16 Example - LRFD for Wide-Flange Column in a Multistory Rigid Frame
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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
BEAM AND GIRDER BRIDGES
12.149
TABLE 12.79 Floorbeam Moments of Inertia and Section Moduli
(a) At midspan
Material
A
Deck 95 ϫ 3⁄8
Web 21 ϫ 3⁄8
Bottom flange 10 ϫ 1⁄2
35.6
7.9
5.0
48.5
d
10.69
Ad
Ad 2
381
4,070
Io
I
4,070
290
580
580
4,940
Ϫ6.73 ϫ 327 ϭ Ϫ2,210
INA ϭ 2,730
290
Ϫ10.75
Ϫ54
327
d ϭ 327 / 48.5 ϭ 6.73 in
Distance from neutral axis to:
Top of deck plate ϭ 10.50 ϩ 0.375 Ϫ 6.73 ϭ 4.15 in
Bottom of rib ϭ 10.50 ϩ 0.50 ϩ 6.73 ϭ 17.73 in
Section moduli
Top of deck plate
Bottom of rib
St ϭ 2,730 / 4.15 ϭ 658 in3
Sb ϭ 2,730 / 17.73 ϭ 154 in3
(b) At supports, gross section
Material
A
Deck 95 ϫ 3⁄8
Web 18 ϫ 3⁄8
Bottom flange 10 ϫ 1⁄2
35.6
6.8
5.0
47.4
d
Ad
Ad 2
9.19
327
3,010
Io
I
3,010
180
430
3,620
Ϫ5.93 ϫ 281 ϭ Ϫ1,670
Gross INA ϭ 1,950
180
Ϫ9.25
Ϫ46
430
281
dg ϭ 281 / 47.4 ϭ 5.93 in
Distance from neutral axis to:
Bottom of rib ϭ 9 ϩ 0.50 ϩ 5.93 ϭ 15.43 in
Section modulus, bottom of rib
Sb ϭ 1,950 / 15.43 ϭ 126 in3
(c) At supports, net section
Material
A
Gross section
Top-flange holes
Bottom-flange holes
Web holes
Ϫ10.2
Ϫ1.0
Ϫ2.3
Ad
9.19
Ϫ9.25
Ϫ94
47.4
33.9
dnet ϭ 196 / 33.9 ϭ 5.77 in
d
Ad 2
Io
281
9
196
I
3,620
Ϫ860
Ϫ90
Ϫ120
Ϫ860
Ϫ90
Ϫ120
2,550
Ϫ5.77 ϫ 196 ϭ Ϫ1,130
Net INA ϭ 1,420
12.150
SECTION TWELVE
TABLE 12.79 Floorbeam Moments of Inertia and Section Moduli (Continued )
Distance from neutral axis to:
Top of deck plate ϭ 9 ϩ 0.375 Ϫ 5.77 ϭ 3.61
Section modulus, top of deck plate
St ϭ 1,420 / 3.61 ϭ 392 in3
Floorbeam Shears. For maximum shear, the truck wheels are placed in each design lane
as indicated in Fig. 12.64. 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. Thus, with the floorbeams assumed
acting as rigid supports for the ribs, the wheel load is 16.5 kips, as for maximum floorbeam
moment. (The effects of floorbeam flexibility can be determined as for bending moments.)
This loading produces a simple-beam reaction of 41.8 kips. It also causes end moments
of Ϫ202 and Ϫ86, which induce a reaction of (Ϫ86 ϩ 202) / 30 ϭ 3.9 kips. Hence, the
maximum live-load reaction and shear equal
VLL ϭ 41.8 ϩ 3.9 ϭ 45.7 kips
Shear due to impact is
VI ϭ 0.30 ϫ 45.7 ϭ 13.7 kips
MAXIMUM FLOORBEAM SHEARS,
KIPS
VDL
VLL
VI
Total V
13.9
45.7
13.7
73.3
Allowable shear stress in the web for Grade 50W steel is 17 ksi. Average shear stress in
the web is
ƒv ϭ
73.3
ϭ 10.9 Ͻ 17 ksi
18 ϫ 3⁄8
Transverse stiffeners are not required.
Flange-to-Web Welds. The web will be connected to the deck plate and the bottom flange
by a fillet weld on opposite sides of the web. These welds must resist the horizontal shear
between flange and web. For the weld to the 10 ϫ 1⁄2-in bottom flange, the minimum size
TABLE 12.80 Bending Stresses in Member III
At midspan:
Top of deck plate
Bottom flange
At supports:
Top of deck plate
Bottom flange
ƒb ϭ 221 ϫ 12 / 658 ϭ 4.03 ksi (compression)
ƒb ϭ 221 ϫ 12 / 154 ϭ 17.2 Ͻ 27 ksi (tension)
ƒb ϭ 224 ϫ 12 / 392 ϭ 6.9 ksi (tension)
ƒb ϭ 224 ϫ 12 / 126 ϭ 21.4 Ͻ 27 ksi
BEAM AND GIRDER BRIDGES
12.151
FIGURE 12.64 Positions of truck wheels for maximum shear in floorbeam.
fillet weld permissible with a 1⁄2-in plate, 1⁄4 in, may be used. Shear, however, governs for
the weld to the deck plate.
For computing the shear v, kips per in, between web and deck plate, the total maximum
shear V is 73.3 kips and the moment of inertia of the floorbeam cross section I is 1,950 in.4
The static moment of the deck plate is
Q ϭ 35.6(4.15 Ϫ 0.19) ϭ 141 in3
Hence, the shear to be carried by the welds is
vϭ
VQ 73.3 ϫ 141
ϭ
ϭ 5.30 kips per in
I
1,950
The allowable stress on the weld is 18.9 ksi. So the allowable load per weld is 18.9 ϫ
0.707 ϭ 13.4 kips per in, and for two welds, 26.7 kips per in. Therefore, the weld size
required is 5.30 / 26.7 ϭ 0.20 in. Use 1⁄4-in fillet welds.
Floorbeam Connections to Girders. Since the bottom flange of the floorbeam is in compression, it can be connected to the inner web of each box girder with a splice plate of the
same area. Use a 10 ϫ 1⁄2-in plate, shop-welded to the girder and field-bolted to the floorbeam. With A325 7⁄8-in-dia. high-strength bolts in slip-critical connections with Class A
surfaces, the allowable load per bolt is 9.3 kips. If the capacity of the 10 ϫ 1⁄2-in flange is
developed at the allowable stress of 27 ksi, the number of bolts required in the connection
is 27 ϫ 5 / 9.3 ϭ 15. Use 16.
The deck plate is spliced to the girder with 7⁄8-in-dia. high-strength bolts. To meet girder
requirements, the pitch may vary from 3 to 51⁄2 in (Fig. 12.60d ). But the bolts also must
transmit the tensile forces from the deck plate to the girder when the plate acts as the top
flange of member III. The shear in the bolts from the girder compression is perpendicular
to the shear from the floorbeam tension. Hence, the allowable load per bolt decreases from
9.3 to 9.3 ϫ 0.707 ϭ 6.6 kips. With an average tensile stress in the deck plate of 6.2 ksi,
and a net area after deduction of holes of 35.6 Ϫ 10.2 ϭ 25.4 in2, the plate carries a tensile
force of 25.4 ϫ 6.2 ϭ 158 kips. Thus, to transmit this force, 158 / 6.6 ϭ 24 bolts are needed.
If a pitch of 3 in is used in the 95-in effective width of the plate, 31 bolts are provided. Use
a 3-in pitch for 4 ft on each side of every floorbeam.
The web connection to the girder must transmit both vertical shear, V ϭ 73.3 kips, and
bending moment. The latter can be computed from the stress diagram for the cross section
(Fig. 12.65a).
12.152
SECTION TWELVE
FIGURE 12.65 (a) Bending stresses in floorbeam at supports. (b)
Bolted web connection of floorbeam to girder.
M ϭ 1⁄2 ϫ 3⁄8(4.23 ϫ 3.07 ϫ 2.05 ϩ 20.7 ϫ 14.93 ϫ 9.95) ϭ 581 in-kips
Assume that the connection will be made with two rows of six bolts each, on each side
of the connection centerline (Fig. 12.65b). The polar moment of inertia of these bolts can
be computed as the sum of the moments of inertia about the x (horizontal) and y (vertical)
axes.
Ix ϭ 4(1.52 ϩ 4.52 ϩ 7.52) ϭ 315
Iy ϭ 12(1.5)2
ϭ 27
ϭ
J ϭ 342
Load per bolt due to shear is
Pv ϭ
73.3
ϭ 6.1 kips
12
Load on the outermost bolt due to moment is
Pm ϭ
581 ϫ 7.63
ϭ 12.95 kips
342
The vertical component of this load is
Pv ϭ
12.95 ϫ 1.5
ϭ 2.5 kips
7.63
and the horizontal component is
Ph ϭ
12.95 ϫ 7.5
ϭ 12.7 kips
7.63
The total load on the outermost bolt is the resultant
P ϭ ͙(6.1 ϩ 2.5)2 ϩ 12.72 ϭ 15.3 Ͻ 2 ϫ 9.3
For the web connection plates, try two plates 171⁄2 ϫ 5⁄16 in. They have a net moment of
inertia
BEAM AND GIRDER BRIDGES
Iϭ2
12.153
(5⁄16)17.53
Ϫ 50 ϭ 228 in4
12
To transmit the 581-in-kip moment in the web, they carry a bending stress of
ƒb ϭ
581 ϫ 8.75
ϭ 22.3 Ͻ 27 ksi
228
The assumed plates are therefore satisfactory if Grade 50W steel is used.
12.15.4
Design of Deck Plate
The deck plate is to be made of Grade 50W steel. This steel has a yield strength Fy ϭ
50 ksi for the 3⁄8-in deck thickness.
Stresses. Bending stresses in the deck plate as the top flange of ribs (member II), floorbeams (member III), and girders (member IV) are relatively low. Combining the stresses of
members II and IV yields 4.75 ϩ 9.73 ϭ 14.48 Ͻ 1.25 ϫ 27 ksi.
Deflection. The thickness of deck plate to limit deflection to 1⁄300 of the rib spacing can
be computed from Eq. 11.72. For a 16-kip wheel, assumed distributed over an area of
26 ϫ 12 ϭ 312 in2, the pressure, including 30% impact, is
p ϭ 1.3 ϫ 16 / 312 ϭ 0.0667 ksi
Required thickness with rib spacing a ϭ e ϭ 12 in is
t ϭ 0.07 ϫ 12(0.0667)1/3 ϭ 0.341 Ͻ 0.375 in
The 3⁄8-in deckplate is satisfactory.
12.16
CONTINUOUS-BEAM BRIDGES
Articles 12.1 and 12.3 recommended use of continuity for multispan bridges. Advantages
over simply supported spans include less weight, greater stiffness, smaller deflections, and
fewer bearings and expansion joints. Disadvantages include more complex fabrication and
erection and often the costs of additional field splices.
Continuous structures also offer greater overload capacity. Failure does not necessarily
occur if overloads cause yielding at one point in a span or at supports. Bending moments
are redistributed to parts of the span that are not overstressed. This usually can take place
in bridges because maximum positive moments and maximum negative moments occur with
loads in different positions on the spans. Also, because of moment redistribution due to
yielding, small settlements of supports have no significant effects on the ultimate strength
of continuous spans. If, however, foundation conditions are such that large settlements could
occur, simple-span construction is advisable.
While analysis of continuous structures is more complicated than that for simple spans,
design differs in only a few respects. In simple spans, maximum dead-load moment occurs
at midspan and is positive. In continuous spans, however, maximum dead-load moment
occurs at the supports and is negative. Decreasing rapidly with distance from the support,
the negative moment becomes zero at an inflection point near a quarter point of the span.
Between the two dead-load inflection points in each interior span, the dead-load moment is
positive, with a maximum about half the negative moment at the supports.
12.154
SECTION TWELVE
As for simple spans, live loads are placed on continuous spans to create maximum stresses
at each section. Whereas in simple spans maximum moments at each section are always
positive, maximum live-load moments at a section in continuous spans may be positive or
negative. Because of the stress reversal, fatigue stresses should be investigated, especially in
the region of dead-load inflection points. At interior supports, however, design usually is
governed by the maximum negative moment, and in the midspan region, by maximum positive moment. The sum of the dead-load and live-load moments usually is greater at supports
than at midspan. Usually also, this maximum is considerably less than the maximum moment
in a simple beam with the same span. Furthermore, the maximum negative moment decreases
rapidly with distance from the support.
The impact fraction for continuous spans depends on the length L, ft, of the portion of
the span loaded to produce maximum stress. For positive moment, use the actual loaded
length. For negative moment, use the average of two adjacent loaded spans.
Ends of continuous beams usually are simply supported. Consequently, moments in threespan and four-span continuous beams are significantly affected by the relative lengths of
interior and exterior spans. Selection of a suitable span ratio can nearly equalize maximum
positive moments in those spans and thus permit duplication of sections. The most advantageous ratio, however, depends on the ratio of dead load to live load, which, in turn, is a
function of span length. Approximately, the most advantageous ratio for length of interior
to exterior span is 1.33 for interior spans less than about 60 ft, 1.30 for interior spans between
about 60 to 110 ft, and about 1.25 for longer spans.
When composite construction is advantageous (see Art. 12.1), it may be used either in
the positive-moment regions or throughout a continuous span. Design of a section in the
positive-moment region in either case is similar to that for a simple beam. Design of a
section in the negative-moment regions differs in that the concrete slab, as part of the top
flange, cannot resist tension. Consequently, steel reinforcement must be added to the slab to
resist the tensile stresses imposed by composite action.
Additionally, for continuous spans with a cast-in-place concrete deck, the sequence of
concrete pavement is an important design consideration. Bending moments, bracing requirements, and uplift forces must be carefully evaluated.
12.17
ALLOWABLE-STRESS DESIGN OF BRIDGE WITH
CONTINUOUS, COMPOSITE STRINGERS
The structure is a two-lane highway bridge with overall length of 298 ft. Site conditions
require a central span of 125 ft. End spans, therefore, are each 86.5 ft (Fig. 12.66a). The
typical cross section in Fig. 12.66b shows a 30-ft roadway, flanked on one side by a 21-inwide barrier curb and on the other by a 6-ft-wide sidewalk. The deck is supported by six
rolled-beam, continuous stringers of Grade 36 steel. Concrete to be used for the deck is
Class A, with 28-day strength ƒЈc ϭ 4,000 psi and allowable compressive stress ƒc ϭ 1,600
psi. Loading is HS20-44. Appropriate design criteria given in Sec. 11 will be used for this
structure.
Concrete Slab. The slab is designed to span transversely between stringers, as in Art. 12.2.
A 9-in-thick, two-course slab will be used. No provision will be made for a future 2-in
wearing course.
Stringer Loads. Assume that the stringers will not be shored during casting of the concrete
slab. Then, the dead load on each stringer includes the weight of a strip of concrete slab
plus the weights of steel shape, cover plates, and framing details. This dead load will be
referred to as DL and is summarized in Table 12.81.