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1、Behavior and Design of Flexibly-Connected Building Frames KURT H. GERSTLE and MICHAEL H. ACKROYD INTRODUCTION Although the behavior of connections in steel construction extends over the full range from near-pinned to almost- rigid, traditional engineering practice has considered only the extreme lim
2、iting cases: either perfectly pinned, as in ideal trusses, or fully rigid, as in rigid-frame construction. The neglect of real connection behavior can lead to unrealistic predictions of the response and strength of steel structures, and less than optimal design in steel construction. This paper was
3、written in order to demonstrate that more realistic connection behavior can be included in analysis without undue pain, and that design of flexibly- connected steel frames is fully within reach of professional office practice. We have tried to explain the concepts and procedures in a simple fashion,
4、 and to demonstrate the benefits of a more realistic approach by means of several examples. EFFECT OF CONNECTION FLEXIBILITY ON STRUCTURE BEHAVIOR Connections which transmit moments M between adjacent members will undergo relative rotation, as shown in Fig. 1a. The relation between these two quantit
5、ies is represented by the moment-rotation (M-) curve, shown in Fig. 1b. The traditional extreme assumptions of ideal-pinned, or perfectly rigid behavior are given by the straight lines along the -axis in the first, and along the M-axis in the latter case. In fact, any connection will have some inter
6、mediate stiffness between these extremes, as shown for several real connections in Fig. 1b. These M- relations are in general non-linear, with decreasing stiffness under increasing moment given by the slope k of the M- curve. We will consider this actual connection behavior later, but for the time b
7、eing we will simplify the situation by assuming a linear M- curve, of representative constant rotational stiffness k. We will show later that such an assumption can capture the structure behavior under service load with reasonable accuracy. We will now consider the interplay between connection behav
8、ior and structure behavior by means of two examples. Floor Framing Under Gravity Loads Floor beams with double web angle connections or shear tabs to floor beams, shown in Fig. 2a, are usually analyzed Kurt H. Gerstle is professor at the University of Colorado, Boulder. Michael H. Ackroyd is preside
9、nt of First Principles ngineering, Acton, Massachusetts. and designed as simply supported. The design moment Mmax, or the deflection , will determine the beam size. In fact, the web angle connections to the girders will have some rotational stiffness and therefore moment resistance which will serve
10、both to diminish the design moment and the beam deflection, as shown in Fig. 2b. It will be worth exploring whether consideration of the actual connection stiffness will allow the use of lighter floor beams. Lindsey and colleagues1 have followed just such an approach in the design of roof purlins, a
11、nd realized a saving of 16 percent of material by relying on the available stiffness of the specified shear tab connections. Multi-Story Under Lateral Load 2 Figure 3a shows an unbraced four-story building frame under lateral loads. Analyses were carried out considering a range of beam-column connec
12、tion stiffneses ranging from near-pinned to near-rigid. The column moment diagrams of Figs. 3b to e indicate moment variations ranging from those of a cantilever beam to those which we generally associate with the shear-type deformations of rigid-jointed frames. Similarly, the sways of the frames ar
13、e also shown in these figures for different connection stiffnesses, and indicate the sensitivity of the deflections to the connection behavior. We observe in particular that the assumption of rigid joints may lead to gross underestimation of both column moments and story sways. In fact, Figs. 2b to
14、e show that the structure deflections seem to depend more on the connection than on the member behavior. In view of this observation it seems inconsistent to expend much loving care on member behavior, and treat the connections in rather cavalier fashion. No doubt we do this because we can express m
15、ember behavior in terms of elegant and attractive theory, but connections are messy and uneducated and do not lend themselves readily to Fig. 1. Moment-rotation characteristics of steel frame connections. 22ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION analysisas we will see shortly
16、. REQUIREMENTS FOR OFFICE PRACTICE To implement flexibly-connected frame design in office practice, we need to consider these factors: 1. Code authorization, 2. Information about connection behavior, 3. Simple analysis procedures, and 4. Office-oriented design methods. We will discuss these aspects
17、in the following: Code Authorization Flexibly-connected frame design has been accepted by the AISC Allowable Stress (ASD) Specifications since many years under the labels Type 2 and Type 3 Construction.3 The former is an approximate method predating computer days. Based more on art than science, it
18、has been widely used to design serviceable buildings, however, of unknown stiffness or strength. Type 3 suggests, but does not explicitly require, a rational analysis which considers the effects of actual connection behavior. No specific guidelines are provided as to how this might be implemented, a
19、nd we hope that our presentation might be useful toward this end. In the 1986 LRFD Manual4 Type PR, or partially restrained, again authorizes flexibly-connected frame design, but without further instructions. Following either steel design procedures, it is clear that codes present no obstacle to a m
20、ore realistic approach to steel frame design. Indeed, LRFD encourages the use of precise procedures, such as second-order inelastic analysis. Connection Behavior It is interesting to note that in spite of various attempts5,6,7 no reliable analytical method for the prediction of connection behavior h
21、as been accepted by the profession. It provides food for thought that, in spite of all analytical progress of recent years, such a longstanding problem still escapes our understanding. Fig. 2. Flexibly-connected floor beam. In the absence of analytical solutions, reliance must be placed on test resu
22、lts. Connection testing has been carried out only sporadically since the 1930s.8 Complete, systematic test programs of specific connection types covering a full range of sizes and conditions are rare.6 Two recent collections9,10 have attempted to gather all available test data on connections for use
23、 by engineers; however, in most cases this information is insufficient to cover the full range of connection types, sizes, kinds of fasteners, and member-connection interplay. For the designer who needs connection data, Refs. 9 and 10, along with considerable imagination and daring, are probably the
24、 best resource. It was such imagination and daring which enabled Frye and Morris5 to develop empirical polynomial moment-rotation relations for a variety of connections in non-dimensional form, of the type shown in Fig 1b, with a scaling factor to account for connection size. Although Ref. 10 shows
25、that agreement between these curves and test results is not perfect, the Frye and Morris formulation has been widely used and can offer great help to the designer. It has been observed that after loading of connections along the non-linear paths shown in Fig. 1b, subsequent unloading and moderate mo
26、ment reversal will take place along a linear path of stiffness similar to that under initial loading. This may provide justification for the linearization which we will advocate for office use in our further discussion. Alternately, a secant modulus from the origin to the point representing the allo
27、wable connection moment under working loads might be used. Because of the variability of actual connection behavior due to fabricating and erection practice, extreme care in the choice of connection stiffness seems unjustified; a fair approximation is sufficient, as will be shown below. ANALYSIS PRO
28、CEDURES FOR FLEXIBLY-CONNECTED FRAMES Working Load Analysis We suggest a linearly elastic analysis to determine forces and deformations at service levels. Accordingly, the Fig. 3. Moments and sways of flexibly-connected steel frame. FIRST QUARTER / 199023 connections can be modeled as linearly elast
29、ic rotational springs of stiffness k, attached to the prismatic beam as shown in Fig. 4a. For use in the displacement method frame analysis programs which are the mainstay of most offices, the standard (44) beam element stiffness matrix can be modified by classical methods of analysis to include the
30、 elastic springs. For springs of equal stiffness k at both ends, the stiffness matrix for the nodal numbering of Fig. 4a is shown in Fig. 4b. Only the parameter EI/kL, defining the ratio of rotational beam to connection stiffness, is needed to include connection rotations. The only additional input
31、data are the connection stiffnesses k. (For unequal connection stiffnesses, somewhat more complex matrices are derived in Ref. 11.) Fixed-end moments for beams with elastic end connections can be derived similarly, and will also depend only on the modifying factor EI/kL.11 In any case, it is a simpl
32、e matter to modify any rigid frame analysis computer program to analyze flexibly- connected frames as well, and we believe that such a program should be among the available tools of any well- equipped structural design office. Strength Analysis Linearly elastic analysis cannot predict the strength o
33、f ductile structures. For this purpose, some form of non- linear analysis is needed. We consider that for office practice, the simplest type of such an approach must serve; accordingly, we suggest the representation of connection behavior by a bilinear, flat-topped, elastic-perfectly plastic M- curv
34、e, as shown in Fig. 7b. With this assumption, it follows that conditions under service loads can be predicted by elastic theory, as previously suggested, and structure strength can be computed by the plastic-hinge method, a well-established technique which has been often used for rigid-frame analysi
35、s.12 BEHAVIOR OF FLEXIBLY-CONNECTED FRAMES To demonstrate the use and results of the suggested analysis procedures, we will offer several examples. Fig. 4. Stiffness matrix for flexibly-connected member. Range of Effective Connection Flexibility 13 The flexibly-connected member stiffnesses in Fig. 4
36、b show that they differ from those for a rigidly connected member only by a factor EI/kL. A plot of the ratio of these stiffnesses as function of EI/kL (plotted logarithmically) is shown for the rotational beam stiffness k33 and k44 as well as for the fixed-end moment in Fig. 5. This ratio varies fr
37、om unity for rigid connections to zero for very soft connections. For values of EI/kL 0.05, this ratio will be within about 20 percent of unity, and perfect rigidity can reasonably be assumed. For values of EI/kl 1.0, the ratio will be within about 20 percent of those for ideal pin-ends, so that thi
38、s condition might well be assumed in analysis. It follows that the effects of connection flexibility should be considered for cases in which 0.05 EI/kL 1.0. A review of typical building frames13 has indicated the ranges of EI/kl for fully welded, and bolted, structures shown below the horizontal axi
39、s of Fig. 5. It seems that field-bolted, or lightly welded frames should be analyzed as flexibly-connected, but frames with fully welded joints might be assumed rigid with good accuracy. Sway of Flexibly-Connected Frames We carried out linearly elastic analyses of a family of frames with various fle
40、xible connections ranging in height from five to 25 stories. The top-story sways from these analyses are plotted non-dimensionally in Fig. 6 versus the frame slenderness H/B. We considered three different connection types: floppy top and seat angle connections, fairly stiff flange plates, and rigid
41、joints representing fully welded construction. The curves of Fig. 6 indicate the importance of connection flexibility on frame sway: The contribution of the flexible connection types considered here varies from one-third to two-thirds of the total sway: elastic member deformations may be responsible
42、 for only a minor amount of the total deflections. By drawing a horizontal line in Fig. 6 at the specified Fig. 5. Effective ranges for flexibly-connected members. 24ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION Fig. 6. Sways of flexibly-connected steel frames. allowable sway ratio,
43、 the permissible frame slenderness can be estimated. It may well be that the widely used sway ratio of 1/400 was adopted in full realization of the possible sway underestimation by rigid-frame analysis, and might be increased in recognition of the more realistic analysis. Behavior and Strength Accor
44、ding to Elastic-Plastic Analysis In order to assess the effects of the elastic-plastic idealization suggested earlier, we considered a flexibly- connected girder under uniform load w, shown in Fig. 7a. Four different types of end connections, of moment- curvature relations shown in Fig. 7a were cons
45、idered. Two different analyses were carried out for each case: a fully non-linear analysis using the curves shown solid in Fig. 7b, and an elastic-perfectly plastic analysis using the bi-linear moment-rotation curves shown dashed in Fig. 7b. Figure 7c shows the resulting load-deflection curves, with
46、 the results of the non-linear analysis solid, those of the bi-linear analysis dashed. The limiting cases of perfectly simply supported, and perfectly fixed ends, are also shown dashed. We see that the bi-linear analysis is capable of capturing the essence of the structure behavior. Other analyses14
47、 confirm this conclusion. OFFICE-ORIENTED DESIGN METHODS Over the course of the past fifteen years, we have looked into many approaches to designing building frames that use flexible beam-to-column connections. We were interested in evaluating existing design procedures with regard to safety, servic
48、eability, and economy. We were also determined to come up with improved simpler approaches Fig. 7. Elastic-plastic analysis of flexibly-connected frames. FIRST QUARTER / 199025 to frame design, if possible. At this point in time, we recommend any of the following three different methods, depending u
49、pon how “computer rich“ your office is. For the Computer Elite If you use a minicomputer or super-microcomputer in your office, we heartily recommend that you consider using a matrix structural analysis program for providing a precise analysis of your frames. You can commit yourself to basically two levels of analysis/design sophistication: linear elastic analysis/allowable stress design, or non-linear analysis/limit state design. 1. Frame Design Based on Non-linear Analysis. In our many evaluation studies, we have developed programs that analyze flexibly-connected frames to mi
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