11 Example - LRFD for Floorbeam with Overhang
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FIGURE 12.58 Half cross section of orthotropic-plate bridge with 120-ft span.
12.131
12.132
SECTION TWELVE
FIGURE 12.59 Moment
girder (member IV).
envelope
for
should be placed at that point. For maximum shear at any point, a 26-kip concentrated load
should be placed there, and the uniform load should extend from the point to the farthest
support. For example, maximum moment occurs at midspan with the 18-kip load placed
there and the uniform load over the entire span:
MT ϭ
0.64(120)2 18 ϫ 120
ϩ
ϭ 1,152 ϩ 540 ϭ 1,692 ft-kips
8
4
Maximum shear occurs at a support with the 26-kip load placed there and the uniform load
over the entire span:
VT ϭ
0.64 ϫ 120
ϩ 26 ϭ 38.4 ϩ 26 ϭ 64.4 kips
2
In distributing the two lanes of live load to the girders, the deck is restrained against rotation
to some extent by the girders. It may be assumed to be somewhere between simply supported
and fixed. In this example, either assumption gives about the same result. For the assumption
of a simple support, with one truck wheel 2 ft from a girder, the load distributed by the deck
to that girder is
Wϭ
28 ϩ 22 ϩ 16 ϩ 10
ϭ 2.53 wheel
30
ϭ 1.27 axles
Then, the maximum moment at midspan due to live load is
MLL Ϫ 1.27 ϫ 1,692 ϭ 2,150 ft-kips
Similarly, maximum live-load shear at the support and maximum reaction is
VLL ϭ 1.27 ϫ 64.4 ϭ 81.8 kips
Maximum live-load moments and shears at 10-ft intervals along the span are listed in Tables
12.68 and 12.69.
Impact for maximum moments and maximum shear and reaction at the supports may be
taken as the following fraction of live-load stress:
BEAM AND GIRDER BRIDGES
FIGURE 12.60 Details of box girder with orthotropic-plate top flange.
FIGURE 12.61 Sections through rib of orthotropic-plate deck. (a)
Rib alone. (b) Rib with part of plate as top flange.
12.133
12.134
SECTION TWELVE
TABLE 12.67 Dead Load on Member IV, kips
per ft
Railing:
Curb: 0.150 ϫ 3.0 ϫ 0.83
Wearing course: 0.150 ϫ 15 ϫ 2⁄12
Deck plate: 0.153 ϫ 15
Ribs: 7 ϫ 0.0329
Floorbeams—assume:
Girder and details—assume:
DL per girder:
Iϭ
0.025
ϭ 0.375
ϭ 0.375
ϭ 0.230
ϭ 0.230
0.050
0.500
1.785, say 1.8
50
50
ϭ
ϭ 0.204
L ϩ 125 120 ϩ 125
Impact moments and shears also are given in Tables 12.68 and 12.69.
These tables, in addition, give the total moments and shears at 10-ft intervals. The moments are used to plot the curve of maximum moments (moment envelope) in Fig. 12.59.
Web Size. Minimum depth-span ratio for a girder is 1:25. For a 120-ft span, therefore, the
girders should be at least 120⁄25 ϭ 4.8 ft ϭ 58 in deep. Use webs 60 in deep.
With an allowable shear stress of 12 ksi for Grade 36 steel, the thickness required for
shear in each web is
tϭ
207
ϭ 0.144 in
2 ϫ 12 ϫ 60
With stiffeners, the web thickness, however, to prevent buckling, should be at least 1⁄165 of
the depth, or 0.364 in. Use two 60 ϫ 3⁄8-in webs, with an area of 45 in2.
Check of Section at Midspan. The assumed girder section at midspan is shown in Fig.
12.58, with the bottom flange taken as 11⁄8 in thick. (See also Fig. 12.60.) The neutral axis
is located by taking moments of areas of the components about middepth of the webs. This
computation and those for moment of inertia and section moduli are given in Table 12.70.
The neutral axis is found to be 12.36 in above middepth of web, based on the gross section.
(Area of bolt holes is less than 15% of the flange area.)
Maximum bending stress occurs in the bottom flange and equals
TABLE 12.68 Moments in Orthotropic-Plate Box Girder, ft-kips
Distance from support
MDL
MLL
MI
Total
10
20
30
40
50
60
990
660
130
1,780
1,800
1,200
240
3,240
2,430
1,610
330
4,370
2,880
1,910
390
5,180
3,150
2,090
430
5,670
3,240
2,150
440
5,830
BEAM AND GIRDER BRIDGES
12.135
TABLE 12.69 Shears in Orthotropic-Plate Box Girder, kips
Distance from support
VDL
VLL
VI
Total
ƒv, ksi
0
10
20
30
40
50
60
108
82
17
207
4.60
90
71
15
176
3.91
72
60
12
144
3.20
54
99
10
113
2.52
36
38
8
82
1.82
18
27
6
51
1.13
0
17
3
20
0.44
ƒb ϭ
5,830 ϫ 12
ϭ 19.9 Ͻ 20 ksi
3,520
The section is satisfactory. Moment capacity supplied is
MC ϭ
20 ϫ 3,520
ϭ 5,870 ft-kips
12
Change in Bottom Flange. At a sufficient distance from midspan, the bending moment
decreases enough to permit reducing the thickness of the bottom flange to 3⁄4 in. As indicated
in Table 12.71, the moment of inertia reduces to 121.400 in4 and the section modulus of the
bottom flange to 2,640 in3.
TABLE 12.70 Maximum Moment at Midspan of Box Girder with Orthotropic-Plate Flange
Material
A
Deck plate 180 ϫ 3⁄8
Seven ribs
Top flange 42 ϫ 1⁄2
Two webs 60 ϫ 3⁄8
Bottom flange 42 ϫ 11⁄8
67.50
65.87
21.00
45.00
47.25
246.62
d
Ad
Ad 2
32.49
25.25
30.25
2,192
1,664
635
71,300
42,100
19,200
Ϫ30.56
Ϫ1,443
44,400
Io
I
13,500
71,300
42,100
19,200
13,500
44,400
190,500
3,048
d ϭ 3,048 / 246.62 ϭ 12.36 in
Ϫ12.36 ϫ 3,048 ϭ Ϫ37,600
INA ϭ 152,900
Distance from neutral axis to:
Top of steel ϭ 30.50 ϩ 0.38 ϩ 0.02 ϫ 180 Ϫ 12.36 ϭ 22.12 in
Bottom of steel ϭ 31.13 ϩ 12.36 ϭ 43.49 in
Bottom of rib ϭ 22.12 Ϫ 0.38 Ϫ 12 Ϫ 0.02 ϫ 12 ϭ 9.50 in
Section moduli
Top of steel
Bottom of steel
Bottom of rib
St ϭ 152,900 / 22.12
ϭ 6,910 in3
Sb ϭ 152,900 / 43.49
ϭ 3,520 in3
Sr ϭ 152,900 / 9.50
ϭ 16,100 in3
12.136
SECTION TWELVE
TABLE 12.71 Section near Supports of Box Girder with Orthotropic-Plate Flange
Material
A
Midspan section
Flange decrease 42 ϫ 3⁄8
Ϫ15.75
d
Ad
Ad 2
I
Ϫ30.94
3,048
487
3,535
Ϫ15,100
Ϫ15,100
246.62
230.87
d ϭ 3,535 / 230.87 ϭ 15.30 in
190,500
175,400
Ϫ15.30 ϫ 3,535 ϭ Ϫ54,000
INA ϭ 121,400
Distance from neutral axis to:
Top of steel ϭ 30.50 ϩ 0.38 ϩ 0.02 ϫ 180 Ϫ 15.30 ϭ 19.18 in
Bottom of steel ϭ 30.75 ϩ 15.30 ϭ 46.05 in
Bottom of rib ϭ 19.18 Ϫ 0.38 Ϫ 12 Ϫ 12 ϫ 0.02 ϭ 6.56 in
Section moduli
Top of steel
Bottom of steel
Bottom of rib
St ϭ 121,400 / 19.18
ϭ 6,330 in3
Sb ϭ 121,400 / 46.05
ϭ 2,640 in3
Sr ϭ 121,400 / 6.56
ϭ 18,500 in3
With the 3⁄4-in bottom flange, the section has a moment capacity of
MC ϭ
20 ϫ 2,640
ϭ 4,400 ft-kips
12
When this is plotted on Fig. 12.59, the horizontal line representing it stays above the moment
envelope until within 30 ft of midspan. Hence. flange size can be decreased at that point.
Length of the 11⁄8-in bottom-flange plate then is 60 ft and of the 3⁄4-in plate, which extends
to the end of the girder, 30 ft.
The flange plates will be spliced with complete-penetration groove welds. If these welds
are ground smooth in the direction of stress and a transition slope of 1 to 21⁄2 is provided
for change in plate thickness, the connection falls in Stress Category B for fatigue calculations. The bridge may be treated as a redundant-load-path structure subjected to 500,000
cycles of loading. Hence, the allowable stress range Fr ϭ 29 ksi. The stress range for live
loads plus impact is
ƒr ϭ
1,940 ϫ 12
ϭ 8.82 ksi Ͻ 29 ksi—OK
2,640
Flange-to-Web Welds. Each flange will be connected to each web by a fillet weld on
opposite sides of the web. These welds must resist the horizontal shear between flange and
web. Computations can be made, as for the plate-girder stringers in Art. 12.4, but minimum
size of weld permissible for the thickest plate at the connection governs. Therefore, use 1⁄4in welds with the 1⁄2-in top flange, the 3⁄4-in bottom flange, and 5⁄16-in welds with the 11⁄8in bottom flange (Fig. 12.60c).
BEAM AND GIRDER BRIDGES
12.137
Bending Stresses in Deck. As part of member IV, the deck plate is subjected to the following maximum bending stresses:
At midspan
At flange change
ƒb ϭ 5,830 ϫ 12 / 6,910 ϭ 10.12 ksi
ƒb ϭ 4,370 ϫ 12 / 6,330 ϭ 8.28 ksi
At midspan, maximum stress at the bottom of a rib is
ƒb ϭ
5,830 ϫ 12
ϭ 4.35 ksi
16,100
Intermediate Transverse Stiffeners. To determine if transverse stiffeners are required, the
allowable shear stress Fv will be compared with the average shear stress ƒv ϭ 4.60 ksi at
the support. Web depth-thickness ratio D / t ϭ 60 / (3⁄8) ϭ 160.
Fv ϭ [270 / (D / t)]2 ϭ (270⁄160)2 ϭ 2.85 ksi Ͻ 4.60 ksi
Therefore, transverse intermediate stiffeners are required.
Maximum spacing of stiffeners may not exceed 3 ϫ 60 ϭ 180 in or D[260 / (D / t)]2 ϭ
60(260 / 160)2 ϭ 158 in. Try a stiffener spacing do ϭ 90 in. This provides a depth-spacing
ratio D / do ϭ 60 / 90 ϭ 0.667. From Eq. (11.24d ), for use in Eq. (11.25a), k ϭ 5[1 ϩ
(0.667)2] ϭ 7.22 and ͙k / Fy ϭ ͙7.22 / 36 ϭ 0.445. Since D / t ϭ 160, C in Eq. (10.30a) is
determined by the parameter 160 / 0.445 ϭ 357 Ͼ 237. Consequently, C is given by
Cϭ
45,000k
45,000 ϫ 7.22
ϭ
ϭ 0.352
(D / t)2Fy
1602 ϫ 36
From Eq. (11.25a), the maximum allowable shear for do ϭ 90 in is
ͫ
ͫ
F vЈ ϭ Fv C ϩ
ϭ
0.87(1 Ϫ C )
͙1 ϩ (do / D)2
ͬ
ͬ
36
0.87(1 Ϫ 0.352)
0.352 ϩ
ϭ 7.98 ksi Ͼ 4.60 ksi
3
͙1 ϩ (90 / 60)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 60 ϭ 90 in and the spacing for which the allowable shear
stress in the end panel does not exceed
Fv ϭ CFy / 3 ϭ 0.352 ϫ 36 / 3 ϭ 4.22 ksi Ͻ 4.60 ksi
Therefore, closer spacing is needed near the supports. Try do ϭ 45 in, for which k ϭ 13.89,
C ϭ 0.674, and Fv ϭ CFy / 3 ϭ 0.674 ϫ 36 / 3 ϭ 8.09 ksi Ͼ 4.60 ksi. Therefore, 45-in
spacing will be used near the supports and 90-in spacing for the rest of the girder (Fig.
12.60c).
Where required, a single, vertical plate stiffener of Grade 36 steel will be welded inside
each box girder to each web. (Longitudinal stiffeners are not required, since the 3⁄8-in web
thickness exceeds D͙ƒb / 727 ϭ 60͙19.1 / 727 ϭ 0.362 in.) Width of transverse stiffeners
should be at least
12.138
SECTION TWELVE
2ϩ
D
60
ϭ2ϩ
ϭ 4 in
30
30
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
ͩͪ
60
90
2
Ϫ 2 ϭ Ϫ0.89—Use 0.5
I ϭ 90(3⁄8)30.5 ϭ 2.37 in4
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. Use a bearing diaphragm. See Art 12.13.
Intermediate Cross Frames. Cross frames should be provided at the floorbeams (Fig.
12.58) to maintain the cross-sectional shape of the box girders.
Longitudinal Splice of Deck and Box Girder. The 3⁄8-in deck plate is to be attached to
the 1⁄2-in top flange of the box girder with A325 7⁄8-in-dia, high-strength bolts in slip-critical
connections with Class A surfaces (Fig. 12.60a). With an allowable stress of 15.5 ksi, each
bolt has a capacity of 9.3 kips. The bolts must be capable of resisting the horizontal shear
between the top flange and the deck plate. For determination of the pitch of the bolts along
the longitudinal splice (Fig. 120.60b), the statical moment Q of the deck-plate area, including
the ribs about the neutral axis of the girder is needed.
For the girder section at midspan,
Q ϭ 67.5 ϫ 20.13 ϩ 65.87 ϫ 12.89 ϭ 1,360 ϩ 850 ϭ 2,210 in3
Also, for this section, Q / I ϭ 2,210 / 152,900 ϭ 0.0145. For the section near the supports,
Q ϭ 67.5 ϫ 17.19 ϩ 65.87 ϫ 9.95 ϭ 1,162 ϩ 656 ϭ 1,818 in3
And for this section, Q / I ϭ 1,818 / 121,400 ϭ 0.0150.
Multiplication of Q / I by the shear V, kips, yields the shear, kips per in, to be resisted by
the bolts. The pitch required then is found by dividing VQ / I into the bolt capacity 9.3. The
shears V can be obtained from Table 12.69. The computed pitch is shown by the dash line
in Fig. 12.60d, while the pitch to be used is indicated by the solid line. For sealing, the
maximum pitch, in, is
4 ϩ 4t ϭ 4 ϩ 4 ϫ 3⁄8 ϭ 51⁄2 in
(See also ‘‘F1oorbeam Connections to Girders’’ in Art. 12.15.3.)
Camber. The girders should be cambered to compensate for dead-load deflection. In computation of this deflection, an average moment of inertia may be used, in this case, 140,000
in4. The deflection can be computed from Eq. (12.5) with wDL ϭ 1.8 kips per ft.
12.139
BEAM AND GIRDER BRIDGES
␦ϭ
22.5 ϫ 1.8(120)4
ϭ 2.1 in
29,000 ϫ 140,000
Live-Load Deflection. Maximum live-load deflection should be checked to ensure that it
does not exceed 12L / 800. This deflection may be computed with acceptable accuracy for
lane loading from
␦ϭ
144ML2
EI
(12.45)
where M ϭ MLL ϩ MI at midspan, ft-kips
L ϭ span, ft
I ϭ average moment of inertia, in4
For MLL ϩ MI ϭ 2,380 ft-kips, from Table 12.68, and an average I ϭ 140,000 in4,
␦ϭ
144 ϫ 2,380(120)2
ϭ 1.21 in
29,000 ϫ 140,000
And the deflection-span ratio is
1.21
1
1
ϭ
Ͻ
120 ϫ 12 1,190 800
Thus, the live-load deflection is acceptable.
Other Details. These may be treated in the same way as for I-shaped plate girders.
12.15.2
Design of Ribs
Select Grade 50W steel for the ribs and deck plate for atmospheric corrosion resistance. This
steel has a yield strength of 50 ksi in thicknesses up to 4 in. The trapezoidal section chosen
for the ribs is shown in Fig. 12.61.
Stresses in the ribs, and in the deck plate as part of the ribs (member II). may be determined by orthotropic-plate theory (Art. 4.12). In the first stage of the calculations, the ribs
are assumed to be continuous and supported by rigid floor-beams. In the second stage,
midspan bending moments are increased, because the floorbeams actually are flexible. The
decrease in rib moments at the supports, however, is ignored.
Rib Dead Load. Each rib supports its own weight and the weights of framing details, a
24-in-wide strip of deck plate. and a 2-in-thick wearing course (Table 12.72). Because ribs
in adjoining spans are subjected to the same uniform loading, each rib may be treated as a
fixed-end beam.
TABLE 12.72 Dead Load, kips per ft, on Rib
Rib: 30.74 ϫ 0.00106
ϭ 0.0327
Deck: 24 ϫ 0.00128
ϭ 0.0306
Wearing course: 0.150 ϫ 2 ϫ 2⁄12 ϭ 0.0500
Details:
0.0032
DL per rib:
0.1165, say 0.12
12.140
SECTION TWELVE
Dead-load moment at the support is
0.12(15)212
MDL ϭ Ϫ
ϭ Ϫ27 in-kips
12
And at midspan,
MDL ϭ
0.12(15)212
ϭ 14 in-kips
24
Shear will not be computed because, with two webs per rib, shear stresses are negligible.
Effective Width of Rib Top Flange. Before live-load moments can be determined for the
ribs, certain properties of member II must be computed:
Effective span se ϭ 0.7s ϭ 0.7 ϫ 15 ϫ 12 ϭ 126 in [Eq. (4.166)]
For determination of the effective width of the deck plate as the top flange of member
II, take the effective width ae at top of rib equal to the actual width a, and the effective rib
spacing ee equal to the actual spacing e between ribs. Then, ae / se ϭ 12⁄126 ϭ 0.1 and ee / se ϭ
12
⁄126 ϭ 0.1. From Table 4.6, ao / ae ϭ 1.08 and eo / ee ϭ 1.08. Hence, the effective width of
the top flange is
aoЈ ϩ eoЈ ϭ 1.08 ϫ 12 ϩ 1.08 ϫ 12 ϭ 26 in
The resulting rib cross section is shown in Fig. 12.61b. The neutral axis can be located
by taking moments of the areas of rib and deck plate about the top of rib. This computation
and those for moment of inertia and section moduli are given in Table 12.73b.
The effective top-flange width for computing the relative rigidities of floorbeam and rib
is 1.1 (a ϩ e) ϭ 26.4 in. The section properties given in Table 12.73b will be used, however,
because the effect on stresses, in this case, is negligible.
Slenderness of Rib. From Table 12.73b, the radius of gyration of the rib r ϭ ͙INA / A ϭ
͙376 / 19.16 ϭ 4.43 in, and the slenderness ratio is L / r ϭ 15 ϫ 12 / 4.43 ϭ 41. The yield
strength Fy of the rib steel is 50 ksi. The maximum compressive stress in the deck plate F
is 10.12 ksi (Art. 12.15.1). The maximum permissible slenderness ratio is
L
ϭ 1000
r
1.5 2.7 ϫ 10.12
ϭ 1000 Ί
Ϫ
ϭ 138 Ͼ 41—OK
Ί1.5F Ϫ 2.7F
F
50
50
y
2
y
2
Torsional Rigidity. The basic differential equation for an orthotropic plate with closed ribs
[Eq. (4.178)] contains two parameters, the flexural rigidity Dy in the longitudinal direction
and the torsional rigidity H. The latter may be computed from Eq. (4.181). For that computation, for the trapezoidal rib, by Eq. (4.183),
Kϭ
(12 ϩ 6)2122
ϭ 357.5 in4
(6 ϩ 2 ϫ 12.37) / 0.3125 ϩ 12 / 0.375
With the shearing modulus G ϭ 11,200 ksi,
GK ϭ 11,200 ϫ 357.5 ϭ 4.01 ϫ 106 kip-in2
Also needed for computing H is the reduction factor v. It can be obtained approximately
from Eq. (4.184), with
BEAM AND GIRDER BRIDGES
12.141
TABLE 12.73 Properties of Ribs
(a) Rib without deck plate
Material
A
d
Ad
Ad 2
Io
Two sides 12.37 ϫ 5⁄16
Flange 5.38 ϫ 5⁄16
7.73
1.68
9.41
6.00
11.84
46.4
19.9
66.3
278
236
79
d ϭ 66.3 / 9.41 ϭ 7.05 in
I
357
236
593
Ϫ7.05 ϫ 66.3 ϭ Ϫ468
INA ϭ 125
(b) Rib with 26-in plate flange
Material
A
d
Ad
Rib alone
Top flange 26 ϫ 3⁄8
9.41
9.75
19.16
Ϫ0.1875
Ϫ1.8
Ad 2
66.3
I
593
0
593
Ϫ3.37 ϫ 64.5 ϭ Ϫ217
INA ϭ 376
0.34
64.5
d ϭ 64.5 / 19.16 ϭ 3.37 in
Distance from neutral axis to:
Top of deck plate ϭ 0.375 ϩ 3.37 ϭ 3.75 in
Bottom of rib ϭ 12 Ϫ 3.37 ϭ 8.63 in
Section moduli
Top of deck plate
Bottom of rib
St ϭ 376 / 3.75 ϭ 100 in3
Sb ϭ 376 / 8.63 ϭ 43.6 in3
EI p ϭ
29,000(0.375)3
ϭ 140 kip-in2
10.92
Thus, the reciprocal of the reduction factor may be taken as
1
v
ϭ1ϩ
ͩ ͪ ͫͩ ͪ ͩ
4.01 ϫ 106
122
140
12(12 ϩ 12)2 145.8
2
ϫ
12
12
3
ϩ
ͪͬ
12 Ϫ 6
6
ϩ
12 ϩ 6 12
2
ϭ 6.61
Then, by Eq. (4.181),
Hϭ
1 4.01 ϫ 106
ϭ 12,600 in-kip
2 6.61(12 ϩ 12)
Flexural Rigidity. The other parameter, the flexural rigidity in the longitudinal direction,
for the basic differential equation for the orthotropic plate [Eq. (4.178)] can be obtained
from Eq. (4.180):
Dy ϭ
29,000 ϫ 376
ϭ 454,000 in-kips
12 ϩ 12