Exact static analysis of partially composite beams and beam-columns

The ordinary differential equations and general solutions for the deflection and internal actions and, especially, the pertaining consistent boundary conditions for partially composite Euler–Bernoulli beams and beam-columns are presented. Static loading conditions, including transverse and axial loading and first- and second-order analyses are considered. The theoretical procedure is applicable to general loading and boundary conditions for uniform composite beams and beam-columns with interlayer slip. Further, the exact closed form characteristic equations and their associated exact buckling length coefficients for composite columns with interlayer slip are derived for the four Euler boundary conditions. It is shown that these coefficients are the same as those for ordinary fully composite (solid) columns, except for the Euler clamped-pinned case. For the clamped-pinned case, the difference between the exact buckling length coefficient and the corresponding value for solid columns is less than 1.8% depending on the so-called composite action parameter and relative bending stiffness parameter. Correspondingly, the maximum deviation between the exact and approximate buckling load is at most 2.5%. These small differences can in most practical cases be neglected. Also, the maximum theoretical range for the relative bending stiffness for partially composite beams and beam-columns is derived. An effective bending stiffness, valuable in the determination of the critical buckling load for partially composite members, is derived. This effective bending stiffness is also suitable for analysing approximate deflections and internal actions or stresses in composite beams with flexible shear connection. The beam-column analysis is applied to a specific case. The difference in the approaches to the first- and second-order analysis is illustrated and the results clearly show the magnification in the actions and displacements due to the second-order effect. The magnification of the internal axial forces is different from magnifications obtained for the other internal actions, since only that portion of an internal axial force that is induced by bending is magnified by the second-order effect.     

A simplified analysis method for composite beams with interlayer slip

A simplified static procedure is proposed for analysing and designing composite beams with interlayer slip. The method is parallel to the Eurocode 5 method, but it is general in nature and can be applied to arbitrary boundary and loading conditions. In contrast with Eurocode 5, a general and correct way of choosing the effective beam length of the problem is given by the present procedure, which is that the effective beam length equals the buckling length that is found in the corresponding column buckling problem. The procedure predicts the deflections and internal actions and stresses, in principle by replacing the fully composite bending stiffness (EI_∞) with the effective (partially) composite bending stiffness (EI_eff) in the expressions for these quantities in the corresponding fully composite beam. This effective bending stiffness depends on two non-dimensional parameters: the composite action parameter (shear connection stiffness) and the relative bending stiffness parameter. The method is applied to a number of simple practical cases and the results obtained have been compared with the exact values. The applicability of the simplified analysis procedure was found to be very good, except for interlayer shear stresses. The error in the Eurocode 5 procedure, as compared with the method proposed in this paper, can in some cases be up to almost 30% depending on the boundary conditions.     

Composite beam–columns with interlayer slip—Approximate analysis

An approximate second order analysis procedure for composite beam–columns with interlayer slip subjected to transverse loading and axial compressive loads is developed. The magnification factors to be applied to the first order solutions in order to estimate the deflections and internal forces obtained by the second order analysis approach are presented. The method of applying magnification factors to internal axial forces is discussed. The approximate second order analysis procedure is developed for the four Euler cases with various transverse load conditions. The procedure is applied to and the accuracy is illustrated for simply supported partially beam–columns of steel and concrete, and timber and concrete with different bending stiffness and interlayer slip properties. The deflections and internal forces obtained by the approximate method compared extremely well, except for slip forces in case of very flexible shear connectors, with those obtained by the more rigorous second order analysis approach for different composite action (partial interaction) parameters (shear connector stiffness values). The study also shows that the magnification factor associated with the deflections can be utilized to estimate also the internal actions, except shear forces in case of very flexible shear connectors, in the second order case with minimal error for simply supported beam–columns. Thus, for members with shear connector stiffness of structural significance the proposed approximate method can be used in general for simply supported beam–columns. For other boundary and loading conditions, the approximate method needs to be re-evaluated. The approach of using one magnification factor greatly simplifies the analysis task for those components.     

Free vibration analysis of composite beams with two non-overlapping delaminations

An analytical solution to the free vibration of composite beams with two non-overlapping delaminations is presented. The delaminated beam is modeled as seven interconnected Euler-Bernoulli beams using the delaminations as their boundaries. The continuity and the equilibrium conditions are satisfied between adjoining beams. The analysis includes the differential stretching between the delaminated layers and the bending-extension coupling. The results of the present model agree well with the analytical and experimental data reported in the literature. Parametric studies show that the sizes and locations of the delaminations have significant effect on the natural frequencies and mode shapes. These results provide useful information in the study of the free vibration of delaminated composite beams.     

Implications of warping restraint on statics and dynamics of elastically tailored thin-walled composite beams

Static response and free vibration of elastically tailored thin-walled beams accounting for the warping restraint effect are investigated via an exact solution methodology within the context of a refined beam model. Analytical results obtained from the restrained warping model are compared with those based on its Saint-Venant model counterpart. It is revealed that the beam slenderness and thickness ratio, as well as the elastic anisotropy, considered in conjunction with the warping restraint have profound effects on the static and dynamic response characteristics. It is also shown that even for anisotropic composite thin-walled beams with high slenderness ratios, warping restraint can still be significant, implying the inadequacy of merely considering the geometric aspects in the modeling of anisotropic composite thin-walled beams.     

Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko's beam theory

The static, dynamic, and buckling behavior of partial interaction composite members is investigated in this paper by taking into account for the influences of rotary inertia and shear deformations. The governing differential equations obtained are very comprehensive, covering and extending the current models for the problems that are based on Euler–Bernoulli beam theory. The analytical solutions of the deflection are then found for the beam with uniformly distributing load under common boundary conditions. The free vibration and buckling behavior are also studied and the analytical expressions of the frequencies of the simply supported beam are obtained explicitly, as are the buckling loads. For other boundary conditions, the eigen-equations are transcendental and thus some numerical examples are presented to demonstrate the effects of the shear deformation and rotary inertia on the resonant frequencies and buckling loads.     

Geometrically nonlinear bending analysis of Metal-Ceramic composite beams under thermomechanical loading

A new method is developed to derive equilibrium equations of Metal-Ceramic beams based on first order shear deformation plate theory which is named first order shear deformation beam theory2(FSDBT2). Equilibrium equations obtained from conventional method (FSDBT1) is compared with FSDBT2 and the case of cylindrical bending of Metal-Ceramic composite plates for non-linear thermomechanical deformations and various loadings and boundary conditions. These equations are solved by using three different methods (analytical, perturbation technique and finite element solution). The through-thickness variation of the volume fraction of the ceramic phase in a Metal-Ceramic beam is assumed to be given by a power-law type function. The non-linear strain-displacement relations in the von-Kármán sense are used to study the effect of geometric non-linearity. Also, four other representative averaging estimation methods, the linear rule, Mori-Tanaka, Self-Consistent and Wakashima-Tsukamoto schemes, by comparing with the power-law type function are also investigated. Temperature distribution through the thickness of the beams in thermal loadings is obtained by solving the one-dimensional heat transfer equation. Finally it is concluded that for Metal-Ceramic composites, these two theories result in identical static responses. Also the displacement field and equilibrium equations in the case of cylindrical bending of Metal-Ceramic plates are the same as those supposed in FSDBT2.     

Active robust vibration control of flexible composite beams with parameter perturbations

In this paper, a time domain approach is presented to treat the problem of active controlling simultaneously the bending and torsional vibration of flexible composite beams under both mode truncations and parameter perturbations. In the proposed approach, the residual model, which is known as unstructured uncertainty, is viewed as an additive perturbation to the controlled model. Based on a state space model, which incorporates both bending and torsional deformation effects, of the flexible composite beam with piezoelectric sensors and actuators, a robust stability condition is derived to guarantee that both bending and torsional vibration of the flexible composite beam, which is subject to both mode truncation and linear time-varying parameter perturbations, can be actively controlled by an observer-based controller. The proposed robust stability condition gives an insight into the relationship between the stability margins of the controlled and residual mode subsystems, spillover effects and additive time-varying parameter perturbations. Finally, an active robust vibration control problem of a cantilevered flexible composite beam with piezoelectric sensors and actuators is provided for illustration.     

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton’s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin’s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.     

Rolling cylinder on an elastic half-plane with harmonic oscillations in normal force and rotational speed. Part I: Solution of the partial slip contact problem

We study the effect of harmonic oscillations during the steady rolling of a cylinder on a plane in partial slip contact conditions in the limit of small oscillations. The solution is an extension of that given in Barber et al. [1] for infinitely large coefficient of friction. Here, the effect of varying normal load and hence contact area is investigated in detail by analyzing the first order variation of the tangential force and the corresponding relative displacements.In particular, the solution is given in terms of an explicit length scale d in the Flamant solution used as a Green function. Appropriate choice of values of d allows to treat both two-dimensional problems and three-dimensional ones having elliptical contact area sufficiently elongated in the direction of the rotation axis.Also, this analysis can be used as starting point for corrugation calculations in railway tracks, where oscillations in time of the normal forces can result in non-uniform wear and hence in amplification of the corrugation.     

Natural frequencies of composite plates with random material properties using higher-order shear deformation theory

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters associated with the manufacturing and fabrication processes. In the present work, the material properties have been taken as random variables for accurate prediction of the system behavior. Higher order shear theory including rotatory inertia effects has been accounted for in the system dynamic equations. A first order perturbation technique has been employed to obtain the solution of the governing equations. An approach has been outlined for obtaining closed form expressions for the variances of eigen solutions. The effects of side to thickness ratio and variation in standard deviation of the material properties have been investigated for cross-ply symmetric and anti-symmetric laminates. The mean and standard deviations of the first five natural frequencies have been worked out for laminated rectangular plates with all edges simply supported. The higher order shear deformation theory results have been validated with Monte Carlo simulation results and compared with the results based on classical laminate and first order shear deformation theories.     

Nonlinear free vibration of laminated composite plates on elastic foundation with random system properties

The present investigation is concerned with free vibration analysis of laminated composite plates resting on elastic foundation undergoing large amplitude oscillation with random system properties. The lamina material properties and foundation stiffness parameters are modeled as basic random variables for accurate prediction of the system behavior. The basic formulation of the problem is based on higher-order shear displacement theory including rotatory inertia effects and von Karman-type nonlinear strain displacement relations. A C⁰ finite element is used for descretization of the laminate. A direct iterative method in conjunction with first-order Taylor series based perturbation technique procedure is developed to solve random nonlinear generalized eigenvalue problem. The developed probabilistic procedure is successfully used for the nonlinear free vibration problem with a reasonable accuracy. Typical numerical results (second-order statistics) are obtained for the composite plates resting on Winkler and Pasternak elastic foundations with different support conditions, side-to-thickness ratio, aspect ratio, oscillation amplitude ratio, stacking sequences and foundation parameters for symmetric and anti-symmetric cross-ply and angle-ply laminates. The results are validated with existing available results and independent Monte Carlo simulation.     

Sensitivity analysis of track parameters on train-track dynamic interaction

Dynamic behavior of railway tracks when trains are running is influenced by several factors, i.e. rolling stock, the components of superstructure and their specifications. Usually, features like the sleeper spacing, rail pad stiffness, ballast damping and stiffness have an effect on the dynamic response of the track. The best method to study the dynamic behavior of the track is to model the track assembly and the train as a whole and carry out an analysis of dynamic interaction. Such analysis makes the identification of the track’s dynamic behavior easer and helps to anticipate the deterioration of the track elements, and determines the effects of increase or decrease of mentioned parameters. This paper presents track-train dynamic interaction without considering irregularity of the rail face. A sensitivity analysis was carried out on the selected model. The analysis was undertaken with the view of varying one of the mentioned parameters and the results were presented to further identify the deterioration of the track elements. The results indicate that reducing sleeper spacing, rail pad stiffness, ballast stiffness, and increasing ballast damping reduces wheel-rail, rail-sleeper, and sleeper - ballast contact forces.     

Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method

The present study is concerned with the vibration analysis of cross-ply laminated beams subjected to different sets of boundary conditions. The analysis is based on a three-degree-of-freedom shear deformable beam theory. The continuity conditions between layers of symmetric cross-ply laminated beams are satisfied by the use of the shape function incorporated into the theory which also unifies the 1D shear deformable beam theories developed previously. The governing equations are obtained by means of Hamilton's principle. Six different combinations of free, clamped and simply supported edge boundary conditions are considered. The free vibration frequencies are obtained by applying the Ritz method where the three displacement components are expressed in a series of simple algebraic polynomials. The numerical results obtained for different length-to-thickness ratios and lay-ups are presented and compared with results available in the literature.     

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.     

Vibration analysis of delaminated composite beams and plates using a higher-order finite element

In order to analyze the vibration response of delaminated composite plates of moderate thickness, a FEM model based on a simple higher-order plate theory, which can satisfy the zero transverse shear strain condition on the top and bottom surfaces of plates, has been proposed in this paper. To set up a C⁰-type FEM model, two artificial variables have been introduced in the displacement field to avoid the higher-order derivatives in the higher-order plate theory. The corresponding constraint conditions from the two artificial variables have been enforced effectively through the penalty function method using the reduced integration scheme within the element area. Furthermore, the implementation of displacement continuity conditions at the delamination front has been described using the present FEM theory. Various examples studied in many previous researches have been employed to verify the justification, accuracy and efficiency of the present FEM model. The influences of delamination on the vibration characteristic of composite laminates have been investigated. Especially the variation of ‘curvature of vibration mode’ (i.e., the second-order differential of deflections in vibration mode) caused by delamination has been studied in detail to provide valuable information for the possible identification of delamination. Furthermore, two approaches have been investigated to detect a delamination in laminates by employing this information.     

Characterization of the near-surface layers of carbon steels modified by interaction with intense pulsed plasma beams: scanning electron microscopy investigations

Modification of materials is a wide area in materials science, especially surface modification. To investigate the results of the modification process, treated and nontreated samples were compared. Intense plasma pulses of argon or nitrogen were used to irradiate the carbon steels. In all samples, the near‐surface layer was melted. Results of scanning electron microscopy investigations of the surface morphology and cross‐sections, as well as the results of tribological tests, are presented. The obtained results allowed us to draw conclusions about changes in material properties and to propose subsequent studies using other investigation techniques.     

Application of 2D-infinite beam elements in dynamic analysis of train-track interaction

Railway structure is a structure with an infinite length for which its modeling is usually carried out by accepting some assumptions for boundary conditions. In practice, there are various methods of modeling by forming equations of motions and dynamic analysis. One of these methods is modeling the railway track by finite element method. Generally in such modeling, a limited length of a track is modeled and executed by implementing some boundary conditions. Assortment of boundary conditions has an effect on dynamic responses. To reduce such effects, usually the length of the track is chosen to such a measure to minimize the dynamic responses of the end points of the track elements to zero. Doing so would cause an increase in the length of the track and hence add to an equation’s degrees of freedom, volume of output and prolongation of analysis time. For this reason, a combination of finite elements and infinite beam elements (two end elements) has been proposed for railway track modeling. Also, matrices of mass, damping and stiffness of one infinite element which has been laid on a visco-elastic bed, have been calculated by implementing selected shape functions. Therefore, by applying two infinite beam elements on either side of the model, a railway track is formed like a beam on an elastic bed which creates the possibility of eliminating the boundary condition effects. This paper was recommended for publication in revised form by Associate Editor Hong Hee Yoo Jabbar Ali Zakeri received a B.S. degree in civil Engineering and MSc. Degree in structural engineering from Tabriz University in 1992 and 1995, respectively. He then went on to receive his Ph.D. degrees in Road and Railway Engineering from Beijing Jiaotong University in 1997. Dr. Zakeri is currently an Assistant Professor at the School of Railway Engineering at Iran University of Science and Technology. Dr. Zakeri’s research interests are in the area of dynamic analysis of train — track interaction, railway track dynamics, track maintenance and construction.     

Dynamic instability characteristics of laminated composite plates subjected to partial follower edge load with damping

The study of vibration and dynamic instability behaviour of laminated composite plates subjected to partially distributed non-conservative follower forces is presented by using the finite element technique. The first-order shear deformation theory is used to model the plate, considering the effects of shear deformation and rotary inertia. The modal transformation technique is employed to the resulting equilibrium equation for subsequent analysis. Structural damping is introduced into the system in terms of equivalent viscous damping to study the significance of damping on stability characteristics. The effects of load width, boundary condition, aspect ratio, ply orientation, direction control of the load and damping parameters are considered for the stability behaviour of the plates. The results show that under follower loading, the system is susceptible to instability due to flutter alone or due to both flutter and divergence, depending on system parameters.     

Buckling analysis of multi-walled carbon nanotubes under combined loading considering the effect of small length scale

The torsional and axially compressed buckling of an individual embedded multi-walled carbon nanotube (MWNTs) subjected to an internal and/or external radial pressure was investigated in this study. The emphasis is placed on new physical phenomena which are due to both the small length scale and the surrounding elastic medium. Multiwall carbon nanotubes which are considered in this study are classified into three categories based on the radius to thickness ratio, namely, thin, thick, and almost solid. Explicit formulas are derived for the van der Waals (vdW) interaction between any two layers of an MWNT based on the continuum cylindrical shell model. In most of the previous studies, the vdW interaction between two adjacent layers was considered only and the vdW interaction among other layers was neglected. Moreover, in these works, the vdW interaction coefficient was treated as a constant that was independent of the radii of the tubes. However, in the present model the vdW interaction coefficients are considered to be dependent on the change of interlayer spacing and the radii of the tubes. The effect of the small length scale is also considered in the present formulation. The results show that there is a unique buckling mode (m,n) corresponding to the critical shear stress. This result is obviously different from what is expected for the pure axially compressed buckling of an individual multi-walled carbon nanotube.