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					Committee for Specifications for the Design of                 Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                                Attachment B
Subcommittee 10, Element Behavior and Direct Strength                      Date: November 3, 2008

1.2.2 Beam 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 Braced Beam (Mne = My)
Committee for Specifications for the Design of               Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                              Attachment B
Subcommittee 10, Element Behavior and Direct Strength                    Date: November 3, 2008

       Mp




            inelastic reserve
            capacity,
            Eq. 1.2.2-11
            and -15




     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 C3.1.2.1-2, -3, -4 and -5). In the Direct Strength Method,
this is converted from a stress to a moment by multiplying by the gross section modulus,
Sf, resulting in the formulas for Mne 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-
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 fully braced beam are
presented in Figure C-1.2.2-1 and compared to the critical elastic buckling curve. While
the strength in both the local and distortional modes exhibit both an inelastic regime and
a post-buckling regime, the post-buckling reserve for the local mode is predicted to be
greater than that of the distortional mode.
Committee for Specifications for the Design of                         Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                                        Attachment B
Subcommittee 10, Element Behavior and Direct Strength                              Date: November 3, 2008

    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


            1.5
                                                                                Local: Eq. 1.2.2-6

                                                                                Distortional: Eq. 1.2.2-9

                                                                                Local
             1
                                                                                Distortional
   M test
   My


            0.5




             0
                  0              1               2                 3              4                         5
                                               max    M y M cr


                      Figure C-1.2.2-2 Direct Strength Method for laterally braced beams

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 (Section 1.2.2.4) were added to take advantage of the inelastic
reserve available in cross-sections which 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 not achievable in common sections). As
Committee for Specifications for the Design of               Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                              Attachment B
Subcommittee 10, Element Behavior and Direct Strength                    Date: November 3, 2008

Figure C-1.2.2-1 shows the inelastic reserve capacity is assumed to linearly increase with
decreasing slenderness.
  1.2.2.1 Lateral-Torsional Buckling
      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.

  1.2.2.2 Local Buckling
      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.

  1.2.2.3 Distortional Buckling
      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.
  1.2.2.4 Inelastic Reserve Capacity
      Any cross-section, which after checking lateral-torsional buckling, local buckling,
  and distortional buckling per sections 1.2.2.1-1.2.2.3, shows no potential reduction in
  strength i.e., My is the predicted capacity in all limit states, may develop inelastic
  reserve capacity. The main Specification recognizes this phenomena through Section
  C3.1.1(b) where the strain (Cyy) that a stiffened or unstiffened element may sustain
  beyond the yield strain (y) is determined based on the element slenderness and the
  engineer is left to calculate the moment capacity in the partially plastified cross-
  section based on the limiting strain of Cyy and basic mechanics.
      Based on the work of Shifferaw and Schafer (2007, 2008, 2009) the provisions of
  this Appendix generalize and simplify the approach of the main Specification.
  Consistent with the Direct Strength Method the cross-section slenderness, not the
  element slenderness, is used to determine the limiting strain (Cyy). Further, rather
Committee for Specifications for the Design of                 Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                                Attachment B
Subcommittee 10, Element Behavior and Direct Strength                      Date: November 3, 2008

  than only consider local buckling of stiffened and unstiffened elements, the Direct
  Strength Method considers inelastic lateral-torsional, inelastic local and inelastic
  distortional buckling of all cross-sections. Finally, it may be shown that the limiting
  strain multiplier Cy is directly related to the inelastic reserve moment; therefore, a
  simple direct relation is provided for calculating the moment capacity.

  1.2.2.4.1 Inelastic Lateral-Torsional Buckling
      The hot-rolled steel standard (AISC 2005) has long provided expressions for
  inelastic lateral-torsional buckling of compact sections. The expression provided in
  Specification Eq. 1.2.2-11 is a conservative extension of the AISC approach: first it may
  be shown that the unbraced length at which Mp may be developed using the AISI
  equation is ½ of that used specified by 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 2009).

  1.2.2.4.1 Inelastic Local Buckling
      For sections symmetric about the axis of bending Specification Eq. 1.2.2-13 is exact
  if the extreme fiber strain that the section can withstand in compression (Cyy) is
  known. For sections with first yield in compression Eq. 1.2.2-13 is approximate, but
  with sufficient accuracy (Shifferaw and Schafer 2009). The compressive strain which
  the cross-section may sustain in local bucking Cyy, is shown to grow as specified in
  Eq. 1.2.2-14 in both back-calculated strains from tested sections and average
  membrane strains from finite element models (Shifferaw and Schafer 2008). The
  engineer should be aware that local 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 2008,2009). As a result, and consistent with the
  main Specification, Cy has been 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 Eq. 1.2.2-16 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 Cyt (3) times the
  yield strain; thus the moment is also limited by this value, i.e., Myt3.
      Note, the slenderness  is still defined in terms of My, not Myc to provide
  continuity with the expressions of Section 1.2.2.2. Further, the elastic buckling
  moment, Mcr, is still determined based on the elastic bending stress distrtibution not
  the plastic stress distribution. These simplifications were shown to be sufficiently
  accurate when comapred with existing tests and a parametric study using rigorous
  nonlinear finite element analysis (Shifferaw and Schafer 2008, 2009).
Committee for Specifications for the Design of               Committee/Subcommittee Ballot: XXX
Cold-Formed Steel Structural Members                                              Attachment B
Subcommittee 10, Element Behavior and Direct Strength                    Date: November 3, 2008

  1.2.2.4 Inelastic Distortional Buckling
      The approach for strength prediction in inelastic distortional buckling is similar to
  that of inelastic local buckling. Use of the same form for the Cyd of Eq. 1.2.2-19 as that
  of Cy of Eq. 1.2.2-14 results in slightly more conservative strength predictions for
  inelastic distortional buckling (Shifferaw and Schafer 2009). 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. (2007). "Inelastic bending capacity in cold-
       formed steel members." Annual Technical Session and Meeting, Structural
       Stability Research Council, New Orleans, LA, 279-299.
Shifferaw, Y., and Schafer, B. W. (2008). "Inelastic bending capacity in cold-
       formed steel members." Report to American Iron and Steel Institute –
       Committee on Specifications, July 2008.
Shifferaw, Y., and Schafer, B. W. (2009). "Inelastic bending capacity in cold-
       formed steel members." to be submitted to ASCE Journal of Structural
       Engineering (BWS note: this reference will need to be finalized before
       publication of the Specification)

				
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