AASHTO (2004) Article 6.10 and its Appendices A and B provide a unified approach for consideration of I-girder major-axis bending, minor-axis bending and torsion from any source. Similar to prior Guide Specifications for curved steel bridge design such as AASHTO (2003), the AASHTO (2004) provisions focus on the flange lateral bending caused by the warping (i.e., cross-bending) of the flanges as the primary response associated with the torsion of I-section members. Significant flange lateral bending may be caused by wind, by eccentric concrete deck overhang loads acting on forming brackets placed along exterior girders, and by the use of discontinuous cross-frame lines in bridges with skew angles larger than about 20o. For the majority of straight non-skewed bridges, flange lateral bending effects tend to be the most significant during construction and tend to be insignificant in the final constructed condition. However, for horizontally curved bridges, in addition to the effects from the above sources, flange lateral bending due to the curvature must be considered at all limit states and during construction. The intent of the Article 6.10 provisions is to provide a straightforward approach for the Engineer to account for the above effects in design in a direct and rational manner whenever these effects are nonnegligible. When the various flange lateral bending effects are judged negligible or incidental, the provisions reduce to the format for the design of I-section members subjected to major-axis bending alone (outlined in Sections 5.3.5 and 5.3.6).
The basic form of the AASHTO (2004) resistance equations that account for the combined effects of major-axis bending and flange lateral bending is
n f bu f F 3 1 f + l ≤φ (5.3.7-1, AASHTO 6.10.3.2.1-2, 6.10.3.2.2-1, 6.10.7.2.1-2, 6.10.8.1.1-1 & 6.10.8.2-1) for members in which the major-axis bending resistance is expressed in terms of the
corresponding flange stress and n f x u f S M 3 1
M + l ≤φ (5.3.7-2, AASHTO 6.10.7.1.1-1, A6.1.1-1 & A6.1.2-1)
for members in which the major-axis bending resistance is expressed in terms of the bending moment, where:
fbu = the elastically-computed flange major-axis bending stress,
fl = the elastically-computed flange lateral bending stress,
φf Fn = the factored flexural resistance in terms of the flange major-axis bending stress,
Sx = the elastic section modulus about the major-axis of the section to the flange under
consideration, taken as the short-term section modulus for composite members in positive bending or the section modulus of the composite section for composite members in negative bending, and
φf Mn = the factored flexural resistance in terms of the member major-axis bending moment.
Equations (1) and (2) are referred to in AASHTO (2004) as the one-third rule. These equations are simple, yet they do an excellent job of characterizing the various strength limit states that can govern the resistance of I-section members. Equations (1) and (2) address the combined major-axis and flange lateral bending effects essentially by handling the flanges as equivalent beam-columns.
Equation (1) is targeted specifically at checking of slender-web noncomposite members, slender-web composite members in negative bending, and noncompact composite members in positive bending. Also, as discussed previously in Section 3, the resistance of I-section members in horizontally curved bridges is limited in general to fbu < φf Fn. In the limit that the flange
lateral bending stress fl is equal to zero, Eq. (1) reduces to this basic member check for major- axis bending only. The maximum potential value of Fn isthe flange yield strength Fyf, butFn can
be less thanFyf due to slender-web bend buckling and/or hybrid-web yielding effects, or due to
compression flange lateral-torsional (LTB) or local buckling (FLB) limit states.
Equation (2) may be used for checking the strength limit states of straight noncomposite members or composite members in negative bending that have compact or noncompact webs, and for checking of compact composite members in positive bending. For these member types, φf Mn can be as large as φf Mp, where Mp is the section plastic moment resistance. The reader is
referred to Sections 5.3.3 through 5.3.5 for definitions of the terms slender, noncompact and compact and for an overview of the calculation of φf Fn and φf Mn. Equation (1) may be used as a
simple conservative resistance check for all types of I-section members. AASHTO (2004) Article 6.10 emphasizes this fact by relegating the use of Eq. (2) for straight compact and noncompact web noncomposite members and composite members in negative bending to its Appendix A. The definition of Sx as the short-term modulus for composite sections in positive
bending, and as the section modulus of the composite section for composite sections in negative bending, is a conservative simplification. This simplification is consistent with the precedent of neglecting the influence of the different types of loading on the resistance for compact composite members in positive bending, and with the limited dependency of the different loading types for compact- and noncompact-web composite members in negative bending, as discussed previously in Sections 5.3.3(B) and 5.3.5(L).
In the application of Eqs. (1) and (2), the stresses fl and fbu, and the moment Mu, are taken as
the largest values throughout the unbraced length when checking against the base flexural resistance φf Fn or φf Mn associated with lateral-torsional buckling. This is consistent with the
application of the AASHTO and AISC interaction equations for a general beam-column
subjected to combined axial load and bending. The stress fbu in Eq. (1) and the moment Mu in Eq.
(2) are analogous to the axial loading in a general beam-column, and the stress fl is analogous to the beam-column bending moment. The moment Mu is analogous to axial loading since it
produces axial stresses in the flanges. When checking compression flange local buckling or tension flange yielding, fl, fbu and Mu may be determined as the corresponding values at the
cross-section under consideration. Generally, Eq. (1) or (2), as applicable, must be checked for each flange, and both the FLB and LTB based resistances must be checked for the compression flange in calculating Fnc or Mnc. The check providing the largest ratio of the left-hand side to the
right-hand side of these equations governs.
The Engineer is permitted to use fl = 0 within the top flange of composite I-girders, once the section is composite, since the composite slab tends to restrain the top flange lateral bending. AASHTO (2004) Article 6.10.7.2.1.1 requires that in shored composite construction, the
concrete slab flexural stress shall be checked along with the use of Eq. (1). However, AASHTO (2004) does not require checking of the slab longitudinal stress in unshored construction, since this stress is typically much less than fc' at the Eq. (1) strength limit.
As noted above, for curved bridges, AASHTO (2004) restricts the I-girder design in all cases to the use of Eq. (1). This restriction is due to the lack of a comprehensive understanding of the implications of significant member yielding and the concomitant inelastic redistribution on the forces and moments in curved bridge structural systems at the time that these provisions were implemented. Otherwise, Eqs. (1) and (2) are valid generally for all types of I-section members that satisfy the limits
Lb/R < 0.1 (5.3.7-3, AASHTO 6.7.4.2-1)
within the final constructed configuration, where Lb is the unsupported length between the cross-
frame locations and R is the horizontal radius of curvature,
Lb < Lr (5.3.7-4, AASHTO 6.7.4.2-1)
where Lr is the unbraced length limit beyond which the base lateral-torsional buckling limit state
is elastic, and
fl < 0.6 Fyf (5.3.7-5, AASHTO 6.10.1.6-1)
The first of these limits is a practical upper bound for the subtended angle between the cross- frame locations (for constant R). It ensures that the I-girder webs will not have a do/R larger than
0.1, where do is the spacing of the transverse stiffeners. Equations (1) and (2) have been observed
to perform adequately in a number of cases with Lb/R larger than 0.2 (White et al. 2001).
However, the development of these equations as well as the validation of the AASHTO (2004) Article 6.10.9.3 tension-field action shear strength equations for curved web panels has focused predominantly on members designed up to the limit specified by Eq. (3). Equation (4) is a
practical upper bound for the unbraced length Lb beyond which the second-order amplification of
the flange lateral bending stresses tends to be particularly severe. The reason for Eq. (5) is discussed in Section 5.3.7(C).
5.3.7(B) Calculation of flange lateral bending stresses
Various methods may be used for calculating the flange elastic lateral bending stresses fl. AASHTO (2004) Article 6.10.1.6 gives simple equations for estimating the first-order lateral bending stresses due to the torsion associated with horizontal curvature (see Eq. (2.2.1-1) and AASHTO (2004) Article 4.6.1.2.4b), the torsion from eccentric concrete deck overhang loads acting on cantilever forming brackets placed along exterior girders (see AASHTO (2004) Article C6.10.3.4), and due to wind load (see AASHTO (2004) Article 4.6.2.7). These equations are based on the assumption of interior unbraced lengths in which the flange is continuous with adjacent unbraced lengths, as well as equal lengths of the adjacent segments such that, due to approximate symmetry boundary conditions, the ends of the unbraced lengths are effectively torsionally and laterally fixed. The Engineer should consider other more appropriate idealizations, or the use of computer analysis methods, when these assumptions do not approximate the actual conditions. Implications of various types of computer analysis on the calculation of fl are addressed by Jung et al. (2005) and Chang et al. (2005).
Similar to the amplification of internal bending moments in beam-column members, flange lateral bending stresses are generally amplified due to stability effects. However, it is impractical to calculate second-order live load stresses for moving live loads. Therefore, when Eq. (1) is applied for checking the compression flange, AASHTO (2004) Article 6.10.1.6 provides the following simple lateral bending amplification equation to account in an approximate fashion for these second-order effects:
1 1 LTB . e bu f f F f 1 85 . 0 fl l ≥ l ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − = (5.3.7-6, AASHTO 6.10.1.6-4 & 6.10.1.6-5) where:
Fe.LTB = the compression flange elastic lateral-torsional buckling resistance from Eq. (5.3.5-19)
for compact- or noncompact-web members or Eq. (5.3.5-23) for slender-web members, fl1 = the first-order compression flange lateral bending stress at the section under
consideration (for checking of FLB), or the largest first-order compression flange lateral bending stress within the unbraced length (for checking of LTB), and
fbu = the largest value of the compression flange major-axis bending stress within the
unbraced length under consideration.
Amplification of the tension flange lateral bending stresses is not required, since this effect tends to be relatively minor compared to the compression flange response. White et al. (2001) show that Eq. (6) gives accurate to conservative estimates of the flange second-order lateral bending stresses. The purpose of Eq. (6) is to guard conservatively against large unbraced lengths in which second-order lateral bending effects are significant. The Engineer should be particularly mindful of the amplified compression flange lateral bending in exterior girders due to eccentric
concrete deck overhang loads during construction. In situations where the amplification given by these equations is large, the Engineer may wish to consider using an effective length factor K < 1 in the calculation of Fe.LTB (using the procedure outlined in Section 5.3.5(M)). In cases where the
amplification of construction stresses is large, a second alternative is to conduct a direct geometric nonlinear analysis to determine the second-order effects within the superstructure more accurately. In the final constructed condition, the above amplification typically is applied only to the bottom flange in negative moment regions of continuous spans. In this case, Fe.LTB is
increased significantly due to the moment gradient in these regions, via the moment gradient modifier Cb (see Section 5.3.5(K)).
5.3.7(C) One-third rule concept
Figure 5.3.7-1 compares the result from Eq. (2) to the theoretical fully plastic resistance for several doubly-symmetric noncomposite compact-flange, compact-web cross-sections. Figure 5.3.7-2 shows a sketch of a typical fully plastic stress distribution on this type of cross-section. The equations for the fully plastic cross-section resistances are based on the original research by Mozer et al. (1971) and are summarized by White and Grubb (2005). The specific stress
distribution shown in Fig. 5.3.7-2 is associated with equal and opposite lateral bending in each of the equal-size flanges (i.e., warping of the flanges due to nonuniform torsion). However, the solution is the same if one considers equal flange lateral bending moments due to minor axis bending.
M
u
/M
y
Fig. 5.3.7-1. Comparison of the AASHTO (2004) one-third rule equation to the theoretical fully-
plastic cross-section resistance for several doubly-symmetric noncomposite compact-flange, compact-web I-sections (adapted from White and Grubb (2005)).
One can observe that, within the limit of Eq. (5), the one-third rule equation (Eq. (2)) gives an accurate to somewhat conservative estimate of the theoretical cross-section resistances for the different web-to-flange area ratios Aw/Af shown in Fig. 5.3.7-1. In the limit that Aw/Af is taken
equal to zero, the same approximation is provided by both Eq. (2) and Eq. (1). The comparison of the theoretical and approximate equations shown in Fig. 5.3.7-1 is useful for gaining a
conceptual understanding of the one-third rule equations in the limit of compact-flange, compact-web, compactly-braced noncomposite members. Also, Schilling (1996) and Yoo and Davidson (1997) present other useful cross-section yield interaction relationships. However, cross-section yield interaction equations are limited in their ability to fully characterize the combined influence of distributed yielding along the member lengths along with the various stability effects (FLB, LTB and web bend buckling). Furthermore, yield interaction equations generally do not reduce to the resistance equations for straight members subjected only to major- axis bending in the limit that fl = 0.
Fig. 5.3.7-2. Stress distribution on a fully-plastified compact doubly-symmetric I-section, from
Mozer et al. (1971).
Equations (1) and (2) are a basic extension of the one-third rule approximation of the above theoretical cross-section resistances to address the influence of general yielding and stability limit states on the member resistance. The basic extension is accomplished simply by changing the flange yield strength Fyf to φfFn in Eq. (1) and by changing the section plastic moment
resistance Mp to φfMn in Eq. (2). The 1/3 coefficient accurately captures the strength interaction
including the various yielding and stability effects (White et al. 2001). The extension from cross- section equations to member equations is ad hoc, but it is similar in many respects to the
development of the AISC (2005) and AASHTO (2004) general beam-column interaction relationships. The shape of the interaction (i.e., the slope of the line relating fbu and fl in Eq. (1)
or Mu and fl in Eq. (2)) is based on curve fitting. Equations (1) and (2) are thus semi-analytical
and semi-empirical. White and Grubb (2005) provide a summary of the correlation of Eqs. (1) and (2) with analytical, numerical and experimental results.
5.3.8 Shear Strength