09/20/2011: Modified Fatigue limit State to include Fatigue I and II Limit States.
Special Unit Definitions:
This worksheet performs the necessary calculations to check the bending capacity of a composite positive moment section for the Strength I, III, IV, V, Service II and Fatigue Limit State at a discrete location.
This worksheet considers compact sections only.
The 10% redistribution of negative moment, as allowed by Article 4.6.4, was not performed.
No longitudinal stiffeners.
No holes in the tension flange.
Homogenous Section.
Stresses computed assuming the deck is in compression for all loading.
Section is further than 10% of the span length away from the support.
Deck Thickness:
Deck Effective Width:
Sacrificial Wearing Surface:
Concrete Compressive Strength:
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):
Top Flange Section Modulus for Short Term Composite Section:
Bottom Flange Section Modulus for Short Term Composite Section:
Hybrid Factor:
Continuous Span? (Y or N)(Capitol Letter):
Unbraced Length:
Field Section Piece Length:
Barrier Rail Height:
Wind Pressure on Structure (Article 3.8.1):
Plastic Moment:
Depth of Web in Compression at Plastic Moment):
Distance From Top Of Deck To Plastic N.A.:
Fatigue Input:
Fatigue Detail Category:
Number of Lanes in One Direction:
Average Daily Traffic:
Percent Trucks:
Span Length:
Applied Moments (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:
Live Load Fatigue Plus Impact (Positive Moment):
Live Load Fatigue Plus Impact (Negative Negative)
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)(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:
Load Modifier Relating to Ductility, Redundancy and Operational Importance:
A design check will be performed for a discrete section located in the positive 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 moments 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.
In addition to the strength limit states above, there are service limit states that are to be checked.
The LRFD specifications allow discrete sections to develop capacities up to the plastic moment at ultimate loads if certain criteria are met.
Top flange stress:
Bottom flange stress:
The forces and stresses will be determined in a later section of these calculations.
Fatigue II Limit State -
The forces and stresses will be determined in a later section of these calculations.
Since the section properties are routine engineering calculations and are independent of the different code specifications, only the properties or capacities that are unique to the LRFD specifications are illustrated herein.
The plastic moment capacity is not unique to the LRFD specification, however, it is an important calculation which defines the maximum capacity (or upper bound) of the section.
Inputted plastic moment:
Depth of web in compression at the plastic moment:
Distance from top of deck to the plastic neutral axis:
The yield moment is the minimum moment which causes first yield in either flange.
In equation form:
where:
M
M
M
M
S
S
Rewrite in Terms of Stress:
Sum moments to obtain the Yield Moment:
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)
(IaDOT Policy)
The flexural capacity of composite sections in positive flexure is essentially determined based on the web slenderness at the plastic moment.
Check if section is considered compact (6.10.6.2.2):
Check the depth of the plastic neutral axis (Dp) relative to the top of deck:
Total section depth
The Nominal Flexural Resistance M
If the spans are continuous, then limit the flexural resistance:
Ductility Check:
The ductility check is required regardless of whether the section is compact or not.
The ultimate moment or stress is compared to the flexural resistance as follows (Article 6.10.7):
Moment format:
(6.10.7.1.1-1)
Stress format:
where:
f
f
S
F
In many cases the lateral component (f
Ultimate moment:
Nominal resistance:
Performance ratio:
Ultimate moment:
Ultimate lateral moment:
Corresponding lateral stress on the bottom flange:
(6.10.1.6-1)
Nominal resistance:
Major axis section modulus with respect to the tension flange (yield moment/Fy):
Calculate the left side of the equation and call it M
Performance ratio:
Ultimate moment:
Nominal resistance:
Performance ratio:
Ultimate moment:
Ultimate lateral moment:
Corresponding lateral stress on the bottom flange:
(6.10.1.6-1)
Nominal resistance:
Calculate the left side of the equation and call it M
Performance ratio:
To guard against possible yielding under service conditions, the Service II Limit State is checked.
Performance ratio top flange:
Performance ratio bottom flange:
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.
The short term composite section is used for both negative and positive bending in these calculations.
Top & bottom flange fatigue stress range under the fatigue limit states:
Top & bottom live load stress due to the fatigue truck (positive indicates tension), using the Fatigue Limit States (short term composite section):
Article 6.6.1.2.1 states that in regions where the unfactored permanent loads (f
If
fatigue shall be considered
Dead load compressive stress in the top flange (Negative indicates compression):
Maximum tensile live load stress in the top flange. (Positive indicates tension):
Check if fatigue should be considered for the top flange:
Dead load compressive stress in the bottom flange (Negative indicates compression):
Since the bottom 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 resistance for Fatigue I (ΔFn) is defined as:
where:
The fatigue resistance for Fatigue II (
Table Lookup For The Constant Amplitude Fatigue Threshold:
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 4.2 above for compact section check and ductility check.
(This worksheet should only be used to check compact sections. In the event the section is not compact, the section should be determined based on first yield.)
See Section 3.3 above for section proportion checks
Fatigue II (top):
Fatigue II (bot):