Curved Sandwich Beams Research Articles

Shear coupling effects of the core in curved sandwich beams

We consider a composite package formed by two curved external Euler-Bernoulli beams, which sandwich an elastic core with negligible bending strength but providing the shear coupling of the external layers. This coupling considerably affects the gross response of the composite structure. There is an extensive literature on straight sandwich beams of this type, but very little attention has been paid to the effects of curvature. Here, an analytical linear elastic model is proposed for beams with arbitrary variable curvature. Equilibrium equations and boundary conditions are obtained through a variational approach. Useful simplifications are possible for the case of moderately curved beams and beams with constant curvature.

The analytical solution for buckling of curved sandwich beams with a transversely flexible core subjected to uniform load

In this paper, linear buckling analysis of a curved sandwich beam with a flexible core is investigated. Derivation of equations for face sheets is accomplished via the classical theory of curved beam, whereas for the flexible core, the elasticity equations in polar coordinates are implemented. Employing the von-Karman type geometrical non-linearity in strain-displacement relations, nonlinear governing equations are resulted. Linear pre-buckling analysis is performed neglecting the rotation effects in pre-buckling state. Stability equations are concluded based on the adjacent equilibrium criterion. Considering the movable simply supported type of boundary conditions, suitable trigonometric solutions are adopted which satisfy the assumed edge conditions. The critical uniform load of the beam is obtained as a closed-form expression. Numerical results cover the effects of various parameters on the critical buckling load of the curved beam. It is shown that, face thickness, core thickness, core module, fiber angle of faces, stacking sequence of faces and openin angle of the beam all affect greatly on the buckling pressure of the beam and its buckled shape.

Bending Analysis of Curved Sandwich Beams with Functionally Graded Core

In this article, bending analysis of curved sandwich beams with transversely and functionally graded (FG) core is studied. The Euler–Bernoulli beam theory is used to model the thin face-sheets and high-order shear theory is used to analyze the core. Equilibrium/field equations, compatibility and boundary conditions are used to derive the set of governing equations. The numerical solution of the governing nonlinear differential equations is based on the series Fourier–Galerkin method. Finally, the effect of geometric properties on radial deflection of core and the effect of core radius and Young's modulus on radial deflection, circumferential displacement, and stresses are investigated.

The effect of temperature variation on the free vibration of a simply supported curved sandwich beam with a flexible core

In this article, the free vibration of a singly curved sandwich beam with a transversely flexible core under various temperature conditions is investigated. The beam is assumed to have a constant width with simply supported end conditions. The core and face sheets are considered to be made of materials with temperature-dependent mechanical properties. In this model, faces of the sandwich beam are treated as thin beams with negligible shear strain and flexural rigidities obeying Bernoulli's assumptions. The analysis is based on the high-order sandwich panel theory. The core is assumed to possess vertical normal and shear stiffness and act as a medium that transfers its inertia load to face sheets. Equations of motion and boundary conditions are derived using Hamilton's principle. The variation of free vibration frequencies and eigenmodes of the beam with temperature variation considering the temperature gradient across the thickness is studied. The effect of geometrical parameters such as the ratios of the length and the thickness to the mean radius of the beam on the vibration response of the beam is investigated. It is found that the free vibration frequency of the beam would decrease when its temperature is increased.

An experimental investigation of free vibration response of curved sandwich beam with face/core debond

The aim of this research was to investigate the curvature and face/core debond influence on the vibration behavior of curved composite sandwich beams made up of carbon/epoxy laminate skins over a polyurethane foam core. Flat and curved sandwich beams with debond were prepared by keeping the arc length of the sandwich beams equal to the length of the flat sandwich beams. Sandwich specimens with three different curvatures and debond between the top and bottom face/core were manufactured. Systematic vibration and static experiments were carried to determine the response and to study the effect of curvature and debond. Flexural stiffness and strength of the sandwich beams were determined by four-point bending tests. The natural frequencies and damping loss factors of the sandwich beams were determined using impulse frequency response technique under free—free boundary conditions. Four-point bending test results indicate that stiffness of the beams decrease with presence of debond. Vibration test results show that face/core debond causes reduction of the natural frequencies whereas damping loss factor values increase with presence of debond. The effects of debond and curvature on damping loss factor is higher than that on natural frequency.

Experimental evaluation of the vibration behavior of flat and curved sandwich composite beams with face/core debond

A systematic experimental study to determine the effect of curvatures and debond on the flexural stiffness and strength of composite sandwich beam structures made from fiber-glass/epoxy laminate skins and polyurethane foam core has been carried out. Flat and curved sandwich beams (with 15°, 30°, 45° and 60° open angle) with debond located at the top (upper) and bottom (lower) face/core interface were manufactured. Flexural stiffness and strength were determined from impulse frequency response technique and four-point bending. Vibration test results indicate that the natural frequencies and apparent stiffness are less affected by the presence of curvature and debond, whereas, the loss factors are significantly affected. Four-point test results indicate a reduction in stiffness due the presence of debonds and curvature, but more predominantly, the results show that the peak load reduces sharply with the presence of upper debond.

Two-dimensional elasticity solutions for temperature-dependent in-plane vibration of FGM circular arches

Abstract Temperature-dependent in-plane vibration of functionally graded (FGM) circular arches based on the two-dimensional theory of elasticity is investigated. An analytical solution using the state space formulation and Fourier series expansion is obtained for a simply supported circular arch. For such functionally graded arches, the state equation has variable coefficients. Because a definite, continuously varying FG model through the thickness is impractical if not impossible, an approximate laminate model is constructed to derive an asymptotic solution through the thickness direction. The transfer relationship between the state vectors at the inner and outer surfaces is ultimately obtained by considering the continuity conditions at the artificial interfaces. The new formulation is validated by comparing some numerical solutions with established results in open literature, such as functionally graded straight beams, curved sandwich beams and laminated thick circular arches. Effective material properties are predicted using the Mori–Tanaka model and taken as temperature-dependent. Effects of the gradient index, temperature and geometric parameters, i.e. the curvature, length-to-thickness ratio, subtended angle, on the vibration frequency are analyzed and discussed.

Characterization of residual stresses in a composite curved sandwich beam

Residual stresses are of great importance in ensuring geometrical stability of sandwich structures. Indeed, creep phenomena occur after manufacturing and can lead to unexpected deformations in sandwich shells. In order to assess these, a first experiment has been performed on the shell to provide an order of magnitude indication of the residual stresses by means of cutting to release the stresses. This approach is then extended to a sandwich beam extracted from the shell. This is completed by the thermo-mechanical characterisation of the skins in order to provide the properties required for modelling. Finally, a three-dimensional finite element model is presented, which includes both thermal and mechanical behaviour. Comparisons are made between the model and a heating experiment on the sandwich beam. The thermo-mechanical response of the beam is found to be particularly sensitive to the thermal properties of the constituents.

Nonlinear analysis of a curved sandwich beam joined with a straight sandwich beam

The buckling behaviour of straight sandwich beams joined with curved sandwich beams loaded in pure bending is investigated using two different models. One is based on a high order sandwich beam theory and the other model is based on finite element analysis. The analyses are applied to a numerical example and the results are compared with experimental results.

Experimental investigation of local bending effects in the vicinity of a junction between a straight sandwich beam and a curved sandwich beam

The junction between a curved and a straight sandwich beam is investigated experimentally using electronic speckle pattern interferometry. This technique facilitates a whole field measurement of the displacements through the thickness of the sandwich beam. The experimental results are compared with results obtained using a high order sandwich theory model. The results generally show good agreement within the accuracy of the measurements, thus indicating that the gross response of the model is predicted accurately by the high order sandwich theory, while the localised bending effects in the vicinity of curvature change in sandwich panels have not been verified experimentally.

Analytical Solutions to Predict Flexural Behavior of Curved Sandwich Beams

This paper is concerned with the analysis of curved sandwich beams with a focus on debonding and buckling/wrinkling of the faces. An elasticity-theory-based approach is used for studying through-thickness tension in curved sandwich beam in the first section. The approach ensures an accurate description of the through-thickness stresses in curved sandwich beam. In the second section, the critical load for instability of a curved beam on an elastic foundation which is correspondent to the skin of sandwich beam, is considered and compared with the result for a flat beam on an elastic foundation. The flexural strength of sandwich beam is then studied with a view to identify debonding and local instability characteristics. The effects of various parameters, such as geometrical configuration, stiffness of the skin and core, on through-thickness tension stress and local instability respectively are also studied.

Curved Sandwich Beams with Face–Core Debond Subjected to Bending Moment

Curved sandwich beams subjected to opening bending moment are studied. Face–core debonds of varying size are introduced at the compressively loaded face sheet and the structural integrity is investigated. Analytical and finite element models are compared in order to identify the governing failure modes of the beams. A simple expression is presented as a tool for getting a quick estimate of the severity of an interface crack in a curved sandwich beam. Five different configurations of beams are tested experimentally in a custom made bending rig.

Free Vibration Analysis of Curved Sandwich Beams

A study of free vibrations of singly curved sandwich beams is presented. The model takes into consideration both radial and circumferential displacements of the beam core with the assumption of linear distribution across the thickness. The faces of the sandwich are treated as thin beams. Linear equations of motion as well as the boundary conditions are derived with the help of calculus of variations. Verification of the model is performed via a number of asymptotic cases. It is shown that there exist four types of eigen modes. Numerical analyses of free vibrations of the simply supported and clamped beams are carried out. Effect of the curvature on the eigen modes and their frequencies is investigated. A coupling coefficient is introduced to study the dynamic coupling of different types of motions in an eigen mode.

Flexural Strength of Curved Sandwich Beams with Face/Core Debond

The influence of a face/core (F/C) debond on the flexural stiffness and strength of curved sandwich beams under opening bending moments has been experimentally examined. The sandwich beams consisted of glass/polyester face sheets over a PVC foam core. Debonds of various sizes were defined by inserting thin non-stick Teflon films at the outer F/C interface in the curved region of the beam prior to processing. The beams were tested in flexure in a specially designed four-point bend test fixture. The load versus displacement and surface strains at the apex of the beam were recorded to examine the response. It was observed that the debonded face sheet lifted off of the core and underwent a bifurcation associated with strain reversal. Beam failure occurred by propagation of the debond. It was observed that the beams suffered marginal reductions in stiffness, but substantial reductions in strength due to the presence of a debond. Linear finite element analysis was performed to examine the initial response of the beams.

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.

Free Vibrations of Curved Sandwich Beams with a Transversely Flexible Core

A high-order model for free vibrations of singly curved sandwich beams is presented. The model takes into account the transverse flexibility of the sandwich core while the faces of the sandwich are treated as thin beams. Linear equations of motion as well as the natural boundary conditions are derived. Verification of the model is performed for straight beams via a number of asymptotic cases. For the straight and curved sandwich beams, it is shown that there exist four eigenmodes. Their nature is clarified, and simple theoretical estimations for the eigenfrequencies are obtained. A numerical analysis of free vibration of the simply supported beams is carried out. Effects of design parameters of the sandwich constituents on the eigenmodes and their appropriate frequencies are investigated. The developed model may find its use in the context of various applications of curved sandwich members in the high-performance vehicles.

Experimental study of a uniformly loaded curved sandwich beam

A sandwich curved beam subjected to a uniform loading is experimentally investigated. Load-deflection and thrust-deflection dependencies are shown to be nonlinear, while load-deformation dependencies for sandwich faces are found to be linear. A technical solution for implementation of the simple support is realized. An actual stiffness of the beam tie is measured experimentally and estimated theoretically.

Bending of curved sandwich beams, a numerical approach

Curved sandwich beams in bending are analysed with the finite element method (FEM) and compared to simple analytical formulae and solutions to Airy's stress function. It is shown that the curvature can have a significant influence on the circumferential stresses in the faces and the radial stress in the core. It is also shown that there is very good correspondence between the exact elasticity solution and numerical results far away from stress concentrations. In design and sizing of curved sandwich members the results show that special considerations have to be taken to the stress concentrations that are present at the interface between the straight and curved parts of sandwich beams with both straight and curved segments. The use of layered sandwich shell elements and layered brick elements on curved sandwich beams in bending is investigated. The results with the layered elements show that poor correlation in stress levels to the analytical results can be expected for low to medium radius of curvature to sandwich thickness ratios.

Bending of curved sandwich beams

Curved sandwich beams in bending are analyzed with analytical elasticity methods and compared to simple analytical formulae and FEM calculations. Solutions to Airy's stress function in polar co-ordinates are used to obtain the stress distributions in the radial and circumferential directions. Plane stress and plane strain solutions are given with isotropic and orthotropic material models. It is shown that the properties of the core can have a significant influence on the circumferential stresses of the faces, but that the radial stress of the core is nearly constant with varying core properties. The radius of curvature can have a large influence on the circumferential stresses of the faces when the radius of curvature to sandwich thickness ratio is small to moderate. It is shown that simple analytical methods can be utilised to calculate the radial stresses in the core and the circumferential stresses of the faces for beams with large to moderate radii of curvature and thin to moderately thick faces.

Prediction of loss factors of curved sandwich beams

In this paper an analytical model for the coupled flexural and longitudinal vibration of a curved sandwich beam system is described. The system consists of a primary beam and a constraining beam with a viscoelastic damping material forming the core. The governing equations of motion for the forced vibration of the system are derived using the energy method and Hamilton's principle. Both shear and thickness deformation in the adhesive layer are included in the analysis. A matrix equation for solving the system resonance frequencies and loss factors is obtained by using the Rayleigh-Ritz method. A parametric study has been conducted to evaluate the effects of curvature, core thickness and adhesive shear modulus on the system resonance frequencies and loss factors. The implications of this parametric study on the damping effectiveness of the system along with some design guidelines are presented in the paper.