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1、Ponding of Two-Way Roof Systems FRANK J. MARINO SEVERAL PAPERS HAVE been published1, 2 analyzing roof beams subject to ponding. However, the scope of these papers has been limited to the one-directional action of beams. That is, the flexural members are considered supported by unyielding knife edges
2、 with no consideration given to surface deflection transverse to the beam span under study. The effect of interaction between members in a roof framing system can be considerable and should not be neglected. The present AISC Specification3 is cognizant of the ponding problem. Chinn1 points out that
3、the Specification provision is arbitrary in nature and could be overly conservative. It is interesting to note here that in all the cases of collapse attributed to the ponding phenomenon that the author has reviewed, the members involved did violate the present Specification provision. However, the
4、provision may actually be unconservative for very large spans. The purpose of this paper is to analyze the ponding of a roof system, accounting for the interaction of members and to develop a design aid suitable for office use. A restriction imposed on the analysis is that the structural system must
5、 consist essentially of two-way framing (i.e., main girders or primary members and secondary sub-members spanning perpendicular to the girders) with the deck contributing negligible deflection to the system. However, if in the absence of other sub-members the deck spans a substantial distance betwee
6、n main members, it should be treated as the secondary system. ANALYSIS Ponding may be defined as a situation caused by the flexible nature of a structural assembly, created when a flat roof retains rain water that causes deflection of the roof system, which in turn increases its volumetric capacity.
7、 This process is iterative in nature and continues until convergence, which is termed the equilibrium position; or, if the system is divergent, until collapse occurs. Frank J. Marino is Assistant Research Engineer, American Institute of Steel Construction, New York, N. Y. Figure 1 In the case of sim
8、ply-supported members, the deflection due to dead and live loads on the structure, or accidental negative camber, can initiate ponding. In the case of continuous members having identical stiffness in adjacent spans, ponding action can be initiated by small accidental differences in the levels of the
9、se spans before loading. In such circumstances the higher level spans unload, causing accelerated deflections in the adjacent lower spans, and the ponding effect will be similar to that which occurs in a simple span. In either case, it is evident that, to prevent collapse, the equilibrium position m
10、ust be reached before the maximum flexural stresses in the member reach yield point. This phenomenon is by no means unique to steel construction. However, due to steels high strength-to-weight ratio, as compared with other building materials, the problem may be more acute. It is further accented by
11、the introduction of high strength steels and the popularity of plastic design. Both these factors tend to produce designs of shallower depths and therefore more flexible systems. Figure 1 shows the system under investigation. The primary member under discussion is interior girder AB. The secondary m
12、ember considered is the beam GH, which frames into the girder at its mid-span. This, of course, is the critical secondary member. Figure 2 shows the deflected position of members AB and GH at the equilibrium position previously defined. Note that the supporting primary members at both ends are 93 JU
13、LY/1966 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. assumed to be proportional in stiffness and loading so as to have the same deflection . Figure
14、3 illustrates the loads imposed on each member by the ponded rain. In each case, the deflected elastic curve for the members is assumed as a half sine wave. The ponding loads on the primary member, due to deflection of both the primary and secondary members, are assumed to vary as the ordinates of a
15、 half sine wave. The reactions of secondary member on the primary member are assumed continuously distributed rather than as discreet concentrated loads. The ponding loads on the secondary member, however, take the shape of a since curve due to the deflection of the secondary member plus a uniform l
16、oad due to the deflection of the primary member. Considering the primary member, the following observations can be drawn from the load diagram: WLL psowp =+ 2 () and WLLLL sspswp =+ ( ) ( ) ()() 22 2 2 1 ww1 The bending moment at mid-span of the primary member is: M L LL L spspww = + + + 2 2 2 1 3 2
17、()() w 2 2 1 L L spw () + 8 The mid-span deflection due to the ponding rain can be calculated by the conjugate beam method as: + w= + L L EI sp p w 4 4 () 2 4 11 ()() www + and letting L L EI C sp p p 4 4 = www =+CCC ppp 2 2 44 11 CCC pp op ww + Solving for w: wwww =+ + + pp 4 2 4 2 11 (1) where p p
18、 p C C = 1 (a)(b) Figure 2 (a)(b) Figure 3 By similar deduction the mid-span deflection of the critical secondary member due to ponded rain can be expressed as: ww =+ + ss 22 88 (2) where s s s C C = 1 Also, lws =(2a) By combining Equations (1), (2) and (2a), = + + + pps ps 44 () 1 4 (3) and ws =+ +
19、 + 22 884 ps + 4 2 4 ss p 1 s (4) 94 AISC ENGINEERING JOURNAL 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. It is evident that as the quantity (/4ps)
20、 approaches unity (or ps approaches 4/) the ponding deflections w and w will approach infinity. Therefore one could conclude that if the parameter (ps) were less than 4/, w and w would have a finite limit, termed the equilibrium position. However, another factor must be considered. The analysis so f
21、ar has assumed elasticity on the part of the structural members. If stresses in any member of the system were to exceed the elastic limit before reaching the point of theoretical equilibrium, a runaway condition could ensue with respect to ponding. Therefore, in addition to complying with the stipul
22、ation concerning the parameter (ps), the design should also ensure that the maximum flexural stress of any member be maintained below the yield point of the material. In order to develop a criterion that is suitable for design office use, the following substitutions were made: = C C s p (5) This is
23、self-evident from the observation that both the deflection and the flexibility constant for a beam are directly proportional to L4/EI. Substituting Equation (5) into Equations (3) and (4) yields: w= + pss ps 1 44 1 4 (1+) (6) and w p = + spps ps 1 328 0185 1 4 32 (1+). (6a) Noting that deflection is
24、 proportional to stress, f f ww = and placing as a limitation on the stress induced by ponding, f F f y w F.S. then w = F f f F f y y F.S. F.S. 1 1 A factor of safety (F.S.) of 1.25 against yielding is suggested for design office use. By substituting into Equation (6): 1 1 1 44 1 4 F.S. F f y o p ps
25、s ps + (1+) (7) Similarly for the secondary member: 1 1 F.S. F f y s spps ps 1 328 0185 1 4 32 + (1+). p (8) Equations (7) and (8) afford a relatively easy method for checking the ponding stability of a two-way roof system. Figures 4 and 5 are design aids which plot the relationship between the para
26、meters Cp and Up, and Cs and Us, for Equations (7) and (8), respectively. The term U represents the left side of these equations. To use these charts, tentatively select member sizes, as usual, on the basis of the design loading. Then from the known characteristics, compute the values of Up, Us, Cp,
27、 and Cs. To check the primary member, enter Fig. 4 at the left with the value of Up. Proceed to the right to the intersection with the curve representing the flexibility constant of the secondary member (Cs). Descend to the abscissa and read the maximum flexibility constant of the primary member to
28、satisfy Equation (7). If the actual Cp is larger than this value, it indicates that the system is potentially unstable and the design should be revised. A similar procedure can be used, employing Fig. 5, to check the secondary member. As a further simplification, the parameters Cp and Cs can be comp
29、uted from the following expressions in which , and E have been replaced by their numerical equivalent: C l l I p s p p = 4 4 32 10 C sl I p s s = 4 4 32 10 where ls, lp and s are in feet and Is and Ip are in in.4 It is important to note that the span involved in this analysis is the distance between
30、 support points (i.e., column spacing) and not between splice points. The flexibility limitation obtained by this analysis should be applied to simple and continuous spans alike. This is because unequal deflections in adjacent continuous spans can result in a greater accumulation in one 95 JULY/1966
31、 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. Figure 4 96 AISC ENGINEERING JOURNAL 2003 by American Institute of Steel Construction, Inc. All rights
32、 reserved. This publication or any part thereof must not be reproduced in any form without the written permission of the publisher. Figure 5 97 JULY/1966 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any for
33、m without the written permission of the publisher. span which, in turn, tends to unload the adjacent spans. This reduces the restraint at the ends of the ponded span, increases the deflection due to ponding in that span, and ultimately causes the continuous beam to act as though it were simply suppo
34、rted. Another consideration to bear in mind relates to the calculation of Up and Us. In these terms, the value of fo is the stress associated with all dead and live loads which are likely to be on the roof at the time that ponding commences. This would include any anticipated water load due to reser
35、voir action of curbs and similar architectural features. DISCUSSION AND CONCLUSIONS The most desirable method to preclude the effect of ponding is to provide sufficient slope to the roof surface along with adequate drainage facilities to prevent the accumulation of rain water in the first instance.
36、For the slope to be sufficient, the upward pitch provided to a roof surface must exceed the downward slope of the beams elastic curve at or near the support point, caused by all gravity loads. Experience as well as theoretical considerations indicate that a pitch of 1/8-in. per ft will suffice for t
37、his purpose under normal conditions of free drainage. However, the hydraulics of roof drainage is actually a very complex problem which requires careful study and is not included in the scope of this paper. In many cases, it is not feasible to drain a roof area without incurring the risk of some acc
38、umulation. This analysis presented has been tested against several cases of roof collapse attributed to ponding. In each case the instability would have been predicated by a significant margin. REFERENCES 1.Chinn, James Failure of Simply-Supported Flat Roofs by Ponding of Rain, AISC Engineering Jour
39、nal, April, 1965. 2.Haussler, R. W. Roof Deflection Caused by Rainwater Pools, Civil Engineering, October, 1962. 3.Specification for the Design, Fabrication and Erection of Structural Steel for Buildings, American Institute of Steel Construction, New York, N. Y., April 17, 1963. NOMENCLATURE CpFlexi
40、bility constant of primary member = L L El sp p 4 4 CsFlexibility constant of secondary member = SL El s s 4 4 EModulus of elasticity (psi) F.S.Factor of safety FyYield point of member considered (psi) IpMoment of inertia, primary member (in.4) IsMoment of inertia, secondary member (in.4) LpSpan of
41、primary member (in.) LsSpan of secondary member (in.) MpBending moment in primary member, at midspan (lb- in.) SSpacing of secondary members (in.) UpStress index of primary member = 1 1 F.S. F f y UsStress index of secondary member = 1 1 F.S. F f y WpTotal water load, on primary member, due to volum
42、etric configuration of primary members (lb) WsTotal water load, on primary member, due to volumetric configuration of secondary members (lb) dpDepth of primary member (in.) dsDepth of secondary member (in.) foExtreme fiber flexural stress in a member at onset of ponding (psi) fpExtreme fiber flexura
43、l stress in primary member (psi) fsExtreme fiber flexural stress in secondary member (psi) fwExtreme fiber flexural stress in a member due to ponding (psi) lpSpan of primary member (ft) lsSpan of secondary member (ft) sSpacing of secondary members (ft) pFlexibility parameter of primary member = C C
44、p p 1 sFlexibility parameter of secondary member = C C s s 1 Unit weight of water (lb/in.3) oDeflection in primary member at onset of ponding (in.) wDeflection in primary member due to ponding effect (in.) oDeflection in secondary member at onset of ponding (in.) wDeflection in secondary member at c
45、enter line of primary member due to ponding effect (in.) 1wDeflection in secondary member at end of primary member due to ponding effect (in.) Initial deflection ratio = o/o APPENDIX Example 1An industrial building has been designed with 50 ft-0 in. x 38 ft-0 in. bays. The structural members of the
46、flat roof have been proportioned by conventional analysis. Check the design for ponding. 98 AISC ENGINEERING JOURNAL 2003 by American Institute of Steel Construction, Inc. All rights reserved. This publication or any part thereof must not be reproduced in any form without the written permission of t
47、he publisher. Given: Girders (50 ft-0 in. span): 21WF55 Ip = 1140.7 in.4 fb = 23.0 ksi Secondary members (38 ft-0 in. span): 20H7 open web joist Is 160 in.4 fb = 28.5 ksi Joist spacing: 6 ft-3 in. o.c. Live load: 20 psf Dead load: 15 psf Solution: Assume that one-quarter of L.L. is on roof at outset of ponding. fo(girder)ksi= + = 23 15 5 35 132 . fo(joist)ksi= + = 285 15 5 35 163 Up= = 1 125 36 132 1118 . Us= = 1 125 50 163 1 145 . Cp= = 3850 321011407 065 4 4 . . Cs= = 625 38 32 10160 026 4 4 . . (a) Check
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