Deflection of a Composite Beam

Transcription

1 COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA SEPTEMBER 2002 COMPOSITE BEAM DESIGN BS Technical Note This Technical Note describes how the program checks deflection when the user selects the BS code. Deflection Check Locations For each design load combination specified for deflection calculations, the program checks deflection at the following locations: All design station locations defined by the user. The point of maximum moment for the load combination. The point load location for the load combination. Deflection of a Composite Beam Deflections are determined under serviceability load combinations specified for deflection calculation in the program (BS 6.1.1, 2.4.1). The deflection is calculated differently for propped (shored) and unpropped (unshored) construction. The effect of partial composite connection is also taken into account. The program uses the following formula for calculating deflection: I eff c +. 1, for propped construction (BS 6.1.4) Ibare = 1 0 5( 1 PCC ) I eff c +. 1, for unpropped construction(bs 6.1.4) Ibare = 1 0 3( 1 PCC ) The preceding two formulas are the simplified version of those given in the code: = c (1 N a /N p ) ( s c ), for propped construction (BS 6.1.4) = c (1 N a /N p ) ( s c ), for unpropped construction (BS 6.1.4) Deflection Check Locations Page 1 of 6

2 In the preceding expressions, = Deflection of a composite beam at a station for a load combination considering partial composite connection, c = Deflection of a composite beam with full shear connection; calculation of c is described in the next section, s = Deflection of a composite beam with 0% shear connection; it is related to moment of inertia of bare steel section (steel shape with cover plate, if present), N a = Actual number of connectors provided between a point of zero moment and a point of maximum moment, N p = Number of shear connections required between a point of zero moment and a point of maximum moment for full composite connection, PCC = Percent composite connection, used as a ratio, I bare = Moment of inertia of steel section, including cover plate if present, and I eff = Effective moment of inertia of composite section. Deflection of Composite Beam for Full Composite Connection When calculating c, the behavior of a composite beam is taken as linear elastic (BS 6.1.4). The program calculates composite beam deflections using a moment-area technique. An M/EI diagram is constructed by calculating M/EI values at each output station along the length of the beam and then connecting the M/EI values at those stations with straight-line segments. In constructing the M/EI diagram, I eff is used for I, moment of inertia. For simply supported or continuous composite beams, I eff is taken as I p, the equivalent moment of inertia for a cracked section in positive moment with 100% composite connection. For cantilever beams, I eff is taken as I n, the equivalent moment of inertia for the cracked section in negative moment. The program assumes that the moment of inertia does not vary along the length of the beam. Deflection of a Composite Beam Page 2 of 6

3 Deflections for the beam are calculated at each output station. The overall deflected shape of the beam is drawn by connecting the computed values of deflection at each output station with straight-line segments. In this program's composite beam design, the reported deflection is the vertical displacement relative to a line drawn between the deflected position of the ends of the beam. For example, refer to the beam shown in Figure 1. Figure 1a shows the original undeformed beam and also shows an arbitrary point along the beam labeled A. Figure 1b shows the beam in its deformed position and illustrates the deflection that the Composite Beam Design postprocessor reports for the beam at point A. A A Original position of beam Line between position of beam shown a) b) Deflected Shape of Deflection reported by Composite Beam postprocess Figure 1: Deflection Results Reported by the Composite Beam Design Postprocessor For cantilever overhangs, the program's Composite Beam Design postprocessor reports the displacement of the beam relative to the deformed position of the supported end. If you use the Display menu > Show Deformed Shape command to review the displacement at the end of the cantilever, the displacement is reported relative to the undeformed position of the end of the cantilever. In that case, the rotation at the supported end of the cantilever overhang is correctly taken into account. However, the displacements displayed are all based on the analysis section properties (non-composite moment of inertias). Deflection of a Composite Beam Page 3 of 6

4 The program considers the effect of propped and unpropped construction methods. For unpropped construction, the imposed load deflection is based on the properties of the composite section, but the dead load deflection, resulting from the self weight of the steel beam and wet concrete, is based on the properties of the bare steel section. For propped construction, all deflections are based on the properties of the composite section (BS 6.1.1, ). Typically, the composite beams are simply supported. For those simply supported composite beams, there is no scope for moment redistribution. Also the effect of pattern loading and shakedown effects can be neglected. The program does not consider moment redistribution, shakedown and pattern loading for calculation of deflection. Those factors may be important for continuous beams, and the user should consider those effects independently (BS 6.1.1, 6.1.3, , ). For simply supported composite beams, the code recommends the use of I g, the gross moment of inertia of the equivalent uncracked section, instead of I p, the moment of inertia of the equivalent cracked section, for calculation of deflection (BS 6.1.2, , 4.2.1). The user should be aware of that there might be a slight difference between I p for 100% PCC and I g. Effective Moment of Inertia, Ieff The program uses the effective moment of inertia of composite section, I eff, for deflection calculation. For simply supported or continuous composite beams, I eff is taken as I p, the equivalent moment of inertia for cracked section in positive moment with 100% composite connection. For cantilever beams, I eff is taken as I n, the equivalent moment of inertia for cracked section in negative moment. For calculation of I p, the width of the concrete slab and ribs (if ribs run parallel to the beam) is scaled down by a factor of E c /E s to make the section equivalent to the steel section in terms of stiffness. Also, the concrete depth that is in tension under elastic moment distribution is neglected. If the steel section is large, the elastic neutral axis lies in the web of the steel section. In such cases, I p becomes the same as I g, the equivalent moment of inertia for gross uncracked section. If the concrete section becomes very large, the elastic neutral axis lies in concrete, and in that case, I p may become slightly smaller than I g. The effect of the short term and long term modular ratio is Deflection of a Composite Beam Page 4 of 6

5 considered for calculation if I p (BS 4.1). See Technical Note Transformed Section Moment of Inertia Composite Beam Design BS for details. For calculating I n, the concrete is neglected. If there is a cover plate, it is considered. I n becomes the moment of inertia for bare steel (I bare ), and it also becomes I p for 0% composite connection. Deflection Limits The deflection limit for total load and live load is taken as follows: L TL,limit = 240 L LL,limit = 360 These are the default deflection limits for total load and live load, represectively, in the program. The user can change those limits (BS 2.4.2; BS , Table 5). Note that camber is subtracted from the total load deflection before the total load deflection is compared to the total load deflection limit. See Technical Note Camber Calculation Composite Beam Design BS for details about camber. Deflection Checks For each service load combination, two deflections are calculated one for live load and the other for total load for every point. The maximum of the total load deflection within the span, TL, is compared with its allowable limit, TL,limit. Similarly, the maximum of the live load deflection within the span, LL, is compared with its allowable limit. LL,limit. The following ratios are calculated. TL camber TL, lim it and LL LL,lim it, where, TL = Maximum total load deflection for a load combination, Deflection Limits Page 5 of 6

6 LL = Maximum live load deflection for a load combination, TL, limit = Maximum allowed total load deflection, LL, limit = Maximum allowed live load deflection, and camber = Camber of the beam. The maximum of the total load deflection ratio and the maximum of the live load deflection ratios considering all of the service load combinations are reported by the program. Note that camber is subtracted from the total load deflection before the total load deflection is compared to the total load deflection limit. See Technical Note Camber Calculation Composite Beam Design BS for details about camber. Deflection Checks Page 6 of 6