Flexural–torsional coupled vibration and buckling of thin-walled open section composite beams using shear-deformable beam theory

A general analytical model based on shear-deformable beam theory has been developed to study the flexural–torsional coupled vibration and buckling of thin-walled open section composite beams with arbitrary lay-ups. This model accounts for all the structural coupling coming from the material anisotropy. The seven governing differential equations for coupled flexural–torsional–shearing vibration are derived from Hamilton's principle. The resulting coupling is referred to as sixfold coupled vibration. Numerical results are obtained to investigate effects of shear deformation, fiber orientation and axial force on the natural frequencies, corresponding mode shapes as well as load–frequency interaction curves.     

Large deflection analysis of a thin walled channel section cantilever beam

For a special application the large deflection behaviour of thin walled channel section beams made of thin sheet steel has been investigated. The experiments consisted of cantilever bending tests with the beam loaded through the shear centre and through the centroid. When loaded through the shear centre the beam buckling took place in the compression flange at the root of the cantilever. When loaded through the centroid however, it was noted that the compression flange buckled at a fixed distance from the fixed end. The general theory of thin walled beams developed by Vlasov was applied to the problem and indicated that the maximum compression stress at the edge of the flange would be at some distance from the fixed end. The value of the maximum compression stress obtained by the general linear theory was small and its position did not coincide with the experimental position. The Vlasov analysis has been modified to include the increase in the twisting moment due to the lateral deformation of the beam along its length. Good agreement between the modified theory and experiment both for the position of the maximum compressive stress and for the twist of the cantilever at three points along its length.

Because of the very low torsional stiffness of thin walled channel sections, the small deflection theory is only applicable for small bending loads applied through the centroid and the modified theory should be used for practical loading cases.     

Improved analysis of the modified split-cantilever beam for mode-III fracture

A closed-form solution for the compliance and the energy release rate of the updated version of the mode-III split-cantilever beam specimen is developed incorporating linear beam theories. Apart from bending and shear of the specimen, the Saint-Venant and free torsion effects are considered. The analytical solution is verified by finite element calculations, leading to the conclusion that the compliance is very accurately determined, while the energy release rate differs only with 5% compared to the finite element calculations. Based on the finite element analysis the recommended crack length range is given in order to design a configuration that gives 98% mode-III contribution to the total energy release rate. At the final stage the analytical model is applied to reduce the data from experiments performed on delaminated glass/polyester composite beams. The results show that the closed-form solution is in a very good agreement compared to the results by experiments, although this requires very accurate measurements.     

A finite thin circular beam element for in-plane vibration analysis of curved beams

In this paper, the stiffness and the mass matrices for the in-plane motion of a thin circular beam element are derived respectively from the strain energy and the kinetic energy by using the natural shape functions of the exact in-plane displacements which are obtained from an integration of the differential equations of a thin circular beam element in static equilibrium. The matrices are formulated in the local polar coordinate system and in the global Cartesian coordinate system with the effects of shear deformation and rotary inertia. Some numerical examples are performed to verify the element formulation and its analysis capability. The comparison of the FEM results with the theoretical ones shows that the element can describe quite efficiently and accurately the in-plane motion of thin circular beams. The stiffness and the mass matrices with respect to the coefficient vector of shape functions are presented in appendix to be utilized directly in applications without any numerical integration for their formulation.     

硅微构件的弹性模量和残余内应力的测试

叙述了利用硅悬臂梁和两端固定梁微构件进行硅微薄膜材料的弹性模量和残余内应力的测试方法。本方法利用简单的光学装置,通过测量悬臂梁和两端固定梁的一阶谐振频率,分别求弹性模量和残余内应力。本方法测量系统简单,试样制作方便。本试验获得（100）面<110>方向单晶硅微薄膜的弹性模量为128GPa,残余内应力为74.7MPa.     

The twisting of a transversely braced thin-walled beam with allowance for localized flange deformations

The torsional stiffness of a channel-section beam braced by evenly spaced transverse beams is studied. Previous work by Vlasov and the writer is extended theoretically to include localized deformations which take place in the flanges. A family of curves, resulting from a finite difference analysis of the flange deformations, is used in conjunction with a simple expression to predict a “reduced” second moment of area for the transverse beams from which a modified St. Venant torsional constant for the braced channel can be calculated. Experiments on a series of braced Perspex channels show good agreement with the theoretical predictions when the ratio of the length of the transverse beams to their depth is large but as the beams become “deep”, the engineer's theory of bending with the usual shear correction inadequately describes their behaviour. A correction factor, obtained from a graph based upon a finite element analysis, is used to obtain closer agreement for all length to depth ratios.     

Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques

Mechanical properties of micro/nanoscale structures are needed to design reliable micro/nanoelectromechanical systems (MEMS/NEMS). Micro/nanomechanical characterization of bulk materials of undoped single-crystal silicon and thin films of undoped polysilicon, SiO(2), SiC, Ni-P, and Au have been carried out. Hardness, elastic modulus and scratch resistance of these materials were measured by nanoindentation and microscratching using a nanoindenter. Fracture toughness was measured by indentation using a Vickers indenter. Bending tests were performed on the nanoscale silicon beams, microscale Ni-P and Au beams using a depth-sensing nanoindenter. It is found that the SiC film exhibits higher hardness, elastic modulus and scratch resistance as compared to other materials. In the bending tests, the nanoscale Si beams failed in a brittle manner with a flat fracture surface. The notched Ni-P beam showed linear deformation behavior followed by abrupt failure. The Au beam showed elastic-plastic deformation behavior. FEM simulation can well predict the stress distribution in the beams studied. The nanoindentation, scratch and bending tests used in this study can be satisfactorily used to evaluate the mechanical properties of micro/nanoscale structures for use in MEMS/NEMS.     

Numerical study on thermal stress cutting of silicon wafers using two-line laser beams

We propose a method of cleaving silicon wafers using two-line laser beams. The base principle is separating the silicon wafer using crack propagation caused by laser-induced thermal stress. Specifically, this method uses two-line laser beams parallel to the cutting line such that the movements of the laser beam along the cutting line can be omitted, which is necessary when using a point beam. To demonstrate the proposed method, 3D numerical analysis of a heat transfer and thermo-elasticity model was performed. Crack propagation was evaluated by comparing the stress intensity factor (SIF) at the crack tip with the fracture toughness of silicon, where crack propagation is assumed begin when the SIF exceeds the fracture toughness. The influences of laser power, line beam width, and distance between two laser beams were also investigated. The simulation results showed that the proposed method is appropriate for cleaving silicon wafers without any thermal damage.

     

拉压式环形梁测力与称重传感器

在中、大容量的应变式测力与称重传感器中,应用较多的是圆柱、圆筒、板环、悬臂梁、双端固支梁和轮辐式等结构,这些结构对同时检测拉伸、压缩对称循环载荷,或拉向、压向灵敏度不对称无法满足计量要求,或需要增加辅助部件才能完成计量任务。本文介绍的多梁串联配置的环形梁结构,弥补了上述弹性元件的不足,它是把产生正应力或切应力的应变梁串联式的相互连接起来,并沿着封闭的圆柱表面进行配置,外载荷以交错排列的形式施加于应变梁上,使得拉向、压向灵敏度的一致性好,易于进行拉、压对称循环栽荷测量。文中重点介绍了矩形截面正应力和工字形截面切应力环形梁弹性元件的结构特点、受力分析和理论计算,并给出了结构设计与理论计算范例。     

Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the crosssection(i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio Independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design Insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.     

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.     

Modeling and analysis of curved beams with debonded piezoelectric sensor/actuator patches

In this paper, a mathematical model for thin-walled curved beams with partially debonded piezoelectric actuator/sensor patches is presented for investigating the effect of debonding of the actuator/sensor on their open- and closed-loop behaviors. The actuator equations and the sensor equations of the curved beam in perfectly bonded and debonded regions are derived. In the perfect bonding region, the adhesive layer is modeled to carry constant peel and shear stresses; while in the debonding area, it is assumed that there is no peel and shear stress transfer between the host beam and the piezoelectric layer. Both displacement continuity and force equilibrium conditions are imposed at the interfaces between the bonded and debonded regions. Based on the model and the sensing equation of the sensor, a closed-loop vibration control for the curved beams is performed. To obtain the frequency response from the presented model, a solution scheme for solving the complex governing equations is given. Using this model and the solution scheme, the effects of the debonding of actuator and sensor patches on open- and closed-loop control are investigated through an example. The results show that edge debonding of the piezoelectric patch has a significant side effect on the closed-loop control of the curved beams.     

Investigation for dimension effect on mechanical behavior of a metallic curved micro-cantilever beam

This paper is aimed at studying the mechanical behaviors such as maximum stress, contact force, and fatigue life of a special designed metallic curved micro-cantilever beam, using analytical, numerical (FEM), and experimental methods. This micro-beam is an applicable specimen generally used in MEMS devices. The focus is at determining the dimensions which are very important in functional conditions, reliability evaluation, and durability of this specimen and at finding a guideline for optimum design. Various types of samples that have different dimensions have been simulated and some specific specimens requested to be fabricated by a manufacturing company. In our research work, the material of the specimen is assumed to be of metallic alloys, which have good mechanical and electrical properties. Also, the structure of the specimen has a special shape and includes some fingers which have effect on mechanical behavior and electrical conductivity. In experimental method, with specified boundary conditions, mechanical loading is applied to the beam by simple but accurate methods. In analytical approach, after some mathematical calculations, for a curved cantilever beam, the equations related to the mechanical behaviors are obtained. Finally, the results obtained from the foregoing methods are compared with each other and the most fundamental effective dimension parameter on mechanical behavior of this special designed metallic curved micro-cantilever beam is determined.     

Enhancing chemi-mechanical transduction in microcantilever chemical sensing by surface modification

The use of chemically selective thin-film coatings has been shown to enhance both the chemical selectivity and sensitivity of microcantilever (MC) chemical sensors. As an analyte absorbs into the coating, the coating can swell or contract causing an in-plane stress at the associated MC surface. However, much of the stress upon absorption of an analyte may be lost through slippage of the chemical coatings on the MC surface, or through relaxation of the coating in a manner that minimizes stress to the cantilever. Structural modification of MC chemical sensors can improve the stress transduction between the chemical coating and the MC. Surfaces of silicon MC were modified with focused ion beam milling. Sub-micron channels were milled across the width of the MC. Responses of the nanostructured, coated MCs to 2,3-dihydroxynaphthalene and a series of volatile organic compounds (VOCs) were compared to smooth, coated MCs. The analytical figures of merit for the nanostructured, coated MCs in the sensing of VOCs were found to be better than the unstructured MCs. A comparison is made with a previously reported method of creating disordered nanostructured MC surfaces.     

Microfluidic channel depth determination with Tywman–Green interferometer

A microfluidic channel is fabricated on a silica wafer using reactive ion etching (RIE). The depth of the microfluidic channel has been measured using a surface profilometer and a Twyman–Green interferometer (TGI) setup. The TGI setup which mainly consists of a 660-nm wavelength He-Ne laser source, glass cube beam splitter and two prisms produced interference fringes based on the optical path difference between two interfering beams when the microfluidic channel is inserted into one of the beams. The TGI setup that was developed has shown high repeatability when measuring microfluidic channel depth and also eliminates back injection into the laser source and alignment criticality. The TGI setup applied a single photodiode to detect the shifting of the bright and dark fringe produced from the interference of the TGI. The depth of microfluidic channel obtained from the TGI is 1.79?±?0.31 μm using fringe shifting and intensity measurements, while according to the surface profilometer the depth of microfluidic channel obtained is 1.67?±?0.07 μm. The resolution of the TGI is 0.25 μm and can still go well below that depending on the wavelength of the laser source. This research describes the capability of the TGI to perform depth measurements on a microfluidic channel of a silica substrate which can also be improvised for other microscale devices and applications.     

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.     

Deflections of Timoshenko beam with varying cross-section

Closed form solutions are presented for bending beams with linearly and (in the binomial form) parabolically varying depth and for bending beams with linearly varying width along the beam's length. The solutions are developed taking into account the shear deformation of the beam. The solutions are achieved, in an original way, by transforming the fourth-order differential equations with variable coefficients into fourth-order differential equations with constant coefficients. Though the solutions presented refer to three recurrent variations in the beam cross-section shape, the procedure outlined can be applied to beams with binomial variation (with any exponent) in the depth or width of the cross-section. Moreover, the solutions can be achieved for polynomial, exponential and sinusoidal load conditions. The solutions can be utilized to obtain the stiffness factors and the flexibility coefficients of beams in the analysis of frames. Closed form solutions for longitudinal displacements are also presented. The analytical solutions are applied to four recurrent beams commonly used in civil engineering practice and a comparison with a numerical procedure is made.     

含有泡沫铝芯的复合板弯曲断裂行为的原位研究

对由泡沫金属铝芯和金属面板组成的三层和多层复合板四点弯曲条件下的变形和断裂行为进行原位观察。研究结果表明：在弯曲条件下,复合板有两种基本的破坏方式,一种是复合板表面凹陷(Indentation, ID),它是表面局部集中塑性变形的结果;另一种是泡沫铝内芯切断 (Core shear, CS),它是内芯在最大切应力作用下的破坏。对一个给定的三层复合板,当凹陷破坏的载荷极限FID大于内芯切断的载荷极限FCS时发生内芯切断式破坏,反之发生表面凹陷式破坏。对于多层复合板,破坏方式受金属面板制约,不能直接应用三层板的破坏判据。若三层板发生凹陷型破坏,具有与三层板相同金属面板厚度的多层复合板发生凹陷加内芯切断的混合型破坏。当三层板只发生内芯切断型破坏时,具有与三层板相同金属面板厚度的多层复合板完全发生内芯切断型破坏。     

Torsional vibration of cracked beams of non-circular cross-section

The equation of motion and associated boundary conditions are derived for a uniform beam of non-circular cross-section which is subjected to torsional vibration and which contains one or more pairs of symmetric cracks. The procedure is to use the Hu-Washizu variational principle which requires plausible assumptions about the displacement, momentum, strain and stress fields to be chosen. The perturbation in the stress and strain distributions of the beam due to the presence of the cracks is incorporated through local functions which have their maximum value at the cracked section and which decay exponentially from the crack location. The rate of exponential decay can be evaluated from experimental tests. The fundamental torsional frequency of the beam is computed for various depths of crack using the theoretical model. These predictions are then compared with results obtained from experiments on beams which contain cuts to simulate the cracks. It is found that the theoretical predictions closely match the experimental results.     

An improved closed-form solution to interfacial stresses in plated beams using a two-stage approach

Shear stress and normal stress in the thickness direction at interfaces (referred to as interfacial shear and transverse normal stresses, respectively, hereafter) have played a significant role in understanding the premature debonding failure of beams strengthened by bonding steel/composite plates at their tension surfaces. Due to the occurrence of dissimilar materials and the abrupt change of cross-section, the stress distribution at plate ends becomes singular and is hence considerably complicated. Extensive experimental and analytical analyses have been undertaken to investigate this problem. Large discrepancies have been found from various studies, particularly from experimental results due to the well-acknowledged difficulty in measuring interfacial stresses. Numerical analyses, e.g. 2-D or 3-D finite-element analysis (FEA), may predict accurate results, but they demand laborious work on meshing and sensitivity analysis. Analytical solutions, in particular those in a closed form, are more desirable by engineering practitioners, as they can be readily incorporated into design equations. This paper reports an improved closed-form solution to interfacial stresses in plated beams using a two-stage approach. In this solution, beams and bonded plates can be further divided into a number of sub-layers to facilitate the inclusion of steel bars or multiple laminae. Thermal effects may also be considered by using equivalent mechanical loads, i.e. equivalent axial loads and end moments. Numerical examples are presented to show interfacial stresses in concrete or cast iron beams bonded with steel or FRP plates under mechanical and/or thermal loads. The effect of including steel reinforcements with various ratios in the RC beam on interfacial stresses is also investigated. Compared with previously published analytical results, this one improves the accuracy of predicting the transverse normal stresses in both adhesive-beam and plate-adhesive interfaces and the solution is in a closed form.