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**Two-Span LRFD Design Example**

Karl Barth and Jennifer Righman West Virginia University

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**Objective The primary focus of this example**

is to demonstrate the use of Appendix A and Appendix B for a two-span continuous structure

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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

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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

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Design Information

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Design Information Framing Plan

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**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

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**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

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**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

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**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

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**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

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**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

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**Interior Girder Distribution Factors**

Moment Varies with girder dimensions due to Kg term One design lane Two or more design lanes

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**Interior Girder Distribution Factors**

Shear One design lane Two or more design lanes (CONTROLS)

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**Exterior Girder Distribution Factors**

AASHTO exterior girder correction factor Moment Shear Empirical formulas for exterior girder will not control

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**Exterior Girder Distribution Factor**

Lever Rule – One Design Lane

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**Exterior Girder Distribution Factor**

Special Analysis One design lane Two or more design lanes Controls for Moment

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**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

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**Unfactored Design Moments**

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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)

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**6.10 Provisions Addressed Cross section proportion limits**

Constructibility Serviceability Fatigue Strength

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**Appendix A Design 63’ 54’ 12 x 3/4 16 x 1-1/4 16 x 1-1/2 16 x 2-1/2**

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**Cross Section Proportion Limits**

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**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

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**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

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**Constructibility Check**

Stresses in compression flange of positive bending section control the allowable cross-frame spacing Strength I Strength IV

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**Service Limit State For top flange For bottom flange**

Bottom flange in positive bending (controls)

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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

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**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:

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**Positive Flexural Capacity**

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

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**Negative Flexural Capacity Appendix A**

Therefore, Appendix A is applicable.

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**Web Plastification Factors**

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

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**Web Plastification Factors (cont.)**

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**Compression Flange Local Buckling Resistance**

Check if flange is compact - YES

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**Lateral Torsional Buckling Resistance**

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**Lateral Torsional Buckling Resistance**

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**Negative Flexural Capacity Summary**

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**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

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**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

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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

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**Appendix B Requirements**

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

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**Appendix B Requirements (cont.)**

Compression flange proportions Lateral Bracing

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**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

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**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

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**Redistribution Moment (cont.)**

Redistribution moment at pier: Redistribution moment varies linearly at other locations along the span Pier 1 Pier 2 Mrd1 Mrd2

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**Redistribution Moments (Strength I)**

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**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

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**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

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**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

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**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

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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

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