Euler Beam Theory Research Articles

Capillary collapse of a micro-double cantilever beam: a rigorous study

High-aspect-ratio microstructures have been found, in the literature, to collapse due to capillary forces of liquids. In this paper, mathematical models are developed to study the collapse of a microstructure represented by a double cantilever beam (DCB) with a liquid droplet located at the free end. Formulations are presented using the classical Bernoulli–Euler beam theory as well as an analysis that accounts for geometrical nonlinearity. The models introduce rigorous coupling between the DCB deformation, the capillary forces, and the meniscus position, and have predicted interesting nonlinear behaviors that previous models could not. Parameters governing the capillary collapse of the DCB are identified, and their influence is discussed. A single dimensionless number that controls the condition for collapse is proposed and validated against numerical results. Comparison between the linear and nonlinear beam analyses shows that linear analysis generally suffices for the description of capillary collapse of microstructures.

Mixed-mode partition theories for one-dimensional delamination in laminated composite beams

Completely analytical theories are presented for the mixed-mode partitioning of one-dimensional delamination in laminated composite beams. The work builds on previous research by the authors on one-dimensional fractures in layered isotropic beams. The partition theories are developed within the contexts of both Euler and Timoshenko beam theories. Two sets of orthogonal pairs of pure modes are found and used to partition mixed modes. Approximate ‘averaged partition rules’ are also established for 2D elasticity. The beam partition theories and averaged rules are extensively validated against numerical simulations using the finite element method (FEM). The contact behavior of double cantilever beams (DCBs) is also investigated. Two types of contact exist: crack tip running contact, which results in a region of pure mode II; and point contact at the DCB tip, which can result in either in mixed modes or pure mode II.

Analytical solution of a bilayer functionally graded cantilever beam with concentrated loads

A general two-dimensional solution for a bilayer functionally graded cantilever beam with concentrated loads at the free end is developed. The beam is treated as a nonhomogeneous plane stress problem, and the elastic modulus of each graded layer varies with the thickness as an arbitrary function, respectively, which makes this analytical solution has a wider application than the solutions only considering FGMs as core materials or modeling the face sheets by Euler beam theory. It can be verified the paper’s close solution coincides with the classical one for beams while let the elastic modulus be constants. The methodology presented here could be easily extended to analyze the functionally graded sandwich beams.

The Improved Mechanics Behavior of the FRP Deck

The objective of this study is to improve the stiffness and control the failure mode of the pultruded glass fiber reinforced polymers (GFRP) deck members to meet the design requirements. In this study, the authors proposed that the GFRP member was affixed by another GFRP deck inside it to form a combined GFRP deck member; an epoxy mortar was filled into some of the hollow sections of the GFRP member; and the basalt/carbon fiber sheets was wrapped around the GFRP member in order to improve the ultimate strength and stiffness and also to change its original failure mode of the GFRP member. Eight GFRP deck specimens where investigated in the three-point bending test results for their ultimate strength, stiffness, and corresponding failure modes. Finally, the Euler beam theory was used to obtain the force-displacement relationships of the GFRP members and then to compare the experimental results.

Dynamic stability and instability of a double-beam system subjected to random forces

On the basis of the Bernoulli–Euler beam theory, the stability and instability of a double-beam system subjected to compressive axial loading is investigated. It is assumed that the two beams of the system are simply supported and continuously joined by a Winkler elastic layer. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. By using the direct Lyapunov method, bounds of the almost sure stability and instability and uniform stochastic stability of a double-beam system as a function of viscous damping coefficient, bending stiffness, stiffness modulus of the Winkler layer, variances of the stochastic forces, and intensity of the deterministic components of axial loading are obtained. When the almost sure stability and instability are investigated, numerical calculations are performed for the Gaussian process with a zero mean as well as a harmonic process with random phase. When axial forces are white noise processes, conditions for uniform stochastic stability are determined.

Elastic modulus and Poisson’s ratio determination of micro-lattice cellular structures by analytical, numerical and homogenisation methods

The mechanical properties of micro-lattice structures made of interconnected metallic struts are calculated analytically, in order to derive their elastic modulus and Poisson’s ratio in the three Cartesian directions. The geometry of the investigated unit cell is a body centered cuboid, which is a more generic case of the well-known body centered cubic geometry. The Bernoulli–Euler and Timoshenko beam theories are used for the analytical solution of the unit cell response under complex loading. The derived elastic constants are compared to the relative numerical ones. The influence of geometrical parameters on the stiffness coefficients of the cellular structure is parametrically determined. The derived elasticity matrices are introduced in a homogenised numerical model of the core and its results are compared to a numerical model with explicit representation of the core geometry indicating an excellent accuracy, while the solution time is considerably reduced.

Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.

The elastic behavior of aluminum alloy foam under uniaxial loading and bending conditions

The elastic behavior of aluminum alloy foam under uniaxial loading and bending conditions is studied by experimental investigations and theoretical analyses. It is found from the experiments that the elastic moduli measured from three- and four-point bending tests as well as flexural vibration tests are significantly larger than the corresponding elastic moduli measured from uniaxial compression tests. This discrepancy cannot be described by the well-known elastic modulus equation introduced by Gibson and Ashby for foam materials. Theoretical analyses are carried out to explain the discrepancy and to improve the elastic modulus equation of Gibson and Ashby. In the analyses, unit-cell models are proposed, and elastic modulus equations for both uniaxial loading and bending conditions are derived. The derived equations offer a good description of the discrepancy as well as reveal the prominent role of cell local deformation under bending condition. Furthermore, the present work offers clear evidence that the classical continuum theories, such as Bernoulli–Euler beam theory, are inadequate for describing the elastic bending behavior of metal foams.

On the dynamic stability of embedded single-walled carbon nanotubes including thermal environment effects

Based on the nonlocal Bernoulli–Euler and Timoshenko beam theories, the dynamic stability of embedded single-walled carbon nanotubes (SWCNTs) under axial compression is studied in a thermal environment. The developed nonlocal models have the capability to interpret small scale effects. A Winkler-type elastic foundation is employed to represent the interaction of the SWCNT and the surrounding elastic medium. The free vibration and axial buckling of SWCNTs are discussed as subset problems. A parametric study is conducted to investigate the influences of the static load factor, temperature change, nonlocal elastic parameter, slenderness ratio and spring constant of the elastic medium on the dynamic stability characteristics of the SWCNTs, with simply-supported boundary conditions. It is found that the difference between instability regions predicted by local and nonlocal beam theories is significant for nanotubes with lower aspect ratios. Moreover, it is observed that in contrast to high temperature environments, at low temperatures, increasing the temperature change moves the origins of the instability regions to higher excitation frequencies and leads to further stability of the system at lower excitation frequencies.

Bending relationships between the modified couple stress-based functionally graded Timoshenko beams and homogeneous Bernoulli–Euler beams

The Bernoulli–Euler and Timoshenko beam theories are reformulated using a modified couple stress theory and through-thickness power-law variation of a two-constituent material [functionally graded material (FGM)]. The model contains a material length scale parameter that can capture the size effect in a FGM. The equations are then used to develop algebraic relationships for the deflections, slopes, stress resultants of the Timoshenko beam theory (TBT) for microstructure-dependent FGM beams in terms of the same quantities of the conventional Bernoulli–Euler beam theory (BET). The relationships allow determination of the solutions of the TBT for microstructure-dependent FGM beams whenever solutions based on the BET are available. Examples of the use of the relationships are presented using straight beams with simply supported and clamped boundary conditions.

Experimental assessment of mixed-mode partition theories

The propagation of mixed-mode interlaminar fractures is investigated using existing experimental results from the literature and various partition theories. These are (i) a partition theory by Williams (1988) based on Euler beam theory; (ii) a partition theory by Suo (1990) and Hutchinson and Suo (1992) based on 2D elasticity; and (iii) the Wang–Harvey partition theories of the authors based on the Euler and Timoshenko beam theories. The Wang–Harvey Euler beam partition theory seems to offer the best and most simple explanation for all the experimental observations. No recourse to fracture surface roughness or new failure criteria is required. It is in excellent agreement with the linear failure locus and is significantly closer than other partition theories. It is also demonstrated that the global partition of energy release rate when using the Wang–Harvey Timoshenko beam or averaged partition theories or 2D elasticity exactly corresponds with the partition from the Wang–Harvey Euler beam partition theory. It is therefore concluded that the excellent performance of the Wang–Harvey Euler beam partition theory is either due to the failure of materials generally being based on global partitions or that for the specimens tested, the through-thickness shear effect is negligibly small. Further experimental investigations are definitely required.

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.

Flexibility Factors for High Density Polyethylene Elbows

A method is proposed to calculate Flexibility Factors for molded and mitered 90 deg polyethylene elbows for both in-plane and out-of-plane loading based on Code Case N-319-3, January 17, 2000. Displacement measurements in the direction of the applied forces were made during the testing of each elbow in addition to measuring the applied forces. By making use of Code Case equations, Flexibility Factors can be directly developed. Appendices to the paper show how these factors can be added to the deformation and rotation equation from curved Euler beam theory. One of the basic assumptions used in the Code Case and in this derivation is that there is no ovalization from torsion. Thus, the Flexibility Factor for the torsional displacement (twisting) of an elbow is one. Results show that an exact solution for Flexibility Factors based on the definition in Section III of the ASME Boiler and Pressure Vessel Code can be obtained for in-plane loading. However, for out-of-plane loading, the Flexibility Factors are geometry dependent if the values are based on measured elbow deflections. The difference is associated with the contributions of both torsion and bending to elbow deflections.

Numerical and analytical study of delamination in composite laminates

A robust numerical method is developed to study delamination in composite beam structures under lateral and axial loads. A tensor symmetrisation technique is used to formulate the beam element based on the Euler beam theory with full geometrical non-linearity to achieve high computational efficiency. Ply interfaces are modelled with high-stiffness springs. It is found that the beam element suffers from membrane locking for non-symmetric laminates. A method is found to overcome it. The model is used to simulate double cantilever composite beam structure tests and end notched flexure tests. Excellent agreement is observed with analytical and existing numerical and experimental data. The model is also used to study the buckling, post-buckling and delamination propagation in moderately thick composite beams. Satisfactory agreement is demonstrated between the present predictions and existing numerical and experimental data. It is noted that the through-thickness shear effect is significant for moderately thick composite laminates.

Mixed mode partition theories for one dimensional fracture

The crack in a double cantilever beam is the most fundamental one-dimensional fracture problem. It has caused considerable confusion due to its in-depth subtleness and complex entanglement with different theories and numerical simulations. The present paper presents completely analytical theories based on Euler and Timoshenko beam theories using a brand new approach which reveals the hidden mechanics of the problem. Orthogonal pairs of pure modes are found and used to partition mixed modes. The developed theories are extensively validated against numerical simulations using finite element methods. Moreover, the fracture mode partition space is thoroughly investigated and crack tip running contact is found which results in a region of pure mode II. The theories are finally applied to general one-dimensional fracture in beams and axisymmetric plates.

FREE VIBRATION ANALYSIS OF DOUBLE-WALLED CARBON NANOTUBES USING THE SMOOTHED FINITE ELEMENT METHOD

This paper studies the resonant frequencies and the associated vibration modes of an individual double-walled carbon nanotube, using gradient smoothing technique. The study uses a double-elastic beam based on the Euler beam theory with the consideration of the intertube van der Waals interactions. A gradient smoothed formulation is deployed to deal with the fourth-order differential equation of the Euler beam problems for dynamic analysis of the double-walled carbon nanotubes. In the double-thin beam problems, a set of linear shape functions are adopted to approximate the displacement field. Smoothing domains are then formed for computing the smoothed curvature and bending moment field. Compared with analysis results of multi-beam theory, numerical examples indicate that very accurate results could be yielded when a reasonable number of nodes were used. The results also showed that non-coaxial intertube resonance would be excited at the higher resonant frequencies of double-walled carbon nanotubes. In addition, free vibration of a double-walled carbon nanotube with inner wall simple supported and outer wall one end fixed and the other end free is studied.

A theory of one-dimensional fracture

A completely analytical theory is developed for the mixed mode partition of one-dimensional fracture in laminated composite beams and plates. Two sets of orthogonal pure modes are determined first. It is found that they are distinct from each other in Euler beam or plate theory and coincide at the Wang-Harvey set in Timoshenko beam or plate theory. After the Wang-Harvey set is proved to form a unique complete orthogonal pure mode basis within the contexts of both Euler and Timoshenko beam or plate theories, it is used to partition a mixed mode. Stealthy interactions are found between the Wang-Harvey pure mode I modes and mode II modes in Euler beam or plate theory, which alter the partitions of a mixed mode. The finite element method is developed to validate the analytical theories.

Dynamic Response of Electromagnetic Launcher’s Rail Subjected to Cosine Pressure

In order to extend the rail life and improve the firing accuracy, the electromagnetic launcher’s rail can be modeled as a beam on elastic foundation with simply supported beam by moving load. Euler beam theory is applied to build the Mechanical model and the analytical solution of the equation subjected to cosine magnetic pressure is derived in detail, which has successfully avoided the errors which are caused by using the uniform pressure to approximately replace the variable force. Numerical analysis of the dynamic response on rail by the MATLAB software shows that as the exciting frequency increases, the peek of the maximal deflection and vibration velocity increase gradually. Taking the same speed of load into account, the dynamic response of rail is obviously smaller than constant force. Therefore the reliable theory basis is provided for the design and control the rail, which promotes the practical application of the electromagnetic Launcher.

Atomistic study of the buckling of gold nanowires

In this work, we present results from atomistic simulations of gold nanowires under axial compression, with a focus on examining the effects of both axial and surface orientation effects on the buckling behavior. This was accomplished by using molecular statics simulations while considering three different crystallographic systems: 〈1 0 0〉/{1 0 0}, 〈1 0 0〉/{1 1 0} and 〈1 1 0〉/{1 1 0}{1 0 0}, with aspect ratios spanning from 20 to 50 and cross-sectional dimensions ranging from 2.45 to 5.91 nm. The simulations indicate that there is a deviation from the inverse square length dependence of critical forces predicted from traditional linear elastic Bernoulli–Euler and Timoshenko beam theories, where the nature of the deviation from the perfect inverse square length behavior differs for different crystallographic systems. This variation is found to be strongly correlated to either stiffening or increased compliance of the tangential stiffness due to the influence of nonlinear elasticity, which leads to normalized critical forces that decrease with decreasing aspect ratio for the 〈1 0 0〉/{1 0 0} and 〈1 0 0〉/{1 1 0} systems, but increase with decreasing aspect ratio for the 〈1 1 0〉/{1 1 0}{1 0 0} system. In contrast, it was found that the critical strains are all lower than their bulk counterparts, and that the critical strains decrease with decreasing cross-sectional dimensions; the lower strains may be an effect emanating from the presence of the surfaces, which are all more elastically compliant than the bulk and thus give rise to a more compliant flexural rigidity.

Interaction of subway LIM vehicle with ballasted track in polygonal wheel wear development

This paper develops a coupled dynamics model for a linear induction motor (LIM) vehicle and a subway track to investigate the influence of polygonal wheels of the vehicle on the dynamic behavior of the system. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom. A Timoshenko beam is used to model the rails which are discretely supported by sleepers. The sleepers are modeled as rigid bodies with their vertical, lateral, and rolling motions being considered. In order to simulate the vehicle running along the track, a moving sleeper support model is introduced to simulate the excitation by the discrete sleeper supporters, in which the sleepers are assumed to move backward at a constant speed that is the same as the train speed. The Hertzian contact theory and the Shen-Hedrick-Elkins’ model are utilized to deal with the normal dynamic forces and the tangential forces between wheels and rails, respectively. In order to better characterize the linear metro system (LMS), Euler beam theory based on modal superposition method is used to model LIM and RP. The vertical electric magnetic force and the lateral restoring force between the LIM and RP are also taken into consideration. The former has gap-varying nonlinear characteristics, whilst the latter is considered as a constant restoring force of 1 kN. The numerical analysis considers the effect of the excitation due to polygonal wheels on the dynamic behavior of the system at different wear stages, in which the used data regarding the polygonal wear on the wheel tread are directly measured at the subway site.