Thermoelastic lateral-torsional buckling of fixed slender beams under linear temperature gradient

The thermal expansions and rotations that result from a linear in-plane temperature gradient field are fully restrained at the ends of a fixed beam. These restrained expansions and rotations will produce internal bending and compressive actions in the beam, and these actions increase with an increase of the temperature differential and average temperature of the linear temperature gradient field. When these actions reach critical values, the fixed beam may bifurcate from its primary equilibrium state to a buckled equilibrium configuration. This paper presents a systematic treatment of classical buckling analysis for thermoelastic lateral-torsional buckling and for in-plane thermoelastic flexural buckling of a fixed beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is shown that the effective centroid and shear centre, rather than the geometric centroid and shear centre, should be used in formulating the thermoelastic prebuckling and buckling analysis and that the effects of temperature on the buckling resistance need to be considered. The thermoelastic lateral-torsional buckling of a fixed beam under a linear temperature gradient field is more complicated than its mechanical counterpart for uniform bending or for uniform compression, and iterative methods are needed to obtain accurate solutions.     

Effects of prebuckling deformations on the elastic flexural-torsional buckling of laterally fixed arches

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial.     

Nonlinear in-plane elastic buckling of shallow circular arches under uniform radial and thermal loading

This paper presents a nonlinear in-plane elastic buckling analysis of circular shallow arches that are subjected both to a uniform temperature field and to a uniform radial load field. A virtual work method is used to establish nonlinear equilibrium equations and buckling equilibrium equations, and analytical solutions for the limit instability and bifurcation buckling loads are obtained. It is found that the temperature influences the limit instability, bifurcation buckling and postbuckling behaviour of shallow arches significantly. The limit instability and bifurcation buckling loads increase with an increase of the temperature. A maximum temperature is shown to exist for the occurrence of bifurcation buckling of shallow arches, and when the temperature is higher than this value, bifurcation buckling of an arch is not possible.An arch geometric parameter is introduced to define switches between the limit instability and bifurcation buckling modes, and between buckling and no buckling. Formulae and methods for the calculation of the limiting values of the arch geometric parameter are developed. It is also found that the limiting values of the arch geometric parameter decrease with an increase of the temperature.     

Application d'un modèle de flambement conditionné à la pose des câbles à fibres optiques

Application of a conditioned buckling solution to the laying of optical cables. Mechanical behaviour of cables submitted to an axial compressive load is subjected to buckling instabilities due to their length compared with their diameter. In some cases rigid walls modify the boundary conditions during the loading or the buckling phenomenon, it will be called conditioned buckling. This paper presents the buckling of a homogeneous cable in a horizontal circular rigid duct subjected to its own weight and an axial compressive load. Buckling load and pitch associated to sinusoidal and helical buckling are determined as transmission of axial load during buckling. A finite element simulation is computed and comparisons are made with the analytical solution. A direct application to the optical fibre cable laying in underground duct is made and experimentation is conducted with a life-sized bench designed at the LM ² S of the ENSAM of Paris and located at the CNET Lannion. Comparisons between experimental and analytical results are presented.     

Out of plane stability of circular arches

An elastic buckling theory is developed for thin-walled arches. Using the principle of minimum total potential energy derives the governing differential equations. An explicit and clear approximation of the curvature effect is made in the derivation process. Closed form solutions are obtained for arches subjected to equal and opposite end moments (uniform bending) and to uniformly distributed radial loads (uniform compression). Also, closed form solutions for the torsional buckling moment of braced arches in uniform bending and for the torsional buckling load of braced arches in uniform compression are obtained. The solutions are compared with previous theoretical results.     

Parametric studies on buckling loads and critical speeds of microdrill bits

Transverse bending vibrations of the spinning microdrill bit subjected to a compressive axial load are developed based on the Timoshenko beam theory. The system equations of motion are discretized into the form of time-dependent ordinary differential equations by the finite element method. Two types of eigenvalue problems are formulated and utilized to study the effects of the drill helix angle, flute length and diameter on the buckling load and critical speed of microdrill bits with different supported ends. Equivalent formulae similar to those of untwisted Euler beams are established to predict critical buckling loads and critical speeds for microdrills and provide results with sufficient accuracy. The effect of rotational speed on the buckling load, and the influence of thrust force on critical speed are also investigated. A Galef-type equation associated with critical speed, thrust force and buckling load is formulated.     

Elastic flexural-torsional instability of structural arches under hydrostatic pressure

This paper addresses the mechanics of the flexural-torsional buckling instability of pin-ended elastic circular arches, which are acted upon by a hydrostatic loading. This loading arrangement differs from the gravity-based loading usually considered in the literature, in that the load changes its direction with the deformation of the elastic arch during its flexural-torsional buckling, always remaining normal to the contour profile of the arch. The previous treatments of the mechanics of the problem, that assume the load direction remains invariant during flexural-torsional buckling, have been motivated by applications in structural engineering in which this loading regime is valid, but there are a number of applications in more general mechanics where this assumption cannot be made and a solution is needed. Both a mathematically based virtual work principle and a mechanical visualisation of the mechanics of the deformation are considered separately, and they are shown to arrive at the same formulation of the linear differential equations of equilibrium of the buckled arch when the buckling deformations are considered infinitesimal. The differential equations for buckling under radial loading that is distributed uniformly around the circumference of the arch are shown to be solvable in analytic form, resulting in a closed form solution for the elastic buckling load of the arch that hitherto has not been formulated. The buckling equation demonstrates that an arch is stiffer under hydrostatic loading than under gravity loading in its resistance to elastic flexural-torsional buckling.     

Energy release rate of postbuckled laminates with a through-width delamination

The strain energy release rate is calculated for buckled one-dimensional delamination (through-width delamination) in composite laminates subjected to in-plane compression. A crack closure method based on plate finite elements is used in this analysis. For some laminates containing a one-dimensional delamination in cylindrical bending, closed form solutions are available. The present finite element solutions show excellent agreement with the analytical solutions. The strain energy release rate for various types of laminates is also calculated using the present finite element method. The results show that the strain energy release rate strongly depends on the type of laminate.     

Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.     

The buckling and post-buckling behaviour of composite plates under biaxial loading

The response to applied load of general composite plates exhibits a coupling between the bending and extensional modes of deformation which may be significant when shear or compressive loads are applied in-plane. The additional deformation modes may affect the nature of the buckling behaviour, reduce the buckling load or change the post-buckled stiffness. This paper considers the stiffness immediately after buckling of a rectangular panel of angle-ply type in which coupling effects occur and which undergoes bifurcational buckling when biaxial load is applied in directions parallel to the panel edges. Expressions are derived for the buckling loads and for the in-plane stiffness of the panel immediately after the instant of buckling, it being found that the coupling terms affect the stiffness at buckling mainly via the associated change in buckling mode shape.     

A post-buckling analysis for isotropic prismatic plate assemblies under axial compression

This paper presents a post-buckling analysis for prismatic plate assemblies made of isotropic materials. The structures are assumed to consist of a series of long flat strips rigidly connected together at their edges, subjected to longitudinal in-plane compressive load. The buckling load and corresponding buckling mode of the structure are first obtained as the results of transcendental eigenvalue problems, which arise when exact solutions to the member differential equations are used to form the stiffness matrix of the plate assemblies. The other post-buckling field functions are also obtained analytically as exact solutions to the member differential equations. Results for the load end-shortening and load–deflection relationships for long prismatic plate assembly examples are obtained and compared with results obtained by other authors.     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

循环热-机械双轴载荷下薄壁圆筒的热棘轮极限

热棘轮失效是薄壁圆筒的主要失效模式之一,现有ASME锅炉及压力容器规范和EN13445等设计标准主要考虑环向应力,而未考虑轴向应力条件,使得设计结果可能偏于不安全。针对循环热-机械双轴载荷下薄壁圆筒热棘轮设计理论的不足,采用非循环分析方法系统研究双轴应力状态下薄壁圆筒热棘轮极限的解析解,重点考虑轴向压缩应力对循环温度梯度和稳定内压组合载荷下薄壁圆筒热棘轮极限的影响,并提出相应的设计方法,并采用有限元法对理论结果进行验证。结果表明,循环热-机械载荷下轴向压缩应力会显著降低薄壁圆筒的热棘轮极限,且理论解与有限元分析结果吻合良好,这说明此方法可用于循环热-机械双轴载荷及类似工况下薄壁圆筒的热棘轮设计限,具有良好的工程价值。     

Elastic stresses for 90° elbows under in-plane bending

Using finite element analysis, this paper extends elastic stress solutions for 90° pipe elbows under in-plane bending, given in Marie et al. (2007) [1], to cases of mean pipe radius-to-thickness ratio up to 50. It is found that for 90° elbows an in-plane bending moment produces not only an axial membrane stress component but also axial and hoop bending stress components. Furthermore, the magnitudes of these stress components depend strongly on the mean radius-to-thickness ratio, the circumferential location and the longitudinal location. Maximum stresses tend to occur in the centre of the elbow at or near the crown.     

Subsurface crack mechanisms under indentation loading

A linear elastic fracture mechanics analysis of plane-strain indentation of a homogeneous half-space with a subsurface horizontal crack was performed using the finite element method. Stress intensity factor results obtained for an infinite plate with a central crack subjected to far-field tension and a half-space with a frictionless subsurface horizontal crack under a moving surface point load are shown to be in good agreement with corresponding analytical results. Crack mechanism maps illustrating the occurrence of separation, forward and backward slip, stick, and separation at the crack interface are presented for different indentation load positions and crack face friction coefficients. Results for the stresses in the vicinity of the crack tips and the mode I and mode II stress intensity factors are given for different indentation positions, crack face friction coefficients, and both concentrated and distributed surface normal tractions. Although indentation produces a predominantly shear and compressive stress field, mode I loading conditions are shown to occur for certain indentation positions. However, the magnitude of the mode I stress intensity factor is significantly smaller than that of mode II, suggesting that in-plane shear mode crack growth is most likely to occur in the absence of microstructural defects. The significance of crack face friction and sharpness of the indenter on the subsurface shear mode crack propagation rate is interpreted in terms of the mode II stress intensity factor range and material behavior.     

Dynamic instability analysis of stiffened shell panels subjected to partial edge loading along the edges

The dynamic instability characteristics of stiffened shell panels subjected to partial in-plane harmonic edge loading are investigated in this paper. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the shell panels and the stiffeners, respectively. As the usual formulation of degenerated beam element is found to overestimate the torsional rigidity, an attempt has been made to reformulate it in an efficient manner. Moreover, the new formulation for the beam element requires five degrees of freedom per node as that of shell element. The method of Hill's infinite determinant is applied to analyze the dynamic instability regions. Numerical results are presented through convergence and comparison with the published results from the literature. The effects of parameters like loading type and shell geometry are considered in the dynamic instability analysis of stiffened panels subjected to non-uniform in-plane harmonic loads along the boundaries. The tension buckling aspect of the stiffened panels are also considered and the dynamic stability behavior due to tensile in-plane edge loading is studied for the concentrated load.     

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.     

Thermal buckling analysis of annular FGM plate having variable thickness under thermal load of arbitrary distribution by finite element method

In this paper, a finite element formulation is developed for analyzing the axisymmetric thermal buckling of FGM annular plates of variable thickness subjected to thermal loads generally distributed nonuniformly along the plate radial coordinate. The FGM assumed to be isotropic with material properties graded in the thickness direction according to a simple power-law in terms of the plate thickness coordinate, and has symmetry with respect to the plate midplane. At first, the pre-buckling plane elasticity problem is developed and solved using the finite element method, to determine the distribution of the pre-buckling in-plane forces in terms of the temperature rise distribution. Subsequently, based on Kierchhoff plate theory and using the principle of minimum total potential energy, the weak form of the differential equation governing the plate thermal stability is derived, then by employing the finite element method, the stability equations are solved numerically to evaluate the thermal buckling load factor. Convergence and validation of the presented finite element model are investigated by comparing the numerical results with those available in the literature. Parametric studies are carried out to cover the effects of parameters including thickness-to-radius ratio, taper parameter and boundary conditions on the thermal buckling load factor of the plates.     

Dynamic relaxation method for nonlinear buckling analysis of moderately thick FG cylindrical panels with various boundary conditions

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

     

Analytical study on elastic transition in short-fiber composites for plane strain case

An analytical approach for short-fiber-reinforced composites is developed for three-dimensional (3D) elastic stress field distribution subjected to an applied axial load. Two sets of exact displacement solutions for matrix and fiber, which are respectively called far-field and transient solutions, are derived based on the theory of elasticity. The superposition state of these solutions are then used to obtain the analytical expressions for the 3D stress field components over the entire composite system, including the fiber end region, through the adding imaginary fiber technique. The fiber/matrix 3D stress field components fully satisfy the equilibrium and compatibility conditions in the theory of elasticity. The stress field components also satisfy the overall boundary, interface continuity, and axial force equilibrium conditions. The analytical results obtained are then validated by finite element method modeling.