, and provide an interesting insight into the dynamics of such shells. The formulation may be applied to cylindrical, spherical, elliptical paraboloidal, and hyperbolic paraboloidal shallow shells. Previous works in this field have not addressed the present boundary conditions in exact analytical form.

Vibration analysis of functionally graded stiffened shallow shells under thermo-mechanical loading

This paper presents a mathematical model to analyse free vibration characteristics of eccentrically stiffened functionally graded shallow shells taking into account the thermo-mechanical loads. The functionally graded shallow shells are simply-supported at all edges and are reinforced by transversal and longitudinal stiffeners on the internal or external surface. The formulation is based on the first-order shear deformation theory with the smeared stiffeners technique for the overall behavior of shells. The equations of motion for eccentrically-stiffened functionally graded shallow shells are obtained using the Galerkin method. The spherical, cylindrical, hyperbolic-paraboloidal shells and plate are modeled and analyzed as special cases by setting components of curvature. Results of the dynamic analysis show that if the number of stiffeners is increased natural frequencies also increases for all the shells but after 10 stiffeners increment is no significant. It is also found that the tendencies of the plate and hyperbolic-paraboloidal shells are similar to each other with the increase in the number of stiffeners and volume fraction index. The effect of volume fraction index is more discernible for spherical shell and least for hyperbolic-paraboloidal.

Studies on Reinforced Concrete Inverted Umbrella Hyperbolic Paraboloid Shells

Hyperbolic paraboloid shell roofs have been extensively used in construction because of impressive appearance and economy in cost. For wide spans, the cost of form work is generally high. By reusing this form work we can reduce the cost of construction. These shells require special care in laying reinforcement, concrete pouring and compaction and hence the labour required for this type of construction is highly skilled. In this study, the design of inverted umbrella type hyperbolic paraboloid shells have been done using IS 2210 - 1988 code and IS 456-2000 code. Concrete of grade M30 and Steel of grade Fe500 are used throughout the investigation. The central dip is taken as variable and it will be taken as a/3, a/4 and a/5. In this study, 21 inverted umbrella type reinforced concrete hyperbolic paraboloid shells have been designed and the estimation of cost has been done. The effect of central dip on the cost per sq.m covered investigated for the inverted umbrella shells.