Exact static analysis of partially composite beams and beam-columns

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

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

Examining the trend in loss of flexural stiffness of simply supported RC beams with various crack severity using model updating

The flexural stiffness of simply supported cracked reinforced concrete beams was determined by model updating. The beams were 150 mm wide, 250 mm deep and 2200 mm long. Different FE models were created which include a datum and models with a single crack at three different locations along the length of the beam. The mode shape equation was obtained by using non-linear regression. The equation used in the regression was the generalized solution of transverse vibration of a prismatic beam. Local flexural stiffness, EI, at each coordinate point was derived by substituting the regressed data by using the centered-finite-divided-difference formula. Experimental modal analysis was performed on a control beam and beams with load-induced cracks at predetermined loading. Results from FE analyses showed the trend in the loss of stiffness was similar to the results obtained on the experimental beams. The more severe the damage, the higher the loss of stiffness and the loss patterns are similar for damage at different locations along the beam. The updating technique is able to indicate the trend in the loss of stiffness as a result of cracks of varying severity in the RC beams showing good agreement with experimental results.     

Thermal buckling behavior of laminated composite plates: a finite-element study

In this paper, the thermal buckling behavior of composite laminated plates under a uniform temperature distribution is studied. A finite element of four nodes and 32 degrees of freedom (DOF), previously developed for the bending and mechanical buckling of laminated composite plates, is extended to investigate the thermal buckling behavior of laminated composite plates. Based upon the classical plate theory, the present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. The numerical implementation of the present finite element allowed the comparison of the numerical obtained results with results obtained from the literature: 1) with element of the same order, 2) the first order shear deformation theory, 3) the high order shear deformation theory and 4) the three-dimensional solution. It was found that the obtained results were very close to the reference results and the proposed element offers a good convergence speed. Furthermore, a parametrical study was also conducted to investigate the effect of the anisotropy of composite materials on the critical buckling temperature of laminated plates. The study showed that: 1) the critical buckling temperature generally decreases with the increasing of the modulus ratio E _L/E _T and thermal expansion ratio α _T/α _L, and 2) the boundary conditions and the orientation angles significantly affect the critical buckling temperature of laminated plates.     

Exact dynamic analysis of composite beams with partial interaction

The partial differential equations and general solutions for the deflection and internal actions and the pertaining consistent boundary conditions are presented for composite Euler-Bernoulli members with interlayer slip subjected to general dynamic loading. Both free and forced vibrations are treated. The solutions are shown to be unique and complete under certain conditions, and valid for all so-called restricted admissible boundary conditions. Specifically, the exact eigenmode length coefficients are derived for the four Euler BC. They differ from those valid for ordinary, fully composite (solid) beams, except for the pinned-pinned case. The maximum deviation for beams with the other three Euler BC is shown to be less than 2-6% with respect to the eigenmode length coefficient and 3-10% with respect to the eigenfrequency, respectively, depending on the two non-dimensional parameters, composite action or shear connector stiffness and relative bending stiffness parameters. However, these deviations occur in a rather narrow range of the determining parameters, so for most practical cases the eigenmode length coefficients given for solid (fully composite) beams can approximately be used also for partially composite beams. The procedures of analysing beam vibrations are applied to a specific case. These solutions illustrate the effect of interlayer connection on the peak velocity of the beam vibrations. The proposed analytical theory is verified by tests and finite element calculations.     

The complex buckling of flexible sheet materials—Part I. Theoretical approach

This paper reviews the technological importance of the problems of complex buckling of textile fabrics and other sheet materials and the reasons why conventional shell theory is of little value. It stresses the importance of the occurrence of membrane strains in double curvature over finite areas, reviews earlier work on the subject, discusses briefly the characterization of the properties of sheet materials and shows the approach to the problem through simple three-fold buckling. A major section of the paper relates to order-of-magnitude estimates of bending, membrane and gravitational energies and introduces two important dimensionless groups, J₁ = Yl²/D and J₂ = γgl³/D, in terms of which deformed shapes can be calculated. A mathematical model for three-fold buckling is described and the contributions of different energies during deformations are presented.     

Composite beam–columns with interlayer slip—Approximate analysis

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

The anisotropy of carbon steels due to compression and the charpy impact value

Materials, which accumulate a deformation history from the fabrication process to which they have been subjected, are deformed to produce the component through many processing operations. It is important therefore, to establish the correlation between predeformation and workability. In this work, the Charpy impact test is carried out on steels which have been pre-strained in simple compression: it is hoped to be able to assess ductility using this method.For annealed steels, a linear relationship has been found between the logarithm of the ratio of crack initiion energy (EI) to yeild stress (σ_y) and impact tensile fracture strain (ϵ_f). In pure iron, the EI in the axial and radial directions decreased remarkably with increasing pre-strain and axial specimens fractured in a brittle manner for pre-strains greater than 0.6. In S55C steel, the EI and crack propagation energy (EP) in the axial direction increased with increase in the pre-strain up to about 0.2, then decreased abruptly at a pre-strain of about 0.3, whilst in S25C steel, the EI and the EP in both directions decreased considerably with increasing pre-strain, but their anisotropy in S25C steel was a little more than that in pure iron. These trends in carbon steels may be chiefly due to compressive deformation in the pearlite.The raio EI/σ_y or maximum displacement δ_m are found to be a measure of ductility in the Charpy impact test and are considered to be sensitive to the change in, and damage of, the micro-structure in the ferrite and pearlite, etc.     

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.     

Estimation of the equivalent complex modulus of laminated glass beams and its application to sound transmission loss prediction

The complex modulus E(jω) characterizes the visco-elastic behavior of a material. Using a system identification approach, this modulus can be measured via broadband modal analysis experiments. The technique is applied to determine the equivalent complex modulus E(jω), with its uncertainty bound, of multilayer glass beams from transversal vibration experiments in free-free boundary conditions. This property is related to the effective complex bending stiffness of the laminated glass specimen, and is further used for predicting the sound transmission loss of a multilayer plate. The data are rationalized in the terms of the linear visco-elastic properties of the polymer interlayer.     

Radial nanofretting behaviors of ultrathin carbon nitride film on silicon substrate

With a nanoindenter, the radial nanofretting behaviors of amorphous ultrathin carbon nitride (a-CN_x) film on the silicon substrate were investigated by a spherical diamond indenter. The experimental results indicate that the radial nanofretting damage on a-CN_x film usually successively experiences the buckling, cracking and detachment of film. These damages can be easily detected by the variation in the apparent contact stiffness. Generally, the initial increase in the contact stiffness indicates the buckling of film; the following sharp decrease in the contact stiffness reveals the initiation and propagation of circular cracks in film; the final stable contact stiffness implies the detachment of film.     

基于新型梁单元模型的薄壁弯梁耐撞性优化

以矩形截面Z型薄壁弯梁为例,引入新型梁单元参数化模型建模方法,通过对薄壁梁进行碰撞分析提取了塑性铰的弯曲刚度特性曲线,将参数化后的弯曲刚度特性曲线赋予梁单元简化模型,实现了梁单元模型对壳单元模型的等效替代。在此基础上,应用均匀拉丁方试验设计方法,选取一定数量的参数样本点,建立了设计变量与弯曲刚度曲线参数之间的响应面近似模型,并采用连续二次规划的优化算法对梁单元模型进行了耐撞性优化。结果表明,采用新型梁单元简化模型代替详细的壳单元模型进行优化分析是可行的。     

Coach simplified structure modeling and optimization study based on the PBM method

For the coach industry, rapid modeling and efficient optimization methods are desirable for structure modeling and optimization based on simplified structures, especially for use early in the concept phase and with capabilities of accurately expressing the mechanical properties of structure and with flexible section forms. However, the present dimension-based methods cannot easily meet these requirements. To achieve these goals, the property-based modeling (PBM) beam modeling method is studied based on the PBM theory and in conjunction with the characteristics of coach structure of taking beam as the main component. For a beam component of concrete length, its mechanical characteristics are primarily affected by the section properties. Four section parameters are adopted to describe the mechanical properties of a beam, including the section area, the principal moments of inertia about the two principal axles, and the torsion constant of the section. Based on the equivalent stiffness strategy, expressions for the above section parameters are derived, and the PBM beam element is implemented in HyperMesh software. A case is realized using this method, in which the structure of a passenger coach is simplified. The model precision is validated by comparing the basic performance of the total structure with that of the original structure, including the bending and torsion stiffness and the first-order bending and torsional modal frequencies. Sensitivity analysis is conducted to choose design variables. The optimal Latin hypercube experiment design is adopted to sample the test points, and polynomial response surfaces are used to fit these points. To improve the bending and torsion stiffness and the first-order torsional frequency and taking the allowable maximum stresses of the braking and left turning conditions as constraints, the multi-objective optimization of the structure is conducted using the NSGA-II genetic algorithm on the ISIGHT platform. The result of the Pareto solution set is acquired, and the selection strategy of the final solution is discussed. The case study demonstrates that the mechanical performances of the structure can be well-modeled and simulated by PBM beam. Because of the merits of fewer parameters and convenience of use, this method is suitable to be applied in the concept stage. Another merit is that the optimization results are the requirements for the mechanical performance of the beam section instead of those of the shape and dimensions, bringing flexibility to the succeeding design.     

夹层阻尼结构抑振特性研究

利用等效复合刚度法,对多夹层阻尼结构的抑振特性进行了理论推导,并对不同夹层阻尼结构的等效复合刚度和损耗因子进行了仿真计算,分析了不同的夹层材料和几何参数对结构抑振量的影响。研究表明,增加多夹层组合结构的夹层层数对组合结构损耗因子的提高量有限;适当增加夹层的厚度可改善组合结构抑振性能;变化夹层的材料,采用较硬的约束层和较软的阻尼层可显著提高组合结构的抑振量。     

Finite element analysis of stress concentrations and failure criteria in composite plates with circular holes

In this study, the stress concentration factors (SCF) in cross-and-angle-ply laminated composite plates as well as in isotropic plates with single circular holes subjected to uniaxial loading is studied. A quadrilateral finite element of four-node with 32 degrees of freedom at each node, previously developed for the bending and mechanical buckling of laminated composite plates, is used to evaluate the stress distribution in laminated composite plates with central circular holes. Based up on the classical plate theory, the present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. The numerical results obtained by the present element compare favorably with those obtained by the analytic approaches published in literature. It is observed that the obtained results are very close to the reference results, which demonstrates the accuracy of the present element. Additionally, to determine the first ply failure (FPF) of laminated plate, several failure criterions are employed. Finally, to show the effect of E ₁/E ₂ ratio on the failure of plates, a number of figures are given for different fiber orientation angles.     

Inelastic buckling of two-layer composite columns with non-linear interface compliance

A mathematical model for a slip-buckling problem has been proposed and its exact solution has been found for the analysis of materially inelastic two-layer composite columns with non-linear interface compliance. The mathematical model has been carried out to evaluate exact critical buckling loads. It has been demonstrated mathematically exactly, that exact critical buckling loads are influenced by the initial stiffness, and hence on linear portion of the interface force-slip relationship. Besides, it has been shown that material inelasticity can reduce the critical buckling loads significantly and that the interlayer stiffness has an important effect on the transition between the elastic and inelastic buckling.     

Residual stresses in TiN, DLC and MoS2 coated surfaces with regard to their tribological fracture behaviour

Thin hard coatings on components and tools are used increasingly due to the rapid development in deposition techniques, tribological performance and application skills. The residual stresses in a coated surface are crucial for its tribological performance. Compressive residual stresses in PVD deposited TiN and DLC coatings were measured to be in the range of 0.03-4 GPa on steel substrate and 0.1-1.3 GPa on silicon. MoS₂ coatings had tensional stresses in the range of 0.8-1.3 on steel and 0.16 GPa compressive stresses on silicon. The fracture pattern of coatings deposited on steel substrate were analysed both in bend testing and scratch testing. A micro-scale finite element method (FEM) modelling and stress simulation of a 2 μm TiN-coated steel surface was carried out and showed a reduction of the generated tensile buckling stresses in front of the sliding tip when compressive residual stresses of 1 GPa were included in the model. However, this reduction is not similarly observed in the scratch groove behind the tip, possibly due to sliding contact-induced stress relaxation. Scratch and bending tests allowed calculation of the fracture toughness of the three coated surfaces, based on both empirical crack pattern observations and FEM stress calculation, which resulted in highest values for TiN coating followed by MoS₂ and DLC coatings, being K_C = 4-11, about 2, and 1-2 MPa m^1/2, respectively. Higher compressive residual stresses in the coating and higher elastic modulus of the coating correlated to increased fracture toughness of the coated surface.     

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.     

Analysis of lateral track buckling using new frame-type equations

In the past several decades, many lateral track buckling studies have been conducted in an effort to determine an allowable safe temperature increase for preventing the occurrence of buckling. The present analysis builds on some of these studies, but uses recently derived frame-type equations that more accurately represent the response of the rail-tie structure in the lateral plane. The presented analysis takes into account the effects of the torsional stiffness of the rail fasteners, the lateral bending stiffness of the cross-ties, and the track gauge to model more accurately the lateral response of the track panel to temperature increases. It determines effective ways to raise the allowable safe temperature increase, whether by increasing the axial and lateral resistances or by increasing the rotational stiffness of the fasteners. Also, the effect of lateral tie-stiffness on the safe temperature increase is examined.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.