Large deflections of metallic sandwich and monolithic beams under locally impulsive loading

A rigid-perfectly plastic model is adopted to predict the dynamic response of fully clamped sandwich and monolithic beams subjected to localized impulse. Large deflection effect is incorporated in analysis by considering interaction between plastic bending and stretching. Based on the principle of energy equilibrium, a membrane factor for metallic sandwich beams with nonuniform cross-section thickness is derived to consider the effect of axial force induced by large deflection. Then, the dynamic response solution is obtained for the large deflection of metallic sandwich beams subjected to localized impulse. In addition, tighter ‘bounds’ of the solutions for sandwich beams are derived by using the inscribed and circumscribed squares of a new yield criterion including the core effect. As a degenerated limit case, solution for the large deflection response of solid monolithic beams is also obtained. The present solutions are in good agreement with finite element (FE) results and lie in the ‘bounds’ of the solutions. It is demonstrated that the axial (membrane) force associated with stretching plays an important role in the dynamic response of large deflections; in comparison with small deflection solutions, the axial (membrane) forces substantially stiffen the metallic sandwich and monolithic beams.     

In-plane and out-of-plane buckling of arches made of FGM

The mechanical buckling of curved beams made of functionally graded materials is studies in this paper. The equilibrium and stability equations of curved beams under mechanical loads are derived. Using proper approximate functions for the displacement components, the stability equations are employed to obtain the related eigenvalues associated with the buckling load of the curved beam. Closed-form solutions are obtained for mechanical buckling of curved beams with doubly symmetric cross section subjected to uniform distributed radial load and pure bending moment. The results are validated with the known data in the literature for beams with isotropic materials.     

Applying the equivalent section method to solve beam subjected to lateral force and bending-compression column with different moduli

Based on elastic theory of different tension-compression moduli, bending beam subjected to lateral force and bending-compression column with different moduli were solved by the equivalent section method. Formulas for the neutral axis, normal stress, shear stress and displacement were developed also. This equivalent section method can turn conveniently different moduli problems into the similar moduli ones, i.e., classical elasticity problems so the existent results aimed beams and columns with similar moduli both can be used indiscriminately without complicated derived process. Compared with the present derived method based on different moduli theory the applicability and efficiency of equivalent section method is demonstrated.     

Dynamic analysis of bending–torsion coupled composite beams using the Flexibility Influence Function Method

In this work the dynamic behaviour of symmetrical laminated beams was studied, taking into account the effect of bending–torsion coupling by a one-dimensional model. This model includes the influence of the shear force and rotatory inertia. To solve the equations of motion, the Flexibility Influence Function Method (FIFM) was used. The dynamic displacements (deflection, bending rotation, and torsional rotation) were calculated for a beam in which the deflection and torsional rotation were restricted at its both ends, allowing the bending rotation. The accuracy of this method was determined by using a Three-Dimensional Finite Element Method (FEM3D) model to compare the dynamic displacements. The need was shown to incorporate coupling in the one-dimensional model in order to calculate the dynamic deflection and bending rotation of a composite beam.     

Influence of pre-buckling displacements on the elastic critical loads of thin walled bars of open cross section

Nonlinear expressions for the strains occurring in thin walled bars of open cross section, when subjected to axial, flexural and torsional displacements, are incorporated in a general instability analysis based on the vanishing of the second variation of the total potential energy. It is shown that the influence of the pre-buckling displacements is automatically included in the analysis. A closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending agrees exactly with a solution based on the governing differential equations. Solutions obtained using numerical methods are also presented. The significance of the second order axial strains induced by rotation about the shear centre, is investigated by considering the instability of an inverted T-beam subjected to uniform bending.     

Establishment of a 3D FE model for the bending of a titanium alloy tube

Establishing and developing a finite element (FE) model is key to the effective study of the complex bending of a titanium alloy tube. This paper focuses on the establishment of a three-dimensional (3D) FE model for the numerically controlled (NC) bending of a titanium alloy tube that considers both bending and springback. The key procedures for establishment of the model are described in detail, including the choice of elements, mesh density control, the choice of single- or double-precision computation, and the choice of the mass scaling factor. Combining explicit and implicit FE methods, a significant amount of springback can be modeled for the bending, which is in good agreement with what is observed in practice. Single-precision computation tends to provide increasingly inadequate results as the mass scaling factor decreases. Analyses performed using double-precision computation can improve the accuracy of the results. A decrease in the mass scaling factor when double-precision computation is used leads to improved results to some degree, but also to an increase in computation time. Taking accuracy and efficiency into consideration, a mass scaling factor of 1600 is considered reasonable. Using our 3D FE model, results for the distributions and variations of the tangential stress, tangential strain, wall thickness, and springback angle for the bending of a TA18 M tube were obtained.     

Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques

Mechanical properties of micro/nanoscale structures are needed to design reliable micro/nanoelectromechanical systems (MEMS/NEMS). Micro/nanomechanical characterization of bulk materials of undoped single-crystal silicon and thin films of undoped polysilicon, SiO(2), SiC, Ni-P, and Au have been carried out. Hardness, elastic modulus and scratch resistance of these materials were measured by nanoindentation and microscratching using a nanoindenter. Fracture toughness was measured by indentation using a Vickers indenter. Bending tests were performed on the nanoscale silicon beams, microscale Ni-P and Au beams using a depth-sensing nanoindenter. It is found that the SiC film exhibits higher hardness, elastic modulus and scratch resistance as compared to other materials. In the bending tests, the nanoscale Si beams failed in a brittle manner with a flat fracture surface. The notched Ni-P beam showed linear deformation behavior followed by abrupt failure. The Au beam showed elastic-plastic deformation behavior. FEM simulation can well predict the stress distribution in the beams studied. The nanoindentation, scratch and bending tests used in this study can be satisfactorily used to evaluate the mechanical properties of micro/nanoscale structures for use in MEMS/NEMS.     

The elastic-plastic pure bending and springback of L-shaped beams

L-shaped beams subjected to pure elastic-plastic bending are investigated thoroughly. The shifting and rotating of the neutral axis of the beam during bending and springback after unloading are described mathematically, while the curvatures of the beam before and after springback are given. The method proposed in this paper is valuable both to the further theoretical studies of the asymmetric elastic-plastic bending of beams with arbitrary cross-section and to its application in practical bending operations.     

The effect of ‘material ratchetting’ on the incremental growth of beams subjected to steady axial loads and cyclic bending

An elastic-plastic material behaviour model capable of predicting material ratchetting, cyclic relaxation and cyclic hardening has been incorporated into finite element programs. The resulting programs have been used to predict the behaviour of beams subjected to the steady axial loads and cyclic bending. The results are compared with experimental data obtained from three tests performed on beams made from a model material. Overall, the predictions were found to be good; discrepancies in the results for the most highly loaded beam were attributed to creep effects in the experiment.The results indicate that an elastic-perfectly plastic material behaviour model, with an equivalent yield stress, does not allow acceptable prediction of the beam behaviour to be obtained.     

Interaction yield hypersurfaces for the plastic behaviour of beams—I. Combining bending, tension and shear

A variational method has been used to construct envelopes of the interaction yield surfaces for elastic, perfectly plastic beams subjected to combined bending moment, axial tension and transverse shear force. A lower bound of the envelope for the interaction yield curves relating bending moment and transverse shear force is obtained using a numerical scheme for beams having rectangular-shaped cross-sections. The result is recast into a simple equation with the aid of the least-squares method. A good upper bound of the envelope for the interaction yield curves which combine transverse shear force and axial tension is also derived. A formula for the interaction yield surfaces for combined bending moment, axial tension and transverse shear force is suggested.     

Local and distortional buckling of cold-formed zed-section beams under uniformly distributed transverse loads

This paper presents a numerical investigation on the local and distortional buckling behaviour of cold-formed steel zed-section beams subjected to uniformly distributed transverse loads. The analysis is performed using a semi-analytical finite strip method. The beams investigated include both detached sections and restrained sections. The results obtained from the present study highlight the differences in the local and distortional buckling behaviours of the thin-walled sections between pure bending and the uniformly distributed loading.     

Examining the trend in loss of flexural stiffness of simply supported RC beams with various crack severity using model updating

The flexural stiffness of simply supported cracked reinforced concrete beams was determined by model updating. The beams were 150 mm wide, 250 mm deep and 2200 mm long. Different FE models were created which include a datum and models with a single crack at three different locations along the length of the beam. The mode shape equation was obtained by using non-linear regression. The equation used in the regression was the generalized solution of transverse vibration of a prismatic beam. Local flexural stiffness, EI, at each coordinate point was derived by substituting the regressed data by using the centered-finite-divided-difference formula. Experimental modal analysis was performed on a control beam and beams with load-induced cracks at predetermined loading. Results from FE analyses showed the trend in the loss of stiffness was similar to the results obtained on the experimental beams. The more severe the damage, the higher the loss of stiffness and the loss patterns are similar for damage at different locations along the beam. The updating technique is able to indicate the trend in the loss of stiffness as a result of cracks of varying severity in the RC beams showing good agreement with experimental results.     

Flexibility of rectangular beams with abrupt changes of section

The usual strength of materials approach to the calculation of deflexions of stepped beams can incur errors if applied to stepped beams with large and abrupt changes of section. After initial comparison with experimental results, the finite element displacement method was used in an investigation into the properties of such beams, particular attention being paid to the relationships between the geometry and the centre line deflexion curves. Data are presented so that for a beam of any geometry and any loading condition, the centre line deflexion curve can be calculated by using simple bending relationships in conjunction with an equivalent variable second moment of area.     

Modelling drawbeads in sheet metal forming

In this paper, we describe the numerical modelling of drawbead forces required to draw a sheet metal through a bead with a constant cross section. The model is formulated using an elasto-plastic large strains finite element method combined with an improved numerical technique taking into account the contact and friction conditions, based on linear programming techniques and fixed point conditions. The numerical simulations were carried out with various drawbead geometries and the results are expressed in terms of drawing restraining force versus drawing movement, as a function of the drawbead geometry, gap conditions and friction conditions. With this procedure it is also possible to determine the main deformation paths in the drawbead region where the bending strain predominates over the membrane strain. The results obtained are compared with experimental data and the agreement proves to be good. The calculated results can be used as a basis to obtain better approximations of the drawbead constitutive equations to be used in general 3D FE simulation codes in cases where it is impossible to exactly determine the drawbead geometry.     

Nonlinear centrifugal effects on a prestressed laminated rotor

Electrical rotors are usually used in rotating machinery such as in ship engine pods. Designing such rotors in bending requires careful modeling of a laminated stack prestressed by tie rods. Theoretically, steel sheets, varnish layers and prestressing make it difficult to homogenize the constitutive properties of a stack both at rest and under operating conditions. Moreover, it is noteworthy that tie rods subjected to centrifugal forces bend until reaching annular clearance. Their nonlinear bending tends to increase their stiffening effects and the axial lamination load, therefore it affects the homogenized constitutive properties of a stack modeled with Timoshenko beams. By combining a fixed-point algorithm and penalty method at each speed of rotation, it is possible to plot the Campbell diagram and unbalance responses by considering updated properties and the axial load of the tie rods generated by centrifugal effects. An industrial application is used to show that centrifugal effects have a slight influence on rotor dynamics.     

Effect of intervertebral disc degeneration on biomechanical behaviors of a lumbar motion segment under physiological loading conditions

The consideration of biomechanical alterations due to intervertebral disc (IVD) degeneration is crucial for the accurate analysis of spine biomechanics. In this study, finite element (FE) models of the L4-L5 motion segment with full coverage of the degeneration grades from healthy IVD to severe degeneration were developed. The effects of IVD degeneration on spine biomechanics were analyzed under physiological loading conditions using compressive forces and bending moments. The FE models of all degeneration grades were consistent with published data in terms of the ranges of motion. Severe IVD degeneration showed lower inter-segmental rotations in flexion-extension and lateral bending, lower intradiscal pressure in all motions, higher facet joint forces in lateral bending and axial rotation, and higher von-Mises stress in annulus ground substance in all motions versus the healthy IVD. These findings could provide fundamental information for understanding the characteristics of the biomechanical behaviors of degenerated lumbar motion segments.     

Large plastic deformation of beams of angle-section under symmetric bending

The large deflection flexural behaviour, particularly in plastic range, of beams of right-angle section under four-point symmetric bending is investigated thoroughly. Experiments were carried out for symmetric bending of aluminium alloy L-beams to observe the global load-deflection characteristics, as well as the distortion of cross-sections. The effect of supporting condition in the four-point bending on the global behaviour of the L-beams was also examined experimentally. By considering the geometrical change of beam and the plastic energy dissipations due to the global bending of beam and the angular distortion of cross-sections, a rigid, linear hardening model is proposed. The model is capable of predicting the crosssectional distortion and the structural softening behaviour observed in the experiments with a reasonable accuracy.     

Mechanical property measurements of nanoscale structures using an atomic force microscope

This paper describes nanometer-scale bending tests of fixed single-crystal silicon (Si) and silicon dioxide (SiO2) nanobeams using an atomic force microscope (AFM). The technique is used to evaluate elastic modulus of the beam materials and bending strength of the beams. Nanometer-scale Si beams with widths ranging from 200 to 800 nm were fabricated on a Si diaphragm using field-enhanced anodization using an AFM followed by anisotropic wet etching. Subsequent thermal oxidation of Si beams was carried out to create SiO2 beams. Results from the bending tests indicate that elastic modulus values are comparable to bulk values. However, the bending strength appears to be higher for these nanoscale structures than for large-scale specimens. Observations of the fracture surface and calculations of the crack length from Griffith's theory appear to indicate that the maximum peak-to-valley distance on the beam top surfaces influence the values of the observed bending strengths.     

The three-point bending of Y-frame and corrugated core sandwich beams

Sandwich beams comprising Y-frame and corrugated cores have been manufactured by assembling and brazing together pre-folded AISI type 304 stainless steel sheets. The longitudinal axis of the cores coincides with the axis of the beams. The quasi-static three-point bending response of both simply supported and clamped beams is measured along with the indentation response of the beams placed on a rigid foundation. The investigation reveals that the initial collapse strength of the beams is governed by the indentation of the Y-frame or corrugated core for all beam geometries considered here. The simply supported beams have a softening response beyond the initial peak load while the clamped beams display a hardening response due to the longitudinal stretching of the face-sheets. The experimental investigation reveals that sandwich beams with Y-frame or corrugated cores have comparable responses for each of the loading situations considered. Additional insight into the deformation modes is obtained by three-dimensional finite element (FE) calculations.