The load and resistance factor design approach implemented in AASHTO (2004) uses the general form
Σ ηi γi Qi < φRn (3-1, AASHTO 3.4.1-1)
for assessment of the adequacy of the structure. The left-hand side of this equation represents a given design load effect, typically calculated by analysis. The right-hand side represents the design resistance corresponding to a given limit state. In the context of strength limit states, the left- and right-hand sides of Eq. (1) can be referred to as the required and the available design strengths respectively. A selected component is adequate for a given limit state if the required strength determined by structural analysis is less than or equal to its available design strength. The design load effect or required strength is determined as the largest value from various sums (or combinations) of appropriate nominal load effects Qi multiplied by the load factors ηi and γi.
The terms γi are scale factors that account for the variability and uncertainty associated with each
of the nominal loads for a given load combination. The various load combinations generally account for a maximum lifetime event for certain loadings taken with appropriate arbitrary point in time values of other loadings. On the right-hand side of Eq. (1), the φ terms are resistance factors, which account for the variability, uncertainty and consequences of failure associated with different limit states. The parameters ηi increase or decrease the nominal loads based on
general considerations of the ductility, redundancy and operational importance of the structure. With the exceptions of inelastic redistribution of pier section moments in certain types of continuous-span stringer bridges and general inelastic analysis for extreme event limit states (i.e., earthquake, ice loads, collision by vessels or vehicles and certain hydraulic events), AASHTO (2004) specifies the use of elastic structural analysis for calculation of the design load effects. Conversely, the nominal resistances Rn in Eq. (1) are based generally on inelastic behavior of the
structural components. For composite stringers, the concrete section is assumed fully effective in positive and negative bending for calculation of the internal forces and moments, but it is
assumed to be fully cracked for calculation of the resistances at strength limit states. These apparent inconsistencies are explicitly addressed in several locations within the AASHTO (2004) Commentary, i.e., Articles C1.3.1, C4.1, C4.5.2.2, and C6.10.6.2.1. Simply put, the Engineer is allowed to neglect the influence of all material nonlinearity on the distribution of forces and moments within the structure up to the limit of resistance of the most critical component, including the effects of residual stresses in the steel, concrete cracking, and various stress contributions that are considered incidental. Numerous physical tests indicate that this
approximation is acceptable. It is assumed that the resistance of the complete structure is reached when the left and right-hand sides of Eq. (1) are equal for the most critically loaded component. As explained in Article 1.3.2.4 of AASHTO (2004), multi-stringer bridges usually have
substantial additional reserve capacity beyond this resistance level. This is because the live load cannot be positioned to maximize the force effects on all parts of the bridge cross-section simultaneously. However, this reserve capacity is not necessary to justify the above elastic analysis assumptions.
There are three situations in steel design where AASHTO (2004) implements specific restrictions to ensure the validity of the above elastic analysis-design approach:
1. For continuous-span girders that are composite in positive bending, AASHTO Article 6.10.7.1 limits the moment capacity to
Mn = 1.3RhMy (3-2, AASHTO 6.10.7.1.2-3)
unless specific Appendix B requirements are satisfied that ensure ductility of the adjacent pier sections. Equation (2) is intended to limit the yielding in positive moment regions of continuous-span girders, where the shape factor Mp/My can be larger than 1.5, when the
inelastic rotation capability of the pier sections is somewhat limited or undefined. In many cases, compact-flange pier sections in straight I-girder bridges with small support skew satisfy the Appendix B requirements without any special modification. However, the skew must be less than 10 degrees and the ratio of the lateral unbraced length to the compression flange width Lb/bfc at the pier sections must be approximately 10 or less in addition to other
requirements. All continuous-span box girders are required to satisfy Eq. (2) or more
restrictive limits. Equation (2) guards against significant partial yielding of the cross-section over a relatively large length within the positive moment region, where the moment diagrams and envelopes are relatively flat. This helps restrict inelastic redistribution of positive
moments to pier sections that have limited ability to sustain these additional uncalculated moments. Also, the Engineer should note that the analysis assumption that the concrete slab is fully effective in tension and compression tends to give a slightly conservative estimate of the true pier section moments, were the cross-sections to remain fully elastic in the positive moment regions.
2. For curved I-girder bridges, all composite sections in positive bending are required to be considered as noncompact sections. Furthermore, the use of AASHTO (2004) Chapter 6 Appendix A is not permitted for curved I-girder sections in negative bending with compact or noncompact webs. Both of these restrictions limit the calculated girder flexural resistance to a maximum potential value of
Mn = RbRhMyc (3-3)
in the absence of any flange lateral bending, where Rb is the web load shedding strength
reduction factor, equal to 1.0 for noncompact and compact webs, Rh is the hybrid factor, and
Myc is the nominal yield moment capacity with respect to the compression flange. These
restrictions are due to the limited data on the influence of partial cross-section yielding on the distribution of forces and moments within curved I-girder bridges.
Beshah and Wright (2006) and Jung and White (2006a) provide extensive results from a full- scale curved composite I-girder bridge test as well as parametric extensions of these test results using refined inelastic finite element analysis. All cases considered indicate that the influence of partial yielding on the internal forces and moments is small in curved I-girder bridges up to the limit of resistance of the most critical bridge component based on the plastic moment Mp with a reduction for flange lateral bending effects. However, the majority of
these studies focus solely on simple-span bridges. Further studies are needed to address the influence of partial cross-section yielding on continuous-span curved I-girder bridges. The benefits of designing positive moment sections using the plastic moment resistance Mp or Eq.
Mp based on ductility considerations (see Section 5.3.3). No studies have been conducted to
date (2006) that address the potential use of a Eq. (2) or other plastic moment-based resistance formulas for curved composite box girders in positive bending.
The Engineer should note that the resistance equations for curved I and box girders generally are based on some partial cross-section yielding at the calculated limit of the resistance. However, Eq. (2) and other Mp-based resistance equations rely on the development of a
larger extent of yielding.
3. For shored composite construction, the maximum compression stress in the concrete deck is limited to 0.6f'c under all strength loading conditions for noncompact composite I-sections
and box-sections in positive bending (see Sections 5.3.3(A) and 5.3.4 for the definition of a noncompact composite I-section). This limit is required to ensure linear behavior of the concrete. However, Article C6.10.1.1 of AASHTO (2004) recommends against the use of shored composite construction. Unshored construction is considered generally more economical. Furthermore, there is limited data on the influence of concrete creep on the response of shored composite I-girders subjected to large dead load (AASHTO 2004). In addition to the above restrictions, the analysis is generally required to consider the separate noncomposite stresses generated in the structure due to self weight and other loadings before composite action is achieved, as well as the short and long-term stresses generated in the composite structure. Moments from these three different loading conditions may not be added for the purpose of calculating stresses, and superposition (based on small-deflection theory) cannot be applied for the analysis of construction processes that include changes in the stiffness of the structure. Long-term loading effects are considered by use of a modular ratio of 3n, where n = Es/Ec is the modular ratio of the composite section for short term loading. Finally, Article
6.10.1.7 of AASHTO (2004) implements specific slab reinforcing steel requirements for regions subjected to negative flexure. These requirements are intended to control concrete cracking, i.e., to ensure distributed cracking with small crack widths. This helps ensure the validity of the assumption of taking the concrete as fully effective in tension for calculation of the elastic internal forces and moments.
AASHTO (2004) Article C4.5.3.1 states that small-deflection theory, or a geometrically linear or first-order analysis, is usually adequate for stringer-type bridges. The terms first-order analysis, geometrically linear analysis or small-deflection theory all indicate that equilibrium of the structure is considered on the undeflected geometry. Article C4.5.3.1 also indicates that bridges that resist loads by a couple whose tensile and compressive forces remain essentially in fixed positions relative to each other while the bridge deflects, such as trusses and tied arches, tend to be insensitive to deformations. However, the internal forces and bending moments can be influenced significantly by second-order effects in structures with members or components subjected to significant axial compression relative to their elastic buckling resistance. Also, the internal forces in form-active structures, such as suspension bridges, are influenced significantly by these effects.
In some stringer-type bridges, construction deflections and stresses (prior to the completion of the full structure) may be influenced significantly by second-order effects. For example, the torsional deformations during construction of some curved and/or skewed I-girder bridges are
sensitive to these effects (Jung and White 2006a; Chang and White 2006). The influence of second-order effects on the flange lateral bending stresses can be significant in straight or curved fascia I-girders subjected to eccentric concrete deck overhang loads acting on cantilever forming brackets. In these cases, an approximate second-order analysis consisting of applying
amplification factors to the internal stresses obtained from first-order methods is essential at a minimum5. As the second-order effects become larger, the use of a refined second-order analysis is prudent.
The term second-order analysis indicates that equilibrium is evaluated on the deflected geometry of the structure. The second-order effects are the changes in the deflections, internal forces and internal moments, relative to those estimated from first-order analysis, due to
considering equilibrium on the deflected geometry. First-order analysis is sufficient generally for calculation of live load effects on all stringer-type bridges in their final constructed configuration.
The component resistance equations in AASHTO (2004), and the strength limit states checks represented by Eq. (1), are based on the assumption that the second-order elastic internal stresses on the initially perfect structure (no consideration of geometric imperfections), are calculated with sufficient accuracy in cases where these effects are important. That is, initial geometric imperfections within fabrication and erection tolerances do not need to be considered in the analysis. These effects are considered in addition to the effects of initial residual stresses within the component resistance equations. Various other incidental contributions to the internal stresses are neglected generally at the discretion of the engineer. These include flange lateral bending stresses in straight non-skewed I-girder bridges, stresses due to restraint of thermal expansion, longitudinal warping stresses in boxes under strength loading conditions, and St. Venant torsional shear stresses in I-section members. Article C6.7.2 of AASHTO (2004) states that the Engineer may need to consider the potential for problematic locked-in stresses in curved I-girder flanges or the cross-frames or diaphragms of curved I-girder bridges when the cross-frames are detailed such that they fit up with the I-girders in an idealized web-plumb position under the steel or total dead load. This article states further, “The decision as to when these stresses should be evaluated is currently a matter of engineering judgment. It is anticipated that these stresses will be of little consequence in the vast majority of cases …” Chang and White (2006) have
developed and applied prototype tools that permit the precise calculation of erection stresses and deflections in curved I-girder bridges. Their results support the above statement, although one of their examples illustrates a curved I-girder bridge where consideration of lack-of-fit and second- order effects is important.
Many of the provisions in AASHTO (2004) Chapter 4 address the appropriate assumptions and limits for the use of approximate analysis methods, with the approximate analysis of stringer-type bridges using line-girder models receiving substantial attention. AASHTO Article 4.1 states:
“The primary objective in the use of more sophisticated methods of analysis is to obtain a better understanding of the structural behavior. Such improved understanding may often, but not always, lead to the potential for saving material…. With rapidly improving computing technology, the more
refined and complex methods of analysis are expected to become commonplace. Hence, this section addresses the assumptions and limitations of such methods. It is important that the Engineer understand the method employed and its associated limitations.”
One of the limitations of general second-order elastic analysis methods is that superposition of the effects from separate loading types is not valid. With these methods, the structure strictly must be analyzed for each load combination and load placement. However, various
simplifications and approximations allow for limited superposition of certain results. For
example, for a curved I-girder bridge that is sensitive to second-order effects in its noncomposite condition but insensitive to these effects after the structure is made composite, second-order analysis can be employed to determine the dead load and construction stresses. The results from a first-order geometrically linear analysis can be subsequently added to these stresses for
evaluation of the composite structure (Jung and White 2006a; Chang and White 2006). For suspension bridges, Podolny and Goodyear (2006) discuss commonly employed approximate linearized solutions that allow the development and use of influence lines.
4. OVERALL SYSTEM BUCKLING VERSUS INDIVIDUAL MEMBER BUCKLING