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1、ENGINEERING JOURNAL / SECOND QUARTER / 2005 / 99 Flexible Moment Connections for Unbraced Frames Subject to Lateral ForcesA Return to Simplicity LOUIS F. GESCHWINDNER and ROBERT O. DISQUE I t seems that there has been confusion among structural engineers about the type of construction referred to in
2、 the AISC Load and Resistance Factor Design Specifi cation for Structural Steel Buildings, since 1986, as Partially Re- strained or PR. The general concept has been of interest to the authors for many years and has been the topic of several of their papers. The purpose of this paper is to reacquaint
3、 the profession with a longstanding and successfully applied approach to structural steel frame design, herein called “Flexible Moment Connections (FMC),” and to compare this approach to the Specifi cation-defi ned PR approach. In addition, the goal is to show that although much has changed in the p
4、rofession, including specifi cations and the tools for their application, FMC design remains an acceptable and economical approach for steel structures. The “Flexible Moment Connections” approach has been permitted in this country and around the world since at least the 1910s (Fleming, 1915). The ba
5、sic principles of the FMC approach are to treat the beams as simply connected under gravity loads but as moment connected under lateral loads. The approach used for these historic designs has been re- ferred to as “Type 2 with wind,” “Semi-rigid,” “Smart Con- nections,” “Flexible Wind Connections,”
6、or with the current British term “Wind-Moment Connections” (Salter, Couch- man, and Anderson, 1999). These historical approaches appear to have fi rst been recognized in United States (U.S.) specifi cations through the AISC Specifi cation in 1946, as Type 2 with wind. Perhaps the fi rst U.S. paper t
7、o address the actual connection moment- rotation capacity and suggest that an approach other than the Type 2 with wind approach be used, was that of Rathbun (1935). What may be the fi rst paper to discuss the actual response of frames designed with the Type 2 with wind approach was that of Sourochni
8、koff (1950), although the method had been in common use for more than 40 years at the time. Another early paper that addressed the seeming paradox of connections knowing when to resist moment and when not to resist moment was presented by Disque (1964). An uncounted number of buildings have been suc
9、cessfully designed with this approach, including such well-known structures as the Empire State Building and the UN Secretariat as well as a large number of unnamed buildings of a common nature. However, current buildings do not exhibit the same level of extra unaccounted for stiffness, such as stif
10、f masonry infi ll walls, that those earlier buildings exhibited. Thus, since the Type 2 with wind approach continues to be used by the profession, it seems appropriate that the approach should be reassessed and, if proven viable in todays world of structural engineering, updated as a tool for todays
11、 designers. It is important to recognize that the proposed approach rests on a signifi cant number of approximations regard- ing connection stiffness, frame behavior, column effective length, and bending moment amplifi cation. As with any de- sign approach, it must be carried out using good judgment
12、 and a thorough understanding of the assumptions made. BASIC UNDERSTANDING Connections In order to understand FMC, it is fi rst necessary to under- stand the general behavior of a beam-to-column connection. Figure 1 illustrates the moment rotation behavior of three generic connections. One that exhi
13、bits a small amount of ro- tation with a large amount of moment is noted as a rigid con- nection. A second connection that exhibits a large amount of rotation with a small amount of moment is noted as simple. The third connection is noted as a semi-rigid connection and provides some moment restraint
14、 while permitting some rotation. Semi-rigid connections can fall anywhere between simple and rigid as shown. In general, connections that are capable of resisting at least 90 percent of the beam fi xed-end Louis F. Geschwindner is vice-president of engineering and research, American Institute of Ste
15、el Construction, Inc., and professor emeritus of architectural engineering, The Pennsyl- vania State University, University Park, PA. Robert O. Disque is a consultant, Milford, CT. 100 / ENGINEERING JOURNAL / SECOND QUARTER / 2005 moment are referred to as rigid. Those that offer enough duc- tility
16、to accommodate beam end rotation while resisting no more than 20 percent of the fi xed-end moment are referred to as simple. Any connection that is capable of resisting a moment between these limits while permitting some rota- tion must be treated as semi-rigid. It should be clear that in order to m
17、ake this distinction, something about the beam to which the connection is attached as well as the details of the connection must be known. Numerous researchers have presented the details of connection behavior and at least two collections of this data have been presented (Goverdhan, 1983; Kishi and
18、Chen, 1986). A mathematical model for the semi-rigid connection will be discussed later. With the introduction of the Load and Resistance Factor Design Specifi cation for Structural Steel Buildings in 1986, new terms were introduced to defi ne connection behavior. The rigid connection became known a
19、s fully restrained or FR and all other connections, both semi-rigid and simple, became known as partially restrained or PR. That is, simple connections were redefi ned as a special case of PR moment connections. In keeping with this, for the remainder of this paper, the term PR will be used to refer
20、 to all connections that are not FR. Attention will fi rst be given to the infl uence of connection behavior on members and then a more detailed discussion of connection behavior will be presented. Beams Carrying Gravity Load Only For a symmetrical, uniformly-loaded beam, connected to rigid supports
21、 and assumed to behave elastically, the end rotation of the member is directly related to the load mag- nitude. This is most easily described through the use of the classic Slope-Defl ection Equation such that where M = end moment = end rotation This equation is shown as a straight line in Figure 2
22、and is referred to as the “beam line.” Superposition of a PR moment connection curve from Figure 1 with the beam line from Figure 2 is shown in Fig- ure 3. Equilibrium is attained at the intersection of these two curves, shown as point a. For the nonlinear connection curve shown, this point is not e
23、asily obtained. However, if the con- nection were to be modeled as a straight line, then a simple mathematical solution could be easily found. The straight- line connection model shown in Figure 3 has a slope of K = M/ and intersects the beam line at the same point as the actual connection curve. So
24、lving for the connection rota- tion from this relationship and substituting into Equation 1, the moment in the connection and on the end of the beam at equilibrium is Fig. 1. Typical moment-rotation curves for the three connection types. M wLEI L = 2 12 2 (1) Fig. 2. Beam line for symmetrically load
25、ed beam. M wL EI KL wL u = + = +() 22 12 1 2 12 12 (2) ENGINEERING JOURNAL / SECOND QUARTER / 2005 / 101 where Due to symmetry, the moment at the center of the beam span can be found by subtracting the end moment from the simple beam moment. A plot of both the end and mid-span moments as a function
26、of the beam/connection stiffness ra- tio, u, is given in Figure 4. It can be seen that as the end mo- ment decreases, the mid-span moment increases. The maxi- mum limits are the fi xed-end moment for the beam end and the simple beam moment at mid-span. At the intersection of these curves, the end an
27、d centerline moments are the same. If a connection could be built with the required moment ro- tation behavior, this would be the most economical case for beam design. This is the same result that would be obtained if the plastic mechanism moments were found for beams with connections capable of att
28、aining the plastic fl exural strength of the member. It is also noted that 90 percent of the fi xed-end moment results when u = 0.055 and 20 percent of the fi xed-end moment occurs when u = 2.0. A more com- plete discussion of the beam/connection relationship can be found in a paper by Geschwindner
29、(1991). Beams Carrying Lateral Load Only Since it is the intention here to also address the infl uence of PR moment connections on lateral systems, it is important to fi rst study these members in a simplifi ed format. A simple rigid portal frame with lateral load H is shown in Figure 5a. Fig. 3. Be
30、am line and connection curveequilibrium. u EI KL = (3) Fig. 4. Beam response as a function of connection stiffness. Fig. 5. Portal frame with lateral and gravity load. 102 / ENGINEERING JOURNAL / SECOND QUARTER / 2005 A statically indeterminate analysis will show that the frame response is indeed as
31、ymmetric and the horizontal shears at the column supports are each H/2 as shown in the fi gure. If the rigid connections are now replaced with linear PR mo- ment connections that exhibit the same behavior for both positive and negative rotation, asymmetry will still prevail and the support shears wi
32、ll again be H/2. By plotting the beam end moment as a function of beam and connection stiffness, similar to what was shown in Figure 4, it can be shown that the end moment remains unchanged, as long as the connections continue to behave the same under positive and negative rotation. However, the rot
33、ation required for moment equilibrium changes as the connection stiffness changes. This rotation will, of course, yield a corresponding lateral displacement for the frame, which will increase as the connection stiffness decreases. Beams Carrying Combined Gravity and Lateral Loads The next step in de
34、veloping a basic understanding is to put the gravity system and the lateral system together. The uniformly-loaded beam is now made part of a simple portal frame shown in Figure 5b. The beam line for a given load magnitude and the linear connection line are shown again in Figure 6. Note that for grav
35、ity load only, equilibrium is given at the intersection of these two lines and is noted as point a for the left end of the beam and point a for the right end. If the lateral load is then applied, the windward connection unloads and the moment in the connection reduces as the rotation reduces, shown
36、as b. For the leeward connection, the moment increases as the rotation increases, shown as b. If the gravity load is increased or decreased while the lateral load is maintained, points b and b move up or down the connection line while maintaining their separation along the connection line. If the la
37、teral load is reduced, points b and b move toward each other, each moving the same amount; and if the lateral load is increased, they move apart. A simi- lar discussion for the nonlinear connection will be presented later. PR MOMENT CONNECTION MODEL As was mentioned earlier, a number of connection m
38、oment- rotation models have been proposed in the literature (Rich- ard, 1961; Kennedy, 1969; Frye and Morris, 1975; Krish- namurthy, Huang, Jeffrey, and Avery, 1979; Mayangarum, 1996). For the discussion here, the Three Parameter Power Model, which has been used for a wide range of connection types
39、and is calibrated for a variety of connection element properties (Kim and Chen, 1998), will be used. Figure 7 shows power model curves for a top- and seat-angle, double web-angle connection with top- and seat-angle thickness in- dicated. It is important to note that the accuracy with which the model
40、 predicts the true connection behavior is quite im- portant. An increase or decrease in thickness of the top and bottom angles yields the curves above and below the original curve. A beam line is superimposed on the connection curves and it is seen that there can be a signifi cant difference in the
41、moment magnitude at the point of equilibrium. Referring back to the curves of Figure 4, it can be seen that there is a potential for signifi cant error in the calculated versus actual beam moments. In addition, a review of the development of any connection behavior prediction equation will reveal th
42、at even for those that are accurate, they are only accurate within some specifi ed range for specifi c parameters. Fig. 6. Beam line with linear connection stiffness. Fig. 7. Infl uence of top- and seat-angle thickness on connection response. ENGINEERING JOURNAL / SECOND QUARTER / 2005 / 103 In addi
43、tion to requiring an accurate model of the connec- tion as load is applied, it is necessary to have a model for unloading and reloading if a detailed analysis of a PR con- nected frame is to be carried out. The normal assumption is that a connection loads along the moment-rotation curve and unloads
44、linearly with a slope equal to the initial slope of the curve. This type of model has been verifi ed in numerous connection tests and has been used in dynamic frame analy- sis by Khudada and Geschwindner (1997) and static frame analysis by Rex and Goverdhan (2002). Since the work of Rex and Goverdha
45、n was applied to a real building structure and followed the analysis through several cycles of load, it will be instructive to review their results. Figure 8 shows the power model connection curve for the top and bottom angle connection similar to that used by Rex and Goverdhan (2002). It should be
46、noted that Rex and Goverdhan used the Mayangarum model (Mayangarum, 1996) for their analysis. Using the general results of their analysis and the presentation by Sourchnikoff (1950), the response of the PR moment connection can be described. Point a, a represents the equilibrium position under gravi
47、ty load for this connection on the end of a uniformly loaded beam in a lateral load resisting system. When lateral load is applied, as presented earlier for the linear connection, the windward connection unloads and the leeward connection loads. Since unloading is expected to follow the initial con-
48、 nection stiffness, point a moves down the straight line to point b. The loading connection continues up the connection curve to point b. The fi rst thing to notice here is that the response is not the symmetric response used earlier for the linear connection model. This is quite important since it
49、sig- nifi cantly complicates the analysis. No longer are the lateral load moments the same on each end of the member and of course no longer will the response take on the simple form of points b and b moving up and down the connection curve as discussed earlier. Many possible sequences of loading could be considered from this stage on but for the current discussion, a few spe- cifi c cases will be presented. Consider that one half of the lateral load is removed. The response moves to points c and c, both represented by a linear change. Now reapplying the half lateral load, t