Structural Steel Designers Handbook Part 11 doc

12.68 SECTION TWELVE TABLE 12.39 Composite Section for G 1 for Maximum Moment Material A d Ad Ad 2 I o I (a) For dead loads, n ϭ 30 Steel section 93.5 Ϫ534 74,550 Concrete 84 ϫ 7/30 19.6 37.0 725 26,830 80 26,900 113.1 191 101,450 d s0 ϭ 191/113.1 ϭ 1.669 in Ϫ1.69 ϫ 191 ϭϪ320 I NA ϭ 101,130 Distance from neutral axis of composite section to: Top of steel ϭ 31.50 Ϫ 1.69 ϭ 29.81 in Bottom of steel ϭ 32.00 ϩ 1.69 ϭ 33.69 in Top of concrete ϭ 29.81 ϩ 2 ϩ 7 ϭ 38.81 in Section moduli Top of steel Bottom of steel Top of concrete S ϭ 101,130 /29.81 st 3 ϭ 3,400 in S ϭ 101,130 /33.69 sb 3 ϭ 3,010 in S ϭ 101,130 /38.81 c 3 ϭ 2,615 in (b) For live loads, n ϭ 10 Materials A d Ad Ad 2 I o I Steel section 93.5 Ϫ534 74,500 Concrete 84 ϫ 7/10 58.8 37.0 2,175 80,510 240 80,750 152.3 1,641 155,300 d 10 ϭ 1,641/152.3 ϭ 10.76 Ϫ10.76 ϫ 1,641 ϭϪ17,700 I NA ϭ 137,600 Distance from neutral axis of composite section to: Top of steel ϭ 31.50 Ϫ 10.76 ϭ 20.74 in Bottom of steel ϭ 32.00 ϩ 10.76 ϭ 42.76 in Top of concrete ϭ 20.74 ϩ 2 ϩ 7 ϭ 29.74 in Section moduli Top of steel Bottom of steel Top of concrete S ϭ 137,600 /20.74 st 3 ϭ 6,630 in S ϭ 137,600 /42.76 sb 3 ϭ 3,210 in S ϭ 137,600 /29.74 c 3 ϭ 4,630 in BEAM AND GIRDER BRIDGES 12.69 TABLE 12.40 Maximum Lateral Bending Moments, ft-kips 2 DL: M ϭϪ0.1 ϫ 12 ϫ 1,767(15) /(300 ϫ 61.75) L 2 SDL: M ϭϪ0.1 ϫ 12 ϫ 838(15) /(3000 ϫ 64.38) L 2 LL ϩ I: M ϭϪ0.1 ϫ 12 ϫ 1,297(15) /(300 ϫ 66.03) L Total: ϭϪ26 ϭϪ12 ϭϪ18 Ϫ56 TABLE 12.41 Steel Stresses in G 1 , ksi Top of steel (compression) Bottom of steel (tension) DL:ƒ ϭ 1.767 ϫ 12 / 1,992 ϭ 11.03 b SLD:ƒ ϭ 838 ϫ 12 /3,400 ϭ 2.95 b LL ϩ I:ƒ ϭ 1,297 ϫ 12/ 6,630 ϭ 2.34 b 12 L:ƒ ϭ 26 ϫ ⁄ 81 ϭ 3.85 b Total: 20.17 Ϸ 20 ƒ ϭ 1,767 ϫ 12/ 2,720 ϭ 7.79 b ƒ ϭ 838 ϫ 12 / 3,010 ϭ 3.34 b ƒ ϭ 1,297 ϫ 12 /3,210 ϭ 4.85 b ƒ ϭ 56 ϫ 12 / 161.5 ϭ 4.16 b 20.14 Ϸ 20 Therefore, the composite section for G 1 is satisfactory. Use for G 1 in the region of max- imum moment the section shown in Fig. 12.27. The procedure is the same for design of other sections and for the other stringers. For design of other elements, see Arts. 12.2 and 12.4. Fatigue design is similar to that for straight girders. 12.8 DECK PLATE-GIRDER BRIDGES WITH FLOORBEAMS For long spans, use of fewer but deeper girders to span the long distance between supports becomes more efficient. With appropriately spaced stringers between the main girders of highway bridges, depth of concrete roadway slab can be kept to the minimum permitted, thus avoiding increase in dead load from the deck. Spans of the longitudinal stringers are kept short by supporting them on transverse floor-beams spanning between the girders. If spacing of the floorbeams is 25 ft or less, additional diaphragms or cross frames between the girders are not required. This type of construction can be used with deck or through girders. Through girders carry the roadway between them. Their use generally is limited to locations where vertical clear- ances below the bridge are critical. Deck girders carry the roadway on the top flange. They generally are preferred for highway bridges where vertical clearances are not severely re- stricted, because the girders, being below the deck, do not obstruct the view from the deck. Structurally, deck girders have the advantage that the concrete deck is available for bracing the top flange of the girders and for composite action. Bracing of the bottom flange is accomplished with horizontal lateral bracing. The design procedure for through plate girders with floor beams is described in Art. 12.10. Article 12.9 presents an example to indicate the design procedure for a deck girder bridge with floorbeams and stringers. In general, design of the stringers is much like that for a stringer bridge (Art. 12.2), and design of the girders is much like that for the girders of a multigirder bridge (Art. 12.4). In the following example, however, the stringers and girders are not designed for composite action. See also Art. 12.3. 12.70 SECTION TWELVE TABLE 12.42 Stresses in G 1 at Top of Concrete, ksi SDL:ƒ ϭ 838 ϫ 12 / (2,615 ϫ 30) c LL ϩ I:ƒ ϭ 1,297 ϫ 12 /(4,630 ϫ 10) c Total: ϭ 0.13 ϭ 0.34 0.47 Ͻ 1.2 FIGURE 12.28 Framing plan for four-lane highway bridge with deck plate girders. 12.9 EXAMPLE—ALLOWABLE-STRESS DESIGN OF DECK PLATE-GIRDER BRIDGE WITH FLOORBEAMS Two simply supported, welded, deck plate girders carry the four lanes of a highway bridge on a 137.5-ft span. The girders are spaced 35 ft c to c. Loads are distributed to the girders by longitudinal stringers and floorbeams (Fig. 12.28). The typical cross section in Fig. 12.29 shows a 48-ft roadway flanked by 3-ft-wide safety walks. Grade 50 steel is to be used for the girders and Grade 36 for stringers, floorbeams, and other components. Concrete to be used for the deck is class A, with 28-day strength ƒ ϭ 4,000 psi and allowable compressiveЈ c stress ƒ c ϭ 1,400 psi. Appropriate design criteria given in Sec. 11 will be used for this structure. 12.9.1 Design of Concrete Slab The slab is designed, to span transversely between stringers, in the same way as for rolled- beam stringers (Art. 12.2). A 7.5-in thick concrete slab will be used. 12.71 FIGURE 12.29 Typical cross section of deck-girder bridge at a floorbeam. 12.72 SECTION TWELVE TABLE 12.43 Dead Load on S2, kips per ft Slab: 0.150 ϫ 8.75 ϫ 7.5/12 ϭ 0.82 Haunch—assume: 0.033 Stringer—assume: 0.068 DL per stringer: 0.923 12.9.2 Design of Interior Stringer Spacing of interior stringers c to c is 8.75 ft. Simply supported, a typical stringer S2 spans 20 ft. Table 12.43 lists the dead loads on S2. Maximum dead-load moment occurs at midspan and equals 2 0.923(20) M ϭϭ46.1 ft-kips DL 8 Maximum dead-load shear occurs at the supports and equals 0.923 ϫ 20 V ϭϭ9.2 kips DL 2 The live load distributed to the stringer with spacing S ϭ 8.75 ft is S 8.75 ϭϭ1.59 wheel loads ϭ 0.795 axle loads 5.5 5.5 Maximum moment induced in a 20-ft span by a standard HS20 truck is 160 ft-kips. Hence, the maximum live-load moment in a stringer is M ϭ 0.795 ϫ 160 ϭ 127.2 ft-kips LL Maximum shear caused by the truck is 41.6 kips. Consequently, maximum live-load shear in the stringer is V ϭ 0.795 ϫ 41.6 ϭ 33.0 kips LL Impact is taken as 30% of live-load stress, because 50 50 I ϭϭ ϭ0.35 Ͼ 0.30 L ϩ 125 20 ϩ 125 So the maximum moment due to impact is M ϭ 0.30 ϫ 127.2 ϭ 38.1 ft-kips I and the maximum shear due to impact is V ϭ 0.30 ϫ 33.0 ϭ 9.9 kips I Maximum moments and shears in S2 are summarized in Table 12.44. With an allowable bending stress F b ϭ 20 ksi for a stringer of Grade 36 steel, the section modulus required is BEAM AND GIRDER BRIDGES 12.73 TABLE 12.44 Maximum Moments and Shears in S2 DL LL I Total Moments, ft-kips 46.1 127.2 38.1 211.4 Shears, kips 9.2 33.0 9.9 52.1 TABLE 12.45 Dead Load on S1, kips per ft Railing: 0.070 ϫ 9.83/ 7 ϭ 0.098 Sidewalk: 0.150 ϫ 1 ϫ 3 ϫ 8/7 ϭ 0.514 Slab: 0.150 ϫ 8 ϫ 7.5/12 ϫ 4/7 ϭ 0.428 Stringers, brackets, framing details—assume: 0.110 DL per stringer: 1.150 M 211.4 ϫ 12 3 S ϭϭ ϭ127 in F 20 b With an allowable shear stress F v ϭ 12 ksi, the web area required is 52.1 2 A ϭϭ4.33 in w 12 Use a W21 ϫ 68. It provides a section modulus of 139.9 in 3 and a web area of 0.43 ϫ 21.13 ϭ 9.1 in 2 . 12.9.3 Design of an Exterior Stringer Simply supported, S1 spans 20 ft. It carries sidewalk as well as truck loads (Fig. 12.28). Dead loads are apportioned between S1 and the girder, 7 ft away, by treating the slab as simply supported at the girder. Table 12.45 lists the dead loads on S1. Maximum dead-load moment occurs at midspan and equals 2 1.15(20) M ϭϭ57.5 ft-kips DL 8 Maximum dead-load shear occurs at the supports and equals 1.15 ϫ 20 V ϭϭ11.5 kips DL 2 The live load from the roadway distributed to the exterior stringer with spacing S ϭ 7ft from the girder is S 7 ϭϭ1.22 wheel loads ϭ 0.61 axle loads 4.0 ϩ 0.25S 4.0 ϩ 0.25 ϫ 7 Maximum moment induced in a 20-ft span by a standard HS20 truck load is 160 ft-kips. Hence, the maximum live-load moment in S1 is 12.74 SECTION TWELVE TABLE 12.46 Maximum Moments and Shears in S1 DL LL I Total Moments, ft-kips 57.5 97.7 29.3 184.5 Shears, kips 11.5 25.4 7.6 44.5 M ϭ 0.61 ϫ 160 ϭ 97.7 ft-kips LL Maximum shear caused by the truck is 41.6 kips. Therefore, maximum live-load shear in S1 is V ϭ 0.61 ϫ 41.6 ϭ 25.4 kips LL Impact for a 20-ft span is 30% of live-load stress. Hence, maximum moment due to impact is M ϭ 97.7 ϫ 0.3 ϭ 29.3 ft-kips I and maximum shear due to impact is V ϭ 25.4 ϫ 0.3 ϭ 7.6 kips I Sidewalk loading at 85 psf on the 3-ft-wide sidewalk imposes a uniformly distributed load w SLL on the stringer. With the slab assumed simply supported at the girder, 0.085 ϫ 3 ϫ 8 w ϭϭ0.29 kip per ft SLL 7 This causes a maximum moment of 2 0.29(20) M ϭϭ14.5 ft-kips SLL 8 and a maximum shear of 0.29 ϫ 20 V ϭϭ2.9 kips SLL 2 Maximum moments and shears in S1 are summarized in Table 12.46. If the exterior stringer has at least the capacity of the interior stringers, the allowable stress may be increased 25% when the effects of sidewalk live load are combined with those from dead load, traffic live load, and impact. In this case, the moments and shears due to sidewalk live load are less than 25% of the moments and shears without that load. Hence, they may be ignored. With an allowable bending stress F b ϭ 20 ksi for Grade 36 steel, the section modulus required for S1 is M 184.5 ϫ 12 3 S ϭϭ ϭ111 in F 20 b With an allowable shear stress F v ϭ 12 ksi, the web area required is BEAM AND GIRDER BRIDGES 12.75 FIGURE 12.30 Dead loads on a floorbeam of the deck-girder bridge. 44.5 2 A ϭϭ3.7 in w 12 Use a W21 ϫ 68, as for S2. 12.9.4 Design of an Interior Floorbeam Floorbeam FB2 is considered to be a simply supported beam with 35-ft span and symmetrical 9.5-ft brackets, or overhangs (Fig. 12.29). It carries a uniformly distributed dead load due to its own weight and that of a concrete haunch, assumed at 0.21 kip per ft. Also, FB2 carries a concentrated load from S1 of 2 ϫ 11.5 ϭ 23.0 kips and a concentrated load from each of three interior stringers S2 of 2 ϫ 9.2 ϭ 18.4 kips (Fig. 12.30). Moments and Shears in Main Span. Because of the brackets, negative moments occur and reach a maximum at the supports. The maximum negative dead-load moment is 2 9.5 M ϭϪ0.21 Ϫ 23.0 ϫ 7 ϭϪ171 ft-kips ͩͪ DL 2 The reaction at either support under the symmetrical dead load is 3 ϫ 18.4 0.21 ϫ 54 R ϭϩ23 ϩϭ56.3 kips DL 22 Maximum dead-load shear in the overhang is V ϭ 23 ϩ 0.21 ϫ 9.5 ϭ 25.0 kips DL Hence, the maximum shear between girders is V ϭ 56.3 Ϫ 25.0 ϭ 31.3 kips DL Maximum positive dead-load moment occurs at midspan and equals 2 0.21(17.5) M ϭ 31.3 ϫ 17.5 Ϫ 18.4 ϫ 8.75 ϪϪ171 ϭ 184 ft-kips DL 2 Maximum live-load stresses in the floorbeam occur when the center truck wheels pass over it (Fig. 12.31). In that position, the wheels impose on FB2 a load 16 ϫ 64ϫ 6 W ϭ 16 ϩϩϭ22 kips 20 20 For maximum positive moment, trucks should be placed in the two central lanes, as close to midspan as permissible (Fig. 12.32). Then, the maximum moment is 12.76 SECTION TWELVE FIGURE 12.31 Positions of loads on a stringer for maximum live load on a floorbeam. FIGURE 12.32 Positions of loads for maximum positive moment in a floor- beam. M ϭ 2 ϫ 22 ϫ 15.5 Ϫ 22 ϫ 6 ϭ 550 ft-kips LL Maximum negative moment occurs at a support with a truck in the outside lane with a wheel 2 ft from the curb (Fig. 12.33). This moment equals M ϭϪ22 ϫ 4.5 ϭϪ99 ft-kips LL Maximum live-load shear between girders occurs at support A with three lanes closest to that support loaded, as indicated in Fig. 12.34. Because three lanes are loaded, the floorbeam need to be designed for only 90% of the resulting shear. The reaction at A is 0.90 ϫ 22(39.5 ϩ 33.5 ϩ 27.5 ϩ 21.5 ϩ 15.5 ϩ 9.5) R ϭϭ83.2 kips LL 35 Subtraction of the shear in the bracket for this loading gives the maximum liveload shear between girders: V ϭ 83.2 Ϫ 0.9 ϫ 22 ϭ 63.4 kips LL The maximum live-load shear in the overhang is produced by the loading in Fig. 12.33 and is V LL in 22 kips. Impact is taken as 30% of live-load stress, because 50 50 I ϭϭ ϭ0.31 Ͼ 0.30 L ϩ 125 35 ϩ 125 Sidewalk loading transmitted by exterior stringers S1 to the floorbeam equals 2 ϫ 2.9 ϭ 5.8 kips. This induces a shear in the overhang V SLL ϭ 5.8 kips. Also, it causes a reaction [...]... computed from Eq (11. 25a) Assume that the stiffener spacing do ϭ 48 in ϭ the web depth D Hence, do / D ϭ 1 From Eq (11. 24d), for use in Eq (11. 25a), k ϭ 5(1 ϩ 12) ϭ 10 and ͙k / Fy ϭ ͙10 / 36 ϭ 0.527 Since D / tw ϭ 128, C in Eq (11. 24a) is determined by the parameter 128 / 0.527 ϭ 243 Ͼ 237 Hence, C is given by Eq (11. 24c): Cϭ 45,000k 45,000 ϫ 10 ϭ ϭ 0.763 (D / t w)2Fy 1282 ϫ 36 From Eq (11. 25a), the maximum... ksi The web depth-thickness ratio D / tw ϭ 110 / (7⁄16) ϭ 251 Maximum spacing of stiffeners is limited to 110 (270 / 251)2 ϭ 127 in Try a stiffener spacing do ϭ 80 in This provides a depth spacing ratio D / do ϭ 110 / 80 ϭ 1.375 From Eq (11. 24d ), for use in Eq (11. 25a), k ϭ 5[1 ϩ (1.375)2] ϭ 14.45 and ͙k / Fy ϭ ͙14.45 / 50 ϭ 0.537 Since D / tw ϭ 251, C in Eq (11. 24a) is determined by the parameter 251... allowable stress of 17 ksi for thin Grade 50 steel, the web thickness required for shear is tϭ 470 ϭ 0.25 in 17 ϫ 110 Without a longitudinal stiffener, according to Table 10.15 thickness must be at least tϭ 110 ͙50 ϭ 0.8 in, say 990 13 ⁄16 in Even with a longitudinal stiffener, however, to prevent buckling, web thickness, from Table 11. 25, must be at least tϭ 110 ͙50 ϭ 0.393 in, say 7⁄16 in 1,980 though... spacing then is Fv ϭ ͫ ͬ 50 0.87(1 Ϫ 0.131) 0.131 ϩ ϭ 10.69 ksi Ͼ 7.28 ksi 3 ͙1 ϩ (120 / 110 )2 The 10-ft spacing is satisfactory Actual spacing throughout the span is shown in Fig 11. 40 The moment of inertia provided by each pair of stiffeners must satisfy Eq (11. 21), with J as given by Eq (11. 22) J ϭ 2.5 ͩ ͪ 110 27 2 Ϫ 2 ϭ 39.5 I ϭ 27(7⁄16)3 39.5 ϭ 89.3 in4 The moment of inertia furnished by a pair... 0.437(6)3 ϭ 31.5 in4 3 With transverse stiffeners spaced 120 in apart, the moment of inertia required by Eq (11. 28a), is ͫ ͩ ͪ Imin ϭ 110 (0.437)3 2.4 120 110 2 Ϫ 0.13 ͬ ϭ 25.1 Ͻ 31.5 in4 Therefore, use a 6 ϫ 7⁄16-in plate for the longitudinal stiffener A 6 ϫ 3⁄8-in plate would also check Bearing Stiffeners A pair of bearing stiffeners of Grade 50 steel is provided at each support They are designed to transmit... required, a pair of transverse stiffeners of Grade 36 steel will be welded to the girder web Minimum width of stiffener is 24 / 4 ϭ 6.0 in Ͼ (2 ϩ 110 / 30 ϭ 5.7 in) Use a 71⁄2-in wide plate Minimum thickness required is 7⁄16 in Try a pair of 71⁄2 ϫ 7⁄16-in stiffeners Maximum spacing of the transverse stiffeners can be computed from Eq (11. 25a) For the 110 ϫ 7⁄16-in girder web and a maximum shear at the... Hence, C is given by Eq (11. 24c): Cϭ 45,000k 45,000 ϫ 14.45 ϭ ϭ 0.206 (D / tw)2Fy 2512 ϫ 50 From Eq (11. 25a), the maximum allowable shear for do ϭ 80 in is ͫ ͫ F Ј ϭ Fv C ϩ v ϭ 0.87(1 Ϫ C ) ͙1 ϩ (do / D)2 ͬ ͬ 50 0.87(1 Ϫ 0.206) 0.206 ϩ ϭ 12.74 ksi Ͼ 9.77 ksi 3 ͙1 ϩ (80 / 110 )2 Since the allowable stress is larger than the computed stress, the stiffeners may be spaced 80 in apart The location of floorbeams... system for deck-girder bridge W2 ϭ 0.61 ϫ 20 ϭ 12.2 kips W1 ϭ 0.61(20 ϩ 18.75) ϭ 11. 8 kips 2 W0 ϭ 0.61 ͩ ͪ 18.75 ϩ 1.5 ϭ 6.6 kips 2 The reaction at each support is R ϭ 2 ϫ 12.2 ϩ 11. 8 ϩ 6.6 ϭ 42.8 kips With the wind considered a moving load, maximum shear in each panel is: V1 ϭ 42.8 Ϫ 6.6 ϭ 36.2 kips V2 ϭ 36.2 Ϫ 11. 8 ϫ 118 .75 ϭ 25.9 kips 137.5 V3 ϭ 25.9 Ϫ 12.2 ϫ 98.75 ϭ 17.1 kips 137.5 V4 ϭ 17.1 Ϫ 12.2... ϫ 20 ϭ 114 kips Compressive capacity with Fa ϭ 8.5 ksi on the gross area is C ϭ 7.80 ϫ 8.5 ϭ 66 Ͻ 114 kips Tensile capacity governs Hence, the number of bolts required is determined by TABLE 12.55 Net Area of Diagonal, in2 Gross area: 7.80 Half web area: Ϫ5.45 ϫ 0.345 / 2 ϭ Ϫ0.94 Two holes: Ϫ2 ϫ 1 ϫ 0.576 ϭ Ϫ1.15 Net area: 5.71 BEAM AND GIRDER BRIDGES 0.75 ϫ 114 ϭ 86 kips Ͼ ͩ 12.97 ͪ 26.6 ϩ 114 ϭ 70... with a longitudinal stiffener Flange Size at Midspan For Grade 50 steel 4 in thick or less, Fy ϭ 50 ksi and the allowable bending stress is 27 ksi With a maximum moment at midspan, from Fig 12.37, of 15,359 ft-kips, and distance between flange centroids of about 113 in, the required area of one flange is about Aƒ ϭ 15,359 ϫ 12 ϭ 60.4 in2 113 ϫ 27 Assume a 24 ϫ 21⁄2-in plate for each flange It provides an . section to: Top of steel ϭ 31.50 Ϫ 1.69 ϭ 29.81 in Bottom of steel ϭ 32.00 ϩ 1.69 ϭ 33.69 in Top of concrete ϭ 29.81 ϩ 2 ϩ 7 ϭ 38.81 in Section moduli Top of steel Bottom of steel Top of concrete S. section to: Top of steel ϭ 31.50 Ϫ 10.76 ϭ 20.74 in Bottom of steel ϭ 32.00 ϩ 10.76 ϭ 42.76 in Top of concrete ϭ 20.74 ϩ 2 ϩ 7 ϭ 29.74 in Section moduli Top of steel Bottom of steel Top of concrete S. 66.03) L Total: ϭϪ26 ϭϪ12 ϭϪ18 Ϫ56 TABLE 12.41 Steel Stresses in G 1 , ksi Top of steel (compression) Bottom of steel (tension) DL:ƒ ϭ 1.767 ϫ 12 / 1,992 ϭ 11. 03 b SLD:ƒ ϭ 838 ϫ 12 /3,400 ϭ 2.95 b LL