Membrane Analogy of Buckling and Vibration of Inhomogeneous Plates

Exact explicit eigenvalues are found for compression buckling, hygrothermal buckling, and vibration of sandwich plates with dissimilar facings and functionally graded plates via analogy with membrane vibration. These results apply to simply supported polygonal plates using the first-order shear deformation theory and the classical theory. A Winkler-Pasternak elastic foundation, a hydrostatic inplane force, hygrothermal effects, and rotary inertias are incorporated. Bridged by the vibrating membrane, exact correspondence is readily established between any pairs of eigenvalues associated with buckling and vibration of sandwich plates, functionally graded plates, and homogeneous plates. Positive definiteness is proved for the critical buckling hydrostatic pressure and, in the range of either tension loading or compression loading prior to occurrence of buckling, for the natural vibration frequency.     

Comparative Modeling Study of Reinforced Beam on Elastic Foundation

In this paper, governing ordinary differential equations are derived for a reinforced Timoshenko beam on an elastic foundation. An analytical solution is obtained for a point load on an infinite Timoshenko beam on elastic foundation. Special attention is drawn to the location, tension, and shear stiffness of reinforcement and its influence on settlement/deflection of the beam and reinforcement tension force. A finite element (FE) model is established for the same infinite beam problem. Results from the TB model (Timoshenko beam on elastic foundation) are compared with results from the FE model and from the PB model (the Winkler model, based on the pure bending beam theory). It is found that results from the proposed TB model are, in general, in good agreement with results from the FE model as compared with results from the PB model. The TB model is better than the PB model in considering the shear deformation of the beam. This TB model is particularly useful in modeling a reinforced beam with or without considering the reinforcement shear stiffness. The TB model has practical applications in modeling geosynthetics∕fiber-glass reinforcement of foundation soils or pavement.     

Effects of the Loading History on the Local Buckling Behavior and Failure Mode of Welded Beam-to-Column Joints in Moment-Resisting Steel Frames

The cyclic behavior of welded beam-to-column joints for moment-resisting steel frames was assessed by constant amplitude cyclic quasi-static tests. Reanalysis of the results showed that the failure mode of the joints strongly depends on cycle amplitudes. A premature brittle failure of the welds may occur if the cycle amplitude is not large enough to cause local buckling of beam flanges. Influence of both flange and web slenderness ratios on local buckling behavior, investigated through an experimental parametric study, is discussed. Four tests, carried out adopting different variable amplitude displacement histories, confirmed that isolated large amplitude cycles have beneficial effects on the joint response, extending its life; on the contrary, many large cycles clustered together endanger the seismic performance of beam-to-column joints. Numerical analyses allowed interpretation of the experimental data in terms of local stresses and strains. For “large amplitude” cycles numerical results indicate a local state of strain causing a progressive collapse for low-cycle fatigue while, in case of “small amplitude” cycles, brittle failure mode is due to the ratcheting of material.     

Finite Elements on Generalized Elastic Foundation in Timoshenko Beam Theory

Using the Vlasov foundation model, a modified approach of the continuous beam on elastic supports, leading to both a mechanical model and the proper foundation parameters of the generalized foundation is shown. Two formulations of the beam finite-element with shear deformation effect, resting on a two-parameter elastic foundation, characterized by distinct contributions of normal and rotary reactions are presented. The behavior of the second foundation parameter in the two formulations is governed by the bending cross section rotation of a beam. The first formulation, yielding a free-of-meshing stiffness matrix and equivalent nodal load vector, is based on the transcendental or “exact” solution of the governing differential equation of the beam resting on the elastic layer of constant thickness. Considering a linear variation of the layer thickness along the beam, the second formulation is based on the assumed polynomial displacement field. Numerical comparisons with the exact approach show that the cubic formulation leads to better results when the foundation parameters are variables. The practical utility of the analogy between a tensile axial force and the second foundation parameter is exemplified, too.     

Dynamic Axial-Moment Buckling of Linear Beam Systems by Power Series Stiffness

The paper applies the power series method to find the dynamic stiffness for the dynamics axial-moment buckling analyses of linear framed structures. Since the formulation is exact in classical sense, one element is good enough for the entire beam. The dynamic stiffness thus obtained can be decomposed into the stiffness, mass and initial stress matrices at a particular frequency, a particular axial force and a particular initial moment. The given axial force and moment can be nonuniformly distributed. The interaction diagrams in classical loading conditions of uniform moment, moment due to concentrated and distributed lateral force are given explicitly. The effects of warping rigidity, torsion rigidity, axial tension and compression are investigated in detail. The static and dynamic interaction buckling of a two-section I-beam structure is studied. Finally, we conclude that the three dimensional interaction diagram of the dynamic biaxial moment buckling can be obtained simply by rotating the three dimensional interaction diagram of the dynamic mono-axial moment buckling about the frequency axis if the bimoments are appropriately scaled. It is shown that application for non-uniform section is not suitable due to convergent problem. The method is very efficient that many interaction diagrams are produced for the first time.     

Forces on Plunge Pool Slabs: Influence of Joints Location and Width

The hydrodynamic pressures due to jets impinging on plunge pools must be taken into account in the stability design of pool floor concrete slabs. The contraction joints between slabs are normally sealed with waterstops, which prevent the transmission to the foundation of the pressures applied on the upper faces of the slabs. However, a waterstop failure will allow pressure transmission to the foundation, inducing uplift forces on the slabs. The use of open joints might also become a feasible solution for the lining of plunge pool floors if the pressure field that develops around each slab could be adequately evaluated. This paper presents an analytical model and experimental research developed to assess the forces on plunge pool slabs, considering either open or closed. The influence of the relative width of the contraction joints and the joint between the slabs and the foundation is analyzed. The mean value and standard deviation of the hydrodynamic vertical force are determined based on point pressure measurements, and their relative importance is discussed.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

Spatial Stability of Shear Deformable Nonsymmetric Thin-Walled Curved Beams: A Centroid-Shear Center Formulation

An improved shear deformable curved beam theory to overcome the drawback of currently available beam theories is newly proposed for the spatially coupled stability analysis of thin-walled curved beams with nonsymmetric cross sections. For this, the displacement field is introduced considering the second order terms of semitangential rotations. Next the elastic strain energy is newly derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. Then the potential energy due to initial stress resultants is consistently derived with accurate calculation of the Wagner effect. Finally, equilibrium equations and force–deformation relations are obtained using a stationary condition of total potential energy. The closed-form solutions for in-plane and out-of-plane buckling of curved beams subjected to uniform compression and pure bending are newly derived. Additionally, finite-element procedures are developed by using curved beam elements with arbitrary thin-walled sections. In order to illustrate the accuracy and the practical usefulness of this study, closed-form and numerical solutions for spatial buckling are compared with results by available references and ABAQUS’ shell elements.     

连退过程带钢热瓢曲指数及其影响因素

 针对带钢连退过程中的热瓢曲问题,充分考虑到连退机组的设备与工艺特点,从内力、外力两方面因素分析了热瓢曲以及横向压应力的产生机理,阐述了热瓢曲的发生规律,首次提出了瓢曲指数的新概念,给出了相应的瓢曲临界条件,建立了一套适合于连退过程的带钢热瓢曲指数的计算模型,定量分析了不同工艺段内带钢瓢曲指数分布规律以及相同工艺段内来料板形、横向温差、炉辊辊型、摩擦因数、设定张力、通板速度等因素对带钢热瓢曲指数的影响,并将相关理论应用到生产实践,开发出了“连退机组通板过程带钢瓢曲预报软件”,实现了对连退过程带钢瓢曲的在线预报,为瓢曲缺陷的预防和控制奠定了坚实的理论基础。     

Global and Local Buckling of a Sandwich Beam

A two-dimensional mechanical model is developed to predict the global and local buckling of a sandwich beam, using classical elasticity. The face sheet and the core are assumed as linear elastic isotropic continua in a state of planar deformation. The core is assumed to have two deformation modes: antisymmetrical and symmetrical with respect to the core geometric midplane. Characteristics of the two deformation modes and the corresponding buckling behavior are shown and it appears that they are identical when the buckling wavelength is short. The present analysis is compared with various previous analytical studies and corresponding experimental results. On the basis of the model developed here, validation and accuracy of several previous theories are discussed for different geometric and material properties of a sandwich beam. The results presented in this paper, verified through finite-element analysis and experiment, are an accurate prediction of the overall buckling behavior of a sandwich beam, for a wide range of material and geometric parameters.     

Wrinkling and Edge Buckling in Orthotropic Sandwich Beams

A sandwich beam buckling problem is studied here using two-dimensional elasticity to model the beam constituents. The global and local instability of such a beam with orthotropic constituents under various boundary conditions are investigated. The face sheet and the core are assumed to be linear elastic orthotropic continua. General buckling deformation modes of the sandwich beam subjected to uniaxial compressive loading are considered. The appropriate incremental stress and conjugate incremental finite-strain measure for the instability problem of the sandwich beam, and the corresponding constitutive model are addressed. It is shown that a sandwich beam having a core with a negligible stiffness compared to the face sheets is prone to fail by edge buckling. The present analysis is compared with several previous analytical studies and corresponding experimental results. Finite-element analyses are carried out for comparison against the theoretical predictions. The formulation used in the finite-element code is discussed in relation to the formulation adopted in the theoretical derivation.     

Initial Compressive Buckling of Clamped Plates Resting on Tensionless Elastic or Rigid Foundations

The unilateral contact buckling problem of thin plates resting on tensionless foundations is investigated. Three different plate models are considered. For a plate of limited length on a tensionless elastic foundation, the plate is first simplified to a one-dimensional mechanical model by assuming a buckling mode in terms of transverse coordinates, after which a new method is employed to determine the initially unknown boundaries of the areas in contact. Based on the continuity condition on the borderline between contact and noncontact regions, the buckling mode displacements of the whole plate may be expressed through the critical load coefficient and the first half-wavelength, reducing the buckling problem to two nonlinear algebraic equations with two unknowns. This procedure has been named the transfer function method. For a very long plate with a symmetric buckling mode, an infinite plate model with two half-waves is presented. For a plate on a rigid foundation, a single half-wave buckling model is shown to be appropriate. Comparison of limiting cases with exact solutions and with ABAQUS results showed good agreement. Finally, the influences of aspect ratio and foundation stiffness are presented.     

Response of a Buckled Beam Constrained by a Tensionless Elastic Foundation

In this paper, we study the deformation and stability of a pinned buckled beam under a point force. The buckled beam is constrained by a tensionless elastic foundation, which is flat before deformation. From static analysis, we found a total of five different deformation patterns: (1)?noncontact, (2)?full contact, (3)?one-sided contact, (4)?isolated contact in the middle, and (5)?two-sided contact. For a specified set of parameters, there may coexist multiple equilibria. To predict the response of the buckled beam foundation system as the point force moves from one end to the other, we have to determine the stability of these equilibrium configurations. To achieve this, a vibration method is adopted to calculate the natural frequencies of the system, taking into account the slight variation of the contact range between the buckled beam and the tensionless foundation during vibration. It is concluded that among all five deformation patterns, Deformations 1, 2, 3, and 4 may become stable for certain loading parameters. In the extreme case in which the foundation is rigid, on the other hand, only Deformations 1 and 3 are stable.     

Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model

The hybrid nonlocal Euler-Bernoulli beam model is applied for the bending, buckling, and vibration analyzes of micro/nanobeams. In the hybrid nonlocal model, the strain energy functional combines the local and nonlocal curvatures so as to ensure the presence of small length-scale parameters in the deflection expressions. Unlike Eringen’s nonlocal beam model that has only one small length-scale parameter, the hybrid nonlocal model has two independent small length-scale parameters, thereby allowing for a more flexible and accurate modeling of micro/nanobeamlike structures. The equations of motion of the hybrid nonlocal beam and the boundary conditions are derived using the principle of virtual work. These beam equations are solved analytically for the bending, buckling, and vibration responses. It will be shown herein that the hybrid nonlocal beam theory could overcome the paradoxes produced by Eringen’s nonlocal beam theory such as vanishing of the small length-scale effect in the deflection expression or the surprisingly stiffening effect against deflection for some classes of beam bending problems.     

空间预应力砼梁预应力分布的数值模拟与计算

文章结合预应力钢筋混凝土梁的非线性特性,以工字形混凝土简支梁为例,分析和计算了梁中预应力的分布情况,并将计算结果与工程实践进行比较,证明该数值方法可有效地模拟钢筋混凝土梁的受力性能,为梁的强度计算和合理设计莫定理论基础.     

Static Stability Formulas of a Weakened Timoshenko Column: Effects of Shear Deformations

The static stability analysis of two-dimensional Timoshenko columns weakened at an arbitrary section is derived in a classic manner. The effects of shear deformations along the column, influenced by the additional shear force induced by the applied axial load as the member deforms according to the modified shear equation proposed by Haringx, are presented and studied in detail. The proposed model also captures: (1) the influence on the buckling load capacity of the column when an arbitrary weakened section is formed at any location; (2) the tension buckling phenomenon due to the low shear stiffness of columns made of composite materials or elastomeric rubbers; and (3) the beneficial effects of an additional lateral bracing located at the weakened section to alleviate the buckling load reduction of the column. Seven classical and nonclassical cases of columns mostly used in civil and mechanical engineering are summarized in condensed formulas which allow the straightforward determination of buckling loads and shapes.     

Unilateral Buckling Restrained by Initial Force Supports

The common computation for understanding the buckling of reinforcement bars is associated with a unilateral rigid foundation with a set of uniformly spaced spring supports. In this technical note we examine the related situation of a set of uniformly spaced supports that require a finite force to initiate the displacement of the support.     

Closed-Form Solution for Reinforced Timoshenko Beam on Elastic Foundation

This paper suggests a method for obtaining closed-form solutions for a reinforced Timoshenko beam on an elastic foundation subjected to any pressure loading. A particular solution is obtained for uniform pressure loading at any location of the beam. This solution can be used to calculate settlement, rotation, tension, bending moment, and shear force of the beam. A parametric study is carried out to investigate the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading. Results are presented and discussed.     

Lateral–Torsional Buckling of Singly Symmetric Tapered Beams: Theory and Applications

A general variational formulation to analyze the elastic lateral–torsional buckling (LTB) behavior of singly symmetric thin-walled tapered beams is presented, numerically implemented, validated and illustrated. It (1) begins with a precise geometrical definition of a tapered beam; (2) extends the kinematical assumptions traditionally adopted to study the LTB of prismatic beams; (3) includes a careful derivation of the beam total potential energy; and (4) employs Trefftz’s criterion to ensure the beam adjacent equilibrium. In order to validate and illustrate the application and capabilities of the proposed formulation, several numerical results are presented, discussed and, when possible, also compared with values reported by other authors. These results (1) are obtained by means of the Rayleigh–Ritz method, using trigonometric functions to approximate the beam critical buckling mode, and (2) concern the critical moments of doubly and singly symmetric web-tapered I-section simply supported beams and cantilevers acted by point loads. In particular, one shows that modeling a tapered beam as an assembly of prismatic beam segments is conceptually inconsistent and may lead to rather inaccurate (safe or unsafe) results. Finally, it is worth mentioning that the paper includes a state-of-the-art review concerning one-dimensional analytical formulations for the LTB behavior of tapered beams.     

Buckling of Variable Section Columns under Axial Loading

In this paper, the static stability of the variable cross section columns, subjected to distributed axial force, is considered. The presented solution is based on the singular perturbation method of Wentzel-Kramers-Brillouin and the column is modeled using Euler-Bernoulli beam theory. Closed-form solutions are obtained for calculation of buckling loads and the corresponding mode shapes. The obtained results are compared with the results in the literature to verify the present approach. Using numerous examples, it is shown that the represented solution has a very good convergence and accuracy for determination of the instability condition.