Two-Span LRFD Design Example

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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
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Design Notes

• 2004 AASHTO LRFD Specifications, 3rd Edition
• Structural steel ASTM A709, Grade 50W
• Normal weight concrete (145 pcf) with fc4 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
• 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

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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

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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)
• I1.15 for fatigue limit state
• I1.33 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

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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 1.75(LLI)
• Strength III 1.25DC 1.5DW 1.4WS
• Strength IV 1.5(DC DW)
• Strength V 1.25DC 1.5DW 1.35(LLI) 0.4WS
• Service
• Service II 1.0(DC DW) 1.3(LLI)
• Fatigue 0.75(LLI)

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6.10 Provisions Addressed

• Cross section proportion limits
• Constructibility
• Serviceability
• Fatigue
• Strength

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Appendix A Design
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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
Constructibility (Strength I) Bottom Flange 0.30
Constructibility (Strength IV) Top Flange 0.93
Constructibility (Strength IV) Bottom Flange 0.36
Service Limit State Top Flange 0.47
Service Limit State Bottom Flange 0.70
Fatigue and Fracture Limit State Bolted Conn. 0.80
Fatigue and Fracture Limit State Welded Conn. 0.98
Strength Limit State (Strength I) Flexure 0.69
Strength Limit State (Strength I) Shear 0.83
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Appendix A Performance Ratios Negative Bending
Region
Constructibility (Strength I) Top Flange 0.46
Constructibility (Strength I) Bottom Flange 0.34
Constructibility (Strength IV) Top Flange 0.55
Constructibility (Strength IV) Bottom Flange 0.39
Service Limit State Top Flange 0.57
Service Limit State Bottom Flange 0.69
Fatigue and Fracture Limit State Bolted Conn. NA
Fatigue and Fracture Limit State Welded Conn. 0.58
Strength Limit State (Strength I) Flexure 0.96
Strength Limit State (Strength I) Shear 0.78
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Appendix B Design

• Moment redistribution procedures are used to
  create a more economical design

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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

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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
Constructibility (Strength I) Bottom Flange 0.30
Constructibility (Strength IV) Top Flange 0.93
Constructibility (Strength IV) Bottom Flange 0.36
Service Limit State Top Flange 0.47
Service Limit State Bottom Flange 0.70
Fatigue and Fracture Limit State Bolted Conn. 0.80
Fatigue and Fracture Limit State Welded Conn. 0.99
Strength Limit State (Strength I) Flexure 0.75
Strength Limit State (Strength I) Shear 0.83
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Appendix B Performance Ratios Negative Bending
Region
Constructibility (Strength I) Top Flange 0.55
Constructibility (Strength I) Bottom Flange 0.42
Constructibility (Strength IV) Top Flange 0.66
Constructibility (Strength IV) Bottom Flange 0.48
Service Limit State Top Flange 0.62
Service Limit State Bottom Flange 0.79
Fatigue Limit State Welded Conn. 0.55
Strength Limit State (Strength I) Flexure 0.48
Strength Limit State (Strength I) 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

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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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QUESTIONS?