6.4.the Load Path and Load Distribution in Bridge Superstructures

CHAPTER 4

THE LOAD PATH AND LOAD DISTRIBUTION IN BRIDGE SUPERSTRUCTURES [4]

4.1 INTRODUCTION

An efficient design of a bridge's superstructure is essential to achieving overall economy in the whole bridge structure, in that the superstructure deadweight may form a significant portion of the total gravity load that the bridge must transmit to the foundation. The low initial cost of a lightweight superstructure will translate into an overall economy resulting from reduced size of both the substructure and the foundation components. A clear understanding of the structural behavior of superstructures under loads is essential for efficient design.

A bridge superstructure is an integrated body of various members of reinforced concrete, prestressed concrete, or steel in the form of slabs, stringers, floor beams, diaphragms, etc.; determination of forces in these components is essential for design purposes. The term "load distribution" is often used in a generic sense to denote superstructural analysis; i.e., the determination of forces in and interaction among its components; these two terms will be used synonymously throughout this text.

The advent of computers has led the analysis of bridge superstructures from hand calculations to methods that have made their complex analysis possible without recourse to complicated mathematical theory. However, a bridge engineer should be familiar with the underlying theories, not only because they provide a necessary background to understanding the physical behavior of superstructures and give a feel for the computer methods that are based on the approximate solutions of these classical methods, but also because they are useful in discerning the merits and applicability of various methods. Accordingly, an overview of these methods is presented in this chapter.

CHAPTER 4: THE LOAD PATH AND LOAD DISTRIBUTION IN BRIDGE SUPERSTRUCTURES

4.2 BRIDGE GEOMETRY

A brief review of the principal types of superstructures from the viewpoint of geometric and behavioral characteristics is presented in this section. Several terms will be used in the context of this discussion. As shown in Fig. 4.1, the term longitudinal is used to denote a direction parallel to traffic, while transverse denotes a direction perpendicular to it. From geometric considerations, bridges are often described as normal (or right), skew, and curved. Normal, or right, bridges are those in which the longitudinal axis of the bridge, which is parallel to the longitudinal axes of the slab, and the supporting, beams (when present) are normal to the centerlines of supports (abutments and/or piers).

Figure 4.1 Definition of right and skew bridges

Often, such a plan configuration may not be feasible because of human-created obstacles, complex intersections, space limitations, mountainous terrain, etc., and the result is a skew bridge. A skew bridge, simple or continuous, is characterized by its longitudinal axis, which forms an acute angle, instead of a right angle, with the centerlines of the supports. Angle of skew (or skew angle) is defined as the angle between the centerline of the supports and the normal to the axis of the bridge. The skew angles at the two end supports may not necessarily be the same. In bridge geometry with skewed but parallel lines of supports at the two opposite ends is known as the standard skew. Bridges with the line of support at one end

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normal to the bridge axis but with the other support skewed are half-skew, and those with different skew angles at the two supports are known as trapezoidal skew. Skew angle is an important parameter affecting the analysis of the bridge structure, whether simple or continuous; with torsionally stiff girders, the skew angle can have a considerable effect on the shear and bending moments in the girders. It has been suggested that, for skew angles not exceeding 20° (30° for slab-on-beam bridges, described later), bridges can be safely designed as right or normal bridges by simplified methods.

Curved bridges also referred to as horizontally curved bridges, have become almost standard features of highway interchanges and urban expressways in recent times. They are characterized by their out-of-straightness alignment, as viewed in the plan. Curved bridges result from several factors, such as the design requirements for interchanges, the need for smooth dissemination of congested traffic, right-of-way limitations, local topography and foundation conditions, and aesthetics. Initially, curved bridges comprised a series of straight girders used as chords in forming a curved alignment. Recent developments, however, have led to the replacement of straight girders by curved girders. Curved geometry introduces considerable complexity in bridge analysis, for curved girders are subjected not only to flexural stresses, but also, due to the eccentricity of the mid span with respect to the supports, to very significant torsional stresses. Accordingly, the methods of analysis used for straight bridges cannot be used for curved bridges.

4.3 DIAPHRAGMS

Diaphragms are short structural members positioned transversely to and between adjacent stringers at various intervals and at abutments. They usually consist of channels, W shapes, cross frames, or solid vertical slabs (in the case of concrete beams). The purpose of providing diaphragms is to ensure lateral distribution of live loads to various adjacent stringers, which depends on both the stiffness of the diaphragms relative to the connected stringers and the method of connectivity. However, the extent of this structural contribution has not been quantified. The diaphragm's action comes into play when a load is placed on the deck and applied at the diaphragm location, a condition seldom realized in practice. Research shows that a diaphragm under the load lessens the load carried by the girder immediately under the vehicle by transferring portions of it to adjacent girders; remote diaphragms do not participate in this distributive action. For full effectiveness (i.e., more uniform distribution of live loads transversely) under highway loadings or railway loadings, several closely spaced diaphragms should be provided; this will cause the deck to act as a two-way slab.

4.4 BASIC CONCEPTS OF LOAD DISTRIBUTION

To understand the meaning of and the concern for load distribution in highway bridges, it is instructive to examine the load path (or the load-transfer mechanism) in buildings with floors supported on longitudinal beams. The dead load of the floor is assumed to be transferred to the supporting beams on the tributary area basis. The live load, such as building occupants, furniture, machines, and fixtures, although not uniform in reality, is assumed to be uniformly distributed over the floor area and is also assumed to be transferred to the supporting beams on the tributary area basis. Equally spaced interior beams, under this assumption, are assumed to share floor loads equally. The same analogy can be used for bridge decks supporting bridge live loads

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(vehicles), with one notable exception: the live loads (machines, fixtures, equipment, etc.) in buildings occupy relatively fixed positions (they are pseudo static), whereas bridge live loads occupy only a partial area of the decks.

Figure 4.2 (a) A slab-on-beam deck (b) Bending of the directly loaded beam in the plane of the loads without twisting (c) Transverse bending of the slab accompanied by twisting of beams remote from the directly loaded beam

In fact, the live load on bridge decks, consisting of concentrated wheel loads, may occupy random positions, both longitudinally and transversely, and thus will affect the live load shared by various beams supporting the deck. This aspect of live-load distribution is one of the primary concerns in the analysis of bridge decks.

Physical reasoning can be used to get a feel for the complexity involved in analysis of bridge decks. For simplicity, a slab-on-beam type deck, shown in (Fig a), may be used to illustrate the general nature of the problem. Again, for simplicity, it may be assumed that the deck is loaded longitudinally by one line of wheel loads. If these concentrated loads are placed on the deck directly over one of the beams, that particular beam will bear a greater share of the total load than the other parallel beams remote from it. The slab and all of the beams will bend longitudinally in

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the plane containing the line of the loads (Fig b). The slab, however, bends transversely also, causing the remote beams (but not the directly loaded beam) to twist along with it, to maintain the overall compatibility of displacements at the slab-beam interfaces (Fig c).The participatory action of these remote beams will depend on the stiffness of the slab, and the span, spacing, and stiffness of beams. The determination of this participatory action of the various superstructure components is referred to as load distribution.

4.5 STRUCTURAL FORMS AND BEHAVIORAL CHARACTERISTICS

A bridge deck is the medium through which all bridge loads are transferred to other components. Figure shows a typical cross section of a bridge over a waterway, in which various components are identified. The general load path (or load transfer mechanism) for most common types of bridge decks is shown in Fig.

Depending on the purpose, bridge superstructures can be classified in several ways, as explained in Chapter 2. The applicability of an analytical method for a particular type of deck depends on the complexity of its structural form and behavioral characteristics; from this standpoint, the most commonly used bridge decks can be classified as follows.

1. Slab decks2. Beam-and-slab decks3. Beam decks

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Figure 4.3 The Load Path in Bridge superstructure

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Figure 4.4 Typical cross section of a bridge over a waterway

4.5.1 SLAB DECKS

The slab deck is the most commonly used type of deck for short-span bridges. The load-carrying mechanism of a slab is analogous to that of a plate, which is characterized by its ability to transfer bending and twisting in its own plane owing to continuity in all directions. Its physical behavior is explained in Fig, which shows the slab divided into several rectangular elements.

Application of a load on a portion of a slab causes it to deflect locally in a "dish," causing a two-dimensional system of bending and twisting moments; through this mechanism the load is transferred to the adjacent parts of the deck, which are less severely loaded. Usually, slabs are poured in place. For analytical purposes, they are said to be isotropic if they have similar stiffness in both the longitudinal and transverse directions, and orthotropic otherwise Slab decks are not economical for spans exceeding 50 ft or so, owing to the excessive deadweight resulting from large depth requirements. Hollow concrete slabs (voided slabs) were developed to overcome this problem, by incorporating voids of circular or rectangular cross section placed symmetrically about the neutral axis.

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Figure 4.5 Plate action in slab; Distribution of forces by bending and twisting in two directions

4.5.2 BEAM AND SLAB DECKS

Figure 4.6 shows cross sections of commonly used beam-and-slab bridge decks, schemes (a), (b), and (c) being the most commonly used for highway bridges in the short-span range. Beam-and-slab decks comprise a number of usually equally spaced (generally 6-12 ft apart) beams spanning longitudinally between supports (hence the design is also referred to as spaced beam-and-slab decks, or parallel girder systems with a thin, structurally continuous slab spanning transversely across the top. The slab serves the dual purpose of supporting the live load on the bridge and acting as the top flange of the longitudinal beams The slab can be non composite or composite, the latter being the obvious choice for economy and structural efficiency, in which case the slab structurally acts as the top flange of the beams.

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Figure 4.6 Cross sections of bridge-and-slab decks: (a) T-beam bridges, (b) Slab-on-steel beams, (c) Slab on prestressed concrete beams

In this scheme, under the load, the slab deflects in a single smooth wave so that the resulting structural behavior is akin to that of an orthotropic slab with longitudinal stiffening. In conjunction with steel beams, the concrete slab may be replaced by stiffened steel "battle deck" fabricated from stiffened thin plate.

A structural characteristic of the spaced beam-and-slab deck schemes of Fig. 4.6 is their use of beams of open cross section, which are inherently torsionally weak. A more efficient type of deck is the multi span deck shown in Fig. 4.8 which comprises girders of closed cross section (concrete or steel) and a continuous structural concrete deck at the top. In its extreme form, this type of configuration can have as few as two spine beams, concrete or steel. Of necessity, concrete spine beams are more closely spaced than steel spine beams. In a two-spine steel beam deck, the spacing of beams can be more than 40 ft, whereas the spacing of solid concrete spines is generally in the 24-ft range.

Beam-and-slab decks of various configurations can be idealized as comprising a series of longitudinally spanning parallel T-beams connected along their edges with full continuity (Fig. 4.7). Under load, the response of a slab is characterized by longitudinal bending as flanges of T-beams, accompanied by transverse bending as a continuous beam.

Figure 4.7 Response of a beam-and-slab deck (a) Longitudinal Bending as flanges of T-beams, (b) Transverse bending as a continuous beam

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4.5.3 BEAM DECKS

A bridge deck can be assumed to behave as a beam when its length-to-width ratio is such that, under loads, its cross sections displace bodily but without any distortion. Because the dominant load in such decks is concentric, distortion of the cross section under eccentric loading has relatively little influence on the principal bending stresses. These decks can be analyzed by classical methods of structural analysis, with stresses determined from simple bending theory and torsion of noncircular cross sections.

4.6 AASHTO METHOD OF LIVE LOAD DISTRIBUTION-SLABS AND BEAMS

With reference to the various superstructure types (Fig. 4.6) and the load path (Fig. 4.3), we now discuss application of the AASHTO method to determine design forces in various components, such as slab, stringers, floor beams, etc.

4.6.1 CONCRETE SLABS

4.6.1.1 SUPPORT CONDITIONS FOR SLAB

Two cases are considered, based on the direction of the span of the slab:

The deck consists only of a reinforced concrete slab supported on abutments and/or piers: The main reinforcement in this case runs parallel to traffic (Fig. 4.8(a)).

The deck slab is supported over a number of parallel steel, concrete, or timber beams. The main reinforcement in this case is oriented perpendicular to traffic (and beams) (Fig. 4.8(b), (c), and (d)).

In both cases, for analytical purposes, the loads are placed on the slab in a specified manner. The two key design forces to be determined are moment and shear in the slab. The empirical formulas used for calculating slab moments are based on the assumed position of the wheel load: one ft from the curb, or one ft from the rail if a sidewalk or curb is not provided.

4.6.1.2 DETERMINATION OF MOMENTS AND SHEAR IN SLABS Case 1: SLAB SUPPORTED ON AN ABUTMENT AND/0R PIERS

This case is covered by AASHTO 3.24.3.2 under "Case B-Main Reinforcement Parallel to Traffic." For moment computations, the span (S) is defined as the distance between the centers of the supports, but S need not exceed the length of the clear span plus the thickness of the slab (AASHTO 3.24.1.1).

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Figure 4.8 Support conditions for a concrete slab: (a) Reinforced concrete slab bridge; (b) Reinforced concrete T-beam bridge; (c) Reinforced concrete slab supported on steel stringers; (d) Reinforced concrete slab supported on prestressed concrete girders

Design moment For simple spans,Dead-load moment = wl2/8

Where w = dead load/ftz of slab and L = span

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Live-load moment (LLM) Two methods are prescribed in AASHTO 3.24.3.2 for determining slab moments.

Method 1 Moment is calculated as if it were caused by concentrated load acting on a simple span. In reality, however, the wheel loads are not true concentrated (or point) loads, but are distributed over the tire contact area. Because of the lateral stiffness of the slab, it is assumed that the distribution area for the wheel load is larger than the actual contact area between the tire and the slab. Thus, a truck wheel load is distributed over a width of E (4 + 0.06S) ft, to a maximum of 7.0 ft. The value of the concentrated load is obtained by dividing the rear wheel load (16 kips for H2O or HS20 loading) by this distribution width. For lane loading, the concentrated load is distributed over twice the distribution width (i.e., 2E ft). Maximum moment is obtained by placing the concentrated load so obtained at the midspan, which gives

M = P’S/4where

S = span (ft)and

P' = P/E for truck loading = P/2E for lane loadingwhere

E = 4 + 0.06S (ft)and

P = P15 = 12,000 lb for H15 truck loading = P20 = 16,000 lb for H2O truck loading = 13,500 lb for H15 lane loading = 18,000 lb for H20 lane loading

Note that in the case of lane loading, the uniform lane loading should also be distributed over the width of 2E and moment (wS2/8), calculated accordingly.

Method 2 Alternatively, for HS20 loading, the maximum live-load moment (LLM) per foot width of slab can be closely approximated by the following empirical formulas:

For spans up to 50 ft, LLM = 900S ft-lbFor spans > 50 to 100 ft, LLM = 1000(1.3S - 20.0) ft-lb

The obvious approximate nature of these values of moments should be recognized. For HS 15 loading, LLM can be taken as three-fourths of the preceding values.

The edges of the deck slab should be stiffened. This can be accomplished by either providing an additionally reinforced slab section, a beam integral with and deeper than the slab, or an integrally reinforced section composed of slab and curb. This portion of the slab, known as the longitudinal edge beam, is designed for a live-load moment equal to 0.1PS, where P = P15 or P20, respectively, for slabs designed for H15 or HS15, or H2O and HS20 loadings. According to AASHTO, the value of moment for the edge beam is not to be increased for impact considerations.The requirement that a wheel load should not be placed closer than 1 ft from the curb or the parapet was noted earlier. These distances are sometimes called the edge distances and

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have an important bearing on the analytical aspects of the slab. The stiffened section at the edges of the slab essentially acts as an L-beam whose neutral axis is higher than that of the slab. For continuous spans, the simple-span moment can be reduced by 20 percent, both for positive and negative moments.

Shear and bond Deck slabs designed according to the preceding methods are considered safe in shear and bond; accordingly, no check is required for this design consideration (AASHTO 3.24.4).

4.6.1.2 DETERMINATION OF MOMENTS AND SHEAR IN SLABS Case 2: SLAB SUPPORTED ON BEAMS AND STRINGERS

This case is covered by AASHTO 3.24.3.1, "Case A-Main Reinforcement Perpendicular to Traffic."

Span In this case, the slab is usually continuous over several parallel stringers or beams, which can be reinforced concrete (T beams), prestressed concrete, steel, or timber. The effective span, S, is different in each case, as follows (AASHTO 3.24.1.2):

When the supporting beams are made from reinforced concrete, the slab is usually cast monolithic with them, resulting in a T -beam section. In such cases, S equals the clear span (Fig. 4.9(a)). This specification also applies in the case of slabs supported on rigid top flange prestressed concrete beams, where the ratio of top flange width to minimum thickness is less than 4.0.

When the slab is supported on steel stringers, S equals the clear span plus half the width of the stringer flange (Fig. 4.9(b)). In a case where the widths of the top and the bottom flanges are different (e.g., in a composite plate girder), the top flange width should be considered in computing S. This specification also applies in the case of slabs supported on thin top flange prestressed concrete beams, where the ratio of top flange width to minimum thickness is 4.0 or larger.

Figure 4.9 Effective spans for stringer-supported slabs

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Design momentDead load moment = MD = wL2/8 or wS2/8

Where w = dead load/ft2 of slab and L = effective span

Live-load moment (LLM) LLM is determined from following equation

LLM = P(S+2)/32 ft-lb

Where Pi = P20 = 16,000 lb (load on one rear wheel of an HS20 truck) = P15 = 12,000 lb (load on one rear wheel of an HS15 truck)

Note that above equation gives design moment per foot width of a simple slab, and the span, S, is in foot units. These moments are to be reduced by 20 percent when the slab is continuous over three or more stringers, which is generally the case for concrete decks. This reduction of moments is specified in AASHTO 3.24.3.1, which permits a continuity factor of 0.8 to be applied to Equation. In a general sense, this is analogous to using, for uniform load, M = wL2/10 for a continuous span instead of M = wL2/8 for a simple span (0.8wL2/8 = wL2/10).

4.6.2 FORCES IN STRINGERS

Stringers are the longitudinal beams supported on abutments and piers. They receive their loads directly from the slab which they support. The two outside stringers on each side of the bridge deck are called the exterior stringers; the remaining stringers are the interior stringers.

4.6.2.1 DEAD LOAD FORCES

Dead load is distributed to various stringers in proportion to their tributary widths. In most cases, the stringers are spaced equally, resulting in equal tributary widths for all interior stringers. Consequently, all interior stringers are assumed to carry equal amounts of dead load. Essentially, the dead load consists of the deck slab, the wearing surface, sidewalks, curbs, parapets, and railings. Usually, the slab is poured first, and after the slab has hardened, the pouring of the curb, sidewalk, and parapet follows. The dead load due to these components, also referred to as superimposed dead load (usually denoted as DL2 in computations), is assumed to be shared equally by all stringers, although, in reality, they are positioned in closer proximity to the exterior stringers (AASHTO 3.23.2.3.1.1). Recent studies suggested that

1. For the right bridges, 80 percent of the sidewalk and parapet loads are taken by the exterior beams, and 20 percent by the interior beams;

2. The asphalt wearing surface load is distributed to each beam in the ratio of its mo-ment of inertia to the total moment of inertia of all the beams.

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Figure 4.10 A typical parapet section

The tributary area of the exterior stringers is usually smaller than that of the interior stringers. Also, as shown in Fig. 4.09, the deck slab, with edges stiffened by an inte grally cast curb, parapet, or sidewalk, generally overhangs from the exterior stringers in the transverse direction. In such cases, the load on the exterior stringer can be calculated as a reaction by considering the slab (with an overhang) as simply supported on both the exterior and the first interior stringer (i.e, by the lever rule method). Figure 4.10 shows one of the commonly used concrete parapet curbs (the mounted aluminum railing not shown), weighing about 505 lb/ft. Other variations are also used.

4.6.2.2 LIVE LOAD FORCES

The problem of determining shears and moments in stringers due to live loads received from the slab is highly indeterminate owing to the fact that moving loads, generally speaking, can literally occupy any position on the deck; the only exception to this is the portion of the slab covered by the sidewalk, curb, and parapet. Although live loading is assumed to occupy and move in the designated lanes, such specification is hardly enforceable. This complexity and uncertainty in application of the live load on the slab has led to simplifying assumptions regarding the transverse position of the moving load on the deck. The conclusion is that the live load shared by the exterior stringer would be different from that shared by the interior stringer. This is evident because the exterior stringers support the portion of the slab covered by the sidewalk, curb, and railing, thereby precluding the possibility of the live load occupying this portion of the slab.

Owing to the presence of the slab, the loads are not applied directly to the stringers; rather, the stiffness of the slab causes lateral distribution of the moving loads to the adjacent stringers. The lateral distribution, however, is a highly indeterminate coupled phenomenon that depends on the stiffness and the type of the deck, the type and spacing of supporting stringers, and the stiffness of diaphragms. In view of the theoretical complexity involved in load distribution to stringers, AASHTO 3.23 provides a simplified, but empirical, method to determine the lateral distribution of moving loads to both exterior and interior stringers. This method is referred to as distribution of loads, and, although simple, it does have some

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limitations (discussed in a later section). According to this method of analysis, shear and moments in stringers are obtained first as if they are directly loaded by the axle (or wheel) loads. These values are then multiplied by the appropriate live-load distribution factors, DF, (listed in AASHTO Table 3.23.1) and by the impact factors (1 + I), to obtain design shear and moments in stringers.

In the case of lane loading, the uniform load is assumed to be distributed over a lane width of 10 ft. This stipulation also applies to the concentrated loads associated with the lane loading. For example, in the case of HS20 lane loading, the 0.64-k/ft lane load is assumed to be distributed over a width of 10 ft on a line normal to the centerline of the lane, resulting in a uniform load intensity of 0.064 k/ft2 on the loaded lane. The concentrated loads associated with HS20 lane loading are 18 kips for moment and 26 kips for shear (AASHTO 3.7.1.3). These concentrated loads are also assumed to be distributed over a width of 10 ft, giving a line load of 1.8 k/ft for moment and 2.6 k/ft for shear (AASHTO 3.7.1.2). Note that, as far as the live-load distribution factors are concerned, no distinction is made in the AASHTO specifications between noncomposite and composite construction.

Shear and moment in a stringer due to moving loads can be easily determined from influence lines. It is incumbent on a designer to apply bridge live loads on the deck in specific manners outlined in the AASHTO specifications. The following points must be noted in this context:

1. In the case of single-span bridges, regardless of the span length of the bridge, only one H or HS truck is assumed to occupy the bridge. The distance between the two rear axles is kept as a variable between 14 and 30 ft, as shown in Fig. 4.15; this dis-tance should be chosen so as to cause maximum stresses in the supporting members.

2. In the case of lane load, the entire span is assumed to be occupied by uniform lane loading in the designated lane. The line load due to concentrated loads is to be posi-tioned, along with the uniform live load, to cause the most critical stress conditions in the member under consideration.

3. In the case of continuous spans, they should be loaded so as to cause maximum effects (stresses and deflections) in the member under consideration.

Note that shears and moments tabulated in the AASHTO specifications have been com-puted for the governing truck or the lane loading. In the case of truck loading, only one truck is assumed to be present on the span, and the tabulated values of moments and shears are computed for the corresponding axle loads (8 kips and 32 kips for H20; and8 kips, 32 kips, and 32 kips for HS20 loading; all spaced 14 ft apart). The effect of multipresence of vehicles (i.e., two or more lanes loaded simultaneously) is not included in these values. The effect of multilane loading is considered by multiplying the single-lane loading by the multipresence reduction factor, as given in AASHTO 3.12. However, it is reiterated that the multipresence reduction factors are not to be used in conjunction with the distribution factors, except where the lever rule is used or where special requirements for the exterior beams in beam-slab bridges as specified in AASHTO-LRFD 4.6.2.2.2d are used.

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6.4.the Load Path and Load Distribution in Bridge Superstructures

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