The following groups represent various combinations of service loads and forces to which a structure may be subjected. Every component of substructure and superstructure should be proportioned to resist all combinations of forces applicable to the type of bridge and its site.

For working-stress design, allowable unit stresses depend on the loading group, as indicated in Table 11.6. These stresses, however, do not govern for members subject to repeated stresses when allowable fatigue stresses are smaller. Note that no increase is permitted in allowable stresses for members carrying only wind loads. When the section required for each loading combination has been determined, the largest should be selected for the member being designed.

The ‘‘Standard Specifications for Highway Bridges’’ of the American Association of State Highway and Transportation Officials specifies for LFD, factors to be applied to the various types of loads in loading combinations. These load factors are based on statistical analysis of loading histories. In addition, in LRFD, reduction factors are applied to the nominal resistance of materials in members and to compensate for various uncertainties in behavior.

To compare the effects of the design philosophies of ASD, LFD, and LRFD, the group loading requirements of the three methods will be examined. For simplification, only D, L, and I of Group I loading will be considered. Although not stated, all three methods can be considered to use the same general equation for determining the effects of the combination of loads:

For a non-compact flexural member subjected to bending by dead load, live load, and impact forces, let D, L, I represent the maximum tensile stress in the extreme surface due to dead load, live load, and impact, respectively. Then, for each of the design methods, the following must be satisfied:

For LFD and LRFD, if the section is compact, the full plastic moment can be developed.

The effect of the applied loads appears to be less for LRFD, but many other factors apply to LRFD designs that are not applicable to the other design methods. One of these is a difference in the design live-load model. Another major difference is that the LRFD specifications require checking of connections and components for minimum and maximum loadings.

(Dead loads of components and attachments are to be varied by using a load factor of 0.9 to 1.25.) LRFD also requires checking for five different strength limit states, three service limit states, a fatigue-and-fracture limit state, and two extreme-event limit states. Although each structure may not have to be checked for all these limit states, the basic philosophy of the LRFD specifications is to assure serviceability over the design service life, safety of the bridge through redundancy and ductility of all components and connections, and survival (prevention of collapse) of the bridge when subjected to an extreme event; e.g., a 500-year flood. (See Art. 11.5.4.)

## Simplified Example of Methods

To compare the results of a design by ASD. LFD, and LRFD, a 100-ft, simple-span girder bridge is selected as a simple example. It has an 8-in-thick, noncomposite concrete deck, and longitudinal girders, made of grade 50 steel, spaced 12 ft c to c. It will carry HS20 live load. The section modulus S, in3, will be determined for a laterally braced interior girder with a live-load distribution factor of 1.0.

The bending moment due to dead loads is estimated to be about 2,200 ft-kips. The maximum moment due to the HS20 truck loading is 1,524 ft-kips (Table 11.7).

If a noncompact section is chosen, this value of S is the required elastic section modulus.

For a compact section, it is the plastic section modulus Z. Figure 11.4 shows a noncompact section supplying the required section modulus, with a 3⁄8-in-thick web and 15⁄8-in-thick flanges. For a compact section, a 5⁄8-in-thick web is required and 11⁄4-in-thick flanges are satisfactory. In this case, the noncompact girder is selected and will weigh 265 lb per ft.

**Load-and-Resistance-Factor Design**. The live-load moment ML is produced by a combination of truck and lane loads, with impact applied only to the truck moment:

The section selected for ASD (Fig. 11.3) is satisfactory for LRFD.For this example, the weight of the girder for LFD is 94% of that required for ASD and 90% of that needed for LRFD. The heavier girder required for LRFD is primarily due to the larger live load specified. For both LFD and LRFD, a compact section is advantageous, because it reduces the need for transverse stiffeners for the same basic weight of girder.

## Limit States

The LRFD Specifications requires bridges ‘‘to be designed for specified limit states to achieve the objectives of constructibility, safety and serviceability, with due regard to issues of inspectability, economy and aesthetics’’. Each component and connection must satisfy Eq.

11.8 for each limit state. All limit states are considered of equal importance. The basic relationship requires that the effect of the sum of the factored loads, Q, must be less than or equal to the factored resistance, R, of the bridge component being evaluated for each limit state. This is expressed as

There are four limit states to be satisfied: Service; Fatigue and Fracture; Strength; and, Extreme Event. The Service Limit State has three different combinations of load factors, which place restrictions on stress, deformation and crack width under regular service conditions.

Service I and III apply to control of prestressed members. Service II, intended to control yielding of steel structures and slip of slip-critical connections, corresponds to what was previously known as the ‘‘overload’’ check.

The Fatigue and Fracture Limit State checks the dynamic effect on the bridge components of a single truck known as the fatigue truck. Restrictions are placed on the range of stress induced by passage of trucks on the bridge. This limit is intended to prevent initiation of fatigue cracking during the design life of the bridge. Article 11.10 provides additional discussion of the Fatigue Limit State.

Fracture is controlled by the requirement for minimum material toughness values included in the LRFD Specification and the AASHTO or ASTM material specifications, and depends upon where the bridge is located. (See Art. 1.1.5.) Section 11.9 provides additional discussion of the Fracture Limit State.

The Strength Limit State has five different combinations of load factors to be satisfied.

This limit state assures the component and/or connection has sufficient strength to withstand the designated combinations of the different permanent and transient loadings that could statistically happen during the life of the structure. This is the most important limit state since it checks the basic strength requirements. Strength I is the basic check for normal usage of the bridge. Strength II is the check for owner specified permit vehicles. Strength III checks for the effects of high winds (>55 mph) with no live load on the bridge, since trucks would not be able to travel safely under this condition. Strength IV checks strength under a possible high dead to live load force-effect ratio, such as for very long spans. This condition governs when the ratio exceeds 7.0. Strength V checks the strength when live load is on the bridge and a 55 mph wind is blowing.

Extreme Event Limit State is intended ‘‘to ensure the structural survival of a bridge during a major earthquake or flood, or when collided by a vessel, vehicle or ice flow possibly under a scoured condition.’’ This design requirement recognizes that structural damage is acceptable under extreme events, but collapse should be prevented.

For the design example included in the Appendix, page 11.78, the engineers provided a summary to illustrate the relative influence for all the LRFD requirements on the design.

The results for each limit state are expressed in terms of a performance ratio, defined as the ratio of a calculated value to the corresponding allowable value. This summary, Table A1, indicates that the Fatigue and Fracture Limit State, Base metal at connection plate weld to bottom flange (at 0.41L) is the governing criteria. In fact, it is slightly overstressed, in that the ratio between actual and allowable value is 1.008. However, this very small excess was accepted. It is recommended that designers develop performance ratios for all designs.

## LRFD Load Combinations

The effects of each of the loads discussed in Art. 11.4, appropriately factored, must be evaluated in various combinations for LRFD as indicated in Tables 11.8 and 11.9. These combinations are statistically based determinations for structure design. Only those applicable to steel bridge superstructure designs are listed. See the LRFD Specification for a complete listing. See the example in the Appendix for a listing of design factors and illustration of application of load combinations and load factors.

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