Closed-form moment solution for continuous beams and bridge structures

Moment distribution continues to be a valuable structural analysis tool for spot-checking member-end-moments of continuous beams and frame structures from computer results. Prior to the widespread use of mainframe and, later, personal computers in the 1980s and 1990s, moment distribution provided a simple hand method for practicing civil engineers to analyze and design statically indeterminate structures, since its public introduction in the early 1930s by Hardy Cross, directly leading to new types of designs and construction. The chief benefits of this method of analysis over the other available methods are that (1) member-end-moments are found directly from the applied loads without first determining various displacement components, (2) no simultaneous equations need to be formed and solved, (3) it is easy to remember and apply, (4) an approximate solution can be obtained at any stage in the analysis and (5) a remarkable variety of statically indeterminate structures can be solved by hand. While the development and introduction of moment distribution represents a great advancement in the history of structural engineering, the distribution process can be tedious, especially if only a few final member-end-moments of a multi-redundant structure are desired. Additionally, when results are wanted to a greater accuracy, the number of distribution cycles required and the number of significant figures that must be carried through the process make it a daunting procedure. In this paper, a new method is presented that gives exact member-end-moments for continuous beams and bridge structures, without the need to distribute moments back and forth as in moment distribution or to set up and solve simultaneous equations, as with the stiffness method. It is anticipated that the primary application of the proposed method will be to spot-check computer results.     

A generalized method for estimating drifts and drift components of tall buildings under lateral loading

A generalized method for estimating the drifts of tall buildings composed of planar moment‐resisting frames and coupled shear walls under lateral loading is presented. This method establishes the stiffness equations at the story levels by assuming that all the nodes in the same floor of a planar lateral‐force‐resisting unit have an identical lateral displacement, an identical rotation component due to the axial deformations of the columns, and an identical rotation component due to the flexural and shear deformations of the beams. By adopting this simplification, the story drifts contributed by different types of deformations, namely, the axial deformations of the columns or wall piers, the flexural and shear deformations of the beams, and the double‐curvature bending and shear deformations of the columns or wall piers, can be identified. In the formulation of the stiffness matrix, the P‐Delta effects were also incorporated. Through comparisons between the lateral displacements and story drifts computed using the proposed method and those computed using the structural analysis software Midas/Gen, the proposed method is proved to have high accuracy in estimating the drifts of tall building structures.     

Sources of elastic deformations in steel frame and framed-tube structures: Part 2: Detailed subassemblage models

This paper examines the accuracy of a set of equations for computing Displacement Participation Factors (DPFs) for beam-column subassemblages of steel moment resisting frame buildings. These factors allow the analyst to determine how the entire subassemblage or individual components of a subassemblage contribute to a given structural displacement. Additionally, the component’s contribution to displacement may be evaluated in terms of sources of axial, flexural, or shear deformation.When applied to a set of 12 isolated subassemblages, it was shown in Part 1 of the paper that deformations in the beam-column joint are very significant, and that flexural deformation in the joint, which is often ignored, should be considered in all analyses. The total displacement predicted through the use of the DPFs correlates extremely well with the results of detailed three dimensional finite element analyses of the same subassemblages. However, it was also shown that there is considerable uncertainty in the bending moments and moments of inertia that are used to compute joint flexural deformations.The objective of this paper, which is the second part of a two-part paper, is to further investigate the accuracy of the DPF expressions developed in Part 1. This is done by computing DPFs from the results of detailed three dimensional finite element analysis, and comparing these to those computed through the use of the simple expressions.The results of the analysis show that the joint flexural deformations are accurately predicted by the simple DPF expressions, but that this accuracy arises from compensating “errors” in the simplified analysis. It is also shown that the use of beam flange continuity plates has a marginal effect on computed displacements. The paper ends with recommendations for using the simplified expressions for computing subassemblage deformations, and for including such deformations in structural analysis of steel frame and tube structures.     

Bestimmung der Wölbspannungen von doppeltsymmetrischen Kastenquerschnitten mit Diagrammen

Determination of warping stresses of box sections by diagrams. For determination of warping torsion for beams with box section it is sufficient to examine only points with external torsional moments. Moreover in the case of statically indeterminate systems the reaction torsional moments may be calculated neglecting warping torsion completely. This contribution allows the determination of normal and shear stresses of warping torsion by diagrams or alternatively by formulae depending only of two parameters of the section. Warping moment and secondary torsional moment must then not be calculated. The applied theory encludes secondary shear deformation, which is always necessary in the case of hollow sections.     

Closed-form expressions for long-term deflections in high-rise composite frames

This paper presents closed-form expressions for rapid prediction of long-term deflections in high-rise steel concrete composite frames subjected to service load. The closed-form expressions predict the inelastic mid-span deflections in beams of frames (typically for 20 years, considering cracking, and time effects, i.e., creep and shrinkage in concrete) from the elastic moments and elastic mid-span deflections (neglecting cracking, and time effects). The expressions also take into account the sagging moments developed in beams due to the substantial differential shortening of adjacent columns in high-rise frames. The expressions can be used for frames with any number of bays and storeys. The expressions have been obtained from trained neural networks. The training, validating, and testing data sets for the neural networks are generated using a hybrid analytical-numerical procedure of analysis. The proposed expressions are verified for example frames of different number of spans and storeys and the errors are shown to be small. The expressions can be used in every day design as they enable a rapid prediction of inelastic deflections with reasonable accuracy for practical purposes without detailed complex analysis and require computational effort that is a fraction of that required for the available methods.     

纤维增强复合筋超静定混凝土梁的抗弯性能

采用改进的局部变形模型,对纤维增强复合筋超静定混凝土梁的抗弯性能进行研究。首先提出一个模型并将其性能与由FRP筋加固的简支及连续混凝土梁的试验结果进行对比。然后将模型应用到FRP筋连续梁中来预测弯矩分布,以及弯曲裂缝的形态,包括裂缝间距和裂缝宽度。尤其是该模型具有在梁中的高应变区确定变形的能力。基于理论分析结果,通过比较FRP筋增强的梁与普通钢筋混凝土梁,阐述了混凝土梁的延性和超载性能。所有的结果和相关解释均取决于某些特定假设和模型中的输入参数,尤其取决于FRP筋和混凝土之间的粘结性能。     

钢筋混凝土梁的弹塑性损伤分析

基于刚度下降建立了力学概念明晰、能涵盖截面、构件和结构三个层次的损伤指标体系,编制了相应的有限元程序,并用于对钢筋混凝土多跨超静定连续梁进行弹塑性损伤评估,表明了该方法的可行性.     

Analytical expressions for preliminary design of dissipative bracing systems in steel frames

In this paper a direct displacement-based design (DDBD) method for seismic design of steel frames equipped with dissipative braces is proposed. Attention is focused on concentric braced steel frames with pinned beam-to-column joints in which the bracing system (with viscoelastic or elastoplastic dissipative devices) is the main seismic resistant component. The proposed design method uses an equivalent continuous model where flexural deformability and shear deformability are related respectively to columns and diagonals of the bracing system. In this way, analytical expressions of the required flexural and shear stiffness distributions are obtained. These expressions are quite simple and can be conveniently used in preliminary design of dissipative diagonal braces and columns. Examples are shown for steel frames with dissipative braces based on elastomeric dampers (viscoelastic devices) and steel frames with buckling-restrained braces (elastoplastic devices). Results of time history analyses are illustrated and discussed in order to evaluate the effectiveness of the proposed DDBD procedure.     

Damage-plasticity model for mixed hinges in steel frames

A damage-plasticity based mixed axial-shear-flexural (PVM) link element for the inelastic analysis of frames is introduced in this paper. The multi-surfaces yield concept is utilized in the definition of the element. The yield surfaces are defined in deformation space and interaction of axial-shear-flexural deformations is considered by defining non-rectangular yield surfaces. The element is capable of considering damage and post-peak softening behavior. The analytical results of introduced element are verified by existing experimental results of steel beam to column connection and it is indicated that the analytical results have reasonable agreement with the test results. Nonlinear dynamic analyses are performed on moment frames with the verified element in two cases: with and without consideration of damage. The results exhibit that damage consideration considerably changes the results. The effect of consideration of the interaction between shear deformation and flexural deformation at the ends of beams of frames under lateral and gravity loads is also investigated. The results for the long beams show that, although in the stage of small plastic deformations the shear-flexural interaction is negligible, for the extensive plastic deformation near the failure point, it could be decisive.     

Steel self-centering moment frames with collector beam floor diaphragms

Following a major earthquake, self-centering moment resisting frames (SC-MRFs) are capable of returning to their original plumb position and sustaining little structural damage. Under earthquake loading, these frames are characterized by a gap that develops between one of the beam flanges and the column face at the connection. This gap causes the frame to expand and therefore imposes special design requirements on the floor diaphragm and on the floor system (e.g. filler beams and slab) that are connected to it. Through nonlinear analyses of several floor diaphragm designs, this paper examines the influence that the floor diaphragm stiffness, strength, and configuration have on the seismic response of SC-MRFs. The floor diaphragm is represented by collector beams that transfer the inertia forces from the building floor to the SC-MRF. This work also presents several equations that are used to predict the effect that the floor diaphragm has on the axial force and moments that develop in the SC-MRF beams. This effect is shown to not be negligible. Finally, practical considerations related to the construction of SC-MRFs that use collector beams as the floor diaphragm are discussed.     

Moment redistribution in continuous plated RC flexural members. Part 1: neutral axis depth approach and tests

It is now common practice to retrofit or rehabilitate existing reinforced concrete beams and slabs by adhesively bonding fibre reinforced polymer (FRP) or metal plates to their surfaces. Advanced design rules are available for quantifying the various plate debonding mechanisms and consequently the shear and flexural capacities of the plated sections. These design rules show that even though the required increase in strength can be obtained by plating, plate debonding can severely reduce the ductility of a flexural member to such an extent that plating guidelines often exclude moment redistribution. This exclusion may reduce the application of plating, in particular to retrofitting buildings where ductility is often a requirement, or it may require the development of a radically different approach to design that does not rely implicitly on ductility. In this paper, it is shown that the commonly used neutral axis depth approach for moment redistribution in RC flexural members cannot be used for most plated structures because plate debonding often occurs before the concrete crushes. Tests on plated flexural members are also reported which show that moment redistribution can occur. In Part 2 of this paper, a moment redistribution analysis procedure is developed that can cope with plate debonding of externally bonded plates.     

剪力滞对斜拉桥中Π形主梁弯曲刚度影响分析

按实际施工动态过程分析剪力滞对Π形结合梁斜拉桥中主梁弯曲刚度的影响。基于能量变分法引入简化的剪力滞翘曲位移函数,导出一种可以考虑剪力滞与Π形结合梁弯曲刚度耦合影响的实用梁单元。该方法每节点采用两个剪力滞自由度,可以处理斜拉桥中集中弯矩引起的剪力滞位移边界条件突变和施工动态过程中荷载和剪力滞位移边界条件不断变化的问题。结合某Π形结合梁斜拉桥进行数值分析并与实测数据比较,验证了本文方法的有效性与准确性。研究表明,剪力滞效应使斜拉桥中主梁的抗弯刚度减小,且在悬臂施工阶段比成桥阶段的影响更大;相对于简支箱梁和连续箱梁而言,剪力滞对Π形结合梁斜拉桥主梁弯曲刚度的影响相对小一些,在悬臂施工阶段其对中跨的影响约为2%~7%,对边跨的影响约为3%~12%。     

钢筋砼框架考虑斜裂缝影响的非线性全过程分析

本文通过对一些约束梁、约束柱及连续梁抗剪试验资料的分析研究，采用斜压场模型模拟有反弯点的剪跨区段当斜裂缝出现后剪力作用下的受力状态，并考虑剪跨比的影响及纵筋的暗销作用，导出了剪力、剪切变形、轴力之间的关系。用此模型对不同条件的５８根剪切破坏的约束梁、柱进行了计算比较。同时，将模型引入框架非线性分析中的单元刚度矩阵及固端力列阵，编写了框架非线性全过程分析程序，对一定数量发生弯曲剪切破坏的两跨连续梁进行了计算比较，计算结果与实测结果符合较好。本文进行了三榀门式框架及二根两跨连续梁的试验，用上述程序进行了框架、连续梁考虑斜裂缝对内力重分布及构件变形影响的比较分析。     

Sources of elastic deformation in steel frame and framed-tube structures: Part 1: Simplified subassemblage models

Three simple models for including the effect of beam-column joint deformation in the analysis of steel moment resisting frame and framed tube structures are presented. The first model, called the Fictitious Joint model, is based on two-dimensional frame analysis and is useful for preliminary analysis only. The second model, called the Krawinkler Joint model, and the third model, known as the Scissors Joint model, use an assembly of rigid links and rotational springs to represent the joint, and may be used in preliminary and final analysis of full structural systems. All derivations are provided in the form of “displacement participation factors”, which allow a detailed breakdown of the various components of subassemblage displacement.When applied to isolated beam-column subassemblages, it is shown that all three modeling approaches produce the same general expression for computing deflections arising from shear deformations in the panel zone region. However, the Krawinkler and Scissors models do not include the effects of flexural deformation within the beam-column joint, whereas the Fictitious Joint model does. While not the dominant source of deformation, it is shown in the paper that the effects of flexural deformations in the beam-column joint should not be ignored.It is also shown in this paper that the overall displacements predicted by the simplified models correlate very well with displacements computed from detailed three dimensional finite element analysis of the same subassemblage. However, the finite element analysis approach, taken alone, is not capable of providing a breakdown of the subassemblage displacements into components, such as panel zone shear, or column joint flexure. Part 2 of the paper presents a method for providing this information from the results of detailed finite element analysis.     

弯矩梯度下梯形波纹板Ⅰ形梁的弯矩修正系数

研究弯矩梯度和不同端部约束条件下,梯形波纹板Ⅰ形梁的弯矩修正系数。通过使用梁单位有限元法得到弯矩修正系数,并给出梯形波纹板I形梁横截面特性的一般公式和一个新的翘曲常数。通过与采用壳单元的商业有限元软件给出的结果进行对比证实了弯矩修正系数理论结果的有效性。对波纹尺寸和长度不同的一系列有限元模型进行分析研究,结果表明,目前的弯矩修正系数公式不能很好地预测各种端部约束条件下端部弯矩不规则的梯形波纹板Ⅰ形梁的屈曲性能。为设计方便,给出弯矩修正系数的封闭表达式并成功得到验证。     

预应力混凝土超静定梁的预应力筋布置分析

针对预应力混凝土超静定梁 ,采用结构力学方法 ,推导了在均布及集中荷载作用下 ,二跨及三跨连续梁的次弯矩表达式 ;在满足混凝土设计规范中规定的预应力混凝土构件在施工阶段预拉区抗裂要求的条件下 ,得到了预应力筋布置偏心距的表达式。可作为超静定梁进行预应力筋布置的依据。     

考虑剪切变形影响的框架结构稳定性

巨型框架结构由于框架梁柱截面的剪切变形较大,在分析巨型结构整体稳定性时,应当同时考虑巨型梁柱的弯曲变形与剪切变形。本文研究了框架梁和框架柱截面的剪切变形对框架结构整体稳定性的影响。分别研究了简单对称框架柱顶作用有相同集中荷载时、简单对称框架柱顶作用有不等集中荷载时、以及不对称框架柱顶作用不等集中荷载时,框架结构的整体稳定性。分析了不对称框架柱剪切变形对框架稳定性的影响及框架发生整体有侧移失稳的内在规律。着重讨论了框架梁截面的剪切变形对框架整体稳定性的影响规律,根据这一规律提出了简化方法以考虑横梁剪切变形的影响。本文提出了计算框架临界荷载的近似计算方法,与有限元法的结果对比,具有很好的精度,近似算法均考虑了框架梁和框架柱截面的剪切变形及弯曲变形。     

Biegedrillknicken von Riegeln und Stielen von Hallenrahmen mit nachgiebigen Rahmenecken

Lateral torsional buckling of beams and columns of frames with semi‐rigid beam‐to‐column joints. Lateral torsional bucking safety of beams and columns of frames depends, among other things, on the ratio of the bending moment of the frame knee to the plastic moment resistance of the frame beam and the frame column respectively; in this context the ratio of the bending moment of the frame knee to the midspan bending moment of the frame beam is also of importance. Large bending moments of frame knees can cause all over “flexural buckling” of the lower flange of the frame beam if the frame beam is laterally supported only at the upper flange. Using semi‐rigid beam‐to‐column joints bending moments of frame knees can be decreased and production cost can be diminished. Considering a frame with welded beam‐to‐column joints the influence of the moment resistance of the frame knee (which depends on the type of joint) is discussed with respect to lateral torsional buckling safety of beams and columns of frames. The determination of the moment resistance and the moment‐rotation‐characteristic of beam‐to‐column joints is based on the rules given in EC 3 Annex J, which represents (outside the application range of finite‐element‐methods) the current level of technology achieved.     

The effect of shear deformations in floorbeams on the moment distribution in orthotropic plated bridge decks

Floorbeams of orthotropic plated bridge decks are generally elements with a low slenderness, especially in the case of railway bridges. When loaded, such elements generate not only flexural deformations but shear deformations as well. Depending on the configuration, these shear deformations can be considerably large and should certainly not be neglected. In a design according to the Pelikan-Esslinger method, this deformation is taken into account in the second stage of the calculation of the orthotropic deck. At this stage, the additional bending moments, shear forces and floorbeam reactions due to the floorbeam flexibility are evaluated. The deflection of a directly loaded floorbeam creates a distribution of the load to adjacent non-loaded floorbeams. In addition, the deflection will affect the longitudinal ribs, increasing the sagging moments at midspan and decreasing the bending moments at the supports of the ribs provided by the floorbeams. In this paper a method is developed to take the shear deformation in the floorbeam into account, thus correcting the Pelikan-Esslinger method. The correction is especially important since the floorbeam web is frequently further weakened by cutouts for the longitudinal ribs and in some cases additional cope holes. The floorbeam web resisting the shear deformation is further reduced by these openings, increasing the importance of shear deformations even for more slender floorbeams. The validity of the proposed method is checked by full finite element calculation using shell elements which inherently comprise shear deformation. The effect of the correction on the calculations in the second stage of the calculation are relatively small for the floorbeam reactions; on the contrary, it can be quite large for the additional bending moments in the longitudinal ribs. In that case, the effect is worth considering.