Zur Entwicklung der Steifigkeitsmatrizen für dünnwandige Stäbe

On the development of stiffness matrices for thin‐walled members. A glance into the early development of element stiffness matrices for thin‐walled members is given in the present paper and their most mature shapes are shown. Those shapes, the most general and later reduced to two particular forms, refer to the specifics of the classical Vlasov theory of thin‐walled bars [1] as well as to the Bornscheuer systematics of the cross‐sectional properties involved [2]. Appropriate modifications of the relevant matrices are performed and their coordinate system dependence is demonstrated.     

Berechnung dünnwandiger Stäbe mit offenem Querschnitt bei Beanspruchung durch Längskraft,zweiachsige Biegung und Torsion – ohne Bestimmung von Schwerpunkt,Schubmittelpunkt und Hauptachsen

Analysis of thin‐walled bars with open cross‐section loaded by normal force, biaxial bending and torsion without use of centroid, shear centre and principal axis. For a generally loaded prismatic bar an analysis is presented, where the relations between stress resultants and deformations are not normalized, but written in a coupled form. This means that any system of coordinates y, z in the section can be chosen and is then maintained for all calculations. On this basis the equation of transfer matrix is developed, which allows the calculation of all stress resultants and displacements for a beam with arbitrary support. Finally at any point of the sections normal and shear stresses can be determined.     

Eine Finite‐Elemente‐Lösung auf der Basis eines erweiterten eindimensionalen Querschnittselementes für die freie Torsion dünnwandiger prismatischer Stäbe

A finite element solution on the basis of an extended one‐dimensional cross‐section‐element for the Saint‐Venant torsion of thin‐walled prismatic beams. It is presented a finite element solution on basis of an extended one‐dimensional cross‐section‐element to the calculation of the warping function, the torsional properties and the shear stresses, dependent on it, for thin‐walled prismatic beams under Saint‐Venant torsion. The formulated finite two‐node‐element with six element degrees of freedom can capture through inclusion of the torsion around the element axis the linear term of the variance of warping function perpendicular to the element axis. Only the shear stresses of the ring shear flows in the closed section parts unchangeable over the wall thickness can be calculated with the simple two‐node‐element with two element degrees of freedom. The extended two‐node‐element supplies in addition also the shear stresses of the cut open cross‐section linearly changeable over the wall thickness.     

Drillknicken einfach‐symmetrischer Stahlstützen mit erzwungener Drehachse

Torsional buckling of mono‐symmetrical columns of steel structures with prescribed axis of rotation. Presented herein solution concerns the problem of elastic, spatial stability of mono‐symmetrical I‐columns of steel structures with prescribed axis of rotation placed in the plane of the cross‐sectional symmetry, parallel to the longitudinal axis. In association with it one‐bay columns with point elastic constraints, along the length, due to torsion and warping have been considered. The point's constraints have been taken into account model using a distributional calculus. The columns are loaded by the combined compressive force and bending moment, acting in the plane of the cross‐sectional symmetry. For the columns under constant bending moment the prob‐lem of stability has been solved in the closed form for the arbitr‐ary conditions of ends supporting. Whereas for columns under non‐uniform moment the solution has been obtained by the Bubnov‐Galerkin orthogonalization method for chosen conditions of the ends supports. The solutions have been depicted by calculations, in which the critical load of the elastic, torsional buckling has been determined for steel columns with prescribed axis of rotation. The computations have been executed by the help of an own software worked out on the basis of the introduced solutions. The experimental investigations carried out on I‐columns, braced laterally and longitudinally, under compression and bend‐ing, in order to determine their critical loads of elastic spatial buckling have been presented as well. These experimental inves‐tigations are to verify the theoretical solution derived by approximation method. General remarks and conclusions of practical nature have finished the work.     

Berechnung von gekrümmten Stäben mit Kreishohlprofil unter Berücksichtigung der Querschnittsverformung

Analysis of tube‐bars with initial curvature including deformation of cross‐section. Curved bars under bending show a nonlinear distribution of stresses with peak at internal side of the section. Bernoulli‐hypothesis is still valid but Naviers bending theory can no more be applied. Additionally the effect of deformation of the thin‐walled tube‐section must be taken into account. This effect is unfavourable for stresses and deformations if the bending moment increases the curvature of the bar. This paper gives formulae for calculation of internal forces, stresses and deformations considering both mentioned effects; examples show the application. Finally the principle of virtual work is formulated and demonstrated at a two‐hinged arch.     

Untersuchungen zur Längssteifigkeit von Stahlstützen mit kastenförmigem Querschnitt

Experimental tests and theoretical analyses of longitudinal stiffness of closed cross‐sections steel columns. In structures made of thin‐walled elements it is of great importance to establish how the compressed elements change their stiffness while loading increases. Nonuniform change of longitudinal stiffness induces change of force and stress redistribution. More ductile elements take less loading and more stiff, tensed, ones are being overloaded. Columns with closed cross‐sections, box cross‐sections and hollow circular cross‐sections need to be distinguished from steel elements with longitudinal stiffness decreasing while loading increases. Shortening of column working in elastic‐plastic phase comes from material compression, cross‐section walls buckling and increased displacement of column as a whole. Imperfections that depend on fabrication process of cross‐section have a great influence on stiffness variation of an element. It can be designated, among other factors, from the equilibrium paths of tested elements. Those equilibrium paths, up to the limit point, were obtained by applying incremental method of numerical integration of equilibrium equations assuming initial displacements and residual stresses of column's walls and initial displacements of column's axes. Some numerical analyses were calculated using FEA software LUSAS. All theoretical analyses were verified by experimental tests carried out by the authors with full scale elements: box cross‐section columns – length 5280 mm, cold‐formed welded cross‐section columns – length 2000 mm. The research results and analyses by others were also used.     

A theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion

Results of a theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion are presented. The local critical bimoment, which generates local buckling of a thin-walled bar and constitutes the limit of the applicability of the classical Vlasov theory, is defined. A method of determining local critical bimoment on the basis of critical warping stress is developed. It is shown that there are two different local critical bimoments with regard to absolute value for bars with an unsymmetrical cross-section depending on the sense of torsion load (sign of bimoment). However, for bars with bisymmetrical and monosymmetrical sections, the determined absolute values of local critical bimoments are equal to each other, irrespective of the sense of torsional load. Critical warping stresses, local critical bimoments and local buckling modes for selected cases of thin-walled bars with open cross-section are determined.     

Interactive instabilities of thin‐walled laminated composite pultrusion box columns

The problem of interactive buckling and post‐buckling of intermediate length thin‐walled columns built of laminated plate elements subjected to compressive load has been proposed and solved analytically. Pultrusion columns have wide‐range applications in high‐rise building due to their low weight and high load carrying capacity. Classic stability theory and laminate theory were implemented to prove the existence of mixed‐mode buckling in thin‐walled pultrusion columns. Interactive stability modes can result in lower loading capacity of most compressive members and affects their post‐buckling behaviour in major proportions. Interactive buckling load analysis has been performed by means of a simplified theoretical model and verified by means of numerical analysis. The calculations were carried out for commonly used square section thin‐walled composite columns dimensions. The post‐buckling performance of selected sections has been investigated and an optimum layup configuration criterion for each section has been extracted according to pre‐ and post‐critical behaviour. Copyright © 2010 John Wiley & Sons, Ltd.     

Ermittlung der Grenztragfähigkeit von Mehrschraubenverbindungen bei Normalkraft‐Biegemoment‐Interaktion

Determination of the ultimate bearing capacity of multi‐bolt connections with normal force – bending moment – interaction. This paper will present a general design method for single or multi‐bolt connections of beams with arbitrary thin‐walled cross sections, suitable for application in computer programs. The design method is based on the classical strain iteration algorithm for the determination of the stress distribution in cross sections. In this case, the ultimate capacity of bolted connections will be obtained using an iterative numerical determination of the elastic‐plastic stress distribution in the connection elements. The numerical method will be derived in two steps – the first step is the numerical determination of the stress distribution in the connection for a given combination of internal forces and the next step is the determination of the ultimate bearing capacity of the connection. Furthermore, an analytical design method for a multi‐bolt tube connection will be derived. Finally, results of numerical and analytical calculations will be compared with corresponding test results.     

L型混凝土薄壁构件在设计中的运用

设计中常见的梯间梁问题的一种解决方法。明确了薄壁构件在实际受力中同普通的矩形梁的区别。进行了薄壁构件受弯、剪、扭应力状态的分析 ,和构件在受竖向荷载下的控制因素的计算 ,并由此导出了构件的内力的计算方法和构件的配筋方法     

FE‐Methoden zur Bestimmung der Lage des Schubmittelpunktes dünnwandiger Querschnitte

Finite element methods for the shear centre determination of thin walled cross‐sections. The position of the shear centre can be determined using different conditions and methods respectively. The calculation for complex cross sections, which consist of many components or hollow parts, is very extensive. For that reason the application of the finite element method (FEM) is suitable. It is shown, that the calculations are similar to FEM‐investigations of beam structures.     

Wölbkrafttorsion von Durchlaufträgern mit konstantem Querschnitt unter Berücksichtigung sekundärer Schubverformungen

Warping torsion of continuous beam with constant cross‐section considering shear deformation. The analogy between the theory of warping torsion and second order theory of a bending member with tensile force is also valid, if shear‐deformations in both cases are additionally included. On the basis of this analogy the three‐moment equation for continuous beam with constant cross‐section is given. It is demonstrated, that shear‐deformations for open sections can normally be neglected but must be encluded in the case of hollow‐sections. It is also shown, that for these sections warping‐torsion occurs only in ranges of discontinuity, but causes stresses which are in the same order as the stresses of primary torsion.     

变截面圆形薄壁压杆的弹性稳定分析

本文推导出轴向荷载作用下变截面圆形薄壁杆弹性稳定问题的基本微分方程,并用解析法求出问题的特征解。根据杆件不同的几何参数,将数值计算结果绘成图表,以供参考。     

锈蚀钢筋截面分布特征及轴向拉伸力学性能

采用半浸泡通电加速锈蚀法获得平均锈蚀率为0%～18%的钢筋,应用三维扫描的实物反求技术获得精确的锈蚀钢筋截面积数据,研究得到了平均锈蚀率与最大截面锈蚀率的关系,统计建立了锈蚀钢筋截面积概率分布模型;通过轴向拉伸试验研究锈蚀钢筋力学性能退化原因,定量分析锈蚀率与钢筋各力学特征值损失率的关系.结果表明:锈蚀钢筋实际材料性能未发生改变,其名义屈服强度、名义极限强度由钢筋最小残余截面积决定;钢筋变形与钢筋残余截面积分布有关,与平均锈蚀率及最大截面锈蚀率的相关程度较低;锈蚀钢筋本构关系仍可采用未锈蚀钢筋的本构关系,并可通过残余截面积概率分布模型来反映锈蚀对钢筋力学性能的影响.     

Coupled flexural–torsional dynamic stiffness matrix of an elastically supported axially loaded beam with shear resistant in‐fill

The dynamic stiffness matrix of an axially loaded elastically supported uniform beam with doubly asymmetric cross‐section that exhibits coupling between flexural and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The beam comprises a thin‐walled outer section that encloses, and works compositely with, a core of shear resistant in‐fill material. The outer layer provides flexure, warping and Saint–Venant rigidity, while the inner layer provides both Saint–Venant and shear rigidity. A three‐parameter Winkler model is used to describe the distributed elastic support. Hamilton's principle is used to derive the partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions. This gives rise to three coupled equations that are subsequently combined into a single, 12th order, ordinary differential equation. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick–Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Finally, in order to verify the accuracy of the presented theory, the numerical solutions are given and compared with the results that are available in the literature and finite element solutions using abaqus software. Copyright © 2014 John Wiley & Sons, Ltd.     

Von der Bauteilfügung zu leichten Tragwerken: Trocken gefügte Flächenelemente aus UHPFRC

Light‐Weight Structures made of thin walled, dry jointed surface elements made of UHPFRC With the development of novel jointed, thin walled and thus light components the resource‐efficient use of ultra‐high performance fiber‐reinforced fine grained concrete in structures can be realized. The manufacturing of modular and efficient UHPFRC components is achieved by the digital workflow as a result of digital design, calculation and CNC‐controlled manufacturing technology. By linking lightweight concrete components with new, high‐precision non standardized joints, such components can be connected material and force flow compatible with each other. The results with dry jointed T‐beams show the manufacturability and the potential of the dry jointing system on a large scale. When checking the load capacity of thin‐walled construction elements, the influence of fiber reinforcement must be especially considered.     

Effect of horizontal members on column shortening of reinforced concrete building structures

An improved analysis method for column shortening of tall buildings has been proposed. The analysis method considers the restraining effects of steel bars and horizontal members. A step‐by‐step method was applied for the exact displacement and internal forces for any elapsed time. The sectional properties of the asymmetrically reinforced and cracked section were also taken into account. Column shortenings of an 80‐story reinforced concrete frame–shear wall building were investigated as numerical examples. The horizontal members reduce the differential shortening between the adjacent vertical members and transfer the axial forces of more shortened vertical members to less shortened members.     

Torsionsversuche an textilbetonverstärkten Stahlbetonbauteilen

Zur Untersuchung des Torsionstragverhaltens von textilbetonverstärkten Stahlbetonbauteilen wurden Torsionsversuche an Bauteilen mit runden und quadratischen Querschnitten durchgeführt. Anhand von Versuchsergebnissen wird gezeigt, dass durch die Textilbetonverstärkung die Tragfähigkeit und Gebrauchstauglichkeit deutlich verbessert werden kann. Die Ergebnisse der experimentellen Untersuchungen werden mit Hilfe von Stabwerksmodellen beschrieben. Torsion Test at Textile Reinforced Concrete Strengthened Components In order to examine the torsion bearing behaviour of textile reinforced concrete components torsion tests have been carried out at building units with round and square shaped cross sections. Test results display that using textile reinforced concrete can considerably improve the load carrying capacity and the serviceability. The results of the experimental research are being described by strut‐and‐tie models.     

配筋钢纤维高强混凝土薄壁箱梁受扭性能分析

本文在配筋钢纤维高强混凝土薄壁箱形截面纯扭构件试验研究的基础上,应用空间软化桁架理论,结合钢纤维高强混凝土本构关系及其软化系数方程,编制了软化桁架模型分析程序对试验构件进行了全过程分析。分析时考虑了钢纤维高强混凝土的抗拉强度作用,不仅能较准确地得到极限扭矩,而且可以获得开裂扭矩,能较好地模拟试验构件的受力-变形全过程。其结论也得到其它相关钢纤维高强混凝土纯扭构件试验结果的证实。在此基础上应用软化桁架模型分析程序对配筋钢纤维高强混凝土薄壁箱形截面纯扭构件进行了参数分析,提出了相应的开裂扭矩和极限扭矩计算公式,为其工程应用提供依据。     

Torsionsträgheitsmoment und Schubmittelpunkt von Winkelprofilen

Torsional cross‐section properties for angles. Using the finite element method (FEM) it is possible to determine accurate torsional cross‐section properties for hot rolled angle‐profiles. These values partly show relative big discrepancies to the approximations used in practice. For this reason the accurate position of the shear centre as well as the St. Venant's torsion constant for the profiles of DIN EN 10056‐1 are specified. In addition the knowledge of these values can be used to improve familiar approximations and to develop new formulae for the torsional constant.