7 Example - LRFD for Double Angles with Axial Compression
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BEAM AND GIRDER BRIDGES
12.123
TABLE 12.64 Composite Section for Maximum Moment in Box Girder
(a) For dead loads, n ϭ 30
Material
A
d
Steel section
Concrete 216 ϫ 7 / 30
163.5
50.4
213.9
37.0
Ad
Ad 2
Io
Ϫ845
1,865
1,020
69,000
d30 ϭ 1,020 / 213.9 ϭ 4.77 in
I
134,000
69,200
203,200
Ϫ4.77 ϫ 1,020 ϭ Ϫ4,900
INA ϭ 198,300
200
Distance from neutral axis of composite section to:
Top of steel ϭ 33.00 Ϫ 4.77 ϭ 28.23 in
Bottom of steel ϭ 33.00 ϩ 4.77 ϭ 37.77 in
Top of concrete ϭ 28.23 ϩ 0.50 ϩ 7 ϭ 35.73 in
Section moduli
Top of steel
Bottom of steel
Top of concrete
Sst ϭ 198,300 / 28.23
ϭ 7,020 in3
Ssb ϭ 198,300 / 37.77
ϭ 5,250 in3
Sc ϭ 198,300 / 35.73
ϭ 5,550 in3
(b) For live loads, n ϭ 10
Material
Steel section
Concrete 216 ϫ 7 / 10
d10
A
163.5
151.2
314.9
ϭ 4,749 / 314.7 ϭ 15.09 in
d
Ad
Ϫ845
37.0
5,594
4,749
Ad 2
Io
I
134,000
207,600
341,600
Ϫ15.09 ϫ 4,749 ϭ Ϫ71,600
INA ϭ 270,000
207,000
600
Distance from neutral axis of composite section to:
Top of steel ϭ 33.00 Ϫ 15.09 ϭ 17.91 in
Bottom of steel ϭ 33.00 ϩ 15.09 ϭ 48.09 in
Top of concrete ϭ 17.91 ϩ 0.50 ϩ 7 ϭ 25.41 in
Section moduli
Top of steel
Bottom of steel
Top of concrete
Sst ϭ 270,000 / 17.91
ϭ 15,070 in3
Ssb ϭ 270,000 / 48.09
ϭ 5,610 in3
Sc ϭ 270,000 / 25.41
ϭ 10,630 in3
12.124
SECTION TWELVE
TABLE 12.65 Stresses in Composite Box Girder, ksi
(a) Steel stresses
Top of steel (compression)
Bottom of steel (tension)
DL: ƒb ϭ 4,210 ϫ 12 / 3,570 ϭ 14.39
SDL: ƒb ϭ 1,280 ϫ 12 / 7,020 ϭ 2.19
LL ϩ I: ƒb ϭ 2,530 ϫ 12 / 15,070 ϭ 2.02
Total:
18.60 Ͻ 20
ƒb ϭ 4,210 ϫ 12 / 4,810 ϭ 10.50
ƒb ϭ 1,280 ϫ 12 / 5,250 ϭ 2.93
ƒb ϭ 2,530 ϫ 12 / 5,610 ϭ 5.41
18.84 Ͻ 20
(b) Stresses at top of concrete
SDL: ƒc ϭ 1,280 ϫ 12 / (5,550 ϫ 30) ϭ 0.09
LL ϩ I: ƒc ϭ 2,530 ϫ 12 / (10,630 ϫ 10) ϭ 0.29
Total:
0.38 Ͻ 1.0
Flange-to-Web Welds. Fillet welds placed on opposite sides of each girder web to connect
it to each flange must resist the horizontal shear between flange and web. In this example,
as is usually the case (see Art 12.4, for example), the minimum size of weld permissible for
the thickest plate at the connection determines the size of weld. For both the 7⁄8in bottom
flange and the 1-in top flanges, the minimum size of weld permitted is 5⁄16 in. Therefore,
use a 5⁄16-in fillet weld on opposite sides of each web at each flange.
Intermediate Transverse Stiffeners. To determine if transverse stiffeners are required, the
allowable shear stress Fv will be computed and compared with the average shear stress ƒv ϭ
5.03 ksi at the support.
Fv ϭ [270(D / t)]2 ϭ (270 / 170)2 ϭ 2.52 ksi Ͻ 5.03 ksi
Therefore, transverse intermediate stiffeners are required.
Maximum spacing of stiffeners may not exceed 3 ϫ 64 ϭ 192 in or D[260 / (D / t)]2 ϭ
64(260 / 170)2 ϭ 150 in. Try a stiffener spacing do ϭ 90 in. This provides a depth-spacing
ratio D / do ϭ 64 / 90 ϭ 0.711. From Eq. (11.24d ), for use in Eq. (11.25a), k ϭ 5[1 ϩ
(0.711)2] ϭ 7.53 and ͙k / Fy ϭ ͙7.53 / 36 ϭ 0.457. Since D / t ϭ 170, C in Eq. (11.24a) is
determined by the parameter 170 / 0.457 ϭ 372 Ͼ 237. Hence, C is given by
TABLE 12.66 Maximum Shear in Composite Box Girder
Distance from support, ft
DL, kips
LL ϩ I, kips
Total, kips
ƒv, ksi
0
10
20
30
40
50
60
183
89
272
5.49
153
81
234
4.73
124
73
197
3.98
93
65
158
3.19
62
57
119
2.40
31
49
80
1.62
0
41
41
0.83
BEAM AND GIRDER BRIDGES
Cϭ
12.125
45,000k
45,000 ϫ 7.53
ϭ
ϭ 0.326
(D / t)2Fy
1702 ϫ 36
From Eq. (11.25a), the maximum allowable shear for do ϭ 90 in is
F vЈ ϭ
ϭ
ͫ
ͫ
Fy
0.87(1 Ϫ C )
Cϩ
3
͙1 ϩ (do / D)2
ͬ
ͬ
36
0.87(1 Ϫ 0.326)
0.326 ϩ
ϭ 7.99 ksi Ͼ 5.03 ksi
3
͙1 ϩ (90 / 64)2
Since the allowable stress is larger than the computed stress, the stiffeners may be spaced
90 in apart.
The AASHTO standard specifications limit the spacing of the first intermediate stiffener
to the smaller of 1.5D ϭ 1.5 ϫ 64 ϭ 96 in and the spacing for which the allowable shear
stress in the end panel does not exceed
Fv ϭ CFy / 3 ϭ 0.326 ϫ 36 / 3 ϭ 3.91 ksi Ͻ 5.03 ksi
Therefore, closer spacing is needed near the supports. Try do ϭ 45 in, for which k ϭ 15.11,
C ϭ 0.654, and Fv ϭ CFy / 3 ϭ 0.654 ϫ 36⁄3 ϭ 7.85 ksi Ͼ 5.03. Therefore, 45-in spacing
will be used near the supports and 90-in spacing in the next 22.5 ft of girder, as shown in
Fig. 12.55. Transverse stiffeners are omitted from the central 60 ft of girder, except at
midspan.
Where required, a single plate stiffener of Grade 36 steel will be welded inside the box
girder to each web. Minimum width of stiffeners is one-fourth the flange width, or 21 / 4 ϭ
5.25 Ͼ 2 ϩ 66 / 30 ϭ 4.2 in. Use a 6-in-wide plate. Minimum thickness required is 6 / 16 ϭ
3
⁄8 in. Try 6 ϫ 3⁄8-in stiffeners.
The moment of inertia provided by each stiffener must satisfy Eq. (11.21), with J as given
by Eq. (11.22).
J ϭ 2.5
ͩͪ
64
90
2
Ϫ 2 ϭ Ϫ0.73
use 0.5
I ϭ 90(3⁄8)30.5 ϭ 2.37
The moment of inertia furnished is
Iϭ
(3⁄8)63
ϭ 27 Ͼ 2.37 in4
3
Hence, the 6 ϫ 3⁄8-in stiffeners are satisfactory. Weld them to the webs with a pair of 1⁄4-in
fillet welds.
Bearing Stiffeners. Instead of narrow-plate stiffeners and a cross frame over the bearings,
a plate diaphragm extending between the webs is specified. The plate diaphragm has superior
resistance to rotation, displacement, and distortion of the box girder. Assume for the diaphragm a bearing length of 20 in at each web, or a total of 40 in. The allowable bearing
stress is 29 ksi. Then, the thickness required for bearing is
tϭ
271.9
ϭ 0.23 in
40 ϫ 29
But the thickness of a bearing stiffener also is required to be at least
12.126
SECTION TWELVE
FIGURE 12.55 Locations of stiffeners, cross frames, and shear connectors for
composite box girder.
tϭ
bЈ
12
Ί33 ϭ 1220 Ί3336 ϭ 1.74 in
Fy
Therefore, use a plate 64 ϫ 13⁄4 in extending between the webs at the supports, with a 30in-square access hole.
The welds to the webs must be capable of developing the entire 271.9-kip reaction.
Minimum-size fillet weld for the 13⁄4-in diaphragm is 5⁄16 in. With two such welds at each
web, their required length, with an allowable stress of 15.7 ksi, is
271.9
ϭ 19.6 in
4(5⁄16)0.707 ϫ 15.7
Weld the full 66-in depth of web.
Shear Connectors. To ensure composite action of concrete deck and box girders, shear
connectors welded to the top flanges of the girders must be embedded in the concrete (Art
11.16). For this structure, 7⁄8-in-dia. welded studs are selected. They are to be installed in
groups of three at specified locations to resist the horizontal shear between the steel section and the concrete slab (Fig. 12.55). With height H ϭ 4 in, they satisfy the requirement
H / d Ն 4, where d ϭ stud diameter, in.
BEAM AND GIRDER BRIDGES
12.127
With ƒ Јc ϭ 2,800 psi for the concrete, the ultimate strength of a 7⁄8-in welded stud is,
from Eq. (12.26),
Sv ϭ 0.4d 2 ͙ƒ cЈEc ϭ 0.4(7⁄8)2 ͙2.8 ϫ 2,900 ϭ 27.6 kips
This value is needed for determining the number of shear connectors required to develop
the strength of the steel girder or the concrete slab, whichever is smaller. With an area As ϭ
163.5 in2, the strength of the girder is
P1 ϭ As Fy ϭ 163.5 ϫ 36 ϭ 5,890 kips
The compressive strength of the concrete slab is
P2 ϭ 0.85ƒ cЈ bt ϭ 0.85 ϫ 2.8 ϫ 216 ϫ 7 ϭ 3,600 Ͻ 5,890 kips
Concrete strength governs. Hence, from Eq. (12.25), the number of studs provided between
midspan and each support must be at least
N1 ϭ
P1
3,600
ϭ
ϭ 153
Sv 0.85 ϫ 27.6
With the studs placed in groups of three on each top flange, there should be at least 153 / 6
ϭ 26 groups on each half of the girder.
Pitch is determined by fatigue requirements. The allowable load range, kips per stud, may
be computed from Eq. (12.4). With ␣ ϭ 10.6 for 500,000 cycles of load (AASHTO specifications),
Zr ϭ 10.6(0.875)2 ϭ 8.12 kips per stud
At the supports, the shear range Vr ϭ 89 kips, the shear produced by live load plus
impact. Consequently, with n ϭ 10 for the concrete, and the transformed concrete area equal
to 151.2 in2, and I ϭ 252,300 in4 from Table 12.64b, the range of horizontal shear stress is
Sr ϭ
Vr Q 89 ϫ 151.2 ϫ 20.70
ϭ
ϭ 1.107 kips per in
I
252,300
Hence, the pitch required for stud groups near the supports is
pϭ
6Zr 6 ϫ 8.12
ϭ
ϭ 44 in
Sr
1.107
Use a pitch of 15 in to satisfy both this requirement and that for 26 groups of studs between
midspan and each support (Fig. 12.55).
Intermediate Cross Frames. Though intermediate cross frames or diaphragms are not required by standard specifications, it is considered good practice by many designers to specify
such interior bracing in box girders to help maintain the shape under torsional loading. So
in addition to the end bearing diaphragm, cross frames will be installed at 30-ft intervals.
Minimum-size angles can be used (Fig. 12.56).
Camber. The girders should be cambered to compensate for dead-load deflections under
DL and SDL. For computation for DL, the moment of inertia I of the steel section alone
should be used. For SDL, I should apply to the composite section with n ϭ 30 (Table 12.64a).
Both deflections can be computed from Eq. (12.5) with wDL ϭ 2.34 kips per ft and wSDL ϭ
0.33 kip per ft.
12.128
SECTION TWELVE
FIGURE 12.56 Intermediate cross frame.
DL: ␦ ϭ 22.5 ϫ 2.34(120)4 / (29,000 ϫ 123,300) ϭ 3.04 in
SDL: ␦ ϭ 22.5 ϫ 0.33(120)4 / (29,000 ϫ 187,500) ϭ 0.29 in
Total:
3.33 in
Live-Load Deflection. Maximum live-load deflection should be checked to ensure that it
does not exceed 12L / 800. This deflection may be obtained with acceptable accuracy from
Eq. (12.6), with
PT ϭ 8 ϫ 1.113 ϩ 0.204 ϫ 8 ϫ 1.113 ϭ 10.73 kips
From Table 12.64b, for n ϭ 10, I ϭ 270,000 in4. Therefore,
␦ϭ
324 ϫ 10.73
(1203 Ϫ 555 ϫ 120 ϩ 4,780) ϭ 0.74 in
29,000 ϫ 270,000
And the deflection-span ratio is
0.74
1
1
ϭ
Ͻ
120 ϫ 12 1,800 800
Thus, the live-load deflection is acceptable.
Other Details. These may be treated in the same way as for I-shaped plate girders.
12.14
ORTHOTROPIC-PLATE GIRDER BRIDGES
In orthotropic-plate construction, a steel-plate deck is used instead of concrete. The plate is
topped with a wearing surface that may or may not be concrete. The steel-plate deck serves
the usual deck function of distributing loads to main carrying members, but it also acts as
the top flange of those members (Art. 11.20). Because the deck provides a large area,
orthotropic-plate construction is very efficient in resisting bending. With a lightweight wearing surface, furthermore, bridges of this type have relatively low dead load, a characteristic
particularly important for keeping down the costs of long spans. Figure 12.57 shows some
examples of cross sections that have been used for orthotropic-plate bridges.
These examples indicate that orthotropic plates often are used with box girders. In addition to low dead weight, this type of construction offers many of the advantages of composite box girders discussed in Art. 12.12. The examples, however, are all long-span bridges.
It may also be economical for medium spans to use orthotropic plates with girders with
inverted-T shapes.
12.129
FIGURE 12.57 Examples of cross sections of orthotropic-plate highway bridges. (a) Fremont Bridge, Portland, Ore, incorporates continuous tied-arch spans of 448.3–255.3–448.3 ft. (b) Poplar St. Bridge over the
Mississippi River at St. Louis has continuous spans of 300–500–600–500–265 ft. (c) San Mateo-Heyward
Bridge over San Francisco Bay, Calif., contains three cantilever–type spans of 375–750–375 ft with a 375-ft
suspended span and 187.5 cantilevers. (d ) San Diego-Coronado Bridge over San Diego Bay, Calif., provides
continuous spans of 600–600–500 ft. (e) Wye River Bridge, England, cable-stayed, spans 285–770–285 ft.
( ƒ ) Severin River Bridge over the Rhine River, Cologne, Germany, also cable-stayed, has spans of 161–292–
157–990–494–172 ft.