A Beginner's Guide to the Steel Construction Manual, 13^th ed. (old)

Chapter 8 - Bending Members

© 2006, 2007, 2008 T. Bartlett Quimby

Section 8.2.4

Lateral Torsional Buckling Limit State

Last Revised: 06/16/2011

As mentioned earlier, Lateral Torsional Buckling (LTB) is a strong axis phenomena.  It need not be considered for weak axis bending.  The equations for each of the cases shown in SCM Table User Note F1.1 are found in the Chapter F sections referenced in the table.  The general form used to compute LTB effects is the same for cases F2, F3, F4, and F5.  The differences are in the computation of the key quantities (L_p, L_r, M_p, and M_r, see Figure 8.2.1.5) and the details of the equations used in various buckling ranges.

Note also that web slenderness is considered in several of the LTB cases.  As a result, Web Local Buckling (WLB) is integrated in the LTB equations, making WLB only a consideration for HSS and other square and rectangular tubes.

The Limit State

The basic limit state follows the standard form.  The statement of the limit states and the associated reduction factor and factor of safety are given here:

The values of M_u and M_a are the LRFD and ASD factored loads, respectively, applied to the flexural member. 

In this case M_n is the nominal LTB flexural strength of the member.  Since this is a buckling phenomena, limits need to be found for the three strength regions:  plastic, inelastic buckling, and elastic buckling as shown in Figure 8.2.1.5.

The General Form

The general form of the LTB limit state follows the typical buckling curves.  The slenderness parameter used is L_b, the laterally unbraced length. 

It is worth noting that each laterally unbraced length will have it's own M_n to be compared to the required M_n (M_u/f or WM_a) on its length.  This means that a beam with several laterally unbraced lengths will have a varying moment capacity along its length.  This will necessitate doing the flexural strength (M_n) calculation on each laterally unbraced length!

The limits of the buckling regions are specified by the terms L_p (the limit of the plastic region) and L_r (the limit of the inelastic buckling region) as shown in Figure 8.2.1.5.  The figure is repeated here as Figure 8.2.4.1 for ease of use.  Hence:

if L_b < L_p then the plastic strength, M_p, controls and LTB does not occur
if L_p < L_b < L_r then inelastic LTB occurs
if L_r < L_b then elastic LTB occurs

Figure 8.2.4.1
General LTB Strength
Click on image for larger view

The unique factor associated with LTB is the beam has non-constant compressive stress along its length.  This behavior is accounted for by the scale factor C_b.  C_b shifts the LTB curve upward as illustrated in Figure 8.2.4.2.  Note that there is an upper limit of M_p on all LTB strength equations.  There is no condition for which a beam's strength exceeds M_p.

Figure 8.2.4.2
LTB Strength Modified by C_b
Click on image for larger view

The equations for determining L_p and L_r are found in the section of chapter F that deals with each situation.  We will look at the equations as they are found in SCM F2.2.

The equation for L_p (SCM equation F2-5) is relatively simple.  The equation for L_r (SCM equation F2-6), on the other hand, is quite complex.  This equation has been substantially reorganized since prior versions of the specification.

Note that both L_p and L_r are functions of F_y and the member section properties.  They are not functions of the member internal forces.  This means that, from the section property data for rolled sections provided with the SCM, tables of L_p and L_r for each section can be created for the different material types.  Such a table for I-shaped sections in ASTM A992 steel has been included in Part 3 of the SCM.  This table, which includes quantities in addition to L_p and L_r, starts on SCM page 3-11.  Take a look at it.

In this text, the SCM equations have been rearranged to fit the general form of the strength curves shown above.  Namely, terms for M_p and M_r are extracted (where necessary) from the SCM equations for use in the general form shown in Figure8.2.4.1. The equations have been rewritten in these terms as well.  The resulting equations are the same as the SCM equations with some algebraic rearrangement. This will help you to see the various sections presented in a form that will be common across the applicable SCM sections.  A summary of the equations can be down loaded at the bottom of this page.

M_p

• SCM F2 & F3:  The plastic moment capacity, M_p, is the plastic strength of the member computed in the yielding limit state.
• SCM F4:  M_p = R_pcM_yc = R_pcF_yS_xc (SCM Equation F4-1).
• SCM F5:  Stresses are used instead of moments in this section.  F_y takes the place of M_p (extracted from SCM equation F5-3) in the general form of the equation for the inelastic buckling region.  M_p in the general form becomes R_pgF_yS_xc (SCM equation F5-1)

M_r

The moment at the interface of the elastic and inelastic ranges, M_r, is found embeded in the linear interpolation function used to compute the strength in the inelastic buckling range. 

• SCM F2 & F3:
 M_r = 0.7F_yS_x  (extracted from SCM equation F2-2)
• SCM F4:
  M_r = F_LS_xc  (extracted from SCM equation F4-2)
• SCM F5:  Stress is used instead of moments.  F_r takes the place of M_r in the general form of the equation for the inelastic buckling region:

F_r = 0.3F_y  (extracted from SCM equation F5-3)

The values for M_p and M_r can also be tabulated for given sections and materials.  SCM Table 3-2 tabulates values for W shapes from which M_p and M_r can be determined with minor computation:

M_px = (M_px / W_b) W_b = (f_bM_px) / f_b

M_rx = (M_rx / W_b) W_b = (f_bM_rx) / f_b

Now that the limits are defined, the equations for each range can be considered.

Plastic Range

As noted above, when L_b < L_p LTB does not happen.  Consequently in the plastic range, M_n equals M_p.

In-elastic Buckling Range

A linear interpolating function is used to compute M_n in the in-elastic buckling range.  The value resulting from the interpolation is then scaled by C_b.  This value is compared with M_p to find the final M_n.

The equation can be written as:

M_n = min [C_b(M_p - (M_p - M_r)*(L_b - L_p)/(L_r - L_p)), M_p]

Elastic Buckling Range

The nominal moment capacity, M_n, in the elastic range is found by computing the elastic moment that creates the critical buckling stress, F_cr, in the compression flange.

M_n = min[F_crS_xc, M_p]

A modified Euler type function is used to the compute the critical buckling stress, F_cr. 

For SCM sections F2, F3, and F4, F_cr is computed with the equation:

Where the difference between SCM chapter F sections is the computation of "r".

• SCM F2 & F3: "r" is computed using SCM equation F2-7. 
• SCM F4: The equation for "r" differs based on how the section is configured.
• SCM F5: The equation for F_cr is similar but different from the one used in F2, F3, and F4.  It is similar in that it is also a form of the Euler buckling equation.

Suggested Procedure for Computing M_n

To minimize hand computations, the following procedure is recommended for computing the nominal moment capacity M_n for each laterally unbraced length on a flexural member.

Locate the unbraced lengths on the flexural member
For each laterally unbraced length
   Compute M_p, L_p
   If (L_b < L_p) then
       M_n = M_p
   Else
       Compute M_r, L_r
       If (L_b < L_r) then
           M_n = min [C_b(M_p - (M_p - M_r)*(L_b - L_p)/(L_r - L_p)), M_p]
       Else
           M_n = min[F_crS_xc, M_p]
       End if
   End if
Next laterally unbraced length

Summary

• Download a summary of LTB requirements here.
• Download a spreadsheet example that produces the LTB curve for any W shape.

The following spreadsheet example computes the major axis flexural capacity, M_nx, including LTB effects, for a typical W section.  The input values are in the grey shaded cells and the result in the yellow highlighted cell.

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