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
considered simply supported for dead load, fixed end for live loads.
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
TABLE 12.77 Dead Load on Floorbeam,
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
Maximum dead-load moment occurs at midspan and equals
MDL ϭ
0.925(30)2
ϭ 104 ft-kips
8
Maximum dead-load shear occurs at the supports and equals
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
For this loading, the bending moment at the support is
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
reaction of the loading on the floorbeam is
R ϭ 0.302 ϫ 0.75 ϩ 2 ϫ 0.073 ϫ 0.99 ϫ 0.14 ϭ 0.247
Now, Q1 ϭ 0.302 corresponds to the loading that produced the live-load moments previously
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
sƒ
(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