In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

钢管混凝土圆弧桁架拱平面内徐变稳定分析

核心混凝土的徐变会增加钢管混凝土拱肋的屈曲前变形,降低结构的稳定承载力,因此只有计入屈曲前变形的影响,才能准确得到钢管混凝土拱的徐变稳定承载力。基于圆弧形浅拱的非线性屈曲理论,采用虚功原理,建立了考虑徐变和剪切变形双重效应的管混凝土圆弧桁架拱的平面内非线性平衡方程,求得两铰和无铰桁架拱发生反对称分岔屈曲和对称跳跃屈曲的徐变稳定临界荷载。探讨了钢管混凝土桁架拱核心混凝土徐变随修正长细比、圆心角和加载龄期对该类结构弹性稳定承载力的影响,为钢管混凝土桁架拱长期设计提供理论依据。     

Elastic responses of underground circular arches considering dynamic soil-structure interaction: A theoretical analysis

Due to the wide applications of arches in underground protective structures, dynamic analysis of circu- lar arches including soil-structure interactions is important. In this paper, an exact solution of the forced vibration of circular arches subjected to subsurface denotation forces is obtained. The dynamic soil-structure interaction is considered with the introduction of an interfacial damping between the structure element and the surrounding soil into the equation of motion. By neglecting the influences of shear, rotary inertia and tangential forces and assuming the arch incompressible, the equations of motion of the buried arches were set up. Analytical solutions of the dynamic responses of theprotective arches were deduced by means of modal superposition. Arches with different opening angles, acoustic impedances and rise-span ratios were analyzed to discuss their influences on an arch. The theoretical analysis suggests blast loads for elastic designs and predicts the potential failure modes for buried protective arches.     

用Excel绘制三铰拱的内力图

Microsoft Excel软件有较强的数据计算能力和图表功能, 利用软 件的内置函数, 设置公式计算三铰拱每一截面的内力, 并应用其图表功能, 绘制三铰拱的内力图. 用Excel软件完 成三铰拱繁琐的内力计算, 准确、快捷地绘制其内力图, 有助于三铰拱的教学, 并提高学生 学习三铰拱的兴趣.     

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.     

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.     

弹性拱静力屈曲的突变行为

应用数学突变理论研究弹性两铰拱的静力屈曲，分析中考虑了拱的挠度变化、轴向压缩变形的影响，得到拱面内失稳的尖点突变模型和临界条件。     

Nonlinear dynamic behaviors of viscoelastic shallow arches

The nonlinear dynamic behaviors of nonlinear viscoelastic shallow arches sub- jected to external excitation are investigated. Based on the d＇Alembert principle and the Euler-Bernoulli assumption, the governing equation of a shallow arch is obtained, where the Leaderman constitutive relation is applied. The Galerkin method and numerical in- tegration are used to study the nonlinear dynamic properties of the viscoelastic shallow arches. Moreover, the effects of the rise, the material parameter and excitation on the nonlinear dynamic behaviors of the shallow arch viscoelastic shallow arches may appear to have a are investigated. The results show that chaotic motion for certain conditions.     

Experimental behavior of thin circular arches subjected to forced excitation

Experimental evidence is presented to verify the steady-state solution for a thin circular arch, pinned at the ends, and subjected to symmetrical and unsymmetrical support excitation. The steady-state solutions consist of a series of the free modes of vibration. It is shown how these solutions are developed when both supports of the arch are moving simultaneously and in phase with one another. The unsymmetrical case, where only one support is moving, is also considered. The arches chosen for testing had a radius-to-thickness ratio of 121 to 179. The arch-opening half-angles varied from 90 to 125 deg. The arches were vibrated on an electrodynamic-shaker table. Dynamic arch amplitudes were measured using a specially designed micrometer probe. Comparison of theory with experiment was considered good; the average error in prediction of resonant frequencies was less than three percent. For the firced excitation, the modal shapes agreed quite closely with that predicted by theory. It was found that experimental arches were quite sensitive to variations in the arch radius and that, in general, for all arches tested, the degree of agreement between theory and experiment was more sensitive to changes in the opening half-angle rather than theR/H value. A further observation was that, for some poorly constructed arches, it was found that out-of-plane vibrations occurred at approximately 16 times the fundamental flexural frequency. Paper was presented at 1972 SESA Fall Meeting held in Seattle, Wash. on October 17–20.     

倒三角形截面板管连接式钢圆弧拱在平面内的稳定承载力研究

采用理论推导与数值模拟相结合的方法,对倒三角形截面板管连接式钢圆弧拱在平面内的弹性屈曲和弹塑性屈曲进行了深入研究。首先,理论推导了拱的截面剪切刚度,并提出了拱在全跨均布径向荷载作用下的弹性屈曲公式。此外,还提出了避免连接板和弦杆在拱发生整体弹性失稳之前发生局部失稳的限制条件。然后,分别研究了在全跨均布径向荷载和全跨均布竖向荷载作用下,拱的整体弹塑性失稳机理。结果表明,在全跨均布径向荷载下,拱在1/4跨和3/4跨附近的弦杆会发生屈服,最终发生拱的整体弹塑性失稳。基于数值结果,建立了拱在全跨均布径向荷载作用下的稳定曲线,并针对拱发生整体弹塑性屈曲提出了相应的稳定承载力设计公式。在全跨均布竖向荷载作用下,钢拱发生整体失稳时,在拱脚两端附近的下弦杆会进入屈服。同样地,本文也提出了拱在全跨均布竖向荷载作用下,发生整体弹塑性失稳时的稳定极限承载力设计公式。本文所建议的公式与有限元结果符合得较好,可供实际工程设计参考。     

In-plane buckling of circular arches and rings with shear deformations

A finite strain formulation is developed for elastic circular arches and rings in which the effects of shear deformations are included. Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for stress and finite strain are based on a hyperelastic constitutive model. Virtual work and equilibrium equations are derived. Closed-form in-plane buckling solutions are developed for circular rings and high arches under hydrostatic pressure. The effects of axial deformation prior to buckling as well as shear deformations are included in the buckling analysis. The formulation developed is compared with solutions in the literature and to the predictions of the finite element package ANSYS. The importance of including the effects of shear deformations for deep arches is investigated.     

动支座对拱结构抗爆承载力的影响

建立了具有动支座拱的计算模型,该模型考虑了竖向的弹性支承和阻尼支承、水平向弹性支承和扭转约束等柔性支承形式。基于大变形动力微分方程并利用有限差分方法,研究了动支座拱在爆炸荷载作用下的动力响应,并分析动支座对结构承载力的影响。研究表明:动支座对拱的抗爆承载能力有较大影响,不同形式的柔性支承对拱承载力的影响截然不同,竖向弹性支承能够使爆炸荷载作用下拱的弯矩峰值减小,并且使到达峰值的时间增加,提高了拱的抗爆或承受瞬态荷载的能力。而水平弹性支承使拱的内力值和相对位移值增大,对结构的承载力不利。     

固支浅圆拱受子弹撞击的实验研究

本文报导了铝合金固支浅圆拱在子弹撞击下动力响应的实验研究,试验中采用应变片测量并由高速摄影得到试件变形的瞬态记录。实验结果表明,此类浅拱的动力响应不存在失稳现象;但在某一撞击速度区间内,拱的中心位移增加较快,拱的响应前期轴力可以忽略不计,而在后期则必须考虑。     

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.     

In-plane free vibration and stability of loaded and shear-deformable circular arches

The first known equations governing vibrations of preloaded, shear-deformable circular arches are derived according to a variational principle for dynamic problems concerning an elastic body under equilibrium initial stresses. The equations are three partial differential equations with variable coefficients. The governing equations are solved for arches statically preloaded with a uniformly distributed vertical loading, by obtaining a static, closed-form solution and an analytical dynamic solution from series solutions and dynamic stiffness matrices. Convergence to accurate results is obtained by increasing the number of elements or by increasing both the number of terms in the series solution and the number of terms in the Taylor expansion of the variable coefficients. Graphs of non-dimensional frequencies and buckling loads are presented for preloaded clamped arches. They clarify the effects of opening angle and thickness-to-radius ratio on vibration frequencies and buckling loads. The effects of static deformations on vibration frequencies are also investigated. This work also compares the results obtained from the proposed governing equations with those obtained from the classical theory neglecting shear deformation.     

A New Matrix Method for Solving Buckling and Free Vibration Problems of Circular Arches with Variable Rigidity

ABSTRACT

ABSTRACT This paper proposes a new matrix method for calculation of critical loads and natural frequencies of circular arches with variable cross section and arbitrary forms of applied load distribution. A difference from other matrix methods is that the coefficient determinant of the corresponding characteristic equation can be reduced to one of third-order, no matter how many elements are discretized in calculation. The essential reason for the important advantage is introduction of correlation matrices associated with discrete nodes into the analysis. The method is applicable to elastic arches of arbitrary geometry.     

In-plane buckling of prismatic funicular arches with shear deformations

The in-plane buckling behavior of funicular arches is investigated numerically in this paper. A finite strain Timoshenko beam-type formulation that incorporates shear deformations is developed for generic funicular arches. The elastic constitutive relationships for the internal beam actions are based on a hyperelastic constitutive model, and the funicular arch equilibrium equations are derived. The problems of a submerged arch under hydrostatic pressure, a parabolic arch under gravity load and a catenary arch loaded by overburden are investigated. Buckling solutions are derived for the parabolic and catenary arch. Subsequent investigation addresses the effects of axial deformation prior to buckling and shear deformation during buckling. An approximate buckling solution is then obtained based on the maximum axial force in the arch. The obtained buckling solutions are compared with the numerical solutions of Dinnik (Stability of arches, 1946) [1] and the finite element package ANSYS. The effects of shear deformation are also evaluated.     

Optimal Forms of Shallow Arches with Respect to Vibration and Stability

Shallow, linearly elastic arches of unspecified form but with given uniform cross section and material are considered. For given span and length of the arch, two different optimization problems are formulated and solved. In the first, we determine the form of the arch which maximizes the fundamental vibration frequency. The corresponding vibration mode turns out to be either symmetric or antisymmetric. In the second, a static load with given spatial distribution is considered, and the critical value of the load magnitude for snap-through instability is maximized. This instability may occur at a limit point or a bifurcation point. Optimal forms are determined for sinusoidal loading, uniform loading, and a central concentrated load. In both types of problems, arches with simply supported or clamped ends are considered, and the maximum frequencies and critical loads obtained are compared to those for a circular arch with similar end conditions. In all the cases with simply supported ends, it is found that a circular arch is almost optimal. For clamped ends, however, it turns out that the optimal arches have zero slope at the ends and that they are much more efficient than a circular arch.     

Visualization of flow and vortex structure around a swimming loach by dynamic stereoscopic PIV

Loach has a unique swimming style of bending the whole body and staying at the bottom of water. We studied the three-dimensional flow field around and behind the loach using stereoscopic-PIV. We captured flow fields in horizontal and vertical plane, and it seems loach leaves vortex tube arches. From the analysis of body motion and flow field, we propose flow structure with vortex tube arches connected along the loach body. After being released, they are separated and flow away and dissipate. This research article was submitted for the special issue on Animal locomotion: The hydrodynamics of swimming (Vol. 43, No. 5).     

Pulsatile flows and wall-shear stresses in models simulating normal and stenosed aortic arches

Pulsatile aqueous glycerol solution flows in the models simulating normal and stenosed human aortic arches are measured by means of particle image velocimetry. Three transparent models were used: normal, 25% stenosed, and 50% stenosed aortic arches. The Womersley parameter, Dean number, and time-averaged Reynolds number are 17.31, 725, and 1,081, respectively. The Reynolds numbers based on the peak velocities of the normal, 25% stenosed, and 50% stenosed aortic arches are 2,484, 3,456, and 3,931, respectively. The study presents the temporal/spatial evolution processes of the flow pattern, velocity distribution, and wall-shear stress during the systolic and diastolic phases. It is found that the flow pattern evolving in the central plane of normal and stenosed aortic arches exhibits (1) a separation bubble around the inner arch, (2) a recirculation vortex around the outer arch wall upstream of the junction of the brachiocephalic artery, (3) an accelerated main stream around the outer arch wall near the junctions of the left carotid and the left subclavian arteries, and (4) the vortices around the entrances of the three main branches. The study identifies and discusses the reasons for the flow physics’ contribution to the formation of these features. The oscillating wall-shear stress distributions are closely related to the featured flow structures. On the outer wall of normal and slightly stenosed aortas, large wall-shear stresses appear in the regions upstream of the junction of the brachiocephalic artery as well as the corner near the junctions of the left carotid artery and the left subclavian artery. On the inner wall, the largest wall-shear stress appears in the region where the boundary layer separates.