09/20/2011: Modified Fatigue limit State to include Fatigue I and II Limit States.
This worksheet performs the necessary calculations to check the bending capacity of a non-composite negative moment section for the Strength I, III, IV, V, Service II and Fatigue Limit States at a discrete location.
Special Unit Definitions:
The 10% redistribution of negative moment, as allowed by Article 4.6.4, was not performed.
Alternate method of Appendix A was not used.
Compression flange is discretely braced.
Longitudinal stiffeners not required.
Deck reinforcement over the pier not included per IaDOT policy.
No holes in the tension flange.
Section is within 10% of the span length of the interior support.
Bottom flange is always in compression.
Fatigue check not valid for cantilever girders.
Cb calculated for the controlling load group only (Strength I).
Deck Thickness:
Width of the Compression Flange:
Thickness of the Compression Flange:
Width of the Tension Flange:
Thickness of the Tension Flange:
Web Depth:
Web Thickness:
Compression Flange Yield Strength:
Tension Flange Yield Strength:
Web Yield Strength:
Steel Modulus of Elasticity:
Haunch (Bottom of Deck To Top of Web):
Depth of Web in Compression at Strength I Limit State:
Hybrid Factor:
Continuous Span? (Y or N)(Capital Letter):
Unbraced Length:
Barrier Rail Height:
Wind Pressure on Structure (Article 3.8.1):
Compression Flange Section Modulus for Non-Composite Section:
Tension Flange Section Modulus for Non-Composite Section:
Fatigue Input:
Fatigue Detail Category:
Number of Lanes in One Direction:
Average Daily Traffic:
Percent Trucks:
Span Length:
Applied Stresses Top Flange (unfactored)(tensile stresses are positive):
Dead Load of Components Applied to Non-Composite Section:
Dead Load of Components Applied to Long Term Section:
Dead Load Wearing Surface and Utilities:
Live Load Plus Impact:
Applied Stresses Bottom Flange (unfactored):
Dead Load of Components Applied to Non-Composite Section:
Dead Load of Components Applied to Long Term Section:
Dead Load Wearing Surface and Utilities:
Live Load Plus Impact:
Applied Moments (unfactored):
Live Load Fatigue Plus Impact (Positive Moment):
Live Load Fatigue Plus Impact (Negative Moment)
Factored Stresses Required To Calculate C
C
Largest Compressive Stress At Either End of The Unbraced Length:
Compressive Stress At Opposite End of The Unbraced Length:
Stress At The Middle of The Unbraced Length:
Load Modifier Relating to Ductility, Redundancy and Operational Importance:
A design check will be performed for a discrete section located in the negative moment region.
Wind forces are considered in Strength III & V Limit States.
Wind pressure for this bridge:
(3.8.1.2)
Wind on Composite Section:
Assume force to the bottom flange is based on the tributary area (mid-height of the web to bottom flange). Since the girder is composite, part of the wind force will be transferred directly into the concrete slab which acts as a large diaphragm.
Calculate the factored stresses for the various limit states considering loads applicable to a tangent bridge.
Wind on Structure (WS) is considered for lateral bending on the flange only.
Wind on Structure (WS) is considered for lateral bending on the flange only. Wind on live load is assumed to be transferred directly into the concrete deck.
The LRFD specifications allow discrete sections to develop capacities up to the plastic moment at ultimate loads if certain criteria are met.
Fatigue I Limit State - 1.5(LL+I)
The forces and stresses will be determined in a later section of these calculations.
Girder proportions must meet basic proportion limits as specified in Article 6.10.2.
(6.10.2.2-1)
(6.10.2.2.2)
(6.10.2.2-3)
The flexural capacity of composite sections in negative flexure is taken as the lesser of the flexural resistance due to flange local buckling or lateral torsional buckling.
The calculation of C
where:
f
f
f
If the member is prismatic and the midpoint stress is less than f
Calculate f
Maximum of f
Calculate C
Check for user input override:
Compression Flange Slenderness Ratio:
Limiting Slenderness Ratio for a Compact Flange:
Compression Flange Slenderness Ratio:
Limiting Slenderness Ratio for a Noncompact Flange
Calculate Nominal Capacity:
Flange Local Buckling For Non-Compact Flange
Determine Flange Compactness
Calculate required variables for resistance equations:
Radius of Gyration:
Limiting unbraced length to achieve nominal flexural resistance under uniform bending:
Limiting unbraced length to achieve nominal yielding in either flange:
Calculate nominal capacity for compact, non-compact and slender unbraced lengths:
Compact Unbraced Length
Non-Compact Unbraced Length
Slender Unbraced Length
Determine Lateral Torsional Capacity based on L
Compression Flange Flexural Resistance (minimum of Flange Local Buckling and Lateral Torsional Buckling)
The ultimate moment or stress is compared to the flexural resistance as follows (6.10.8):
(6.10.8.1.1-1)
Ultimate Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Lateral Stress:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Lateral Stress:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
To guard against possible yielding under service conditions, the Service II Limit State is checked.
Since the section is being designed non-composite, Equation 6.10.4.2.2-3 is calculated as the flexural resistance.
(6.10.4.2.2-3)
Calculate the web bend buckling resistance of the compression flange.
Check F
Service II Stress:
Nominal resistance:
Performance ratio:
Ultimate Stress:
Nominal resistance:
Performance ratio:
Check the fatigue limit states for load induced fatigue.
The properties that were used to calculate the fatigue stresses assume that the deck is not able to resist tensile stresses.
Top & bottom flange fatigue stress range under the fatigue limit state (non-composite section):
Top & bottom live load stress due to the fatigue truck (positive indicates tension), using the fatigue limit state (non-composite section):
The assumption in this Mathcad sheet is that the bottom flange will be in compression and the top flange will be in tension for the unfactored permanent loads. This means the top flange must be considered for fatigue, but consideration of the bottom flange for fatigue depends on Article 6.6.1.2.1. This article states that in regions where the unfactored permanent loads (f
If
fatigue shall be considered
Dead load compressive stress in the bottom flange (Negative indicates compression):
Maximum tensile live load stress in the bottom flange (Positive indicates tension):
Check if fatigue should be considered for the top flange:
Dead Load compressive stress in the Top flange (Negative indicates compression)
Since the top flange is subjected to a net applied tensile stress, fatigue shall be considered
If fatigue is not needed for consideration, this worksheet will still compute the fatigue limit states below.
The input detail category is:
Given traffic data:
Average daily truck traffic in both directions:
The fatigue I resistance (ΔFn.I) is defined as:
where:
ΔFTH = constant-amplitude fatigue threshold taken from table 6.6.1.2.5-3
Table Lookup For The Constant Amplitude Fatigue Threshold:
The fatigue II resistance (
where:
Determine the constant amplitude fatigue threshold from Table 6.6.1.2.5-3.
Table Lookup
ADTT
(from Table 3.6.1.4.2-1)
Convert the ADTT (both directions) to one direction.
therefore,
Determine the cycles per truck passage (n) from Table 6.6.1.2.5-2.
The Detail Category Constant, A, is taken from Table 6.6.1.2.5-1
Note: (Article 6.6.1.2.3)
Except as specified herein for fracture critical members, where the projected (ADTT)SL is less than or equal to that specified in Table 6.6.1.2.3-2 for the component or detail under consideration, that component or detail should be designed for finite life using the fatigue II load combination specified in Table 3.4.1-1. Otherwise, the component or detail should be designed for inifinite life using the Fatigue I load combination.
This worksheet will calculate the limit states for both infinite life (Fatigue I) and finite life (Fatigue II). It is up to the engineer to decide which limit state is applicable to their design.
Fatigue I Limit State
Check Top flange:
Performance ratio:
Check Bottom Flange:
Performance ratio:
Fatigue II Limit State:
Check Top Flange:
Performance ratio:
Check Bottom Flange:
Performance ratio:
(Fatigue range need only be checked if the fatigue stress is tensile and the fatigue stress exceeds 1/2 of the dead load compressive stress)
See Section 3.0 above for section proportion checks
Fatigue II:
Fatigue II: