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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

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Page 1: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Two-Span LRFD Two-Span LRFD Design ExampleDesign Example

Karl Barth and Jennifer Karl Barth and Jennifer RighmanRighman

West Virginia UniversityWest Virginia University

Page 2: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Objective

The primary focus of this example is to demonstrate the use of Appendix A and Appendix B

for a two-span continuous structure

Page 3: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 4: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 5: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Design Information

Page 6: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Design Information

Framing Plan

Page 7: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 8: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 9: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 10: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 11: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 12: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 13: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 14: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 15: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 16: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Exterior Girder Distribution Factor Lever Rule – One Design Lane

84.02.17.0DF

MPF10

6105.05.0DF

Page 17: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 18: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 19: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Unfactored Design Moments

Page 20: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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)

Page 21: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

6.10 Provisions Addressed Cross section proportion limits Constructibility Serviceability Fatigue Strength

Page 22: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 23: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 24: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 25: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 26: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 27: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 28: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 29: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 30: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 31: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Negative Flexural Capacity Appendix A

Therefore, Appendix A is applicable.

ksi70ksi50Fyf

3.13750

290007.5FE7.528.61

5.032.152

tD2

ycw

c

Page 32: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 33: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 34: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 35: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 36: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 37: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

Negative Flexural Capacity Summary

ncfxcu MSf31M l

ytptfu MRM

.ftkips6415.ftkips5992

.ftkips6218381563.10.1.ftkips5992

Page 38: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 39: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 40: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 41: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 42: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 43: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 44: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 45: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 46: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 47: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 48: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 49: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 50: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 51: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD

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

Page 52: Vigas Continuas Compuestas Barth Nsba Wsbs LRFD