Two-Span LRFD Two-Span LRFD Design Example Design Example Karl Barth and Jennifer Karl Barth and Jennifer Righman Righman West Virginia University West Virginia University
Two-Span LRFD Two-Span LRFD Design ExampleDesign Example
Karl Barth and Jennifer Karl Barth and Jennifer RighmanRighman
West Virginia UniversityWest Virginia University
Objective
The primary focus of this example is to demonstrate the use of Appendix A and Appendix B
for a two-span continuous structure
Appendix A Overview Accounts for the ability of compact
and non-compact sections to resist moments greater than My
Economy gained by Appendix A provisions increases with decreasing web slenderness
Effects of St. Venant torsion are incorporated
Appendix B Overview Traditional AASHTO specifications have
permitted up to 10% of the maximum pier section bending moment to be redistributed to positive bending regions
Appendix B provisions explicitly compute the level of redistribution based on an effective plastic moment concept for sections meeting prescribed geometric criteria
Design Information
Design Information
Framing Plan
Design Notes 2004 AASHTO LRFD Specifications, 3rd
Edition Structural steel: ASTM A709, Grade 50W Normal weight concrete (145 pcf) with
fc’=4 ksi Fyr = 60 ksi for reinforcing steel Operational importance, redundancy,
and ductility factors = 1.0
Design Loads – DC1
DC1 loads are equally distributed to all girders Slab =0.983 k/ft Haunch (average wt/length) =0.017 k/ft Overhang taper =0.019 k/ft Girder (average wt/length, varies) =0.200 k/ft Cross-frames and misc. steel =0.015 k/ft Stay-in-place forms =0.101 k/ft =1.335 k/ft
Design Loads – DC2 and DW DC2
Barrier weight = 520 lb/ft Weight/girder = (0.520)x(2)/(4) = 0.260
k/ft
DW Future wearing surface = 25 psf DW = (0.025 ksf)x(34 ft)/4 = 0.213 k/ft
Design Loads – WS and WL WS
Wind forces are calculated assuming bridge is located 30’ above water in open country
Wind on upper half of girder, deck, and barrier assumed to be resisted by diaphragm action of the deck
WS = 0.081 k/ft (on bottom flange) WL
Assumed to be transmitted by diaphragm action
WL is neglected
Design Loads – Live Load Controlling case of:
Truck + Lane Tandem + Lane 0.9 (Double Truck + Lane) (in negative
bending)
Impact factors used for all vehicular live loads (excluding lane load) I=1.15 for fatigue limit state I=1.33 for all other limit states
Design Loads – Live Load Live load effects are approximated
using distribution factors Interior girder
AASHTO empirical equations are used Exterior girder
AASHTO empirical equation correction factor
Lever rule Special analysis
Interior Girder Distribution Factors Moment
Varies with girder dimensions due to Kg term
One design lane
Two or more design lanes
0.523(8)(90)(12.0)
(702025)9010
14100.06
tL12.0K
LS
14S0.06
0.1
3
0.30.40.1
3s
g0.30.4
0.756(8)(90)(12.0)
(702025)9010
9.5100.075
tL12.0K
LS
9.5S0.075
0.1
3
0.20.60.1
3s
g0.20.6
0007000004002 ,to,eAInK gg
Interior Girder Distribution Factors Shear
One design lane
Two or more design lanes (CONTROLS)
0.7600.250.1036.0
0.25S36.0
0.952
0.20.2
3510
12102.0
35S
12S2.0
Exterior Girder Distribution Factors
AASHTO exterior girder correction factor
Moment
Shear
Empirical formulas for exterior girder will not control
interiorgeg
1.00.990.
..
d.e e 192770
19770
1.00.800.d.e e 10260
1060
Exterior Girder Distribution Factor Lever Rule – One Design Lane
84.02.17.0DF
MPF10
6105.05.0DF
Exterior Girder Distribution Factor
Special Analysis
One design lane
Two or more design lanes
B
L
N
NEXT
B
L
x
ex
NNDF 2
73202151521215
41
22 ..)(
))((MPFDF
0.860
01515201215
42
22 .)())((MPFDF Controls for
Moment
Distribution Factors for Fatigue Based on one design lane No multiple presence factor
applied Maximum one lane distribution
factor results from the lever rule, i.e., EXTERIOR GIRDER CONTROLS
DF = 0.70
Unfactored Design Moments
Limit States All applicable limits states for steel
structures were considered Strength
Strength I controls in this example Strength I = 1.25DC + 1.5DW + 1.75(LL+I) Strength III = 1.25DC + 1.5DW + 1.4WS Strength IV = 1.5(DC + DW) Strength V = 1.25DC + 1.5DW + 1.35(LL+I) + 0.4WS
Service Service II = 1.0(DC + DW) + 1.3(LL+I)
Fatigue = 0.75(LL+I)
6.10 Provisions Addressed Cross section proportion limits Constructibility Serviceability Fatigue Strength
Appendix A Design
63’ 63’54’
12 x 3/4 16 X 1-1/4 12 x 3/4
16 x 1-1/2 16 x 2-1/2 16 x 1-1/2
36 x 7/16 36 x 1/2 36 X 7/16
63’ 63’54’
12 x 3/4 16 x 1-1/4 12 x 3/4
16 x 1-1/2 16 x 2-1/2 16 x 1-1/2
36 x 7/16 36 x 1/2 36 x 7/16
Cross Section Proportion Limits
150tD
w
15082167
36tD
w
0.12t2
b
f
f 0.12875.02
12t2
b
f
f
wf t1.1t 55.0)5.0(1.175.0t f
10II
1.0yt
yc
1021.0165.112112431211.0 3
3
6Db f 6
636
6D12b f
Constructibility For discretely braced compression
flanges
Fnc may be computed using Appendix A which accounts for increased torsional resistance
For discretely braced tension flanges and continuously braced flanges
ksi50500.10.1FRff ychfbu l
ksi 49.8 varies,Fff ncfbu l31
ksi50500.10.1FRff yfhfbu l
Constructibility - Loads Vertical DC1 loads
are determined considering deck casting sequence
Lateral flange bending stresses are induced by the overhang form brackets Construction dead
and live loads considered
Constructibility Check Stresses in compression flange of
positive bending section control the allowable cross-frame spacing Strength I
Strength IV
ksi50ksi8.4697.1947.2125.1ffbu l
ksi50ksi3.4613.1447.215.1ffbu l
Service Limit State For top flange
For bottom flange
Bottom flange in positive bending (controls)
ksi5.47500.195.0FR95.0f yhf
ksi5.47500.195.0FR95.02ff yhf l
ksi5.47ksi1.332012
121916153.1
1131111135
843692
2fff
l
Fatigue Limit State Fatigue requirements significantly impact
the design of the positive bending region Bolted stiffener to flange connections
employed at locations of maximum stress range, i.e., cross-frames at midspan
Bolted connections / Category B details
Welded connections / Category C’ details ksi0.8ksi36.6F max
ksi0.6ksi92.5F max
Fatigue Limit State (cont.) Use of bolted cross-frame connections
requires that net section fracture requirements are satisfied
Assuming one 7/8” diameter bolt hole is used:
ytug
nt FF
AA84.0f
2n in5.22)5.1)(8
18
7()5.1(16A
2g in0.24)5.1(16A
50Ff51650.245.2284.0f yttt
OK506.44ft
Positive Flexural Capacity If , then
Otherwise
Unless certain geometric conditions are satisfied
Ductility check:
tp DD 1.0 pn MM
.in75.4)5.13628(1.0D1.0.in709.7D tp
.inkips58255.47
709.77.007.16091DD
7.007.1MMt
ppn
inkips606746671.01.3M1.3RM yhn
ftkip5825Mftkips4026Sf31M nfxtu l
.95.195.4742.042.0.709.7 inDinD tp
Negative Flexural Capacity Appendix A
Therefore, Appendix A is applicable.
ksi70ksi50Fyf
3.13750
290007.5FE7.528.61
5.032.152
tD2
ycw
c
Web Plastification Factors Check if web is compact - NO
Noncompact web plastification factors are used
80.37
1.0MR
M54.0
FE
92.415.0
)48.10(2tD2
2
yh
p
ycDpw
w
cpcp
Web Plastification Factors (cont.)
28.61tD2
w
cw
28.5548.1032.158.37
DD
cp
cDpwDpw cpc
3.137FE7.5yc
rw
yc
p
yc
p
Dpwrw
Dpww
p
ychpc M
MMM
MMR
11Rc
c
yt
p
yt
p
Dpwrw
Dpww
p
ythpt M
MMM
MMR
11Rc
c
64.1Rpt
04.1Rpc
Compression Flange Local Buckling Resistance
Check if flange is compact - YES
15.950
2900038.0FE38.020.3
5.2216
t2b
ycfc
fcf
ftkips6415M
616804.1MRM
FLBnc
ycpcFLBnc
Lateral Torsional Buckling Resistance
437.4
tbtD
31112
br
fcfc
wc
fct
180L8.10750
29000437.4FErL byc
tp
8.575J
hSEF
76.611hS
JFEr95.1L
2
xcyr
xcyrtr
lengthunbracedNoncompactLLL rbp
Lateral Torsional Buckling Resistance
ycpcycpcpr
pb
ycpc
xcyrbLTBnc MRMR
LLLL
MRSF
11CM
ftkips6415M LTBnc
.ftkips6415M
M,MminM
nc
LTBncFLBncnc
ksi50,ksi95.301480916500.1,ksi35)50(7.0min
F,SSFR,F7.0minF yw
xc
xtythycyr
Negative Flexural Capacity Summary
ncfxcu MSf31M l
ytptfu MRM
.ftkips6415.ftkips5992
.ftkips6218381563.10.1.ftkips5992
Appendix A Performance Ratios Positive Bending Region
Constructibility (Strength I)
Top Flange 0.94Bottom Flange 0.30
Constructibility(Strength IV)
Top Flange 0.93Bottom Flange 0.36
Service Limit StateTop Flange 0.47Bottom Flange 0.70
Fatigue and Fracture Limit State
Bolted Conn. 0.80Welded Conn. 0.98
Strength Limit State(Strength I)
Flexure 0.69Shear 0.83
Appendix A Performance Ratios Negative Bending Region
Constructibility (Strength I)
Top Flange 0.46Bottom Flange 0.34
Constructibility(Strength IV)
Top Flange 0.55Bottom Flange 0.39
Service Limit StateTop Flange 0.57Bottom Flange 0.69
Fatigue and Fracture Limit State
Bolted Conn. NAWelded Conn. 0.58
Strength Limit State(Strength I)
Flexure 0.96Shear 0.78
Appendix B Design Moment redistribution procedures
are used to create a more economical design
63’ 63’54’
12 x 3/4 16 x 1 12 x 3/4
16 x 1-1/2 16 x 2 16 x 1-1/2
36 x 7/16 36 x 1/2 36 x 7/16
Appendix B Requirements Appendix B is valid for girders meeting
certain geometric and material limits Web Proportions
150725.0
36tD
w
8.163FE8.62.64
tD2
ycw
c
27D75.048.14Dcp
Appendix B Requirements (cont.) Compression flange
proportions
Lateral Bracing
15.9FE38.00.4
t2b
ycfc
fc
47.825.4D16bfc
191FEr
MM06.01.0180L
yc
t
2
1b
Appendix B Requirements (cont.) Shear
Section Transitions No section transitions are permitted within the
first cross-frame spacing on each side of the pier
Bearing Stiffeners Bearing stiffeners are required to meet projecting
width, bearing resistance, and axial resistance requirements
crvVV
Redistribution Moment Amount of moment redistributed to positive bending region
is a function of the effective plastic moment, Mpe
Higher Mpe values are permitted for girders with either: Transverse stiffeners placed at D/2 or less on each side of the pier “Ultra-compact” webs such that
Alternative Mpe equations are given for strength and service limit states
ycw
cp
FE3.2
tD2
Redistribution Moment (cont.)
Redistribution moment at pier:
Redistribution moment
varies linearly at otherlocations along the span
nnfc
yc
fc
fc
fc
yc
fc
fcpe MM
bD
EF
tb39.0
bD35.0
EF
tb3.263.2M
ftkip4951Mpe
epeerd M2.0MMM
epeferd M%13.ftkips75349515704MMM
Pier 1 Pier 2
Mrd1 Mrd2
Redistribution Moments (Strength I)
-6000
-4000
-2000
0
2000
4000
6000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Length along span, x/L
Mom
ent,
kips
-ft.
M+M+ + MrdM-M- + Mrd
Appendix B Design Checks Positive bending capacity
Evaluated for positive bending moment plus redistribution moment (at strength and service limit states)
Negative bending capacity within one lateral brace spacing on each side of the pier Not evaluated
Negative bending capacity at other locations Evaluated for negative bending moment minus
redistribution moment Otherwise, same as before
Appendix B Performance Ratios Positive Bending Region
Constructibility (Strength I)
Top Flange 0.94Bottom Flange 0.30
Constructibility(Strength IV)
Top Flange 0.93Bottom Flange 0.36
Service Limit StateTop Flange 0.47Bottom Flange 0.70
Fatigue and Fracture Limit State
Bolted Conn. 0.80Welded Conn. 0.99
Strength Limit State(Strength I)
Flexure 0.75Shear 0.83
Appendix B Performance Ratios Negative Bending Region
Constructibility (Strength I)
Top Flange 0.55Bottom Flange 0.42
Constructibility(Strength IV)
Top Flange 0.66Bottom Flange 0.48
Service Limit StateTop Flange 0.62Bottom Flange 0.79
Fatigue Limit State Welded Conn. 0.55
Strength Limit State(Strength I)
Flexure* 0.48Shear 0.78
* Design of negative bending region controlled by 20% limit
Appendix A / Appendix B Design Comparisons Positive moment region same in both
designs (controlled by fatigue) Cross-frame spacing the same
(controlled by constructibility) Appendix B negative moment region 18%
lighter Appendix B girder 6% lighter overall
63’ 63’54’
12 x 3/4 16 x 1 12 x 3/4
16 x 1-1/2 16 x 2 16 x 1-1/2
36 x 7/16 36 x 1/2 36 x 7/16
63’ 63’54’
12 x 3/4 16 x 1-1/4 12 x 3/4
16 x 1-1/2 16 x 2-1/2 16 x 1-1/2
36 x 7/16 36 x 1/2 36 x 7/16
APPENDIX A DESIGN APPENDIX B DESIGN
Concluding Comments Fatigue requirements significantly impact the
design of the positive moment region due to the relatively high distribution factor for the exterior girder
Constructibility and Appendix B requirements led to the use of a 15 ft cross-frame spacing throughout
Use of Appendix A leads to increasing economy with decreasing web slenderness (that is a section with a noncompact web at the upper limit will gain very little from Appendix A)
Appendix B provides even greater economy