Improvements of the smearing technique for cross-stiffened thin rectangular plates

New developments in the simplified smearing technique for modeling vibrations of cross-stiffened, thin rectangular plates are presented. The computationally efficient smearing technique has been known for many years, but so far the accuracy of, say, predicted natural frequencies has been inadequate. The reason is that only the stiffeners at a right angle to the axis of angular motion are taken into account when calculating the bending stiffness, whereas the stiffeners that are parallel to this axis of angular motion are neglected. To improve predictions, the parallel stiffeners are taken into account in this paper. The improved smearing technique results in better accuracy for predicted natural frequencies of flat stiffened plates, as demonstrated for both simply supported and clamped boundary conditions. The improved prediction accuracy is demonstrated by comparing results from a numerical model based on the current development with results from finite element (FE) simulations that include the exact cross-sectional geometries of the stiffened panel. In order to demonstrate applications of the improved smearing technique, the predicted forced response is compared with both experimental and FE results. Another improvement concerns the orientation of the stiffeners. The original smearing technique presupposes that the stiffeners are parallel to the edges of the plate, but simple considerations make it possible to relax this requirement. To test the validity of the resulting technique a series of plates are examined for stiffeners angled relative to the plate edges.     

Free vibration analysis of doubly curved shallow shells using the Superposition-Galerkin method

This paper demonstrates the applicability of the Superposition Method for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of simply supported and clamped edges. The same building block yields the natural frequencies for 55 combinations of edge conditions. The natural frequency parameters of the shells were obtained using the Superposition-Galerkin Method (SGM) for seven sets of boundary conditions, several different curvature ratios and two aspect ratios. The SGM uses approximate steady state solutions as building blocks but the method proves to be accurate and efficient. It has also been shown that even with approximate building blocks, the monotonic nature of convergence of the natural frequencies with respect to the number of driving coefficients holds, as long as the number of admissible functions in the steady state solution is kept constant. The results for natural frequencies of the seven boundary conditions may be considered as benchmarks.     

Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core

In this paper, the Ritz minimum energy method, based on the use of the Principle of Virtual Displacements (PVD), is combined with refined Equivalent Single Layer (ESL) and Zig Zag (ZZ) shell models hierarchically generated by exploiting the use of Carrera's Unified Formulation (CUF), in order to engender the Hierarchical Trigonometric Ritz Formulation (HTRF). The HTRF is then employed to carry out the free vibration analysis of doubly curved shallow and deep functionally graded material (FGM) shells. The PVD is further used in conjunction with the Gauss theorem to derive the governing differential equations and related natural boundary conditions. Donnell–Mushtari's shallow shell-type equations are given as a particular case. Doubly curved FGM shells and doubly curved sandwich shells made up of isotropic face sheets and FGM core are investigated. The proposed shell models are widely assessed by comparison with the literature results. Two benchmarks are provided and the effects of significant parameters such as stacking sequence, boundary conditions, length-to-thickness ratio, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements are discussed.     

Flexural response of doubly curved laminated composite shells

In the present work, analytical solutions for laminated composite doubly curved panels on rectangular plan form undergoing small deformations and subjected to uniformly distributed transverse load have been obtained. The problem is formulated using first order shear deformation theory. The spatial descretization of the linear differential equations is carried out using fast converging finite double Chebyshev series. The effect of panel thickness, curvature, boundary conditions, lamination scheme as well as material property on the static response of panel has been investigated in detail.     

Nonlinear vibrations of functionally graded doubly curved shallow shells

Nonlinear forced vibrations of FGM doubly curved shallow shells with a rectangular base are investigated. Donnell’s nonlinear shallow-shell theory is used and the shell is assumed to be simply supported with movable edges. The equations of motion are reduced using the Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Using the multiple scales method, primary and subharmonic resonance responses of FGM shells are fully discussed and the effect of volume fraction exponent on the internal resonance conditions, softening/hardening behavior and bifurcations of the shallow shell when the excitation frequency is (i) near the fundamental frequency and (ii) near two times the fundamental frequency is shown. Moreover, using a code based on arclength continuation method, a bifurcation analysis is carried out for a special case with two-to-one internal resonance between the first and second doubly symmetric modes with respect to the panel’s center (ω₁₃≈2ω₁₁). Bifurcation diagrams and Poincaré maps are obtained through direct time integration of the equations of motion and chaotic regions are shown by calculating Lyapunov exponents and Lyapunov dimension.     

Vibration of doubly curved shallow shells with arbitrary boundaries

The first comprehensive study of shallow shell vibrations subjected to as many as 21 possible boundary conditions is presented. Thin shallow shell theory is used. Relatively accurate results for natural frequencies of doubly-curved shallow shells have been obtained. These can be used for benchmarking by researchers as well as reference data for practicing engineers. The Ritz method is used to solve for natural vibrations of these shells with arbitrary boundary conditions. Natural frequencies are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells.     

Vibration of simply supported cylindrical shells with longitudinal stiffeners

The vibration of simply supported cylindrical shells stiffened by discrete longitudinal stiffeners is investigated by using an energy method. Vlasov's thin walled beam theory is used for stringers. Shell theories based on Donnell's approximate theory and Flügge's more exact theory are used for the skin and numerical results indicate that Donnell's approximate theory gives excellent results for the stiffened shells. Sinusoidal wave form is considered in the longitudinal direction, and mode shapes in the circumferential direction are represented by Fourier series. Numerical results on frequencies and mode shapes computed for a shell stiffened by various number of stiffeners are presented and compared favorably with existing experimental results and other analytical solutions.     

Solutions of the Lévy type for the free vibration analysis of diagonally supported rectangular plates

A Lévy type solution is developed for the vibratory response of a simply supported rectangular plate subjected to a harmonic force distributed along the diagonal. The solution is then extended to determine the free vibration response of the same rectangular plate with inelastic lateral support on the diagonal. It is found that there is an excellent agreement between computed eigenvalues obtained here and those obtained by the author in an earlier paper in which a Navier type solution was utilized. The significant advantages inherent in the present Lévy type solution are discussed.     

Flutter analysis of periodically supported curved panels

This paper presents the one-dimensional axial wave propagation in an infinitely long periodically supported cylindrically curved panel subjected to supersonic airflow. The aerodynamic forces are based on piston theory. For this study the structure is considered as an assemblage of a number of identical cylindrically curved panels each of which will be referred to as a periodic element. A high precision triangular finite element with certain wave boundary conditions (Floquet's principle) is introduced in flutter problems of the proposed structure for the first time. The airflow is assumed in the direction of the straight edges of the panel. It is assumed that the deflection function accounts for a phase lag term only and does not consider any attenuation terms. Aerodynamic damping has been neglected for brevity. For a given geometry a three-dimensional plot related to the phase constant, flutter frequency and pressure parameter has been obtained corresponding to the optimum periodic angle. The “flutter line”(line of instability) has been identified. The limiting values of flutter frequencies and pressure parameters of the “flutter line” are compared with the critical flutter condition of a single curved panel, using two methods—an exact approach and a finite element method. The critical flutter results for multi-supported (1-span, 2-span and 3-span) curved panels are obtained using the band discretization principle.     

Natural frequencies of rectangular orthotropic plates with a pair of parallel edges simply supported

Solutions of the exact characteristic equations for the title problem derived earlier by an extension of Bolotin's asymptotic method are considered. These solutions, which correspond to flexural modes with frequency factor, R, greater than unity, are expressed in convenient forms for all combinations of clamped, simply supported and free conditions at the remaining pair of parallel edges. As in the case of uniform beams, the eigenvalues in the CC case are found to be equal to those of elastic modes in the FF case provided that the Kirchoff's shear condition at a free edge is replaced by the condition $\text{?M}{}_{\mathrm{n}}\text{?n = 0}$. The flexural modes with frequency factor less than unity are also investigated in detail by introducing a suitable modification in the procedure. When Poisson's ratios are not zero, it is shown that the frequency factor corresponding to the first symmetric mode in the free-free case is less than unity for all values of side ratio and rigidity ratios. In the case of one edge clamped and the other free it is found that modes with frequency factor less than unity exist for certain dimensions of the plate—a fact hitherto unrecognized in the literature.     

Free vibrations of a simply supported,rectangular plate: An exact elasticity solution

An exact, three-dimensional solution for the free vibrations of simply supported, rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper. The solution, obtained in a semi-inverse fashion as was the solution of the elastostatic problem for such plates, satisfies all of the boundary conditions of the problem in a pointwise manner. It is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi-inverse solution. Other modes of oscillation may exist in the three-dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes. The two types of modes found are analogous to the flexural modes of classical plate theory and the thickness-twist modes, here called breathing modes, of Mindlin plate theory. Some numerical results are given which indicate that the predictions of Mindlin plates are uncannily good approximations to the flexural frequencies given by the present, three-dimensional analysis even for very thick plates. However, the predictions of Mindlin plate theory for the thickness-twist, or breathing, frequencies are not nearly so good. These discrepancies are discussed briefly in an appendix.     

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.     

Free vibration analysis of continuous rectangular plates

Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.