Structural Steel Designers Handbook Part 12 doc

12.148 SECTION TWELVE TABLE 12.78 Moments, ft-kips, in Floorbeam with Orthotropic-Plate Flange M DL M LL M I Total M Midspan 104 90 27 221 Supports 0 Ϫ712 Ϫ52 Ϫ224 be found by proportion from those previously calculated. Thus, the moment at the support is 210 ϫ 0.247 M ϭϪ ϭϪ172 ft-kips LL 0.0302 and the moment at midspan is 110 ϫ 0.247 M ϭϭ90 ft-kips LL 0.302 Impact. For a 30-ft span, impact is taken as 30%. At midspan, M I ϭ 0.30 ϫ 90 ϭ 27 ft-kips. At supports, M I ϭ 0.30(Ϫ172) ϭϪ52 ft-kips. Total Floorbeam Moments. The design moments previously calculated are summarized in Table 12.78. Properties of Floorbeam Sections. For stress computations, an effective width s o of the deck plate is assumed to act as the top flange of member III. For determination of s o , the effective spacing of floorbeams s ƒ is taken equal to the actual spacing, 180 in. The effective span l e , with the floorbeam ends considered fixed, is taken as 0.7 ϫ 30 ϫ 12 ϭ 252 in. Hence, s 180 ƒ ϭϭ0.715 l 252 e From Table 4.6 for this ratio, s o ϭ 0.53, and s ϭ 0.53 ϫ 180 ϭ 95 in o s ƒ (Fig. 12.63b and c). The neutral axis of the floorbeam sections at midspan and supports can be located by taking moments of component areas about middepth of the web. This computation and those for moments of inertia and section moduli are given in Table 12.79. Floorbeam Stresses. These are determined for the total moments in Table 12.78 with the section properties given in Table 12.79. Calculations for the stresses at midspan and the supports are given in Table 12.80. Since the stresses are well within the allowable, the floorbeam sections are satisfactory. BEAM AND GIRDER BRIDGES 12.149 TABLE 12.79 Floorbeam Moments of Inertia and Section Moduli (a) At midspan Material A d Ad Ad 2 I o I Deck 95 ϫ 3 ⁄ 8 35.6 10.69 381 4,070 4,070 Web 21 ϫ 3 ⁄ 8 7.9 290 290 Bottom flange 10 ϫ 1 ⁄ 2 5.0 Ϫ10.75 Ϫ54 580 580 48.5 327 4,940 d ϭ 327/ 48.5 ϭ 6.73 in Ϫ6.73 ϫ 327 ϭϪ2,210 I NA ϭ 2,730 Distance from neutral axis to: Top of deck plate ϭ 10.50 ϩ 0.375 Ϫ 6.73 ϭ 4.15 in Bottom of rib ϭ 10.50 ϩ 0.50 ϩ 6.73 ϭ 17.73 in Section moduli Top of deck plate Bottom of rib S t ϭ 2,730/ 4.15 ϭ 658 in 3 S b ϭ 2,730/ 17.73 ϭ 154 in 3 (b) At supports, gross section Material A d Ad Ad 2 I o I Deck 95 ϫ 3 ⁄ 8 35.6 9.19 327 3,010 3,010 Web 18 ϫ 3 ⁄ 8 6.8 180 180 Bottom flange 10 ϫ 1 ⁄ 2 5.0 Ϫ9.25 Ϫ46 430 430 47.4 281 3,620 d g ϭ 281/ 47.4 ϭ 5.93 in Ϫ5.93 ϫ 281 ϭϪ1,670 Gross I NA ϭ 1,950 Distance from neutral axis to: Bottom of rib ϭ 9 ϩ 0.50 ϩ 5.93 ϭ 15.43 in Section modulus, bottom of rib S b ϭ 1,950/ 15.43 ϭ 126 in 3 (c) At supports, net section Material A d Ad Ad 2 I o I Gross section 47.4 281 3,620 Top-flange holes Ϫ10.2 9.19 Ϫ94 Ϫ860 Ϫ860 Bottom-flange holes Ϫ1.0 Ϫ9.25 9 Ϫ90 Ϫ90 Web holes Ϫ2.3 Ϫ120 Ϫ120 33.9 196 2,550 d net ϭ 196/ 33.9 ϭ 5.77 in Ϫ5.77 ϫ 196 ϭϪ1,130 Net I NA ϭ 1,420 12.150 SECTION TWELVE TABLE 12.79 Floorbeam Moments of Inertia and Section Moduli (Continued ) Distance from neutral axis to: Top of deck plate ϭ 9 ϩ 0.375 Ϫ 5.77 ϭ 3.61 Section modulus, top of deck plate S t ϭ 1,420/ 3.61 ϭ 392 in 3 TABLE 12.80 Bending Stresses in Member III At midspan: Top of deck plate ƒ b ϭ 221 ϫ 12/658 ϭ 4.03 ksi (compression) Bottom flange ƒ b ϭ 221 ϫ 12/154 ϭ 17.2 Ͻ 27 ksi (tension) At supports: Top of deck plate ƒ b ϭ 224 ϫ 12/392 ϭ 6.9 ksi (tension) Bottom flange ƒ b ϭ 224 ϫ 12/126 ϭ 21.4 Ͻ 27 ksi Floorbeam Shears. For maximum shear, the truck wheels are placed in each design lane as indicated in Fig. 12.64. The 16-kip wheels are placed over the floorbeam. A 4-kip wheel is located 14 ft away on each of the adjoining rib spans. Thus, with the floorbeams assumed acting as rigid supports for the ribs, the wheel load is 16.5 kips, as for maximum floorbeam moment. (The effects of floorbeam flexibility can be determined as for bending moments.) This loading produces a simple-beam reaction of 41.8 kips. It also causes end moments of Ϫ202 and Ϫ86, which induce a reaction of (Ϫ86 ϩ 202)/30 ϭ 3.9 kips. Hence, the maximum live-load reaction and shear equal V ϭ 41.8 ϩ 3.9 ϭ 45.7 kips LL Shear due to impact is V ϭ 0.30 ϫ 45.7 ϭ 13.7 kips I M AXIMUM F LOORBEAM S HEARS , KIPS V DL V LL V I Total V 13.9 45.7 13.7 73.3 Allowable shear stress in the web for Grade 50W steel is 17 ksi. Average shear stress in the web is 73.3 ƒ ϭϭ10.9 Ͻ 17 ksi v 3 18 ϫ ⁄ 8 Transverse stiffeners are not required. Flange-to-Web Welds. The web will be connected to the deck plate and the bottom flange by a fillet weld on opposite sides of the web. These welds must resist the horizontal shear between flange and web. For the weld to the 10 ϫ 1 ⁄ 2 -in bottom flange, the minimum size BEAM AND GIRDER BRIDGES 12.151 FIGURE 12.64 Positions of truck wheels for maximum shear in floorbeam. fillet weld permissible with a 1 ⁄ 2 -in plate, 1 ⁄ 4 in, may be used. Shear, however, governs for the weld to the deck plate. For computing the shear v, kips per in, between web and deck plate, the total maximum shear V is 73.3 kips and the moment of inertia of the floorbeam cross section I is 1,950 in. 4 The static moment of the deck plate is 3 Q ϭ 35.6(4.15 Ϫ 0.19) ϭ 141 in Hence, the shear to be carried by the welds is VQ 73.3 ϫ 141 v ϭϭ ϭ5.30 kips per in I 1,950 The allowable stress on the weld is 18.9 ksi. So the allowable load per weld is 18.9 ϫ 0.707 ϭ 13.4 kips per in, and for two welds, 26.7 kips per in. Therefore, the weld size required is 5.30/26.7 ϭ 0.20 in. Use 1 ⁄ 4 -in fillet welds. Floorbeam Connections to Girders. Since the bottom flange of the floorbeam is in com- pression, it can be connected to the inner web of each box girder with a splice plate of the same area. Use a 10 ϫ 1 ⁄ 2 -in plate, shop-welded to the girder and field-bolted to the floor- beam. With A325 7 ⁄ 8 -in-dia. high-strength bolts in slip-critical connections with Class A surfaces, the allowable load per bolt is 9.3 kips. If the capacity of the 10 ϫ 1 ⁄ 2 -in flange is developed at the allowable stress of 27 ksi, the number of bolts required in the connection is 27 ϫ 5 /9.3 ϭ 15. Use 16. The deck plate is spliced to the girder with 7 ⁄ 8 -in-dia. high-strength bolts. To meet girder requirements, the pitch may vary from 3 to 5 1 ⁄ 2 in (Fig. 12.60d ). But the bolts also must transmit the tensile forces from the deck plate to the girder when the plate acts as the top flange of member III. The shear in the bolts from the girder compression is perpendicular to the shear from the floorbeam tension. Hence, the allowable load per bolt decreases from 9.3 to 9.3 ϫ 0.707 ϭ 6.6 kips. With an average tensile stress in the deck plate of 6.2 ksi, and a net area after deduction of holes of 35.6 Ϫ 10.2 ϭ 25.4 in 2 , the plate carries a tensile force of 25.4 ϫ 6.2 ϭ 158 kips. Thus, to transmit this force, 158/6.6 ϭ 24 bolts are needed. If a pitch of 3 in is used in the 95-in effective width of the plate, 31 bolts are provided. Use a 3-in pitch for 4 ft on each side of every floorbeam. The web connection to the girder must transmit both vertical shear, V ϭ 73.3 kips, and bending moment. The latter can be computed from the stress diagram for the cross section (Fig. 12.65a). 12.152 SECTION TWELVE FIGURE 12.65 (a) Bending stresses in floorbeam at supports. (b) Bolted web connection of floorbeam to girder. 13 M ϭ ⁄ 2 ϫ ⁄ 8 (4.23 ϫ 3.07 ϫ 2.05 ϩ 20.7 ϫ 14.93 ϫ 9.95) ϭ 581 in-kips Assume that the connection will be made with two rows of six bolts each, on each side of the connection centerline (Fig. 12.65b). The polar moment of inertia of these bolts can be computed as the sum of the moments of inertia about the x (horizontal) and y (vertical) axes. 222 I ϭ 4(1.5 ϩ 4.5 ϩ 7.5 ) ϭ 315 x 2 I ϭ 12(1.5) ϭ 27 y ϭ J ϭ 342 Load per bolt due to shear is 73.3 P ϭϭ6.1 kips v 12 Load on the outermost bolt due to moment is 581 ϫ 7.63 P ϭϭ12.95 kips m 342 The vertical component of this load is 12.95 ϫ 1.5 P ϭϭ2.5 kips v 7.63 and the horizontal component is 12.95 ϫ 7.5 P ϭϭ12.7 kips h 7.63 The total load on the outermost bolt is the resultant 22 P ϭ ͙(6.1 ϩ 2.5) ϩ 12.7 ϭ 15.3 Ͻ 2 ϫ 9.3 For the web connection plates, try two plates 17 1 ⁄ 2 ϫ 5 ⁄ 16 in. They have a net moment of inertia BEAM AND GIRDER BRIDGES 12.153 3 5 (⁄ 16 )17.5 4 I ϭ 2 Ϫ 50 ϭ 228 in 12 To transmit the 581-in-kip moment in the web, they carry a bending stress of 581 ϫ 8.75 ƒ ϭϭ22.3 Ͻ 27 ksi b 228 The assumed plates are therefore satisfactory if Grade 50W steel is used. 12.15.4 Design of Deck Plate The deck plate is to be made of Grade 50W steel. This steel has a yield strength F y ϭ 50 ksi for the 3 ⁄ 8 -in deck thickness. Stresses. Bending stresses in the deck plate as the top flange of ribs (member II), floor- beams (member III), and girders (member IV) are relatively low. Combining the stresses of members II and IV yields 4.75 ϩ 9.73 ϭ 14.48 Ͻ 1.25 ϫ 27 ksi. Deflection. The thickness of deck plate to limit deflection to 1 ⁄ 300 of the rib spacing can be computed from Eq. 11.72. For a 16-kip wheel, assumed distributed over an area of 26 ϫ 12 ϭ 312 in 2 , the pressure, including 30% impact, is p ϭ 1.3 ϫ 16/312 ϭ 0.0667 ksi Required thickness with rib spacing a ϭ e ϭ 12 in is 1/3 t ϭ 0.07 ϫ 12(0.0667) ϭ 0.341 Ͻ 0.375 in The 3 ⁄ 8 -in deckplate is satisfactory. 12.16 CONTINUOUS-BEAM BRIDGES Articles 12.1 and 12.3 recommended use of continuity for multispan bridges. Advantages over simply supported spans include less weight, greater stiffness, smaller deflections, and fewer bearings and expansion joints. Disadvantages include more complex fabrication and erection and often the costs of additional field splices. Continuous structures also offer greater overload capacity. Failure does not necessarily occur if overloads cause yielding at one point in a span or at supports. Bending moments are redistributed to parts of the span that are not overstressed. This usually can take place in bridges because maximum positive moments and maximum negative moments occur with loads in different positions on the spans. Also, because of moment redistribution due to yielding, small settlements of supports have no significant effects on the ultimate strength of continuous spans. If, however, foundation conditions are such that large settlements could occur, simple-span construction is advisable. While analysis of continuous structures is more complicated than that for simple spans, design differs in only a few respects. In simple spans, maximum dead-load moment occurs at midspan and is positive. In continuous spans, however, maximum dead-load moment occurs at the supports and is negative. Decreasing rapidly with distance from the support, the negative moment becomes zero at an inflection point near a quarter point of the span. Between the two dead-load inflection points in each interior span, the dead-load moment is positive, with a maximum about half the negative moment at the supports. 12.154 SECTION TWELVE As for simple spans, live loads are placed on continuous spans to create maximum stresses at each section. Whereas in simple spans maximum moments at each section are always positive, maximum live-load moments at a section in continuous spans may be positive or negative. Because of the stress reversal, fatigue stresses should be investigated, especially in the region of dead-load inflection points. At interior supports, however, design usually is governed by the maximum negative moment, and in the midspan region, by maximum pos- itive moment. The sum of the dead-load and live-load moments usually is greater at supports than at midspan. Usually also, this maximum is considerably less than the maximum moment in a simple beam with the same span. Furthermore, the maximum negative moment decreases rapidly with distance from the support. The impact fraction for continuous spans depends on the length L, ft, of the portion of the span loaded to produce maximum stress. For positive moment, use the actual loaded length. For negative moment, use the average of two adjacent loaded spans. Ends of continuous beams usually are simply supported. Consequently, moments in three- span and four-span continuous beams are significantly affected by the relative lengths of interior and exterior spans. Selection of a suitable span ratio can nearly equalize maximum positive moments in those spans and thus permit duplication of sections. The most advan- tageous ratio, however, depends on the ratio of dead load to live load, which, in turn, is a function of span length. Approximately, the most advantageous ratio for length of interior to exterior span is 1.33 for interior spans less than about 60 ft, 1.30 for interior spans between about 60 to 110 ft, and about 1.25 for longer spans. When composite construction is advantageous (see Art. 12.1), it may be used either in the positive-moment regions or throughout a continuous span. Design of a section in the positive-moment region in either case is similar to that for a simple beam. Design of a section in the negative-moment regions differs in that the concrete slab, as part of the top flange, cannot resist tension. Consequently, steel reinforcement must be added to the slab to resist the tensile stresses imposed by composite action. Additionally, for continuous spans with a cast-in-place concrete deck, the sequence of concrete pavement is an important design consideration. Bending moments, bracing require- ments, and uplift forces must be carefully evaluated. 12.17 ALLOWABLE-STRESS DESIGN OF BRIDGE WITH CONTINUOUS, COMPOSITE STRINGERS The structure is a two-lane highway bridge with overall length of 298 ft. Site conditions require a central span of 125 ft. End spans, therefore, are each 86.5 ft (Fig. 12.66a). The typical cross section in Fig. 12.66b shows a 30-ft roadway, flanked on one side by a 21-in- wide barrier curb and on the other by a 6-ft-wide sidewalk. The deck is supported by six rolled-beam, continuous stringers of Grade 36 steel. Concrete to be used for the deck is Class A, with 28-day strength ϭ 4,000 psi and allowable compressive stress ƒ c ϭ 1,600ƒЈ c psi. Loading is HS20-44. Appropriate design criteria given in Sec. 11 will be used for this structure. Concrete Slab. The slab is designed to span transversely between stringers, as in Art. 12.2. A 9-in-thick, two-course slab will be used. No provision will be made for a future 2-in wearing course. Stringer Loads. Assume that the stringers will not be shored during casting of the concrete slab. Then, the dead load on each stringer includes the weight of a strip of concrete slab plus the weights of steel shape, cover plates, and framing details. This dead load will be referred to as DL and is summarized in Table 12.81. BEAM AND GIRDER BRIDGES 12.155 FIGURE 12.66 (a) Spans of a continuous highway bridge. (b) Typ- ical cross section of bridge. TABLE 12.81 Dead Load, kips per ft, on Continuous Steel Beams Stringers S 1 and S 3 Stringers S 2 Slab 0.618 0.630 Haunch and SIP forms: 0.102 0.047 Rolled beam and details—assume: 0.320 0.320 DL per stringer 1.040 0.997 Sidewalks, parapet, and barrier curbs will be placed after the concrete slab has cured. Their weights may be equally distributed to all stringers. Some designers, however, prefer to calculate the heavier load imposed on outer stringers by the cantilevers by taking moments of the cantilever loads about the edge of curb, as shown in Table 12.82. In addition, the six composite beams must carry the weight, 0.016 ksf, of the 30-ft-wide latex-modified concrete wearing course. The total superimposed dead load will be designated SDL. The HS20-44 live load imposed may be a truck load or lane load. For these spans, truck loading governs. With stringer spacing S ϭ 6.5 ft, the live load taken by outer stringers S 1 and S 3 is S 6.5 ϭϭ1.155 wheels ϭ 0.578 axle 4 ϩ 0.25S 4 ϩ 0.25 ϫ 6.5 The live load taken by S 2 is 12.156 SECTION TWELVE TABLE 12.82 Dead Load, kips per ft, on Composite Stringers SDL x Moment Barrier curb: 0.530/6 ϭ 0.088 1.33 0.117 Sidewalk: 0.510/6 0.085 3.50 0.298 Parapet: 0.338 /6 0.056 6.50 0.364 Railing: 0.015/ 6 0.002 6.50 0.013 0.231 0.675 1 1 ⁄ 4 -in LMC course 0.078 SDL for S 2 : 0.309 Eccentricity for S 1 ϭ 0.117/ 0.088 ϩ 6.5 ϩ 1.38 ϭ 9.21 ft Eccentricity for S 3 ϭ 0.675/ 0.143 ϩ 6.5 Ϫ 3.88 ϭ 7.34 ft SDL for S 1 ϭ 0.309 ϫ 9.21/6.5 ϭ 0.438 SDL for S 3 ϭ 0.309 ϫ 7.34/6.5 ϭ 0.349 S 6.5 ϭϭ1.182 wheels ϭ 0.591 axle 5.5 5.5 Sidewalk live load (SLL) on each stringer is 0.060 ϫ 6 w ϭϭ0.060 kip per ft SLL 6 The impact factor for positive moment in the 86.5-ft end spans is 50 50 I ϭϭ ϭ0.237 L ϩ 125 86.5 ϩ 125 For positive moment in the 125-ft center span, 50 I ϭϭ0.200 125 ϩ 125 And for negative moments at the interior supports, with an average loaded span L ϭ (86.5 ϩ 125)/2 ϭ 105.8 ft, 50 I ϭϭ0.217 105.8 ϩ 125 Stringer Moments. The steel stringers will each consist of a single rolled beam of Grade 36 steel, composite with the concrete slab only in regions of positive moment. To resist negative moments, top and bottom cover plates will be attached in the region of the interior supports. To resist maximum positive moments in the center span, a cover plate will be added to the bottom flange of the composite section. In the end spans, the composite section with the rolled beam alone must carry the positive moments. For a precise determination of bending moments and shears, these variations in moments of inertia of the stringer cross sections should be taken into account. But this requires that [...]... 12. 84 Stresses, ksi, in End Span for Maximum Positive Moment (a) Steel stresses Top of steel (compression) Bottom of steel (tension) DL: ƒb ϭ 434 ϫ 12 / 1,030 ϭ 5.06 SDL: ƒb ϭ 183 ϫ 12 / 2,370 ϭ 0.93 LL ϩ I: ƒb ϭ 786 ϫ 12 / 7,220 ϭ 1.31 Total: 7.30 Ͻ 20 ƒb ϭ 434 ϫ 12 / 1,030 ϭ 5.06 ƒb ϭ 183 ϫ 12 / 1,264 ϭ 1.74 ƒb ϭ 786 ϫ 12 / 1,445 ϭ 6.53 13.33 Ͻ 20 (b) Stresses at top of concrete SDL: ƒc ϭ 183 ϫ 12. .. Steel stresses Top of steel (compression) Bottom of steel (tension) DL: ƒb ϭ 773 ϫ 12 / 1,062 ϭ 8.73 SDL: ƒb ϭ 325 ϫ 12 / 2,387 ϭ 1.63 LL ϩ I: ƒb ϭ 907 ϫ 12 / 6,936 ϭ 1.56 Total: 11.92 Ͻ 20 ƒb ϭ 773 ϫ 12 / 1,159 ϭ 8.00 ƒb ϭ 325 ϫ 12 / 1,414 ϭ 2.76 ƒb ϭ 907 ϫ 12 / 1,617 ϭ 6.70 17.46 Ͻ 20 (b) Stresses at top of concrete SDL: ƒc ϭ 325 ϫ 12 / (1,398 ϫ 24) ϭ 0 .12 LL ϩ I: ƒc ϭ 907 ϫ 12 / (2,774 ϫ 8) ϭ 0.49... on both sides of the web Table 12. 89 presents the calculations for the net area of the splice plates The plates can be considered satisfactory See Fig 12. 71 TABLE 12. 88 Tensile Stresses, ksi, 12. 5 ft from Midspan DL: ƒb ϭ 694 ϫ 12 / 1,030 ϭ 8.09 SDL: ƒb ϭ 293 ϫ 12 / 1,264 ϭ 2.78 LL ϩ I: ƒb ϭ 863 ϫ 12 / 1,445 ϭ 7.17 Total: 18.04 Ͻ 20 BEAM AND GIRDER BRIDGES 12. 167 FIGURE 12. 71 Cover plates and field splice... 3,375 ϭ Ϫ65,240 INA ϭ 128 ,260 145,520 d8 ϭ 3375 / 174,6 ϭ 19.33 in 640 Distance from neutral axis of composite section to: Top of steel ϭ 30.75 Ϫ 19.33 ϭ 11.42 in Bottom of steel ϭ 31.50 ϩ 19.33 ϭ 50.83 in Top of concrete ϭ 11.42 ϩ 2 ϩ 8.50 ϭ 21.92 in Section Modulus: Top of steel Bottom of steel Top of concrete Sst ϭ 128 ,260 / 11.42 ϭ 11,230 in3 Ssb ϭ 128 ,260 / 50.83 ϭ 2,520 in3 Sc ϭ 128 ,260 / 21.92 ϭ... to: Top of steel ϭ 18.26 Ϫ 11.26 ϭ 7.00 in Bottom of steel ϭ 18.26 ϩ 0.50 ϩ 11.26 ϭ 30.02 in Top of concrete ϭ 7.00 ϩ 2 ϩ 8.5 ϭ 17.50 in Section moduli Top of steel Bottom of steel Top of concrete Sst ϭ 48,550 / 7.00 ϭ 6,936 in3 Ssb ϭ 48,550 / 30.02 ϭ 1,617 in3 Sc ϭ 48,550 / 17.50 ϭ 2,774 in3 12. 166 SECTION TWELVE TABLE 12. 87 Stresses, ksi, in Center Span for Maximum Positive Moment (a) Steel stresses... concrete ϭ 12. 70 ϩ 2 ϩ 7.75 ϭ 22.45 in Section moduli Top of steel Bottom of steel Top of concrete Sst ϭ 30,100 / 12. 70 ϭ 2,370 in3 Ssb ϭ 30,100 / 23.82 ϭ 1,264 in3 Sc ϭ 30,100 / 22.45 ϭ 1,341 in3 (b) For live loads, n ϭ 8 Material Steel section Concrete 76.5 ϫ 8.5 / 8 A d Ad Ad 2 Io I 82.4 81.3 24.51 1,993 48,840 18,900 490 18,900 49,330 163.7 1,993 d8 ϭ 1993 / 163.7 ϭ 12. 17 in 68,230 12. 17 ϫ 1,993... 43,970 Distance from neutral axis of composite section to: Top of steel ϭ 18.26 Ϫ 12. 17 ϭ 6.09 in Bottom of steel ϭ 18.26 ϩ 12. 17 ϭ 30.43 in Top of concrete ϭ 6.09 ϩ 2 ϩ 8.5 ϭ 16.59 in Section moduli Top of steel Bottom of steel Top of concrete Sst ϭ 43,970 / 6.09 ϭ 7,220 in3 Ssb ϭ 43,970 / 30.43 ϭ 1,445 in3 Sc ϭ 43,970 / 16.59 ϭ 2,650 in3 12. 162 SECTION TWELVE MAXIMUM POSITIVE MOMENTS CENTER SPAN, FT-KIPS... bridge (Fig 12. 13) similar to that considered in Art 12. 4 and 12. 5 by ASD and LFD methods The span length is 100 ft and the girder spacing is 8 ft 4 in The HL-93 live load for LRFD will be used (Art 11.4) FIGURE 12. 73 Camber of girder to offset dead-load deflections 12. 170 SECTION TWELVE 12. 18.1 Stringer Design Procedure The design procedure for LRFD in most cases resembles that discussed in Art 12. 5.1... trial steel section and composite section in Tables 12. 90 and 12. 91 The plastic moment capacity of the composite section will be determined by force equilibrium Concrete haunch and deck reinforcement will be neglected Assume plastic neutral axis (PNA) is at top of top flange: FIGURE 12. 76 Cross section assumed for plate girder for LRFD example BEAM AND GIRDER BRIDGES 12. 179 TABLE 12. 90 Properties of Steel. .. GIRDER BRIDGES 12. 161 TABLE 12. 83 End-Span Composite Section (a) For dead loads, n ϭ 24 Material A d Ad Ad 2 Io I Steel section Concrete 76.5 ϫ 7.75 / 24 82.4 24.7 24.14 596 14,400 18,900 120 18,900 14,520 107.1 596 33,420 d24 ϭ 596 / 107.1 ϭ 5.56 in Ϫ5.56 ϫ 596 ϭ Ϫ3,320 INA ϭ 30,100 Distance from neutral axis of composite section to: Top of steel ϭ 18.26 Ϫ 5.56 ϭ 12. 70 in Bottom of steel ϭ 18.26 ϩ . ϫ 12 / (1,398 ϫ 24) ϭ 0 .12 c LL ϩ I:ƒ ϭ 907 ϫ 12 /(2,774 ϫ 8) ϭ 0.49 c Total: 0.61 Ͻ 1.6 TABLE 12. 88 Tensile Stresses, ksi, 12. 5 ft from Midspan DL:ƒ ϭ 694 ϫ 12 /1,030 ϭ 8.09 b SDL:ƒ ϭ 293 ϫ 12. section to: Top of steel ϭ 18.26 Ϫ 12. 17 ϭ 6.09 in Bottom of steel ϭ 18.26 ϩ 12. 17 ϭ 30.43 in Top of concrete ϭ 6.09 ϩ 2 ϩ 8.5 ϭ 16.59 in Section moduli Top of steel Bottom of steel Top of concrete S. Positive Moment (a) Steel stresses Top of steel (compression) Bottom of steel (tension) DL:ƒ ϭ 434 ϫ 12 /1,030 ϭ 5.06 b SDL:ƒ ϭ 183 ϫ 12 /2,370 ϭ 0.93 b LL ϩ I:ƒ ϭ 786 ϫ 12/ 7,220 ϭ 1.31 b Total: