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)

90

10

14

100.06

tL12.0

K

L

S

14

S0.06

0.1

3

0.30.40.1

3s

g0.30.4

0.756(8)(90)(12.0)

(702025)

90

10

9.5

100.075

tL12.0

K

L

S

9.5

S0.075

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)

0.7600.25

0.1036.0

0.25

S36.0

0.952

0.20.2

35

10

12

102.0

35

S

12

S2.0

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

19

2770

19770

1.00.800.d

.e e 10

260

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

N

NDF

2

7320215152

1215

4

122

..)(

))((MPFDF

0.860

015152

01215

4

222

.)(

))((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 2-1/216 x 1-1/2

36 x 7/16 36 x 1/2 36 x 7/16

Cross Section Proportion Limits

150t

D

w

15082167

36

t

D

w

0.12t2

b

f

f 0.12875.02

12

t2

b

f

f

wf t1.1t 55.0)5.0(1.175.0t f

10I

I1.0

yt

yc

1021.0165.1121

12431211.0

3

3

6

Dbf 6

6

36

6

D12bf

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 l3

1

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

ksi5.47500.195.0FR95.02

ff yhf l

ksi5.47ksi1.332

012

1219

16153.1

1131

111135

843

692

2

fff

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

A

A84.0f

2n in5.22)5.1)(8

18

7()5.1(16A

2g in0.24)5.1(16A

50Ff51650.24

5.2284.0f 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

709.77.007.16091

D

D7.007.1MM

t

ppn

inkips606746671.01.3M1.3RM yhn

ftkip5825Mftkips4026Sf3

1M nfxtu l

.95.195.4742.042.0.709.7 inDinD tp

Negative Flexural Capacity Appendix A

Therefore, Appendix A is applicable.

ksi70ksi50Fyf

3.137

50

290007.5

F

E7.528.61

5.0

32.152

t

D2

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(2

t

D22

yh

p

yc

Dpww

cp

cp

Web Plastification Factors (cont.)

28.61t

D2

w

cw

28.5548.10

32.158.37

D

D

cp

cDpwDpw cpc

3.137F

E7.5

ycrw

yc

p

yc

p

Dpwrw

Dpww

p

ychpc M

M

M

M

M

MR11R

c

c

yt

p

yt

p

Dpwrw

Dpww

p

ythpt M

M

M

M

M

MR11R

c

c

64.1Rpt

04.1Rpc

Compression Flange Local Buckling Resistance

Check if flange is compact - YES

15.950

2900038.0

F

E38.020.3

5.22

16

t2

b

ycfc

fcf

ftkips6415M

616804.1MRM

FLBnc

ycpcFLBnc

Lateral Torsional Buckling Resistance

437.4

tbtD

31

112

br

fcfc

wc

fct

180L8.10750

29000437.4

F

ErL b

yctp

8.575J

hS

E

F76.611

hS

J

F

Er95.1L

2

xcyr

xcyrtr

lengthunbracedNoncompactLLL rbp

Lateral Torsional Buckling Resistance

ycpcycpcpr

pb

ycpc

xcyrbLTBnc MRMR

LL

LL

MR

SF11CM

ftkips6415M LTBnc

.ftkips6415M

M,MminM

nc

LTBncFLBncnc

ksi50,ksi95.301480

916500.1,ksi35)50(7.0min

F,S

SFR,F7.0minF yw

xc

xtythycyr

Negative Flexural Capacity Summary

ncfxcu MSf3

1M l

ytptfu MRM

.ftkips6415.ftkips5992

.ftkips6218381563.10.1.ftkips5992

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

Service Limit StateTop Flange 0.47

Bottom Flange

0.70

Fatigue and Fracture Limit State

Bolted Conn. 0.80

Welded Conn.

0.98

Strength Limit State(Strength I)

Flexure 0.69

Shear 0.83

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

Service Limit StateTop Flange 0.57

Bottom Flange

0.69

Fatigue and Fracture Limit State

Bolted Conn. NA

Welded Conn.

0.58

Strength Limit State(Strength I)

Flexure 0.96

Shear 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

36

t

D

w

8.163F

E8.62.64

t

D2

ycw

c

27D75.048.14Dcp

Appendix B Requirements (cont.)

Compression flange proportions

Lateral Bracing

15.9F

E38.00.4

t2

b

ycfc

fc

47.825.4

D16bfc

191F

Er

M

M06.01.0180L

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

F

E3.2

t

D2

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

b

D

E

F

t

b39.0

b

D35.0

E

F

t

b3.263.2M

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

Mo

men

t, k

ips-

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 0.94

Bottom Flange

0.30

Constructibility(Strength IV)

Top Flange 0.93

Bottom Flange

0.36

Service Limit StateTop Flange 0.47

Bottom Flange

0.70

Fatigue and Fracture Limit State

Bolted Conn. 0.80

Welded Conn.

0.99

Strength Limit State(Strength I)

Flexure 0.75

Shear 0.83

Appendix B Performance Ratios Negative Bending Region

Constructibility (Strength I)

Top Flange 0.55

Bottom Flange

0.42

Constructibility(Strength IV)

Top Flange 0.66

Bottom Flange

0.48

Service Limit StateTop Flange 0.62

Bottom Flange

0.79

Fatigue Limit StateWelded Conn.

0.55

Strength Limit State(Strength I)

Flexure* 0.48

Shear 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 overall63’ 63’54’

12 x 3/4 16 x 1 12 x 3/4

16 x 1-1/216 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 1-1/216 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