Tải bản đầy đủ - 0 (trang)
11 Example - LRFD for Floorbeam with Overhang

# 11 Example - LRFD for Floorbeam with Overhang

Tải bản đầy đủ - 0trang

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

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

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:

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

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

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

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

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

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

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

(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

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

␦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.

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

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),

(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

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

Rib alone

Top flange 26 ϫ 3⁄8

9.41

9.75

19.16

Ϫ0.1875

Ϫ1.8

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),

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

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

11 Example - LRFD for Floorbeam with Overhang

Tải bản đầy đủ ngay(0 tr)

×