# Two-Span LRFD Design Example

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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 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 S =1.335 k/ft

Design Loads – DC2 and DW DC2 DW Barrier weight = 520 lb/ft
Weight/girder = (0.520)x(2)/(4) = k/ft DW Future wearing surface = 25 psf DW = (0.025 ksf)x(34 ft)/4 = k/ft

Design Loads – WS and WL 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 = k/ft (on bottom flange) WL Assumed to be transmitted by diaphragm action WL is neglected

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

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

Interior Girder Distribution Factors
Shear One design lane Two or more design lanes (CONTROLS)

Exterior Girder Distribution Factors
AASHTO exterior girder correction factor Moment Shear Empirical formulas for exterior girder will not control

Exterior Girder Distribution Factor
Lever Rule – One Design Lane

Exterior Girder Distribution Factor
Special Analysis One design lane Two or more design lanes 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 (LL+I) Strength III = 1.25DC + 1.5DW + 1.4WS Strength IV = 1.5(DC + DW) Strength V = 1.25DC + 1.5DW (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’ 54’ 12 x 3/4 16 x 1-1/4 16 x 1-1/2 16 x 2-1/2

Cross Section Proportion Limits

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

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

Service Limit State For top flange For bottom flange
Bottom flange in positive bending (controls)

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

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:

Positive Flexural Capacity
If , then Otherwise Unless certain geometric conditions are satisfied Ductility check:

Negative Flexural Capacity Appendix A
Therefore, Appendix A is applicable.

Web Plastification Factors
Check if web is compact - NO Noncompact web plastification factors are used

Web Plastification Factors (cont.)

Compression Flange Local Buckling Resistance
Check if flange is compact - YES

Lateral Torsional Buckling Resistance

Lateral Torsional Buckling Resistance

Negative Flexural Capacity Summary

Appendix A Performance Ratios Positive Bending Region
Constructibility (Strength I) Top Flange 0.94 Bottom Flange 0.30 (Strength IV) 0.93 0.36 Service Limit State 0.47 0.70 Fatigue and Fracture Limit State Bolted Conn. 0.80 Welded Conn. 0.98 Strength Limit State Flexure 0.69 Shear 0.83

Appendix A Performance Ratios Negative Bending Region
Constructibility (Strength I) Top Flange 0.46 Bottom Flange 0.34 (Strength IV) 0.55 0.39 Service Limit State 0.57 0.69 Fatigue and Fracture Limit State Bolted Conn. NA Welded Conn. 0.58 Strength Limit State Flexure 0.96 Shear 0.78

Appendix B Design Moment redistribution procedures are used to create a more economical design 63’ 54’ 12 x 3/4 16 x 1 16 x 1-1/2 16 x 2 36 x 7/16 36 x 1/2

Appendix B Requirements
Appendix B is valid for girders meeting certain geometric and material limits Web Proportions

Appendix B Requirements (cont.)
Compression flange proportions Lateral Bracing

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

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

Redistribution Moment (cont.)
Redistribution moment at pier: Redistribution moment varies linearly at other locations along the span Pier 1 Pier 2 Mrd1 Mrd2

Redistribution Moments (Strength I)

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 (Strength IV) 0.93 0.36 Service Limit State 0.47 0.70 Fatigue and Fracture Limit State Bolted Conn. 0.80 Welded Conn. 0.99 Strength Limit State Flexure 0.75 Shear 0.83

Appendix B Performance Ratios Negative Bending Region
Constructibility (Strength I) Top Flange 0.55 Bottom Flange 0.42 (Strength IV) 0.66 0.48 Service Limit State 0.62 0.79 Fatigue Limit State Welded Conn. Strength Limit State Flexure* 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 overall 63’ 54’ 12 x 3/4 16 x 1 16 x 1-1/2 16 x 2 36 x 7/16 36 x 1/2 16 x 1-1/4 16 x 2-1/2 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

QUESTIONS?