Problem Description

Cb calculation for back-span of a beam with a cantilever can be unconservative due to skip load patterns. 

Reason

For beams with a cantilever on one or both ends the program automatically skip loads (i.e., pattern loads) the Live Load to determine the maximum negative (hogging) moment – at the support, when the Live Load is place only on the cantilever as shown for Case 1 in the Figure, and the maximum positive (sagging) moment – when the Live Load is placed only on the span as shown for Case 2 in the Figure. The beam is then designed for these maximum positive and negative moments. Where beams are braced at intermediate points, the maximum positive and negative moments are determined for each segment. In the calculation of the member capacity for the limit state of lateral-torsional buckling there is a modification factor for nonuniform moment term (called Cb in AISC, mLT in BS 5950, and am in AS 4100) based on moments at quarter points within the unbraced segment. These moments are determined from the same skip load condition that produced the maximum moment within the segment (either Case 1 or Case 2). This works efficiently and satisfactorily, except potentially for the unbraced segment within the main span directly adjacent to the support on the end with the cantilever. For that unbraced segment the maximum negative moment is the same as the maximum negative moment from the cantilever (see the moment diagram for Case 1). In the calculation of the Cb factor used in the calculation of the lateral-torsional buckling capacity the program determines the moments used in the Cb equation from the skip load condition with Live Load only on the cantilever (Case 1, since that is the skip condition that results in the largest negative moment). However, it is possible that by considering the Live Load applied to both the cantilever and the span (i.e., not skipping the Live Load as shown for Case 3 in the Figure), a smaller Cb factor may be obtained, potentially resulting in a smaller capacity. In that end segment the maximum negative moment is the same as the cantilever moment, even if the live load isn’t skipped, so the impact can be that while the design moment is the same whether the live load is skipped or not, the calculated capacity is smaller if it is not skipped, resulting in a beam that should fail but is not detected by the program.

Note that this is a rare condition.

• It is highly unlikely to happen with a girder, braced at intermediate points by the supported beams and carrying large loads in the span.
• The most susceptible configuration is a beam (not a girder) that is unbraced for the entire span, with a short cantilever with a heavy load.
• It is unlikely to happen with longer cantilevers, since Cb is only equal to 1.0 and hence the cantilever will often control the design.
• The error does not impact the design if the size is not controlled by lateral-torsional buckling (e.g., deflection controls).
• The error does not impact the design if the lateral-torsional buckling capacity is limited to the plastic capacity (i.e., Cb doesn’t influence the capacity or only marginally influences it to the extent that it brings the capacity up to the limit of the plastic capacity).
• The error does not occur for beams without cantilevers.

Note that to some degree the impact of the defect is mitigated because the design methodology is conservative for the case of a beam supporting a deck; it doesn’t consider the contribution of the deck in bracing the top flange, even for the portion of the “unbraced length” where the top flange is in compression, in the determination of the design capacity of the section. And in Case 3 much more of the top compression flange is braced by the deck than for Case 1, but that contribution would not have been considered by the Code if the Case 3 condition had been investigated.

This defect can occur if the following design codes are selected:

• AISC 360
• AISC LRFD 3rd
• BS 5950 (m_LT   p54)
• AS 4100  (am  Clause 5.6.1.1)

Solution

None.