SECTION 4: STRUCTURAL ANALYSIS AND EVALUATION TABLE...

SECTION 4: STRUCTURAL ANALYSIS AND EVALUATION

TABLE OF CONTENTS

4-i

4 4.1 SCOPE..................................................................................................................................... 4-1 4.2 DEFINITIONS ........................................................................................................................ 4-1 4.3 NOTATION ............................................................................................................................ 4-1 4.4 ACCEPTABLE METHODS OF STRUCTURAL ANALYSIS ........................................ 4-3

4.4.1 General ............................................................................................................................ 4-3 4.4.2 Amplification of Live Load Effects ................................................................................ 4-3

4.6.1.3 Structure Skewed in Plan ....................................................................................... 4-5 4.6.2 Approximate Methods of Analysis ................................................................................ 4-6

4.6.2.2 Beam-Slab Bridges and Box-Beam Bridges ......................................................... 4-7 4.6.2.2.1 Application................................................................................................... 4-7

4.6.2.2.2 Slab-on-Beam and Box-Beam Bridges ............................................................. 4-13 4.6.2.2.2a Shear and Reaction.................................................................................... 4-14 4.6.2.2.2b Bending Moment....................................................................................... 4-15 4.6.2.2.3 Open Steel Boxes (c from 4.6.2.2-1)........................................................... 4-17 4.6.2.2.4 Wood Decks on Wood or Steel Beams....................................................... 4-17 4.6.2.2.6 Beams with Corrugated Steel Decks........................................................... 4-20 4.6.2.2.7 Special Loads with other Traffic................................................................ 4-21 4.6.2.2.8 Curved Steel Bridges .................................................................................. 4-21

A4 DECK SLAB DESIGN TABLE ............................................................................................ 4-24 B4 LEVER RULE FORMULA................................................................................................... 4-25 C4 ALTERNATE MOMENT DISTRIBUTION FACTOR APPROACH.................................. 4-27

SECTION 4 (US)

STRUCTURAL ANALYSIS AND EVALUATION

4.1 SCOPE

…

C4.1

… 4.2 DEFINITIONS

… Adjusted equal distribution – an approximate distribution factor method that is based upon a uniform distribution (equal to all girders) with increases to better approximate rigorous analysis.

… Calibrated lever rule – an adjustment to the results of a lever rule distribution factor analysis to better approximate rigorous analysis.

… Lever rule – An approximate distribution factor method that assumes no transverse deck moment continuity at interior beams, rendering the transverse deck cross section statically determinate. The method uses direct equilibrium to determine the load distribution to a beam of interest. The centerline of box girders may be assumed to be the center of the girder and used for lever rule computations. Lever rule formulae – formulae that facilitate the use of the lever rule. Provided in Appendix B, these formulae are functionally equivalent to the lever rule.

… 4.3 NOTATION

… am = calibration constant for the lever rule for moment (4.6.2.2.2) av = calibration constant for the lever rule for shear (4.6.2.2.2)

… bv = calibration constant for the lever rule for shear (4.6.2.2.2) bm = calibration constant for the lever rule for moment, effective flange width for normal forces acting at

anchorage zones (in.) (4.6.2.6.2) (4.6.2.2.2)

… COVS/R = coefficient of variation of ratio of the samples of simplified (specification) to rigorous methods for live

load distribution factor computation (C4.4.2)

… Fl = optional modification factor based on span length in 4.6.2.2.2b. (C4.6.2.2.2) Fst = modification factor for structure type in Table 4.6.2.2.2b-3. (4.6.2.2.2)

… gm = multiple lane live load distribution factor (4.6.2.2.7)

… glever rule = distribution factor computed with the lever rule. (4.6.2.2.2) (4.6.2.2.4) (4.6.2.2.6)

… mg = distribution factor, adjusted for the variability of the simplified method with an analysis factor, including

the effects of multiple presence. (4.6.2.2.1) (4.6.2.2.2) (4.6.2.2.3) (4.6.2.2.4) (4.6.2.2.6)

4-1

4-2 AASHTO LRFD BRIDGE DESIGN SPECIFICATIONS

mgm = distribution factor, adjusted for the variability of the simplified method with an analysis factor, including the effects of multiple presence, moment. (4.6.2.2.1) (4.6.2.2.2) (4.6.2.2.3) (4.6.2.2.4) (4.6.2.2.6)

mgv = distribution factor, adjusted for the variability of the simplified method with an analysis factor, including the effects of multiple presence, shear. (4.6.2.2.1) (4.6.2.2.2) (4.6.2.2.3) (4.6.2.2.4) (4.6.2.2.6)

… Ng = number of girders, beams, or stringers in the bridge cross section. (4.6.2.2.1) (4.6.2.2.2) (4.6.2.2.3) NL = maximum number of design lanes considered in an analysis (4.6.2.2.2, 4.6.2.2.3) Nlanes = number of lanes considered in an analysis, e.g., used for lever rule (4.6.2.2.2, 4.6.2.2.3)

… S = girder spacing; if splayed use the largest spacing within the span, ft. If skewed, S is measured

perpendicular to the girder or support elements. (4.6.2.2.1) (4.6.2.2.3) (4.6.2.2.4) (4.6.2.2.6) (Appendix B4)

… Wc = clear roadway width, ft. (4.6.2.2.2)

… za = number of standard deviations that the simplified distribution method approach is adjusted above the

mean rigorous analysis results; set to 0.5 (a conservative adjustment) (C4.4.2)

… γs = live load distribution simplification factor from Table 4.6.2.2.1-2. (4.4.2) (4.6.2.2.1) (4.6.2.2.2)

(4.6.2.2.3) (4.6.2.2.4) (4.6.2.2.6)

… σS/R = standard deviation of the samples of simplified (specification) to rigorous methods for live load

distribution factor computation (C4.4.2)

…

SECTION 4 (US): STRUCTURAL ANALYSIS AND EVALUATION 4-3

4.4 ACCEPTABLE METHODS OF STRUCTURAL

ANALYSIS 4.4.1 General

…

C4.4

…

4.4.2 Amplification of Live Load Effects C4.4.2 Load effects based upon simplified methods of

Articles 4.6.2.2 shall be amplified by the live load distribution simplification factors, γs. Bridges rigorouslyanalyzed with methods of Article 4.4.1 may use a liveload analysis factor of 1.0. Bridges analyzed usingarticles 4.6.2.3, 4.6.2.4, 4.6.2.5, 4.6.2.6, 4.6.2.7, and4.6.3 may use an distribution simplification factor of 1.0. Rigorous analysis results shall not be less than 70percent of those determined by the simplified methodsof 4.6.2.2, 4.6.2.3, 4.6.2.4, 4.6.2.5, 4.6.2.6, 4.6.2.7, and4.6.2.3 without consent of the owner.

When a rigorous analysis is performed, the distribution factors shall be calculated by the simplified method outlined in the specification as well. The codified method is expected to provide a reasonable estimate of rigorous results. Therefore, when results vary significantly the owner shall be advised. This article is to encourage rigorous analysis by providing the owner an approach to help ensure that results from advanced methods will be reasonable. Distribution factors below 70 percent of the simplified methods are possible, especially in the cases of significant skew and unusual geometries. The lower bound provides owners impetus to encourage advanced methods with underlying warning that requires additional review. The analysis factors in Article 4.6.2.2.1 indicate the situations where larger variability between rigorous and simple methods is expected. Higher factors indicate higher variability with respect to rigorous analyses. All rigorous analysis should be compared with independent hand checks as standard practice.

The procedure used to compute the analysis factor is outlined below.

( )/ /1s S R a S Rzγ μ σ= +

where: γs is the distribution simplification factor μS/R is the mean of the ratio of the simple to rigorous for

each sample i

i

SR

⎛ ⎞⎜ ⎟⎝ ⎠

σS/R is the standard deviation of the ratio of the simple to rigorous for each sample

za is the number of standard deviations that the method

is above the mean of the rigorous results

Solve for γs to give


( ) ( )//

/ /

1 1a S Rs a S

S R S R

zz COV

σγ

μ μ+ ⎛ ⎞

= = +⎜ ⎟⎝ ⎠

R

where COVS/R is the coefficient of variation.

For one-half standard deviation, i.e., za = 0.5,

substitution gives

( ) ( )//

/ /

1 0.5 1 0.5S Rs S R

S R S R

COVσ

γμ μ

+ ⎛ ⎞= = +⎜ ⎟

⎝ ⎠

For details, see Puckett, et al. (2006)

…


. 4.6.1.3 Structure Skewed in Plan

When the line supports are skewed and the

difference between skew angles of two adjacent lines ofsupports does not exceed 30 degrees, the effect of skew may be ignored.

Shear in the exterior beam at the obtuse corner ofthe bridge shall be adjusted when the line of support isskewed more than 30 degrees. The value of theadjustment factor shall be obtained from Table 1 andapplied to the distribution factors specified in 4.6.2.2. Indetermining the end shear in multibeam bridges, theskew correction at the obtuse corner shall be applied toall beams. The adjustment for shear shall be applied forreactions. A rigorous analysis shall be performed for skews exceeding 60 degrees.

The maximum bending moment in the beams of askewed bridge is less than in a similar straight bridge. Areduction for such may be ignored.

C4.6.1.3 The effect of skew is relatively small when the

skew angle is less than 30 degrees. However, the adjustments provided for small skew may be used. The moment reduction for skew may be conservatively ignored in design.

The skew adjustment for shear also applies to reactions at interior and exterior supports. NCHRP 20-7 Report 107 (Modjeski and Masters, 2002) outlines a linear interpolation in the transverse and longitudinal direction for the skew adjustment factor. This refinement may be used.

The skew adjustment factor shall be applied to shears and reactions near all supports, both interior and exterior.


Table 4.6.1.3-1 Adjustment Factors for Load Distribution Factors for Support Shear of the Obtuse Corner.

Type of Superstructure Applicable Cross-Section

from Table 4.6.2.2.1-1 Correction Factor

Range of

Applicability

Concrete Deck, Filled Grid, Partially Filled Grid, or Unfilled Grid Deck Composite with Reinforced Concrete Slab on Steel or Concrete Beams; Concrete T-Beams, T- and Double T-Section

a, e and also i, j

if sufficiently connected to act as a unit

1.0 0.20 tan+ θ 0 603.5 16.020 240

4b

SL

N

° ≤ θ ≤ °≤ ≤≤ ≤≥

Precast Concrete I, Bulb Tee Beams, and Precast channel sections with Shear Keys

h, k 1.0 0.09 tan+ θ

0 603.5 16.020 240

4b

SL

N

° ≤ θ ≤ °≤ ≤≤ ≤≥

Cast-in-Place Concrete Multicell Box

d 12.01.0 0.25 tan70

Ld

⎛ ⎞+ + θ⎜ ⎟⎝ ⎠

0 606.0 13.020 24035 110

3c

SLd

N

° < θ ≤ °< ≤≤ ≤≤ ≤≥

Concrete Deck on Spread Concrete Box Beams

b, c

12.01.0 tan6

Ld

S+ θ

0 606.0 11.520 14018 65

3b

SLd

N

° < θ ≤ °≤ ≤≤ ≤≤ ≤≥

Concrete Box Beams Used in Multibeam Decks

f, g 12.01.0 tan90

Ld

+ θ 0 6020 12017 6035 605 20b

Ldb

N

° < θ ≤ °≤ ≤≤ ≤≤ ≤

≤ ≤

4.6.2 Approximate Methods of Analysis

C4.6.2 Articles 4.6.2 are largely based upon the research

work of NCHRP 12-26 and 12-62 (Zokaie et al 1991, Puckett et al 2006).

… ….


4.6.2.2 Beam-Slab Bridges and Box-Beam Bridges

4.6.2.2.1 Application The provisions of this Article may be applied to

straight girder bridges and horizontally curved concrete bridges, as well as horizontally curved steel girder bridges complying with the provisions of Article 4.6.1.2.4. The provisions of this Article may also be used to determine a starting point for some methods of analysis to determine force effects in curved girders of any degree of curvature in plan.

Except as specified in Article 4.6.2.2.7, the

provisions of this article shall be taken to apply tobridges being analyzed for:

• A single lane of loading, or

• Multiple lanes of live load yieldingapproximately the same force effect per lane.

If one lane is loaded with a special vehicle orevaluation permit vehicle, the design force effect pergirder resulting from the mixed traffic may bedetermined as specified in Article 4.6.2.2.7.

C4.6.2.2.1 The V-load method is one example of a method of

curved bridge analysis which starts with straight girder distribution factors (United States Steel 1984).

For two- and three-girder bridges, the live load oneach beam shall be the reaction of the loaded lanesbased on the lever rule unless specified otherwise herein.

The lever rule involves summing moments about one support to find the reaction at another support by assuming that the supported component is hinged at interior supports.

When using the lever rule on a three-girder bridge, the notional model should be taken as shown in Figure C1. Moments should be taken about the assumed, or notional, hinge in the deck over the middle girder to find the reaction on the exterior girder.

Figure C4.6.2.2.1-1 Notional Model for Applying Lever Rule to Three-Girder Bridges.

The provisions of this article shall be taken to apply

to bridges being analyzed for single and multiple lanesof live load, except as specified in Article 4.6.2.2.7.Bridges not meeting the requirements of this article shallbe analyzed as specified in Article 4.4.1.

As specified in Article 3.6.1.1.2, multiple presence factors shall be independently applied based upon thenumber of loaded lanes or as specified herein.

The multiple presence and analysis simplification factors are applied after the computation of the distribution factors. This independence keeps the three


The distribution of live load, specified in Article4.6.2.2.2, may be used for girders, beams, and stringersthat meet the following conditions and any otherconditions identified in tables of distribution factors asspecified herein:

• Width of deck is constant,

• Beams are parallel and have approximately thesame stiffness and spacing,

• Unless otherwise specified, the number ofbeams is not less than four,

• Unless otherwise specified, the position of theoutside wheel, absolute value of de, does notexceed the girder spacing S,

• Curvature in plan is less than the limit specifiedin Article 4.6.1.2.4, or where distributionfactors are required in order to implement anacceptable approximate or refined analysismethod satisfying the requirements of Article4.4 for bridges of any degree of curvature inplan;

• Cross-section is consistent with one of thecross-sections shown in Table 1.

aspects of a simplified live load distribution estimate distinct. The procedure is outlined below:

• Article 4.6.2.2 is used to estimate the live load

effect calibrated near the mean of rigorous analyses. For example, see Figure C4.6.2.2.1-1 from Puckett et al (2006) in whichapproximately 1500 bridges were rigorously analyzed to provide the basis for the simplified method outlined herein.

• Some simplified approaches better approximate rigorous analysis than others for various bridge types and loading conditions. For cases where the simplified method predicts the rigorous analysis results well, the distribution simplification factor is near unity. As an example, the coefficient of variation is 13.4% for the ratio of the simple method (specification) divided by the distribution factor based upon rigorous analysis for the data illustrated in Figure C4.6.2.2.1-1. For this case, the analysis factor is 1.04 and shifts this ratio approximately one-half standard deviation to the conservative side. In cases exhibiting more variability, the distribution simplification factor is larger, e.g. cross section types d, f, and g in Table 4.6.2.2.1-2 (two or more loaded lanes).Numerous examples are given in Puckett et al (2006).

• Finally, the multiple presence factors are applied.

Clearly, all three of these factors could be combined into one coefficient for further computational simplification. However, with this approach the Designer is apprised of the effect of variability of the simple procedure (distribution simplification factor) and the effect of multiple presence. In previous specifications, these effects were masked by embedding the factors into the distribution factor equations.


Figure C4.6.2.2.1-1 Example Specification-based Distribution vs. Rigorous Analysis for Moment

Where moderate deviations from a constant deck

width or parallel beams exist, the equations in the tables of distribution factors may be used in conjunction with asuitable value for beam spacing. The distribution factormay either be varied at selected locations along the spanor else a single distribution factor may be used.

Most of the equations for distribution factors were derived for constant deck width and parallel beams. Past designs with moderate exceptions to these two assumptions have performed well when the S/Ddistribution factors were used. While the distribution factors specified herein are more representative of actual bridge behavior, common sense indicates that some exceptions are still possible, especially if the parameter S is chosen with prudence.

Cast-in-place multicell concrete box girder bridgetypes may be designed as whole-width structures. Suchcross-sections shall be designed for the live loaddistribution factors in Articles 4.6.2.2.2 for interiorgirders, multiplied by the number of girders, i.e., webs.

Whole-width design is appropriate for torsionally-stiff cross-sections where load-sharing between girders is extremely high and torsional loads are hard to estimate. Prestressing force should be evenly distributed between girders. Cell width-to-height ratios should be approximately 2:1.

If one lane is loaded with heavy vehicles (for example, a special vehicle or evaluation permit vehicle)with routine traffic in adjacent lanes, the design forceeffect per girder may be determined as specified inArticle 4.6.2.2.7. In such cases, the multiple presencefactor is prescribed by the owner.

The bridge types indicated in tables in Article4.6.2.2.2, with reference to Table 1, may be taken asrepresentative of the type of bridge to which eachapproximate method applies.

Except as permitted by Article 2.5.2.7.1, regardlessof the method of analysis used, i.e., approximate orrefined, exterior girders of multibeam bridges shall notbe designed for a total load effect less than an interior beam.


Table 4.6.2.2.1-1 Common Deck Superstructures Covered in Articles 4.6.2.2.2 through 4.6.2.2.6.

SUPPORTING COMPONENTS TYPE OF DECK TYPICAL CROSS-SECTION Steel Beam

Cast-in-place concrete slab, precast concrete slab, steel grid, glued/spiked panels, stressed wood

Closed Steel or Precast Concrete Boxes

Cast-in-place concrete slab

Open Steel or Precast Concrete Boxes

Cast-in-place concrete slab, precast concrete deck slab

Cast-in-Place Concrete Multicell Box

Monolithic concrete

Cast-in-Place Concrete Tee Beam

Monolithic concrete


Precast Solid, Voided or Cellular Concrete Boxes with Shear Keys

Cast-in-place structural concrete overlay

Precast Solid, Voided, or Cellular Concrete Box with Shear Keys and with or without Transverse Post-Tensioning

Integral concrete

Precast Concrete Channel Sections with Shear Keys

Cast-in-place structural concrete overlay

Precast Concrete Double Tee Section with Shear Keys and with or without Transverse Post-Tensioning

Integral concrete

Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning

Integral concrete

Precast Concrete I or Bulb-Tee Sections

Cast-in-place concrete, precast concrete

Wood Beams

Cast-in-place concrete or plank, glued/spiked panels or stressed wood


Live load effects determined with the simplified

methods of Article 4.6.2.2.2 shall be increased by thelive load distribution simplification factor as specified inTable 2,

s simplifiedmg mgγ= (4.6.2.2.1-1) where: γs = live load distribution simplification factor of Table

4.6.2.2.1-2. m = multiple presence factor as specified in Article

3.6.1.1.2 gsimplified = distribution factor per 4.6.2.2.2. mg = distribution factor considering the analysis factor,

see Table 4.6.2.2.1-2, and effect of multiplepresence.


Table 4.6.2.2.1-2. Distribution Simplification Factors

One-laneTwo or

more lanes One-laneTwo or

more lanesShear 1.02 1.02 1.02 1.04

Moment 1.04 1.07 1.11 1.04Shear 1.03 1.02 1.02 1.04

Shear 1.03 1.03 1.05 1.08Moment 1.02 1.05 1.13 1.05

Shear 1.05 1.02 1.04 1.03Moment 1.17 1.04 1.18 1.06

Shear 1.02 1.03 1.03 1.03Moment 1.14 1.04 1.08 1.07

Shear 1.03 1.06 1.15 1.10

Shear 1.00 1.00 1.00 1.00Moment 1.00 1.00 1.00 1.00

Precast Concrete I-Beam,Precast Concrete Bulb-Tee Beam,

Precast Concrete Tee Section with Shear Keys and with or

without Transverse Post-Tensioning, Precast Concrete Double Tee with Shear Keys

with or without Post-Tensioning, Precast Concrete Channel with

Shear Keys

1.041.081.101.04

Interior

Steel I-Beam a

h, k, i, j

Structure Type

AASHTO LRFD Cross

Section Type Action

Exterior

Moment

Cast-in-Place Concrete Tee Beam e

Cast-in-Place Concrete Multicell Box d

1.21

Precast Concrete Spread Box Beam b

Adjacent Box Beam with Cast-in-Place Concrete Overlay, Adjacent Box Beam with

Integral Concrete

f, g 1.05

Open Steel Box Beam c

Moment 1.15 1.10

C4.6.2.2.2 Article 4.6.2.2.2 is summarized below. See Puckett

et al (2005 a,b) , Huo et al (2003), and Patrick et al (2005) for details.

One loaded lane for moment, and for one and

multiple loaded lanes for shear:

mg = γs m[a(glever rule) + b]

4.6.2.2.2 Slab-on-Beam and Box-Beam Bridges The live load distribution for the bridge cross

sections “a to h” and “k” from Table 4.6.2.2.1-1 may be determined as specified herein. Cross section and decktypes include: concrete deck, filled grid, partially filledgrid or unfilled grid deck composite with reinforcedconcrete on steel or concrete beams, concrete T-beams, double T-sections and spread boxes. Steel open sections“c” may be determined per Article 4.6.2.2.3.

where a and b are constants, m is the multiple presence factor, and

γs is the live load distribution simplification factor. The lower bound is number of lanes divided by number


of girders. A modified uniform distribution approach is used for

multiple loaded lanes for moment:

( )10

Cs m m

g

Wmg m a b FN

γ⎡ ⎤⎛ ⎞

= +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦L

In Appendix 4B, lever rule formulae are provided

for simplification and are based on a 6-ft gage and 4-ft vehicle spacing. The designer could compute the lever rule distribution factors in the usual manner and then adjust the result by the calibration constants a and b as outlined. For adjacent or spread box beams, the center of the beam may used as the girder location for lever rule computations.

The equation for multiple loaded lanes for moment

is based upon Henry’s method (Huo et al, 2003), which starts with a uniform distribution (same for all girders, see above), using a 10-ft design lane, and then adjusts the result upward, typically 10 to 15 percent. The amand bm are recalibrations that vary from Huo’s original work. Note that the multiple presence factor for multiple lanes loaded for moment is equal to 0.85 or higher. The four or more lanes loaded case (with m = 0.65) will not control.

The 2006 Specifications are provided in Appendix

D4 as an alternative method.

4.6.2.2.2a Shear and Reaction The live load distributions for shear and reactions

are determined using distribution factor based upon thelever rule adjusted per Equation 1 with coefficients fromTable 1. Cross section types “i”, “j” may also bedetermined using Equation 1 if sufficiently connected toensure continuity of transverse deck bending moments.

( ) lanesv s v lever rule v

g

Nmg m a g b mN

γ⎡

⎡ ⎤= + ≥ ⎢⎣ ⎦ ⎢ ⎥⎣ ⎦

where am, bm and FL are constants, Wc is the curb-to-curb distance, ft. and Ng is the number of girders. The FL term is excluded from the specification equation for simplicity. Its use is optional. For span lengths less than or equal to 100 ft, FL is 1.0. For span lengths greater than 100 ft, FL is 0.95. Average span lengths may be used for negative moment regions.

⎤⎥ (4.6.2.2.2a-1)

where: mg = distribution factor including multiple presence


glever rule = distribution factor based upon the lever rule av, bv = calibration constants for shear and reactions

defined in Table 1 m = multiple presence factor as specified in Article

3.6.1.1.2 γs = live load distribution simplification factor of Table

4.6.2.2.1-2. Nlanes = number of design lanes considered in the lever

rule analysis. Ng = number of girders in the cross section

For the concrete beams, other than box beams usedin multibeam decks with shear keys: Deep, rigid enddiaphragms shall be provided to ensure proper loaddistribution; and if the stem spacing of stemmed beamsis less than 4.0 ft. or more than 10.0 ft., a refinedanalysis complying with Article 4.4 shall be used. Table 4.6.2.2.2a-1. Live Load Shear Calibration Factors (per Equation 4.6.2.2.2a-1)

a v b v a v b v a v b v a v b v

Steel I-Beam a 0.70 0.13 0.83 0.11 1.04 -0.12 0.99 0.01Precast Concrete I-Beam,Precast Concrete Bulb-Tee Beam, Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning, Precast Concrete Double Tee with Shear Keys with or without Post-Tensioning, Precast Concrete Channel with Shear Keys

h, i, j, k 0.83 0.07 0.92 0.06 1.08 -0.13 0.94 0.03

Cast-in-Place Concrete Tee Beam e 0.79 0.09 0.94 0.05 1.24 -0.22 1.21 -0.17Cast-in-Place Concrete Multicell Box Beam d 0.85 0.00 0.82 0.04 1.19 -0.20 0.71 0.23

Adjacent Box Beam with Cast-in-Place Concrete Overlay f

Adjacent Box Beam with Integral Concrete g

Precast Concrete Spread Box Beam b 0.61 0.15 0.78 0.12 1.00 -0.11 0.83 0.07Open Steel Box Beam c

AASHTO LRFD Cross

Section TypeStructure Type Lever Rule

Use Article 4.6.2.2.3

0.87 -0.051.00-0.101.050.03

Two or More Lanes

ShearExterior Interior

One Loaded Lane Two or More Lanes One Loaded Lane

0.910.03

4.6.2.2.2b Bending Moment The live load distribution for bending moment is

determined using either lever rule or the adjusteduniform method per Tables 1 and 2. Cross section types“i” and “j” may also be determined using Tables 1 and 2if sufficiently connected to ensure continuity oftransverse deck bending moments. Wc is the travel way width (curb-to-curb distance), ft.

The procedures of Appendix 4C may be used as an alternative method for computing distribution factors for bending moment. This method is provided primarily to achieve better correlation with rigorous analysis for one loaded lane.

Nlanes is the number of lanes consider in the analysis

using the lever rule and NL is the maximum number of design lanes.


Table 4.6.2.2.2b-1. Distribution of Live Loads per Lane for Moment in Longitudinal Beams for Slab-on-Beam and Box-Beam Bridges (a-h, k, and possibly i, j from Table 4.6.2.2-1)

Number of

Loaded Lanes

Girder Distribution Factor Multiple Presence Factor

Use integer part of

m shall be greater than or equal to 0.85.

m = 1.2Interior and ExteriorOne

to determine number of loaded lanes N L for multiple presence.

Interior and Exterior

Two or more

Loaded Lanes

10c L

m s m mg g

W Nmg m a b mN N

γ⎡ ⎤⎛ ⎞ ⎡ ⎤

= + ≥⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

12cW

( ) lanesm s m lever rule m

g

Nmg m a g b mN

γ⎡ ⎤

⎡ ⎤= + ≥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

Table 4.6.2.2.2b-2. Live Load Moment Calibration Factors (per Table 4.6.2.2.2b-1)

a m b m a m b m a m b m a m b m


Open Steel Box Beam c

0.62

Structure Type

AASHTO LRFD Cross

Section Type

0.77-0.061.00-0.08

-0.201.250.140.68 -0.191.39-0.411.33

0.53 -0.081.17-0.240.97-0.121.140.19

Lever Rule Uniform Lever Rule Uniform

MomentExterior Interior

One Loaded Lane Two or More Lanes One Loaded Lane Two or More Lanes

Steel I-Beam a

-0.041.14-0.411.40-0.141.110.150.65

Precast Concrete I-Beam,Precast Concrete Bulb-Tee Beam, Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning, Precast Concrete Double Tee with Shear Keys with or without Post-Tensioning, Precast Concrete Channel with Shear Keys

h, i, j, k

-0.100.93-0.821.71-0.070.65-0.090.54

0.020.26 0.050.64-0.150.59

Cast-in-Place Concrete Multicell Box Beam d

Cast-in-Place Concrete Tee Beam e

Adjacent Box Beam with Cast-in-Place Concrete Overlay f

Use Article 4.6.2.2.3

Precast Concrete Spread Box Beam b 0.000.90-0.17

-0.010.53


4.6.2.2.3 Open Steel Boxes The live load distribution for open steel box beams

is determined per Equation 1. For systems that exceedthe range of applicability, use lever rule or rigorousanalysis. For purposes of this article, the live loaddistribution simplification factors per 4.4.2 may be takenas 1.0.

For multiple steel box girders with a concrete deck,the live load flexural moment may be determined usingthe distribution factor determined by the followingexpression:

( )0.4250.05 0.85 L

v or m sg L

NmgN N

γ⎡ ⎤

= + +⎢ ⎥⎢ ⎥⎣ ⎦

(4.6.2.2.3-1)

where: NL = number of design lanes Ng = number of girders in the cross section

For:

5.15.0 ≤≤g

L

NN

When the spacing of the box girders varies along

the length of the bridge, the value of NL shall be determined, as specified in Article 3.6.1.1.1, using the width, W, taken at midspan.

For multiple steel box girders with a concrete deck

in bridges satisfying the requirements of Article 6.11.2.3, the live load flexural moment may be determined using the appropriate distribution factor specified in Equation 1. Where the spacing of the box girders varies along the length of the bridge, the distribution factor may either be varied at selected locations along the span or else a single distribution factor may be used in conjunction with a suitable value of NL. In either case, the value of NL shall be determined as specified in Article 3.6.1.1.1, using the width, w, taken at the section under consideration.

The results of analytical and model studies of simple span multiple box section bridges, reported in Johnston and Mattock (1967), showed that folded plate theory could be used to analyze the behavior of bridges of this type. The folded plate theory was used to obtain the maximum load per girder, produced by various critical combinations of loading on 31 bridges having various spans, numbers of box girders, and numbers of traffic lanes.

Multiple presence factors, specified in Table 3.6.1.1.2-1, are not applied because the multiple factors in past editions of the Standard Specifications were considered in the development of Equation 1 for multiple steel box girders.

The lateral load distribution obtained for simple spans is also considered applicable to continuous structures.

The bridges considered in the development of the equations had interior end diaphragms only, i.e., no interior diaphragms within the spans, and no exterior diaphragms anywhere between boxes. If interior or exterior diaphragms are provided within the span, the transverse load distribution characteristics of the bridge will be improved to some degree. This improvement can be evaluated, if desired, using the analysis methods identified in Article 4.4.

4.6.2.2.4 Wood Decks on Wood or Steel Beams The live load distribution for shear and bending

moment for the cross section types “a” and “l” fromTable 4.6.2.2.1-1 may be determined by applying the


distribution factors of Table 1. For purposes of thisarticle, the live load analysis factors per 4.4.2 may betaken as 1.0.

The live load flexural moment and shear for interior

beams with transverse wood decks may be determinedby applying the lane fraction specified in Table 1 andEq. 1. For exterior beams, use the lever rule and Eq. 1.

When investigation of shear parallel to the grain inwood components is required, the distributed live loadshear shall be determined by the following expression:

( )0.50 0.60LL LU LDV = V V+⎡⎣ ⎤⎦ (4.6.2.2.4-1) where: VLL = distributed live load vertical shear (kips) VLU = maximum vertical shear at 3d or L/4 due to

undistributed wheel loads (kips) VLD = maximum vertical shear at 3d or L/4 due to

wheel loads distributed laterally a specifiedherein (kips)

For undistributed wheel loads, one line of wheels is

assumed to be carried by one bending member.


Table 4.6.2.2.4-1. Distribution of Live Load per Lane for Bending Moment and Shear in Beams with Wood Decks.

Type of Deck

Application Cross Section from Table 4.6.2.2.1-1

Girder Location

One Design Lane Loaded (Use One-Lane Multiple Presence Factor)

Two Or More Design Lanes Loaded (Use Two Loaded Lanes Multiple Presence Factor, i.e., m = 1)

Range of Applicability

Interior mg = m(S/8) mg = m(S/7.5) Plank a, l Exterior mg = m(glever rule) mg = m(glever rule)

S ≤ 5.0 ft

Interior mg = m(S/11) mg = m(S/9.0) Stress Laminated

a, l

Exterior mg = m(glever rule) mg = m(glever rule)

S ≤ 6.0 ft

Interior mg = m(S/10) mg = m(S/8.5) Spike Laminated

a, l


S ≤ 6.0 ft

Interior mg = m(S/12) mg = m(S/10.0) Glued Laminated Panels on Glued Laminated Stringers

a, l


S ≤ 6.0 ft

Interior mg = m(S/11) mg = m(S/9.0) Glue Laminated Panel on Steel Stringers

a, l


S ≤ 6.0 ft

4.6.2.2.5 Flexural Moments and Shear in Transverse Floorbeams If the deck is supported directly by transverse floorbeams, the

floorbeams may be designed for loads in accordance with Table 1.For purposes of this article, the live load analysis factors per 4.4.2may be taken as 1.0.

The fractions provided in Table 1 shall be used in conjunction

with the 32.0-kip design axle load alone. For spacings of floorbeams outside the given ranges of applicability, all of the design live loadsshall be considered, and the lever rule may be used.


Table 4.6.2.2.5-1 Distribution of Live Load per Lane for Transverse Beams for Moment and Shear.

Fraction of Wheel Load to

Each Floorbeam Range of

Applicability Type of Deck

4S N/A Plank

Laminated Wood Deck 5

S S ≤ 5.0

Concrete 6S S ≤ 6.0

Steel Grid and Unfilled Grid Deck Composite with Reinforced Concrete Slab

4.5S tg ≤ 4.0 S ≤ 5.0

Steel Grid and Unfilled Grid Deck Composite with Reinforced Concrete Slab

6S tg > 4.0 S ≤ 6.0

Steel Bridge Corrugated Plank 5.5

S tg ≥ 2.0

4.6.2.2.6 Beams with Corrugated Steel Decks The live load flexural moment for interior beams with corrugated

steel plank deck may be determined by applying the distributionfactor, g, specified in Table 1. For purposes of this article, the live load analysis factors per Article 4.4.2 may be taken as 1.0.

Table 4.6.2.2.6-1 Distribution of Live Load Per Lane for Bending Moment Beams with Corrugated Steel Plank Decks

Beam Location One Design Lane Loaded

Two or More Design Lanes Loaded

Range of Applicability

(Use One-Lane Multiple Presence Factor)

(Use Two Loaded Lanes Multiple Presence Factor, i.e., m = 1)

mgm = m(S/11) mgm = m(S/9.0) Interior 5.52.0g

St

≤≥

Exterior mgm = m(glever rule) mgm = m(glever rule)


4.6.2.2.7 Special Loads with other Traffic

The provisions of this article may be applied wherethe approximate methods of analysis for beam-slab bridges specified in Article 4.6.2.2 and slab-type bridges specified in Article 4.6.2.3 are used.

( ) ( )p 1 D m 1G G g G g g= + − (4.6.2.2.4-1)

where: G = final force effect applied to a girder Gp = force effect due to special vehicle g1 = one-lane live load distribution factor GD = force effect due to design loads gm = multiple-lane live load distribution factor

C4.6.2.2.7

In Equation 1, the multiple presence factors are not included in the distribution factors. The owner should provide a multiple presence factor for the combined case if it is assumed different than unity.

A similar formula was developed from a similar

formula presented without investigation by Modjeski and Masters, Inc. (1994) in a report to the PennsylvaniaDepartment of Transportation in 1994, as was examinedin Zokaie (1998).

4.6.2.2.8 Curved Steel Bridges Approximate analysis methods may be used for

analysis of curved steel bridges. The Engineer shallascertain that the approximate analysis method used isappropriate by confirming that the method satisfies the requirements stated in Article 4.4. In curved systems,consideration should be given to placing parapets,sidewalks, barriers and other heavy line loads at theiractual location on the bridge. Wearing surface and otherdistributed loads may be assumed uniformly distributedto each girder in the cross-section

C4.6.2.2.8 The V-load method (United States Steel 1984) has

been a widely used approximate method for analyzing horizontally curved steel I-girder bridges. The method assumes that the internal torsional load on the bridge— resulting solely from the curvature—is resisted by selfequilibrating sets of shears between adjacent girders. The V-load method does not directly account for sources of torque other than curvature and the method does not account for the horizontal shear stiffness of the concrete deck. The method is only valid for loads such as normal highway loadings. For exceptional loadings, a more refined analysis is required. The method assumes a linear distribution of girder shears across the bridge section; thus, the girders at a given cross-section should have approximately the same vertical stiffness. The Vload method is also not directly applicable to structures with reverse curvature or to a closed-framed system with horizontal lateral bracing near, or in the plane of one or both flanges. The V-load method does not directly account for girder twist; thus, lateral deflections, which become important on bridges with large spans and/or sharp skews and vertical deflections, may be significantly underestimated. In certain situations, the Vload method may not detect uplift at end bearings. The method is best suited for preliminary design, but may also be suitable for final design of structures with radial supports or supports skewed less than approximately 10°. The M/R method provides a means to account for the


effect of curvature in curved box girder bridges. The method and suggested limitations on its use are discussed by Tung and Fountain (1970). Vertical reactions at interior supports on the concave side of continuous-span bridges may be significantly underestimated by both the V-load and M/R methods. Live load distribution factors for use with the Vload and M/R methods may be determined using the appropriate provisions of Article 4.6.2.2. Strict rules and limitations on the applicability of both of these approximate methods do not exist. The Engineer must determine when approximate methods of analysis are appropriate.

… …

SECTION 4 (US)

REFERENCES

… Huo, X.S., Conner, S.O., and Iqbal, R., “Re-examination of the Simplified Method (Henry’s Method) of Distribution Factors for Live Load Moment and Shear,” Final Report, Tennessee DOT Project No. TNSPR-RES 1218, Tennessee Technological University, Cookeville, TN (June 2003).

…. Modjeski and Masters, Inc., 2002. “Shear in Skewed Multi-Beam Bridges,” NCHRP 20-07/Task 107 Final Report, Transportation Research Board, National Research Council, Washington, D.C.

… Patrick, M.D., Huo, X.S., Puckett, J.A., Jablin, M.C., and Mertz, D., “Sensitivity of Live Load Distribution Factors to Vechile Spacing,” Journal of Bridge Engineering, ASCE, in review.

… Puckett, J. A., Huo, X. S., Patrick, M.D., Jablin, M.C., Mertz, D., and Peavy, M.D., “Simplified Live Load Distribution Factor Equations for Bridge Design,” Proceedings of International Bridge Engineering Conference, Transportation Research Board, Paper Number 113, in review. Puckett, J. A., Huo, X. S., Mertz, D., Jablin, M.C., Patrick, M.D., and Peavy, M.D., 2006 “Simplified Live Load Distribution Factor Equations.” NCHRP Report for 12-62. TRB, National Research Council, Washington, DC. (Tentative).

…

…

4-23


APPENDIX A4 DECK SLAB DESIGN TABLE

…

SECTION 4: STRUCTURAL ANALYSIS AND EVALUATION 4-25

B4 LEVER RULE FORMULA

Tables B4-1 and B4-2 may be used to perform lever rule computations. The following assumptions were used in developing these tables and should be considered when using the listed values for analysis:

• two-foot spacing between the outer wheel centroid and the curb/barrier

• six-foot axle gage

• four-foot between vehicles.

The limitations of the equations are illustrated in the tables. Where the limitations are not met, conventional lever

rule computations apply.

Table B4-1. Lever Rule Distribution of Live Loads per Lane Longitudinal Beams for Slab-on-Beam and Box-Beam Bridges for Interior Girders

Girder Location

Number of Loaded

Lanes Distribution Factor Range of Application Loading Diagram

Number of Wheels to Beam

1

Use lever rule and manually place the vehicle for critical effect on the first interior beam

Interior

varies

2 or more

4

3

1

2

2

112

60e

S ftd

≤≥

31S

−60e

S ftd

>≥

12

40e

S ftd

≤≥

21S

−4 6

0e

S ftd

< ≤≥

3 52 S

−6 10

0e

S ftd

< ≤≥

102S

−10 16

0e

S ftd

< ≤≥

0ed <

6'6' 4'

S S

ed S

4'6' 6'

S S

6'6' 4'

d Se SS

S

6'

S

ed S

6'

6'6'

S S

4'


Table B4-2. Lever Rule Distribution of Live Loads per Lane Slab-on-Beam and Box-Beam Bridges for Exterior Girders

Girder Location

Number of Loaded

Lanes Distribution Factor Range of Application Loading Diagram

Number of Wheels to Beam

2

2

3

4

1

1

2 or more

Exterior

12 2

edS

+( ) 6e

e

d S ft

d S

+ ≤

<

31 edS S

+ − ( ) 6ed S ft+ >

31 edS S

+ − ( ) 10ed S ft+ ≤

33 82 2

edS S

+ − ( )10 16ed S ft< + ≤

( )16 20ed S ft< + ≤ed S

6'6' 4'

ed S

6' 6'4'

ed S

6'6' 4'

S

ed S

6'

Sde

6'

2 162 edS S

+ −


)

C4 ALTERNATE MOMENT DISTRIBUTION FACTOR APPROACH

Better one-lane loaded estimates may be achieved with the combined use of the lever rule (LR), uniform distribution (Uniform), or parametric formula method (PF) as summarized in Table 4C-1. The value of the distribution factor determined by using the methods prescribed in Table 4C-1 and shall be modified by the calibration factors provided in Tables 4C-2 and analysis factors provided in Table 4C-3. Shear computations are as prescribed in Article 4.6.2.2 and do not change with the methods outlined in this appendix. The moment distribution factor may be computed as:

( , ,m s m mmg m a LR Uniform or PF bγ= +⎡ ⎤⎣ ⎦ (4C-1)

where: guniform=

( ) ( )10 g

Travelway widthN

⎛⎜ ⎟⎜ ⎟⎝ ⎠

⎞ (4C-2)

and the parametric distribution is

31 2 1ExpExp Exp

PFg

S SgD L N

⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (4C-3)

where Exp1, Exp2, Exp3, and D are constants that vary with bridge type, S = the girder spacing in feet, L = the span length in feet, and Ng = the number of girders or (number of cells + 1) for the bridge.

NCHRP 12-62 provides the basis for Article 4.6.2.2. Within that work, compromises were made to accommodate simplification. In most cases, the simplified approach works well and provides reasonable live load distribution factors with good correlation relative to rigorous results. However, in some cases decreased accuracy relative to rigorous analysis resulted, specifically for flexure distribution factors for one loaded lane. To better estimate the live load effects for the one loaded lane case, a parametric power equation may be used. See Equation 4C-3. For the one-lane loading case, the transverse deflections (and curvatures) are much more localized than the multiple-lanes loaded case. Therefore, a parameter that represents the longitudinal to transverse stiffness provides a better live load estimate. The quantification of this was achieved by the geometric ratio (S/L). The span length, L, may be used where the point of interest resides within the interior portion of the span. The average of the adjacent span lengths may be used for the regions near the pier. All moments should carry the same distribution factors, i.e., whether of positive or negative sense. See Table 4C-4 for guidance. The correlation coefficients for the flexure cases for the lever rule, uniform distribution, and parametric equation methods are presented in Puckett et al (2006). In some cases, the parametric equation provides a significant improvement in accuracy, and this may be of particular importance to cases where a bridge is being load rated.


Table 4C-1. Simplified Method

h, k, i, j a d f, g e b1

Multiple Uniform Uniform PF Uniform PF Uniform1 LR LR PF PF PF PF

Multiple LR LR PF Uniform LR LRh, k, i, j a d fg e b

exp1 0.34 0.40 0.00 0.00 0.20 0.00exp2 0.26 0.18 0.35 0.27 0.30 0.26exp3 0.00 0.00 0.32 0.45 0.00 0.30

D 24.30 42.30 1.00 1.00 44.60 1.00

PF Constants

Moment

LanesBridge Types

Interior PF

Exterior

Table 4C-2 – Calibration Factors for Moment

a m b m a m b m a m b m a m b m

0.53 0.19 0.71 0.23 1.53 -0.17 1.17 -0.08

0.68 0.14 0.85 0.15 1.17 -0.07 1.39 -0.19

0.65 0.15 0.84 0.13 2.44 -0.58 1.14 -0.04

1.25 -0.12 1.8 -0.12 1.13 -0.04 2.03 -0.05


2.26 -0.35 0.54 0.19 1.85 -0.30 0.90 0.00Open Steel Box Beam c Use Article 4.6.2.2.3

bPrecast Concrete Spread Box Beam UniformParametric FormulaLever RuleParametric Formula

Adjacent Box Beam with Cast-in-Place Concrete Overlay f UniformParametric FormulaUniformParametric Formula

1.20 -0.05 0.64 0.05

Parametric FormulaParametric FormulaParametric FormulaParametric Formula

UniformParametric FormulaLever Rule

e UniformParametric FormulaLever RuleLever Rule

Precast Concrete I-Beam,Precast Concrete Bulb-Tee Beam, Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning, Precast Concrete Double Tee with Shear Keys with or without Post-Tensioning, Precast Concrete Channel with Shear Keys

h, i, j, k

Lever Rule

Cast-in-Place Concrete Multicell Box Beam d

Cast-in-Place Concrete Tee Beam

1.31 -0.12 0.53 -0.01

Steel I-Beam

Structure Type

Cross Section Type

MomentExterior Interior

One Loaded Lane Two or More Lanes Loaded

Two or More Lanes Loaded One Loaded Lane

a Lever Rule Lever Rule UniformParametric Formula


Table 4C-3 – Distribution Simplification Factors

One-laneTwo or

more lanes One-laneTwo or

more lanesSteel I-Beam a 1.04 1.06 1.04 1.04

Cast-in-Place Concrete Tee Beam e 1.02 1.03 1.13 1.05

Cast-in-Place Concrete Multicell Box d 1.07 1.05 1.06 1.03

Precast Concrete Spread Box Beam b 1.10 1.07 1.11 1.07

Open Steel Box Beam c 1.00 1.00 1.00 1.00

Structure Type

AASHTO LRFD Cross

Section Type

Exterior Interior

Adjacent Box Beam with Cast-in-Place Concrete Overlay, Adjacent Box Beam with

Integral Concrete

f, g

Precast Concrete I-Beam,Precast Concrete Bulb-Tee Beam,

Precast Concrete Tee Section with Shear Keys and with or

without Transverse Post-Tensioning, Precast Concrete Double Tee with Shear Keys

with or without Post-Tensioning, Precast Concrete Channel with

Shear Keys

h, i, j, k

1.051.16 1.10 1.08

1.041.061.051.04

Table 4C-4 – L for Use in Live Load Distribution Factor Equations.

L (ft.) FORCE EFFECT Positive Moment The length of the span for

which moment is being calculated The average length of the two adjacent spans

Negative Moment—Near interior supports of continuous spans from point of contraflexure to point of contraflexure under a uniform load on all spans Negative Moment—Other than near interior supports of continuous spans

The length of the span for which moment is being calculated

Shear The length of the span for which shear is being calculated

Exterior Reaction The length of the exterior span Interior Reaction of Continuous Span The average length of the two

adjacent spans

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