Beam Of Rectangular Cross-section Research Articles

Piezoresistive Sensing of Strain and Damage in Carbon/Epoxy Rectangular and I-shaped Cross-section Beams in Flexure

The strain and damage sensing capabilities of a carbon fiber/epoxy composite subjectedto monotonic and incremental cyclic flexural loading is investigated using theelectrical resistance approach. The piezoresistive sensitivity of the carbon fiber/epoxycomposites were first characterized using rectangular cross-section specimens andthen application of the electrical resistance approach was extended to an I-shapedcross-section laminated composite. The change in the electrical resistance of the specimenswas simultaneously monitored on both the tensile and compressive sides whilethe specimens were tested under flexural loading. On the tensile side, the positive electricalresistance change (ΔR) showed a linear trend for small deformations (less than0.3% strain) and increased in a nonlinear fashion for larger deformations. In contrast tothe tensile side, ΔR on the compressive side showed a nonlinear and non-monotonicbehavior. Initially, ΔR decreased for low values of deformation (less than 0.3% strain),leveled off at the onset of the specimen’s compressive failure, and then increased upto complete failure. Correlations between abrupt changes in ΔR and indications ofcomposite damage are confirmed with scanning electron microscopy. Carbon fibersin both rectangular and I-shaped cross-section carbon/epoxy specimens are capableof sensing tensile and compressive strain, and adequately capture the onset and progressionof failure by fiber micro-buckling, formation of kink bands, and delamination.The cross-section properties (e.g. stiffness and layup sequence) affect the piezoresistiveand damage sensitivity, with the stiffer I-shaped cross-section being less sensitiveto ΔR than standard specimens of rectangular cross-section. ΔR under cyclic flexuralloading for both tensile and compressive sides showed a permanent change with applieddeformation after each cycle, indicating first the release of residual stresses andthen damage.

Lateral Torsional Buckling of Rectangular Reinforced Concrete Beams

This paper reports the results of a study aimed at investigating the lateral stability of rectangular reinforced concrete slender beams. A total of nine reinforced concrete beams of rectangular cross-section with d/b ratios ranging from 10.20 to 12.45 and Lb ratios varying between 96 and 156 were tested. A gravity load simulator that allowed the load to remain vertical when the beam reinforcing bars placed out of plane was used for loading the beams. The test results were compared with various lateral buckling solutions in the literature and the solution developed in this study, which accounts for the inelastic and non-homogeneous nature and cracking of reinforced concrete and the initial geometric imperfections. Tests have shown that sweep increases the lateral deformations of a concrete beam before buckling and reduces the buckling load. The proposed analytical solution was found to closely estimate the experimental buckling loads of the specimens.

Mode shapes analysis of a cracked beam and its application for crack detection

Abstract In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.

Energy harvesting from the tail beating of a carangiform swimmer using ionic polymer–metal composites

In this paper, we study energy harvesting from the beating of a biomimetic fish tail using ionic polymer–metal composites. The design of the biomimetic tail is based on carangiform swimmers and is specifically inspired by the morphology of the heterocercal tail of thresher sharks. The tail is constituted of a soft silicone matrix molded in the form of the heterocercal tail and reinforced by a steel beam of rectangular cross section. We propose a modeling framework for the underwater vibration of the biomimetic tail, wherein the tail is assimilated to a cantilever beam with rectangular cross section and heterogeneous physical properties. We focus on base excitation in the form of a superimposed rotation about a fixed axis and we consider the regime of moderately large-amplitude vibrations. In this context, the effect of the encompassing fluid is described through a hydrodynamic function, which accounts for inertial, viscous and convective phenomena. The model is validated through experiments in which the base excitation is systematically varied and the motion of selected points on the biomimetic tail tracked in time. The feasibility of harvesting energy from an ionic polymer–metal composite attached to the vibrating structure is experimentally and theoretically assessed. The response of the transducer is described using a black-box model, where the voltage output is controlled by the rate of change of the mean curvature. Experiments are performed to elucidate the impact of the shunting resistance, the frequency of the base excitation and the placement of the ionic polymer–metal composite on energy harvesting from the considered biomimetic tail.

Finite amplitude vibrations of cantilevers of rectangular cross sections in viscous fluids

In this paper, we study finite amplitude vibrations of a cantilever beam of rectangular cross section immersed in a viscous fluid under harmonic base excitation. Fluid–structure interactions are modeled through a complex hydrodynamic function that describes added mass and damping effects in response to moderately large oscillation amplitudes. The hydrodynamic function is identified from the analysis of the two-dimensional flow physics generated by a rigid rectangle undergoing harmonic oscillations in a quiescent fluid. Computational fluid dynamics is used to investigate the effects of three salient non-dimensional parameters on the flow physics and inform the formulation of a tractable expression for the hydrodynamic function. Theoretical results are validated against experimental findings on underwater vibration of compliant beams of varying cross sections.

Electrical and Acoustic Emissions in cement mortar beams subjected to mechanical loading up to fracture

The application of mechanical loading on cement-based materials generates weak electrical currents due to the formation and propagation of microfractures. This paper introduces the simultaneous recording of electrical signal emissions know as Pressure Stimulated Currents (PSCs) and Acoustic Emissions (AEs) detected in cement mortar beams of rectangular cross-section subjected to mechanical loading using the Three Point Bending tests. The rate of the applied load during the experiments was constant up to the fracture of the specimen. The characteristics of the AE and PSC were put in contrast in order to better understand the evolution of the microfracture processes up to fracture. Specifically, the recorded PSC and AE were studied in terms of their time series, cumulative energy along with AE rate and AE interevent times that may provide information about the upcoming specimen fracture. Moreover, non-extensive statistical physics modeling was attempted in terms of studying the Tsallis entropy. Specifically, the entropic index q was calculated and its dependence on the applied mechanical load was investigated.

Vibration analysis of cracked frame structures

In this study, the effects of crack depth and crack location on the in-plane free vibration of cracked frame structures have been investigated numerically by using the Finite Element Method. For the rectangular cross-section beam, a crack element is developed by using the principles of fracture mechanics. The effects of crack depth and location on the natural frequency of multi-bay and multi-store frame structures are presented in 3D graphs. The comparison between the present work and the results obtained from ANSYS shows a very good agreement.

Conditions for Failure of Normal Section in Flexural Reinforced Concrete Beams of Rectangular Cross-Section

The stress in longitudinal tensile reinforcement is one of the main important parameters while examining the technical state of under- reinforced concrete structures. The most important issue is to determine whether the external loads cause the close to yield stress in the main reinforcement. Yield stress in tensile reinforcement could be treated as the start of incipient failure of the flexural structure. The state of tensile reinforcement of flexural reinforced concrete structures could be examined by observing the properties of the normal cracks. The application of fracture mechanics of solids could be used for determining the actual damage to the structure by knowing only the measured height of the normal crack.

Simplified description of out-of-plane waves in thin annular elastic plates

Dispersion relations are derived for the out-of-plane wave propagation in planar elastic plates with constant curvature using the classical Kirchhoff thin plate theory. The dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges. Elementary Bernoulli–Euler theory for a beam of rectangular cross-section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out-of-plane deformation. The applicability range of the elementary beam theory is validated. The wave finite element method in the formulation of the three-dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range, where the classical thin plate theory is valid. Thus, the paper summarizes the effects brought to the propagation of out-of-plane waves in thin elastic plates by their constant curvature and the models of these plates.

Thermoelastic dissipation in stepped-beam resonators

In recent years, stepped beam resonators have found broad application in MEMS/NEMS devices. A beam resonator with an undercut at the support, produced due to isotropic etching of the supporting substrate during fabrication, has also been characterized as stepped beam in the literature. The present study deals with thermoelastic dissipations of clamped–clamped stepped beam resonators under adiabatic surface thermal conditions having j (j = 1, 2, …., n) number of sections defined by (j − 1) number of steps along the length. Numerical results are obtained for three different types of stepped beams of rectangular cross-section having single step such as beams with cross-sectional change at the step only in lateral direction (type-1), in bending direction (type-2), and in both lateral and bending directions (type-3) where the section on the right of the step possesses smaller cross-sectional size compared to the other. The obtained results show that Q-factors vary significantly with step positions in all the three types of stepped beams. For constant length, the Q-factor increases in the type-1 while it decreases in other two types of stepped beams as the step position moves from the left support to the right along the length. Moreover, Q-factors in a type-1 stepped beam depend on the widths of different sections and can be higher than a uniform beam of same thickness for some particular step positions. For most common lengths of stepped beams in real applications with the step close to the left support, type-1 stepped beams provide higher quality factors than the other stepped beams provided that they have the same cross-sectional area.

Effect of cross-sectional shape on thermoelastic dissipation of micro/nano elastic beams

In spite of practical relevance, little effort has been made to analyze the effect of geometrical shape of cross-section on thermoelastic dissipation of micro/nano beams under different surface thermal condition. The present paper aims to study thermoelastic dissipation of micro/nano beams of elliptical, triangular or arbitrary rectangular cross-section with accurate satisfaction of the surface thermal condition. Detailed formulas are derived for quality (Q-) factor of beams of the above-mentioned cross-sections. Our results show that for all cross-section discussed, thermoelastic dissipation is a non-monotonic function of the absolute size of the cross-section provided the beam length is fixed, and the maximum dissipation appears at a specific size which is of the order of a few hundreds of nanometers for examples discussed. These results suggest that thermoelastic dissipation could increase with decreasing cross-sectional size within the micron scale, while it could decrease with decreasing cross-sectional size within the nano scale. In general, for all beams of the above-mentioned cross-sections, the Q-factor for isothermal surface is always higher than the Q-factor for adiabatic surface under otherwise identical conditions at micro/nano scales. The present analysis also indicates that, to achieve a high quality factor, beam resonators with elliptical or triangular cross-sections are best to operate at higher frequencies while beams of rectangular cross-sections are best to operate at lower frequencies.

Radial breathing modes of multi-walled carbon nanotubes by an atomic beam-spring model

Based on molecular force fields, a new finite element model is constructed for multi-walled carbon nanotubes where the interlayer interactions and C—C bonds are simulated by the elements of piece-wise linear spring and rectangular cross section beam, respectively. For high computation efficiency and atomic reification, the radial breathing modes of multi-walled carbon nanotubes are studied systemically using this model. The results show the correspondence between carbon nanotube structures and vibrational modes, which provide unequivocal data for the experimental characterization of carbon nanotubes. An empirical relationship of radial breathing modes frequencies with the nanotube radius are also obtained for two-layer carbon nanotubes.

Vibration modeling and testing of bilayer beams for determination of film elastic modulus

Analysis of the main parameters affecting the fundamental vibrating frequency of film/substrate bilayer beams of rectangular cross-section is discussed based on modeling and testing. Initially, the limits of validity of two analytical models to obtain the fundamental frequency of perfectly-bonded bilayer beams in cantilever configuration are determined by comparing the predicted frequencies to a finite element model developed herein. Using a selected analytical formulation, a modeling-assisted methodology is employed to investigate the parameters that are most influential on the determination of the elastic modulus of the film using a vibratory technique. Modeling suggests the use of thin compliant substrates for extracting the modulus of stiff (metallic) films. If the substrate is stiffer than the film, a thicker film is required to yield measurable shifts in the resonant frequency. The elastic modulus of a millimeter-thick thermosetting polymer extracted by this method agrees with the results obtained from conventional tensile testing of the polymer. Measurements carried out on a gold (100 nm)/polysulfone (130 µm) system yield an average elastic modulus of the gold film similar to the values reported in the literature.

Wave propagation characteristics of a twisted micro scale beam

Twisted structural elements are inherently complex and their use in engineering systems requires deeper understanding. In this paper, starting with the modified couple stress theory, the elastodynamics governing partial differential equations of motion for the transverse dynamics of a twisted micro scale beam is derived. A micro scale beam of rectangular cross-section, for which the rate of twist introduced a bending–bending coupling effect, is considered. The presented governing equation of motion is used to address the effects of the rate of twist and the material length scale on the bending wave propagation characteristics of the micro scale beam. Results are presented for the spectrum curve, the cut-off frequency, the phase speed and the group velocity of a propagating harmonic wave profile in the twisted micro scale beam. It is observed that the rate of twist bifurcates the spectrum curve of the micro scale beam within a given frequency range, while the material length scale improves the dispersion of the traveling wave. The cut-off frequency is found to be independent of the material length scale, but proportional to the fourth power of the rate of twist. Increasing the material length scale is further observed to increase the group velocity of the wave, while a high rate of twist lowers the wave speeds.

Bending moments and stress state in the protective roof shelf of exploitation field, left between different mining technologies

This paper analyzes the protective roof shelf as a continuous loaded carrier beam of rectangular cross section. In this case, the statically undetermined sizes are moments in the supports. Support moments are determined using the equation of 'three moments' by Clapeyron. Stress control in the carrier beam is carried out starting from the differential equation of elastic line that has a form per the second statement.

The tones of the kalimba (African thumb piano)

The acoustic spectrum of the kalimba (African thumb piano) is measured and analyzed for tonal structure. The frequency f(1) of the fundamental tone of each tine (key) is investigated in relation to the frequencies of its two dominant overtones, f(2) and f(3). These frequencies are identified as the first three modes of transverse vibration of a beam of rectangular cross section. As is typical for vibrating-beam instruments, the overtone sequence is inharmonic, that is, the sequence f(1), f(2), f(3),[ellipsis (horizontal)] is unevenly spaced and the frequency ratios f(2)/f(1) and f(3)/f(1) are not integers. The kalimba tines are modeled by applying the Euler-Bernoulli beam equation with one end clamped, the other end free, and an intermediate point (the bridge) simply supported. Unlike the cases of free-free and clamped-free beams, it is found that the clamped-supported-free frequency ratios f(2)/f(1) and f(3)/f(1) are not fixed values, but depend uniquely upon where the bridge supports and subdivides the tine. The model solution is more thoroughly investigated analytically for the special case in which the beam segment ratio is unity, which has some analytic solutions. Numerically computed mode frequencies agree well with acoustic measurements, validating the model. Mode shapes are computed for the first three modes of a typical tine.

Assessment of Fatigue Cracks in Rails

The fatigue behaviour of cracks at the foot region of a rail subjected to bending load has been investigated. Depending on the position of initiation of a small semi-elliptical surface crack in a rail, the crack front changes shape during propagation to failure due to the large variation of the stress intensity factor from point to point round the crack front due to differences in the local stress field as each point round the crack front lies at different distances from the neutral axis of the rail. This condition implies a variable crack growth rate that transforms the crack front shape during fatigue crack propagation. SIF values have been estimated by means of both the finite element method and analytical solutions derived for a semi-elliptical crack in a finite rectangular cross-section beam. The SIF value predictions obtained with the two methods show good agreement suggesting that the analytical solutions can be used for a rapid assessment of the severity of a flaw in a rail. A predictive model for crack growth has been derived for an initial small crack at an initiation point at the foot/web corner of a rail tested under four point bending fatigue in the laboratory, showing a reasonably good prediction of both the shape and size of the crack at failure when compared with experiment.

Comparison studies between two wavelet based crack detection methods of a beam subjected to a moving load

In this paper, the performances of two wavelet based damage detection approaches to find the location and the size of a crack in a beam subjected to a moving load are compared. In the first approach, designated as ‘Fixed Sensor Approach’, a sensor is assumed to be located at the mid span of the beam. Thus in this approach, Continuous Wavelet Transform (CWT) coefficient of the time varying deflection attributed to the beam at mid span is used. In the second approach, the sensor is attached to the moving load and CWT coefficient of moving sensor is analyzed. The crack is modeled as a rotational spring which its stiffness is obtained from fracture mechanics, and modal expansion theory is used to find the deflection of cracked beam. The two approaches are compared for several cases of moving load speeds, crack sizes and crack locations, and also several scales of CWT. In the both approaches, wavelet Gaussian 4 is employed. It is shown that, the highest magnitude of CWT coefficient occurs at the exact location of crack and its value depends on damage size. As a result, it can be considered as a damage index. It is found that the moving sensor approach is more effective than the fixed sensor. Thereafter, the effect of beam parameters such as length, width, modulus of elasticity and density and also, crack size and location on the damage index were investigated based on the moving sensor approach. The obtained damage index has two significant features: (a) it has an explicit expression and shows the effects of all parameters clearly, (b) it is not dependent upon the response of the undamaged beam. From the obtained explicit relation for damage index, it is illustrated that the damage index does not depend on material properties of a homogeneous beam. Moreover in a rectangular cross section beam, the damage index is independent of width and has a linear relationship with the height to length ratio. It is shown that using this damage index, small cracks with a depth more than 10% of the beam height can be detected.

Flexural Behavior and Ductility of Concrete Beams Strengthened with Near-Surface Mounted GFRP Bars

The objective of this study is to investigate the flexural behaviors and ductility of reinforced concrete beam with near-surface mounted (NSM) GFRP bars. A total of 5, simply supported rectangular cross-section beams strengthened with NSM GFRP bars and two control beams are tested in this paper. The mode of failure, load-deflection relationship, strain distributions along the height of the beam on mid-span cross-section, and ductility of the beams are analyzed. The studies indicate that the yield and ultimate load have a significant increase compared with the control specimen. Especially, the ultimate load has a more measurable improvement than the yield load. The strain distribution along the height of specimen shows a clear nonlinear characteristic as a result of the crack propagating and steel yield. Also, the studies show that GFRP bars has played an important role in the flexural capacity of the strengthened beam during the stage from the yield to the failure of the beam. Besides, the strengthened beam is considered to be ductile using the displacement ductility index.

Dynamic Deformation Behavior Experiments of Alloy Galvanized Deep Drawing Steel Sheet for Automobile and Simulation

Quasi-static and dynamic tensile tests are carried out on alloyed galvanized deep drawing steel sheet for automobile(DC53D+ZF) to investigate its tensile mechanical behavior under different strain rates.Stress-strain relationships are obtained in strain rate range of 0～500 s–1.Johnson-Cook constitutive model of the steel is derived from the experiment results.Model without considering strain rate,Johnson-Cook,Cowper-Symonds model and experiment data are compared by simulation model of rectangular cross section beam collision.The results show that the J-C constitutive model derived is in good agreement with the test results,which proves the parameters obtained can well describe the dynamic deformation behavior of DC53D+ZF under high strain rate.