Simply-supported Shell Research Articles

Exact solutions for clamped spherical and cylindrical panels via a unified formulation and boundary discontinuous method

Numerous works in both recent and historical literature have concentrated on formulating theories to perform static analysis on simply-supported shell structures. However, it is worth noting that obtaining analytical solutions for clamped boundary conditions presents a strong challenge. In this paper, closed-form solutions for clamped cross-ply laminated and sandwich shells are achieved by employing a robust and hybrid methodology not previously reported in the literature. The high versatility of the Carrera Unified Formulation (CUF), based on the Equivalent-Single-Layer (ESL) description, is utilized to implement several refined shell theories. The Principle of Virtual Displacements (PVD) is utilized to derive the strong form of the governing equations in terms of displacement variables. As the main novelty, these equations are solved by the Boundary Discontinuous Fourier-based method (BDM) which provides highly accurate analytical solutions. The validity and robustness of the proposed methodology are assessed through a detailed comparison with references available in the open literature, as well as with FEM 3D results obtained with commercial software. Furthermore, the stress recovery technique is exploited to fulfill zero-stress and interlaminar continuity (IC) conditions. The findings might be useful in training artificial intelligence (AI) models, which, for instance, could facilitate the development of digital twin structures.

Analytical analysis of sound transmission in passive damped multilayered shells

An analytical shell model is here proposed for the sound transmission analysis of composite multilayered structures embedding viscoelastic sheets. The present analytical solution, based on the Navier approach, is developed starting from the Principle of Virtual Displacements, through the use of advanced layer-wise higher-order models and by considering real viscoelastic frequency-dependent materials with fractional derivative constitutive models. Several simply-supported shell structures with isotropic and cross-ply composite layers embedding viscoelastic frequency-dependent sheets are considered and the results of the passive noise reduction analysis are given in terms of acoustic parameters based on Rayleigh integral method. Moreover a comprehensive study on different types of time loads is taken into account, considering impulse time load, time-window load histories, space–time dependent loads.

Dynamic analysis of a laminated cylindrical shell with piezoelectric layer and clamped boundary condition

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to exponential dynamic load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity partial differential equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in axial direction. The resulting equations are reduced to a system of ODE with constant coefficients by means of Galerkin׳s integral method and finally the governing equations are solved by using linear finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The ratio effect of the radius to thickness on dynamic behaviour is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented.

Elasticity Solution of a Laminated Clamped Cylindrical Shell with Piezoelectric Layer under Dynamic Load

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to impulse load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity PDE equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in the axial direction. The resulting equations are transferred into state space form and reduced to an eigenvalue problem by using Galerkin's finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The radius to thickness ratio effect on dynamic behavior is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented and natural frequencies are compared with simply-supported shell results.

Cylindrical thin-shell model based on modified strain gradient theory

In this paper, the cylindrical thin-shell model is developed based on modified strain gradient theory. For this purpose, the study develops the thin shell theory, having considered size effects through modified strain gradient theory. Besides, partial equations of shell motion with classical and non-classical corresponding boundary conditions are derived from Hamilton principle. Finally, by way of example, the free vibration of the single-walled carbon nanotube (SWCNT) is investigated. The study models the SWCNT as a simply-supported shell. Besides, the Navier procedure is used to solve the vibration problem. The results of the new model are compared with those of the couple stress model and the classical theory, leading to the conclusion that the mentioned models are special cases of the modified strain gradient theory. The findings also indicate that the rigidity of the nanoshell in the modified strain gradient theory is greater than that in couple stress model and the classical theory, which leads to the increase in natural frequencies. Furthermore, the effect of the material length scale parameter on the vibration of the nanoshell for different lengths is taken into account.

Locating damage in circular cylindrical composite shells based on frequency sensitivities and mode shapes

Abstract This paper presents a feasibility study on locating damage in circular cylindrical fiber-reinforced composite shells based on natural frequencies and mode shape information at specific locations. The axial position of the damage is located by comparing the measured and the predicted frequency changes due to introducing damage. The theoretical frequency changes are approximated by frequency sensitivities. The advantage is that a damaged shell model is not required, as frequency sensitivities are readily obtained from a defect-free finite element model. As the shells are axisymmetric, additional information on mode shape is employed to locate the exact damage position. Results from simulations showed that the proposed detection scheme can easily locate damage in free–free and simply-supported shells. For a clamped-free shell, a successful detection scheme is secured by excluding those vibration modes with a highly nonlinear response to the severity of damage. The robustness of the proposed method in the presence of both modelling and measurement errors is demonstrated by simulation.

341 FRP 積層円筒殻の振動における最適積層形態について

This paper studies on the optimal stacking sequence for vibration of FRP laminated circular cylindrical shells. For this purpose, the frequency equation for a simply-supported shell is derived based on Flugge-type classical lamination theory, and then two types of optimization problems for the shell are formulated by employing continuous design variables and by discrete design variables, respectively. In optimization, two types of genetic algorithms are also employed to solve these optimization problems. Optimal design solutions are obtained for the laminated shells with different stacking sequence from numerical calculations, and the maximization for fundamental frequency of the shell is also studied by comparing with numerical results obtained.

Numerical fluid loading coefficients for the modal velocities of a cylindrical shell

The generalized fluid loading coefficients for the modal velocities of a simply-supported shell section are defined and formulated. The simplified nature of infinite cylindrical coordinates is employed in the geometry of interaction by assuming the predominance radial effects in a baffled extension of the finite shell. An efficient numerical procedure for the computational evaluation of the integrals which define the direct and cross mode components of the fluid impedance is presented and applied. The approach and illustrated results are directly applicable in the combined solution of shell and fluid interaction problems.

Free Vibration of Hyperbolic Cooling Towers

A modified finite difference technique is used to determine the natural frequencies and mode shapes of hyperbolic cooling tower shells. The influence of the meridional curvature and the boundary conditions on the vibration characteristics of the tower is investigated. In all cases, changes in frequency are found to be essentially due to changes in membrane energy. It is shown that, for a fixed-free shell, the increased meridional curvature leads to an increase in the natural frequency. The lack of axial restraint results in a large reduction in the membrane energy and consequently the natural frequency. For simply-supported shells, a critical meridional curvature at which the membrane energy effectively vanishes is shown to exist.

On the non-symmetric vibrations of thin cylindrical shells with clamped-clamped edges

This paper presents another approach toward the analytical investigation of the free vibrations of thin cylindrical shells. Using shallow shell equations, clamped-clamped cylinders are simulated by simply-supported shells with harmonic moments applied along both edges. By setting slopes at these edges to zero, the characteristic frequency equations are obtained. The natural frequencies are then searched between every two consecutive frequencies of the corresponding simply-supported shell. A comparison is made of the frequencies obtained according to the present technique of solution to those based on other methods of solution and experimental results. Present results are in good agreement with the experiments in the over-all frequency range.