Stresses and strains in a race track strap with a radial clearance

In certain situations it is desirable to transfer only tensile loads between two points in a structure whilst minimising any stress concentrations. In such circumstances a so-called race track strap can be utilised. The strap consists of two semi-circular elements that transfer the load, from bolts at each end, to two parallel flexible elements joining the whole together. The strap is loaded in tension by means of these bolts. This paper considers the situation where there is clearance between the bolts and the semi-circular elements of the strap. To develop an analytical model, it is assumed that engineering beam theory is applicable, that the influence of the membrane strains can be ignored and that the clearance is small as compared with the bolt and strap radii. It was found that the simple analytical model compared well with both finite element calculations and experiments.     

典型连接结构螺栓的附加弯矩形成机理与承载能力研究及结构优化设计

针对典型连接结构中, 高强螺栓在受拉工况下因产生附加弯矩而极大削弱其承载能力的问题, 开展了螺栓附加弯矩产生的机理研究, 并提出了一种有效降低螺栓附加弯矩的结构优化设计方法. 首先, 建立典型连接结构的等效力学模型, 推导出螺栓附加弯矩的解析解, 进一步开展数值仿真分析, 验证了解析方法的正确性. 考虑螺栓同时承受拉弯耦合载荷工况, 引入梁塑性弯曲理论, 研究了不同拉弯组合下的螺栓截面各类应力分布的交互关系, 并给出了考虑轴力影响的弯矩塑性折减系数. 基于最大应力破坏准则, 开展了考虑附加弯矩和弯曲塑性影响的螺栓载荷失效判据研究, 该判据更加具有工程应用价值. 从机理出发, 开展典型连接结构优化设计以降低螺栓的附加弯矩进而提高其承载能力, 进一步采用解析方法, 阐述了铰支球头的工作机理. 采用数值仿真方法, 开展了螺栓附加弯矩灵敏度分析, 验证了优化设计方法的有效性. 进一步开展试验研究, 获得不同连接状态下螺栓的附加弯矩, 验证了优化设计方法的正确性和可行性. 该方法能够极大降低高强螺栓的附加弯矩, 最大程度发挥螺栓的承载能力, 提高连接结构的可靠性.      

A CW ultrasonic bolt-strain monitor

There exists a need for a relatively inexpensive system for measuring strain in bolts. The torque wrench is one technique for straining bolts which has been widely applied. Unfortunately, friction in the bolt threads and between the nut and the work tend to make such a simple system inaccurate. In practice, a torque wrench is unacceptable for many situations where strain is critical. In this article, an ultrasonic technique is described which can indicate changes in bolt strain to better than one part in 10⁴. The technique is based on the one-dimensional propagating-ultrasonic-wave model and uses a new ultrasonic instrument called a Reflection Oscillator Ultrasonic Spectrometer which is a closed-loop feedback marginal-oscillator system that frequency locks the device to the peak of a mechanical resonance in the bolt. The instrument indicates a shift in the bolt resonance frequency due to elongation and changes in velocity of sound due to strain. Data are presented comparing a standard torque wrench to the ultrasonic monitor for different measured stresses on the bolt as well as for different bolt conditions. The strain instrument can be used to monitor changing stresses, to measure material properties and may be applied as a strain gage or load cell.     

Base strain effects on force measurements

Force transducers directly interact with their environment. In this study, a force transducer is attached to the midpoint of a free-free beam and is used to measure the force on a rigid mass that vibrates with the beam. The ratio of force to mass times acceleration is measured for several different masses over a frequency range that includes the first four odd natural frequencies of the beam. Then the force transducer is mounted so that the beam's strain is isolated from the transducer. The tests are repeated. The results dramatically illustrate the effects of base strain on the force measurements. A simple theoretical model is developed that explains the vertical axis shift in the calibration curve. Paper was presented at SEM 1994 Spring Conference and Exhibits.     

Photoelastic analysis of a thin-walled compressor housing

The fabrication, testing, and analysis of a full-sized photoelastic model of a thin-walled compressor housing are presented. The conical model and connecting bolts were stress frozen under axial and bending loads to simulate severe in-flight manoeuver loads of a jet aircraft sufficient to cause flange separation. Results and discussion are presented for stresses in and near the flange, for stresses experienced by the bolts, and for the fabrication techniques used.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

点支撑预应力中厚矩形板的弯曲

基于Reissner-Mindlin一阶剪切变形板理论，讨论在预加面内机械荷载或温度场作用下，点支撑中厚矩形板的弯曲问题。温度场假定沿板表面为均布，沿板厚方向为线性分布的。利用考虑剪切变形影响的Timoshenko梁函数，采用Rayleigh-Ritz法给出不同边界条件下点支撑中厚板在横向荷载作用下的挠度和弯矩分布。结果表明，均匀温度场与预加面内压力将使板的挠度和弯矩增加。支撑点位置的变化、边界约束条件和横向剪切变形效应都对板的内力大小和分布有显著影响。     

Polychromatic X-ray method for residual-stress measurements in a subsurface layer

The classical sin² method with characteristic X-rays is widely used to measure residual stress nondestructively in the steel members of a structure or a machine. With this method it is, however, difficult to measure the three-dimensional stress distribution with a steep gradient that occurs along the depth direction in a subsurface layer of the material after surface treatment such as grinding or cold rolling. This paper presents a new polychromatic X-ray method for residual-stress measurements in a subsurface layer. The relationship between the diffracted beam peak of the polychromatic X-ray and the strain along the depth direction in a subsurface layer was obtained by theoretical analysis. It was modeled by numerical simulation to obtain probable values of the parameters, and these were used along with experimental X-ray data to derive an experimental value for the stress gradient. This was compared with the values predicted from plate bending theory.Paper was presented at the 1994 SEM Spring Conference on Experimental Mechanics held in Baltimore, MD on June 6–9.     

功能梯度变曲率曲梁的几何非线性模型及其数值解

基于弹性曲梁平面问题的精确几何非线性理论,建立了功能梯度变曲率曲梁在机械和热载荷共同作用下的无量纲控制方程和边界条件,其中基本未知量均被表示为变形前的轴线坐标的函数。以椭圆弧曲梁为例,采用打靶法求解非线性常微分方程的两点边值问题,获得了两端固定功能梯度椭圆弧曲梁在横向非均匀升温下的热弯曲变形数值解,分析了材料梯度指数、温度参数、结构几何参数等对曲梁受力及变形的影响。     

Application of ultrasonic-pulse-spectroscopy measurements to experimental stress analysis

We apply the technique of ultrasonic pulse spectroscopy to measure the interference effects between two shear waves propagating in specimens loaded in uniaxial compression. We show that the power spectrum of an echo containing both fast and slow components of a shear wave will exhibit periodic minima. The periodicity exhibited in the spectrum is 1/Δτ, where Δτ is the difference in arrival time between the fast and slow waves. A change in the state of stress which produces a change in the two shear velocities results in a stress-dependent change in wave-arrival times. Because of this velocity change, the frequency at which a particular minimum occurs in the spectrum changes, and this can be used to indicate the state of stress in the material. Our results indicate that, if the spectrum minima frequencies could be resolved to within 10 kHz, the principal-stress differences within 36 psi (0.251 MPa) could be measured in specimens of aluminum 1 in. (2.54 cm) thick. Inherent in analyzing and measuring echo-interference effects is a single-echo requirement. Thus, transducer coupling effects are minimized and measurements in highly attenuating materials or at high frequencies in normal attenuating materials are possible. This technique shows considerable promise as a means of measuring and monitoring the applied stresses in materials.     

Stability analysis of gradient elastic beams by the method of initial value

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.     

大桥螺栓施拧轴力的超声测量

本文利用纵横波相结合的方法对九江大桥联接螺栓的施拧轴力进行了群检测量。该方法无需预知螺栓的初始长度，更为实用。     

Stress intensification in cracked shank of tightened bolt

Defects or cracks in the shank of bolts can degrade their load carrying capacity. The ways with which loading and residual stress intensify the crack border stress field can be reflected through the stress intensity factor quantity as defined in the linear elastic fracture mechanics theory. Use is made of the stiffness derivative method where quarter-point singular finite elements are used in the numerical calculation. Improved accuracy is achieved by considering the displacements not only of the main nodes but also of those quarter-point nodes in plane normal and adjacent to the crack.Numerical results are obtained for a semi-elliptical shaped crack in the bolt shank owing to tension, bending, residual stress and stress caused by tightening of the bolt. Maximum value of the Mode I stress intensity factor K_a due to tension or bending could prevail either at the deepest point on the crack border or at the root of the shank where the crack border terminates depending on the aspect ratio of the ellipse. In general, K₁ at the deepest point of crack penetration is larger than that at the free surface for tension and bending for a fixed crack depth with reference to the bolt diameter. Tightening of the bolt tends to increase K₁ at the free surface if the crack depth is small. The opposite is obtained for deeper cracks. Assumed residual stress effect obtained from experimental data is found to have negligible influence on the stress intensity factor when compared with that arising from tensile load.     

NEW STRAIN GRADIENT THEORY AND ANALYSIS

A new strain gradient theory which is based on energy nonlocal model is proposed in this paper, and the theory is applied to investigate the size effects in thin metallic wire torsion, ultra-thin beam bending and micro-indentation of polycrystalline copper. First, an energy nonlocal model is suggested. Second, based on the model, a new strain gradient theory is derived. Third, the new theory is applied to analyze three representative experiments.     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

Elasto-optic technique for measurement of elastic wave propagation in solids

The modulation of the optical path of the beam of a laser vibrometer in a specimen under acoustic excitation is measured at two planes, separated by a precisely known distance. The phase shift and the decrease in magnitude are used to calculate the phase velocity and attenuation, respectively. The method is demonstrated for a homogeneous specimen, and the results compare favorably with those obtained by a conventional ultrasonic technique. The method is then applied to measure specular and first diffraction-order reflection from a coplanar periodic array of particles in an elastic matrix and phase velocity spectra in a tetragonal periodic particulate composite. As expected, in a periodic composite the establishment of dispersive Floquet-type waves is observed throughout the entire periodic particulate composite.     

温度对螺栓紧固应力超声测量的影响研究

本文从理论和实验两方面研究了温度对螺栓紧固应力超声测量的影响。推导了考虑温度影响时,测量螺栓轴力的应力—声关系.同时,通过实验测定了纵、横波在螺栓中的传播时间与温度的关系,获得了温度修正曲线,其结果与理论分析相吻合.     

Propagation of a stress pulse in a viscoelastic rod

Experimental work is reported on the propagation of a stress pulse in a viscoelastic waveguide. The data obtained are compared with results of analysis using one-dimensional wave-propagation theory. The waveguide used in this work is a low-density polyethylene rod 1/2 in. in diameter and 30-in. long. Stress input to the waveguide and the resulting particle velocity at three stations are measured using a crystal stress transducer, two Faraday-principle velocity transducers and a capacitor transducer. The experiment is described mathematically as a boundary-value problem formulated in terms of the one-dimensional equation of motion, the strain-displacement relationship, a hereditary constitutive equation and the stress-boundary condition. Fourier transform and inversion yield an integral expression for velocity which is evaluated numerically at three stations using measured values for the stress-boundary condition, material attenuation and phase velocity. The analytical results compare favorably with the experimental data. The one-dimensional theory appears adequate to describe pulse propagation of this type. The attenuation and phase velocity used here are found to be a linear function and a logarithmic increasing function of frequency respectively.     

大功率电力机车车体与变压器关键部位强度计算及设计

采用有限元方法对世界最大功率货运电力机车的车体和变压器关键部件强度进行了计算,车体结构用壳元和空间梁单元进行模拟,对作为支撑的高圆簧用刚度等效的圆柱薄壳进行模拟,并根据计算结果以及拓扑优化结果对车体关键受力部位进行了重新设计,设计之后的车体结构基本满足相关要求.车体计算模型共划分了约34万个节点,自由度总数超过200万.对主变压器的关键受力部件进行了进一步计算,变压器拉螺杆使用3节点轴对称单元、吊装螺栓使用四面体单元进行模拟.结果显示,对于拉螺杆以及厚薄螺母的接触,第一匝接触螺纹的应力最高,而后每一匝螺纹上的应力逐渐降低.吊装工况下变压器高应力区域主要发生在吊装螺栓附近,且同一安装座上的吊装螺栓应力分布是不同的.