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14 Example - LRFD for Composite Beam with Concentrated Loads and End Moments

# 14 Example - LRFD for Composite Beam with Concentrated Loads and End Moments

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BEAM AND GIRDER BRIDGES

12.143

Q3 ϭ Q5 ϭ 0

For n ϭ 7,

Q 7 ϭ 0.178(Ϫ0.588)(0.866 ϩ 0.866) ϭ Ϫ0.181

For this example, only two terms of the Fourier series will be used. In general, however,

additional terms are required for accuracy, because the computations require subtraction of

nearly equal numbers.

Parameters for Influence Coefficients. The ribs at this stage are considered continuous,

with span s ϭ 180 in, over rigid supports. For calculation of moment-influence coefficients,

the parameter ␣n given by Eq. (4.187) is needed.

␣n ϭ

n␲

l

n␲

0.166͙2 ϭ 0.00205n

ΊDH ͙2 ϭ 360

y

Values of functions of ␣n for n ϭ 1 and n ϭ 7 needed for moment-influence coefficients

are tabulated in Table 12.74. Also required is the carry-over factor ␬ n given by Eq. (4.188).

For this calculation, ␤n and kn are computed in Table 12.74a.

␬1 ϭ ͙1.8232 Ϫ 1 Ϫ 1.823 ϭ Ϫ0.30

␬7 ϭ ͙2.662 Ϫ 1 Ϫ 2.66 ϭ Ϫ0.20

Moment-Influence Coefficients. Substitution of the computed values in Eq. (4.189) yields

the influence coefficients for bending moment at an unyielding support:

mO1 ϭ Ϫ2.500(3.61 sinh 0.00205y Ϫ cosh 0.00205y Ϫ 0.00722y ϩ 1)

mO7 ϭ Ϫ61.5(1.04 sinh 0.01435y Ϫ cosh 0.01435y Ϫ 0.00667y ϩ 1)

Substitution of the computed values in Eq. (4.190) gives the influence coefficients for

bending moment at midspan when supports are rigid:

mC1 ϭ 240 sinh 0.00205y Ϫ 858(0.1730 sinh 0.00205y Ϫ cosh 0.00205y ϩ 1)

mC7 ϭ 17.8 sinh 0.01435y Ϫ 12.6(0.859 sinh 0.01435y Ϫ cosh 0.01435y ϩ 1)

Rib Live-Load Moments, Rigid Floorbeams. For maximum moment in a rib at a support

with H20 loading, place the 16-kip truck wheels 7 ft from the support on one span ( y ϭ

84) and the 4-kip wheels 7 ft from the same support on the adjoining span (y ϭ 84). With

values tabulated in Table 12.74b, the moment-influence coefficients become, for y ϭ 84,

mO1 ϭ Ϫ8.25 and mO7 ϭ Ϫ12.35.

The bending moments at the support will be computed at the centerline of the bridge,

x ϭ 180 in. Then, by Eq. (4.169),

Q nx ϭ Q 1 Ϫ Q 7 ϩ ⅐ ⅐ ⅐ ϭ 0.293 ϩ 0.181 ϩ ⅐ ⅐ ⅐

By Eq. (4.191), the moment at the support due to the 16-kip wheel at y ϭ 84 is

MO ϭ Ϫ24(8.25 ϫ 0.293 ϩ 12.35 ϫ 0.181) ϭ Ϫ112 in-kips

Because of the 4-kip wheel in the adjoining span, also at y ϭ 84,

MO ϭ Ϫ112 ϫ 4 / 16 ϭ Ϫ28 in-kips

Thus, the live-load moment at the support is MLL ϭ Ϫ112 Ϫ 28 ϭ Ϫ140 in-kips.

12.144

SECTION TWELVE

TABLE 12.74 Values of Functions for Computing Influence Coefficients

(a) Functions of rib span s

sinh

cosh

tanh

coth

␣1

␣1s

␣1s

␣1s

␣1s

␣1s

ϭ

ϭ

ϭ

ϭ

ϭ

ϭ

0.00205

␣1s / 2 ϭ 0.185

0.370

0.379

cosh ␣1s / 2 ϭ 1.017

1.069

tanh ␣1s / 2 ϭ 0.1730

0.354

2.82

0.379 Ϫ 0.370

␤1 ϭ

ϭ 0.0238

0.379

0.370 ϫ 2.82 Ϫ 1

k1 ϭ

ϭ 1.823

0.0238

sinh

cosh

tanh

coth

␣7

␣7s

␣7s

␣7s

␣7s

␣7s

ϭ

ϭ

ϭ

ϭ

ϭ

ϭ

0.1435

␣7s / 2 ϭ 1.29

2.59

6.63

cosh ␣7s / 2 ϭ 1.954

6.70

tanh ␣7s / 2 ϭ 0.859

0.989

1.011

6.63 Ϫ 2.59

␤7 ϭ

ϭ 0.608

6.63

2.59 ϫ 1.011 Ϫ 1

k1 ϭ

ϭ 2.66

0.608

(b) Functions for y ϭ 84

0.00205 ϫ 84 ϭ 0.172

sinh 0.172 ϭ 0.1728

cosh 0.172 ϭ 1.014

0.01435 ϫ 84 ϭ 1.206

sinh 1.206 ϭ 1.520

cosh 1.206 ϭ 1.820

(c) Functions for y ϭ 90

0.00205 ϫ 90 ϭ 0.185

sinh 0.185 ϭ 0.1861

cosh 0.185 ϭ 1.017

0.01435 ϫ 90 ϭ 1.292

sinh 1.292 ϭ 1.683

cosh 1.292 ϭ 1.957

(d ) Functions for y ϭ 78

0.00205 ϫ 78 ϭ 0.160

sinh 0.160 ϭ 0.1607

cosh 0.160 ϭ 1.013

0.01435 ϫ 78 ϭ 1.12

sinh 1.12 ϭ 1.369

cosh 1.12 ϭ 1.696

For maximum moment at midspan, place the 16-kip wheels there ( y ϭ 90).

The 4-kip wheels will be on the adjoining span 6.5 ft from the support ( y ϭ 78). The midspan

moments also will be computed for x ϭ 180. The moment-influence coefficients become,

with values from Table 12.74 for y ϭ 90, mC1 ϭ 31.8 and mC7 ϭ 23.9. Then, for the 16-kip

wheels, by Eq. (4.191),

MC ϭ 24(31.8 ϫ 0.293 ϩ 23.9 ϫ 0.181) ϭ 328 in-kips

The effect of the 4-kip wheels on the midspan moment is found in several steps. First,

the moment MO at the support is computed. This requires determination of the momentinfluence coefficients for y ϭ 78. Then, the carry-over factors are used to calculate the

midspan moment:

MC ϭ

MO(1 Ϫ ␬ n)

2

(12.46)

With values from Table 12.74d, the moment-influence coefficients become mO1 ϭ Ϫ5.00 and

mO7 ϭ Ϫ12.6. By Eq. (4.191), for the 4-kip wheels,

BEAM AND GIRDER BRIDGES

12.145

MO1 ϭ Ϫ4⁄16 ϫ 24 ϫ 5.00 ϫ 0.293 ϭ Ϫ8.8 in-kips

MO7 ϭ Ϫ4⁄16 ϫ 24 ϫ 12.6 ϫ 0.181 ϭ Ϫ13.7 in-kips

With Eq. (12.45), the midspan moment due to the 4-kip wheels is found to be

MC ϭ Ϫ8.8

1 ϩ 0.300

1 ϩ 0.200

Ϫ 13.7

ϭ Ϫ14 in-kips

2

2

Thus, the live-load moment at midspan with rigid supports is

MLL ϭ 328 Ϫ 14 ϭ 314 in-kips

Rib Live-Load Moments, Flexible Floorbeams. Because the floorbeams actually are not

rigid and deflect under live loading, end moments in ribs are less than they would be with

rigid supports, and midspan moments are larger. The decrease in support moments in this

case will be ignored. The increase in midspan moment, however, will be computed from Eq.

(4.176).

From Table 4.4, the influence coefficient for reaction due to a load at midspan is

0.601. The flexibility coefficient has previously been computed to be ␥ ϭ 0.0477. From

Table 4.7, the influence coefficient for midspan moment in a continuous beam on elastic

supports, with load at support 0, is 0.027. Taking into account only the effects of the two supports on either side of midspan, the influence coefficient for change in midspan moment is

2(0.601 ϫ 0.027) ϭ 0.0324. By Eq. (4.176), the change in midspan moment is

⌬ MC ϭ 0.293 ϫ 180 ϫ 24 ϫ 0.0324 ϭ 41 in-kips

Therefore, the live-load moment at midspan with flexible supports is

MLL ϭ 314 ϩ 41 ϭ 355 in-kips

Impact. For the 15-ft rib span, impact should be taken as 30% of live-load stresses.

At midspan, MI ϭ 0.30 ϫ 355 ϭ 106 in-kips.

At supports, MI ϭ 0.30(Ϫ140) ϭ Ϫ42 in-kips.

Maximum Rib Moments. The design moments previously calculated for member II are

summarized in Table 12.75.

In a similar way, stress reversals can be computed for investigation of fatigue stresses.

Rib Stresses. Section properties for determination of member II stresses are given in Table

12.74b. Computations for maximum rib stresses at midspan and supports are given in Table

12.76. The compressive stress at the bottom of member II is augmented by the compressive

stress induced when the rib acts as part of the top flange of member IV. This stress was

TABLE 12.75 Rib Moments, in-kips

MDL

Midspan

Supports

MLL

MI

Total M

14

355

106

475

Ϫ27

Ϫ140

Ϫ42

Ϫ209

12.146

SECTION TWELVE

TABLE 12.76 Bending Stresses in Ribs

At midspan:

Top of deck plate

Bottom of rib

At supports:

Bottom of rib

ƒb ϭ 475 / 100 ϭ 4.75 ksi (compression)

ƒb ϭ 475 / 43.6 ϭ 10.9 Ͻ 27 ksi (tension)

ƒb ϭ 209 / 43.6 ϭ 4.80 ksi (compression)

previously computed to be 4.26 ksi. Hence, the total compressive stress at the bottom of the

rib is

ƒb ϭ 4.80 ϩ 4.26 ϭ 9.06 Ͻ 1.25 ϫ 27 ksi

Rib Stability. Closed ribs of the dimensions usually used have high resistance to buckling.

In this case, therefore, there is no need to check the stability of the ribs, especially since

compressive bending stresses are low.

12.15.3

Design of Floorbeam with Orthotropic-Plate Flange

Select Grade 50W steel for the floorbeams. This steel provides atmospheric corrosion resistance and has a yield strength Fy ϭ 50 ksi in thicknesses up to 4 in.

The floorbeams are tapered. Web depth ranges from 21 in at midspan to 18 in at the boxgirder supports (Fig. 12.63). The span is 30 ft. Spacing is 15 ft c to c. The floorbeams are

Floorbeam Dead Load. Each beam supports its own weight and the weights of framing

details, a 15-ft-wide strip of deck, including 14 ribs, and a 2-in-thick wearing course (Table

12.77).

FIGURE 12.63 Floorbeam with orthotropic-plate top flange.

BEAM AND GIRDER BRIDGES

12.147

kips per ft, with Orthotropic-Plate Flange

Floorbeam

Deck plate: 0.153 ϫ 15

Ribs: 14 ϫ 0.036 ϫ 15⁄30

Wearing course: 0.150 ϫ 15 ϫ 2⁄12

Details:

DL per beam:

ϭ 0.042

ϭ 0.230

ϭ 0.252

ϭ 0.375

0.026

0.925

MDL ϭ

0.925(30)2

ϭ 104 ft-kips

8

VDL ϭ

0.925 ϫ 30

ϭ 13.9 kips

2

Floorbeam Live-Load Moments. Bending moments are computed in two stages. In the

first, the floorbeams are assumed to act as rigid supports for the continuous ribs. In the

second stage, the changes in moments due to flexibility of the floorbeams are calculated.

For maximum moments in a floorbeam in the first stage, the H20 truck live loads are

placed in each of the two design lanes as indicated in Fig. 12.62. 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. From Table 4.4, the reaction influence coefficient for this location is estimated to be

0.06. Hence, the wheel loads on the floorbeam equal

P ϭ 16 ϫ 1.00 ϩ 2 ϫ 4 ϫ 0.06 ϭ 16.5 kips

16.5

MLL ϭ Ϫ 2 [7(23)2 ϩ 13(17)2 ϩ 17(13)2 ϩ 23(7)2] ϭ Ϫ210 ft-kips

30

and the midspan moment is

MLL ϭ Ϫ210 ϩ 2 ϫ 16.5 ϫ 13 Ϫ 16.5 ϫ 6 ϭ 110 ft-kips

The effects on these moments of floorbeam flexibility may be approximated in the following way, using the first term of the Fourier series for the loading in Fig. 12.62: For the

16-kip wheels, Q1 ϭ 0.293. Hence, Q1 ϭ 0.293 ϫ 16.5 / 16 ϭ 0.302 for 16.5-kip wheels,

and for 4-kip wheels, Q1 ϭ 0.293 ϫ 4 / 16 ϭ 0.073. From Table 4.7, for ␥ ϭ 0.0477, the

reaction influence coefficient for unit load at the support is 0.75, and for unit load at an

adjacent support, 0.14. From Table 4.4, the reaction influence coefficient for the 4-kip wheels

1 ft from the adjacent support is 0.99. Thus, for the first term of the Fourier series, the

R ϭ 0.302 ϫ 0.75 ϩ 2 ϫ 0.073 ϫ 0.99 ϫ 0.14 ϭ 0.247

calculated on the assumption that the floorbeams were rigid. Also, Q1 ϭ 0.247 corresponds

to loading with the same distribution on the floorbeam. Therefore, the bending moments can

12.148

SECTION TWELVE

be found by proportion from those previously calculated. Thus, the moment at the support

is

210 ϫ 0.247

MLL ϭ Ϫ

ϭ Ϫ172 ft-kips

0.0302

and the moment at midspan is

MLL ϭ

110 ϫ 0.247

ϭ 90 ft-kips

0.302

Impact. For a 30-ft span, impact is taken as 30%.

At midspan, MI ϭ 0.30 ϫ 90 ϭ 27 ft-kips.

At supports, MI ϭ 0.30(Ϫ172) ϭ Ϫ52 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 so of the

deck plate is assumed to act as the top flange of member III. For determination of so , the

effective spacing of floorbeams sƒ is taken equal to the actual spacing, 180 in. The effective

span le , with the floorbeam ends considered fixed, is taken as 0.7 ϫ 30 ϫ 12 ϭ 252 in.

Hence,

sƒ 180

ϭ

ϭ 0.715

le

252

From Table 4.6 for this ratio,

so

ϭ 0.53, and so ϭ 0.53 ϫ 180 ϭ 95 in

(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.

TABLE 12.78 Moments, ft-kips, in Floorbeam

with Orthotropic-Plate Flange

Midspan

Supports

MDL

MLL

MI

Total M

104

0

90

27

221

Ϫ712

Ϫ52

Ϫ224

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