Cold-Formed Steel Structural Membersattachment B

Committee for Specifications for the Design of Committee/Subcommittee Ballot: CS08-310C-finalized

Cold-Formed Steel Structural MembersAttachment B

Subcommittee 10, Element Behavior and Direct StrengthDate: May July 1628, 2010

COMMENTARY

1.2.2Beam Design

Commentary Section C3 provides a complete discussion on the behavior of cold-formed steel beams as it relates to the main Specification. This commentary addresses the specific issues raised by the use of the Direct Strength Method of Appendix 1 for the design of cold-formed steel beams.

The thin-walled nature of cold-formed beams complicates behavior and design. Elastic buckling analysis reveals at least three buckling modes: local, distortional, and lateral-torsional buckling (for members in strong-axis bending) that must be considered in design. The Direct Strength Method of this Appendix emerged through the combination of more refined methods for local and distortional buckling prediction, improved understanding of the post-buckling strength and imperfection sensitivity in distortional failures, and the relatively large amount of available experimental data.

Figure C-1.2.2-1 Local and Distortional Direct Strength Curves
for a Beam Braced against Lateral-Torsional Buckling (Mne = My)

The lateral-torsional buckling strength, Mne, follows the same practice as the main Specification. The main Specification provides the lateral-torsional buckling strength in terms of a stress, Fc (Equations C3.1.2.1-8). In the Direct Strength Method, this is converted from a stress to a moment by multiplying by the gross section modulus, Sf, resulting in Eq. 1.2.2-2 for Mne as given in Appendix 1.

In the main Specification, for beams that are not fully braced and locally unstable, beam strength is calculated by multiplying the predicted stress for failure in lateral-torsional buckling, Fc, by the effective section modulus, Sc, determined at stress Fc. This accounts for local buckling reductions in the lateral-torsional buckling strength (i.e., local-global interaction). In the Direct Strength Method, this calculation is broken into two parts: the lateral-torsional buckling strength without any reduction for local buckling (Mne) and the strength considering local-global interaction (Mn).

The strength curves for local and distortional buckling of a beam braced against lateral-torsional buckling are presented in Figure C-1.2.2-1 and compared to the critical elastic buckling curve. The post-buckling reserve for the local mode is predicted to be greater than that of the distortional mode. As depicted in the figure, in 2010, provisions for inelastic reserve capacity in bending, i.e. where Mn>My, were added.

The reliability of the beam provisions was determined using the test data of Section 1.1.1.2 and the provisions of Chapter F of the main Specification. Based on a target reliability, , of 2.5, a resistance factor, , of 0.90 was calculated for all the investigated beams. Based on this information the safety and resistance factors of Appendix Section 1.2.2 were determined for the pre-qualified members. For the United States and Mexico = 0.90; while for Canada  = 0.85 because Canada employs a slightly higher reliability, , of 3.0. The safety factor, , is back calculated from  at an assumed dead to live load ratio of 1 to 5. Since the range of pre-qualified members is relatively large, extensions of the Direct Strength Method to geometries outside the pre-qualified set is allowed. However, given the uncertain nature of this extension, increased safety factors and reduced resistance factors are applied in that case, per the rational analysis provisions of Section A1.2(b) of the main Specification.

The provisions of Appendix 1, applied to the beams of Section 1.1.1.2, are summarized in Figure C-1.2.2-2. The controlling strength is determined either by Section 1.2.2.2, which considers local buckling interaction with lateral-torsional buckling, or by Section 1.2.2.3, which considers the distortional mode alone. The controlling strength (minimum predicted of the two modes) is highlighted for the examined members by the choice of marker. Overall performance of the method can be judged by examination of Figure C-1.2.2-2. The scatter shown in the data is similar to that of the main Specification.

In 2010, provisions were added (Sections 1.2.2.1.2, 1.2.2.2.2, and 1.2.2.3.2) to take advantage of the inelastic reserve strength for members that are stable enough to allow partial plastification of the cross-section. Such sections have capacities in excess of My and potentially as high as Mp (though practically this upper limit is rarely achievable). As Figure C-1.2.2-1 shows, the inelastic reserve capacity is assumed to linearly increase with decreasing slenderness.

1.2.2.1Lateral-Torsional Buckling

1.2.2.1.1Lateral-Torsional Buckling Strength

As discussed in detail above, the strength expressions for lateral-torsional buckling of beams follow directly from Section C3 of the main Specification and are fully discussed in Section C3 of the Commentary. The lateral-torsional buckling strength, Mne, calculated in this section represents the upperbound capacity for a given beam. Actual beam strength is determined by considering reductions that may occur due to local buckling and performing a separate check on the distortional mode. See Section 1.1.2 for information on rational analysis methods for calculation of Mcre.

1.2.2.1.2Inelastic Lateral-Torsional Buckling Strength

The hot-rolled steel design specification (AISC 2005) has long provided expressions for inelastic lateral-torsional buckling of compact sections. The expression provided in Specification Equation 1.2.2-5 is a conservative extension of the AISC approach: first, the My/Mcre required to develop Mp may be shown equivalent to 1/2 Lp, as employed in AISC; second, the moment gradient factor (Cb) is only used in the elastic buckling approximation (for Mcre) and not to linearly increase the strength, as in the AISC Specification (Shifferaw and Schafer 2010).

1.2.2.2Local Buckling

1.2.2.2.1Local Buckling Strength

The expression selected for local buckling of beams is shown in Figures C-1.2.2-1 and C-1.2.2-2 and is discussed in Section 1.2.2. The use of the Direct Strength Method for local buckling and the development of the empirical strength expression is given in Schafer and Peköz (1998). The potential for local-global interaction is presumed; thus, the beam strength in local buckling is limited to a maximum of the lateral-torsional buckling strength, Mne. For fully braced beams, the maximum Mne value is the yield moment, My. See Section 1.1.2 for information on rational analysis methods for calculation of Mcr.

1.2.2.2.2Inelastic Local Buckling Strength

Unique expressions were derived for inelastic bending reserve in local buckling. This reserve is only allowed in cross-sections that are predicted to have inelastic bending reserve in lateral-torsional buckling (i.e., Mne>My). The compressive strain which the cross-section may sustain in local bucking Cyy, is shown to grow increase as specified in Specification Equation 1.2.2-13 in both back-calculated strains from tested sections and average membrane strains from finite element models (Shifferaw and Schafer 2010). The engineer should be aware that lLocal strains in the corners and at the surface of the plates (comprising the cross-section) as they undergo bending may be significantly in excess of Cyy, (Shifferaw and Schafer 2010). As a result, and consistent with the main Specification, Cy is limited to 3.

For sections with first yield in tension, the potential for inelastic reserve capacity is great, but the design calculations are more complicated. Specification Equation 1.2.2-5 only applies after the cross-section begins to yield in compression, i.e., when the moment reaches Myc. Calculation of Myc requires the use of basic mechanics to determine the moment strength in the partially plastfied cross-section; My may be used in place of Myc, but this is conservative (excessively so if the tensile strain demands are much higher than the compressive strain demands). Based on experience and past practice it has also been determined that the tensile strain should not exceed 3 times the yield strain; thus the moment is also limited by this value, i.e., Myt3.

Note, the slenderness utilizes My instead of Mne for simplicity in the inelastic reserve regime and is still defined in terms of My, not Myc to provide continuity with the expressions of Section 1.2.2.2.1. Further, the elastic buckling moment, Mcr, is still determined based on the elastic bending stress distribution, not the plastic stress distribution. These simplifications were shown to be sufficiently accurate when compared with existing tests and a parametric study using rigorous nonlinear finite element analysis (Shifferaw and Schafer 2010).

1.2.2.3Distortional Buckling

1.2.2.3.1Distortional Buckling Strength

The expression selected for distortional buckling of beams is shown in Figures C-1.2.2-1 and C-1.2.2-2 and is discussed in Section 1.2.2. Based on experimental test data and on the success of the Australian/New Zealand code (see Hancock, 2001 for discussion) the distortional buckling strength is limited to My instead of Mne. This presumes that distortional buckling failures are independent of lateral-torsional buckling behavior, i.e., little if any distortional-global interaction exists. See Section 1.1.2 for information on rational analysis methods for calculation of Mcrd.

1.2.2.3.2Inelastic Distortional Buckling Strength

The approach for strength prediction in inelastic distortional buckling is similar to that of inelastic local buckling. Use of the same form for Cyd in Specification Equation 1.2.2-21 as that of Cy in Equation 1.2.2-13 results in slightly more conservative strength predictions for inelastic distortional buckling (Shifferaw and Schafer 2010). Specification simplicity and greater concern with post-collapse response in distortional buckling is used as justification for this additional conservatism.

APPENDIX 1 References

Shifferaw, Y., and Schafer, B. W. (2010). "Inelastic bending capacity in cold-formed steel members." submitted to ASCE Journal of Structural Engineering (BWS note: this reference will need to be finalized before publication of the Specification)