lecture 4 - loads ii 4.1 objective of the …ctgttp.edu.free.fr/trungweb/tc tk cau 22 tcn 272 -...

printed on June 24, 2003

Lecture - 4-1

LECTURE 4 - LOADS II

4.1 OBJECTIVE OF THE LESSON

The purpose of this lesson is to continue the developmentsof the loads for bridge design required by the LRFD Specification byproviding information on ice loads, earth pressures.

4.2 ICE LOADS

4.2.1 General

The Specification requires consideration of the followingtypes of ice action.

• Dynamic pressure due to moving sheets or floes of ice beingcarried by stream flow, wind or currents and striking a pier.

• Static pressure due to thermal movements of ice sheets.Static forces may be caused by the thermal expansion of icein which a pier is embedded, or by irregular growth of the icefield.

• Pressure resulting from hanging dams or jams of ice.Hanging dams are the phenomenon of frazil ice passingunder the surface layer of ice and accumulating under thesurface ice at the bridge site. The frazil ice comes typicallyfrom rapids or waterfalls upstream. The hanging dam cancause a back-up of water, which exerts pressure on the pier,and can also cause scour around or under the piers as thewater flows at an increased velocity.

• Static uplift or vertical load resulting from adhering ice inwaters of fluctuating level.

The behavior of ice and the forces that it generates is a verycomplex issue and is not yet fully understood. When undergoinglong-term changes in temperature and sustained loadings, ice canbehave in a relatively plastic manner. Ice which is moving with acurrent can either create a substantial impact load on a pier orbreakup when it hits the pier reducing the load. For purposes ofclassification, the commentary of the Specification uses the typesof ice failure developed by Montgomery in 1984. The reference toMontgomery is given in the Specification. Thus, the following typesof ice failure are considered:


Lecture - 4-2

• crushing - the ice fails by local crushing across the width ofthe pier. The crushed ice is continually cleared from a zonearound the pier as the floe moves past,

• bending - for piers with inclined noses, a vertical reactioncomponent acts on the impinging ice floe. This reactioncauses the floe to rise up the pier nose and fail as flexuralcracks form,

• splitting - where a comparatively small floe strikes a pier,stress cracks propagating from the pier into the floe split thefloe into smaller parts,

• impact - if the floe is small, it is brought to a halt whenimpinging on the nose of the pier before it has failed bycrushing over the full-width of the pier, by bending or bysplitting, and

• buckling - for very wide piers, where a large floe cannotclear the pier as it fails, compressive forces cause the floeto fail by buckling in front of the pier nose.

4.2.2 Design for Ice

The design for ice typically starts with the determination ofthe effective crushing strength of the ice (p). The following valuesare specified when cite specific information is not available.

• 0.38 MPa where break-up occurs at melting temperaturesand the ice is substantially disintegrated in its structure,

• 0.77 MPa where break-up occurs at melting temperaturesand the ice is somewhat disintegrated in its structure,

• 1.15 MPa where break-up or major ice movement occurs atmelting temperatures, but the ice moves in large pieces andis internally sound, and

• 1.53 MPa where break-up or major ice movement occurswith the ice temperature, averaged over its depth,measurably below the melting point.

As indicated in the commentary to the Specification, the valuesidentified above are considerably less than ice values which may bemeasured under laboratory conditions. The lowest value isappropriate for piers where experience with similar sites indicatesthat ice forces are very low, but ice should be considered in thedesign at that location. The maximum value of 1.53 MPa is basedon an observed history of bridges that have survived significant


Lecture - 4-3

icing conditions, and these are detailed in the reference list inSection 3 of the Specification.

Once the effective ice strength, P, has been determined forthe given site, breakup conditions, the next step is to determine thehorizontal force, F, resulting from the pressure of moving ice. Thishorizontal force is related to the effective strength by a series ofexpressions in S3.9.2.2, which are also a function of the piergeometry. A distinction is made as to whether the horizontal forceis caused by an ice flow failing by compression over the full-widthof the pier, or whether the ice flow fails by flexure as it rides up aninclined ice breaking pier nose.

The design expressions require an estimate of the icethickness. The preferred method to determine the ice thickness ishistorical records of actual ice thickness at a potential bridge siteover some number of years. When this data is not available, anempirical expression is provided in the commentary to S3.9.2.2,which provides a means of estimating ice thickness, depending onthe accumulation of freezing days at the site, per year, and theparticular conditions of wind and snow apt to occur at the site.Snow cover is found to be an important factor in determining icethickness. The snow cover in excess of 150 mm in thickness hasbeen shown to reduce ice thickness by almost one-half. Thespecification permits the reduction in ice forces on small streamsnot conducive to the formation of large ice flows. The reduction islimited to not more than 50% of the design force.

Once the ice force, F, has been determined, it is necessaryto consider combinations of longitudinal and transverse forcesacting on a pier. It would be unrealistic to expect ice to moveexactly parallel to a pier, so that a minimum lateral component of15% of the longitudinal force is specified. Piers are usually alignedin the direction of stream flow, usually assumed to be the directionof ice movement.

Two design cases are investigated as follows:

• a longitudinal force equal to F shall be combined with atransverse force of 0.15F, or

• a longitudinal force of 0.5F shall be combined with atransverse force of Ft.

The transverse force, Ft, shall be taken as:

(4.2.2-1)

Ft 'F

2 tan(β/2 % θf)


Lecture - 4-4

where:

β = nose angle in a horizontal plane, for a round nosetaken as 100E (DEG)

θf = friction angle between ice and pier nose (DEG)

Both the longitudinal and transverse forces shall be assumed to actat the pier nose.

Where the longitudinal axis of a pier is not parallel to theprincipal direction of ice action, or where the direction of ice actionmay shift, the total force on the pier shall be determined on thebasis of the projected pier width and resolved into components.Under such conditions, forces transverse to the longitudinal axis ofthe pier is taken to be at least 20% of the total force.

4.2.3 Static Ice Loads on Piers

Ice pressures on piers frozen into ice sheets shall beinvestigated where the ice sheets are subject to significant thermalmovements relative to the pier where the growth of shore ice is onone side only, or other situations which may produce substantialunbalanced forces on the pier. Unfortunately, little guidance isavailable for predicting static ice loads on piers. Under normalcircumstances, the effects of static ice forces on piers may be strainlimited, but expert advice should be sought if there is reason forconcern.

4.2.4 Hanging Dams and Ice Jams

The frazil accumulation in a hanging dam may be taken toexert a pressure of 0.0096 to 0.096 MPa as it moves by the pier.An ice jam may be taken to exert a pressure of 0.96x10-3 to 9.6x10-3

MPa.

The wide spread of pressures quoted reflects both thevariability of the ice and the lack of firm information on the subject.

4.2.5 Vertical Forces due to Ice Adhesion

The vertical force on a bridge pier due to rapid water levelfluctuation is given by:

• for a circular pier, in N:

(4.2.5-1)Fv ' 0.3t 2 % 0.0169Rt 1.25


Lecture - 4-5

• for an oblong pier, in N/mm of pier perimeter:

(4.2.5-2)Fv ' 2.3x10 &3t 1.25

where:

t = ice thickness (mm)

R = radius of circular pier (mm)

Equations 4.2.5-1 and 4.2.5-2 neglect creep and are,therefore, conservative for fluctuations occurring over more than afew minutes, but they are also based on the assumption that failureoccurs on the formation of the first crack, which is non-conservative.

4.2.6 Ice Accretion and Snow Loads on Superstructures

No specific ice accretion or snow loads are specified in theLRFD Specification. However, Owners in areas where uniqueaccumulations of snow and/or ice are possible should specifyappropriate loads for that condition.

The following discussion of snow loads is taken from Ritter(1991).

Snow loads should be considered where a bridge is locatedin an area of potentially heavy snowfall. This can occur at highelevations in mountainous areas with large seasonalaccumulations. Snow loads are normally negligible in areas of theUnited States that are below 600 000 mm elevation and east oflongitude 105EW, or below 300 000 mm elevation and west oflongitude 105EW. In other areas of the country, snow loads aslarge as 0.034 MPa may be encountered in mountainous locations.

The effects of snow are assumed to be offset by anaccompanying decrease in vehicle live load. This assumption isvalid for most structures, but is not realistic in areas where snowfallis significant. When prolonged winter closure of a road makes snowremoval impossible, the magnitude of snow loads may exceedthose from vehicular live loads. Loads also may be notable whereplowed snow is stockpiled or otherwise allowed to accumulate. Theapplicability and magnitude of snow loads are left to Designer'sjudgment.

Snow loads vary from year to year and depend on the depthand density of snow pack. The depth used for design should bebased on a mean recurrence interval or the maximum recordeddepth. Density is based on the degree of compaction. The lightestaccumulation is produced by fresh snow falling at coldtemperatures. Density increases when the snow pack is subjected


Lecture - 4-6

to freeze-thaw cycles or rain. Probable densities for several snowpack conditions are as follows, ASCE (1980):

Table 4.2.6-1 - Snow Density

CONDITION OFSNOW PACK

PROBABLEDENSITY (kg/m3)

Freshly Fallen 96

Accumulated 300

Compacted 500

4.3 EARTH LOADS

4.3.1 General

The Specification defines several broad classifications ofwalls which are also referred to herein. For reference, thesedefinitions are repeated below:

Abutment - A structure that supports the end of a bridge span, andprovides lateral support for fill material on which the roadway restsimmediately adjacent to the bridge.

Anchored Wall - An earth retaining system typically composed ofthe same elements as non-gravity cantilevered walls, and whichderive additional lateral resistance from one or more tiers ofanchors.

Mechanically Stabilized Earth Wall - A soil retaining system,employing either strip or grid-type, metallic or polymeric tensilereinforcements in the soil mass, and a discrete modular precastconcrete facing which is either vertical or nearly vertical.

Non-Gravity Cantilever Wall - A soil retaining system whichderives lateral resistance through embedment of vertical wallelements and support retained soil with facing elements. Verticalwall elements may consist of discrete elements, e.g., piles,caissons, drilled shafts or auger-cast piles spanned by a structuralfacing, e.g., lagging, panels or shotcrete. Alternatively, the verticalwall elements and facing may be continuous, e.g., diaphragm wallpanels, tangent piles or tangent drilled shafts.

Prefabricated Modular Wall - A soil retaining system employinginterlocking soil-filled timber, reinforced concrete or steel modules


Lecture - 4-7

or bins to resist earth pressures by acting as gravity retaining walls.Prefabricated modular walls consist of individual structural unitsassembled at the site into a series of hollow bottomless cells knownas cribs. The cribs are filled with soil, and their stability depends notonly on the weight of the units and their filling, but also on thestrength of the soil used for the filling. The units themselves mayconsist of reinforced concrete, fabricated metal, or timber.

Rigid Gravity, Semi-Gravity and Cantilever Retaining Walls - Astructure that provides lateral support for a mass of soil and thatowes its stability primarily to its own weight and to the weight of anysoil located directly above its base. This classification of wallsincludes:

• A gravity wall depends entirely on the weight of the stoneor concrete masonry and of any soil resting on the masonryfor its stability. Only a nominal amount of steel is placednear the exposed faces to prevent surface cracking due totemperature changes.

• A semi-gravity wall is somewhat more slender than agravity wall and requires reinforcement consisting of verticalbars along the inner face and dowels continuing into thefooting. It does not rely on the weight of the overlying soilfor stability. It is provided with temperature steel near theexposed face.

• A cantilever wall consists of a concrete stem and aconcrete base slab, both of which are relatively thin and fullyreinforced to resist the moments and shears to which theyare subjected.

• A counterfort wall consists of a thin concrete face slab,usually vertical, supported at intervals on the inner side byvertical slabs or counterforts that meet the face slab at rightangles. Both the face slab and the counterforts areconnected to a base slab, and the space above the baseslab and between the counterforts is backfilled with soil. Allthe slabs are fully reinforced.

Several of these types of walls are illustrated in Figure 4.3.1-1.


Lecture - 4-8

Figure 4.3.1-1 - Illustration of Several Wall Types (from Das, B. M.,Principles of Foundation Engineering, Brooks/Cole EngineeringDivision, 1984)

Retained earth exerts lateral pressure on retaining walls andabutments. In general, the magnitude and distribution of the lateralearth pressure on such structures is a function of the compositionand consistency of the retained earth and the magnitude of externalloads applied to the retained soil mass. Typically, development ofa design earth pressure considers the following:

• The type, unit weight, shear strength and creepcharacteristics of the retained earth;

• The anticipated or permissible magnitude anddirection of lateral deflection at the top of the wall orabutment;

• The degree to which backfill soil retained by the wallis to be compacted;


Lecture - 4-9

• The location of the groundwater table within theretained soil;

• The magnitude and location of surcharge loads onthe retained earth mass; and

• The effects of horizontal acceleration of the retainedearth mass during an earthquake.

The degree to which a wall (or abutment) is permitted todeflect laterally, and the characteristics of the retained earth are thetwo most significant factors in the development of lateral earthpressure distributions. Walls which are permitted to tilt or movelaterally away from the retained soil permit the development of anactive state of stress in the retained soil mass and should bedesigned for the active earth pressure. Walls which are restrainedagainst movement (e.g., integral abutments) or walls for whichlateral deflection and associated ground movements may adverselyimpact adjacent facilities (typically within a distance behind the wallless than about one-half the wall height) should be designed toresist the at-rest earth pressure, which may be 50% greater thanthe magnitude of the active pressure. Walls which may deflectlaterally into the retained soil should be designed to resist thepassive earth pressure, which can be 10 to 20 times greater thanthe active pressure. (The passive state of stress is limited, for allpractical purposes, to lateral deflection of the embedded portions offlexible cantilever retaining walls into the supporting soil.)

The lateral wall movement required to permit developmentof the minimum active earth pressure or maximum passive earthpressure is affected by the type of soil retained, as shown in Table4.3.1-1

where:

∆ = lateral movement at top of wall(achieved through rotation ortranslation) required for developmentof active or passive earth pressure(mm)

H = wall height (mm)


Lecture - 4-10

Table 4.3.1-1 - Approximate Values of Relative MovementsRequired to Reach Minimum Active or Maximum Passive EarthPressure Conditions, Clough (1991)

Type of Backfill

Values of ∆/H

Active Passive

Dense sand 0.001 0.01

Medium dense sand 0.002 0.02

Loose sand 0.004 0.04

Compacted silt* 0.002 0.02

Compacted lean clay* 0.010 0.05

Compacted fat clay* 0.010 0.05

*Not typically used to backfill highway structures

Nearly all conventional retaining walls of typical proportions,except very short walls, deflect sufficiently to permit developmentof active earth pressures. Gravity and semi-gravity walls designedwith a sufficient mass to support only active earth pressures willdeflect (tilt or translate) in response to more severe loadingconditions (e.g., at-rest earth pressures) until stresses in theretained soil mass are relieved sufficiently to permit development ofan active state of stress in the retained soil. The most significantpotential for development of at-rest earth pressures on such wallsis on the stems of cantilevered retaining walls, where a rigid stem-to-base connection may prevent lateral deflection of the stem withrespect to the base in response to the lateral pressure of backfill soilretained above the base. For such a condition, excessive lateralearth pressures on the stem could conceivably lead to a structuralfailure of the stem or stem-to-base connection.

A comparison of estimated actual lateral deflections todeflections required to mobilize active earth pressure on the stemof a cantilevered retaining wall backfilled with dense sand isprovided in Tables 4.3.1-2 and 4.3.1-3. Table 4.3.1-2 assumes thatthe full section modulus of the stem is effective in resisting bending,and that the base of the stem is fixed to the foundation slab. Table4.3.1-3 assumes that a reduced section modulus is effective inresisting bending to account for cracking and creep of the concrete.Neither Table 4.3.1-2 nor Table 4.3.1-3 includes lateral deflectionsthat would occur due to differential settlement of the wall base slab.


Lecture - 4-11

Table 4.3.1-2 Cantilever Wall Stem Deflections Using Full SectionModulus

Estimated StemDeflection

WallHeightH(mm)

AverageWall StemThicknesst(mm)

UnderActiveEarthPressure∆/H(dim)

Under At-RestEarthPressure∆/H(dim)

DeflectionRequired toMobilizeActive EarthPressure∆/H(dim)

1500 120 0.00013 0.00021 0.001

4500 300 0.00074 0.00119 0.001

7600 490 0.00159 0.00255 0.001

9100 610 0.00151 0.00241 0.001

Note: Assumes Backfill φ = 37E, ka = 0.25, ko = 0.40 and f'c = 27.6MPa

Table 4.3.1-3 - Cantilever Wall Stem Deflections Using ReducedSection Modulus

Estimated Stem Deflection

WallHeight

H(mm)

AverageWall StemThickness

t(mm)

Under ActiveEarth

Pressure∆/H

(dim)

Under At-Rest EarthPressure∆/H

(dim)

DeflectionRequired to

MobilizeActive Earth

Pressure∆/H

(dim)

1500 120 0.00066 0.00106 0.001

4500 300 0.00371 0.00594 0.001

7600 490 0.00795 0.01273 0.001

9100 610 0.00486 0.00779 0.001

Note: Assumes Backfill φ = 37E, ka = 0.25, ko = 0.40 and f'c = 27.6MPa

Tables 4.3.1-2 and 4.3.1-3 indicate that, with the exceptionof very short walls, cantilever wall stems will generally deflect, crack


Lecture - 4-12

and creep sufficiently to permit mobilization of active earthpressures in backfill soils composed of relatively dense sand, whichis the typical backfill material for such walls. For walls bearing onyielding (soil) foundation materials, tilting of the base slab due tosettlement of the foundation soils will result in even greaterdeflections, such that active earth pressures will be mobilized evenon the stems of very short walls. It is likely that, unless walls areotherwise restrained against rotation or translation or have amassive cross-section, active earth pressure conditions will beachieved on the stems of nearly all cantilever retaining wallsbackfilled with compacted granular soil, with the exception of veryshort walls (less than 1.5 m tall) bearing directly on rock.

For walls retaining cohesive soils, the effects of soil creepmay prevent the permanent establishment of active and passiveearth pressure. Under stress conditions producing the minimumactive or maximum passive earth pressure, cohesive soilscontinually creep, such that shear stresses within the soil mass arepartially received. As a result, the movements indicated in Table4.3.1-1 are produced only temporarily. Without further movement,the lateral earth pressure exerted by a cohesive soil initially in theactive stress state will increase eventually to a value approachingthe at-rest earth pressure. Likewise, the lateral earth pressureexerted by a cohesive soil initially in the passive stress state willdecrease eventually to a value approaching approximately 40% ofthe passive earth pressure.

4.3.2 Compaction

When mechanical compaction equipment is operated withina distance behind a retaining wall equal to about one-half of the wallheight, additional lateral earth pressures are induced on the walldue to the compaction effort. Excessive backfill compaction canincrease lateral earth pressures to values significantly greater thanthe active or even at-rest lateral earth pressure. Such compaction-induced pressures continue to act, even after the compactionequipment has been removed due to the inelastic behavior of thesoil.

A typical lateral earth pressure distribution for an unyieldingwall, including compaction-induced residual pressures, is shown inFigure 4.3.2-1.


Lecture - 4-13

Figure 4.3.2-1 - Residual Earth Pressure after Compaction ofBackfill Behind an Unyielding Wall (after Clough and Duncan, 1991)

The induced residual pressures would be somewhat less ona flexible or unrestrained wall subjected to the same compactionloading conditions since lateral deflection or movement of the wallwould permit partial relief of the stress in the retained soil.

The heavier the compaction equipment and the closer itoperates to the wall, the greater are the compaction-inducedpressures. Therefore, the use of soils which are difficult to compact(e.g., fine-grained, moisture sensitive soils) and heavy compactionequipment immediately behind earth retaining structures is likely tocause unacceptably large lateral soil pressures and should beavoided.Use of free-draining granular soils and light compactionequipment within a distance of H/2 behind retaining walls is usuallyspecified to preclude development of excessive compaction-induced lateral earth pressures. If the use of heavy static orvibratory compaction equipment behind a retaining wall cannot beavoided, the residual compaction-induced lateral earth pressuresshould be estimated by available procedures (e.g., Clough andDuncan, 1991).


Lecture - 4-14

4.3.3 Earth Pressure

As described in Article 4.3.1, the magnitude and distributionof lateral earth pressure on a retaining structure is primarily afunction of the retained soil characteristics and the degree to whichthe wall tilts or translates in response to the loading. For mostabutments and conventional retaining walls, the earth pressuredistribution is assumed to increase linearly with depth in accordancewith the following:

p = kh γ g z 10-9 (4.3.3-1)

where:

γ = unit density of soil (kg/m3)

p = lateral earth pressure (MPa)

kh = lateral earth pressure coefficient taken as ka or ko,depending on the magnitude of lateral deflection(dim) (see Article 4.3.1)

z = depth below backfill surface (mm)

g = gravitational constant (m/sec2)

Although the lateral earth pressure due to the retained soilis assumed to increase linearly with depth, the resultant lateral loaddue to the earth pressure is assumed to act at a height of 0.4Habove the base of the wall for conventional gravity retaining walls(where H is the total wall height measured from the top of thebackfill to the base of the footing) rather than 0.33H, as would beexpected for a linearly proportional (triangular) distribution. As aconventional gravity wall deflects laterally (translates) in responseto lateral earth loading, the backfill behind the wall must slide downalong the back of the wall for the retained soil mass to achieve theactive state of stress. Experimental results indicate that the backfillarches against the upper portion of the wall as the wall translates,causing an upward shift in the location at which the resultant of thelateral earth load is transferred to the wall (Terzaghi, 1934; Clausen,1972, and Sherif, 1982)

For non-gravity cantilever retaining walls or other flexiblewalls which tilt or deform laterally in response to lateral loading,significant arching of the backfill against the wall does not occur,and the resultant lateral load due to earth pressure is assumed toact at a height of 0.33H above the base of the wall.


Lecture - 4-15

4.3.3.1 AT-REST PRESSURE COEFFICIENT, ko

When a retaining wall is restrained against lateral movementor lateral movement of the wall is unacceptable, the lateral earthpressure coefficient (kh) in Equation 4.3.3.1-1 is taken as ko.

For a normally consolidated soil, the at-rest lateral earthpressure coefficient, ko, can be computed by the following:

ko = 1 - sin φf (4.3.3.1-1)

where:

φf = effective stress angle of internal friction of thedrained soil

For overconsolidated soils, the at-rest lateral earth pressurecoefficient is generally considered to vary as a function of the stresshistory, the value of ko increasing with increasing degree ofoverconsolidation in accordance with the following:

ko = (1 - sin φf)(OCR) sin φf (4.3.3.1-2)

where:

OCR = overconsolidation ratio (dim)

Values of ko for various soil types and degrees ofoverconsolidation are presented in Table 4.3.3.1-1.


Lecture - 4-16

Table 4.3.3.1 - Typical Coefficients of At-Rest Lateral EarthPressure

Soil Type

Coefficient of Lateral Earth Pressure, ko

OCR = 1 OCR = 2 OCR = 5 OCR = 10

Loose Sand 0.45 0.65 1.10 1.60

MediumSand

0.40 0.60 1.05 1.55

Dense Sand 0.35 0.55 1.00 1.50

Silt (ML) 0.50 0.70 1.10 1.60

Lean Clay(CL)

0.60 0.80 1.20 1.65

HighlyPlastic Clay(CH)

0.65 0.80 1.10 1.40

4.3.3.2 ACTIVE PRESSURE COEFFICIENT, ka

When a retaining wall deflects laterally in response toloading by the retained earth, a wedge of the retained soil moveslaterally and downward along the back of the wall, as shown inFigure 4.3.3.2-1.


Lecture - 4-17

Figure 4.3.3.2-1 - Active Failure Wedge for Conventional Gravityand Cantilever Retaining Walls, Coulomb Analysis

As the soil wedge moves, the shear strength of the soil isgradually mobilized along the failure plane shown in Figure 4.3.3.2-1. When the full shear strength of the soil is mobilized, theadditional force required to maintain the stability of the wedge (andthe corresponding force acting on the wall) reaches a minimumvalue equal to the active earth pressure, Pa. Prior to wall movementand mobilization of shear strength in the soil, the wall must supportthe at-rest earth pressure (Article 4.3.3.1).

One of two theories is generally used to estimate the activeearth pressure on retaining walls. Rankine earth pressure theoryneglects the vertical friction force applied to the surface of the wallby the retained soil wedge as it moves downward along the back ofthe wall. For the Rankine earth pressure theory, the active earthpressure resultant is assumed to have a line of action parallel to thebackfill surface. Coulomb earth pressure theory accounts for thefriction force exerted on the wall by the retained earth, which resultsin an inclination of the earth pressure resultant of δ with respect tothe back face of the wall (for a gravity wall) or to the verticalpressure surface extending up from the heel of the wall (for acantilever wall), as shown in Figure 4.3.3.2-1. Typical values of δ,


Lecture - 4-18

the friction angle between the wall and backfill, are presented inTable 4.3.3.2-1.

Table 4.3.3.2-1 - Friction Angles Between DissimilarMaterials

Interface Materials FrictionAngle, δ

(deg)

Mass concrete on the following foundationmaterials:

• Clean sound rock• Clean gravel, gravel-sand mixtures, coarse sand• Clean fine to medium sand, silty medium to coarse sand, silty or clayey gravel• Clean fine sand, silt or clayey fine to medium sand• Fine sandy silty, non-plastic silt• Very stiff and hard residual or preconsolidated clay • Medium stiff and stiff clay and silty clay

Masonry on foundation materials has samefriction factors

3529 to 31

24 to 29

19 to 24

17 to 1922 to 26

17 to 19

Steel sheet piles against the following soils:

• Clean gravel, gravel-sand mixtures, well-graded rock fill with spalls• Clean sand, silty sand-gravel mixtures, single-size hard rock fill• Silty, sand, gravel, or sand mixed with silty or clay• Fine sandy silt, non-plastic silt

22

17

14

11


Interface Materials FrictionAngle, δ

(deg)

Lecture - 4-19

Formed or precast concrete or concretesheet piling against the following soils:

• Clean gravel, gravel-sand mixtures, well-graded rock fill with spalls• Clean sand, silty sand-gravel mixtures, single-size hard rock fill• Silty sand, gravel or sand mixture with silt or clay• Find sandy silt, non-plastic silt

22 to 26

17 to 22

17

14

Various structural materials:

• Masonry on masonry, igneous and metamorphic rocks: • dressed soft rock on dressed soft rock • dressed hard rock on dressed soft rock • dressed hard rock on dressed hard rock• Masonry on wood in direction of cross grain• Steel on steel at sheet pile interlocks

35

33

29

26

17

Both AASHTO Standard Specification (AASHTO 1992) andthe LRFD Specification (AASHTO 1993) employ the Coulomb earthpressure theory. For the typical case when a retaining wall ispermitted to deflect sufficiently to develop the active state of stressin the retained soil, the lateral earth pressure coefficient (kh) inEquation 4.8-1 is, therefore, taken as the Coulomb active earthpressure coefficient, ka. For the case of a vertical retaining wall anda horizontal backfill surface, the value of ka can be obtained fromFigure 4.3.3.2-2.

For theoretical solutions like those shown in Table 1 andFigure 1, the angle of internal friction is denoted simply as φ. Thevalue of φ, shown in these solutions, is to be interpreted as theeffective stress friction angle, φf, determined from a drained sheartest, when analyses are performed using effective stresses, and thetotal stress friction angle φ, determined from an undrained sheartest, when analyses are performed using total stresses. For long-term conditions, the earth pressures should be calculated usingeffective stresses, and adding water pressures as appropriate.


Lecture - 4-20

Figure 4.3.3.2-2 - Active and Passive Pressure Coefficients forVertical Wall and Horizontal Backfill - Based on Log Spiral FailureSurfaces

For the more general case of an inclined wall face andsloping backfill surface, the value of the ka can be obtained fromTable 4.3.3.2-2.


Lecture - 4-21

Table 4.3.3.2-2 - Value of ka for Log Spiral Failure Surface

δ(DEG)

i(DEG)

β(DEG)

φ (DEG)

20 25 30 35 40 45

-15 -10 0 10

0.370.420.45

0.300.350.39

0.240.290.34

0.190.240.29

0.140.190.24

0.110.160.21

0 0 -10 0 10

0.420.490.55

0.340.410.47

0.270.330.40

0.210.270.34

0.160.220.28

0.120.170.24

15 -10 0 10

0.550.650.75

0.410.510.60

0.320.410.49

0.230.320.41

0.170.250.34

0.130.200.28

-15 -10 0 10

0.310.370.41

0.260.310.36

0.210.260.31

0.170.230.27

0.140.190.25

0.110.170.23

φ 0 -10 0 10

0.370.440.50

0.300.370.43

0.240.300.38

0.190.260.33

0.150.220.30

0.120.190.26

15 -10 0 10

0.500.610.72

0.370.480.58

0.290.370.46

0.220.320.42

0.170.250.35

0.140.210.31

Studies have shown that the failure surface defining the soilwedge loading the wall is approximated more closely by a log spiralcurve than a straight line. The values of ka, provided in Figure4.3.3.2-1 and Table 4.3.3.2-2, were, therefore, obtained fromanalyses using log spiral surfaces (Caquot and Kerisel, 1948).

Both the AASHTO Standard and LRFD Specifications alsoprovide guidance for estimating earth pressures on special types ofearth retaining structures (e.g., non-gravity cantilevered walls,anchored walls and mechanically stabilized earth walls) and wallssubjected to unusual loading conditions (e.g., passive earthpressures). Where unusual backfill geometries or surchargeconditions exist, the active pressure may be estimated using agraphical trial wedge procedure.


Lecture - 4-22

4.3.3.3 EQUIVALENT FLUID PRESSURE

For simplicity, lateral earth pressure is often estimated as anequivalent fluid pressure, wherein the resultant of the earthpressure is equivalent to the resultant of a fictitious fluid exertinghydrostatic pressure on the wall. Where equivalent fluid pressureis used, the unit earth pressure (in MPa) at any depth is taken as:

p = γeq g Z 10-9 (4.3.3.3-1)

where:

γeq = equivalent fluid unit density of soil, not less than 480 (kg/m3)

Typical values of γeq for design of walls up to 6.5 m in heightare shown in Table 4.3.3.3-1.

Table 4.3.3.3-1 - Typical Values for Equivalent Fluid Unit Densitiesof Soil

Type of SoilLevel Backfill

Backfill with i = 25E

At-Restγeq

(kg/m3)

Active(∆/H =1/240)γeq

(kg/m3)

At-Restγeq

(kg/m3)

Active(∆/H =1/240)γeq

(kg/m3)

Loose sandor gravel

880 640 1040 800

Mediumdense sandor gravel

800 560 960 720

Dense sandor gravel

720 480 880 640

Compactedsilt (ML)

960 640 1120 800

Compactedlean clay(CL)

1120 720 1280 880

Compactedfat clay(CH)

1280 880 1440 1040

The resultant lateral earth load due to the equivalent fluidpressure is assumed to act at a height of 0.4H above the base of


Lecture - 4-23

the wall footing for conventional gravity and semi-gravity retainingwalls.

4.3.4 Presence of Water

If soil mass retained by a retaining wall or abutment containsgroundwater and the groundwater is not eliminated throughdrainage, a water table will develop behind and exert lateralpressure on the structure above the water table, the horizontalpressure is given as:

P = kh γ g z 10-9 (4.3.4-1)

The presence of the water table behind the wall has two additionaleffects, as indicated below and in Figure 4.3.4-1.

• The unit weight of the retained soil is reduced to itssubmerged or buoyant value:

γNs = γs-γw (4.3.4-2)

As a result, the lateral earth pressure below the water tableis reduced to:

P = (kh γs (z-zw) + kh γ's z) g 10-9 (4.3.4-3)

• The retained water exerts a horizontal hydrostaticwater pressure equal to:

Pw = γw g zw 10-9 (4.3.4-4)

where:

γNs = submerged unit density of soil (kg/m3)

γs = total unit density of soil (kg/m3)

γw = unit density of water (kg/m3)

kh = horizontal earth pressure coefficient (dim)

z = depth below backfill surface (mm)

zw = depth below water table (mm)

P = horizontal earth pressure (MPa)

Pw = hydrostatic water pressure (MPa)


Lecture - 4-24

Figure 4.3.4-1 - Effect of Groundwater Table on Earth Pressure

With some algebra, the equations for pressure below thegroundwater surface can be rearranged for ease of computation as:

P = [kh γ z + γw zw - kh γw zw] g x 10-9 (4.3.4-5)

The location of the resultant corresponding terms containing"kh" is taken at 0.4 z or 0.4 zw above the design section defined byz and zw. The location of the resultant of the term γw zw is taken aszw/3 above the design section.

Where possible, the development of hydrostatic waterpressures should be prevented through the use of free-draininggranular backfill and/or by providing a positive means of backfilldrainage, such as weep holes, perforated and solid pipe drains,gravel drains or geofabric drains.

When groundwater levels differ on opposite sides of aretaining wall, seepage occurs beneath the wall. The effect ofseepage forces is to increase the load on the back of the wall (anddecrease any passive resistance in front of the wall). Porepressures in the backfill soil can be approximated throughdevelopment of a flow net or using various analytical methods, andmust be added to the effective horizontal earth pressures todetermine the total lateral pressures on the wall.

4.3.5 Surcharge

Surcharge loads on the retained earth surface produceadditional lateral earth pressure on retaining walls. Where thesurcharge is uniform over the retained earth surface, the additionallateral earth pressure due to the surcharge is assumed to remainconstant with depth and has a magnitude, ∆p, of:

∆p = ks qs (4.3.5-1)

where:


Lecture - 4-25

∆p = constant horizontal earth pressure due to uniformsurcharge (MPa)

ks = coefficient of earth pressure due to surcharge

qs = uniform surcharge applied to the upper surface of theactive earth wedge (MPa)

For active earth pressure conditions, ks is taken as ka, andfor at-rest conditions, ks is taken as ko.

Where vehicular traffic is anticipated within a distancebehind a wall equal to about the wall height, a live load surchargeis assumed to act on the retained earth surface. The uniformincrease in horizontal earth pressure due to live load surcharge istypically estimated as:

∆p = ks γsg heq 10-9 (4.3.5-2)

where:

∆p = constant horizontal earth pressure due to uniformsurcharge (MPa)

γs = unit density of soil (kg/m3)

k = coefficient of earth pressure

heq = equivalent height of soil for the design live load (mm)

Equivalent heights of soil, heq, for highway loading onretaining walls of various heights can be taken from Table 4.3.5-1.

Table 4.3.5-1 - Equivalent Heightof Soil for Vehicular Loading

Wall Height (mm) heq (mm)

# 1500 1700

3000 1200

6000 760

$ 9000 610

The tabulated values of heq were determined based onevaluation of horizontal pressure distributions produced on retainingwalls by the vehicular live loads specified in the LRFD Specification(AASHTO 1993). The pressure distributions were obtained from


Lecture - 4-26

elastic half-space solutions with Poisson's ratio of 0.5, doubled toaccount for the non-deflecting wall.

Alternatively, the increase in horizontal earth pressure, ∆p,on a retaining wall resulting from a live load surcharge, p, can betaken as:

(4.3.5-3)∆p '2pπ (α&sinαcos(α % 2δ))

where:

p = live load intensity (MPa)

α = angle illustrated in Figure 4.3.5.1 (RAD)

δ = angle illustrated in Figure 4.3.5.1 (RAD)

Figure 4.3.5-1 - Horizontal Pressure on Wall Caused by UniformlyLoaded Strip

4.3.6 Effect of Earthquake

Lateral earth pressures on retaining structures are amplifiedduring an earthquake due to horizontal acceleration of the retainedearth mass. The Mononobe-Okabe method of analysis is apseudo-static method which is commonly used to develop an


Lecture - 4-27

equivalent static fluid pressure to model seismic earth pressure onretaining walls. The Mononobe-Okabe method is contingent uponthe following assumptions (Barker, et al, 1991):

• The wall is unrestrained and capable of deflectingsufficiently to mobilize the active earth pressurecondition in the backfill;

• The backfill is cohesionless and unsaturated;

• The failure surface defining the active wedge of soilloading the wall is planar; and

• Accelerations are uniform throughout the soil mass.

As indicated above, the Mononobe-Okabe method assumesthat backfill soils are unsaturated and, as such, not susceptible toliquefaction. The potential for liquefaction should be evaluatedwhere saturated soils may be subjected to earthquake or othercyclical or instantaneous dynamic loadings.

The Mononobe-Okabe method (AASHTO 1992 andAASHTO 1993) considers equilibrium of the soil wedge retained bythe wall as shown in Figure 4.3.6-1.

Figure 4.3.6-1 - Definition Sketch for Seismic Loading (after Barker,1991)

The result of the combined active static and seismic earthpressures is taken as:

PAE = 0.5 x KAE γ g (1-kv) H2 10-9 (4.3.6-1)

where the seismic active earth pressure coefficient, KAE, is:


Lecture - 4-28

s2(φ&θ&β)2βcos(δ%β%θ)

× 1% sin(φ%δ)sicos(δ%β%θ)

(4.3.6-2)

where:

γ = unit density of soil (kg/m3)

H = height of soil face (mm)

φ = angle of internal of soil (deg)

θ = arc tan (kh/(1-kv))

δ = angle of friction between soil and wall (deg) (refer toArticle 4.3.6.2)

kh = horizontal acceleration coefficient (dim)

kv = vertical acceleration coefficient (dim)

ι = slope of backfill surface (deg)

β = slope of wall back face (deg)

g = gravitational constant (m/sec2)

For estimation of lateral earth pressures under earthquakeconditions, the vertical acceleration coefficient, kv, is commonlyassumed equal to zero, and the horizontal acceleration coefficient,kh, is taken as:

kh = 0.5α (4.3.6-3)

for walls designed to move horizontally 250 α (mm), and

kh = 1.5 (4.3.6-4)

for walls designed for zero horizontal displacement

where:

α = A/100

A = horizontal earthquake acceleration (percent of g)

g = acceleration of gravity (m/sec2)


Lecture - 4-29

Values of the earthquake acceleration coefficient, A, areshown in Figures S3.10.2-1, S3.10.2-2 and S3.10.2-3.

Although the Mononobe-Okabe method of analysis does notspecify the point of application of the horizontal seismic earthpressure resultant, the resultant of the dynamic component of theearth pressure (∆PAE) is significantly higher on the wall than thestatic active earth pressure resultant (Pa), as indicated in Figure4.3.6-1. Both the AASHTO Standard Specifications (AASHTO1992) and the LRFD Specifications (AASHTO 1993) indicate thatthe combined static and seismic lateral earth pressure can beassumed to be uniformly distributed with a resultant (PAE) acting atthe mid-height of the wall.

The AASHTO Standard Specification and the LRFDSpecification also provide guidelines for determination of passiveseismic earth pressure on retaining structures which are beingforced horizontally into the backfill, and methods for design ofretaining structures for a limited tolerable displacement underearthquake loading rather than for zero permanent displacement asassumed in the Mononobe-Okabe method.

As with many current methods of seismic analysis, theMononobe-Okabe method neglects inertial forces due to the massof the retaining structure, concentrating only on the inertia of theretained soils mass. For gravity structures which rely solely on theirmass for stability, this assumption is unconservative. The effectsof wall inertia on the behavior of gravity retaining walls is discussedfurther by Richards and Elms (1979), who conclude that thestructure inertia forces should not be neglected. Richards and Elmssuggest a design approach based on limiting wall displacements totolerable levels, rather than designing for no movement, andcomputing the weight of wall required to prevent movementsgreater than specified. The work of Richards and Elms isincorporated into the tolerable displacement procedure presentedin the AASHTO Specifications.

4.3.7 Reduction due to Earth Pressure

In some cases, lateral earth pressures may reduce theeffects of other loads and forces on culverts, bridges and theircomponents. One such case is that of the top slab of a box culvert,for which the maximum bending moment in the top slab is reduceddue to the effect of lateral earth pressure on the side walls. Suchreductions in loading should be limited to the extent that the appliedearth pressures can be expected to be permanent. In lieu of moreprecise methods of estimating the lateral earth pressure forceeffects, the AASHTO Standard Specification (AASHTO 1992) andthe LRFD Specification (AASHTO 1993) permit a load effectreduction of 50% where earth pressures are present.


Lecture - 4-30

4.3.8 Downdrag

When a point bearing pile or drilled shaft penetrates a softlayer subject to settlement (e.g., where an overlying embankmentmay cause settlement of the layer), the soil settling around the shaftexerts a frictional force, or downdrag, around the perimeter of thepile or shaft. This frictional force acts as an additional axial load onthe pile or shaft and, if sufficient in magnitude, could cause astructural failure of the foundation element or a bearing capacityfailure at the tip.

The methods used to estimate downdrag loads are the sameas those used to estimate the side resistance of shafts and pilesdue to skin friction, as the same load transfer mechanism isresponsible for both.

4.3.9 Design of a Cantilever Retaining Wall

The purpose of this example is to illustrate the application ofthe various load factors and load combinations. In order to fullydevelop the example, references to provisions for wall and footings,and references to geotechnical textbooks are required and sonoted.

Some rounding of numbers has been done so it may not bepossible to exactly duplicate all values to the full precision shown.

The cantilever retaining wall below has been proposed tosupport an embankment.


Lecture - 4-31

Figure 4.3.9-1 - Schematic of Example

During the subsurface exploration, it was determined that thefoundation soils are predominantly clay to a depth of 6 m below theproposed bottom of footing, and, therefore, a 150 mm blanket ofcompacted granular material was placed below the footing. Densesand and gravel underlies the clayey foundation soils. Assumeelastic settlement of the dense sand and gravel to be negligible.The proposed wall backfill will consist of a free draining granular fill.Assume the seasonal high water table to be at the bottom of thefooting.

Apply the vehicular live load surcharge (LS) on the backfillas indicated in the figure above.

Determine the lateral pressure distribution on the wall andestimate the bearing capacity for the proposed design. Check thedesign for adequate protection against sliding and estimate theconsolidation settlement of the underlying clay.

Solution:

Step 1: Calculate the Unfactored Loads with η = 1.0

(A) Dead Load of Structural Components and NonstructuralAttachments (DC)

Unit Density of Concrete = 2,400 kg/m3


Lecture - 4-32

Figure 4.3.9-2 - Retaining Wall Area DesignationWeight of Concrete

W1 = (0.3)(4.5)(2,400)g = 31,784 N per m of lengthW2 = (0.5)(4.5)(0.2)(2,400)g = 10,595 N per m of lengthW3 = (0.5)(3.0)(2,400)g = 35,316 N per m of length

DC = Sum of W1 - W3 = 77,695 N per m of length

(B) Vertical Earth Pressure (EV)

Weight of Soil

PEV = W = (2.0)(4.5)(1,920)g = 169,517 N per m of length

(C) Live Load Surcharge (LS)

The live load surcharge shall be applied where vehicularload is expected to act on the backfill within a distance equal to thewall height behind the wall (S3.11.6.2).

An equivalent height of soil for the design vehicular loading(heq) is estimated using Table 4.3.9-1, repeated below, and a wallheight of 5 m.


Lecture - 4-33

Table 4.3.9-1 - Equivalent Heightof Soil for Vehicular Loading

Wall Height(mm)

heq (mm)

# 1500 01700

3000 1200

6000 760

heq = 907 mm

Use soil density of the backfill = 1920 kg/m3

Vertical Component of LS = (1920)(907)(9.81) x 10-9 m3/mm3 =0.0171 MPa

For a width of 2m, PLSV = 34,167 N per m of length

Assume an active earth pressure coefficient ka using Figure4.3.9-3 with a wall friction angle, δ = φ, repeated below.


Lecture - 4-34

Figure 4.3.9-3 - Active and Passive Pressure Coefficients forVertical Wall and Horizontal Backfill - Based on Log Spiral FailureSurfaces

k = ka = 0.29

using Equation S3.11.6.2-1:


Lecture - 4-35

∆p = (k) (γ'1) (g) (heq) (10-9)

∆p = 0.0050 MPa

Using a rectangular distribution, the live load horizontal earthpressure acting over the entire wall will be:

PLSH = (0.0050)(5)(106) = 24,771 N per m of length

(D) Horizontal Earth Pressure (EH)

The basic earth pressure should be assumed to be linearlyproportional to the depth of earth and is given by EquationS3.11.5.1-1:

p = k(γ'1)gz(10-9)

Use k = ka (as above)

At the base of the footing:

p = (0.29)(1,920)(9.81)(5000)(10-9)p = 0.0273 MPa

The horizontal earth pressure acting over the entire wallwill be:

PEH = (0.5)(0.0273)(5)(106) = 68,278 N per m of length

Note triangular pressure distribution

(E) Summary of Unfactored Loads

Table 4.3.9-2 - Vertical Loads and Resisting Moments - Unfactored

ItemV

N/m Arm about 0 mMoment about 0

N-m/m

W1 31,784 0.85 27,017

W2 10,595 0.63 6,710

W3 35,316 1.5 52,974

PEV 169,517 2.0 339,034

PLSV 34,167 2.0 68,334

TOTAL 281,379


Lecture - 4-36

Table 4.3.9-3 - Horizontal Loads and Overturning MomentsMoment - Unfactored

ItemH

N/m Arm about 0 mMoment about

0 N-m/m

PLSH 24,771 2.5 61,928

PEH 68,278 2.0 136,555

For location of resultant for lateral earth pressure seeS3.11.5.1.

Step 2: Determine the Appropriate Load Factors

Using Tables S3.4.1-1 and S3.4.1-2 given in this manual as Tables2.4.1.2-1 and 2.4.1.2-2, respectively:

Table 4.3.9-4 - Load Factors

Group DC EV LSv PLSH

EH(active) Probable Use

StrengthI-a

0.9 1.0 1.75 1.75 1.5 Bearing Capacity(eccent.)&Sliding

StrengthI-b

1.25 1.35 1.75 1.75 1.5 Bearing Capacity(max. value)

StrengthIV

1.5 1.35 - - 1.5 Bearing Capacity(max. value)

Service I 1.0 1.0 1.0 1.0 1.0 Settlement

By inspection, the following conclusions can be drawn for the caseof bearing capacity (maximum) Strength I-b will probably govern,however, the factored loads will also be checked for Strength IV.


Lecture - 4-37

Step 3: Calculate the Factored Loads

Table 4.3.9-5 - Factored Vertical Loads

Group/ItemUnits

W1N/m

W2N/m

W3N/m

PEVN/m

PLSVN/m

TotalN/m

V (Unf.) 31,784 10,595 35,316 169,517 34,167 281,379

StrengthI-a

28,606 9,535 31,784 169,517 59,792 299,235

StrengthI-b

39,731 13,244 44,145 228,848 59,792 385,759

StrengthIV

47,677 15,892 52,974 228,848 0 345,390

Service I 31,784 10,595 35,316 169,517 34,167 281,379

Table 4.3.9-6 - Factored Moments Mv

Group/ItemUnits

W1N-m/m

W2N-m/m

W3 N-m/m

PEVN-m/m

PLSVN-m/m

TotalN-m/m

Mv (Unf.)27,017 6,710 52,974 339,034 68,334 494,068

StrengthI-a

24,315 6,039 47,677 339,034 119,585 536,649

StrengthI-b

33,771 8,388 66,218 457,695 119,585 685,656

StrengthIV

40,525 10,065 79,461 457,695 0 587,747

Service I 27,017 6,710 52,974 339,034 68,334 494,068


Lecture - 4-38

Table 4.3.9-7 - Factored Horizontal Loads

Group/ItemUnits

PLSHN/m

PEHN/m

TotalN/m

H (Unf.) 24,771 68,278 93,049

Strength I-a 43,349 102,416 145,766

Strength I-b 43,349 102,416 145,766

Strength IV 0 102,416 102,416

Service I 24,771 68,278 93,049

Table 4.3.9-8 - Factored Moments Mh

Group/ItemUnits

PLSHN-m/m

PEHN-m/m

TotalN-m/m

Mv (Unf.) 61,928 136,555 198,483

Strength I-a 108,374 204,833 313,206

Strength I-b 108,374 204,833 313,206

Strength IV 0 204,833 204,833

Service I 61,928 136,555 198,483

This example will be continued in Lecture 15 in which afoundation design, eccentricity check, sliding check, and settlementcalculation will be added.


Lecture - 4-39

REFERENCES

Clough, G.W., and J.M. Duncan, 1991, "Earth Pressures," Chapter6, Foundation Engineering Handbook, 2nd Edition, edited by H.Y.Fang, Van Nostrand Reinhold, New York, NY.

Clausen, C.J.F. and S. Johansen, 1972, "Earth PressuresMeasured Against a Section of a Basement Wall," Proceedings, 5thEuropean Conference on Soil Mechanics and FoundationEngineering, Madrid, pp. 515-516.

Sherif, M.A., I. Ishibashi and C.D. Lee, 1982, "Earth PressuresAgainst Rigid Retaining Walls," Journal of GeotechnicalEngineering, ASCE, Vol. 108, GT5, pp. 679-695.

Terzaghi, K., 1934, "Retaining Wall Design for Fifteen-Mile FallsDam, Engineering News Record, May, pp. 632-636.

Barker, R.M., J.M. Duncan, K.B. Rojiani, P.S.K. Ooi, C.K. Tan andS.G. Kim, 1991, "Manuals for the Design of Bridge Foundations,"NCHRP Report 343, Transportation Research Board, Washington,D.C.

American Association of State Highway and Transportation Officials(AASHTO), 1993, "Draft LRFD Bridge Design Specifications,"prepared by Modjeski and Masters, Inc. Consulting Engineers, Inc.

American Association of State Highway and TransportationOfficials, 1992, "Standard Specifications for Highway Bridges,"Fifteenth Edition, AASHTO, Washington, D.C., 1992.

Richards, R., and D.G., Elms, 1979, "Seismic Behavior of GravityRetaining Walls," Journal of the Geotechnical Engineering Division,ASCE, Volume 105, No. GT4.

Caquot, A. and J. Kerisel, 1948, "Tables for the Calculation ofPassive Pressure, Active Pressure and Bearing Capacity ofFoundations," Gauthier-Villars, Imprimeur-Libraire, Libraire duBureau des Longitudes, de L'Ecole Polytechnique, Paris, 120 pp.


Lecture - 5-1

LECTURE 5 - LOADS III


The objective of this lesson is to continue to provide a studentwith the background for the provisions in Section 3, Loads and LoadFactors, of the AASHTO LRFD Specification.

The lesson includes:

• the forces due to superimposed deformations resulting fromuniform and nonuniform temperature changes, differentialsettlement, creep and shrinkage,

• discussion of braking and centrifugal forces,

• discussion of wind loads, and

• discussion of water loads.

5.2 FORCE EFFECTS DUE TO SUPERIMPOSED DEFORMATIONS

Internal force effects in a component due to creep andshrinkage shall be considered. The effect of temperature gradientshould be included where appropriate. Force effects resulting fromresisting component deformation, displacement of points of loadapplication and support movements is included in the analysis.

5.2.1 Uniform Temperature

In the absence of more precise information, the ranges oftemperature shall be as specified in Table 5.2.1-1. The differencebetween the extended lower or upper boundary and the baseconstruction temperature assumed in the design is used to calculatethermal deformation effects.

Table 5.2.1-1 - Temperature Ranges

CLIMATESTEEL ORALUMINUM CONCRETE WOOD

Moderate -18E to 50E C -12E to 27E C -12E to 24E C

Cold -35E to 50E C -18E to 27E C -18E to 24E C

A moderate climate may be determined by the number offreezing days per year. If the number of freezing days is less than 14,the climate is considered to be moderate. Freezing days are dayswhen the average temperature is less than 0E C.


Lecture - 5-2

The actual air temperature averaged over the 24 hour periodimmediately preceding the setting event can be used in installingexpansion bearings and deck joints.

5.2.2 Temperature Gradient

The load factor for temperature gradient should be determinedbased on:

• the type of structure, and

• the limit state being investigated.

There is general agreement that in situ measurements oftemperature gradients have yielded a realistic distribution oftemperatures through the depth of some types of bridges, mostnotably concrete box girders. There is very little agreement on thesignificance of the effect of that distribution. It is generallyacknowledged that cracking, yielding, creep and other non-linearresponses diminish the effects. Therefore, load factors of less than1.0 should be considered, and there is some basis for lower loadfactors at the strength and extreme event limit states than at theservice limit state.

Similarly, open girder construction and multiple steel boxgirders have traditionally, but perhaps not necessarily correctly, beendesigned without consideration of temperature gradient, i.e., γTG = 0.0.

Temperature gradient is included in various load combinationsin Table S3.4.1-1. This does not mean that it need be investigated forall types of structures. If experience has shown that neglectingtemperature gradient in the design of a given type of structure has notlead to structural distress, the Owner may choose to excludetemperature gradient. Multi-beam bridges are an example of a typeof structure for which judgment and past experience should beconsidered.

The vertical temperature gradient in concrete and steelsuperstructures with concrete decks may be taken as shown in Figure5.2.2-1.

The dimension "A" in Figure 5.2.2-1 shall be taken as:

• 300 mm for concrete superstructures, that are 400 mm ormore in depth

• for concrete sections shallower than 400 mm, "A" shall be 100mm less than the actual depth

• 300 mm for steel superstructures, where t = depth of theconcrete deck


Lecture - 5-3

Temperature value T3 shall be taken as 0E C, unless a site-specific study is made to determine an appropriate value, but shall notexceed 3E C.

Figure 5.2.2-1 - Positive Vertical Temperature Gradient in Concreteand Steel Superstructures

This temperature gradient given herein is a modification of thatproposed in Imbsen (1985) which was based on studies of concretesuperstructures. The addition for steel superstructures is patternedafter the temperature gradient for that type of bridge in the Australianbridge specifications, Austroads (1992).

Temperatures for use with Figure 5.2.2-1 may be taken fromTable 5.2.2-1.


Lecture - 5-4

Table 5.2.2-1 - Basis ofTemperature Gradients

Zone T1 (EC) T2 (EC)

1 30 7.8

2 25 6.7

3 23 6

4 21 5

Figure 5.2.2-2 - Solar Radiation Zones for the United States

Positive temperature values for the zones shall be taken asspecified for various deck surface conditions in Table 5.2.2-1.Negative temperature values shall be obtained by multiplying thevalues specified in Table 5.2.2-1 by -0.3 for decks with the concretetop surface exposed and -0.2 for decks with an asphalt overlay.

The temperatures given in Table 5.2.2-1 form the basis forcalculating the change in temperature with depth in the cross-section,not absolute temperature.

Where temperature gradient is considered, internal stressesand structure deformations due to both positive and negativetemperature gradients may be determined by dividing into threeeffects as follows:

• AVERAGE AXIAL EXPANSION - This is due to the uniformcomponent of the temperature distribution which should be


Lecture - 5-5

considered simultaneously with the uniform temperaturespecified in Article S3.12.2. It may be calculated as:

(5.2.2-1)TUG '1Ac m m

TG dw dz

The corresponding total uniform axial strain from both Tu andTG is:

(5.2.2-2)εu ' α [TUG % Tu ]

• FLEXURAL DEFORMATION - Since plane sections remainplane, a curvature is imposed on the superstructure so as toaccommodate the linearly variable component of thetemperature gradient. The rotation per unit lengthcorresponding to this curvature may be determined as:

(5.2.2-3)φ ' αIc m m

TG z dw dz '1R

If the structure is externally unrestrained, i.e., simply supportedor cantilevered, no external force effects are developed due tothis superimposed deformation.

The axial strain and curvature may be used in both flexibilityand stiffness formulations. In the former, εu may be used inplace of P/AE, and φ may be used in place of M/EI intraditional displacement calculations. In the latter, the fixed-end force effects for a prismatic frame element may bedetermined as:

N = EAcεu (5.2.2-4)

M = EIcφ (5.2.2-5)

An expanded discussion with examples may be found in Ghali(1989).

Strains induced by other effects such as shrinkage and creepmay be treated in a similar manner.

• INTERNAL STRESS - Internal stresses in addition to thosecorresponding to the restrained axial expansion and/or rotationmay be calculated as:

σE = E [α TG - α TUG - φz] (5.2.2-6)

where:

TG = temperature gradient (∆EC)

TUG = temperature averaged across the cross-section(EC)


Lecture - 5-6

Tu = uniform specified temperature (EC)

Ac = cross-section area - transformed for steelbeams (mm2)

Ic = inertia of cross-section - transformed for steelbeams (mm4)

α = coefficient of thermal expansion (mm/mm/EC)

E = modulus of elasticity (MPa)

R = radius of curvature (mm)

w = width of element in cross-section (mm)

z = vertical distance from center of gravity of cross-section (mm)

Note that a positive value of σE in Equation 5.2.2-6 denotescompression.

For example, the flexural deformation part of the gradientflexes a prismatic superstructure into a segment of a circle in thevertical plane. For a two-span structure with span length L, in mm, theunrestrained beam would lift off from the central support by ∆ = L2/2Rmm. Forcing the beam down to eliminate ∆ would develop a momentwhose value at the pier would be:

(5.2.2-7)Mc '32EIcφ

Therefore, the moment is the function of the beam rigidity andimposed flexure. As rigidity approaches 0.0 at the strength limit state,Mc tends to disappear. This behavior also indicates the need forductility which ensures structural integrity as rigidity decreases.

A "common sense" explanation of the stress given by Equation5.2.2-6 follows.

Consider the effect of a temperature gradient on a set ofdisconnected layers within the structure. Each layer will expand anamount required by the temperature gradient.


Lecture - 5-7

Equation 5.2.2-6 can be developed in two steps.

Step 1: Consider only the layer restraint stresses. Forillustration, consider the top layer:

∆1 ' TGLα

But, the layers are not really disconnected, so apply a force to eachlayer to make the displacement equal to zero, with compression takenas positive.

P1 '∆1A1EL

' TG1A1Eα

σ ' TGαE

But, P1 + P2 + P3 ... cannot result in a net force on the unrestrainedend.

Similarly, ΣPiYiCG cannot result in a net external moment.

Now apply as notional external loads toP ' &ΣPi & M ' &ΣPiYiCGresult in a net external axial load and moment is zero. The total effectis then:

So far, even for a simply supported beam:

σi ' &PiAi%ΣPiA

%yΣPi yiI

σ ' % TGαE &1AmTGαEdA &

1I mTGαEydA

Since these are notional force effects, they appear to exist even at thefree edge of the beam.

Step 2: Add the effect of restrained indetermanent reactions,if any, caused by the global end rotation in theredundant structure.

! Apply end rotations as shown below:


Lecture - 5-8

! The moment diagram with redundant reactions removed is arectangle, i.e., constant moment from end-to-end. Thismoment results in a displaced shape as shown below.

! Calculate reactions required to have no displacement atredundant supports

! The moment diagram due to only the redundnat reactions isshown below:

! The force effects due to redundant reactions is added to theforce effects obtained in Step 1.

Alternatively, if the axial force and moment given by Equations5.2.2-4 and 5.2.2-5 are used as input into a structural analysispackage, the stresses from those two factors and the redundantreaction are calculated directly, and only the first term of Equation5.2.2-6 need be added to those results to get a complete solution.

5.2.3 Differential Shrinkage

Where appropriate, differential shrinkage strains betweenconcretes of different age and composition, and between concrete andsteel or wood, are determined in accordance with the provisions ofSection S5.

The designer may specify timing and sequence of constructionin order to minimize stresses due to differential shrinkage betweencomponents.

5.2.4 Creep

Creep strains for concrete and wood are given by theprovisions of Section S5 and Section S8, respectively. Traditionally,only creep of concrete is considered. Creep of wood is addressedonly because it applies to prestressed wood decks. In determiningforce effects and deformations due to creep, dependence on time andchanges in compressive stresses shall be taken into account.


Lecture - 5-9

5.2.5 Settlement

Force effects due to extreme values of differential settlementsamong substructure and within individual substructures units isconsidered. Estimates of settlement may be made in accordance withthe provision of Article S10.7.2.3. Force effects due to settlement maybe reduced by considering creep.

5.3 OTHER LIVE LOAD EFFECTS

5.3.1 General

Lecture 3 contained information on the development of thenotional live load model. This lesson focuses on other aspects of thelive load, other than the weight and axle configuration of the notionalload, per se.

5.3.2 Centrifugal Force

The only addition to centrifugal force, compared to previouseditions of Standard Specification, is the inclusion of the factor 4/3 inEquation S3.6.3-1. As explained in the commentary, the group ofexclusion vehicles, described in detail in Lecture 3, produced forceeffects which are generally at least 4/3 of that caused by the designtruck alone on short- to medium-span bridges. This is the origin of the4/3 factor in Equation S3.6.3-1. It is an approximation used to accountfor vehicles whose gross vehicle weight is greater than the 325 kNdesign truck.

Centrifugal force is applied to the design truck or to the tandemaxle, but not to the lane load on the basis that the lane load is used toaccount for disbursed traffic, which normally does not contributesignificantly to a single pier, except on longer spans. Since centrifugalforce is applied to all loaded lanes, the multiple presence factorsapply.

5.3.3 Braking Force

The braking force requirement of Article S3.6.4 is 25% of theaxle weights or the design tandem in each lane with a multiplepresence factor applied. This is substantially greater fraction thanprevious editions of the standard specifications. However, it is onlyapplied to the design load or design tandem, not the uniform load.The 25% specified is based on improved braking capability of moderntrucks. The commentary to this article indicates the set of parametersfrom which the 25% was developed, an exercise in engineeringjudgment. The braking force is not applied to the lane load on thebasis that vehicles are apt to break out-of-phase.


Lecture - 5-10

5.3.4 Vehicular Collision Forces

Portions of structures deemed to be protected are not requiredto be designed for vehicular collision loads. Structures which fall intothis category are those which are protected by an embankment, astructurally independent crashworthy ground-mounted barrier, asdefined in Article S3.6.5.1, or a properly designed barrier positionedto protect the structural elements. Barriers are to be designed for theforces indicated in Section 13, Railings. Vehicular and railroadcollision loads are defined as a static equivalent force of 1800 kNacting at a height 1200 mm above the ground. This force wasdeveloped from information from full-scale tests involving theredirection of tractor trailer trucks by barriers and from analytical workidentified in the commentary.

5.4 WATER LOADS

There are no substantially new requirements for staticpressure, buoyancy or stream pressure. The issue of debris movingin a stream and impacting piers, or even elements of superstructure,is the subject of an NCHRP Research Project, just getting underwayat this writing (Spring 1994). Wherever possible, the amount offreeboard on superstructure and the spacing of piers should beestablished to permit reasonable floating debris, determined on thebasis of review of the flood plain, from impacting the superstructure.

Floating debris does tend to collect around piers, forming whatis called a debris raft. Debris rafts tend to increase the apparent areaof the pier subject to stream pressure. Pending future U. S. researchon the design parameters for debris rafts, the commentary contains aprovision from the New Zealand Highway Bridge Design Specificationwhich may be used until better data becomes available.

In addition to the pressure, identified above, the Specificationcontains a requirement to design for wave loads where structures areexposed to that type of environment. Reference is given in thecommentary to the Shore Protection Manual as a source for wavedesign information.

Also, new to the Specification is a requirement to design thefoundations for scour. Scour is a change in foundation conditionresulting from the design flood for scour and is applicable as both arequirement for strength and service limit states, but also as anextreme event limit state under the check flood for scour, usually a500-year flood. In this case, the requirement is that the structureshould remain stable and intact at its full nominal resistance.


Lecture - 5-11

5.5 WIND LOADS

5.5.1 General Wind Provisions

Article S3.8 establishes wind loads which are consistent withthe format and presentation currently used in meteorology. Windpressures are established with consideration to a base wind velocity(Vb) of 160 km per hour corresponding to the 100 mph wind commonin past specifications. If no better information is available, the windvelocity at 10 000 mm above the ground, a distance presumed to beabove the immediate effects of the ground in open terrain, may betaken as the base wind Vb. Alternatively, the base wind speed may betaken from the Basic Wind Speed Charts available in the literature orsite specific wind surveys may be used to establish V10. Usingcharacteristics available in Table S3.8.1.1-1, repeated below, used todescribe the type of terrain over which approach winds move, twocharacteristic values, V0 and Z0, can be determined. These aremeteorological terms known as the friction velocity and friction length,respectively.

CONDITION OPEN COUNTRY SUBURBAN CITY

V0 (km/hr) 13.2 15.2 19.4

Z0 (mm) 70 1000 2500

Using the information determined above and the height of thestructure above ground or water, if it is over 10 000 mm, it is possibleto calculate a design wind velocity, Vdz, using Equation S3.8.1.1-1,which is repeated below.

(5.5.1-1)VDZ ' 2.5 V0

V10

VBln Z

Zo

Equation S3.8.1.1-1 provides a correction for structureelevation with similar intent to that used by the 1/7 power rule used bydesigners in the past, but agrees with the current meteorologicaltheories.

Given the design wind speed, now corrected for approachconditions and height above the referenced datum of 10 000 mm, it ispossible to calculate the design wind pressure, PD, based on basepressure, PB, given in Table S3.8.1.2-1, repeated below, for variousstructural components. The base wind pressures, specified in thattable, are established for the case where VB is equal to 160 km perhour. The base wind velocity and the wind velocity at the elevation inquestion are used to customize the base wind pressures given inTable S3.8.1.2-1 for the particular site conditions. Additionally, certainminimum design wind pressures, comparable to those in past editionsof the Standard Specification, are also required.


Lecture - 5-12

STRUCTURALCOMPONENT

PBWINDWARD LOAD,

MPa

PBLEEWARD LOAD,

MPa

Trusses,Column andArches

0.0024 0.0012

Beams 0.0024 NA

Large Flat Surfaces 0.0019 NA

Figure 5.5.1-1 shows the variation in design pressure withheight for the three upwind conditions.

00.0010.0020.0030.004

8000

1200

0

1600

0

2000

0

2400

0

2800

0

Z(mm)

PD -

MPa country PD

SuburbanCity

Figure 5.5.1-1 Design Wind Pressure, PD, Vs. Height for PB = 0.0024MPa

Consider the following example data point in Figure 5.5.1-1:

• Assume VB = V10 = 160 km/hr

• Assume suburban setting for which Vo = 15.2 km/hr and Zo =300 mm

• Assume windward pressure on a beam for which PB = 0.0024MPa

• Assume Z = 19 000 mm

VDZ ' 2.5 VoV10

VBRn Z

Zo

VDZ ' 2.5(15.2) 160160

Rn 19 000300


Lecture - 5-13

VDZ ' 158 kmhr

PD ' 0.0024 (158)2

(160)2' 0.002 34MPa

Wind pressure is also applied to vehicular live load, as was thecase in past editions of the Specification. No substantial changes inthis provision have been made.

5.5.2 Wind Blowing at an Angle

Where the wind is not taken as normal to the structure, thebase wind pressures, PB, for various angles of wind direction may betaken as specified in the table below and shall be applied to a singleplace of exposed area. The skew angle is measured from aperpendicular to the longitudinal axis of the bridge. The wind directionfor design shall be that which produces the extreme force effect on thecomponent under investigation. The transverse and longitudinalpressures are to be applied simultaneously. For trusses, columns,and arches, the base wind pressures specified in the table are thesum of the pressures applied to both the windward and leeward areas.

Base Wind Pressures, PB, for Various Angles of Attack andVB = 160 km/hr

Columns and Arches Girders

Skew Angleof Wind

LateralLoad

LongitudinalLoad

LateralLoad

LongitudinalLoad

Degrees MPa MPa MPa MPa

0 0.0036 0 0.0024 0

15 0.0034 0.0006 0.0021 0.0003

30 0.0031 0.0013 0.0020 0.0006

45 0.0023 0.0020 0.0016 0.0008

60 0.0011 0.0024 0.0008 0.0009

5.5.3 Wind Forces Applied Directly to the Substructure

The transverse and longitudinal forces to be applied directly tothe substructure shall be calculated from an assumed base windpressure of 0.0019 MPa. For wind directions taken skewed to thesubstructure, this force shall be resolved into componentsperpendicular to the end and front elevations of the substructure. Thecomponent perpendicular to the end elevation shall act on theexposed substructure area as seen in end elevation, and thecomponent perpendicular to the front elevation shall act on the


Lecture - 5-14

exposed areas and shall be applied simultaneously with the windloads from the superstructure.

5.5.4 Wind Pressure on Vehicles

When vehicles are present, the design wind pressure shall beapplied to both structure and vehicles. Wind pressure on vehiclesshall be represented by an interruptible, moving force of 1.46 N/mmacting normal to, and 1800 mm above, the roadway and shall betransmitted to the structure.

When wind on vehicles is not taken as normal to the structure,the components of normal and parallel force applied to the live loadmay be taken as specified in the table below with the skew angletaken as referenced normal to the surface.

Wind Components on Live Load

Skew AngleNormal

Component

ParallelComponent

Degrees N/mm N/mm

0 1.46 0

15 1.28 0.18

30 1.20 0.35

45 0.96 0.47

5.5.5 Vertical Wind Pressure

Standard Specification has required the use of a windward 1/4point vertical load, and this requirement is continued in the currentSpecification. The purpose of this requirement is to account for thechange in pressure caused by the interruption of a horizontal windstream created by the bridge superstructure.

5.5.6 Aeroelastic Stability

The provisions of this article require that components whosespan to width or depth ratio exceeds 30 be considered wind-sensitiveand that aeroelastic effects should be taken into account for this typeof member or structure. The choice of the value of 30 is somewhatarbitrary and was set so that most conventional composite girderconstruction would not be affected by this provision. This is basedsolely on experience. There have been cases where girders havevibrated during construction, but this has been relatively rare.

All flexible structures or structural components should beinvestigated for resistance to vortex shedding excitation, wake


Lecture - 5-15

buffeting and divergence and flutter, as appropriate. In the past, it hastypically been assumed that suspension bridges and cable-stayedbridges were the type of structures for which aeroelastic effects couldbe significant. However, other relatively flexible modern structuresmay also be susceptible. Similarly, components of structures, notablylong, unbraced structural members and cables, may also be subjectto wind-induced vibrations. In at least one case, a low amplitudevibration of a tied-arch appeared to be driving distortion-inducedfatigue cracking of welds inside the tie girder.

In response to these phenomena, the Specification requiresthat bridges should be designed to be free of divergences and flutterfor wind speeds up to 1.2 times the design wind velocity at the bridgedeck height, and that structural components and bridges should befree of fatigue damage due to vortex shedding or galloping.

It is of some interest to note that the same problems seem toappear over and over again. Consider the four structures discussedbelow, each of which had hanger vibrations due to vortex shedding.

The Tacony-Palmyra Bridge, designed in Circa 1928, hashangers for the deck system which are structural members. Thesehangers vibrated in the wind, probably to vortex shedding. In thiscase, the solution was to reduce the length of the member by puttinga horizontal strut across the bridge intercepting each of the hangers.The Robert Moses Causeway Bridge, Circa 1958, is a similarconfiguration with long structural-shape hangers supporting theroadway. These too vibrated in the wind, probably also from vortexshedding. In this case, the members were studies in the wind tunnel,and it was decided to use an aerodynamic solution by attaching sheetmetal deflectors to the corners of the I-shaped hangers, thus,streamlining the shape. The deflectors contained a projection at 45Ebent into the body of the hanger. This retrofit has been quite effective.

The Commodore Barry Bridge, Circa 1970, also has longvertical members supporting the deck which vibrated in the wind andcaused significant fatigue damage to these members. In this case,tuned mass dampers were added to the members and theireffectiveness was verified in the wind tunnel prior to installation. Thisretrofit also appears to have been successful.

The I-470 Bridge at Wheeling, Circa 1982, used structuralstrand hangers in groups of four to support the roadway. All thesehangers vibrated. On the longer hangers, it was possible to clearlysee 7th and 8th load vibrations in moderate winds. This vibrationcaused damage to some of the wires resulting in wire breaks wherethe structural strands were supported at the deck level. Woodenwedges driven between the strands and their structural membersupports was found to add sufficient damping to stop these vibrations.In this case, retrofit consisted of a steel and neoprene collar added tothe ends of the cables to simulate the effect of the wooden wedges,and spacers between the cables and a set of four to add brace pointsand to create damping by the action of one cable vibrating againstanother.


Lecture - 5-16

The frequency of the vortex shedding and, hence, the pulsatingpressure, is given by:

(5.5.3-1)f ' VSD

where:

V = the wind speed in mm/sec

D = a characteristic dimension, in mm

S = the Strouhal Number

A table of Strouhal Numbers for sections is given in "Wind Forces onStructures", Transactions of the ASCE, Volume 126, Part II, Page1180, and is repeated below.

Table 5.5.3-1 - Strouhal Number for Various SectionsWind Profile Proportion Value of S Profile Proportion Value of S

0.120 0.200

0.137

0.144

b/d2.52.01.51.00.70.5

0.0600.0800.1030.1330.1360.138

0.145

0.147


Lecture - 5-17

Self-exciting oscillations of the member in the directionperpendicular to the wind stream may result when the frequency ofvortex shedding coincides with a natural frequency of the obstruction.Thus, determining the torsional frequency and bending frequency inthe plane perpendicular to the wind and substituting those frequenciesinto the Strouhal equation leads to an estimate of wind speeds atwhich resonance may occur. This motion has led to fatigue crackingof some truss and arch members, particularly cable hangers and I-shaped members. The vortex shedding design approach, describedherein, is oriented towards providing sufficient stiffness to reasonablypreclude vibrations. It does not directly lead to a solution for theamplitude of vibration and, hence, it does not directly lead to a solutionfor vibratory stresses. Solutions for amplitude are available in theliterature.

The following approximate procedure for estimating bendingand torsional frequencies is an excerpt from "Natural Frequencies ofAxially Loaded Bridge Members" by C. C. Ulstrup, Journal of theStructural Division, ASCE, 1978.

The general approximate formula for members whose shearcenter and centroid coincide is as follows:

(5.5.3-2)fn 'a

2πkn ll

2

1 % εpKlπ

2 12

in which:

fn = natural frequency of member for each modecorresponding to n = 1, 2, 3, etc.

knl = eigenvalue for each mode (see table below)

K = effective length factor (see table below)

l = length of the member

a = coefficient dependent on the physical properties of themember, given as ab or at

ep = coefficient dependent on the physical properties of themember and on the axial force, i.e., positive fortension, negative for compression, given as epb or ept


Lecture - 5-18

Table 5.5.3-2 - Eigenvalue knl and Effective Length Factor K

Support Condition knl K

n = 1 n = 2 n = 3 n = 1 n = 2 n = 3

π 2π 3π 1.000 0.500 0.333

3.927 7.069 10.210 0.700 0.412 0.292

4.730 7.853 10.996 0.500 0.350 0.259

1.875 4.694 7.855 2.000 0.667 0.400

For bending:

(5.5.3-3)ab 'EIgγA

12

(5.5.3-4)epb 'PEI

For torsion:

(5.5.3-5)at 'ECwgγ Ip

12

(5.5.3-6)ept 'GJ % PIp A

&1

ECw

in which:

E = Young's modulusG = shear modulusγ = weight density of memberg = gravitational accelerationP = axial force (tension is positive)I = moment of inertia about relevant axisA = area of member cross-sectionCw = warping constantJ = torsion constantIp = polar moment of inertia

In the design of a member, the frequency of vortex sheddingfor the section is set equal with the bending and torsional frequencyand the resulting equation solved for the wind speed V. This is thewind speed at which resonance occurs and the design should be suchthat V exceeds the velocity at which the wind is expected to occur bya reasonable margin.


Lecture - 5-19

REFERENCES

Committee on Ship-Bridge Collisions, "Ship Collisions with Bridges -The Nature of the Accident, their Prevent and Mitigation", MarineBoard, Commission on Engineering and Technical Systems, NationalResearch Council, National Academic Press, Washington, D.C., 1983

Modjeski and Masters, Inc., Consulting Engineers, "Criteria for: TheDesign of Bridge Piers with Respect to Vessel Collision in LouisianaWaterways", Prepared for the Louisiana Department of Transportationand Development and the Federal Highway Administration, NewOrleans, Louisiana, November 1984

American Association of State Highway and Transportation Officials(AASHTO), "Guide Specification and Commentary for Vessel CollisionDesign of Highway Bridges", 1991, Washington, D. C.

Larsen, O. D., "Ship Collision with Bridges - The Interaction betweenVessel Traffic and Bridge Structures", International Association forBridge and Structural Engineering (IABSCE), Structural EngineeringDocuments, 1993


Lecture - 6-1

LECTURE 6 - ANALYSIS I


The objectives of this lesson are to acquaint the student with:

• the various analysis techniques required and/or recommendedfor determining the force effects and components of bridges,and

• the use of approximate and refined methods for thedetermination for force effects in conventional girder-typestructures.

The background on the development of new, improved,distribution factors which were developed under NCHRP Project 12-26has been included for reference in an Appendix.

The use of grid and finite element types of analysis for multi-beam bridges is also recommended in the LRFD Specification. Thesemethods require considerable care in structural modeling, and severalexamples of the large effects of seemingly small errors in structuralmodels will be presented.

6.2 ACCEPTABLE METHODS OF STRUCTURAL ANALYSIS

Article S4.4 contains a list of methods of analysis that areconsidered suitable for analysis of bridges. These include theclassical force and displacement methods, such as virtual work,moment distribution, slope deflection, the so-called general method,the more modern finite element, finite strip and plate analogy-typemethods, analysis based on series expansions and the yield-linemethod for the non-linear analysis of plates and railings. Some ofthese methods of analysis are suitable for hand calculations, but forany problem of large size, some sort of computer solution will almostalways be required for practical design purposes. This is becausealmost all of these methods, with the possible exception of the seriesmethods and the yield-line methods, will eventually require thesolution of large sets of simultaneous equations. The series method,while elegant from a mathematical point of view, will typically requirea computer program to expand the series sufficiently to yield goodresults in a practical time frame. Yield-line methods, which could beconsidered the extension of plastic design to two-dimensionalsurfaces, are typically a hand calculation procedure.

The use of computer programs in bridge design brings up aphilosophical problem as to the responsibility for error. Almost allvendors of commercial computer programs disavow any responsibilityfor error. A release from liability is usually implicit in their use and mayeven be an explicit part of obtaining a license. This means that anorganization using a computer program must be relatively certain of


Lecture - 6-2

the results that it obtains. It is not necessary for every engineer in alarge design section to have personally confirmed every computerprogram, but it is necessary that some verification testing be done orthat the results of previous verification testing be obtained in order toproduce the required level of confidence. Computer programs can beverified against universally accepted closed- form solutions, othercomputer programs which have been previously verified, or the resultsof testing.

Many computer programs for design use also contain codechecking capabilities. Others have portions of the applicable designspecification embedded in the coding of the program. In order toidentify the specification edition which may have been tied to a givenrelease of a program and also to provide a means for determiningwhich structures may have been designed with a version of a programlater found to contain errors, the specification requires that a name,version and a release date of software be identified in the contractdrawing.

6.3 PRINCIPLES OF MATHEMATICAL MODELING

6.3.1 Structural Material Behavior

The LRFD Specification recognizes both elastic and inelasticbehavior of materials for analysis purposes. Inelastic materialbehavior is implicit in many of the equations and procedures specifiedfor the calculation of cross-sectional resistance, such as calculatingthe nominal resistance of a concrete beam or column, the nominalplastic moment resistance of a compact and adequately braced steelcross-section, or the bearing capacity of a spread footing. Often, theforce effects to which this resistance will be compared will becalculated on a basis of a linear structural analysis with elasticmaterial properties having been assumed. This dichotomy has existedin the bridge specification since the introduction of load factor designin the early 1970's. It continues through the LRFD Specification.

On the other hand, there are certain assumed failure modes atextreme events and the use of mechanism and unified autostressdesign procedures for steel girders, where permitted, which requireanalysis based on non-linear behavior. Many times, this analysis willtake a form analogous to plastic design of steel frames. For example,seismic design may be based on the formation of plastic hinges at thetop and bottom of the columns of a bent. Ship collision forces may beabsorbed in a comparable inelastic manner. Furthermore, it isanticipated that future seismic design provisions will be based onextensive research currently underway to develop a step-by-step non-linear force displacement relationship for the lateral displacement ofpiers.

Where inelastic analysis is used, the Designer must be certainthat a ductile failure mode is obtained through proper detailing. Rules


Lecture - 6-3

for achieving this are presented in the sections for steel and concretedesign.

6.3.2 Geometry

6.3.2.1 GENERAL

Most analyses done for the purpose of designing bridges arebased on the assumption that the displacements caused by the loadsare relatively small and, therefore, it is suitably accurate to base thecalculations on the undeformed shape. This is typically referred to asthe small deflection theory, and it is routinely used in the design ofbeam-type structures and bridges which resist loads through a couplewhose tensile and compressive forces remain essentially in fixedpositions relative to each other while the bridge deflects. This will bethe case for a truss and for tied-arches.

For other types of structures and components and for certaintypes of analysis, the effect of the deflections must be considered inthe development of the force equilibrium equations, i.e., the equationsof equilibrium are written for the displaced shape. Almost allengineers are aware that the study of structural stability requiresconsideration of the displaced shape, in fact, if the displaced shape isnot part of the original formulation of the problem, one would never beable to determine that a column, shell or plate can buckle. Considerfor a moment the simple pin-ended column. Unless the deflectedshape of the column is taken into account, the moment caused by theaxial load acting on the displaced shape would not be accounted for.It is this moment which causes the column to move laterally, i.e., tobuckle.

Almost a century ago, it was found that the only reasonablyaccurate way to calculate force effects in suspension bridges of anysize was to include the deflection of the cable in the formulation of theproblem and, therefore, the displacement of a stiffening truss orstiffening girder. As conventional, i.e., not tied, arches became longerand more slender, an effect directly analogous to that observed insuspension bridges can become significant enough that it must beaccounted for in the design of the arch rib. In fact, because the archrib is in compression and can buckle, the effect of large deflectionscan be especially important.

Finally, the compression members of frames in bents can alsobe susceptible to this phenomenon.

Where non-linear effects arriving either out of material non-linearity or large deflections become significant, then super position offorces does not apply. This means that each load case underinvestigation must be studied separately under the full effect of all ofthe factored loads that make up the load combination under study.This is a very significant effect on most practical design calculations.Commonly, a Bridge Engineer calculates the force effect from avariety of individual loads and then combines, or superimposes, the


Lecture - 6-4

force effects calculated for each individual load to make up whatevergroup combination of loadings are needed. For non-linear analysis,each combination must be investigated, i.e., analyzed, individually.

6.3.2.2 APPROXIMATE METHODS

To simplify analysis and to partially bypass the need to analyzeeach load combination separately, as identified above, certainapproximate methods have been developed to allow the designer toadd a correction to a set of force effects calculated in a linear manner.These are sometimes called single-step adjustment methods, themost commonly used of which is moment magnification for beamcolumns, which has been part of the AASHTO Specifications since theearly 1970's.

For beam columns, the moment magnification process is givenby the equations below.

Mc = δb M2b + δs M2s (6.3.2.2-1)

fc = δb f2b + δs f2s (6.3.2.2-2)

for which:

(6.3.2.2-3)δb '

Cm

1 &Pu

φPe

$ 1.0

(6.3.2.2-4)δs '

1

1 &ΣPu

φΣPe

where:

Pu = factored axial load (N)

Pe = Euler buckling load (N)

φ = resistance factor for axial compression as specified inSpecification Sections 5, 6 and 7, as applicable

M2b = moment on compression member due to factoredgravity loads that result in no appreciable sideswaycalculated by conventional first order elastic frameanalysis, always positive (N@mm)

f2b = stress corresponding to M2b (MPa)

M2s = moment on compression member due to factoredlateral or gravity loads that result in sidesway, ∆,


Lecture - 6-5

greater than Ru/1500, calculated by conventional firstorder elastic frame analysis, always positive (N@mm)

f2s = stress corresponding to M2s (MPa)

It may appear that the moment magnification factor contains the Eulerbuckling load, Pe. However, Pe is only a convenient substitution for agroup of terms related to the displacement of the beam column.

A derivation of the moment magnification equation can befound in many textbooks on steel or concrete design.

For cases where the shape of the beam column is expected tobe radically different from that given by the simply-supported case, orthe loads significantly different from those indicated above, then it ispossible to make an adjustment to account for a different initial elasticshape through the factor cm.

The moment magnification procedure has also been extendedto arches, and this has been available in the AASHTO Specificationsfor many years and is reproduced as Article S4.5.3.2.2c with no furtherrefinement.

6.3.2.3 REFINED METHODS

The effect of large deflections can also be rigorouslyaccounted for through iterative solutions of equilibrium equations,taking into account updated positions of the structure, or by usinggeometric stiffness terms. In some cases, e.g., the suspension bridge,solutions are available to the differential equations of equilibrium whichcan be solved in a trial and error fashion, or through series expansion.

6.3.3 Modeling Boundary Conditions

Points of expansion or other forms of articulation in thestructure are commonly idealized as frictionless units. Where pastpractice indicates that this has been a reasonable conservativeapproach, continued use is warranted. There are other instanceswhere the potential for nonfunctional expansion devices and/or thepossibility that joints may close should also be investigated. Thismight be the case, for example, in a seismic analysis where analysismay be made, assuming that expansion joints are operable and open,and then another analysis might be made, assuming that they areclosed and nonfunctional in order to simulate, or bound, the effects ofjoints reaching the limits of travel during the earthquake. Thepossibility of reaching the limit of expansion travel should also beinvestigated when evaluating non-linear effects on substructureelements. It may be possible that the amount of momentmagnification may be reduced because expansion dams will close,jamming the structure against the abutments before the full movementimplicit in the moment magnification factor can be reached. This willreduce the moment magnification factor and, hence, the designmoment.


Lecture - 6-6

Similarly, the effect of boundary conditions at foundation unitsshould also be evaluated. Foundation units are seldom fully fixed orfully pinned, and an evaluation of the potential movement of afoundation unit may be necessary in order to properly assessresponse, as well as secondary moments caused by change ingeometry. Here again, bounding of the range of probable movementmay be the only practical way to attack such a problem.

6.4 STATIC ANALYSIS

6.4.1 The Influence of Plan Geometry

Article S4.6.1 deals with two simplifications which can be madebased on the plan geometry of the superstructure.

The first simplification involves the possibility of replacing thesuperstructure for analysis purposes with a single-line element calleda spine beam. This may be done when the transverse distortion of thesuperstructure is small in comparison with the longitudinaldeformation. Generally, if the superstructure is a torsionally stiffclosed section or sections whose length exceeds 2.5 times their width,it may be idealized as a line element whose dimensions may bedetermined as given in the Specification. This can be used tosignificantly simplify analysis models.

The second simplification deals with when it is possible toconsider curved superstructures as straight for the purpose ofanalysis. If the superstructure is a torsionally stiff closed section andthe central angle of a segment between piers is less than 12E, thenthe segment may be considered straight. If the superstructure ismade of torsionally weak open sections, then the effects of curvaturemay be neglected when the subtended angle is less than that given inTable 6.4.1-1.

Table 6.4.1-1 - Limiting Central Anglefor Neglecting Curvature in DeterminingPrimary Bending Moments

No. of BeamsAngle for One

SpanAngle for

Multiple Spans

2 2E 3E

3 or 4 3E 4E

5 or more 4E 5E


Lecture - 6-7

6.4.2 Approximate Methods for Load Distribution

6.4.2.1 DECK SLABS AND SLAB-TYPE BRIDGES

The Specification permits the approximate analysis of deckslabs by analyzing a strip of deck as a continuous beam. Provisionsare made for determining the width of that strip at the unsupported edge of the slab and at points interior from the edges.

If the spacing of supporting components in the secondarydirection exceeds 1.5 times the spacing in the primary direction, thenall of the wheel loads applied to the deck are considered to be appliedto the primary strip. The secondary strip is designed on a basis ofpercentage of reinforcement in the primary strip.

If the spacing of the supporting components on the secondarydirection is less than 1.5 times that in the primary direction, then acrossed sticks analogy is used. The width of the equivalent strips ineach direction is provided by Table S4.6.2.1.3-1 and the wheel loadis distributed between two idealized intersecting strips according to therelative stiffness of each strip.

Once the wheel loads have been assigned to the strips, foreither case identified above, then the force effects are calculatedbased on a continuous beam. For the purpose of analyzing thecontinuous beam, the span length of each span is taken as a center-to-center of supporting components. For the purpose of calculatingmoment and shear at a design section, some offset from thetheoretical center of support is permitted as given in the Specification.

Decks which form an integral part of a cellular cross-sectionare supported on webs which are monolithic with the deck. Therefore,when the deck rotates, the web of the box girder rotates giving rise tobending stresses throughout the cross-section. For the purpose ofanalyzing this effect, a cross-sectional frame action procedure isidentified in the Specification.

In the case of fully filled and partially filled grids, the results ofrecent research are incorporated in LRFD Article S4.6.2.1.8 to givenbending moments per unit length of grid.

6.4.2.2 BEAM SLAB BRIDGES

6.4.2.2.1 General

The Specification provides a series of empirical rules forassigning portions of a design lane to a supporting component. Theseare commonly called distribution factors. It is important to rememberthat the approximate distribution factors, specified in the LRFDSpecification, are on a lane, i.e., axle basis, not a wheel basis. Thedistribution factors are given for the various kinds of bridges shown inFigure 6.4.2.2.1-1.


Lecture - 6-8

SUPPORTINGCOMPONENTS TYPE OF DECK

TYPICALCROSS-SECTION

Steel Beam - RevisedFactors

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

Closed Steel or PrecastConcrete Boxes - RevisedFactors

Cast-in-place concrete slab

Open Steel or PrecastConcrete Boxes - RevisedFactors

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

Cast-in-Place Concrete Multi-cell Box - Revised Factors

Monolithic concrete

Cast-in-Place Concrete TeeBeam - Revised Factors

Monolithic concrete

Precast Solid, Voided orCellular Concrete Boxes withShear Keys - RevisedFactors

Cast-in-place concrete overlay

Precast Solid, Voided orCellular Concrete Box withShear Keys and with orwithout TransversePost-Tensioning - RevisedFactors (in some cases)

Integral concrete

Precast Concrete ChannelSections with Shear Keys


Precast Concrete Double TeeSection with Shear Keys andwith or without TransversePost-Tensioning

Integral concrete

Precast Concrete TeeSection with Shear Keys andwith or without TransversePost-Tensioning

Integral concrete

Concrete I or Bulb-TeeSections - Revised Factors

Cast-in-place concrete, precastconcrete

Wood Beams - RevisedFactors

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

Figure 6.4.2.2.1-1 - Common Deck Superstructures Covered in LRFD SpecificationArticles 4.6.2.2.2 and 4.6.2.2.3


Lecture - 6-9

Some of the distribution factors are new to the Specification as a resultof an extensive project on load distribution known as NCHRP Project12-26. Where the distribution factors for a given type of cross-sectionhave been developed under that project and are new to theSpecification, the words "revised factors" appear in the columnidentified as "supporting components". Where those words do notappear, the distribution factors have been retained from earliereditions of the AASHTO Standard Specifications.

Some simplifications have been made in utilizing thedistribution factors from NCHRP 12-26. In particular, correctionfactors for various aspects of structural action, typically involvingcontinuity, which were less than 5%, were omitted from the LRFDSpecifications. Similarly, an increase in moments over piers, thoughtto be on the order of 10%, was not included because stresses at ornear internal bearings have been shown to be reduced below thatcalculated by simple analysis techniques due to an action known as"fanning". The distribution factors, given in the LRFD Specification,are also different from those given in NCHRP 12-26, because themultiple presence factors, given in Lecture 3, are built into thedistribution factors, whereas, the multiple presence factors in earliereditions of the AASHTO Standard Specifications are built into theNCHRP 12-26 factors. Additionally, the factors appropriate for theLRFD Specification are based on a lane of live load, rather than a "lineof wheels". Finally, when the SI version of the LRFD Specification isused, conversion to that system of units has also been accounted for.

Various limits on span, spacing and other characteristics areprovided in the Specifications for each of the distribution factors.These parameters identify the range for which the factors weredeveloped. They were not evaluated for factors beyond the rangesindicated. Therefore, for structures which do not comply with theselimitations, a rigorous analysis by grid or finite elements should beused. Furthermore, the distribution factors usually apply for structureswhich are:

• essentially constant in deck width,

• have four or more beams, unless noted,

• have beams which are parallel and approximately of the samestiffness,

• have overhangs that do not exceed 0.9 m, unless specificallynoted,

• have in-plan curvatures less than those specified above, and

• have a cross-section consistent with one of the cross-sectionsidentified in Table 6.4.2.2.1-1.

Since the distribution factors, developed under NCHRP 12-26,are new to the Specification, it is appropriate to review the background


Lecture - 6-10

and development of the new distribution factors. The discussionbelow was taken from the NCHRP Research Results Digest No. 187,a summary of the Final Report of Project 12-26 as summarized byIan M. Friedland, NCHRP Project Coordinator.

Live load distribution on highway bridges is a key responsequantity in determining member size and, consequently, strength andserviceability. It is of critical importance both in the design of newbridges and in the evaluation of the load-carrying capacity of existingbridges.

Using live load distribution factors, engineers can predictbridge response by treating the longitudinal and transverse effects oflive loads as uncoupled phenomena. Empirical live load distributionfactors for stringers and longitudinal beams have appeared in theAASHTO Standard Specifications for Highway Bridges with only minorchanges since 1931. Findings of recent studies suggest a need toupdate these specifications in order to provide improved predictionsof live load distribution.

Live load distribution is a function of the magnitude andlocation of truck live loads and the response of the bridge to theseloads. The NCHRP 12-26 study focused on the second factormentioned above: the response of the bridge to a predefined set ofloads, namely, the HS family of trucks.

In Project 12-26, three levels of analysis were considered foreach bridge type. The most accurate level, Level 3, involves detailedmodeling of the bridge deck. Level 2 includes either graphicalmethods, nomographs and influence surfaces, or simplified computerprograms. Level 1 methods provide simple formulas to predict lateralload distribution, using a wheel load distribution factor applied to atruck wheel line to obtain the longitudinal response of a single girder.

The major part of the research in Project 12-26 was devotedto the Level 1 analysis methods because of its ease of application,established use, and the surprisingly good correlation with the higherlevels of analysis in their application to a majority of bridges. Theformulas presented in the current AASHTO specifications wereevaluated, and alternative formulas were developed that offerimproved accuracy, wider range of applicability, and in some cases,easier application than the current AASHTO formulas. Theseformulas were developed for interior and exterior girder moment andshear load distribution for single or multiple lane loadings. In addition,correction factors for continuous superstructures and skewed bridgeswere developed.

The formulas presented in previous AASHTO Specifications,although simpler, do not present the degree of accuracy demanded bytoday's Bridge Engineers. In some cases, these formulas can resultin highly unconservative results (more than 40%), in other cases theymay be highly conservative (more than 50%). In general, the formulasdeveloped in Project 12-26 are within 5% of the results of an accurate


Lecture - 6-11

analysis. Table 6.4.2.2.1-1 shows comparisons with momentdistribution factors obtained from AASHTO, Level 1, Level 2 and Level3 methods for simple span bridges.

Table 6.4.2.2.1-1 - Comparison of interior girder moment distributionfactors by varying levels of accuracy using the "average bridge" foreach bridge type

Bridge Type AASHTO

NCHRP 12-26(Level 1)

Grillage(Level 2)

Finite Element(Level 3)

Beam-and-slaba 1.413(S/1700) 1.458 1.368 1.378

Box girdera 1.144 1.143 0.970 1.005Slabb 1820 1710 1900 1890Multi-box beama 0.646 0.597 0.540 0.552Spread box beama 1.564 1.282 1.248 1.241

aNumber of wheel lines per girderbWheel line distribution width, in mm

In addition, the study resulted in recommendations for use ofcomputer programs to achieve more accurate results. Therecommendations focus on the use of plane grid analysis, as well asdetailed finite element analysis, where different truck types and theircombinations may be considered.

6.4.2.2.2 Influence of Truck Configuration

The formulas developed in Project 12-26 for the Level 1analysis were based on the standard AASHTO "HS" trucks. A limitedparametric study conducted as part of the research showed thatvariations in the truck axle configuration or truck weight do notsignificantly affect the wheel load distribution factors. The group ofaxle trains used for this study are shown in Figure 6.4.2.2.2-1. It isanticipated that smaller gage widths would result in larger distributionfactors, and larger gage widths would result in smaller distributionfactors. Table 6.4.2.2.1-1 gives the variation of wheel load distributionfactors with different axle configurations applied to a number of beam-and-slab bridges. The differences were below 1% in many cases and,in all cases, the formulas resulted in good predictions. Therefore, withsome caution, these formulas may be applied to other truck types.Obviously, Levels 2 and 3 analyses may also be applied for truckssignificantly different from the AASHTO family of trucks.


Lecture - 6-12

Figure 6.4.2.2.2-1 - Axle Configurations for Truck Types Consideredin Study

Table 6.4.2.2.1-1 - Effect of Load Configuration on Distribution Factor

DISTRIBUTIONFACTOR (g)

PERCENT DIFFERENCEWITH HS-20

HS-20 HTL-57 4A-66 B-141 NCHRP12-26 HTL-57 4A-66 B-141

NCHRP 12-

26Average* 1.293 1.261 1.285 1.268 1.304 -2.4 -0.6 -1.9 +0.9Max. S

5000 mm 2.220 2.162 2.205 2.178 2.308 -2.6 -0.7 -1.9 +4.0

Min. S1000 mm 0.713 0.717 0.713 0.715 0.755 +0.6 0.0 +0.3 +5.9

Max. L60 000 mm 0.982 0.958 0.983 0.952 1.033 -2.4 +0.1 -3.1 +5.2

Min. L6000 mm 1.630 1.625 1.624 1.623 1.807 -0.3 -0.3 -0.4 +10.9

*S = 2200 mmL = 20 000 mmts = 185 mmKg = 2.33 x 1011 mm4


Lecture - 6-13

6.4.2.2.3 Simplified Methods

6.4.2.2.3a Simplified Formulas for Beam-and-Slab Bridges

This type of bridge has been the subject of many previousstudies, and many simplified methods and formulas were developedby previous researchers for multi-lane loading moment distributionfactors. The AASHTO formula, the formulas presented by otherresearchers, and the formulas developed in the study are discussedin the following according to their application.

Table 6.4.2.2.3a-1 is taken from the specifications andsummarized criteria for moment in interior beams or elements forvarious types of cross-sections. Similar tables exist for moment inexterior griders, for shear in interior girders and shear in exteriorgirders.

Table 6.4.2.2.3a-2 describes how the term L (length) may bedetermined for use in the live load distribution factor equations givenin Table 6.4.2.2.3a-1.

In the rare occasion when the continuous span arrangementis such that an interior span does not have any positive uniform loadmoment (i.e., no uniform load points of contraflexure), the region ofnegative moment near the interior supports would be increased to thecenterline of the span, and the L used in determining the live loaddistribution factors would be the average of the two adjacent spans.


Lecture - 6-14

Table 6.4.2.2.3a-1 - Distribution of Live Loads Per Lane for Moment in Interior Beams

Type of Beams ApplicableCross-Section

from Table4.6.2.2.1-1

Distribution Factors Range of Applicability

Wood Deck on Woodor Steel Beams

a, l See Table S4.6.2.2.2a-1

Concrete Deck onWood Beams

l One Design Lane Loaded: S/3700Two or More Design Lanes Loaded: S/3000

S # 1800

Concrete Deck, FilledGrid, or Partially FilledGrid on Steel orConcrete Beams;Concrete T-Beams, T-and Double T-Sections

a, e, k andalso i, j

if sufficientlyconnected toact as a unit

One Design Lane Loaded:

0.06 % S4300

0.4 SL

0.3 Kg

Lt 3s

0.1

Two or More Design Lanes Loaded:

0.075 % S2900

0.6 SL

0.2 Kg

Lt 3s

0.1

1100 # S # 4900110 # ts # 3006000 # L # 73 000Nb $ 4

use lesser of the values obtained from theequation above with Nb = 3 or the lever rule

Nb = 3

Multicell Concrete BoxBeam

d One Design Lane Loaded:

1.75 % S1100

300L

0.35 1Nc

0.45


13Nc

0.3 S430

1L

0.25

2100 # S # 400018 000 # L # 73 000Nc $ 3

If Nc > 8 use Nc = 8

Concrete Deck onConcrete Spread BoxBeams

b, c One Design Lane Loaded:

S910

0.35 SdL2

0.25


S1900

0.6 SdL 2

0.125

1800 # S # 35006000 # L # 43 000450 # d # 1700Nb $ 3

Use Lever Rule S > 3500





Lecture - 6-15

Concrete Beams usedin Multibeam Decks

f One Design Lane Loaded:

k b2.8L

0.5 IJ

0.25

where: k ' 2.5(Nb )&0.2 $ 1.5


k b7600

0.6 bL

0.2 IJ

0.06

900 # b # 15006000 # L # 37 0005 # Nb # 20

gif sufficientlyconnected toact as a unit

h Regardless of Number of Loaded Lanes: S/D

where:

C = K (W/L) # K

D = 300 [11.5 - NL + 1.4NL (1 - 0.2C)2] when C# 5

D = 300 [11.5 - NL] when C > 5

Skew # 45°

NL # 6

g, i, jif connected

only enough toprevent relative

verticaldisplacement at

the interface

K = (1 % µ) IJ

for preliminary design, the following values ofK may be used:

Beam Type KNonvoided rectangular beams 0.7Rectangular beams with circular voids: 0.8Box section beams 1.0Channel beams 2.2T-beam 2.0Double T-beam 2.0

Steel Grids on SteelBeams

a One Design Lane Loaded:S/2300 If tg< 100 mmS/3050 If tg$ 100 mmTwo or More Design Lanes Loaded:S/2400 If tg< 100 mmS/3050 If tg$ 100 mm

S # 1800 mm

S # 3200 mm

Concrete Deck on Multiple Steel BoxGirders

b, c Regardless of Number of Loaded Lanes:

0.05 % 0.85NL

Nb

%0.425

NL

0.5 #NL

Nb

# 1.5


Lecture - 6-16

Table 6.4.2.2.3a-2 - L for Use in Live Load Distribution Factor Equations

FORCE EFFECT L (mm)

Positive Moment The length of the span for whichmoment is being calculated.

Negative Moment - End spans of continuous spans,from end to point of contraflexure under a uniformload on all spans

The length of the span for whichmoment is being calculated.

Negative Moment - Near interior supports ofcontinuous spans, from point of contraflexure to pointof contraflexure under a uniform load on all spans

The average length of the twoadjacent spans.

Positive Moment - Interior spans of continuousspans, from point of contraflexure to point ofcontraflexure under a uniform load on all spans


Shear The length of the span for whichshear is being calculated.

Exterior Reaction The length of the exterior span.

Interior Reaction of Continuous Span The average length of the twoadjacent spans.

Moment Distribution to Interior Girders, Multi-Lane Loading

The AASHTO formula for moment distribution for multi-laneloading is given as S/1800 for reinforced concrete T-beam bridgeswith girder spacing up to 3000 mm, and as S/1700 for steel girderbridges and prestressed concrete girder bridges with girder spacingup to 4300 mm, where S is the girder spacing. When the girderspacing is larger than the specified limit, simple beam distribution is tobe used to calculate the load distribution factors.

Marx, et al, at the University of Illinois, developed a formula forwheel load distribution for moment which included multiple lanereduction factors and is applicable to all beam-and-slab bridges. Theformula is based on girder spacing, span length, slab thickness andbridge girder stiffness.

A formula which does not consider a reduction for multi-laneloading was developed at Lehigh University. The Lehigh formulaincludes terms for the number of traffic lanes, number of girders, girderspacing, span length and total curb-to-curb deck width.

Sanders and Elleby (NCHRP Report 83) developed a simpleformula based on orthotropic plate theory for moment distribution onbeam-and-slab bridges. Their formula includes terms for girder


Lecture - 6-17

spacing, number of traffic lanes and a stiffness parameter based onbridge type, bridge and beam geometry and material properties.

A full-width design approach, known as Henry's Method, isused by the State of Tennessee. Henry's Method includes factors fornumber of girders, total curb-to-curb bridge deck width and a reductionfactor based on number of lanes.

A formula developed as part of NCHRP Project 12-26 includesthe effect of girder spacing, span length, girder inertia and slab thick-ness. The multiple lane reduction factor is built into the formula. Thisformula, applicable to cross-sections with four or more beams, is givenby:

(6.4.2.2.3a-1)g ' 0.075 % S2900

0.6 SL

0.2 Kg

Lt 3s

0.1

where:

S = girder spacing (1100 mm # S # 4900 mm)

L = span length (6000 mm # L # 73 000 mm)

Kg = n(I+Aeg2) (4 x 108 # Kg # 3 x 1012 mm4)

n = modular ratio of girder material to slab material

I = girder moment of inertia

eg = eccentricity of the girder (i.e., distance from centroid ofgirder to mid-point of slab)

ts = slab thickness (110 mm # ts # 300 mm)

This formula is dependent on the inertia of the girder and, thus, avalue for Kg must be assumed for initial design. For this purpose,Kg/LtS

3 may be taken as unity.

All of the above formulas were evaluated using direct finiteelement analysis with the GENDEK-5 program and a database of 30bridges; subsequently, they were evaluated using the MSI method anddatabase of more than 300 bridges. It was found that Equation 6 andthe Illinois formulas are accurate and produce results that are asaccurate as the Level 2 methods.

Moment Distribution to Exterior Girders, Multi-Lane Loading

Previous AASHTO Specifications recommend a simple beamdistribution of wheel loads in the transverse direction for calculatingwheel load distribution factors in edge girders. Any load that falls


Lecture - 6-18

outside the edge girder is assumed to be acting on the edge girder,and any load that is between the edge girder and the first interiorgirder is distributed to these girders by assuming that the slab acts asa simple beam in that region. Any wheel load that falls inside of thefirst interior girder is assumed to have no effect on the edge girder.

Marx, et al, at the University of Illinois, developed a formula forthe exterior girder based on certain assumptions in the placement ofloads and may not be applicable to all bridges. This formula includesterms similar to those used in their formula for moment distribution tointerior girders.

A formula, depending on wheel position, alone was developedas part of this study which results in a correction factor for the edgegirder. The factor must be applied to the distribution factor for theinterior girder to obtain a distribution factor for the edge girder. Thisformula is given by:

(6.4.2.2.3a-2)e ' 0.77 %de

2800

where:

de = distance from edge of the roadway, usually the face ofcurb, to the center of the exterior web of the exteriorcell, in mm

If the edge of the lane is outside of the exterior girder, thedistance is positive; if the edge of the lane is to the interior side of thegirder, the distance is negative.

It was found that the formula developed in Project 12-26resulted in accurate correction factors and was simpler than theprevious AASHTO procedure.

Moment Distribution to Interior Girders, Single-Lane Loading

The literature search performed in this study did not reveal anysimplified formula for single-lane loading of beam-and-slab bridges.The formula developed as part of the study is as follows:

(6.4.2.2.3a-3)g ' 0.06 % S4300

0.4 SL

0.3 Kg

Lt 3s

0.1

where the parameters are the same as those given for Equation6.4.2.2.3a-1.

This formula is applicable to interior girders only.


Lecture - 6-19

Moment Distribution to Exterior Girders

Simple beam distribution in the transverse direction should beused for single-lane loading of edge girders.

One other investigation, applicable to load distribution for bothshear and moment, is required for exterior beams of beam-slabbridges with diaphragms or cross-frames. This addition was notdeveloped as part of NCHRP 12-26, but was added by the NCHRP12-33 Editorial Committee. This distribution is based on treating thecross-section as a transversely rigid unit which deflects and rotates asa straight line. The live load is positioned for maximum effect on anexterior beam (one lane, two lane, three lane, etc., each with itsappropriate multiple presence factor). The total vertical force andmoment about the centroid of the cross-section is applied to the areaof the cross-section, i.e., the number of beams, and the sectionmodulus, i.e., the sum of the square of the distances of each beamfrom the centroid of the beams divided by the distance to the exteriorbeam. The specification puts this in equation form as:

(6.4.2.2.3a-4)R 'NL

Nb

%

Xext jNL

ev

jNb

x 2

where:

R = reaction on exterior beam in terms of lanes

NL = number of loaded lanes under consideration

ev = eccentricity of a design truck or a design laneload from the center of gravity of the pattern ofgirders (mm)

x = horizontal distance from the center of gravity ofthe pattern of girders to each girder (mm)

Xext = horizontal distance from the center of gravity ofthe pattern of girders to the exterior girder (mm)

Nb = number of beams or girders

Shear Distribution

No formula was found from previous research for thecalculation of wheel load distribution factors for shear. Therefore, theformulas developed as part of the 12-26 study are reported fordifferent cases as follows.

The formula for multi-lane loading of interior girders is:


Lecture - 6-20

(6.4.2.2.3a-5)g ' 0.2 % S3600

&S

10 700

2

The correction formula for multi-lane loading edge girder shearis:

(6.4.2.2.3a-6)e ' 0.6 %de

3000

The formula for shear distribution factor due to single-laneloading is:

(6.4.2.2.3a-7)g ' 0.36 % S7600

Equation 6.4.2.2.3a-7 is applicable to interior girders only.Simple beam distribution in the transverse direction should be used forsingle-lane loading of edge girders.

Correction for Skew Effects

Previous AASHTO Specifications did not include approximateformulas to account for the effect of skewed supports. However, someresearchers have developed correction factors for such effects onmoments in interior girders.

Marx, et al, at the University of Illinois, developed fourcorrection formulas for skew, one each for skew angles of 0, 30, 45and 60 degrees. Corrections for other values of skew are obtained bystraight-line interpolation between the two enveloping skew values.These correction formulas are based on girder spacing, span length,slab thickness and bridge girder stiffness.

A formula for a correction factor for prestressed concreteI-girders was developed as part of the research performed at LehighUniversity. This formula is based on the number of traffic lanes,number of girders, girder spacing, span length and total curb-to-curbdeck width, and includes a variable term for skew angle.

A correction factor for moment in skewed supports was alsodeveloped as part of Project 12-26. This formula is:

(6.4.2.2.3a-8)r ' 1 & c1 (tanθ)1.5

where, for θ > 30E,:


Lecture - 6-21

(6.4.2.2.3a-9)c1 ' 0.25Kg

Lt 3s

0.25SL

0.5

If θ # 30E, c1 is taken as zero. In calculating c1, θ should not be takenas greater than 60E. The other parameters are as defined previously.

From the literature review, no correction formulas wereobtained for shear effects due to skewed supports. In Project 12-26,it was found that shear in interior girders need not be corrected forskew effects; that is, the shear distribution to interior girders is similarto that of a straight bridge. A correction formula for shear at theobtuse corner of the exterior girder of two girder systems and allgirders of a multi-girder bridge was developed as part of this study andis given as:

(6.4.2.2.3a-10)r ' 1 % 0.2Lt 3

s

kg

0.3

tanθ

where the parameters are defined in Equation 6.4.2.2.3a-1.

Equation 6.4.2.2.3a-7 is to be applied to the shear distributionfactor in the exterior girder of non-skewed bridges. Therefore, theproduct of factors g, e and r must be obtained to find the obtuse cornershear distribution factor in a beam-and-slab bridge.

The distribution factors calculated for moments are plotted asa function of girder spacing for Spans 9, 18, 27, 36 and 60 m in Figure6.4.2.2.3a-1. For comparisons, AASHTO (1989) distribution factorsare also shown. Girder distribution factors, specified by AASHTO(1989), are conservative for larger girder spacing. For shorter spansand girder spacings, AASHTO (1989) produces smaller distributionfactors than calculated values.


Lecture - 6-22

Figure 6.4.2.2.3a - Calculated Distribution Factors

6.4.2.2.3b Simplified Formulas for Box Girder Bridges

Research on box girder bridges has been performed byvarious researchers in the past. Bridge deck behavior has been wellstudied and many recommendations have been made for detailedanalysis of these bridges. However, there is a limited amount ofinformation on simplified wheel load distribution formulas in theliterature.

In this context, a “girder” is a notional I-shape consisting of oneweb of a multi-cell box and the associated half-flanges on each sideof the web.

Moment Distribution to Interior Girders

Scordelis, at the University of California, Berkeley, presenteda formula for prediction of wheel load distribution for moment distribu-tion in prestressed and reinforced concrete box girder bridges. Theformula is based on modification of distribution factors obtained for arigid cross-section. The formula predicts load distribution factors inreinforced concrete box girders with high accuracy and for prestressedconcrete box girders with acceptable accuracy.

Sanders and Elleby also presented a simple formula formoment distribution factors which is similar to their formula for beam-and-slab bridges.

The following formulas, developed as part of NCHRP 12-26,may be used to predict the moment load distribution factors in theinterior girders of concrete box girder bridges due to single-lane andmulti-lane loadings. These formulas are applicable to both reinforced


Lecture - 6-23

and prestressed concrete bridges, and the multiple presence factor isaccounted for.

For single-lane loading:

(6.4.2.2.3b-1)g ' 1.75 % S1100

300L

0.35 1Nc

0.45

For multi-lane loading:

(6.4.2.2.3b-2)g ' 13Nc

0.3 S430

1L

0.25

where:

S = girder spacing, in mm

L = span length, in mm

Nc = number of cells


The factor for load distribution for exterior girders shall beWe/4300 mm, where We is the width of the exterior girder, taken as thetop slab width measured from the mid-point between girders to theedge of the slab.

Shear Distribution

No formula for shear load distribution was obtained fromprevious research for box girder bridges, but the following weredeveloped as part of NCHRP 12-26.

The shear distribution factor for interior girder multi-laneloading of reinforced and prestressed concrete box girder bridges is:

(6.4.2.2.3b-3)g ' S2200

0.9 dL

0.1

where:


d = girder depth, in mm



Lecture - 6-24

The distribution factor for shear in the interior girders due tosingle-lane loading may be obtained from:

(6.4.2.2.3b-4)g ' S2900

0.6 dL

0.1

where the parameters are as defined in Equation 6.4.2.2.3b-3.

A correction formula for shear in the exterior girder for multi-lane loading is:

(6.4.2.2.3b-5)e ' 0.64 %de

3800

where:



The following formula was developed for correction of momentdue to skewed supports for values of θ from 0E to 60E:

(6.4.2.2.3b-6)r ' 1.05 & 0.25 (tanθ) # 1.0

If θ > 60E, use 60E in Equation 6.4.2.2.3b-6.

Another formula was developed in Project 12-26 for correctionof shear at the obtuse corner of an edge girder. It must be applied tothe shear distribution factor for the edge girder of a non-skewed bridgeand must, therefore, be used in conjunction with the edge girdercorrection factor of Equation 6.4.2.2.3b-5. This formula, applicable forvalues of θ up to 60E, is:

(6.4.2.2.3b-7)r ' 1 % c1 (tanθ)

where:

c1 = 0.25 + L/(70d)

d = bridge depth, in mm



Lecture - 6-25

6.4.2.2.3c Simplified Formulas for Slab Bridges

The literature search did not reveal any simplified formulas forwheel load distribution in slab bridges other than those recommendedby AASHTO. Therefore, the following are formulas that weredeveloped as part of NCHRP 12-26.

Moment Distribution, Multi-Lane Loading

Equation 6.4.2.2.3c-1 was developed to predict wheel loaddistribution (distribution design width) for moment in slab bridges dueto multi-lane loading. Multiple presence factors are already accountedfor in the formula:

(6.4.2.2.3c-1)E ' 2100 % 0.12 L1W10.5 # W

NL

where:

E = the transverse distance over which a wheel line isdistributed

L1 = L # 18 000 mm

W1 = W # 18 000 mm


W = bridge width, in mm, edge-to-edge

Moment Distribution, Single-Lane Loading

The equation below predicts wheel load distribution formoment due to single-lane loading.

(6.4.2.2.3c-2)E ' 250 % 0.42 (L1W1 )0.5

where the parameters are as defined in Equation 6.4.2.2.3c-1.


Equation 6.4.2.2.3b-6 may be used to account for the reductionof moment in skewed bridges.

According to previous AASHTO Specifications, slab bridgesare adequate for shear if they are designed for moment. A quickcheck of this assumption was made and it was concluded that it is avalid assumption. Therefore, no formula or method is presented forcalculation of shear in slab bridges.


Lecture - 6-26

6.4.2.2.3d Simplified Formulas for Multi-Beam Decks which areSufficiently Interconnected to Act as a Unit

Only one formula, other than those presented in the previousAASHTO Specifications, was obtained for load distribution in multi-beam decks. This formula, developed by Arya at the University ofIllinois, is applicable to both box and open section multi-beam bridgesand predicts interior beam moment responses due to single-lane andmulti-lane loading. However, a number of simplified formulasdeveloped in the study are valid only for multi-box beam decks and donot apply to open sections. Therefore, the response of multi-beamdecks made of open members, such as channels, may or may not beaccurately predicted by the formulas developed in that study.


The formula developed by Arya for interior girder loaddistribution in multi-beam decks includes terms for the maximumnumber of wheels that can be placed on a transverse section of thebridge, number of beams, beam width and span length. A variation ofthe formula was also proposed for multi-beam decks made ofchannels, which includes consideration of the overall depth of thechannel section and its average thickness, defined as its area dividedby its length along the centerline of the thickness.

The following formula was developed in Project 12-26 topredict load distribution factors for interior beam moment due to multi-lane loading. The multiple presence reduction factor is alreadyaccounted for in the formula.

(6.4.2.2.3d-1)g ' k b7600

0.6 bL

0.2 IJ

0.06

where:

k = 2.5(Nb)!0.2 $ 1.5

b = beam width, in mm


Nb = number of beams

I = moment of inertia of a beam (mm)4

J = torsional constant of a beam (mm)4

This formula is dependant on the inertia and torsional constant of abeam; an estimated value for these properties must, therefore, beused in preliminary design. The term I/J may be taken as unity for thiscase.


Lecture - 6-27


Arya also presented a load distribution formula for multi-beamdecks designed for one traffic lane. The formulation and parameterswere similar to those presented for multi-lane loading. A variation ofthat equation was also presented for calculation of the interior beammoment distribution factor for a single-lane, channel section, multi-beam deck. It should be noted that Arya's equations are notapplicable to cases of only one-lane loading with more than one trafficlane.

A formula for wheel load distribution for moment in the interiorgirders due to single-lane loading was also developed in NCHRP 12-26. This formula is as follows:

(6.4.2.2.3d-2)g ' k b2.8L

0.5 IJ

0.25

All parameters are defined in Equation 6.4.2.2.3d-1. Equation6.4.2.2.3c-2 is also dependent on inertia and torsional constants, anda value of 1.0 may be used as an approximation for the term I/J duringpreliminary design.


The moment in the edge girder due to multi-lane loading inmulti-beam decks comprised of box units is obtained by using acorrection factor applied to the interior girder distribution factors formulti-lane loading. This correction factor may be found from thefollowing formula:

(6.4.2.2.3d-3)g ' 1.04 %de

7600

where:

de = distance from edge of the lane to the center of the exterior webof the exterior girder, in mm

For exterior beams of sufficiently interconnected multi-beambridge decks comprised of T-shaped units subjected to multi-laneloading, Equation 6.4.2.2.3b-2 applies.

For single-lane loading and for multi-beam decks comprised ofeither box units or units other than box units, the lever rule is used.


Lecture - 6-28

Shear Distribution

Distribution factors for shear in interior girders of multi-beamdecks in "Bridge Decks Comprised of Box Units" due to multi-laneloading may be calculated from the following formula:

(6.4.2.2.3d-4)g ' b4000

0.4 bL

0.1 IJ

0.05

where the parameters are as defined in Equation 6.4.2.2.3d-1.

Distribution factors for shear in the interior girders of multi-beam decks in "Bridge Decks Comprised of Box Units" due to single-lane loading are obtained from the following formula:

(6.4.2.2.3d-5)g ' 0.70 bL

0.15 IJ

0.05

where the parameters are again as defined in Equation 6.4.2.2.3d-1.

Note that Equations 6.4.2.2.3d-4 and 6.4.2.2.3d-5 aredependent on inertia and torsional constants, and a value of 1.0 maybe used as an approximation for the term I/J during preliminarydesign.

The shear in the edge girder of multi-beam deck in "BridgeDecks Comprised of Box Units" due to multi-lane loading can be foundusing a correction factor applied to interior girder distribution factors.This correction factor is obtained from the formula:

(6.4.2.2.3d-6)e ' 1.02 %de

15 000

where:

de = distance from edge of lane to the center of exterior webof the exterior girder, in mm

For shear in exterior beams of sufficiently interconnected multi-beam bridge decks comprised of T-shaped units, Equations6.4.2.2.3b-4 through 6.4.2.2.3b-5 and the lever rule should be used,where appropriate.


The moment in any beam in a skewed bridge may be obtainedby using a skew reduction factor given by Equation 6.4.2.2.3b-6.


Lecture - 6-29

The shear in the interior beams of a skewed multi-beam bridgecomprised of box beams is usually of the same order as that of theshear in the obtuse corner and must be obtained by applying acorrection factor to the response of the edge girder in a straight bridge.This correction factor may be calculated from the formula:

(6.4.2.2.3d-7)r ' 1 % c1 (tanθ)0.5

where:

(6.4.2.2.3d-8)c1 'L

90d

6.4.2.2.3e Simplified Formulas for Multi-Beam Decks which are notSufficiently Interconnected to Act as a Unit

The LRFD Specification contains the same provisions for loaddistribution in this type of bridge superstructure as appeared in recenteditions of the Standard Specification, and as repeated below forcompleteness.

The key difference between bridges treated herein, ascompared to Section 6.4.2.2.3d, is the degree of transverseinterconnection of units. If box, T, channel or other precast units areinterconnected through a structural slab, or sufficiently transverselypost-tensioned to produce a similar level of continuity, then thediscussion of Section 6.4.2.2.3d applies. If the interconnectionbetween the units is expected to transmit shear, but relatively littlemoment over the bridge service life, then the provisions herein apply.

The Specification provides for the computation of a bendingmoment distribution factor, regardless of the number of lanes, givenby:

g ' S300D

for which:

D = 300 [11.5 - NL + 1.4 NL (1 - 0.2C)2] when C # 5

D = 300 [11.5 - NL] when C > 5

C = K (W/L)

K = (1 % µ)IJ


Lecture - 6-30

where:

µ = Poisson ratio

I = moment of inertia (mm)4

J = St. Venant's constant (mm)4

L = span length (mm)

NL = number of lanes

S = spacing of units (mm)

W = edge-to-edge width of bridge (mm)

6.4.2.2.3f Simplified Formulas for Spread Box Beam Bridges

Only one formula, other than those recommended byAASHTO, was obtained from previous research for determining loaddistribution factors in spread box beam bridges. This formula wasdeveloped at Lehigh University for predicting the response of interiorbeams due to multi-lane loading and was later adopted by AASHTO.A correction factor for skewed bridges was also presented. Inaddition, a number of simple formulas were developed as part ofNCHRP Project 12-26.

Moment Distribution to Interior Beams, Multi-Lane Loading

A formula developed in Project 12-26 for moment in interiorspread box beams due to multi-lane loading is as follows:

(6.4.2.2.3f-1)g ' S1900

0.6 SdL 2

0.125

where

S = girder spacing (mm)


d = beam depth (mm)

Moment Distribution to Interior Beams, Single-Lane Loading

A similar formula was developed for distribution to interiorbeams due to single-lane loading:


Lecture - 6-31

(6.4.2.2.3f-2)g ' S910

0.35 SdL 2

0.25

where the parameters are as defined in Equation 6.4.2.2.3f-1.


The moment in edge girders due to multi-lane loading may becalculated by applying a correction factor to the interior girderdistribution factor:

(6.4.2.2.3f-3)e ' 0.97 %de

8700

where:

de = distance from edge of lane to the center of exterior webof the exterior girder (mm)

The distribution factor for moment in the edge girder due tosingle-lane loading may be obtained by simple-beam distribution, i.e.,the lever rule, in the same manner as was described for beam-and-slab bridges.

Shear Distribution

The distribution factor for shear in the interior girders due tomulti-lane loading may be calculated from the following:

(6.4.2.2.3f-4)g ' S2250

0.8 dL

0.1

where the parameters are as defined previously.


(6.4.2.2.3f-5)g ' S3050

0.6 dL

0.1

where the parameters are again as defined previously.

The shear in the edge girder due to multi-lane loading can befound by applying a correction factor to the interior girder equation.This correction factor is:


Lecture - 6-32

(6.4.2.2.3f-6)e ' 0.8 %de

3050

where:


The wheel load distribution factor for shear in the edge girderdue to single-lane loading may be obtained by simple-beamdistribution in the same manner as was described for beam-and-slabbridges, i.e., the lever rule.


Research at Lehigh University also resulted in a formula forcorrection of wheel load distribution for moment in interior girders dueto multi-lane loading in skewed bridges. NCHRP 12-26 concludes thatEquation 6.4.2.2.3b-6 was also applicable to this case.

The shear in the interior beams of a skewed bridge is the sameas that of a straight bridge. However, the shear in the obtuse cornermust be obtained by applying a correction factor to the distributionfactor for the edge girder in a straight bridge, given by Equation6.4.2.2.3b-7, which C1 is taken as:

(6.4.2.2.3f-7)c1 '(Ld )0.5

6S

6.4.2.2.3g Response of Continuous Bridges

The response of continuous bridges was studied by modelinga number of two-span continuous bridges where each span is similarto the average bridge. The wheel load distribution factor for each casewas compared to that of a simple bridge and correction factors forcontinuity were obtained. In the case of beam-and-slab bridges, acomplete parameter study was performed, and it was found that thecorrection factor is generally independent of bridge geometry. Thesefactors are given in the table below.

When the NCHRP 12-26 factors were incorporated into theLRFD Specification, it was decided that 5% corrections wereunwarranted given that the distribution factors are an approximationof actual behavior and are, therefore, subject to some variability. Thecontinuity correction for negative moment, a 10% increase, was alsoneglected on the basis that experimentally observed "fanning" of thereaction tends to reduce the negative moment as compared to atypical beam calculation.


Lecture - 6-33

Table 6.4.2.2.3f-1 - Continuity CorrectionFactors

Beam-and-Slab Bridges

Positive momentNegative momentShear at simply-supported endShear at continuous bent

c = 1.05c = 1.10c = 1.00c = 1.05

Box Girder Bridges


c = 1.00c = 1.10c = 1.00c = 1.00

Slab Bridges

Positive momentNegative moment

c = 1.00c = 1.10

Multi-Beam Bridges


c = 1.00c = 1.10c = 1.00c = 1.05

Spread box beam bridges


c = 1.00c = 1.10c = 1.00c = 1.05

6.4.2.3 TRUSS AND ARCH BRIDGES

6.4.2.3.1 General

The approximate method for load distribution to lines of trussesand arches is a so-called "lever rule", which is simply a matter ofsumming moments about one line of trusses or arches to find thereaction on the other line. This approach is illustrated by calculationsin Lecture 7.


Lecture - 6-34

6.5 REFINED METHODS

6.5.1 Deck Slabs

Where refined analysis of deck slabs is desirable, finiteelement analysis is recommended. Elements should be chosen tosimulate both bending and in-plane or membrane effects. If theanalysis utilizes only plate/membrane or shell elements and has onlyone or two elements through the thickness of the deck, then therefined analysis will report an essentially bending-type response in thedeck. There has been much experimental and analytic work thatsuggests that bending is not the primary source of strength in bridgedecks, but that the development of membrane action, analogous to ashallow arch or dome load path within the deck is the primary sourceof strength. This type of action will only be determined through a veryrigorous modeling of the deck.

6.5.2 Beam Slab Bridges

Relatively rigorous models of beam slab bridges can bedeveloped using general purpose commercial finite element programs,finite strip programs or special purpose greater finite element-basedcomputer programs which have been specifically developed to simplifythe analysis of bridge-type structures. These more custom-orientedprograms often contain mesh generating capabilities, automatic loadplacement capabilities and code checking.

Detailed bridge deck analysis using a finite element computerprogram may be used to produce accurate results. However, extremecare must be taken in preparation of the model, or inaccurate resultswill be obtained. Important points to consider are selection of aprogram capable of accurately modeling responses being investi-gated, calculation of element properties, mesh density and supportconditions. Every model should be thoroughly checked to ensure thatnodes and elements are generated correctly.

Another important point is the loading. Truck loads should beplaced at positions that produce the maximum response in thecomponents being investigated. In many cases, the truck location isnot known before preliminary analysis is performed and, therefore,many loadings should be investigated. This problem is morepronounced in skewed bridges.

Many computer programs have algorithms that allow loads tobe placed at any point on the elements. If this feature is not present,equivalent nodal loads must be calculated. Distribution of wheel loadsto various nodes must also be performed with care, and the meshshould be fine enough to minimize errors that can arise because ofload approximations.

Many computer programs, especially the general purpose finiteelement analysis programs, report stresses and strains, not shear andmoment values. Calculation of shear and moment values from the


Lecture - 6-35

stresses must be carefully performed, usually requiring an integrationover the beam cross-section. Some programs report stresses at nodepoints rather than Gaussian integration points. Integration of stressesat node points is normally less accurate and may lead to inaccurateresults.

Many graphical and computer-based methods are available forcalculating wheel load distribution. One popular method consists ofdesign charts based on the orthotropic plate analogy, similar to thatpresented in the Ontario Highway Bridge Design Code. As computersbecome readily available to designers, simple computer-basedmethods, such as SALOD, become more attractive than nomographsand design charts. Also, grillage analysis presents a good alternativeto other simplified bridge deck analysis methods and will generallyproduce more accurate results.

The grillage analogy may be used to model any one of the fivebridge types studied in this research. Each bridge type requiresspecial modeling techniques. A major advantage of plane gridanalysis is that shear and moment values for girders are directlyobtained and integration of stresses is not needed. Loads normallyneed to be applied at nodal points, and it is recommended that simple-beam distribution be used to distribute wheel loads to individualnodes. If the loads are placed in their correct locations, the results willbe close to those of detailed finite element analysis.

As indicated previously, the designer has to be responsible forconstructing a suitable model and determining that the results areaccurate. It is possible to make seemingly small errors in computermodels which can have dramatic effects on the results which areobtained.

6.5.3 Example of Modeling Errors

The modeling of diaphragms and boundary conditions atsupports and bearings is vital to obtaining the proper results whenusing these sophisticated programs. The burden of correctly handlingthese factors rests with the designer. Consider the following examplewhich shows how a very small modeling error produced veryerroneous results.

The framing plan shown on Figure 6.5.3-1 represents an actualbridge that was designed using a grid-type approach. The designerhad a good model for this structure, except that the rotational degreeof freedom corresponding to the global "x" axis was fixed at all of thebearings. This did not allow the diaphragms at the piers andabutments to respond correctly to the imposed loadings anddeformations, and also had the effect of producing artificially stiff endson the girders by virtue of vector resolution between global and localsystems. The effect of this condition on the reactions obtained at theabutments and piers was dramatic. Modest uplift was reported at theacute corner along the near abutment shown in Figure 6.5.3-1, and avery substantial uplift was reported at the acute angle at the far


Lecture - 6-36

abutment. This is shown in the table in Figure 6.5.3-1, as is a momentdiagram for non-composite dead loads which reflects the incorrectreactions. Also shown in the table of reactions on Figure 6.5.3-1 arethe correct reactions determined when the structure was modeledusing the generic STRESS Computer Program with proper boundaryconditions at the supports. In this case, a positive reaction is found atall bearings, and a significantly different moment diagram for non-composite dead load also resulted. The correct reactions and momentdiagram are also shown on Figure 6.5.3-1.

The modeling of the degrees of freedom at the lines of supporton this structure was also investigated utilizing a relatively completethree-dimensional finite element analysis and the SAPIV ComputerProgram. The model used is illustrated in Figure 6.5.3-2, which showshow the deck slab, girders and cross-frames were modeled in theirproper relative positions in the cross-section which extended along thebridge from end-to-end. Also shown on this figure is a comparison ofthe reactions obtained from STRESS and from SAPIV by applying allof the non-composite loads in a single loading. The comparisonbetween these reactions is excellent.

In order to verify that the order of pouring the deck slab unitswould not contribute to an uplift situation, the pouring sequence wasreplicated in a three-dimensional SAPIV analysis. The results of theanalysis of the three stages of the pouring sequence are also shownin Figure 6.5.3-2, as well as the total accumulated load at the end ofthe pour. Comparison of the sequential loading with the application ofa single loading of non-composite dead load also showed relativelygood agreement in this case.

The important point demonstrated in the example of Figures6.5.3-1 and 6.5.3-2 is that seemingly small errors in modeling of thestructure can result in very substantial changes in the reactions,shears and moments. The designer must be aware of this potentialwhen utilizing the more refined analysis techniques.

Incidentally, there are cases in which an uplift reaction due toskew and/or curvature is possible. The simple span bridge shown inFigure 6.5.3-3 and reported on in the November 1, 1984, issue ofEngineering News-Record, was analyzed at the request of the owner.In this case, the uplift reactions computed by the designer wereverified.

Sometimes modeling problems occur because User's Manualsare not clear, or a "bug" exists, of which the program's author/vendoris not aware. Such a case is illustrated for the simply-supported,partially-curved and skewed bridge in Figure 6.5.3-4. Initially, thisbridge was modeled with extra joints at locations other thandiaphragms in an effort to improve live load determination. As aresult, the number of points along each girder were not equal, butthere was no indication in the program's descriptive literature that thistype of arrangement of program input could potentially cause aproblem. The resulting moment envelopes for the middle and two


Lecture - 6-37

exterior girders are shown in Figure 6.5.3-4, and are obviouslyunusual in shape and also in their order of maximum moment, i.e., #4,#5 and #3.

It was found that the live load processor was not respondingproperly to the unequal number of nodes per girder, that nodes shouldbe essentially "radial", and that it was not certain that nodes to whichdiaphragms were not connected were legitimate. The revised model,shown in Figure 6.5.3-4, produced clearly better results, as shown inthe indicated moment envelopes.


Lecture - 6-38

Figure 6.5.3-1 - Framing plan, comparative live load reactions andmoment envelopes showing effect of proper and improper rotationalboundary condition, as reflected in grid analysis.


Lecture - 6-39

Figure 6.5.3-2 - Finite element idealization and reactions obtained forstructure shown in Figure 6.


Lecture - 6-40

Figure 6.5.3-3 - Framing Plan of Curved Span with Skewed Piers

Figure 6.5.3-4 - Comparative moment envelopes for the middle andtwo outside girders of curved skewed system showing the results ofapparent "bug" in algorithm for applying live load.

6.5.4 Other Types of Bridges

The Specification contains additional requirements for therigorous analysis of cellular-type structures, truss bridges, archbridges, cable-stayed bridges and suspension bridges. Generallyspeaking, refined analysis will involve a computer model whichaccurately affects the geometry, relative component stiffnesses,


Lecture - 6-41

boundary conditions and load supply to the structure. Suspensionbridges will almost always be analyzed using a large deflection theory.The deflection theory may also be applied to arches and cable-stayedbridges. In the case of the cable-stayed bridge of moderate span, itmay be sufficiently accurate to evaluate the second order effects onthe deck system of the tower by supplementary calculations inproviding a correction factor, developed on a bridge-specific basis.The change in stiffness of the cables caused by change in sag as thecable load changes can be accounted for using the so-called "Ernst"equations, given in the Specification, for modified modulus ofelasticity.


Lecture - 6-42

REFERENCES

Jones, 1976, "A Simple Algorithm for Computing Load Distribution inMulti-Beam Bridge Decks", Proceedings, 8th ARRB Conference, 1976


APPENDIX A

The Load Distribution Problem and its Solution in NCHRP 12-26


Lecture - 6-A1

Older editions of the AASHTO Specifications allow for simplified analysis of bridgesuperstructures using the concept of a load distribution factor for bending moment in interior girdersof most types of bridges, i.e., beam-and-slab, box girder, slab, multi-box beam and spread-boxbeam. This distribution factor is given by:

(A-1)g ' S300D

where:

g = a factor used to multiply the total longitudinal response of the bridge due to a singlelongitudinal line of wheel loads in order to determine the maximum response of asingle girder

S = the center-to-center girder spacing (mm)

D = a constant that varies with bridge type and geometry

A major shortcoming of the previous specifications is that the piecemeal changes that havetaken place over the last 55 years have led to inconsistencies in the load distribution criteriaincluding: inconsistent consideration of a reduction in load intensity for multiple lane loading;inconsistent changes in distribution factors to reflect the changes in design lane width; and,inconsistent approaches for verification of live load distribution factors for various bridge types.

The past AASHTO simplified procedures were developed for non-skewed, simply-supportedbridges. Although it was stated that these procedures apply to the design of normal (i.e., supportsoriented perpendicular to the longitudinal girders) highway bridges, there are no other guidelinesfor determining when the procedures are applicable. Because modern highway and bridge designpractice requires a large number of bridges to be constructed with skewed supports, on curvedalignments, or continuous over interior supports, it is increasingly important that the limitations ofload distribution criteria be fully understood by designers.

Advanced computer technology has become available in recent years which allows detailedfinite element analysis of bridge decks. However, many computer programs exist which employdifferent formulations and techniques. It is important that the computer methodology andformulation that produces the most accurate results be used to predict the behavior of bridge decks.In order to identify the most accurate computer programs, data from full-scale and prototype bridgeload tests were compiled. The bridge tests were then modeled by different computer programs andthe experimental and computer results were compared. The programs that produced the mostaccurate results were then considered as the basis for evaluation of the other method levels, i.e.,Levels 2 and 1 methods.

An important part of the development or evaluation of simplified methods is range ofapplicability. In order to ensure that common values of various bridge parameters were considered,a database of actual bridges was compiled. Bridges from various states were randomly selectedin order to achieve national representation. This resulted in a database of 365 beam-and-slabbridges, 112 prestressed concrete and 121 reinforced concrete box girder bridges, 130 slabbridges, 67 multi-box beam bridges and 55 spread-box beam bridges. This bridge database wasstudied to identify the common values of various parameters, such as beam spacing, span length,slab thickness, and so on. The range of variation of each parameter was also identified. Ahypothetical bridge that has all the average properties obtained from the database, referred to as


Lecture - 6-A2

the "average bridge" was created for each of the beam-and-slab, box girder, slab, multi-box beamand spread-box beam bridge types. For the study of moment responses in box girder bridges,separate reinforced concrete and prestressed concrete box girder average bridges were alsoprepared.

In evaluating simplified formulas, it is important to understand the effect of various bridgeparameters on load distribution. Bridge parameters were varied one at a time in the average bridgefor the bridge type under consideration. Load distribution factors for both shear and moment wereobtained for all such bridges. Variation of load distribution factors with each parameter shows theimportance of each parameter. Simplified formulas can then be developed to capture the variationof load distribution factors with each of the important parameters. A brief description of the methodused to develop such formulas is as follows.

In order to derive a formula in a systematic manner, certain assumptions must be made.First, it is assumed that the effect of each parameter can be modeled by an exponential functionof the form axb, where x is the value of the given parameter, and a and b are coefficients to bedetermined based on the variation of x. Second, it is assumed that the effects of differentparameters are independent of each other, which allows each parameter to be considered sep-arately. The final distribution factor will be modeled by an exponential formula of the form: g =(a)(Sb1)(Lb2)(tb3)(...) where g is the wheel load distribution factor; S, L, and t are parameters includedin the formula; a is the scale factor; and b1, b2, and b3 are determined from the variation of S, L,and t, respectively. Assuming that for two cases, all bridge parameters are the same, except forS, then:

g1 = (a)(S1b1)(Lb2)(tb3)(...) (A-2)

g2 = (a)(S2b1)(Lb2)(tb3)(...) (A-3)

therefore:

(A-4)g1

g2

'S1

S2

b1

or:

(A-5)b1 'ln

g1

g2

lnS1

S2

If n different values of S are examined and successive pairs are used to determine the valueof b1, n!1 different values for b1 can be obtained. If these b1 values are close to each other, anexponential curve may be used to accurately model the variation of the distribution factor with S.In that case, the average of n!1 values of b1 is used to achieve the best match. Once all the powerfactors, i.e., b1, b2, and so on, are determined, the value of "a" can be obtained from the averagebridge, i.e.,


Lecture - 6-A3

(A-6)a 'go

S b1o L b2

o t b3o (...)

This procedure was followed during the entire course of the NCHRP 12-26 study to developnew formulas as needed. In certain cases where an exponential function was not suitable to modelthe effect of a parameter, slight variation from this procedure was used to achieve the requiredaccuracy. However, this procedure worked quite well in most cases and the developed formulasdemonstrate high accuracy.

Because certain assumptions were made in the derivation of simplified formulas and somebridge parameters were ignored altogether, it is important to verify the accuracy of these formulaswhen applied to real bridges. The database of actual bridges was used for this purpose. Bridgesto which the formula can be applied were identified and analyzed by an accurate method. Thedistribution factors obtained from the accurate analysis were compared to the results of thesimplified methods. The ratio of the approximate to accurate distribution factors was calculated andexamined to assess the accuracy of the approximate method. Average, standard deviation, andminimum and maximum ratio values were obtained for each formula or simplified method. Themethod or formula that has the smallest standard deviation is considered to be the most accurate.However, it is important that the average be slightly greater than unity to assure slightlyconservative results. The minimum and maximum values show the extreme predictions that eachmethod or formula produced when a specific database was used. Although these values maychange slightly if a different set of bridges is used for evaluation, the minimum and maximum valuesallow identification of where shortcomings in the formula may exist that are not readily identified bythe average or standard deviation values.

It was previously mentioned that different subsets of the database of bridges were used toevaluate different formulas. When a subset included a large number of bridges (100 or more), aLevel 2 method was used as the basis of comparison. When it included a smaller number ofbridges (less than 100), a Level 3 method was used. As a result, LANELL (an influence surfacemethod) was used for verification of formulas for moment distribution in box girder bridges, and aMulti-dimensional Space Interpolation (MSI) method was used for verification of formulas forstraight beam-and-slab and slab bridges.

Findings

Level 3 Methods: Detailed Bridge Deck Analysis

Recent advances in computer technology and numerical analysis have led to thedevelopment of a number of computer programs for structural analysis. Programs that can beapplicable to bridge deck analysis can be divided into two categories. One includes generalpurpose structural analysis programs such as SAP, STRUDL and FINITE. The other category isspecialized programs for analysis of specific bridge types, such as GENDEK, CURVBRG andMUPDI.

In the search for the best available computer program for analysis of each bridge type, allsuitable computer programs (general and specific) that were available at the time of the 12-26research were evaluated. In order to achieve meaningful comparisons and assess the level ofaccuracy of the programs, a number of field and laboratory tests were modeled by each program.The results were then compared in three ways:

• by visual comparison of the results plotted on the same figure,


Lecture - 6-A4

• by comparison of the averages and standard deviations of the ratios of analytical toexperimental results, and

• by comparison of statistical differences of analytical and experimental results. Five bridgetypes were considered: beam-and-slab, box girder, slab, multi-box beam, and spread-boxbeam.

For analysis of beam-and-slab bridges, the following computer programs and models wereevaluated: GENDEK-PLATE, GENDEK-3, GENDEK-5, CURVBRG, SAP and MUPDI. It was foundthat, in general, GENDEK-5 analysis using plate elements for the deck slab and eccentric beamelements for the girders is very accurate. This program is also general enough to cover all typicalcases, i.e., straight, skew, moment and shear. However, for analysis of curved open girder steelbridges, CURVBRG was the most accurate program. MUPDI was also found to be a very accurateand fast program; however, skewed bridges cannot be analyzed with this program and shear valuesnear the point of application of load, or near supports, lack accuracy. GENDEK-5 was, therefore,selected to evaluate Level 2 and Level 1 methods.

For analysis of box girder bridges, computer programs MUPDI, CELL-4 and FINITE wereevaluated. MUPDI was the fastest and most practical program for analysis of straight bridges formoment, but FINITE was found to be the most practical program for skewed bridges and forobtaining accurate shear results. Therefore, MUPDI was selected for the evaluation of LANELL (aLevel 2 method for moment in straight bridges which was, in turn, used for evaluation of Level 1methods) and FINITE was selected for other cases.

For the analysis of slab bridges, computer programs MUPDI, FINITE, SAP and GENDEKwere evaluated. Shear results cannot be obtained accurately in slab bridges and, therefore, werenot considered. The GENDEK-5 program, without beam elements, proved to be very accurate.However, MUPDI was found to be the most accurate and practical method for non-skewedprismatic bridges and was selected to evaluate Level 2 and Level 1 methods.

For the analysis of multi-beam bridges, the following computer programs were evaluated:SAP, FINITE and a specialized program developed by Professor Powell at the University ofCalifornia, Berkeley, for analysis of multi-beam bridges (referred to as the POWELL programherein). Various modeling techniques were studied using different grillage models and differentplate elements. The program that is capable of producing the most accurate results in all cases,i.e., straight and skewed for shear and moment, was the FINITE program. This program was laterused in evaluation of more simplified methods. POWELL is also very accurate in reportingmoments in straight bridges, but it uses a finite strip formulation, similar to MUPDI, and, therefore,is incapable of modeling skewed supports, and shear results near supports and load locations can-not be accurately obtained. This program was used to evaluate simplified methods for straightbridges.

For analysis of spread-box beam bridges, computer programs SAP, MUPDI, FINITE andNIKE-3D were evaluated. FINITE produced the most accurate results, especially when shear wasconsidered. MUPDI was selected to evaluate simplified methods for calculation of moments instraight bridges, and FINITE was selected for all other cases.

Level 2 Methods: Graphical and Simple Computer-Based Analysis

Nomographs and influence surface methods have traditionally been used when computermethods have been unavailable. The Ontario Highway Bridge Design Code uses one such methodbased on orthotropic plate theory. Other graphical methods have also been developed andreported. A good example of the influence surface method is the computer program SALOD


Lecture - 6-A5

developed by the University of Florida for the Florida Department of Transportation. This programuses influence surfaces, obtained by detailed finite element analysis, which are stored in adatabase accessed by SALOD. One advantage of influence surface methods is that the responseof the bridge deck to different truck types can be readily computed.

A grillage analysis using plane grid models can also be used with minimal computerresources to calculate the response of bridge decks in most bridge types. However, the propertiesfor grid members must be calculated with care to assure accuracy. Level 2 methods used toanalyze the five bridge types (beam-and-slab, box girder, slab, multi-beam and spread-box beambridges) are discussed below.

The following methods were evaluated for analysis of beam-and-slab bridges: plane gridanalysis, the nomograph-based method included in the Ontario Highway Bridge Design Code(OHBDC), SALOD and Multi-dimensional Space Interpolation (MSI). All of these methods areapplicable for single- and multi-lane loading for moment. The OHBDC curves were developed fora truck other than HS-20, and using the HS-20 truck in the evaluation process may have introducedsome discrepancies. The method presented in OHBDC was also found to be time consuming, andinaccurate interpolation between curves was probably a common source of error. SALOD can beused with any truck and, therefore, the "HS" truck was used in its evaluation. The MSI method wasdeveloped based on HS-20 truck loading for single- and multiple-lane loading. MSI was found tobe the fastest and most accurate method and was, therefore, selected for the evaluation of Level1 methods. This method produces results that are generally within 5% of the finite element(GENDEK) results.

In the analysis of box girder bridges, OHBDC curves and the LANELL program wereevaluated. The comments made about OHBDC for beam-and-slab bridges are valid for box girderbridges as well. As LANELL produced results that were very close to those produced by MUPDI,it was selected for evaluation of Level 1 methods for moment.

OHBDC, SALOD and MSI were evaluated for the analysis of slab bridges. MSI was foundto be the most accurate method and, thus, was used in the evaluation of Level 1 methods. SALODalso produced results that were in very good agreement with the finite element (MUPDI) analysis.Results of OHBDC were based on a different truck and, therefore, do not present an accurateevaluation.

In the analysis of multi-beam bridges, a method presented in Jones, 1976, was evaluated.The method is capable of calculating distribution factors due to a single concentrated load and wasmodified for this study to allow wheel line loadings. The results were found to be in very goodagreement with POWELL. However, because this method was only applicable for momentdistribution in straight single-span bridges, it was not used for verification of Level 1 methods.

In the analysis of spread-box beam bridges, only plane grid analysis was considered as aLevel 2 method.

In general, Level 2 graphical and influence surface methods generated accurate anddependable results. While these methods are sometimes difficult to apply, a major advantage ofsome of them is that different trucks, lane widths, and multiple presence live load reduction factorsmay be considered. Therefore, if a Level 2 procedure does not provide needed flexibility, its useis not warranted because the accuracy of it is on the same order as a simplified formula. MSI is anexample of such a method for calculation of load distribution factors in beam-and-slab bridges.

A plane grid analysis would require computer resources similar to those needed for someof the methods mentioned above. In addition, a general purpose plane grid analysis program is


Lecture - 6-A6

available to most bridge designers. Therefore, this method of analysis is considered a Level 2method. However, the user has the burden of producing a grid model that will produce sufficientlyaccurate results. As part of NCHRP Project 12-26, various modeling techniques were evaluated,and it was found that a proper plane grid model may be used to accurately produce load distributionfactors for each of the bridge types studied.

Level 1 Methods: Simplified Formulas

The current AASHTO Specifications recommend use of simplified formulas for determiningload distribution factors. Many of these formulas have not been updated in years and do notprovide optimum accuracy. A number of other formulas have been developed by researchers inrecent years. Most of these formulas are for moment distribution for beam-and-slab bridgessubjected to multi-lane truck loading. While some have considered correction factors for edgegirders and skewed supports, very little has been reported on shear distribution factors ordistribution factors for bridges other than beam-and-slab.

The sensitivity of load distribution factors to various bridge parameters was also determinedas part of the study. In general, beam spacing is the most significant parameter. However, spanlength, longitudinal stiffness and transverse stiffness also affect the load distribution factors.Figures 6.4.2.2.3-2 through 6.4.2.2.3-6 show the variation of load distribution factors with variousbridge parameters for each bridge type. Ignoring the effect of bridge parameters, other than beamspacing, can result in highly inaccurate (either conservative or unconservative) solutions.

A major objective of the research in Project 12-26 was to evaluate older AASHTOSpecifications and other researchers' published work to assess their accuracy and developalternate formulas whenever a more accurate method could be obtained. The formulas that wereevaluated and developed are briefly described below, according to bridge type; i.e., beam-and-slab,box girder, slab, multi-beam and spread-box beam.

Figure A-1 - Effect of Parameter Variation on Beam-and-Slab Bridges


Lecture - 6-A7

Figure A-2 - Effect of Parameter Variation on Box Girder Bridges

Figure A-3 - Effect of Parameter Variation on Slab Bridges

Figure A-4 - Effect of Parameter Variation on Multi-Box Beam Bridge


Lecture - 6-A8

Figure A-5 - Effect of Parameter Variation on Spread-Box Beam Bridges


Lecture - 6-1

LECTURE 6 - ANALYSIS I


The objectives of this lesson are to acquaint the student with:

• the various analysis techniques required and/or recommendedfor determining the force effects and components of bridges,and

• the use of approximate and refined methods for thedetermination for force effects in conventional girder-typestructures.

The background on the development of new, improved,distribution factors which were developed under NCHRP Project 12-26has been included for reference in an Appendix.

The use of grid and finite element types of analysis for multi-beam bridges is also recommended in the LRFD Specification. Thesemethods require considerable care in structural modeling, and severalexamples of the large effects of seemingly small errors in structuralmodels will be presented.

6.2 ACCEPTABLE METHODS OF STRUCTURAL ANALYSIS

Article S4.4 contains a list of methods of analysis that areconsidered suitable for analysis of bridges. These include theclassical force and displacement methods, such as virtual work,moment distribution, slope deflection, the so-called general method,the more modern finite element, finite strip and plate analogy-typemethods, analysis based on series expansions and the yield-linemethod for the non-linear analysis of plates and railings. Some ofthese methods of analysis are suitable for hand calculations, but forany problem of large size, some sort of computer solution will almostalways be required for practical design purposes. This is becausealmost all of these methods, with the possible exception of the seriesmethods and the yield-line methods, will eventually require thesolution of large sets of simultaneous equations. The series method,while elegant from a mathematical point of view, will typically requirea computer program to expand the series sufficiently to yield goodresults in a practical time frame. Yield-line methods, which could beconsidered the extension of plastic design to two-dimensionalsurfaces, are typically a hand calculation procedure.

The use of computer programs in bridge design brings up aphilosophical problem as to the responsibility for error. Almost allvendors of commercial computer programs disavow any responsibilityfor error. A release from liability is usually implicit in their use and mayeven be an explicit part of obtaining a license. This means that anorganization using a computer program must be relatively certain of


Lecture - 6-2

the results that it obtains. It is not necessary for every engineer in alarge design section to have personally confirmed every computerprogram, but it is necessary that some verification testing be done orthat the results of previous verification testing be obtained in order toproduce the required level of confidence. Computer programs can beverified against universally accepted closed- form solutions, othercomputer programs which have been previously verified, or the resultsof testing.

Many computer programs for design use also contain codechecking capabilities. Others have portions of the applicable designspecification embedded in the coding of the program. In order toidentify the specification edition which may have been tied to a givenrelease of a program and also to provide a means for determiningwhich structures may have been designed with a version of a programlater found to contain errors, the specification requires that a name,version and a release date of software be identified in the contractdrawing.

6.3 PRINCIPLES OF MATHEMATICAL MODELING

6.3.1 Structural Material Behavior

The LRFD Specification recognizes both elastic and inelasticbehavior of materials for analysis purposes. Inelastic materialbehavior is implicit in many of the equations and procedures specifiedfor the calculation of cross-sectional resistance, such as calculatingthe nominal resistance of a concrete beam or column, the nominalplastic moment resistance of a compact and adequately braced steelcross-section, or the bearing capacity of a spread footing. Often, theforce effects to which this resistance will be compared will becalculated on a basis of a linear structural analysis with elasticmaterial properties having been assumed. This dichotomy has existedin the bridge specification since the introduction of load factor designin the early 1970's. It continues through the LRFD Specification.

On the other hand, there are certain assumed failure modes atextreme events and the use of mechanism and unified autostressdesign procedures for steel girders, where permitted, which requireanalysis based on non-linear behavior. Many times, this analysis willtake a form analogous to plastic design of steel frames. For example,seismic design may be based on the formation of plastic hinges at thetop and bottom of the columns of a bent. Ship collision forces may beabsorbed in a comparable inelastic manner. Furthermore, it isanticipated that future seismic design provisions will be based onextensive research currently underway to develop a step-by-step non-linear force displacement relationship for the lateral displacement ofpiers.

Where inelastic analysis is used, the Designer must be certainthat a ductile failure mode is obtained through proper detailing. Rules


Lecture - 6-3

for achieving this are presented in the sections for steel and concretedesign.

6.3.2 Geometry

6.3.2.1 GENERAL

Most analyses done for the purpose of designing bridges arebased on the assumption that the displacements caused by the loadsare relatively small and, therefore, it is suitably accurate to base thecalculations on the undeformed shape. This is typically referred to asthe small deflection theory, and it is routinely used in the design ofbeam-type structures and bridges which resist loads through a couplewhose tensile and compressive forces remain essentially in fixedpositions relative to each other while the bridge deflects. This will bethe case for a truss and for tied-arches.

For other types of structures and components and for certaintypes of analysis, the effect of the deflections must be considered inthe development of the force equilibrium equations, i.e., the equationsof equilibrium are written for the displaced shape. Almost allengineers are aware that the study of structural stability requiresconsideration of the displaced shape, in fact, if the displaced shape isnot part of the original formulation of the problem, one would never beable to determine that a column, shell or plate can buckle. Considerfor a moment the simple pin-ended column. Unless the deflectedshape of the column is taken into account, the moment caused by theaxial load acting on the displaced shape would not be accounted for.It is this moment which causes the column to move laterally, i.e., tobuckle.

Almost a century ago, it was found that the only reasonablyaccurate way to calculate force effects in suspension bridges of anysize was to include the deflection of the cable in the formulation of theproblem and, therefore, the displacement of a stiffening truss orstiffening girder. As conventional, i.e., not tied, arches became longerand more slender, an effect directly analogous to that observed insuspension bridges can become significant enough that it must beaccounted for in the design of the arch rib. In fact, because the archrib is in compression and can buckle, the effect of large deflectionscan be especially important.

Finally, the compression members of frames in bents can alsobe susceptible to this phenomenon.

Where non-linear effects arriving either out of material non-linearity or large deflections become significant, then super position offorces does not apply. This means that each load case underinvestigation must be studied separately under the full effect of all ofthe factored loads that make up the load combination under study.This is a very significant effect on most practical design calculations.Commonly, a Bridge Engineer calculates the force effect from avariety of individual loads and then combines, or superimposes, the


Lecture - 6-4

force effects calculated for each individual load to make up whatevergroup combination of loadings are needed. For non-linear analysis,each combination must be investigated, i.e., analyzed, individually.

6.3.2.2 APPROXIMATE METHODS

To simplify analysis and to partially bypass the need to analyzeeach load combination separately, as identified above, certainapproximate methods have been developed to allow the designer toadd a correction to a set of force effects calculated in a linear manner.These are sometimes called single-step adjustment methods, themost commonly used of which is moment magnification for beamcolumns, which has been part of the AASHTO Specifications since theearly 1970's.

For beam columns, the moment magnification process is givenby the equations below.

Mc = δb M2b + δs M2s (6.3.2.2-1)

fc = δb f2b + δs f2s (6.3.2.2-2)

for which:

(6.3.2.2-3)δb '

Cm

1 &Pu

φPe

$ 1.0

(6.3.2.2-4)δs '

1

1 &ΣPu

φΣPe

where:

Pu = factored axial load (N)

Pe = Euler buckling load (N)

φ = resistance factor for axial compression as specified inSpecification Sections 5, 6 and 7, as applicable

M2b = moment on compression member due to factoredgravity loads that result in no appreciable sideswaycalculated by conventional first order elastic frameanalysis, always positive (N@mm)

f2b = stress corresponding to M2b (MPa)

M2s = moment on compression member due to factoredlateral or gravity loads that result in sidesway, ∆,


Lecture - 6-5

greater than Ru/1500, calculated by conventional firstorder elastic frame analysis, always positive (N@mm)

f2s = stress corresponding to M2s (MPa)

It may appear that the moment magnification factor contains the Eulerbuckling load, Pe. However, Pe is only a convenient substitution for agroup of terms related to the displacement of the beam column.

A derivation of the moment magnification equation can befound in many textbooks on steel or concrete design.

For cases where the shape of the beam column is expected tobe radically different from that given by the simply-supported case, orthe loads significantly different from those indicated above, then it ispossible to make an adjustment to account for a different initial elasticshape through the factor cm.

The moment magnification procedure has also been extendedto arches, and this has been available in the AASHTO Specificationsfor many years and is reproduced as Article S4.5.3.2.2c with no furtherrefinement.

6.3.2.3 REFINED METHODS

The effect of large deflections can also be rigorouslyaccounted for through iterative solutions of equilibrium equations,taking into account updated positions of the structure, or by usinggeometric stiffness terms. In some cases, e.g., the suspension bridge,solutions are available to the differential equations of equilibrium whichcan be solved in a trial and error fashion, or through series expansion.

6.3.3 Modeling Boundary Conditions

Points of expansion or other forms of articulation in thestructure are commonly idealized as frictionless units. Where pastpractice indicates that this has been a reasonable conservativeapproach, continued use is warranted. There are other instanceswhere the potential for nonfunctional expansion devices and/or thepossibility that joints may close should also be investigated. Thismight be the case, for example, in a seismic analysis where analysismay be made, assuming that expansion joints are operable and open,and then another analysis might be made, assuming that they areclosed and nonfunctional in order to simulate, or bound, the effects ofjoints reaching the limits of travel during the earthquake. Thepossibility of reaching the limit of expansion travel should also beinvestigated when evaluating non-linear effects on substructureelements. It may be possible that the amount of momentmagnification may be reduced because expansion dams will close,jamming the structure against the abutments before the full movementimplicit in the moment magnification factor can be reached. This willreduce the moment magnification factor and, hence, the designmoment.


Lecture - 6-6

Similarly, the effect of boundary conditions at foundation unitsshould also be evaluated. Foundation units are seldom fully fixed orfully pinned, and an evaluation of the potential movement of afoundation unit may be necessary in order to properly assessresponse, as well as secondary moments caused by change ingeometry. Here again, bounding of the range of probable movementmay be the only practical way to attack such a problem.

6.4 STATIC ANALYSIS

6.4.1 The Influence of Plan Geometry

Article S4.6.1 deals with two simplifications which can be madebased on the plan geometry of the superstructure.

The first simplification involves the possibility of replacing thesuperstructure for analysis purposes with a single-line element calleda spine beam. This may be done when the transverse distortion of thesuperstructure is small in comparison with the longitudinaldeformation. Generally, if the superstructure is a torsionally stiffclosed section or sections whose length exceeds 2.5 times their width,it may be idealized as a line element whose dimensions may bedetermined as given in the Specification. This can be used tosignificantly simplify analysis models.

The second simplification deals with when it is possible toconsider curved superstructures as straight for the purpose ofanalysis. If the superstructure is a torsionally stiff closed section andthe central angle of a segment between piers is less than 12E, thenthe segment may be considered straight. If the superstructure ismade of torsionally weak open sections, then the effects of curvaturemay be neglected when the subtended angle is less than that given inTable 6.4.1-1.

Table 6.4.1-1 - Limiting Central Anglefor Neglecting Curvature in DeterminingPrimary Bending Moments

No. of BeamsAngle for One

SpanAngle for

Multiple Spans

2 2E 3E

3 or 4 3E 4E

5 or more 4E 5E


Lecture - 6-7

6.4.2 Approximate Methods for Load Distribution

6.4.2.1 DECK SLABS AND SLAB-TYPE BRIDGES

The Specification permits the approximate analysis of deckslabs by analyzing a strip of deck as a continuous beam. Provisionsare made for determining the width of that strip at the unsupported edge of the slab and at points interior from the edges.

If the spacing of supporting components in the secondarydirection exceeds 1.5 times the spacing in the primary direction, thenall of the wheel loads applied to the deck are considered to be appliedto the primary strip. The secondary strip is designed on a basis ofpercentage of reinforcement in the primary strip.

If the spacing of the supporting components on the secondarydirection is less than 1.5 times that in the primary direction, then acrossed sticks analogy is used. The width of the equivalent strips ineach direction is provided by Table S4.6.2.1.3-1 and the wheel loadis distributed between two idealized intersecting strips according to therelative stiffness of each strip.

Once the wheel loads have been assigned to the strips, foreither case identified above, then the force effects are calculatedbased on a continuous beam. For the purpose of analyzing thecontinuous beam, the span length of each span is taken as a center-to-center of supporting components. For the purpose of calculatingmoment and shear at a design section, some offset from thetheoretical center of support is permitted as given in the Specification.

Decks which form an integral part of a cellular cross-sectionare supported on webs which are monolithic with the deck. Therefore,when the deck rotates, the web of the box girder rotates giving rise tobending stresses throughout the cross-section. For the purpose ofanalyzing this effect, a cross-sectional frame action procedure isidentified in the Specification.

In the case of fully filled and partially filled grids, the results ofrecent research are incorporated in LRFD Article S4.6.2.1.8 to givenbending moments per unit length of grid.

6.4.2.2 BEAM SLAB BRIDGES

6.4.2.2.1 General

The Specification provides a series of empirical rules forassigning portions of a design lane to a supporting component. Theseare commonly called distribution factors. It is important to rememberthat the approximate distribution factors, specified in the LRFDSpecification, are on a lane, i.e., axle basis, not a wheel basis. Thedistribution factors are given for the various kinds of bridges shown inFigure 6.4.2.2.1-1.


Lecture - 6-8

SUPPORTINGCOMPONENTS TYPE OF DECK

TYPICALCROSS-SECTION

Steel Beam - RevisedFactors

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

Closed Steel or PrecastConcrete Boxes - RevisedFactors

Cast-in-place concrete slab

Open Steel or PrecastConcrete Boxes - RevisedFactors

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

Cast-in-Place Concrete Multi-cell Box - Revised Factors

Monolithic concrete

Cast-in-Place Concrete TeeBeam - Revised Factors

Monolithic concrete

Precast Solid, Voided orCellular Concrete Boxes withShear Keys - RevisedFactors


Precast Solid, Voided orCellular Concrete Box withShear Keys and with orwithout TransversePost-Tensioning - RevisedFactors (in some cases)

Integral concrete

Precast Concrete ChannelSections with Shear Keys


Precast Concrete Double TeeSection with Shear Keys andwith or without TransversePost-Tensioning

Integral concrete

Precast Concrete TeeSection with Shear Keys andwith or without TransversePost-Tensioning

Integral concrete

Concrete I or Bulb-TeeSections - Revised Factors

Cast-in-place concrete, precastconcrete

Wood Beams - RevisedFactors

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

Figure 6.4.2.2.1-1 - Common Deck Superstructures Covered in LRFD SpecificationArticles 4.6.2.2.2 and 4.6.2.2.3


Lecture - 6-9

Some of the distribution factors are new to the Specification as a resultof an extensive project on load distribution known as NCHRP Project12-26. Where the distribution factors for a given type of cross-sectionhave been developed under that project and are new to theSpecification, the words "revised factors" appear in the columnidentified as "supporting components". Where those words do notappear, the distribution factors have been retained from earliereditions of the AASHTO Standard Specifications.

Some simplifications have been made in utilizing thedistribution factors from NCHRP 12-26. In particular, correctionfactors for various aspects of structural action, typically involvingcontinuity, which were less than 5%, were omitted from the LRFDSpecifications. Similarly, an increase in moments over piers, thoughtto be on the order of 10%, was not included because stresses at ornear internal bearings have been shown to be reduced below thatcalculated by simple analysis techniques due to an action known as"fanning". The distribution factors, given in the LRFD Specification,are also different from those given in NCHRP 12-26, because themultiple presence factors, given in Lecture 3, are built into thedistribution factors, whereas, the multiple presence factors in earliereditions of the AASHTO Standard Specifications are built into theNCHRP 12-26 factors. Additionally, the factors appropriate for theLRFD Specification are based on a lane of live load, rather than a "lineof wheels". Finally, when the SI version of the LRFD Specification isused, conversion to that system of units has also been accounted for.

Various limits on span, spacing and other characteristics areprovided in the Specifications for each of the distribution factors.These parameters identify the range for which the factors weredeveloped. They were not evaluated for factors beyond the rangesindicated. Therefore, for structures which do not comply with theselimitations, a rigorous analysis by grid or finite elements should beused. Furthermore, the distribution factors usually apply for structureswhich are:

• essentially constant in deck width,

• have four or more beams, unless noted,

• have beams which are parallel and approximately of the samestiffness,

• have overhangs that do not exceed 0.9 m, unless specificallynoted,

• have in-plan curvatures less than those specified above, and

• have a cross-section consistent with one of the cross-sectionsidentified in Table 6.4.2.2.1-1.

Since the distribution factors, developed under NCHRP 12-26,are new to the Specification, it is appropriate to review the background


Lecture - 6-10

and development of the new distribution factors. The discussionbelow was taken from the NCHRP Research Results Digest No. 187,a summary of the Final Report of Project 12-26 as summarized byIan M. Friedland, NCHRP Project Coordinator.

Live load distribution on highway bridges is a key responsequantity in determining member size and, consequently, strength andserviceability. It is of critical importance both in the design of newbridges and in the evaluation of the load-carrying capacity of existingbridges.

Using live load distribution factors, engineers can predictbridge response by treating the longitudinal and transverse effects oflive loads as uncoupled phenomena. Empirical live load distributionfactors for stringers and longitudinal beams have appeared in theAASHTO Standard Specifications for Highway Bridges with only minorchanges since 1931. Findings of recent studies suggest a need toupdate these specifications in order to provide improved predictionsof live load distribution.

Live load distribution is a function of the magnitude andlocation of truck live loads and the response of the bridge to theseloads. The NCHRP 12-26 study focused on the second factormentioned above: the response of the bridge to a predefined set ofloads, namely, the HS family of trucks.

In Project 12-26, three levels of analysis were considered foreach bridge type. The most accurate level, Level 3, involves detailedmodeling of the bridge deck. Level 2 includes either graphicalmethods, nomographs and influence surfaces, or simplified computerprograms. Level 1 methods provide simple formulas to predict lateralload distribution, using a wheel load distribution factor applied to atruck wheel line to obtain the longitudinal response of a single girder.

The major part of the research in Project 12-26 was devotedto the Level 1 analysis methods because of its ease of application,established use, and the surprisingly good correlation with the higherlevels of analysis in their application to a majority of bridges. Theformulas presented in the current AASHTO specifications wereevaluated, and alternative formulas were developed that offerimproved accuracy, wider range of applicability, and in some cases,easier application than the current AASHTO formulas. Theseformulas were developed for interior and exterior girder moment andshear load distribution for single or multiple lane loadings. In addition,correction factors for continuous superstructures and skewed bridgeswere developed.

The formulas presented in previous AASHTO Specifications,although simpler, do not present the degree of accuracy demanded bytoday's Bridge Engineers. In some cases, these formulas can resultin highly unconservative results (more than 40%), in other cases theymay be highly conservative (more than 50%). In general, the formulasdeveloped in Project 12-26 are within 5% of the results of an accurate


Lecture - 6-11

analysis. Table 6.4.2.2.1-1 shows comparisons with momentdistribution factors obtained from AASHTO, Level 1, Level 2 and Level3 methods for simple span bridges.

Table 6.4.2.2.1-1 - Comparison of interior girder moment distributionfactors by varying levels of accuracy using the "average bridge" foreach bridge type

Bridge Type AASHTO

NCHRP 12-26(Level 1)

Grillage(Level 2)

Finite Element(Level 3)

Beam-and-slaba 1.413(S/1700) 1.458 1.368 1.378

Box girdera 1.144 1.143 0.970 1.005Slabb 1820 1710 1900 1890Multi-box beama 0.646 0.597 0.540 0.552Spread box beama 1.564 1.282 1.248 1.241

aNumber of wheel lines per girderbWheel line distribution width, in mm

In addition, the study resulted in recommendations for use ofcomputer programs to achieve more accurate results. Therecommendations focus on the use of plane grid analysis, as well asdetailed finite element analysis, where different truck types and theircombinations may be considered.

6.4.2.2.2 Influence of Truck Configuration

The formulas developed in Project 12-26 for the Level 1analysis were based on the standard AASHTO "HS" trucks. A limitedparametric study conducted as part of the research showed thatvariations in the truck axle configuration or truck weight do notsignificantly affect the wheel load distribution factors. The group ofaxle trains used for this study are shown in Figure 6.4.2.2.2-1. It isanticipated that smaller gage widths would result in larger distributionfactors, and larger gage widths would result in smaller distributionfactors. Table 6.4.2.2.1-1 gives the variation of wheel load distributionfactors with different axle configurations applied to a number of beam-and-slab bridges. The differences were below 1% in many cases and,in all cases, the formulas resulted in good predictions. Therefore, withsome caution, these formulas may be applied to other truck types.Obviously, Levels 2 and 3 analyses may also be applied for truckssignificantly different from the AASHTO family of trucks.


Lecture - 6-12

Figure 6.4.2.2.2-1 - Axle Configurations for Truck Types Consideredin Study

Table 6.4.2.2.1-1 - Effect of Load Configuration on Distribution Factor

DISTRIBUTIONFACTOR (g)

PERCENT DIFFERENCEWITH HS-20

HS-20 HTL-57 4A-66 B-141 NCHRP12-26 HTL-57 4A-66 B-141

NCHRP 12-

26Average* 1.293 1.261 1.285 1.268 1.304 -2.4 -0.6 -1.9 +0.9Max. S

5000 mm 2.220 2.162 2.205 2.178 2.308 -2.6 -0.7 -1.9 +4.0

Min. S1000 mm 0.713 0.717 0.713 0.715 0.755 +0.6 0.0 +0.3 +5.9

Max. L60 000 mm 0.982 0.958 0.983 0.952 1.033 -2.4 +0.1 -3.1 +5.2

Min. L6000 mm 1.630 1.625 1.624 1.623 1.807 -0.3 -0.3 -0.4 +10.9

*S = 2200 mmL = 20 000 mmts = 185 mmKg = 2.33 x 1011 mm4


Lecture - 6-13

6.4.2.2.3 Simplified Methods

6.4.2.2.3a Simplified Formulas for Beam-and-Slab Bridges

This type of bridge has been the subject of many previousstudies, and many simplified methods and formulas were developedby previous researchers for multi-lane loading moment distributionfactors. The AASHTO formula, the formulas presented by otherresearchers, and the formulas developed in the study are discussedin the following according to their application.

Table 6.4.2.2.3a-1 is taken from the specifications andsummarized criteria for moment in interior beams or elements forvarious types of cross-sections. Similar tables exist for moment inexterior griders, for shear in interior girders and shear in exteriorgirders.

Table 6.4.2.2.3a-2 describes how the term L (length) may bedetermined for use in the live load distribution factor equations givenin Table 6.4.2.2.3a-1.

In the rare occasion when the continuous span arrangementis such that an interior span does not have any positive uniform loadmoment (i.e., no uniform load points of contraflexure), the region ofnegative moment near the interior supports would be increased to thecenterline of the span, and the L used in determining the live loaddistribution factors would be the average of the two adjacent spans.


Lecture - 6-14

Table 6.4.2.2.3a-1 - Distribution of Live Loads Per Lane for Moment in Interior Beams




Wood Deck on Woodor Steel Beams

a, l See Table S4.6.2.2.2a-1

Concrete Deck onWood Beams

l One Design Lane Loaded: S/3700Two or More Design Lanes Loaded: S/3000

S # 1800

Concrete Deck, FilledGrid, or Partially FilledGrid on Steel orConcrete Beams;Concrete T-Beams, T-and Double T-Sections

a, e, k andalso i, j

if sufficientlyconnected toact as a unit

One Design Lane Loaded:

0.06 % S4300

0.4 SL

0.3 Kg

Lt 3s

0.1


0.075 % S2900

0.6 SL

0.2 Kg

Lt 3s

0.1

1100 # S # 4900110 # ts # 3006000 # L # 73 000Nb $ 4

use lesser of the values obtained from theequation above with Nb = 3 or the lever rule

Nb = 3

Multicell Concrete BoxBeam

d One Design Lane Loaded:

1.75 % S1100

300L

0.35 1Nc

0.45


13Nc

0.3 S430

1L

0.25

2100 # S # 400018 000 # L # 73 000Nc $ 3

If Nc > 8 use Nc = 8

Concrete Deck onConcrete Spread BoxBeams

b, c One Design Lane Loaded:

S910

0.35 SdL2

0.25


S1900

0.6 SdL 2

0.125

1800 # S # 35006000 # L # 43 000450 # d # 1700Nb $ 3

Use Lever Rule S > 3500





Lecture - 6-15

Concrete Beams usedin Multibeam Decks

f One Design Lane Loaded:

k b2.8L

0.5 IJ

0.25

where: k ' 2.5(Nb )&0.2 $ 1.5


k b7600

0.6 bL

0.2 IJ

0.06

900 # b # 15006000 # L # 37 0005 # Nb # 20

gif sufficientlyconnected toact as a unit

h Regardless of Number of Loaded Lanes: S/D

where:

C = K (W/L) # K

D = 300 [11.5 - NL + 1.4NL (1 - 0.2C)2] when C# 5

D = 300 [11.5 - NL] when C > 5

Skew # 45°

NL # 6

g, i, jif connected

only enough toprevent relative

verticaldisplacement at

the interface

K = (1 % µ) IJ

for preliminary design, the following values ofK may be used:

Beam Type KNonvoided rectangular beams 0.7Rectangular beams with circular voids: 0.8Box section beams 1.0Channel beams 2.2T-beam 2.0Double T-beam 2.0

Steel Grids on SteelBeams

a One Design Lane Loaded:S/2300 If tg< 100 mmS/3050 If tg$ 100 mmTwo or More Design Lanes Loaded:S/2400 If tg< 100 mmS/3050 If tg$ 100 mm

S # 1800 mm

S # 3200 mm

Concrete Deck on Multiple Steel BoxGirders

b, c Regardless of Number of Loaded Lanes:

0.05 % 0.85NL

Nb

%0.425

NL

0.5 #NL

Nb

# 1.5


Lecture - 6-16

Table 6.4.2.2.3a-2 - L for Use in Live Load Distribution Factor Equations

FORCE EFFECT L (mm)

Positive Moment The length of the span for whichmoment is being calculated.

Negative Moment - End spans of continuous spans,from end to point of contraflexure under a uniformload on all spans


Negative Moment - Near interior supports ofcontinuous spans, from point of contraflexure to pointof contraflexure under a uniform load on all spans

The average length of the twoadjacent spans.

Positive Moment - Interior spans of continuousspans, from point of contraflexure to point ofcontraflexure under a uniform load on all spans


Shear The length of the span for whichshear is being calculated.

Exterior Reaction The length of the exterior span.

Interior Reaction of Continuous Span The average length of the twoadjacent spans.


The AASHTO formula for moment distribution for multi-laneloading is given as S/1800 for reinforced concrete T-beam bridgeswith girder spacing up to 3000 mm, and as S/1700 for steel girderbridges and prestressed concrete girder bridges with girder spacingup to 4300 mm, where S is the girder spacing. When the girderspacing is larger than the specified limit, simple beam distribution is tobe used to calculate the load distribution factors.

Marx, et al, at the University of Illinois, developed a formula forwheel load distribution for moment which included multiple lanereduction factors and is applicable to all beam-and-slab bridges. Theformula is based on girder spacing, span length, slab thickness andbridge girder stiffness.

A formula which does not consider a reduction for multi-laneloading was developed at Lehigh University. The Lehigh formulaincludes terms for the number of traffic lanes, number of girders, girderspacing, span length and total curb-to-curb deck width.

Sanders and Elleby (NCHRP Report 83) developed a simpleformula based on orthotropic plate theory for moment distribution onbeam-and-slab bridges. Their formula includes terms for girder


Lecture - 6-17

spacing, number of traffic lanes and a stiffness parameter based onbridge type, bridge and beam geometry and material properties.

A full-width design approach, known as Henry's Method, isused by the State of Tennessee. Henry's Method includes factors fornumber of girders, total curb-to-curb bridge deck width and a reductionfactor based on number of lanes.

A formula developed as part of NCHRP Project 12-26 includesthe effect of girder spacing, span length, girder inertia and slab thick-ness. The multiple lane reduction factor is built into the formula. Thisformula, applicable to cross-sections with four or more beams, is givenby:

(6.4.2.2.3a-1)g ' 0.075 % S2900

0.6 SL

0.2 Kg

Lt 3s

0.1

where:

S = girder spacing (1100 mm # S # 4900 mm)

L = span length (6000 mm # L # 73 000 mm)

Kg = n(I+Aeg2) (4 x 108 # Kg # 3 x 1012 mm4)

n = modular ratio of girder material to slab material

I = girder moment of inertia

eg = eccentricity of the girder (i.e., distance from centroid ofgirder to mid-point of slab)

ts = slab thickness (110 mm # ts # 300 mm)

This formula is dependent on the inertia of the girder and, thus, avalue for Kg must be assumed for initial design. For this purpose,Kg/LtS

3 may be taken as unity.

All of the above formulas were evaluated using direct finiteelement analysis with the GENDEK-5 program and a database of 30bridges; subsequently, they were evaluated using the MSI method anddatabase of more than 300 bridges. It was found that Equation 6 andthe Illinois formulas are accurate and produce results that are asaccurate as the Level 2 methods.

Moment Distribution to Exterior Girders, Multi-Lane Loading

Previous AASHTO Specifications recommend a simple beamdistribution of wheel loads in the transverse direction for calculatingwheel load distribution factors in edge girders. Any load that falls


Lecture - 6-18

outside the edge girder is assumed to be acting on the edge girder,and any load that is between the edge girder and the first interiorgirder is distributed to these girders by assuming that the slab acts asa simple beam in that region. Any wheel load that falls inside of thefirst interior girder is assumed to have no effect on the edge girder.

Marx, et al, at the University of Illinois, developed a formula forthe exterior girder based on certain assumptions in the placement ofloads and may not be applicable to all bridges. This formula includesterms similar to those used in their formula for moment distribution tointerior girders.

A formula, depending on wheel position, alone was developedas part of this study which results in a correction factor for the edgegirder. The factor must be applied to the distribution factor for theinterior girder to obtain a distribution factor for the edge girder. Thisformula is given by:

(6.4.2.2.3a-2)e ' 0.77 %de

2800

where:


If the edge of the lane is outside of the exterior girder, thedistance is positive; if the edge of the lane is to the interior side of thegirder, the distance is negative.

It was found that the formula developed in Project 12-26resulted in accurate correction factors and was simpler than theprevious AASHTO procedure.


The literature search performed in this study did not reveal anysimplified formula for single-lane loading of beam-and-slab bridges.The formula developed as part of the study is as follows:

(6.4.2.2.3a-3)g ' 0.06 % S4300

0.4 SL

0.3 Kg

Lt 3s

0.1

where the parameters are the same as those given for Equation6.4.2.2.3a-1.

This formula is applicable to interior girders only.


Lecture - 6-19


Simple beam distribution in the transverse direction should beused for single-lane loading of edge girders.

One other investigation, applicable to load distribution for bothshear and moment, is required for exterior beams of beam-slabbridges with diaphragms or cross-frames. This addition was notdeveloped as part of NCHRP 12-26, but was added by the NCHRP12-33 Editorial Committee. This distribution is based on treating thecross-section as a transversely rigid unit which deflects and rotates asa straight line. The live load is positioned for maximum effect on anexterior beam (one lane, two lane, three lane, etc., each with itsappropriate multiple presence factor). The total vertical force andmoment about the centroid of the cross-section is applied to the areaof the cross-section, i.e., the number of beams, and the sectionmodulus, i.e., the sum of the square of the distances of each beamfrom the centroid of the beams divided by the distance to the exteriorbeam. The specification puts this in equation form as:

(6.4.2.2.3a-4)R 'NL

Nb

%

Xext jNL

ev

jNb

x 2

where:

R = reaction on exterior beam in terms of lanes

NL = number of loaded lanes under consideration

ev = eccentricity of a design truck or a design laneload from the center of gravity of the pattern ofgirders (mm)

x = horizontal distance from the center of gravity ofthe pattern of girders to each girder (mm)

Xext = horizontal distance from the center of gravity ofthe pattern of girders to the exterior girder (mm)

Nb = number of beams or girders

Shear Distribution

No formula was found from previous research for thecalculation of wheel load distribution factors for shear. Therefore, theformulas developed as part of the 12-26 study are reported fordifferent cases as follows.

The formula for multi-lane loading of interior girders is:


Lecture - 6-20

(6.4.2.2.3a-5)g ' 0.2 % S3600

&S

10 700

2

The correction formula for multi-lane loading edge girder shearis:

(6.4.2.2.3a-6)e ' 0.6 %de

3000

The formula for shear distribution factor due to single-laneloading is:

(6.4.2.2.3a-7)g ' 0.36 % S7600

Equation 6.4.2.2.3a-7 is applicable to interior girders only.Simple beam distribution in the transverse direction should be used forsingle-lane loading of edge girders.


Previous AASHTO Specifications did not include approximateformulas to account for the effect of skewed supports. However, someresearchers have developed correction factors for such effects onmoments in interior girders.

Marx, et al, at the University of Illinois, developed fourcorrection formulas for skew, one each for skew angles of 0, 30, 45and 60 degrees. Corrections for other values of skew are obtained bystraight-line interpolation between the two enveloping skew values.These correction formulas are based on girder spacing, span length,slab thickness and bridge girder stiffness.

A formula for a correction factor for prestressed concreteI-girders was developed as part of the research performed at LehighUniversity. This formula is based on the number of traffic lanes,number of girders, girder spacing, span length and total curb-to-curbdeck width, and includes a variable term for skew angle.

A correction factor for moment in skewed supports was alsodeveloped as part of Project 12-26. This formula is:

(6.4.2.2.3a-8)r ' 1 & c1 (tanθ)1.5

where, for θ > 30E,:


Lecture - 6-21

(6.4.2.2.3a-9)c1 ' 0.25Kg

Lt 3s

0.25SL

0.5

If θ # 30E, c1 is taken as zero. In calculating c1, θ should not be takenas greater than 60E. The other parameters are as defined previously.

From the literature review, no correction formulas wereobtained for shear effects due to skewed supports. In Project 12-26,it was found that shear in interior girders need not be corrected forskew effects; that is, the shear distribution to interior girders is similarto that of a straight bridge. A correction formula for shear at theobtuse corner of the exterior girder of two girder systems and allgirders of a multi-girder bridge was developed as part of this study andis given as:

(6.4.2.2.3a-10)r ' 1 % 0.2Lt 3

s

kg

0.3

tanθ

where the parameters are defined in Equation 6.4.2.2.3a-1.

Equation 6.4.2.2.3a-7 is to be applied to the shear distributionfactor in the exterior girder of non-skewed bridges. Therefore, theproduct of factors g, e and r must be obtained to find the obtuse cornershear distribution factor in a beam-and-slab bridge.

The distribution factors calculated for moments are plotted asa function of girder spacing for Spans 9, 18, 27, 36 and 60 m in Figure6.4.2.2.3a-1. For comparisons, AASHTO (1989) distribution factorsare also shown. Girder distribution factors, specified by AASHTO(1989), are conservative for larger girder spacing. For shorter spansand girder spacings, AASHTO (1989) produces smaller distributionfactors than calculated values.


Lecture - 6-22

Figure 6.4.2.2.3a - Calculated Distribution Factors

6.4.2.2.3b Simplified Formulas for Box Girder Bridges

Research on box girder bridges has been performed byvarious researchers in the past. Bridge deck behavior has been wellstudied and many recommendations have been made for detailedanalysis of these bridges. However, there is a limited amount ofinformation on simplified wheel load distribution formulas in theliterature.

In this context, a “girder” is a notional I-shape consisting of oneweb of a multi-cell box and the associated half-flanges on each sideof the web.

Moment Distribution to Interior Girders

Scordelis, at the University of California, Berkeley, presenteda formula for prediction of wheel load distribution for moment distribu-tion in prestressed and reinforced concrete box girder bridges. Theformula is based on modification of distribution factors obtained for arigid cross-section. The formula predicts load distribution factors inreinforced concrete box girders with high accuracy and for prestressedconcrete box girders with acceptable accuracy.

Sanders and Elleby also presented a simple formula formoment distribution factors which is similar to their formula for beam-and-slab bridges.

The following formulas, developed as part of NCHRP 12-26,may be used to predict the moment load distribution factors in theinterior girders of concrete box girder bridges due to single-lane andmulti-lane loadings. These formulas are applicable to both reinforced


Lecture - 6-23

and prestressed concrete bridges, and the multiple presence factor isaccounted for.

For single-lane loading:

(6.4.2.2.3b-1)g ' 1.75 % S1100

300L

0.35 1Nc

0.45

For multi-lane loading:

(6.4.2.2.3b-2)g ' 13Nc

0.3 S430

1L

0.25

where:



Nc = number of cells


The factor for load distribution for exterior girders shall beWe/4300 mm, where We is the width of the exterior girder, taken as thetop slab width measured from the mid-point between girders to theedge of the slab.

Shear Distribution

No formula for shear load distribution was obtained fromprevious research for box girder bridges, but the following weredeveloped as part of NCHRP 12-26.

The shear distribution factor for interior girder multi-laneloading of reinforced and prestressed concrete box girder bridges is:

(6.4.2.2.3b-3)g ' S2200

0.9 dL

0.1

where:


d = girder depth, in mm



Lecture - 6-24


(6.4.2.2.3b-4)g ' S2900

0.6 dL

0.1

where the parameters are as defined in Equation 6.4.2.2.3b-3.

A correction formula for shear in the exterior girder for multi-lane loading is:

(6.4.2.2.3b-5)e ' 0.64 %de

3800

where:



The following formula was developed for correction of momentdue to skewed supports for values of θ from 0E to 60E:

(6.4.2.2.3b-6)r ' 1.05 & 0.25 (tanθ) # 1.0

If θ > 60E, use 60E in Equation 6.4.2.2.3b-6.

Another formula was developed in Project 12-26 for correctionof shear at the obtuse corner of an edge girder. It must be applied tothe shear distribution factor for the edge girder of a non-skewed bridgeand must, therefore, be used in conjunction with the edge girdercorrection factor of Equation 6.4.2.2.3b-5. This formula, applicable forvalues of θ up to 60E, is:

(6.4.2.2.3b-7)r ' 1 % c1 (tanθ)

where:

c1 = 0.25 + L/(70d)

d = bridge depth, in mm



Lecture - 6-25

6.4.2.2.3c Simplified Formulas for Slab Bridges

The literature search did not reveal any simplified formulas forwheel load distribution in slab bridges other than those recommendedby AASHTO. Therefore, the following are formulas that weredeveloped as part of NCHRP 12-26.

Moment Distribution, Multi-Lane Loading

Equation 6.4.2.2.3c-1 was developed to predict wheel loaddistribution (distribution design width) for moment in slab bridges dueto multi-lane loading. Multiple presence factors are already accountedfor in the formula:

(6.4.2.2.3c-1)E ' 2100 % 0.12 L1W10.5 # W

NL

where:

E = the transverse distance over which a wheel line isdistributed

L1 = L # 18 000 mm

W1 = W # 18 000 mm


W = bridge width, in mm, edge-to-edge

Moment Distribution, Single-Lane Loading

The equation below predicts wheel load distribution formoment due to single-lane loading.

(6.4.2.2.3c-2)E ' 250 % 0.42 (L1W1 )0.5

where the parameters are as defined in Equation 6.4.2.2.3c-1.


Equation 6.4.2.2.3b-6 may be used to account for the reductionof moment in skewed bridges.

According to previous AASHTO Specifications, slab bridgesare adequate for shear if they are designed for moment. A quickcheck of this assumption was made and it was concluded that it is avalid assumption. Therefore, no formula or method is presented forcalculation of shear in slab bridges.


Lecture - 6-26

6.4.2.2.3d Simplified Formulas for Multi-Beam Decks which areSufficiently Interconnected to Act as a Unit

Only one formula, other than those presented in the previousAASHTO Specifications, was obtained for load distribution in multi-beam decks. This formula, developed by Arya at the University ofIllinois, is applicable to both box and open section multi-beam bridgesand predicts interior beam moment responses due to single-lane andmulti-lane loading. However, a number of simplified formulasdeveloped in the study are valid only for multi-box beam decks and donot apply to open sections. Therefore, the response of multi-beamdecks made of open members, such as channels, may or may not beaccurately predicted by the formulas developed in that study.


The formula developed by Arya for interior girder loaddistribution in multi-beam decks includes terms for the maximumnumber of wheels that can be placed on a transverse section of thebridge, number of beams, beam width and span length. A variation ofthe formula was also proposed for multi-beam decks made ofchannels, which includes consideration of the overall depth of thechannel section and its average thickness, defined as its area dividedby its length along the centerline of the thickness.

The following formula was developed in Project 12-26 topredict load distribution factors for interior beam moment due to multi-lane loading. The multiple presence reduction factor is alreadyaccounted for in the formula.

(6.4.2.2.3d-1)g ' k b7600

0.6 bL

0.2 IJ

0.06

where:

k = 2.5(Nb)!0.2 $ 1.5

b = beam width, in mm


Nb = number of beams

I = moment of inertia of a beam (mm)4

J = torsional constant of a beam (mm)4

This formula is dependant on the inertia and torsional constant of abeam; an estimated value for these properties must, therefore, beused in preliminary design. The term I/J may be taken as unity for thiscase.


Lecture - 6-27


Arya also presented a load distribution formula for multi-beamdecks designed for one traffic lane. The formulation and parameterswere similar to those presented for multi-lane loading. A variation ofthat equation was also presented for calculation of the interior beammoment distribution factor for a single-lane, channel section, multi-beam deck. It should be noted that Arya's equations are notapplicable to cases of only one-lane loading with more than one trafficlane.

A formula for wheel load distribution for moment in the interiorgirders due to single-lane loading was also developed in NCHRP 12-26. This formula is as follows:

(6.4.2.2.3d-2)g ' k b2.8L

0.5 IJ

0.25

All parameters are defined in Equation 6.4.2.2.3d-1. Equation6.4.2.2.3c-2 is also dependent on inertia and torsional constants, anda value of 1.0 may be used as an approximation for the term I/J duringpreliminary design.


The moment in the edge girder due to multi-lane loading inmulti-beam decks comprised of box units is obtained by using acorrection factor applied to the interior girder distribution factors formulti-lane loading. This correction factor may be found from thefollowing formula:

(6.4.2.2.3d-3)g ' 1.04 %de

7600

where:

de = distance from edge of the lane to the center of the exterior webof the exterior girder, in mm

For exterior beams of sufficiently interconnected multi-beambridge decks comprised of T-shaped units subjected to multi-laneloading, Equation 6.4.2.2.3b-2 applies.

For single-lane loading and for multi-beam decks comprised ofeither box units or units other than box units, the lever rule is used.


Lecture - 6-28

Shear Distribution

Distribution factors for shear in interior girders of multi-beamdecks in "Bridge Decks Comprised of Box Units" due to multi-laneloading may be calculated from the following formula:

(6.4.2.2.3d-4)g ' b4000

0.4 bL

0.1 IJ

0.05

where the parameters are as defined in Equation 6.4.2.2.3d-1.

Distribution factors for shear in the interior girders of multi-beam decks in "Bridge Decks Comprised of Box Units" due to single-lane loading are obtained from the following formula:

(6.4.2.2.3d-5)g ' 0.70 bL

0.15 IJ

0.05

where the parameters are again as defined in Equation 6.4.2.2.3d-1.

Note that Equations 6.4.2.2.3d-4 and 6.4.2.2.3d-5 aredependent on inertia and torsional constants, and a value of 1.0 maybe used as an approximation for the term I/J during preliminarydesign.

The shear in the edge girder of multi-beam deck in "BridgeDecks Comprised of Box Units" due to multi-lane loading can be foundusing a correction factor applied to interior girder distribution factors.This correction factor is obtained from the formula:

(6.4.2.2.3d-6)e ' 1.02 %de

15 000

where:

de = distance from edge of lane to the center of exterior webof the exterior girder, in mm

For shear in exterior beams of sufficiently interconnected multi-beam bridge decks comprised of T-shaped units, Equations6.4.2.2.3b-4 through 6.4.2.2.3b-5 and the lever rule should be used,where appropriate.


The moment in any beam in a skewed bridge may be obtainedby using a skew reduction factor given by Equation 6.4.2.2.3b-6.


Lecture - 6-29

The shear in the interior beams of a skewed multi-beam bridgecomprised of box beams is usually of the same order as that of theshear in the obtuse corner and must be obtained by applying acorrection factor to the response of the edge girder in a straight bridge.This correction factor may be calculated from the formula:

(6.4.2.2.3d-7)r ' 1 % c1 (tanθ)0.5

where:

(6.4.2.2.3d-8)c1 'L

90d

6.4.2.2.3e Simplified Formulas for Multi-Beam Decks which are notSufficiently Interconnected to Act as a Unit

The LRFD Specification contains the same provisions for loaddistribution in this type of bridge superstructure as appeared in recenteditions of the Standard Specification, and as repeated below forcompleteness.

The key difference between bridges treated herein, ascompared to Section 6.4.2.2.3d, is the degree of transverseinterconnection of units. If box, T, channel or other precast units areinterconnected through a structural slab, or sufficiently transverselypost-tensioned to produce a similar level of continuity, then thediscussion of Section 6.4.2.2.3d applies. If the interconnectionbetween the units is expected to transmit shear, but relatively littlemoment over the bridge service life, then the provisions herein apply.

The Specification provides for the computation of a bendingmoment distribution factor, regardless of the number of lanes, givenby:

g ' S300D

for which:

D = 300 [11.5 - NL + 1.4 NL (1 - 0.2C)2] when C # 5

D = 300 [11.5 - NL] when C > 5

C = K (W/L)

K = (1 % µ)IJ


Lecture - 6-30

where:

µ = Poisson ratio

I = moment of inertia (mm)4

J = St. Venant's constant (mm)4


NL = number of lanes

S = spacing of units (mm)

W = edge-to-edge width of bridge (mm)

6.4.2.2.3f Simplified Formulas for Spread Box Beam Bridges

Only one formula, other than those recommended byAASHTO, was obtained from previous research for determining loaddistribution factors in spread box beam bridges. This formula wasdeveloped at Lehigh University for predicting the response of interiorbeams due to multi-lane loading and was later adopted by AASHTO.A correction factor for skewed bridges was also presented. Inaddition, a number of simple formulas were developed as part ofNCHRP Project 12-26.

Moment Distribution to Interior Beams, Multi-Lane Loading

A formula developed in Project 12-26 for moment in interiorspread box beams due to multi-lane loading is as follows:

(6.4.2.2.3f-1)g ' S1900

0.6 SdL 2

0.125

where

S = girder spacing (mm)


d = beam depth (mm)

Moment Distribution to Interior Beams, Single-Lane Loading

A similar formula was developed for distribution to interiorbeams due to single-lane loading:


Lecture - 6-31

(6.4.2.2.3f-2)g ' S910

0.35 SdL 2

0.25

where the parameters are as defined in Equation 6.4.2.2.3f-1.


The moment in edge girders due to multi-lane loading may becalculated by applying a correction factor to the interior girderdistribution factor:

(6.4.2.2.3f-3)e ' 0.97 %de

8700

where:


The distribution factor for moment in the edge girder due tosingle-lane loading may be obtained by simple-beam distribution, i.e.,the lever rule, in the same manner as was described for beam-and-slab bridges.

Shear Distribution

The distribution factor for shear in the interior girders due tomulti-lane loading may be calculated from the following:

(6.4.2.2.3f-4)g ' S2250

0.8 dL

0.1

where the parameters are as defined previously.


(6.4.2.2.3f-5)g ' S3050

0.6 dL

0.1

where the parameters are again as defined previously.

The shear in the edge girder due to multi-lane loading can befound by applying a correction factor to the interior girder equation.This correction factor is:


Lecture - 6-32

(6.4.2.2.3f-6)e ' 0.8 %de

3050

where:


The wheel load distribution factor for shear in the edge girderdue to single-lane loading may be obtained by simple-beamdistribution in the same manner as was described for beam-and-slabbridges, i.e., the lever rule.


Research at Lehigh University also resulted in a formula forcorrection of wheel load distribution for moment in interior girders dueto multi-lane loading in skewed bridges. NCHRP 12-26 concludes thatEquation 6.4.2.2.3b-6 was also applicable to this case.

The shear in the interior beams of a skewed bridge is the sameas that of a straight bridge. However, the shear in the obtuse cornermust be obtained by applying a correction factor to the distributionfactor for the edge girder in a straight bridge, given by Equation6.4.2.2.3b-7, which C1 is taken as:

(6.4.2.2.3f-7)c1 '(Ld )0.5

6S

6.4.2.2.3g Response of Continuous Bridges

The response of continuous bridges was studied by modelinga number of two-span continuous bridges where each span is similarto the average bridge. The wheel load distribution factor for each casewas compared to that of a simple bridge and correction factors forcontinuity were obtained. In the case of beam-and-slab bridges, acomplete parameter study was performed, and it was found that thecorrection factor is generally independent of bridge geometry. Thesefactors are given in the table below.

When the NCHRP 12-26 factors were incorporated into theLRFD Specification, it was decided that 5% corrections wereunwarranted given that the distribution factors are an approximationof actual behavior and are, therefore, subject to some variability. Thecontinuity correction for negative moment, a 10% increase, was alsoneglected on the basis that experimentally observed "fanning" of thereaction tends to reduce the negative moment as compared to atypical beam calculation.


Lecture - 6-33

Table 6.4.2.2.3f-1 - Continuity CorrectionFactors

Beam-and-Slab Bridges


c = 1.05c = 1.10c = 1.00c = 1.05

Box Girder Bridges


c = 1.00c = 1.10c = 1.00c = 1.00

Slab Bridges

Positive momentNegative moment

c = 1.00c = 1.10

Multi-Beam Bridges


c = 1.00c = 1.10c = 1.00c = 1.05

Spread box beam bridges


c = 1.00c = 1.10c = 1.00c = 1.05

6.4.2.3 TRUSS AND ARCH BRIDGES

6.4.2.3.1 General

The approximate method for load distribution to lines of trussesand arches is a so-called "lever rule", which is simply a matter ofsumming moments about one line of trusses or arches to find thereaction on the other line. This approach is illustrated by calculationsin Lecture 7.