Plates In Cylindrical Bending Research Articles

Correction of local deformations in free vibration analysis of ship deck structures by equivalent single layer elements

ABSTRACTEquivalent single layer (ESL) elements provide an easy and computationally effective way to model stiffened plates in finite element analysis of ship structures. Secondary stiffeners are incorporated into the plate or shell formulation. In the free vibration analysis, these elements ignore inertia induced local deformation of plating between the secondary stiffeners. Oscillating motion causes inertia induced body load that locally deforms the plate. This local deformation may have a significant effect on the global modal frequencies of a deck structure. This paper presents a method for correcting ESL modal frequencies by modifying generalised mass and stiffness of the modes. The modification is based on the kinetic and strain energies of the local deformations. Energy components are derived from local consideration of plate in cylindrical bending under enforced support vibration. The method is validated in a case study of ship deck structure against shell mesh results, and good agreement is found.

Exact analysis of imperfect angle-ply laminated panels with surface-bonded piezoelectric layers

The electromechanical behavior of an angle-ply laminated plate in cylindrical bending with surface-bonded piezoelectric layers is investigated. The interlaminar bonding of this smart laminate is described by a Kelvin–Voigt viscoelastic model. Besides, a matrix reduction method is employed to construct the transfer relations between the piezoelectric layer and the elastic strip. Based on the state-space approach, a state differential equation of the interfacial sliding displacements with respect to time variable is derived, from which the exact solutions are obtained in the time domain. Comparison with the existing techniques validates the high efficiency and excellent accuracy of the present analysis. Furthermore, the numerical results indicate that the intelligent laminate may lose partly or completely the load-bearing capacity and the function of sensor/actuators with time elapsing because of the viscoelastic interfaces.

Transient responses of laminated anisotropic piezothermoelastic plates and cylindrical shells with interfacial diffusion and sliding in cylindrical bending

With aid of the Stroh-type formalism and the state-space approach, a simple and elegant method is presented to obtain an exact solution for the time-dependent problem of a simply-supported laminated anisotropic piezothermoelastic plate in cylindrical bending with interfacial diffusion and sliding. The Stroh-type formalism is used to obtain the general solution of the thermoelectroelastic field in a certain layer. The state-space equation is then constructed for a laminated plate. The relaxation times and the transient thermoelectroelastic response of the laminated plate can be determined by solving the state-space equation via an analysis of a generalized eigenvalue problem. By using a similar method, we also derive an exact solution for a simply-supported laminated anisotropic piezothermoelastic cylindrical shell with interfacial diffusion and sliding under cylindrical bending.

Static and dynamic analysis of plates in cylindrical bending through displacement and mixed approaches

ABSTRACTThis work analyzes the influence of edge-boundary conditions on the static and dynamic behavior of isotropic and cross-ply plates in cylindrical bending conditions. The main concern of the analysis is addressed to dynamic problems (free vibrations) along with certain boundary conditions that seem to be mainly responsible for a low convergence of the eigenvalues to the exact values. The interest on the static behavior comes from the inherent sensitivity of related quantities, which can both be compared with previous studies and can better elucidate such a low convergence. The analysis is carried out within the frame of the three-dimensional theory by using different variational formulations based both on assumed displacements and/or stresses along with the attempt to unify the comparison of the different formulations. The numerical results/performance of such formulations are also compared to those achieved through alternative models.

On perturbation method in mechanical, thermal and thermo‐mechanical loadings of plates: cylindrical bending of FG plates

AbstractThe performance of perturbation method in nonlinear analyses of plates subjected to mechanical, thermal, and thermo‐mechanical loadings is investigated. To this end, cylindrical bending of FG plates with clamped and simply‐supported edges is considered. The governing equations of Mindlin's first‐order shear deformation theory with von Kármán's geometric nonlinearity are solved using one‐ and two‐parameter perturbation methods and the results are compared with the results of an analytical solution. The material properties are assumed to vary continuously through the thickness of the plate according to a power‐law distribution of the volume fraction of the constituents. It is shown that the accuracy of any‐order expansion in perturbation method depends not only on the perturbation parameter, but also on the location chosen for the perturbation parameter and, in general, the solution becomes more accurate when the perturbation parameter is specified at the location where its corresponding response quantity is a maximum. Under thermal loading the possibility of using different parameters as the perturbation parameter for various boundary conditions is investigated. It is observed that, instead of a one‐parameter perturbation method, a two‐parameter perturbation method must be used in the thermal analysis of FG plates. Also, buckling and post‐buckling behavior of FG plates in cylindrical bending is investigated. It is shown that under thermal loading, a bifurcation‐type buckling occurs in clamped FG plates. In addition, a snap‐through buckling may occur in simply‐supported FG plates under thermo‐mechanical loading.

Assessment and correction of theories for multilayered plates with imperfect interfaces

The response of multilayered plates with imperfectly bonded interfaces and delaminations to quasi static and dynamic loadings is conveniently studied by assuming the displacements as two length-scales fields, given by global displacements, which are continuous in the thickness, and local perturbations or enrichments, which account for the multilayered structure and the jumps at the interfaces. The a priori imposition of interfacial continuity conditions then leads to a homogenized field defined by the global variables only. The number of displacement unknowns is independent of the number of layers and is equal to five or six, for sliding or mixed mode interfaces, respectively. Dynamic equilibrium equations are derived using variational principles. This effective idea, which was originally proposed in the zigzag theories for fully bonded plates, was first applied to plates with imperfect interfaces by Cheng et al. (J Appl Mech, 63:1019–1026, 1996), Schmidt and Librescu (Nova J Math Game Theory Algebra, 5:131–147, 1996), and Di Sciuva (AIAA J, 35(11):1753–1759, 1997). These pioneering models, as all subsequent models which extend the theories to various problems, suffer from a common omission in the treatment of the interfacial energy in the derivation of the equilibrium equations, which leads to solutions which are accurate only for fully bonded/debonded plates. In this paper a linear, first order model which resolves the inaccuracy and extends the theories to more general interfacial traction laws is presented along with the corrected formulations of the original models. The formulations are validated using exact 2D elasticity solutions for multilayered plates in cylindrical bending.

Extended Kantorovich method for coupled piezoelasticity solution of piezolaminated plates showing edge effects

The powerful extended Kantorovich method (EKM) originally proposed by Kerr in 1968 is generalized to obtain a three-dimensional coupled piezoelasticity solution of smart piezoelectric laminated plates in cylindrical bending. Such solutions are needed to accurately predict the edge effects in these laminates under electromechanical loading. The Reissner-type mixed variational principle extended to piezoelasticity is used to develop the governing equations in terms of displacements, electric potential as well as stresses and electric displacements. It allows for exact satisfaction of the boundary conditions, including the non-homogeneous ones at all points. An n -term solution generates a set of 11 n algebraic ordinary differential equations in the inplane direction and a similar set in the thickness direction for each lamina, which are solved in closed form. The multi-term EKM is shown to predict the coupled electromechanical response, including the edge effects, of single-layer piezoelectric sensors as well as hybrid laminated panels accurately, for both pressure and electric potential loadings. This work will facilitate development of accurate semi-analytical solutions of many other unresolved problems in three-dimensional piezoelasticity, such as the free-edge stresses in hybrid laminates under bending, tension and twisting.

Analytical Solution of a Functionally Graded Plate in Cylindrical Bending

The elastic problem of a functionally graded plate in cylindrical bending is studied in this paper. The plate is assumed to be simply supported at lateral edges and subjected to normal and shear tractions on top and bottom surfaces. A general solution procedure is developed using Fourier series expansion and variables separation method. Analytical solutions are obtained for some specific variations of material modulus. The influence of different functionally graded models and plate configurations on the stress and displacement fields is investigated through numerical examples.

Assessment of third order smeared and zigzag theories for buckling and vibration of flat angle-ply hybrid piezoelectric panels

A recently developed improved third order theory (ITOT) for angle-ply hybrid piezoelectric plates in cylindrical bending is extended to include geometric non-linearity in the Von Karman sense. The transverse deflection is approximated non-uniformly to explicitly account for the transverse strain due to temperature and electric potential. The coupled non-linear equations of motion and the boundary conditions are derived using the extended Hamilton’s principle. The non-linear theory is used to obtain the buckling and free vibration response of symmetrically laminated hybrid angle-ply panels under inplane electro-thermomechanical loading. This theory and the third order zigzag theory with additional layerwise terms for inplane displacements are assessed in direct comparison with the exact 2D piezothermoelasticity solutions for forced harmonic response, buckling and free vibration response under initial inplane electro-thermomechanical loading. The comparison establishes the accuracy of the results of the zigzag theory and its superiority over the ITOT for the dynamic and buckling response of angle-ply hybrid panels.

Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.

Non-linear analysis of functionally graded plates in cylindrical bending under thermomechanical loadings based on a layerwise theory

Abstract A layerwise theory is used to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The non-linear strain–displacement relations in the von Karman sense are used to study the effect of geometric non-linearity. The equilibrium equations are solved exactly and also by using a perturbation technique. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.

Large deflection of thin plates in cylindrical bending – Non-unique solutions

Considered in this paper is the large deflection of a thin beam. One end of the beam is fixed (clamped) to a rigid wall, while the other end is placed on a flat surface of arbitrary orientation. It is shown that under certain conditions, the solution to the deflected shape of the plate is not unique. Conditions for the existence of multiple solutions are identified. Numerical methodologies are developed to obtain the multiple solutions. Experiments were conducted to verify the numerical predictions. Excellent agreements are found between the predicted deflection and the experimental measurements.

Semi-analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending

Elasticity solutions for free vibration of angle-ply laminates subjected to cylindrical bending are obtained using a newly developed semi-analytical approach. The thickness domain is solved analytically using the transfer matrix method based on the state space concept, while the in-plane domain is solved approximately via the technique of differential quadrature. The present method is applicable to arbitrarily thick laminates and for treating arbitrary edge conditions. The method is verified by comparisons with the exact solutions of Pagano's problem. Effects of variation of ply angle on the vibration properties of laminates are investigated; mode shape switching is observed when ply angle varies. Numerical results for fully clamped thick laminates are presented for future references.

A Computational Procedure for the Determination of Fracture Parameters at Interfacial Contact Cracks in Composite Laminates

This article presents a method to calculate the fracture parameters at interfacial cracks in laminated composites. Depending on the applied loading and crack length, a contact region may develop that changes the nature of the resulting stress intensity factors and energy release rates with respect to the open crack. A combined analytical and numerical model based on frictionless contact is used to eliminate the overlapping of the crack faces where applicable to obtain the correct results. In cases where the contact region is very small, an approximate method to obtain the fracture parameters is derived by consideration of the J-integral. The laminates studied include cross-ply and angle-ply laminates in three-point bending containing a central delamination. The present results are compared with results in the literature obtained by the virtual crack closure integral technique for the special case of cross-ply laminates and a close agreement is found. In addition, results are given for the extent of the contact zone as well as Mode I and II stress intensity factors and energy release rates as a function of the lamination angle. Overall, the method used provides an effective way to analyze interfacial cracks and delaminations in composite plates in cylindrical bending subjected to transverse loads or low velocity impacts.

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.

Benchmark solution of imperfect angle-ply laminated rectangular plates in cylindrical bending with surface piezoelectric layers as actuator and sensor

The static and dynamic responses of simply supported adaptive angle-ply laminates in cylindrical bending are considered. The interlaminar bonding of the host elastic laminate is assumed to be imperfect, described by a spring-layer model, while the bonding between the host elastic laminate and the surface piezoelectric actuator and sensor layers is perfect. The state-space approach, which is directly based on the three-dimensional exact elasticity (piezoelasticity) equations and very effective in analyzing laminated structures, is employed. The numerical results should provide a useful means of comparison in the development of simplified analyses or numerical methods.

Free vibration of laminated piezoelectric composite plates based on an accurate theory

An accurate theory for laminated piezoelectric composite plates in cylindrical bending is developed for free vibration analysis. The displacement and electric potential fields are depicted approximately by the accurate displacement and electric potential distribution functions through thickness, respectively. The two functions are formulated according to particular solutions to the three-dimensional elasticity equilibrium equations and the electrostatics charge equation. The complicated electromechanical coupling relations and the interfacial continuity conditions are enforced. Accordingly the two functions are coupled and make the displacement and potential fields coupled. The governing equations use only four displacement and potential variables, the number of which is independent of the number of layers involved. A corresponding finite element model is also developed. Natural frequencies of piezoelectric laminates subjected to different sets of boundary conditions are given and parameter studies are conducted in numerical examples. The high accuracy of this theory is demonstrated by comparing the present results with the existing exact three-dimensional solutions.

Time-dependent behaviors of angle-ply laminates with viscous interfaces in cylindrical bending

A simply supported angle-ply laminated plate in cylindrical bending with viscous interfaces under static loading is studied. Because of the presence of viscous interfaces, the behavior of the laminate depends on the time variable, which complicates the analysis. A highly accurate analysis is proposed, in which no assumption on deformation is introduced. Concretely, the state-space formulations, which have been used extensively in laminated structural analyses, are employed. The variations of field variables with time are approximated using the power series expansions. Recurrence formulae that make for easier program are constructed. Numerical examples are finally presented and comparison with existing results shows a good agreement. Results also indicate that the behavior of the laminate is very sensitive to the presence of viscous interfaces.

Boley’s method for two-dimensional thermoelastic problems applied to piezoelastic structures

We study a composite piezoelastic plate in cylindrical bending. The plate is composed of perfectly bonded substrate and piezoelastic layers. We assume a plane state of strain; the plate mid-plane deforms into a cylindrical surface perpendicular to, and the electric field vector lies in, the ( x, z)-plane. We utilize Bruno Boley’s method for two-dimensional thermoelastic problems: Boley introduced an expansion of the Airy stress function and a step-by-step solution for each term. Furthermore, he discussed relations to strength-of-material theories. For the piezoelastic problem we apply Boley’s method to the charge equations of electrostatics. The electric potential is expanded and each term is calculated using a step-by-step procedure. Use of Boley’s method is facilitated by the capabilities of modern symbolic computer codes. We solve the problem for an arbitrary distribution of strain first; then we consider the variation of displacements in the form of a third order power series expansion in the z-direction. Strength-of-material theories of different approximation levels are finally extracted, for which the level of approximation for the mechanical and electric field is not independent.

RC beams strengthened with composite material: a limit analysis approach and experimental study

This paper deals with the flexural strengthening of reinforced concrete beams by means of thin carbon fibre reinforced plastic (CFRP) plates. A simplified laminated plate model is used to describe the behaviour of three-layered plate in cylindrical bending which were subjected to third-point line loads. The upper bound theorem of limit analysis is used to approximate the ultimate load capacity for multi-layered plates and identify different collapse mechanisms. A reinforced concrete beam strengthened by CFRP is designed as a three-layered plate. Experimental results are obtained and a comparison with theoretical predictions made.