Transcript Two-Span LRFD Design Example Karl Barth and Jennifer Righman West Virginia University

Two-Span LRFD Design
Example
Karl Barth and Jennifer Righman
West 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
Haunch (average wt/length)
Overhang taper
Girder (average wt/length, varies)
Cross-frames and misc. steel
Stay-in-place forms
=0.983
=0.017
=0.019
=0.200
=0.015
=0.101
k/ft
k/ft
k/ft
k/ft
k/ft
k/ft
S =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


K g  n I  A eg2  400,000 to 700,000

One design lane
0.4
0.3

 S   S   K g

0.06      
3 
 14   L   12.0 L t s 

0.1
0.4
 10   10 
 0.06     
 14   90 
0.3
 (702025) 


3 
 (12.0) (90) (8) 
0.1
 0.523
Two or more design lanes
0.6
0.2

 S   S   K g

0.075  
   
3 
 9.5   L   12.0 L t s 
0.1
0.6
 10   10 
 0.075  
  
 9.5   90 
0.2
 (702025) 


3 
 (12.0) (90) (8) 
0.1
 0.756
Interior Girder Distribution Factors

Shear

One design lane
S
10.0
0.36 
 0.36 
 0.760
25.0
25.0

Two or more design lanes (CONTROLS)
S S
0.2 
 
12  35 
2. 0
10  10 
 0 .2 
 
12  35 
2.0
 0.952
Exterior Girder Distribution Factors

AASHTO exterior girder correction factor
g  e ginterior

Moment
de
2
e  0.77 
 0.77 
 0.990  1.0
9.1
9.1

Shear
de
2
e  0.6 
 0.6   0.800  1.0
10
10

Empirical formulas for exterior girder will not control
Exterior Girder Distribution Factor

Lever Rule – One Design Lane

 10  6  
DF   0.5  0.5
   MPF
 10  

DF  0.7  1.2  0.84
Exterior Girder Distribution Factor

Special Analysis
DF 
NL

NB
xEXT  e
NL
2
x
NB

One design lane
1
(15)(12) 
  1.2  0.732
DF  MPF   
2
2 
 4 2(15  5 ) 

Two or more design lanes
 2 (15)(12  0) 
  1.0  0.860
DF  MPF   
2
2 
 4 2(15  5 ) 
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
2000
1500
1000
Moment, kip-ft
500
0
0
0.2
0.4
0.6
-500
-1000
-1500
-2000
DC1
DC2
DW
LL+IM
-2500
Distance Along Span, x/L
0.8
1
Limit States

All applicable limits states for steel
structures were considered

Strength






Strength
Strength
Strength
Strength
I = 1.25DC + 1.5DW + 1.75(LL+I)
III = 1.25DC + 1.5DW + 1.4WS
IV = 1.5(DC + DW)
V = 1.25DC + 1.5DW + 1.35(LL+I) + 0.4WS
Service


Strength I controls in this example
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
12 x 3/4
63’
12xx7/16
3/4
36
16
36 xx 1-1/2
7/16
16 x 1-1/2
63’
16 X 1-1/4
54’
16
36 x
x 1-1/4
1/2
x 1/2
1636
x 2-1/2
16 x 2-1/2
54’
12 x 3/4
63’
x 3/4
36 12
X 7/16
1636
x 1-1/2
x 7/16
16 x 1-1/2
63’
Cross Section Proportion Limits

D
 150
tw
D
36

 82  150
t w 7 16 

bf
 12.0
2t f
bf
12

 8  12.0
2t f 20.75 

bf 

t f  1.1t w

0 .1 
D
6
Iy c
Iy t
b f  12 
 10
D 36

6
6
6
t f  0.75  1.1(0.5)  0.55
3

1 123 412
0.1 
1 121.5163
 0.21  10
Constructibility

For discretely braced compression flanges


fbu  fl  f RhFy c  1.01.050  50 ksi
1
fbu  fl  f Fnc  varies, 49.8 ksi
3
Fnc may be computed using Appendix A which
accounts for increased torsional resistance
For discretely braced tension flanges and
continuously braced flanges
fbu  fl  f RhFy f  1.01.050  50 ksi
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
fbu  fl  1.2521.47  19.97  46.8 ksi  50 ksi

Strength IV
fbu  fl  1.521.47  14.13  46.3 ksi  50 ksi
Service Limit State

For top flange
ff  0.95RhFy  0.951.050  47.5 ksi

For bottom flange
fl
ff   0.95RhFy  0.951.0 50   47.5 ksi
2

Bottom flange in positive bending (controls)
fl  692 135  111 1.31615
0
12   33.1 ksi  47.5 ksi
ff   



2  843
1131
1219 
2
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
Fmax  6.36 ksi  8.0 ksi

Welded connections / Category C’ details
Fmax  5.92 ksi  6.0 ksi
Fatigue Limit State (cont.)

Use of bolted cross-frame connections requires
that net section fracture requirements are
satisfied
 An 
Fu  Fy t
ft  0.84
A 
 g

Assuming one 7/8” diameter bolt hole is used:
A n  16(1.5)  (7  1 )(1.5)  22.5 in2
8
8
A g  16(1.5)  24.0 in2
 22.5 
ft  0.84
65  51 ft  Fy t  50
 24.0 
ft  44.6  50  OK
Positive Flexural Capacity

If Dp  0.1Dt, then Mn  Mp
Dp  7.709 in.  0.1Dt  0.1(8  2  36  1.5)  4.75 in.

Otherwise
D 


 7.709  
Mn  Mp 1.07  0.7 p   60911.07  0.7
   5825 kips  in.
Dt 
 47.5  



Unless certain geometric conditions are satisfied
Mn  1.3R hMy  1.31.0 4667  6067 kips  in
1
fl S xt  4026 kips  ft  f Mn  5825 kip  ft
3

Mu 

Ductility check:
Dp  7.709 in.  0.42Dt  0.4247.5  19.95 in.
Negative Flexural Capacity Appendix A

Fy f  50 ksi  70 ksi

2Dc 215.32
E
29000

 61.28  5.7
 5.7
 137.3
tw
0.5
Fy c
50

Therefore, Appendix A is applicable.
Web Plastification Factors

Check if web is compact - NO
2Dcp
tw


E
Fy c
2(10.48)
 41.92   pw Dcp  
 37.80
2
0 .5


 Mp 
 0.54
  0.1
R M 


 h y


Noncompact web plastification factors are used
Web Plastification Factors (cont.)

w 
2Dc
 61.28
tw
 Dc 
  37.8 15.32   55.28
  pw Dcp  
D 
 10.48 
 cp 

 pw Dc 

 rw  5.7
E
 137.3
Fy c
  RhMy c   w  pw D   Mp
Mp
c




 1  1 






Mp  rw  pw Dc   My c My c
 

Rpc

  RhMy t   w  pw D   Mp Mp
c


Rpt  1  1 





Mp  rw  pw Dc   My t My t
 
Rpc  1.04
Rpt  1.64
Compression Flange Local Buckling
Resistance

Check if flange is compact - YES
f 

bf c
16
E
29000

 3.20  0.38
 0.38
 9.15
2t f c 22.5
Fy c
50
Mnc FLB   RpcMy c  1.04 6168 
Mnc FLB   6415 kips  ft
Lateral Torsional Buckling Resistance


rt 
bf c
 1 Dc t w 

121 
 3 b f ct f c 
 4.437
E
29000
Lp  rt
 4.437
 107.8  Lb  180
Fy c
50
E
Fy r
2
 Fy r S xch 
J
  575.8
1  1  6.76
S xch
E J 

Lr  1.95rt

Lp  Lb  Lr  Noncompact unbraced length
Lateral Torsional Buckling Resistance



S xt
Fy r  min 0.7Fy c, RhFy t
, Fy w 
S xc




 916 
 min 0.7(50)  35 ksi, 1.0 50 
  30.95 ksi, 50 ksi 
 1480 



Mnc LTB 
 
Fy rS xc  Lb  Lp 


RpcMy c  RpcMy c
 Cb 1  1 



  RpcMy c  Lr  Lp 
Mnc LTB   6415 kips  ft

Mnc  minMnc FLB , Mnc LTB   
Mnc  6415 kips  ft.
Negative Flexural Capacity Summary

1
Mu  fl S xc  f Mnc
3
5992 kips  ft.  6415 kips  ft.

Mu  f RptMy t
5992 kips  ft.  1.01.633815  6218 kips  ft.
Appendix A Performance Ratios
Positive Bending Region
Constructibility
(Strength I)
Top Flange
0.94
Bottom Flange
0.30
Constructibility
(Strength IV)
Top Flange
0.93
Bottom Flange
0.36
Top Flange
0.47
Bottom Flange
0.70
Bolted Conn.
0.80
Welded Conn.
0.98
Flexure
0.69
Shear
0.83
Service Limit State
Fatigue and Fracture Limit
State
Strength Limit State
(Strength I)
Appendix A Performance Ratios
Negative Bending Region
Constructibility
(Strength I)
Top Flange
0.46
Bottom Flange
0.34
Constructibility
(Strength IV)
Top Flange
0.55
Bottom Flange
0.39
Top Flange
0.57
Bottom Flange
0.69
Service Limit State
Fatigue and Fracture Limit
State
Strength Limit State
(Strength I)
Bolted Conn.
NA
Welded Conn.
0.58
Flexure
0.96
Shear
0.78
Appendix B Design

Moment redistribution procedures are
used to create a more economical
design
63’
54’
63’
12 x 3/4
16 x 1
12 x 3/4
36 x 7/16
36 x 1/2
36 x 7/16
16 x 1-1/2
16 x 2
16 x 1-1/2
Appendix B Requirements

Appendix B is valid for girders meeting
certain geometric and material limits

Web Proportions

D 36

 72  150
t w 0 .5

2Dc
E
 64.2  6.8
 163.8
tw
Fy c

Dcp  14.48  0.75D  27
Appendix B Requirements (cont.)


Compression flange proportions

bf c
E
 4.0  0.38
 9.15
2t f c
Fy c

b f c  16 
D
 8.47
4.25
Lateral Bracing


 M1  rtE

Lb  180  0.1  0.06
 191
 M2  Fy c

Appendix B Requirements (cont.)

Shear


Section Transitions


V  v Vcr
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
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
2Dcp
E

2
.
3
“Ultra-compact” webs such that t
F
w

Alternative Mpe equations are given for
strength and service limit states
yc
Redistribution Moment (cont.)


b f c Fy c
b f c Fy c D 
D
Mpe  2.63  2.3
 0.35
 0.39
Mn  Mn
tfc E
bf c
t f c E b f c 


Mpe  4951 kip  ft

Redistribution moment at pier:
Mrd  Me  Mpe  0.2 Me
 Mrd  Me  f Mpe  5704  4951  753 kips  ft.  13%Me

Redistribution moment
varies linearly at other
locations along the span
Pier 1
Mrd1
Pier 2
Mrd2
Redistribution Moments (Strength I)
6000
Moment, kips-ft.
4000
2000
0
0
0.2
0.4
0.6
0.8
1
1.2
-2000
-4000
M+
M+ + Mrd
MM- + Mrd
-6000
Length along span, x/L
1.4
1.6
1.8
2
Appendix B Design Checks

Positive bending capacity


Negative bending capacity within one lateral
brace spacing on each side of the pier


Not evaluated
Negative bending capacity at other locations


Evaluated for positive bending moment plus
redistribution moment (at strength and service
limit states)
Evaluated for negative bending moment minus
redistribution moment
Otherwise, same as before
Appendix B Performance Ratios
Positive Bending Region
Constructibility
(Strength I)
Top Flange
0.94
Bottom Flange
0.30
Constructibility
(Strength IV)
Top Flange
0.93
Bottom Flange
0.36
Top Flange
0.47
Bottom Flange
0.70
Bolted Conn.
0.80
Welded Conn.
0.99
Flexure
0.75
Shear
0.83
Service Limit State
Fatigue and Fracture Limit
State
Strength Limit State
(Strength I)
Appendix B Performance Ratios
Negative Bending Region
Top Flange
Bottom Flange
Top Flange
Bottom Flange
0.55
0.42
0.66
0.48
Fatigue Limit State
Top Flange
Bottom Flange
Welded Conn.
0.62
0.79
0.55
Strength Limit State
(Strength I)
Flexure*
Shear
0.48
0.78
Constructibility
(Strength I)
Constructibility
(Strength IV)
Service Limit State
* 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’
54’
63’
63’
54’
63’
12 x 3/4
16 x 1-1/4
12 x 3/4
12 x 3/4
16 x 1
12 x 3/4
36 x 7/16
36 x 1/2
36 x 7/16
36 x 7/16
36 x 1/2
36 x 7/16
16 x 1-1/2
16 x 1-1/2
16 x 1-1/2
16 x 2-1/2
APPENDIX A DESIGN
16 x 2
APPENDIX B DESIGN
16 x 1-1/2
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