Elastic plates under bending solved by pseudocaustics

The method pf pseudocaustics was applied to the study of out-of-plane bending in elastic plates. It is shown that for bending problems where the loading mode is given, the method determines experimentally the complex potential function at selected points along the boundaries. A conformal mapping of the closed smooth curves of each boundary of the plate on to a unit circle allows the determination of the complex potential ϕ (ζ), expressed in the form of a Laurent series. This in turn yields the complete solution of the bent plate. In order to show the efficiency of the method it was applied to two typical examples of thin infinite plates in cylindrical bending, having either a circular central hole, or a square hole. The experimental results corroborate the theoretical results, thus proving that this combined theoretical and experimental method is suitable for solving elastic problems in applications with high accuracy, where other methods fail to yield satisfactory results.     

Prevention of thermal bending of multilayered beams and plates

A method is described to prevent bending in multilayered beams and plates of different isotropic materials with uniform and nonuniform temperature distribution through the thickness. The method involved the addition of an extra layer to the multilayered beams or plates. With the proper selection of the thermoelastic properties, the added layer would eliminate the curvature produced prior to this addition. A complete analysis for the determination of the various thermoelastic parameters of the extra layer was made. In addition, to ensure that the multilayered beams and plates actually remained straight, a thermoelastic analysis was performed for the solution of thermal stresses and strains in the laminate. The results gave assurance to the straightness of the laminate since the calculated strains have the same value throughout the thickness. The solutions are valid for any given uniform temperature change and for any given nonlinear temperature distribution through the thickness of the multilayered beams and plates. Several numerical examples are presented that illustrate the application of the method for various temperature distributions. A simple experiment was conducted that showed the validity of the analytical method. A brass strip was added to a bimetalic strip made of aluminum and steel at room temperature. The thickness of the brass strip was calculated from the theory to prevent bending. The trimetal strip was placed in a furnace and, as expected, it remained straight for varying temperatures.     

Elastic bending analysis of bilayered beams by an alternative two-variable method

An alternative two-variable method is used to reanalyze thermoelastic bending problems of bilayered beams subjected to external moments and internal stresses. The differences among zero-stress axis, zero-strain axis (i.e. neutral axis), bending axis, centroidal axis, and the parameter conditions for null/single/dual zero-stress axes are investigated analytically and numerically. Comparisons of thermoelastic stress predictions by the present model with Stoney's model and Hsueh's model are discussed in a representative case of GaAs top coat/Si substrate wafers. Results showed that the neutral axis does not coincide with the zero-stress axis in the general case, and the numbers and the locations of zero-strain or zero-stress axes depend on not only elastic modulus, thickness and/or thermal expansion coefficient ratios between the film and the substrate but also mechanical/thermal loading ratio.     

Elastic interaction of multiple delaminations in plates subject to cylindrical bending

This paper deals with the elastic interaction of multiple through-width delaminations in laminated plates subject to static out of plane loading and deforming in cylindrical bending. A model has been formulated utilizing the classical theory of the bending of beams and plates and accounting for non-frictional contact along the delamination faces. Strong interaction effects arise between the delaminations including shielding and amplification of the energy release rate and modification of the mode ratio as compared to a structure with only a single delamination. Such behavior has been summarized in maps that completely characterize the response of a system of two delaminations in a cantilever beam. The quasi-static propagation of the system of delaminations is also strongly controlled by the delamination interactions, which lead to local snap-back and snap-through instabilities, crack arrest and crack pull-along. The results show similarity to those for cracked infinite bodies, but the finite-thickness of the plate plays an important role and gives rise to more complex behaviors. The stability of the equality of length of a system of n delaminations is controlled by their spacing. Finite element calculations confirm that the model proposed here is accurate, except when the difference in the length of the interacting delaminations is less than a few times the separation of their planes.     

The bending moment and springback in pure bending of anisotropic sheets

Finite deformation rigid plastic and elastic–plastic analyses of plane strain pure bending of a plastically anisotropic sheet is presented. An efficient method for finding the exact solution is proposed by extending the previously developed method to the stage of unloading. Using this method the solutions are obtained in closed form or reduced to a numerical treatment of ordinary integrals, or an ordinary differential equation, or transcendental equations. An effect of plastic anisotropy and elastic properties on the bending moment is analyzed. The distribution of residual stresses is illustrated and an effect of material and process parameters on springback is investigated.     

Failure criterion for reinforced concrete beams and plates subjected to membrane force,bending and shear

In this paper, a failure criterion for reinforced concrete plates is derived through the kinematic method in the framework of the limit analysis theory. This criterion is expressed in terms of the stress resultant variables: membrane force, shear force and bending moment at once. The aim of the authors is to be able to predict the failure of reinforced concrete plate structures in statics or in slow dynamics using directly the internal forces (membrane and shear forces and moment) resulting from a finite-element computation.In a first step, a beam criterion is derived. The closed form expression of the criterion shows that it is made up of two parts, one independent of the moment (i.e. depending only on the normal force and the shear force) and one depending on the normal force, the shear force and the bending moment. This structure of the criterion allows to determine two failure modes: shear failure and bending failure.Then in a second step, the beam criterion is extended to the case of reinforced concrete plates. The obtained criterion is partly numerical and partly a close form expression. It gives an upper bound of the load, and when this limit load is reached, the criterion is able to supply, on one hand, the failure mode (as seen in the beam case) and, on the other hand, the angles of the failure plane in the reinforced concrete plate section.Thirdly, the criterion is implemented in the finite element software Europlexus and validated with respect to punching experimental tests. We show that the criterion must be used with an effectiveness factor applied on the concrete compressive strength.     

Transient bending waves in plates studied by hologram interferometry

Propagating bending waves are studied in plates made of aluminum and wood. The waves are generated by the impact of a ballistic pendulum. Hologram interferometry, with a double pulsed ruby laser as the light source, is used to record the out of plane motion of the waves. Elliptic-like fringes visualize differences in wave speed for different directions in the anisotropic plate and circular ones are obtained for the isotropic plate. The experimental data for the isotropic plate compare favorably with analytical results derived from the Kirchhoff-plate equation with a point impact of finite duration. A similarity variable is found when starting conditions are modeled as a Dirac pulse in space and time, that brings new understanding to the importance of specific parameters for wave propagation in plates. A formal solution is obtained for a point force with an arbitrary time dependence. For times much larger than the contact time, the plate deflection is shown to be identical to that from a Dirac pulse applied at the mean contact time. A method for determining material parameters, and the mean contact time, from the interferograms is hence developed.     

Shear deformable bending solutions for nonuniform beams and plates with elastic end restraints from classical solutions

Summary The relationships of bending solutions between Timoshenko beams and Euler-Bernoulli beams are derived for uniform and non-uniform beams with elastic rotationally restrained ends. Extensions of these relationships for the cylindrical bending of Mindlin and Kirchhoff plates and for the bending of symmetrically laminated beams are also discussed. The new set of general relationships is useful because the more complex Timoshenko beam and Mindlin plate solutions may be readily obtained from their simpler Euler-Bernoulli beam and Kirchhoff plate solutions respectively, without much tedious mathematics. Received 16 March 1997; accepted for publication 26 November 1997     

Curvature patterns of bending plates by speckle-shearing interferometry

This paper describes a method to obtain the curvature and twist patterns of a bending plate by combining optical carrier technique and digital image processing. A carrier consisting of parallel fringes is obtained in double-exposure by changing the illumination form so that the deflection of the plate is recorded as the modulation to the carrier frequency. After optical filtering the modulated carrier is input into a digital image processing system to generate a grey image of ‘deformed grating’, whose digital differentiation can be used to obtain the curvature patterns or the twist patterns.     

Shear deformation effect in plates stiffened by parallel beams

In this paper a general solution for the analysis of shear deformable stiffened plates subjected to arbitrary loading is presented. According to the proposed model, the arbitrarily placed parallel stiffening beams of arbitrary doubly symmetric cross section are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. Their unknown distribution is established by applying continuity conditions in all directions at the interfaces. The utilization of two interface lines for each beam enables the nonuniform distribution of the interface transverse shear forces and the nonuniform torsional response of the beams to be taken into account. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second-order effects. The analysis of the plate is based on Reissner’s theory, which may be considered as the standard thick plate theory with which all others are compared, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the analog equation method (AEM), a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled, is achieved using iterative numerical methods. The adopted model permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The effectiveness, the range of applications of the proposed method and the influence of shear deformation effect are illustrated by working out numerical examples with great practical interest.     

Supersonic flow in an angle formed by intersecting plates

A method and results of a numerical solution of the problem of supersonic flow in an angle formed by two perpendicular plates are presented. The computations are performed by the method of build-up. Shocks are obtained as domains with a strong change in the flow parameters.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1972.     

Strength and stiffness of sandwich beams in bending

This investigation is concerned with the experimental versus analytical correlation of the mechanical properties of sandwich-beam specimens. Such sandwich structures are commonly employed in the aircraft industry. Four-point and three-point load tests were conducted on a large number of sandwich-beam specimens, fabricated by using fiber-glass reinforced plastics (both unidirectional and woven-glass cloth) and DTD 685 aluminum alloy for the facings with aluminum honeycomb core and polyurethane foam cores and the indigenously available Araldite as the bonding medium between the core and the facings.The flexural stiffness of the composite sandwich specimens used in this investigation compared favorably with theoretical predictions. The shear stiffness was found to be about 55 percent and 45 percent of the theoretically predicted values for FRP (fiberglass-reinforced-plastic) cloth and FRP unidirectional laminates with aluminum honeycomb core sandwich, respectively. The failure load as determined by experiments was less than the theoretically predicted safe load. There was a loss of strength as well as a steep decrease in the failure load in the case of low density foam core.It was concluded that FRP facing plates with aluminum honeycomb core sandwich structure may be preferred to similar aluminum-alloy facing sandwich construction if high flexural stiffness and shear stiffness properties are required at less cost and weight. Indigenously available Araldite was quite satisfactory for bonding the core to the facings.This investigation has confirmed the importance of experiments in the field of sandwich structures which can effectively replace other conventional uneconomical structural or machine members which are currently in use.     

Vibration in sets of beams and plates induced by traveling loads

We study vibrations of structures composed of beams and plates subjected to moving constant or harmonic loads. Such structures occur in civil engineering, e.g. classical tracks or maglev guideways, highways or tarmacs of airports. We discuss the influence of speed and frequencies as well as parameters describing the construction itself.     

Multiple fractures produced by the bending of brittle beams

This paper describes experiments where the bending of beams results in two or more fractures being formed, apparently simultaneously. This is explained in terms of the stress waves emitted by the initial fracture process. It is shown that three separate types of secondary fracture may occur as a result of the interaction between the stress pulses produced by the initial fracture and the loading stresses already present in the beam. In treating these problems it has been found helpful to use an analytical solution for the bending wave propagated when a semi-infinite beam, which is subjected to a constant bending moment, is suddenly unloaded at the free end. In modelling the longitudinal stress pulse produced by the fracture we have used a simplified model which assumes that the forcing function on the fracture plane is a force field equal to the resultant force acting on the unbroken portion of the fracture surface prior to the onset of fracture.     

Fracture calculation of bending plates by boundary collocation method

IntroductionPlatesarecommonstructuresinaerospace ,civil,mechanicalandchemicalengineering .Duringtheproductionanduseofthestructures,flawsandcracksmayappearintheplates.Inordertoassurethesafetyofplatestructures,toavoidcleavagefractureandfatiguefailure,andtodesignsafetyandmakerationaluseofmaterials,itisnecessarytodofractureanalysisfortheplatestructures.Fractureproblemofabendingplateismorecomplexthanthatoftheplaneelasticity[1,2 ].Firstofall,therearedifferenttheoriesforplatebending ,suchasKirchhoff…     

Creep transition in bending of rectangular plates

Transitional stresses of a rectangular plate bent into the form of a circular cylinder have been derived in closed form. The effect of compressibility is presented graphically. The result indicate that for n (measure index)>1, the circumferential stress is maximum at the inner surface for an incompressible material and not for the compressible material, while for $\text{n=}\text{1}\text{N}$,(N ? 1), it is found to be maximum at some inner point and not at the inner surface. The neutral surface alters with compressibility of the material and n.