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.     

The vibration and post-buckling of geometrically imperfect,simply supported,rectangular plates under uni-axial loading,part II: Experimental investigation

The paper describes the experimental part of a theoretical and experimental study of the post-buckling and free vibrational behaviour of thin, rectangular, simply supported plates having initial geometrical imperfection and subject to uni-axially applied, in-plane, compressive loads. The experimental apparatus and procedure used are described. The fundamental natural frequency and central deflection of several plates of different thickness and degree of initial imperfection, subject to loads varying from zero to several times the critical buckling value, are compared with values predicted by using the Rayleigh-Ritz solution described in the companion paper. For one plate, comparisons of theoretically predicted and experimentally measured strains are given. Close agreement is shown to exist between the theoretical and experimental results. An approximate linear relationship between a load-frequency parameter and the central deflection, discussed in the theoretical study, is also shown to exist for the experimental plates.     

The vibration and post-buckling of geometrically imperfect,simply supported,rectangular plates under uni-axial loading,part I: Theoretical approach

The work described in this paper constitutes the theoretical part of a theoretical and experimental study of the post-buckling and vibration of simply supported rectangular plates having slight initial curvature (geometrical imperfection) and subject to uni-axially applied, in-plane, compressive loads. The experimental part, and the comparison with theoretical predictions, is given in a second paper. The Rayleigh-Ritz approach, with a deflection function formulation for both the in- and out-of-plane behaviour of the plates, is used since this permits the convenient modelling of various types of in-plane boundary conditions, including those encountered in the experimental study. A concept of connection coefficients, introduced to reduce the computational effort involved, is described. In order to illustrate the applicability of the theoretical approach, a number of square plates having various sets of in-plane boundary conditions and degrees of initial imperfection are treated. Graphical results are presented showing the variation of the lateral central deflection and the fundamental natural frequency of vibration with applied in-plate loads varying from zero to several times the lowest critical buckling load. Where possible, comparison is made with values available in the literature and excellent agreement is achieved. The results presented appear to suggest that an approximately linear relationship exists between a load-frequency parameter and the central deflection of the plates considered, for a substantial in-plane loading range.     

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.     

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.