Elastic springback in plates and beams formed by bending

A plane-strain solution is presented to predict the springback of plates formed by bending. The solution assumes that the strains are small, that the material is an isotropichardening von Mises material, and either that the stress-strain diagrams for tension and compression are identical or that an average diagram can be employed. The plastic portion of each stress-strain diagram is approximated by a finite number of straight lines. Both analytical solutions and experimental evidence indicate that the much simpler plane-stress solution can be used to predict springback for both beams and plates. Experimental data confirm that the type of die has a pronounced effect on the springback for beams and plates if the material exhibits a yield point.     

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.     

Circumferentially traveling radial loads on rings and cylinders

The problem of circumferentially traveling radial loads on rings and infinitely long cylindrical shells will be considered in this paper. Since strong transitional excitations are considered, the response involves moderately large rotations. Hence the governing shell equations are non-linear. To facilitate their solution, a modified version of the Linstedt-Poincaré perturbation procedure is employed. Based on this solution, several numerical results are presented. In addition to considering the effects of displacement induced non-linearity, special emphasis is given to the response behavior in load speed zones which mark transitions from sub- to supercritical waveforms.     

Complex modes and traveling waves in axially moving Timoshenko beams

Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.     

Vibration analysis of plates by real-time stroboscopic holography

The application of time-averaged holography to the analysis of vibrating surfaces is now a widespread and very powerful experimental technique. However, the realtime method of exactly superimposing the reconstructed static image on the vibrating object and observing the resulting fringe patterns under stroboscopic illumination is becoming more common. In fact, by introducing an initial family of interference fringes by the rotation of the object (or image) before vibration begins, the advantages of both methods are combined.     

Vibration of continuous circular plates

The general development of the theory given here considers the material to be orthotropic and continuous over (n ? 1) elastic or rigid supports. The effect of rotatory inertia and in-plane loads are also included while formulating the equations of motion. Double and triple series solutions are given for orthotropic continuous plates. By matching the continuity conditions at the intermediate supports and satisfying the boundary conditions at the outer edge, the frequency determinant is obtained. For the purpose of numerical computations, an isotropic plate continuous over an intermediate-rigid or elastic-support and free and with no in-plane loads at the outer edge is considered. It is found that the influence of Poisson's ratio on the frequency parameter is significant only for the first symmetric or asymmetric modes. The rotatory inertia influences the frequency parameter when the radius to thickness ratio is less than 80, viz, when the plate is thick. Moreover, the elasticity of the support influences considerably the free vibration of plates.     

Vibration of rectangular orthotropic plates

Summary An approximate analytical procedure has been given to solve the problem of a vibrating rectangular orthotropic plate, with various combinations of simply supported and clamped boundary conditions. Numerical results have been given for the case of a clamped square plate.Nomenclature 2a, 2b sides of the rectangular plate - h plate thickness - E _x , E _y , E, G elastic constants of te orthotropic material - D _x E _x h ³/12 - D _y E _y h ³/12 - H _xy Eh ³/12+Gh ³/6 D _x , D _y and H _xy are rigidity constants of the orthotropic plate - mass of the plate per unit area - Poisson's ratio - W deflection of the plate - p circular frequency - b/a ratio - X _m , Y _n characteristic functions of the vibrating beam problem - p ² a ² b ²/H _xy the frequency parameter.     

Vibration of thick auxetic plates

This short communication investigates the effect of negative Poisson's ratio on the natural frequency of thick plates of arbitrary shape. Using the Mindlin plate theory, it was generally found that as the plate's Poisson's ratio becomes more negative, the Mindlin-to-Kirchhoff natural frequency ratio increases with decreasing rate. Upon comparing (a) the use of the simplified constant shear correction factor and the more accurate variable shear correction factor, (b) with and without rotary inertia, it was found that all the four combinations stated in (a) and (b) do not give appreciable difference when the Poisson's ratio of the plate is positive. However in the case of plates with negative Poisson's ratio, results reveal that when at least one of the simplifying assumptions is used, the Mindlin-to-Kirchhoff natural frequency ratio is overestimated, and that the overestimation further increases when both the simplifying assumptions are used. When benchmarked against Reddy plate theory, the use of variable shear correction factor has almost the same effect as the inclusion of rotary inertia. Hence the use of either variable shear correction factor or rotary inertia is proposed for modeling the vibrational frequencies of conventional and auxetic isotropic plates.     

Bending of multilayer transversally isotropic plates and beams

Ivano-Frankovskii Sector, IPPMM, National Academy of Sciences, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 30, No. 5, pp. 55–62, May, 1994.     

Non-linear bending of beams with uniformly distributed loads

The non-linear bending of both cantilever and simply supported beams subjected to a uniformly distributed load has been studied. The exact solutions for the slopes have been obtained and the solution for the maximum deflection and the horizontal projection of the beam length for the cantilever case are compared with a known approximate solution.     

Minimum-volume plastic design of beams for movable loads

Summary Minimum-weight designs of a perfectly plastic continuous sandwich beam with two equal spans are obtained analytically for a variety of movable loads, on the basis of the condition of constant integrated specific dissipation rate.

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Sommario Mediante la condizione di costanza dell'integrale della velocità di dissipazione specifica si decudono per via analitica e per una varietà di carichi mobili, progetti di minimo peso di una trave continua a due luci uguali perfettamente plastica e di tipo sandwich.

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Text of a lecture given by the senior author at the Politecnico di Milano and at the Facoltà di Ingegneria dell'Università di Roma in May 1970.     

用加权余量法分析固支圆板和环板在Mises屈服条件下的极限荷载

应用加权余量法求出了承受线性荷载的固支圆板和承受均布荷载的内边界支承环板在 Mises屈服条件下的极限荷载的近似值 ,并与最大弯矩极限条件结果进行了比较 ,说明本文计算结果较合理。     

Optimum plastic design for multiple sets of loads

Summary We study optimum plastic design of structures made up, or conceived as assemblies of finite elements, each having an elemental piece-wise linear rigid-plastic behaviour. Since cost function linearly dependent on design variables are considered, optimization problems in linear programming are encountered. Allowance is made for design dependent mass forces, and for some technological constraints. The design growing process is studied in the case of various sets of alternative applied loads, and the optimality conditions are written in a proper geometrical form which leads to a generalization of the concept of Foulkes mechanism.

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Sommario Si studia il progetto plastico ottimale di strutture formate di elementi finiti, ciascuno avente un comportamento globale rigido-plastico linearizzato a tratti. Si considerano funzioni costo dipendenti linearmente dalle variabili di progetto, e pertanto si va incontro a problemi di ottimizzazione nell'ambito della programmazione lineare. Tenendo conto delle forze di massa e di eventuali vincoli tecnologici, viene analizzato il processo di crescita del progetto nel caso in cui la struttura è sottoposta a diversi sistemi alternativi di carichi, e si scrivono le condizioni di ottimalità in una appropriata forma geometrica, la quale conduce a una generalizzazione del concetto di meccanismo di Foulkes.

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Capital bold-face letters are matrices, small bold-face letters are vectors. 0 is a vector having only zero entries. The tilde posed upon a vector or a matrix means “traspose of”. Other meanings are: