printed on June 24, 2003 LECTURE 4 - LOADS II
4.1 OBJECTIVE OF THE LESSON LESSON The purpose of this lesson is to continue the developments of the loads for bridge design required by the LRFD Specification by providing information on ice loads, earth pressures.
4.2 ICE LOADS 4.2.1 General The Specification requires consideration of the following types of ice action. •
Dyna Dynam mic pres pressu sure re due due to to mov movin ing g she sheet ets s or or flo floes es of ice ice bei being ng carried by stream flow, wind or currents and striking a pier.
•
Stati tatic c pre press ssur ure e due due to ther therma mall mov movem emen entts of of ic ice she sheet ets s. Static forces may be caused by the thermal expansion of ice in which a pier is embedded, or by irregular growth of the ice field.
•
Pres ressure sure resu result ltin ing g from rom hang hangiing dams dams or jams jams of ice. ice. Hanging dams are the phenomenon of frazil ice passing under the surface layer of ice and accumulating under the surface ice at the bridge site. site. The frazil ice comes typically typically from rapids or waterfalls upstream. The hanging dam can cause a back-up of water, which exerts pressure on the pier, and can also cause scour around or under the piers as the water flows at an increased velocity.
•
Stati tatic c upl upliift or verti ertica call loa load d res resul ulti ting ng from from adhe adheri ring ng ice ice in waters of fluctuating level.
The behavior of ice and the forces that it generates is a very complex issue and is not yet fully understood. When undergoing long-term changes in temperature and sustained loadings, ice can behave in a relatively plastic manner. Ice which is moving with a current can either create a substantial impact load on a pier or breakup when it hits the pier reducing the the load. For purposes of classification, the commentary of the Specification uses the types of ice failure developed by Montgomery in in 1984. The reference to Montgomery is given in the Specification. Specification. Thus, the following types of ice failure are considered:
Lecture - 4-1
printed on June 24, 2003 •
crushing - the ice fails by local crushing across the width of the pier. The crushed ice is continually cleared from a zone around the pier as the floe moves past,
•
bending - for piers with inclined noses, a vertical reaction component acts on the impinging ice floe. This reaction causes the floe to rise up the pier nose and fail as flexural cracks form,
•
splitting - where a comparatively small floe strikes a pier, stress cracks propagating from the pier into the floe split the floe into smaller parts,
•
impact - if the floe is small, it is brought to a halt when impinging on the nose of the pier before it has failed by crushing over the full-width of the pier, by bending or by splitting, and
•
buckling - for very wide piers, where a large floe cannot clear the pier as it fails, compressive forces cause the floe to fail by buckling in front of the pier nose.
4.2.2 Design for Ice The design for ice typically starts with the determina tion of the effective crushing strength of the ice (p). The following values are specified when cite specific information is not available. •
0.38 MPa where break-up occurs at melting temperatures and the ice is substantially disintegrated in its structure,
•
0.77 MPa where break-up occurs at melting temperatures and the ice is somewhat disintegrated in its structure,
•
1.15 MPa where break-up or major ice movement occurs at melting temperatures, but the ice moves in large pieces and is internally sound, and
•
1.53 MPa where break-up or major ice movement occurs with the ice temperature, averaged over its depth, measurably below the melting point.
As indicated in the commentary to the Specification, the values identified above are considerably less than ice values which may be measured under laboratory conditions. The lowest value is appropriate for piers where experience with similar sites indicates that ice forces are very low, but ice should be considered in the design at that location. The maximum value of 1.53 MPa is based on an observed history of bridges that have survived significant
Lecture - 4-2
printed on June 24, 2003 icing conditions, and these are detailed in the reference list in Section 3 of the Specification. Once the effective ice strength, P, has been determined for the given site, breakup conditions, the next step is to determine the horizontal force, F, resulting from the pressure of moving ice. This horizontal force is related to the effective strength by a series of expressions in S3.9.2.2, which are also a function of the pier geometry. A distinction is made as to whether the horizontal force is caused by an ice flow failing by compression over the full-width of the pier, or whether the ice flow fails by flexure as it rides up an inclined ice breaking pier nose. The design expressions require an estimate of the ice thickness. The preferred method to determine the ice thickness is historical records of actual ice thickness at a potential bridge site over some number of years. When this data is not available, an empirical expression is provided in the commentary to S3.9.2.2, which provides a means of estimating ice thickness, depending on the accumulation of freezing days at the site, per year, and the particular conditions of wind and snow apt to occur at the site. Snow cover is found to be an important factor in determining ice thickness. The snow cover in excess of 150 mm in thickness has been shown to reduce ice thickness by almost one-half. The specification permits the reduction in ice forces on small streams not conducive to the formation of large ice flows. The reduction is limited to not more than 50% of the design force. Once the ice force, F, has been determined, it is necessary to consider combinations of longitudinal and transverse forces acting on a pier. It would be unrealistic to expect ice to move exactly parallel to a pier, so that a minimum lateral component of 15% of the longitudinal force is specified. Piers are usually aligned in the direction of stream flow, usually assumed to be the direction of ice movement. Two design cases are investigated as follows: •
a longitudinal force equal to F shall be combined with a transverse force of 0.15F, or
•
a longitudinal force of 0.5F shall be combined with a transverse force of Ft.
The transverse force, Ft, shall be taken as:
F t
F 2 tan( β / 2
θf )
(4.2.2-1)
Lecture - 4-3
printed on June 24, 2003 where: β
=
nose angle in a horizontal plane, for a round nose taken as 100 (DEG)
θf
=
friction angle between ice and pier nose (DEG)
Both the longitudinal and transverse forces shall be assumed to act at the pier nose. Where the longitudinal axis of a pier is not parallel to the principal direction of ice action, or where the direction of ice action may shift, the total force on the pier shall be determined on the basis of the projected pier width and resolved into components. Under such conditions, forces transverse to the longitudinal axis of the 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 be investigated where the ice sheets are subject to significant thermal movements relative to the pier where the growth o f shore ice is on one side only, or other situations which may produce substantial unbalanced forces on the pier. Unfortunately, little guidance is available for predicting static ice loads on piers. Under normal circumstances, the effects of static ice forces on piers may be strain limited, but expert advice should be sought if there is reason for concern. 4.2.4 Hanging Dams and Ice Jams The frazil accumulation in a hanging dam may be taken to exert 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 the variability 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 level fluctuation is given by: •
for a circular pier, in N: F v
0.3 t 2
0.0169 R t 1.25
(4.2.5-1)
Lecture - 4-4
printed on June 24, 2003 •
for an oblong pier, in N/mm of pier perimeter: F v
2.3x10
3
t 1.25
(4.2.5-2)
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 a few minutes, but they are also based on the assumption that failure occurs 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 the LRFD Specification. However, Owners in areas where unique accumulations of snow and/or ice are possible should specify appropriate loads for that condition. The following discussion of snow loads is taken from Ritter (1991). Snow loads should be considered where a bridge is located in an area of potentially heavy snowfall. This can occur at high elevations in mountainous areas with large seasonal accumulations. Snow loads are normally negligible in areas of the United States that are below 600 000 mm elevation and east of longitude 105 W, or below 300 000 mm elevation and west of longitude 105 W. In other areas of the country, snow loads as large as 0.034 MPa may be encountered in mountainous locations. The effects of snow are assumed to be offset by an accompanying decrease in vehicle live load. This assumption is valid for most structures, but is not realistic in areas where snowfall is significant. When prolonged winter closure of a road makes snow removal impossible, the magnitude of snow loads may exceed those from vehicular live loads. Loads also may be notable where plowed snow is stockpiled or otherwise allowed to accumulate. The applicability and magnitude of snow loads are left to Designer's judgment. Snow loads vary from year to year and depend on the depth and density of snow pack. The depth used for design should be based on a mean recurrence interval or the maximum recorded depth. Density is based on the degree of compaction. The lightest accumulation is produced by fresh snow falling at cold temperatures. Density increases when the snow pack is subjected Lecture - 4-5
printed on June 24, 2003 to freeze-thaw cycles or rain. Probable densities for several snow pack conditions are as follows, ASCE (1980): Table 4.2.6-1 - Snow Density CONDITION OF PROBABLE SNOW PACK DENSITY (kg/m3) Freshly Fallen
96
Accumulated
300
Compacted
500
4.3 EARTH LOADS 4.3.1 General The Specification defines several broad classifications of walls which are also referred to herein. For reference, these definitions are repeated below: Abutment - A structure that supports the end of a bridg e span, and provides lateral support for fill material on which the roadway rests immediately adjacent to the bridge. Anchored Wall - An earth retaining system typically composed of the same elements as non-gravity cantilevered walls, and which derive additional lateral resistance from one or more tiers of anchors. Mechanically Stabilized Earth Wall - A soil retaining system, employing either strip or grid-type, metallic or polymeric tensile reinforcements in the soil mass, and a discrete modular precast concrete facing which is either vertical or nearly vertical. Non-Gravity Cantilever Wall - A soil retaining system which derives lateral resistance through embedment of vertical wall elements and support retained soil with facing elements. Vertical wall elements may consist of discrete elements, e.g., piles, caissons, drilled shafts or auger-cast piles spanned by a structural facing, e.g., lagging, panels or shotcrete. Alternatively, the vertical wall elements and facing may be continuous, e.g., diaphragm wall panels, tangent piles or tangent drilled shafts. Prefabricated Modular Wall - A soil retaining system employing interlocking soil-filled timber, reinforced concrete or steel modules Lecture - 4-6
printed on June 24, 2003 or bins to resist earth pressures by acting as gravity retaining walls. Prefabricated modular walls consist of individual structural units assembled at the site into a series of hollow bottomless cells known as cribs. The cribs are filled with soil, and their stability depends not only on the weight of the units and their filling, but also on the strength of the soil used for the filling. The units themselves may consist of reinforced concrete, fabricated metal, or timber. Rigid Gravity, Semi-Gravity and Cantilever Retaining Walls - A structure that provides lateral support for a mass of soil and that owes its stability primarily to its own weight and to the weight of any soil located directly above its base. This classification of walls includes: •
A gravity wall depends entirely on the weight of the stone or concrete masonry and of any soil resting on the masonry for its stability. Only a nominal amount of steel is placed near the exposed faces to prevent surface cracking due to temperature changes.
•
A semi-gravity wall is somewhat more slender than a gravity wall and requires reinforcement consisting of vertical bars along the inner face and dowels continuing into the footing. It does not rely on the weight of the overlying soil for stability. It is provided with temperature steel near the exposed face.
•
A cantilever wall consists of a concrete stem and a concrete base slab, both of which are relatively thin and fully reinforced to resist the moments and shears to which they are subjected.
•
A counterfort wall consists of a thin concrete face slab, usually vertical, supported at intervals on the inne r side by vertical slabs or counterforts that meet the face slab at right angles. Both the face slab and the counterforts are connected to a base slab, and the space above the base slab and between the counterforts is backfilled with soil. All the slabs are fully reinforced. Several of these types of walls are illustrated in Figure 4.3.1-
1.
Lecture - 4-7
printed on June 24, 2003
Figure 4.3.1-1 - Illustration of Several Wall Types ( from Das, B. M., Principles of Foundation Engineering, Brooks/Cole Engineering Division, 1984) Retained earth exerts lateral pressure on re taining walls and abutments. In general, the magnitude and distribution of the lateral earth pressure on such structures is a function of the composition and consistency of the retained earth and the magnitude of external loads applied to the retained soil mass. Typically, development of a design earth pressure considers the following: •
The type, unit weight, shear strength and creep characteristics of the retained earth;
•
The anticipated or permissible magnitude and direction of lateral deflection at the top of the wall o r abutment;
•
The degree to which backfill soil retained by the wall is to be compacted;
Lecture - 4-8
printed on June 24, 2003 •
The location of the groundwater table within the retained soil;
•
The magnitude and location of surcharge loads on the retained earth mass; and
•
The effects of horizontal acceleration of the retained earth mass during an earthquake.
The degree to which a wall (or abutment) is permitted to deflect laterally, and the characteristics of the retained earth are the two most significant factors in the development of lateral earth pressure distributions. Walls which are permitted to tilt or move laterally away from the retained soil permit the development of an active state of stress in the retained soil mass and should be designed for the active earth pressure. Walls which are restrained against movement (e.g., integral abutments) or walls for which lateral deflection and associated ground movements may adversely impact adjacent facilities (typically within a distance behind the wall less than about one-half the wall height) should be designed to resist the at-rest earth pressure, which may be 50% greater than the magnitude of the active pressure. Walls which may deflect laterally into the retained soil should be designed to resist the passive earth pressure, which can be 10 to 20 times greater than the active pressure. (The passive state of stress is limited, for all practical purposes, to lateral deflection of the embedded po rtions of flexible cantilever retaining walls into the supporting soil.) The lateral wall movement required to permit development of the minimum active earth pressure or maximum passive earth pressure is affected by the type of soil retained, as shown in Table 4.3.1-1 where: ∆
=
lateral movement at top of wall (achieved through rotation or translation) required for development of active or passive earth pressure (mm)
H
=
wall height (mm)
Lecture - 4-9
printed on June 24, 2003 Table 4.3.1-1 - Approximate Values of Relative Movements Required to Reach Minimum Active or Maximum Passive Earth Pressure Conditions, Clough (1991) Values of ∆/H Type of Backfill
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 development of active earth pressures. Gravity and semi-gravity walls designed with a sufficient mass to support only active earth pressures will deflect (tilt or translate) in response to more severe loading conditions (e.g., at-rest earth pressures) until stresses in the retained soil mass are relieved sufficiently to permit de velopment of an active state of stress in the retained soil. The most significant potential for development of at-rest earth pressures on such w alls is on the stems of cantilevered retaining walls, where a rigid stemto-base connection may prevent lateral deflection of the stem with respect to the base in response to the lateral pressure of backfill soil retained above the base. For such a condition, excessive lateral earth pressures on the stem could conceivably lead to a structural failure of the stem or stem-to-base connection. A comparison of estimated actual lateral deflections to deflections required to mobilize active earth pressure on the stem of a cantilevered retaining wall backfilled with dense sand is provided in Tables 4.3.1-2 and 4.3.1-3. Table 4.3.1-2 assumes that the full section modulus of the stem is e ffective in resisting bending, and that the base of the stem is fixed to the foundation slab. Table 4.3.1-3 assumes that a reduced section modulus is effective in resisting bending to account for cracking and creep o f the concrete. Neither Table 4.3.1-2 nor Table 4.3.1-3 includes lateral deflections that would occur due to differential settlement of the wall base slab.
Lecture - 4-10
printed on June 24, 2003 Table 4.3.1-2 Cantilever Wall Stem Deflections Using Full Section Modulus Estimated Stem Deflection
Wall Height H (mm)
Average Wall Stem Thickness t (mm)
Under Active Earth Pressure ∆/H (dim)
Under AtRest Earth Pressure ∆/H (dim)
Deflection Required to Mobilize Active Earth Pressure ∆/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 = 37 , ka = 0.25, ko = 0.40 and f' c = 27.6 MPa Table 4.3.1-3 - Cantilever Wall Stem Deflections Using Reduced Section Modulus Estimated Stem Deflection
Wall Height H (mm)
Average Wall Stem Thickness t (mm)
Under Active Earth Pressure ∆/H (dim)
Under AtRest Earth Pressure ∆/H (dim)
Deflection Required to Mobilize Active 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 = 37 , ka = 0.25, ko = 0.40 and f'c = 27.6 MPa Tables 4.3.1-2 and 4.3.1-3 indicate that, with the exception of very short walls, cantilever wall stems will generally deflect, crack Lecture - 4-11
printed on June 24, 2003 and creep sufficiently to permit mobilization of active earth pressures in backfill soils composed of relatively dense sand, which is the typical backfill material for such walls. For walls bearing on yielding (soil) foundation materials, tilting of the base slab due to settlement of the foundation soils will result in even greater deflections, such that active earth pressures will be mobilized even on the stems of very short walls. It is likely that, unless walls are otherwise restrained against rotation or translation or have a massive cross-section, active earth pressure conditions will be achieved on the stems of nearly all cantilever retaining walls backfilled with compacted granular soil, with the exception of very short walls (less than 1.5 m tall) bearing directly on rock. For walls retaining cohesive soils, the effects of soil creep may prevent the permanent establishment of active and passive earth pressure. Under stress conditions producing the minimum active or maximum passive earth pressure, cohesive soils continually creep, such that shear stresses within the soil mass are partially received. As a result, the movements indicated in Table 4.3.1-1 are produced only temporarily. Without further movement, the lateral earth pressure exerted by a cohesive soil initially in the active stress state will increase eventually to a value app roaching the at-rest earth pressure. Likewise, the lateral earth pressure exerted by a cohesive soil initially in the passive stress state will decrease eventually to a value approaching approximately 40% of the passive earth pressure. 4.3.2 Compaction When mechanical compaction equipment is operated within a distance behind a retaining wall equal to about one- half of the wall height, additional lateral earth pressures are induced on the wall due to the compaction effort. Excessive backfill compaction can increase lateral earth pressures to values significantly greater than the active or even at-rest lateral earth pressure. Such compactioninduced pressures continue to act, even after the compaction equipment has been removed due to the inelastic behavior of the soil. A typical lateral earth pressure distribution for an unyielding wall, including compaction-induced residual pressures, is shown in Figure 4.3.2-1.
Lecture - 4-12
printed on June 24, 2003
Figure 4.3.2-1 - Residual Earth Pressure after Compaction of Backfill Behind an Unyielding Wall (after Clough and Duncan, 1991) The induced residual pressures would be somewhat less on a flexible or unrestrained wall subjected to the same compaction loading conditions since lateral deflection or movement of the wall would permit partial relief of the stress in the retained soil. The heavier the compaction equipment and the closer it operates to the wall, the greater are the compaction-induced pressures. Therefore, the use of soils which are difficult to compact (e.g., fine-grained, moisture sensitive soils) and heavy compaction equipment immediately behind earth retaining structures is likely to cause unacceptably large lateral soil pressures and should be avoided.Use of free-draining granular soils and light compaction equipment within a distance of H/2 behind retaining walls is usually specified to preclude development of excessive compactioninduced lateral earth pressures. If the use of heavy static or vibratory compaction equipment behind a retaining wall cannot be avoided, the residual compaction-induced lateral earth pressures should be estimated by available procedures (e.g., Clough and Duncan, 1991).
Lecture - 4-13
printed on June 24, 2003 4.3.3 Earth Pressure As described in Article 4.3.1, the magnitude and distribution of lateral earth pressure on a retaining structure is primarily a function of the retained soil characteristics and the degree to which the wall tilts or translates in response to the loading. For most abutments and conventional retaining walls, the earth pressure distribution is assumed to increase linearly with depth in accordance with the following: p = kh γ g z 10-9
(4.3.3-1)
where: γ
=
unit density of soil (kg/m 3)
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 soil is assumed to increase linearly with depth, the resultant lateral load due to the earth pressure is assumed to act at a height of 0.4H above the base of the wall for conventional gravity retaining walls (where H is the total wall height measured from the top of the backfill to the base of the footing) rather than 0.33H, as would be expected for a linearly proportional (triangular) distribution. As a conventional gravity wall deflects laterally (translates) in response to lateral earth loading, the backfill behind the wall must slide down along the back of the wall for the retained soil mass to achieve the active state of stress. Experimental results indicate that the backfill arches against the upper portion of the wall as the wall translates, causing an upward shift in the location at which the resultant of the lateral earth load is transferred to the wall (Terzaghi, 1934; Clausen, 1972, and Sherif, 1982) For non-gravity cantilever retaining walls or other flexible walls 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 to act at a height of 0.33H above the base of the wall.
Lecture - 4-14
printed on June 24, 2003 4.3.3.1 AT-REST PRESSURE COEFFICIENT, ko When a retaining wall is restrained against lateral movement or lateral movement of the wall is unacceptable, the lateral earth pressure coefficient (kh) in Equation 4.3.3.1-1 is taken as k o. For a normally consolidated soil, the at-rest lateral earth pressure coefficient, k o, can be computed by the following: ko = 1 - sin φf
(4.3.3.1-1)
where: φf
=
effective stress angle of internal friction of the drained soil
For overconsolidated soils, the at-rest lateral earth pressure coefficient is generally considered to vary as a function of the stress history, the value of ko increasing with increasing degree of overconsolidation 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 of overconsolidation are presented in Table 4.3.3.1-1.
Lecture - 4-15
printed on June 24, 2003 Table 4.3.3.1 - Typical Coefficients of At-Rest Lateral Earth Pressure Coefficient of Lateral Earth Pressure, k o OCR = 1
OCR = 2
OCR = 5
OCR = 10
Loose Sand
0.45
0.65
1.10
1.60
Medium Sand
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
Highly Plastic Clay (CH)
0.65
0.80
1.10
1.40
Soil Type
4.3.3.2 ACTIVE PRESSURE COEFFICIENT, ka When a retaining wall deflects laterally in response to loading by the retained earth, a wedge of the retained soil moves laterally and downward along the back of the wall, as shown in Figure 4.3.3.2-1.
Lecture - 4-16
printed on June 24, 2003
Figure 4.3.3.2-1 - Active Failure Wedge for Conventional Gravity and Cantilever Retaining Walls, Coulomb Analysis As the soil wedge moves, the shear strength of the soil is gradually mobilized along the failure plane shown in Figure 4.3.3.21. When the full shear strength of the soil is mobilized, the additional force required to maintain the stability of the wedge (and the corresponding force acting on the wall) reaches a minimum value equal to the active earth pressure, P a. Prior to wall movement and mobilization of shear strength in the soil, the wall must support the at-rest earth pressure (Article 4.3.3.1). One of two theories is generally used to estimate the active earth pressure on retaining walls. Rankine earth pressure theory neglects the vertical friction force applied to the surface of the wall by the retained soil wedge as it moves downward along the back of the wall. For the Rankine earth pressure theory, the active earth pressure resultant is assumed to have a line of action parallel to the backfill surface. Coulomb earth pressure theory accounts for the friction force exerted on the wall by the retained earth, which results in an inclination of the earth pressure re sultant of δ with respect to the back face of the wall (for a gravity wall) or to the vertical pressure surface extending up from the heel of the wall (for a cantilever wall), as shown in Figure 4.3.3.2-1. Typical values of δ,
Lecture - 4-17
printed on June 24, 2003 the friction angle between the wall and backfill, are presented in Table 4.3.3.2-1.
Table 4.3.3.2-1 - Friction Angles Between Dissimilar Materials Interface Materials
Friction Angle, δ (deg)
Mass concrete on the following foundation materials: • 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
35 29 to 31 24 to 29
19 to 24 17 to 19 22 to 26 17 to 19
Masonry on foundation materials has same friction factors 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
Lecture - 4-18
printed on June 24, 2003 Interface Materials
Friction Angle, δ (deg)
Formed or precast concrete or concrete sheet 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) and the LRFD Specification (AASHTO 1993) employ the Coulomb ear th pressure theory. For the typical case when a retaining wall is permitted to deflect sufficiently to develop the active state of stress in the retained soil, the lateral earth pressure coefficient (k h) in Equation 4.8-1 is, therefore, taken as the Coulomb active earth pressure coefficient, k a. For the case of a vertical retaining wall and a horizontal backfill surface, the value of k a can be obtained from Figure 4.3.3.2-2. For theoretical solutions like those shown in Table 1 and Figure 1, the angle of internal friction is denoted simply as φ. The value of φ, shown in these solutions, is to be interpreted as the effective stress friction angle, φf , determined from a drained shear test, when analyses are performed using effective stresses, and the total stress friction angle φ, determined from an undrained shear test, when analyses are performed using total stresses. For longterm conditions, the earth pressures should be calculated using effective stresses, and adding water pressures as appropriate.
Lecture - 4-19
printed on June 24, 2003
Figure 4.3.3.2-2 - Active and Passive Pressure Coefficients for Vertical Wall and Horizontal Backfill - Based on Log Spiral Failure Surfaces For the more general case of an inclined wall face and sloping backfill surface, the value of the k a can be obtained from Table 4.3.3.2-2.
Lecture - 4-20
printed on June 24, 2003 Table 4.3.3.2-2 - Value of k a for Log Spiral Failure Surface φ (DEG) δ (DEG)
0
φ
i (DEG)
β (DEG)
20
25
30
35
40
45
-15
-10 0 10
0.37 0.42 0.45
0.30 0.35 0.39
0.24 0.29 0.34
0.19 0.24 0.29
0.14 0.19 0.24
0.11 0.16 0.21
0
-10 0 10
0.42 0.49 0.55
0.34 0.41 0.47
0.27 0.33 0.40
0.21 0.27 0.34
0.16 0.22 0.28
0.12 0.17 0.24
15
-10 0 10
0.55 0.65 0.75
0.41 0.51 0.60
0.32 0.41 0.49
0.23 0.32 0.41
0.17 0.25 0.34
0.13 0.20 0.28
-15
-10 0 10
0.31 0.37 0.41
0.26 0.31 0.36
0.21 0.26 0.31
0.17 0.23 0.27
0.14 0.19 0.25
0.11 0.17 0.23
0
-10 0 10
0.37 0.44 0.50
0.30 0.37 0.43
0.24 0.30 0.38
0.19 0.26 0.33
0.15 0.22 0.30
0.12 0.19 0.26
15
-10 0 10
0.50 0.61 0.72
0.37 0.48 0.58
0.29 0.37 0.46
0.22 0.32 0.42
0.17 0.25 0.35
0.14 0.21 0.31
Studies have shown that the failure surface defining the soil wedge loading the wall is approximated more closely by a log spiral curve than a straight line. The values of ka, provided in Figure 4.3.3.2-1 and Table 4.3.3.2-2, were, therefore, obtained from analyses using log spiral surfaces (Caquot and Kerisel, 1948). Both the AASHTO Standard and LRFD Specifications also provide guidance for estimating earth pr essures on special types of earth retaining structures (e.g., non-gravity cantilevered walls, anchored walls and mechanically stabilized earth walls) and walls subjected to unusual loading conditions (e.g., passive earth pressures). Where unusual backfill geometries or surcharge conditions exist, the active pressure may be estimated using a graphical trial wedge procedure.
Lecture - 4-21
printed on June 24, 2003 4.3.3.3 EQUIVALENT FLUID PRESSURE For simplicity, lateral earth pressure is often estimated as an equivalent fluid pressure, wherein the resultant of the earth pressure is equivalent to the resultant of a fictitious fluid exerting hydrostatic pressure on the wall. Where equivalent fluid pressure is 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/m 3)
Typical values of γeq for design of walls up to 6.5 m in height are shown in Table 4.3.3.3-1. Table 4.3.3.3-1 - Typical Values for Equivalent Fluid Unit Densities of Soil Type of Soil
Backfill with i = 25 Level Backfill
At-Rest γeq (kg/m3)
Active ( ∆/H = 1/240) γeq (kg/m3)
At-Rest γeq (kg/m3)
Active ( ∆/H = 1/240) γeq (kg/m3)
Loose sand or gravel
880
640
1040
800
Medium dense sand or gravel
800
560
960
720
Dense sand or gravel
720
480
880
640
Compacted silt (ML)
960
640
1120
800
Compacted lean clay (CL)
1120
720
1280
880
Compacted fat clay (CH)
1280
880
1440
1040
The resultant lateral earth load due to the equivalent fluid pressure is assumed to act at a height of 0.4H above the base of Lecture - 4-22
printed on June 24, 2003 the wall footing for conventional gravity and semi-gravity retaining walls. 4.3.4 Presence of Water If soil mass retained by a retaining wall or abutment contains groundwater and the groundwater is not eliminated through drainage, a water table will develop behind and exert lateral pressure on the structure above the water table, the horizontal pressure is given as: P = kh γ g z 10 -9
(4.3.4-1)
The presence of the water table behind the wa ll has two additional effects, as indicated below and in Figure 4.3.4-1. •
The unit weight of the retained soil is reduced to its submerged or buoyant value: γ s = γs-γw
(4.3.4-2)
As a result, the lateral earth pressure below the water table is reduced to: P = (kh γs (z-zw) + kh γ's z) g 10 -9 •
(4.3.4-3)
The retained water exerts a horizontal hydrostatic water pressure equal to: Pw = γw g zw 10-9
(4.3.4-4)
where: =
submerged unit density of soil (kg/m3)
γs
=
total unit density of soil (kg/m 3)
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)
γ
s
Lecture - 4-23
printed on June 24, 2003
Figure 4.3.4-1 - Effect of Groundwater Table on Earth Pressure With some algebra, the equations for pressure below the groundwater 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 z w above the design section defined by z and zw. The location of the resultant of the term γw zw is taken as zw/3 above the design section. Where possible, the development of hydrostatic water pressures should be prevented through the use of free-draining granular backfill and/or by providing a positive means of backfill drainage, such as weep holes, perforated and solid pipe drains, gravel drains or geofabric drains. When groundwater levels differ on opposite sides of a retaining wall, seepage occurs beneath the wall. The effect of seepage forces is to increase the load on the back of the wall (and decrease any passive resistance in front of the wall). Pore pressures in the backfill soil can be approximated through development of a flow net or using various analytical methods, and must be added to the effective horizontal earth pressures to determine the total lateral pressures on the wall. 4.3.5 Surcharge Surcharge loads on the retained earth surface produce additional lateral earth pressure on retaining walls. Where the surcharge is uniform over the retained earth surface, the additional lateral earth pressure due to the surcharge is assumed to remain constant with depth and has a magnitude, ∆p, of: ∆p = ks qs
(4.3.5-1)
where:
Lecture - 4-24
printed on June 24, 2003 ∆p
=
constant horizontal earth pressure due to uniform surcharge (MPa)
ks
=
coefficient of earth pressure due to surcharge
qs
=
uniform surcharge applied to the upper surface of the active earth wedge (MPa)
For active earth pressure conditions, ks is taken as k a, and for at-rest conditions, k s is taken as k o. Where vehicular traffic is anticipated within a distance behind a wall equal to about the wall height, a live load surcharge is assumed to act on the retained earth surface. The uniform increase in horizontal earth pressure due to live load surcharge is typically estimated as: ∆p = ks γsg heq 10-9
(4.3.5-2)
where: ∆p
=
constant horizontal earth pressure due to uniform surcharge (MPa)
γs
=
unit density of soil (kg/m 3)
k
=
coefficient of earth pressure
heq
=
equivalent height of soil for the design live load (mm)
Equivalent heights of soil, h eq, for highway loading on retaining walls of various heights can be taken from Table 4.3.5-1. Table 4.3.5-1 - Equivalent Height of Soil for Vehicular Loading Wall Height (mm)
heq (mm)
1500
1700
3000
1200
6000
760
9000
610
The tabulated values of h eq were determined based on evaluation of horizontal pressure distributions produced on retaining walls by the vehicular live loads specified in the LRFD Specification (AASHTO 1993). The pressure distributions were obtained from Lecture - 4-25
printed on June 24, 2003 elastic half-space solutions with Poisson's ratio of 0.5, doubled to account 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 be taken as:
∆ p
2 p π
(α
sin α cos(α
2 δ))
(4.3.5-3)
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 Uniformly Loaded Strip 4.3.6 Effect of Earthquake Lateral earth pressures on retaining structures are amplified during an earthquake due to horizontal acceleration of the retained earth mass. The Mononobe-Okabe method of analysis is a pseudo-static method which is commonly used to develop an Lecture - 4-26
printed on June 24, 2003 equivalent static fluid pressure to model seismic earth pressure on retaining walls. The Mononobe-Okabe method is contingent upon the following assumptions (Barker, et al, 1991): •
The wall is unrestrained and capable of deflecting sufficiently to mobilize the active earth pressure condition in the backfill;
•
The backfill is cohesionless and unsaturated;
•
The failure surface defining the active wedge of soil loading the wall is planar; and
•
Accelerations are uniform throughout the soil mass.
As indicated above, the Mononobe-Okabe method assumes that backfill soils are unsaturated and, as such, not susceptible to liquefaction. The potential for liquefaction should be evaluated where saturated soils may be subjected to earthquake or other cyclical or instantaneous dynamic loadings. The Mononobe-Okabe method (AASHTO 1992 and AASHTO 1993) considers equilibrium of the soil wedge retained by the 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 earth pressures is taken as: P AE = 0.5 x K AE γ g (1-kv) H2 10-9
(4.3.6-1)
where the seismic active earth pressure coefficient, K AE, is:
Lecture - 4-27
printed on June 24, 2003 2 2
(φ θ β)
×1
β cos(δ β θ)
sin(φ δ )s cos(δ β θ
(4.3.6-2) where: γ
=
unit density of soil (kg/m 3)
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 to Article 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 earthquake conditions, the vertical acceleration coefficient, k v, is commonly assumed equal to zero, and the horizontal acceleration coefficient, kh, is taken as: kh = 0.5 α
(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 - 4-28
printed on June 24, 2003 Values of the earthquake acceleration coefficient, A, are shown in Figures S3.10.2-1, S3.10.2-2 and S3.10.2-3. Although the Mononobe-Okabe method of analysis does not specify the point of application of the horizontal seismic earth pressure resultant, the resultant of the dynamic component of the earth pressure ( ∆P AE) is significantly higher on the wall than the static active earth pressure resultant (P a), as indicated in Figure 4.3.6-1. Both the AASHTO Standard Specifications (AASHTO 1992) and the LRFD Specifications (AASHTO 1993) indicate that the combined static and seismic lateral earth pressure can be assumed to be uniformly distributed with a resultant (P AE) acting at the mid-height of the wall. The AASHTO Standard Specification and the LRFD Specification also provide guidelines for determination of passive seismic earth pressure on retaining structures which are being forced horizontally into the backfill, and methods for design of retaining structures for a limited tolerable displacement under earthquake loading rather than for zero permanent displacement as assumed in the Mononobe-Okabe method. As with many current methods of seismic analysis, the Mononobe-Okabe method neglects inertial forces due to the mass of the retaining structure, concentrating only on the inertia of the retained soils mass. For gravity structures which rely solely on their mass for stability, this assumption is unconservative. The effects of wall inertia on the behavior of gravity retaining walls is discussed further by Richards and Elms (1979), who conclude that the structure inertia forces should not be neglected. Richards and Elms suggest a design approach based on limiting wall displacements to tolerable levels, rather than designing for no movement, and computing the weight of wall required to prevent movements greater than specified. The work of Richards and Elms is incorporated into the tolerable displacement procedure presented in the AASHTO Specifications. 4.3.7 Reduction due to Earth Pressure In some cases, lateral earth pressures may reduce the effects of other loads and forces on culverts, bridges and their components. One such case is that of the top slab of a box culvert, for which the maximum bending moment in the top slab is redu ced due to the effect of lateral earth pressure on the side walls. Such reductions in loading should be limited to the extent that the applied earth pressures can be expected to be permanent. In lieu of more precise methods of estimating the lateral earth pressure force effects, the AASHTO Standard Specification (AASHTO 1992) and the LRFD Specification (AASHTO 1993) permit a load effect reduction of 50% where earth pressures are present. Lecture - 4-29
printed on June 24, 2003 4.3.8 Downdrag When a point bearing pile or drilled shaft penetrates a soft layer subject to settlement (e.g., where an overlying embankment may cause settlement of the layer), the soil settling around the shaft exerts a frictional force, or downdrag, around the perimeter of the pile or shaft. This frictional force acts as an additional axial load on the pile or shaft and, if sufficient in magnitude, could cause a structural failure of the foundation element or a bearing capacity failure at the tip. The methods used to estimate downdrag loads are the same as those used to estimate the side resistance of shafts and piles due to skin friction, as the same load transfer mechanism is responsible for both. 4.3.9 Design of a Cantilever Retaining Wall The purpose of this example is to illustrate the application of the various load factors and load combinations. In order to fully develop the example, references to provisions for wall and footings, and references to geotechnical textbooks are required and so noted. Some rounding of numbers has been done so it may not be possible to exactly duplicate all values to the full precision shown. The cantilever retaining wall below has been proposed to support an embankment.
Lecture - 4-30
printed on June 24, 2003
Figure 4.3.9-1 - Schematic of Example During the subsurface exploration, it was determined that the foundation soils are predominantly clay to a depth of 6 m below the proposed bottom of footing, and, therefore, a 150 mm blanket of compacted granular material was placed below the footing. Dense sand and gravel underlies the clayey foundation soils. Assume elastic settlement of the dense sand and gravel to be negligible. The proposed wall backfill will consist of a free draining gr anular fill. Assume the seasonal high water table to be at the bottom of the footing. Apply the vehicular live load surcharge (LS) on the backfill as indicated in the figure above. Determine the lateral pressure distribution on the wall and estimate the bearing capacity for the proposed design. Check the design for adequate protection against sliding and estimate the consolidation settlement of the underlying clay. Solution: Step 1: Calculate the Unfactored Loads with η = 1.0 (A) Dead Load of Structural Components and Nonstructural Attachments (DC) Unit Density of Concrete = 2,400 kg/m 3
Lecture - 4-31
printed on June 24, 2003
Figure 4.3.9-2 - Retaining Wall Area Designation Weight of Concrete W1 = (0.3)(4.5)(2,400)g = 31,784 N per m of length W2 = (0.5)(4.5)(0.2)(2,400)g = 10,595 N per m of length W3 = (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 vehicular load is expected to act on the backfill within a distance equal to the wall 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 wall height of 5 m.
Lecture - 4-32
printed on June 24, 2003 Table 4.3.9-1 - Equivalent Height of Soil for Vehicular Loading Wall Height (mm) 1500
heq (mm) 01700
3000
1200
6000
760
heq = 907 mm Use soil density of the backfill = 1920 kg/m 3 Vertical Component of LS = (1920)(907)(9.81) x 10-9 m3/mm3 = 0.0171 MPa For a width of 2m, P LSV = 34,167 N per m of length Assume an active earth pressure coefficient ka using Figure 4.3.9-3 with a wall friction angle, δ = φ, repeated below.
Lecture - 4-33
printed on June 24, 2003
Figure 4.3.9-3 - Active and Passive Pressure Coefficients for Vertical Wall and Horizontal Backfill - Based on Log Spiral Failure Surfaces k = ka = 0.29 using Equation S3.11.6.2-1:
Lecture - 4-34
printed on June 24, 2003 ∆p = (k) (γ'1) (g) (heq) (10-9) ∆p = 0.0050 MPa
Using a rectangular distribution, the live load horizontal earth pressure 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 linearly proportional to the depth of earth and is given by Equation S3.11.5.1-1: 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 wall will 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 V N/m
Arm about 0 m
Moment 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
Item
TOTAL 281,379
Lecture - 4-35
printed on June 24, 2003 Table 4.3.9-3 - Horizontal Loads and Overturning Moments Moment - Unfactored H N/m
Arm about 0 m
Moment about 0 N-m/m
PLSH
24,771
2.5
61,928
PEH
68,278
2.0
136,555
Item
For location of resultant for lateral earth pressure see S3.11.5.1. Step 2: Determine the Appropriate Load Factors Using Tables S3.4.1-1 and S3.4.1-2 given in this manual as Tables 2.4.1.2-1 and 2.4.1.2-2, respectively: Table 4.3.9-4 - Load Factors
Group
DC
EV
LSv
PLSH
EH (active)
Strength I-a
0.9
1.0
1.75
1.75
1.5
Bearing Capacity (eccent.)&Sliding
Strength I-b
1.25
1.35
1.75
1.75
1.5
Bearing Capacity (max. value)
Strength IV
1.5
1.35
-
-
1.5
Bearing Capacity (max. value)
Service I
1.0
1.0
1.0
1.0
1.0
Settlement
Probable Use
By inspection, the following conclusions can be drawn for the case of bearing capacity (maximum) Strength I-b will probably govern, however, the factored loads will also be checked for Strength IV.
Lecture - 4-36
printed on June 24, 2003 Step 3: Calculate the Factored Loads Table 4.3.9-5 - Factored Vertical Loads Group/ Item Units
W1 N/m
W2 N/m
W3 N/m
PEV N/m
PLSV N/m
Total N/m
V (Unf.)
31,784
10,595
35,316
169,517
34,167
281,379
Strength I-a
28,606
9,535
31,784
169,517
59,792
299,235
Strength I-b
39,731
13,244
44,145
228,848
59,792
385,759
Strength IV
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 M v Group/ Item Units
W1 N-m/m
W2 N-m/m
W3 N-m/m
PEV N-m/m
PLSV N-m/m
Total N-m/m
27,017
6,710
52,974
339,034
68,334
494,068
Strength I-a
24,315
6,039
47,677
339,034
119,585
536,649
Strength I-b
33,771
8,388
66,218
457,695
119,585
685,656
Strength IV
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
Mv (Unf.)
Lecture - 4-37
printed on June 24, 2003 Table 4.3.9-7 - Factored Horizontal Loads Group/Item Units
PLSH N/m
PEH N/m
Total N/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
24,771
68,278
93,049
Service I
Table 4.3.9-8 - Factored Moments M h Group/Item Units
PLSH N-m/m
PEH N-m/m
Total N-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
61,928
136,555
198,483
Service I
This example will be continued in Lecture 15 in which a foundation design, eccentricity check, sliding check, and settlement calculation will be added.
Lecture - 4-38
printed on June 24, 2003 REFERENCES Clough, G.W., and J.M. Duncan, 1991, "Earth Pressures," Chapter 6, 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 Pressures Measured Against a Section of a Basement Wall," Proceedings, 5th European Conference on Soil Mechanics and Foundation Engineering , Madrid, pp. 515-516. Sherif, M.A., I. Ishibashi and C.D. Lee, 1982, "Earth Pressures Against Rigid Retaining Walls," Journal of Geotechnical Engineering , ASCE, Vol. 108, GT5, pp. 679-695. Terzaghi, K., 1934, "Retaining Wall Design for Fifteen-Mile Falls Dam, Engineering News Record , May, pp. 632-636. Barker, R.M., J.M. Duncan, K.B. Rojiani, P.S.K. Ooi, C.K. Tan and S.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 Transportation Officials, 1992, "Standard Specifications for Highway Bridges," Fifteenth Edition, AASHTO, Washington, D.C., 1992. Richards, R., and D.G., Elms, 1979, "Seismic Behavior of Gravity Retaining Walls," Journal of the Geotechnical Engineering Division , ASCE, Volume 105, No. GT4. Caquot, A. and J. Kerisel, 1948, "Tables for the Calculation of Passive Pressure, Active Pressure and Bearing Capacity of Foundations," Gauthier-Villars, Imprimeur-Libraire, Libraire du Bureau des Longitudes, de L'Ecole Polytechnique, Paris, 120 pp.
Lecture - 4-39
printed on June 24, 2003 LECTURE 5 - LOADS III
5.1 OBJECTIVE OF THE LESSON The objective of this lesson is to continue to provide a student with the background for the provisions in Section 3, Loads and Load Factors, of the AASHTO LRFD Specification. The lesson includes: •
the forces due to superimposed deformations resulting from uniform and nonuniform temperature changes, differential settlement, 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 and shrinkage shall be considered. The effect of temperature gradient should be included where appropriate. Force effects resulting from resisting component deformation, displacement of points of load application and support movements is included in the analysis. 5.2.1 Uniform Temperature In the absence of more precise information, the ranges of temperature shall be as specified in Table 5.2.1-1. The difference between the extended lower or upper boundary and the base construction temperature assumed in the design is used to calculate thermal deformation effects. Table 5.2.1-1 - Temperature Ranges STEEL OR CLIMATE ALUMINUM
CONCRETE
WOOD
Moderate
-18 to 50 C
-12 to 27 C
-12 to 24 C
Cold
-35 to 50 C
-18 to 27 C
-18 to 24 C
A moderate climate may be determined by the number of freezing days per year. If the number of freezing days is less than 14, the climate is considered to be moderate. Freezing days are days when the average temperature is less than 0 C.
Lecture - 5-1
printed on June 24, 2003 The actual air temperature averaged over the 24 hour period immediately preceding the setting event can be used in installing expansion bearings and deck joints. 5.2.2 Temperature Gradient The load factor for temperature gradient should be determined based on: •
the type of structure, and
•
the limit state being investigated.
There is general agreement that in situ measurements of temperature gradients have yielded a realistic distribution of temperatures through the depth of some types of bridges, most notably concrete box girders. There is very little agreement on the significance of the effect of that distribution. It is generally acknowledged that cracking, yielding, creep and other non-linear responses diminish the effects. Therefore, load factors of less than 1.0 should be considered, and there is some basis for lower load factors at the strength and extreme event limit states than at the service limit state. Similarly, open girder construction and multiple steel box girders have traditionally, but perhaps not necessarily correctly, been designed without consideration of temperature gradient, i.e., γTG = 0.0. Temperature gradient is included in various load combinations in Table S3.4.1-1. This does not mean that it need be investigated for all types of structures. If experience has shown that neglecting temperature gradient in the design of a given type of structure has n ot lead to structural distress, the Owner may choose to exclude temperature gradient. Multi-beam bridges are an example of a type of structure for which judgment and past experience should be considered. The vertical temperature gradient in concrete and steel superstructures with concrete decks may be taken as shown in Figure 5.2.2-1. The dimension "A" in Figure 5.2.2-1 shall be taken as: •
300 mm for concrete superstructures, that are 400 mm or more in depth
•
for concrete sections shallower than 400 mm, "A" shall be 100 mm less than the actual depth
•
300 mm for steel superstructures, where t = depth of the concrete deck
Lecture - 5-2
printed on June 24, 2003 Temperature value T3 shall be taken as 0 C, unless a sitespecific study is made to determine an appropriate value, but shall not exceed 3 C.
Figure 5.2.2-1 - Positive Vertical Temperature Gradient in Concrete and Steel Superstructures This temperature gradient given herein is a modif ication of that proposed in Imbsen (1985) which was based on studies of concrete superstructures. The addition for steel superstructures is patterned after the temperature gradient for that type of bridge in the Australian bridge specifications, Austroads (1992). Temperatures for use with Figure 5.2.2-1 may be taken from Table 5.2.2-1.
Lecture - 5-3
printed on June 24, 2003 Table 5.2.2-1 - Basis of Temperature Gradients
Zone
T1 ( C)
T2 ( C)
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 as specified for various deck surface conditions in Table 5.2.2-1. Negative temperature values shall be obtained by multiplying the values specified in Table 5.2.2-1 by -0.3 for decks with the concrete top surface exposed and -0.2 for decks with an asphalt overlay. The temperatures given in Table 5.2.2-1 form the basis for calculating the change in temperature with depth in the cross-sect ion, not absolute temperature. Where temperature gradient is considered, internal stresses and structure deformations due to both positive and negative temperature gradients may be determined by dividing into three effects as follows: •
AVERAGE AXIAL EXPANSION - This is due to the uniform component of the temperature distribution which should be
Lecture - 5-4
printed on June 24, 2003 considered simultaneously with the uniform temperature specified in Article S3.12.2. It may be calculated as: 1 Ac
T UG
T G dw dz
(5.2.2-1)
The corresponding total uniform axial strain from both Tu and TG is: εu
•
α [T UG
T u ]
(5.2.2-2)
FLEXURAL DEFORMATION - Since plane sections remain plane, a curvature is imposed on the superstructure so as to accommodate the linearly variable component of the temperature gradient. The rotation per unit length corresponding to this curvature may be determined as: φ
α I c
T G z d w d z
1 R
(5.2.2-3)
If the structure is externally unrestrained, i.e., simply supported or cantilevered, no external force effects are developed due to this superimposed deformation. The axial strain and curvature may be used in both flexibility and stiffness formulations. In the former, εu may be used in place of P/AE, and φ may be used in place of M/EI in traditional displacement calculations. In the latter, the fixedend force effects for a prismatic frame element may be determined 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 creep may be treated in a similar manner. •
INTERNAL STRESS - Internal stresses in addition to those corresponding to the restrained axial expansion and/or rotation may be calculated as: σE = E [α TG - α TUG - φz]
(5.2.2-6)
where: TG
=
temperature gradient ( ∆ C)
TUG
=
temperature averaged across the cross-section ( C)
Lecture - 5-5
printed on June 24, 2003 Tu
=
uniform specified temperature ( C)
Ac
=
cross-section area - transformed for steel beams (mm2)
Ic
=
inertia of cross-section - transformed for steel beams (mm4)
α
=
coefficient of thermal expansion (mm/mm/ C)
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 crosssection (mm)
Note that a positive value of σE in Equation 5.2.2-6 denotes compression. For example, the flexural deformation part of the gradient flexes a prismatic superstructure into a segment of a circle in the vertical plane. For a two-span structure with span length L, in mm, the unrestrained beam would lift off from the central support by ∆ = L2/2R mm. Forcing the beam down to eliminate ∆ would develop a moment whose value at the pier would be:
M c
3 E I c φ 2
(5.2.2-7)
Therefore, the moment is the function of the beam rigidity and imposed flexure. As rigidity approaches 0.0 at the strength limit state, Mc tends to disappear. This behavior also indicates the need for ductility which ensures structural integrity as rigidity decreases. A "common sense" explanation of the stress given by Equation 5.2.2-6 follows. Consider the effect of a temperature gradient on a set of disconnected layers within the structure. Each layer will expand an amount required by the temperature gradient.
Lecture - 5-6
printed on June 24, 2003 Equation 5.2.2-6 can be developed in two steps. Step 1:
∆1
Consider only the layer restraint stresses. illustration, consider the top layer:
For
T G L α
But, the layers are not really disconnected, so apply a force to each layer to make the displacement equal to zero, with compression taken as positive. P 1 σ
∆1 A1 E
L T G α E
T G1 A1 E α
But, P1 + P2 + P3 ... cannot result in a net force on the unrestrained end. Similarly, ΣPiYiCG cannot result in a net external moment. Now apply P
ΣP i & M
ΣP i Y iCG as notional external loads to
result in a net external axial load and moment is zero. The total effect is then:
So far, even for a simply supported beam: P i ΣP i y ΣP i y i σi Ai A I σ
T GαE
1 T GαEdA A
1 T GαEydA I
Since these are notional force effe cts, they appear to exist even at the free edge of the beam. Step 2:
!
Add the effect of restrained indetermanent reactions, if any, caused by the global end rotation in the redundant structure.
Apply end rotations as shown below:
Lecture - 5-7
printed on June 24, 2003 !
The moment diagram with redundant reactions removed is a rectangle, i.e., constant moment from end-to-end. This moment results in a displaced shape as shown below.
!
Calculate reactions required to have no displacement at redundant supports
!
The moment diagram due to only the redundnat reactions is shown below:
!
The force effects due to redundant reactions is added to the force effects obtained in Step 1.
Alternatively, if the axial force and moment given by Equations 5.2.2-4 and 5.2.2-5 are used as input into a structural analysis package, the stresses from those two factors and the redundant reaction are calculated directly, and only the first term of Equation 5.2.2-6 need be added to those results to get a complete solution. 5.2.3 Differential Shrinkage Where appropriate, differential shrinkage strains between concretes of different age and composition, and between concrete and steel or wood, are determined in accordance with the provisions of Section S5. The designer may specify timing and sequence of construction in order to minimize stresses due to differential shrinkage between components. 5.2.4 Creep Creep strains for concrete and wood are given by the provisions of Section S5 and Section S8, respectively. Traditionally, only creep of concrete is considered. Creep of wood is addressed only because it applies to prestressed wood decks. In determining force effects and deformations du e to creep, dependence on time and changes in compressive stresses shall be taken into account.
Lecture - 5-8
printed on June 24, 2003 5.2.5 Settlement Force effects due to extreme values of differential settlements among substructure and within individual substructures units is considered. Estimates of settlement may be made in accordance with the provision of Article S10.7.2.3. Force effects due to settlement may be reduced by considering creep.
5.3 OTHER LIVE LOAD EFFECTS 5.3.1 General Lecture 3 contained information on the development of the notional live load model. This lesson focuses on other aspects of the live load, other than the weight and axle configuration of the notional load, per se. 5.3.2 Centrifugal Force The only addition to centrifugal force, compared to previous editions of Standard Specification, is the inclusion of the factor 4/3 in Equation S3.6.3-1. As explained in the commentary, the group of exclusion vehicles, described in detail in Lecture 3, produced force effects which are generally at least 4/3 of that caused by the design truck alone on short- to medium-span bridges. This is the origin of the 4/3 factor in Equation S3.6.3-1. It is an approximation used to account for vehicles whose gross vehicle weight is greater than the 325 kN design truck. Centrifugal force is applied to the design truck or to the tandem axle, but not to the lane load on the basis that the lane load is used to account for disbursed traffic, which normally does not contribute significantly to a single pier, except on longer spans. Since centrifugal force is applied to all loaded lanes, the multiple presence factors apply. 5.3.3 Braking Force The braking force requirement of Article S3.6.4 is 25% of the axle weights or the design tandem in each lane with a multiple presence factor applied. This is substantially greater fraction than previous editions of the standard specifications. However, it is only applied to the design load or design tandem, not the uniform load. The 25% specified is based on improv ed braking capability of modern trucks. The commentary to this article indicates the set of parameters from which the 25% was developed, an exercise in engineering judgment. The braking force is not applied to the lane load on the basis that vehicles are apt to break out-of-phase.
Lecture - 5-9
printed on June 24, 2003 5.3.4 Vehicular Collision Forces Portions of structures deemed to be protected are not required to be designed for vehicular collision loads. Structures which fall into this category are those which are protected by an embankment, a structurally independent crashworthy ground-mounted barrier, as defined in Article S3.6.5.1, or a properly designed barrier positioned to protect the structural elements. Barriers are to be designed for the forces indicated in Section 13, Railings. Vehicular and railroad collision loads are defined as a static equivalent force of 1800 kN acting at a height 1200 mm above the ground. This force was developed from information from full-scale tests involving the redirection of tractor trailer trucks by barriers and from analytical work identified in the commentary.
5.4 WATER LOADS There are no substantially new requirements for static pressure, buoyancy or stream pressure. The issue of debris moving in a stream and impacting piers, or even elements of superstructure, is the subject of an NCHRP Research Project, just getting underway at this writing (Spring 1994). Wherever possible, the amount of freeboard on superstructure and the spacing of piers should be established to permit reasonable floating debris, determined on the basis of review of the flood plain, from impacting the superstructure. Floating debris does tend to collect around piers , forming what is called a debris raft. Debris rafts tend to increase the apparent area of the pier subject to stream pressure. Pending future U. S. research on the design parameters for debris rafts, the commentary contains a provision from the New Zealand Highway Bridge Design Specification which may be used until better data becomes available. In addition to the pressure, identified above, the Specification contains a requirement to design for wave loads where structures are exposed to that type of environment. Reference is given in the commentary to the Shore Protection Manual as a source for wave design information. Also, new to the Specification is a requirement to design the foundations for scour. Scour is a change in foundation condition resulting from the design flood for scour and is applicable as both a requirement for strength and service limit states, but also as an extreme event limit state under the check flood for scour, usually a 500-year flood. In this case, the requirement is that the structure should remain stable and intact at its full nominal resistance.
Lecture - 5-10
printed on June 24, 2003 5.5 WIND LOADS 5.5.1 General Wind Provisions Article S3.8 establishes wind loads which are consistent with the format and presentation currently used in meteorology. Wind pressures are established with consideration to a base wind velocity (Vb) of 160 km per hour corresponding to the 100 mph wind common in past specifications. If no better information is available, the wind velocity at 10 000 mm above the ground, a distance presumed to be above the immediate effects of the ground in open terrain, may be taken as the base wind Vb. Alternatively, the base wind speed may be taken from the Basic Wind Speed Charts available in the literature or site specific wind surveys may be used to establish V10. Using characteristics available in Table S3.8.1.1-1, repeated below, used to describe the type of terrain over which approach winds move, two characteristic values, V0 and Z0, can be determined. These are meteorological 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 t he height of the structure above ground or water, if it is over 10 000 mm, it is possible to calculate a design wind velocity, Vdz, using Equation S3.8.1.1-1, which is repeated below. V DZ 2.5 V 0
V 10 V B
ln
Z Z o
(5.5.1-1)
Equation S3.8.1.1-1 provides a correction for structure elevation with similar intent to that used by the 1/7 power rule used by designers in the past, but agrees with the current meteorological theories. Given the design wind speed, now corrected for approach conditions and height above the referenced datum o f 10 000 mm, it is possible to calculate the design wind pressure, PD, based on base pressure, PB, given in Table S3.8.1.2-1, repeated below, for various structural components. The base wind pressures, specified in that table, are established for the case where VB is equal to 160 km per hour. The base wind velocity and the wind velocity at the elevation in question are used to customize the base wind pressures given in Table S3.8.1.2-1 for the particular site conditions. Additionally, certain minimum design wind pressures, comparable to those in past editions of the Standard Specification, are also required.
Lecture - 5-11
printed on June 24, 2003 PB PB WINDWARD LOAD, LEEWARD LOAD, MPa MPa
STRUCTURAL COMPONENT Trusses,Column and Arches
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 with height for the three upwind conditions.
0.004 a 0.003 P M0.002 D0.001 P
country PD Suburban City
0 0 0 0 8
0 0 0 2 1
0 0 0 6 1
0 0 0 0 2
0 0 0 4 2
0 0 0 8 2
Z(mm)
Figure 5.5.1-1 Design Wind Pressure, PD, Vs. Height for PB = 0.0024 MPa
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 = 0.0024 MPa
•
Assume Z = 19 000 mm V DZ 2.5 V o
V 10 V B
V DZ 2.5(15.2)
n
160 160
Z Z o n
19 000 300
Lecture - 5-12
printed on June 24, 2003 V DZ 158
P D
km hr
0.0024
(158)2 (160)
2
0.002 34 MPa
Wind pressure is also applied to vehicular live load, as was the case in past editions of the Specification. No substantial changes in this provision have been made. 5.5.2 Wind Blowing at an Angle Where the wind is not taken as normal to the structure, the base wind pressures, PB, for various angles of wind direction may be taken as specified in the table below and shall be applied to a single place of exposed area. The skew angle is measured from a perpendicular to the longitudinal axis of the bridge. The wind direction for design shall be that which produces the ext reme force effect on the component under investigation. The transverse and longitudinal pressures are to be applied simultaneously. For trusses, columns, and arches, the base wind pressures specified in the table are the sum of the pressures applied to bot h the windward and leeward areas. Base Wind Pressures, PB, for Various Angles of Attack and VB = 160 km/hr Columns and Arches
Girders
Skew Angle of Wind
Lateral Load
Longitudinal Load
Lateral Load
Longitudinal Load
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 to the substructure shall be calculated from an assumed base wind pressure of 0.0019 MPa. For wind directions taken skewed to the substructure, this force shall be resolved into components perpendicular to the end and front elevations of the substructure. The component perpendicular to the end elevation shall act on the exposed substructure area as seen in end elevation, and the component perpendicular to the front elevation shall act on the
Lecture - 5-13
printed on June 24, 2003 exposed areas and shall be applied simultaneously with the wind loads from the superstructure. 5.5.4 Wind Pressure on Vehicles When vehicles are present, the design wind pres sure shall be applied to both structure and vehicles. Wind pressure on vehicles shall be represented by an interruptible, moving force of 1.46 N/mm acting normal to, and 1800 mm above, the roadway and shall be transmitted 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 load may be taken as specified in the table below with the skew angle taken as referenced normal to the surface. Wind Components on Live Load
Skew Angle
Normal Componen t
Parallel Component
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 1/4 point vertical load, and this requirement is continued in the current Specification. The purpose of this requirement is to account for the change in pressure caused by the interruption of a horizontal wind stream created by the bridge superstructure. 5.5.6 Aeroelastic Stability The provisions of this article require that components whose span to width or depth ratio exceeds 30 be consi dered wind-sensitive and that aeroelastic effects should be taken into account for this type of member or structure. The choice of the value of 30 is somewhat arbitrary and was set so that most conventional composite girder construction would not be affected by this provision. This is based solely on experience. There have been cases where girders have vibrated during construction, but this has been relatively rare. All flexible structures or structural components should be investigated for resistance to vortex shedding excitation, wake
Lecture - 5-14
printed on June 24, 2003 buffeting and divergence and flutter, as appropriate. In the past, it has typically been assumed that suspension bridges and cable-stayed bridges were the type of structures for which aeroelastic eff ects could be significant. However, other relatively flexible modern structures may also be susceptible. Similarly, components of structures, notably long, unbraced structural members and cables, may also be subject to wind-induced vibrations. In at least one case, a low amplitude vibration of a tied-arch appeared to be driving distortion-induced fatigue cracking of welds inside the tie girder. In response to these phenomena, the Specification requires that bridges should be designed to be free of divergences and flutter for wind speeds up to 1.2 times the design wind velocity at the bridge deck height, and that structural components and bridges should be free of fatigue damage due to vortex shedding or galloping. It is of some interest to note that the same problems seem to appear over and over again. Consider the four structures discussed below, each of which had hanger vibrations due to vortex shedding. The Tacony-Palmyra Bridge, designed in Circa 1928, has hangers for the deck system which are structural members. These hangers vibrated in the wind, probably to vortex shedding. In this case, the solution was to reduce the length of the member by putting a horizontal strut across the bridge intercepting each of the hangers. The Robert Moses Causeway Bridge, Circa 1958, is a similar configuration with long structural-shape hangers supporting the roadway. These too vibrated in the wind, probably also from vortex shedding. In this case, the members were studies in the wind tunnel, and it was decided to use an aerodynamic solution by attaching sheet metal deflectors to the corners of the I-shaped hangers, thus, streamlining the shape. The deflectors contained a projection at 45 bent into the body of the hanger. This retrofit has been quite effective. The Commodore Barry Bridge, Circa 1970, also has long vertical members supporting the deck which vibrated in the wind and caused significant fatigue damage to these members. In this case, tuned mass dampers were added to the members and their effectiveness was verified in the wind tunnel prior to installation. This retrofit also appears to have been successful. The I-470 Bridge at Wheeling, Circa 1982, used structural strand hangers in groups of four to support the roadway. All these hangers vibrated. On the longer hangers, it was possible to clearly see 7th and 8th load vibrations in moderate winds. This vibration caused damage to some of the wires resulting in wire breaks where the structural strands were supported at the deck level. Wooden wedges driven between the strands and their structural member supports was found to add sufficient damping to stop these vibrations. In this case, retrofit consisted of a st eel and neoprene collar added to the 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 points and to create damping by the action of one cable vibrating against another.
Lecture - 5-15
printed on June 24, 2003 The frequency of the vortex shedding and, henc e, the pulsating pressure, is given by: f
VS D
(5.5.3-1)
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 on Structures", Transactions of the ASCE, Volume 126, Part II, Page 1180, and is repeated below. Table 5.5.3-1 - Strouhal Number for Various Sections Wind
Profile Proportion
Value of S
Profile Proportion
0.120
Value of S 0.200
0.137
0.144
0.145
0.147
Lecture - 5-16
b/d 2.5 2.0 1.5 1.0 0.7 0.5
0.060 0.080 0.103 0.133 0.136 0.138
printed on June 24, 2003 Self-exciting oscillations of the member in the direction perpendicular to the wind stream may result when the frequency of vortex shedding coincides with a natural frequency of the obstruction. Thus, determining the torsional frequency and bending frequency in the plane perpendicular to the wind and substituting those frequencies into the Strouhal equation leads to an estimate of wind speeds at which resonance may occur. This motion has led to fatigue cracking of some truss and arch members, particularly cable hangers and Ishaped members. The vortex shedding design approach, described herein, is oriented towards providing sufficient stiffness to reasonably preclude vibrations. It does not directly lead to a solution for the amplitude of vibration and, hence, it does not directly lead to a solution for vibratory stresses. Solutions for amplitude are available in the literature. The following approximate procedure for estimating bending and torsional frequencies is an excerpt from "Natural Frequencies of Axially Loaded Bridge Members" by C. C. Ulstrup, Journal of the Structural Division, ASCE, 1978. The general approximate formula for members whose shear center and centroid coincide is as follows:
f n
a 2π
k n l l
2
1
ε p
Kl π
2
1 2
(5.5.3-2)
in which: f n
=
natural frequency of member corresponding to n = 1, 2, 3, etc.
for
each
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 the member, given as ab or at
ep
=
coefficient dependent on the physical properties of the member and on the axial force, i.e., positive for tension, negative for compression, given as epb or ept
Lecture - 5-17
mode
printed on June 24, 2003 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: ab e pb
1 2
EIg γ A P EI
(5.5.3-3) (5.5.3-4)
For torsion: at e pt
EC w g
1 2
(5.5.3-5)
γ I p
GJ
PI p A
1
(5.5.3-6)
EC w
in which: E G γ g P I A Cw J Ip
= = = = = = = = = =
Young's modulus shear modulus weight density of member gravitational acceleration axial force (tension is positive) moment of inertia about relevant axis area of member cross-section warping constant torsion constant polar moment of inertia
In the design of a member, the frequency of vortex shedding for the section is set equal with the bending and torsional frequency and the resulting equation solved for the wind speed V. This is the wind speed at which resonance occurs and the design should be such that V exceeds the velocity at which the wind is expected to occur by a reasonable margin.
Lecture - 5-18
printed on June 24, 2003 REFERENCES Committee on Ship-Bridge Collisions, "Ship Collisions with Bridges The Nature of the Accident, their Prevent and Mitigation", Marine Board, Commission on Engineering and Technical Systems, Nati onal Research Council, National Academic Press, Washington, D.C., 1983 Modjeski and Masters, Inc., Consulting Engineers, "Criteria for: The Design of Bridge Piers with Respect to Vessel Collision in Louisiana Waterways", Prepared for the Louisiana Department of Transportation and Development and the Federal Highway Administration, New Orleans, Louisiana, November 1984 American Association of State Highway and Transportation Officials (AASHTO), "Guide Specification and Commentary for Vessel Collision Design of Highway Bridges", 1991, Washington, D. C. Larsen, O. D., "Ship Collision with Bridges - The Interaction between Vessel Traffic and Bridge Structures", International Association for Bridge and Structural Engineering (IABSCE), Structural Engineering Documents, 1993
Lecture - 5-19
printed on June 24, 2003 LECTURE 6 - ANALYSIS I
6.1 OBJECTIVE OF THE LESSON The objectives of this lesson are to acquaint the student with: •
the various analysis techniques required and/or recommended for determining the force effects and components of bridges, and
•
the use of approximate and refined methods for the determination for force effects in conventional girder-type structures.
The background on the development of new, improved, distribution factors which were developed under NCHRP Project 12-26 has been included for reference in an Appendix. The use of grid and finite element types of analysis for multibeam bridges is also recommended in the LRFD Specification. These methods require considerable care in structural modeling, and several examples of the large effects of seemingly small errors in structural models will be presented.
6.2 ACCEPTABLE METHODS OF STRUCTURAL ANALYSIS Article S4.4 contains a list of methods of analysis that are considered suitable for analysis of bridges. These include the classical 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-type methods, analysis based on series expansions and the yield-line method for the non-linear analysis of plates and railings. Some of these methods of analysis are suitable for hand calculations, but for any problem of large size, some sort of computer solution will almost always be required for practical design purposes. This is because almost all of these methods, with the possible exception of the series methods and the yield-line methods, will eventually require the solution of large sets of simultaneous equations. The series method, while elegant from a mathematical point of view, will typically require a computer program to expand the series sufficiently to yield good results in a practical time frame. Yield-line methods, which could be considered the extension of plastic design to two-dimensional surfaces, are typically a hand calculation procedure. The use of computer programs in bridge design brings up a philosophical problem as to the responsibility for error. Almost all vendors of commercial computer programs disavow any responsibilit y for error. A release from liability is usually implicit in their use and may even be an explicit part of obtaining a license. This means that an organization using a computer program must be relatively certain of
Lecture - 6-1
printed on June 24, 2003 the results that it obtains. It is not necessary for every engineer in a large design section to have personally confirmed every computer program, but it is necessary that some verification testing be done or that the results of previous verification testing be obtained in order to produce the required level of confidence. Computer programs can be verified against universally accepted closed- form solutions, other computer programs which have been previously verified, or the results of testing. Many computer programs for design use also contain code checking capabilities. Others have portions of the applicable design specification embedded in the coding of the program. In order to identify the specification edition which may have been tied to a given release of a program and also to provide a means for determining which structures may have been designed with a version of a program later found to contain errors, the specification requires that a name, version and a release date of software be identified in the contract drawing.
6.3 PRINCIPLES OF MATHEMATICAL MODELING 6.3.1 Structural Material Behavior The LRFD Specification recognizes both elastic and inelastic behavior of materials for analysis purposes. Inelastic material behavior is implicit in many of the equations and procedures specified for the calculation of cross-sectional resistance, such as calculating the nominal resistance of a concrete beam or column, the nominal plastic moment resistance of a compact and adequately braced steel cross-section, or the bearing capacity of a spread footing. Often, the force effects to which this resistance will be compared will be calculated on a basis of a linear structural analysis with elastic material properties having been assumed. This dichotomy has existed in the bridge specification since the introduction of load factor design in the early 1970's. It continues through the LRFD Specification. On the other hand, there are certain assumed failure modes at extreme events and the use of mechanism and unified autostress design procedures for steel girders, where permitted, which require analysis based on non-linear behavior. Many times, this analysis will take a form analogous to plastic design of steel frames. For example, seismic design may be based on the formation of plastic hinges at the top and bottom of the columns of a bent. Ship collision forces may be absorbed in a comparable inelastic manner. Furthermore, it is anticipated that future seismic design provisions will be based on extensive research currently underway to develop a step-by-step nonlinear force displacement relationship for the lateral displacement of piers. Where inelastic analysis is used, the Designer must be cert ain that a ductile failure mode is obtained through proper detailing. Rules
Lecture - 6-2
printed on June 24, 2003 for achieving this are presented in the sections for steel and conc rete design. 6.3.2 Geometry 6.3.2.1 GENERAL Most analyses done for the purpose of designing bridges are based on the assumption that the displacements caus ed by the loads are relatively small and, therefore, it is suitably accurate to base the calculations on the undeformed shape. This is typically referred to as the small deflection theory, and it is routinely used in the design of beam-type structures and bridges which resis t loads through a couple whose tensile and compressive forces remain essentially in fixed positions relative to each other while the bridge deflects. This will be the case for a truss and for tied-arches. For other types of structures and components and for certain types of analysis, the effect of the deflections must be considered in the development of the force equilibrium equations, i.e., the equations of equilibrium are written for the displaced shape. Almost all engineers are aware that the study of structural stability requires consideration of the displaced shape, in fact, if the displaced shape is not part of the original formulation of the problem, one would never be able to determine that a column, shell or plate can buckle. Consider for a moment the simple pin-ended column. Unless the deflected shape of the column is taken into account, the moment caused by the axial load acting on the displaced shape would not be accounted for. It is this moment which causes the column to move laterally, i.e., to buckle. Almost a century ago, it was found that the only reasonably accurate way to calculate force effects in suspension bridges of any size was to include the deflection of the cable in the formulation of the problem and, therefore, the displacement of a stiffening truss or stiffening girder. As conventional, i.e., not tied, arches became longer and more slender, an effect directly analogous to that observed in suspension bridges can become significant enough that it must be accounted for in the design of the arch rib. In fact, because the arch rib is in compression and can buckle, the effect of large deflections can be especially important. Finally, the compression members of frames in bents can also be susceptible to this phenomenon. Where non-linear effects arriving either out of material nonlinearity or large deflections become significant, then super position of forces does not apply. This means that each load case under investigation must be studied separately under the full effect of all of the 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 a variety of individual loads and then combines, or superimposes, the
Lecture - 6-3
printed on June 24, 2003 force effects calculated for each individual load to make up whatever group 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 analyze each load combination separately, as identified above, certain approximate methods have been developed to allow the designer to add a correction to a set of force effects calc ulated in a linear manner. These are sometimes called single-step adjustment methods, the most commonly used of which is moment magnification for beam columns, which has been part of the AASHTO Specifications sinc e the early 1970's. For beam columns, the moment magnification proces s is given by the equations below. Mc = δb M2b + δs M2s
(6.3.2.2-1)
f c = δb f 2b + δs f 2s
(6.3.2.2-2)
for which: C m
δb
1
δs
1
1.0
P u
(6.3.2.2-3)
φP e
1 ΣP u
(6.3.2.2-4)
φΣP e
where: Pu
=
factored axial load (N)
Pe
=
Euler buckling load (N)
φ
=
resistance factor for axial compression as specified in Specification Sections 5, 6 and 7, as applicable
M2b
=
moment on compression member due to factored gravity loads that result in no appreciable sidesway calculated by conventional first order elastic frame analysis, always positive (N mm)
f 2b
=
stress corresponding to M2b (MPa)
M2s
=
moment on compression member due to factored lateral or gravity loads that result in sidesway, ∆,
Lecture - 6-4
printed on June 24, 2003 greater than u/1500, calculated by conventional first order elastic frame analysis, always positive (N mm) f 2s
=
stress corresponding to M2s (MPa)
It may appear that the moment magnification f actor contains the Euler buckling load, Pe. However, Pe is only a convenient substitution for a group of terms related to the displacement of the beam column. A derivation of the moment magnification equation can be found in many textbooks on steel or concrete design. For cases where the shape of the beam column is expected to be radically different from that given by the simply-supported case, or the loads significantly different from those indicated above, then it is possible to make an adjustment to account for a different init ial elastic shape through the factor cm. The moment magnification procedure has also been extended to arches, and this has been available in the AASHTO Specifications for many years and is reproduced as Article S4.5.3.2.2c with no further refinement. 6.3.2.3 REFINED METHODS The effect of large deflections can also be rigorously accounted for through iterative solutions of equilibrium equations, taking into account updated positions of the structure, or by using geometric stiffness terms. In some cases, e.g., the suspension bridge, solutions are available to the differential equations of equilibrium which can be solved in a trial and error fashion, or through series expan sion. 6.3.3 Modeling Boundary Conditions Points of expansion or other forms of articulation in the structure are commonly idealized as frictionless units. Where past practice indicates that this has been a reasonable conservative approach, continued use is warranted. There are other instances where the potential for nonfunctional expansion devices and/or the possibility that joints may close should also be investigated. This might be the case, for example, in a seismic analysis where analysis may be made, assuming that expansion joints are operable and open, and then another analysis might be made, assuming that they are closed and nonfunctional in order to simulate, or bound, t he effects of joints reaching the limits of travel during the earthquake. The possibility of reaching the limit of expansion travel should also be investigated when evaluating non-linear effects on substructure elements. It may be possible that the amount of moment magnification may be reduced because expansion dams will close, jamming the structure against the abutments before the full movement implicit in the moment magnification factor can be reached. This will reduce the moment magnification factor and, hence, the design moment.
Lecture - 6-5
printed on June 24, 2003 Similarly, the effect of boundary conditions at foundatio n units should also be evaluated. Foundation units are seldom fully fixed or fully pinned, and an evaluation of the potential movement of a foundation unit may be necessary in order to properly assess response, as well as secondary moments caused by change in geometry. Here again, bounding of the range of probable movement may 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 made based on the plan geometry of the superstructure. The first simplification involves the possibility of replacing the superstructure for analysis purposes wit h a single-line element called a spine beam. This may be done when the transverse distortion of the superstructure is small in comparison with the longitudinal deformation. Generally, if the superstructure is a torsionally stiff closed section or sections whose length exceeds 2.5 times their width, it may be idealized as a line element whose dimensions may be determined as given in the Specification. This can be used to significantly simplify analysis models. The second simplification deals with when it is possible to consider curved superstructures as straight for the purpose of analysis. If the superstructure is a torsionally stiff closed section and the central angle of a segment between piers is less than 12 , then the segment may be considered straight. If the superstructure is made of torsionally weak open sections, then the effects of curvature may be neglected when the subtended angle is less than that given in Table 6.4.1-1. Table 6.4.1-1 - Limiting Central Angle for Neglecting Curvature in Determining Primary Bending Moments
No. of Beams
Angle for One Span
Angle for Multiple Spans
2
2
3
3 or 4
3
4
5 or more
4
5
Lecture - 6-6
printed on June 24, 2003 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 deck slabs by analyzing a strip of deck as a continuous beam. Provisions are 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 secondary direction exceeds 1.5 times the spacing in the primary direction, then all of the wheel loads applied to the deck are considered to be applied to the primary strip. The secondary strip is designed on a basis of percentage of reinforcement in the primary strip. If the spacing of the supporting components on the secondary direction is less than 1.5 times that in the primary direction, then a crossed sticks analogy is used. The width of the equivalent strips in each direction is provided by Table S4.6.2.1.3-1 and the wheel load is distributed between two idealized intersecting strips according to the relative stiffness of each strip. Once the wheel loads have been assigned to the strips, for either case identified above, then the force effects are calculated based on a continuous beam. For the purpose of analyzing the continuous beam, the span length of each span is taken as a centerto-center of supporting components. For the purpose of calculating moment and shear at a design section, some offset from the theoretical center of support is permitted as given in the Specification. Decks which form an integral part of a cellular cross-section are supported on webs which are monolithic with the deck. Therefore, when the deck rotates, the web of the box girder rotates giving rise to bending stresses throughout the cross-section. For the purpose of analyzing this effect, a cross-sectional frame action procedure is identified in the Specification. In the case of fully filled and partially filled grids, the results of recent research are incorporated in LRFD Article S4.6.2.1.8 to given bending 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 for assigning portions of a design lane to a supporting component. These are commonly called distribution factors. It is important to remember that the approximate distribution factors, specified in the LRFD Specification, are on a lane, i.e., axle basis, not a wheel basis. The distribution factors are given for t he various kinds of bridges shown in Figure 6.4.2.2.1-1.
Lecture - 6-7
printed on June 24, 2003 SUPPORTING COMPONENTS
TYPE OF DECK
Steel Beam - Revised Factors
Cast-in-place concrete slab, precast concrete slab, steel grid, glued/spiked panels, stressed wood
Closed Steel or Precast Concrete Boxes - Revised Factors
Cast-in-place concrete slab
Open Steel or Precast Concrete Boxes - Revised Factors
Cast-in-place concrete slab, precast concrete deck slab
Cast-in-Place Concrete Multicell Box - Revised Factors
Monolithic concrete
Cast-in-Place Concrete Tee Beam - Revised Factors
Monolithic concrete
Precast Solid, Voided or Cellular Concrete Boxes with Shear Keys - Revised Factors
Cast-in-place concrete overlay
Precast Solid, Voided or Cellular Concrete Box with Shear Keys and with or without Transverse Post-Tensioning - Revised Factors (in some cases)
Integral concrete
Precast Concrete Channel Sections with Shear Keys
Cast-in-place concrete overlay
Precast Concrete Double Tee Section with Shear Keys and with or without Transverse Post-Tensioning
Integral concrete
Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning
Integral concrete
Concrete I or Bulb-Tee Sections - Revised Factors
Cast-in-place concrete, precast concrete
Wood Beams - Revised Factors
Cast-in-place concrete or plank, glued/spiked panels or stressed wood
TYPICAL CROSSSECTION
Figure 6.4.2.2.1-1 - Common Deck Superstructures Covered in LRFD Specification Articles 4.6.2.2.2 and 4.6.2.2.3
Lecture - 6-8
printed on June 24, 2003 Some of the distribution factors are new to the Specification as a result of an extensive project on load distribution known as NCHRP Project 12-26. Where the distribution factors for a given type of cross-section have been developed under that project and are new to the Specification, the words "revised factors" appear in the column identified as "supporting components". Where those words do not appear, the distribution factors have been retained from earlier editions of the AASHTO Standard Specifications. Some simplifications have been made in utilizing the distribution factors from NCHRP 12-26. In particular, correction factors for various aspects of structural action, typically involving continuity, which were less than 5%, were omitted from the LRFD Specifications. Similarly, an increase in moments over piers, thought to be on the order of 10%, was not included because stresses at or near internal bearings have been shown to be reduced below that calculated 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 the multiple presence factors, given in Lecture 3, are built into the distribution factors, whereas, the multiple presence factors in earlier editions of the AASHTO Standard Specifications are built into the NCHRP 12-26 factors. Additionally, the factors appropriate for the LRFD Specification are based on a lane of live load, rather than a "line of wheels". Finally, when the SI version of the LRFD Specification is used, conversion to that system of units has also been accou nted for. Various limits on span, spacing and other characteristics are provided in the Specifications for each of the distribution factors. These parameters identify the range for which the factors were developed. They were not evaluated for factors beyond the ranges indicated. Therefore, for structures which do not comply with these limitations, a rigorous analysis by grid or finite elements should be used. Furthermore, the distribution factors usually apply for structures which are: •
essentially constant in deck width,
•
have four or more beams, unless noted,
•
have beams which are parallel and approximately of the same stiffness,
•
have overhangs that do not exceed 0.9 m, unless specifically noted,
•
have in-plan curvatures less than those specified above, and
•
have a cross-section consistent with one of the cross-sections identified 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 - 6-9
printed on June 24, 2003 and development of the new distribution factors. The discussion below was taken from the NCHRP Research Resul ts Digest No. 187, a summary of the Final Report of Project 12-26 as summarized by Ian M. Friedland, NCHRP Project Coordinator. Live load distribution on highway bridges is a key response quantity in determining member size and, consequently, strength and serviceability. It is of critical importance both in the design of new bridges and in the evaluation of the load-carrying capacity of existing bridges. Using live load distribution factors, engineers can predict bridge response by treating the longitudinal and transverse effects of live loads as uncoupled phenomena. Empirical live load distribution factors for stringers and longitudinal beams have appeared in the AASHTO Standard Specifications for Highway Bridges with only minor changes since 1931. Findings of recent studies suggest a need to update these specifications in order to provide improved predictions of live load distribution. Live load distribution is a function of the magnitude and location of truck live loads and the response of the bridge to these loads. The NCHRP 12-26 study focused on the second factor mentioned above: the response of the bridge to a predefined set of loads, namely, the HS family of trucks. In Project 12-26, three levels of analysis were considered for each bridge type. The most accurate level, Level 3, involves detailed modeling of the bridge deck. Level 2 includes either graphical methods, nomographs and influence surfaces, or simplified computer programs. Level 1 methods provide simple formulas to predict lateral load distribution, using a wheel load distribution factor applied to a truck wheel line to obtain the longitudinal response of a single girder. The major part of the research in Project 12-26 was devoted to the Level 1 analysis methods because of its ease of application, established use, and the surprisingly good correlation with the higher levels of analysis in their application to a majority of bridges. The formulas presented in the current AASHTO specifications were evaluated, and alternative formulas were developed that offer improved accuracy, wider range of applicability, and in some cases, easier application than the current AASHTO formulas. These formulas were developed for interior and exterior girder moment and shear load distribution for single or multiple lane loadings. In addition, correction factors for continuous superstructures and skewed bridges were developed. The formulas presented in previous AASHTO Specifications, although simpler, do not present the degree of acc uracy demanded by today's Bridge Engineers. In some cases, these formulas can result in highly unconservative results (more than 40%), in other cases they may be highly conservative (more than 50%). In general, the formulas developed in Project 12-26 are within 5% of the result s of an accurate
Lecture - 6-10
printed on June 24, 2003 analysis. Table 6.4.2.2.1-1 shows comparisons with moment distribution factors obtained from AASH TO, Level 1, Level 2 and Level 3 methods for simple span bridges. Table 6.4.2.2.1-1 - Comparison of interior girder moment distribution factors by varying levels of accuracy using the "average bridge" for each bridge type
NCHRP 12-26 (Level 1)
Grillage (Level 2)
Finite Element (Level 3)
1.413 (S/1700)
1.458
1.368
1.378
Box girder a
1.144
1.143
0.970
1.005
Slabb
1820
1710
1900
1890
Multi-box beama
0.646
0.597
0.540
0.552
Spread box beama
1.564
1.282
1.248
1.241
Bridge Type Beam-and-slaba
AASHTO
a
Number of wheel lines per girder
b
Wheel line distribution width, in mm
In addition, the study resulted in recommendations for use of computer programs to achieve more accurate results. The recommendations focus on the use of plane grid analysis, as well as detailed finite element analysis, where different truck types and their combinations may be considered. 6.4.2.2.2 Influence of Truck Configuration The formulas developed in Project 12-26 for the Level 1 analysis were based on the standard AASHTO "HS" trucks. A limited parametric study conducted as part of the research showed that variations in the truck axle configuration or truck weight do not significantly affect the wheel load distribution factors. The group of axle trains used for this study are shown in Figure 6.4.2.2.2-1. It is anticipated that smaller gage widths would result in larger distribution factors, and larger gage widths would result in smaller distribution factors. Table 6.4.2.2.1-1 gives the variation of wheel load distribution factors with different axle configurations applied to a number of beamand-slab bridges. The differences were below 1% in many cases and, in all cases, the formulas resulted in good predictions. Therefore, with some caution, these formulas may be applied to other truck types. Obviously, Levels 2 and 3 analyses may also be applied for trucks significantly different from the AASHTO family of trucks.
Lecture - 6-11
printed on June 24, 2003
Figure 6.4.2.2.2-1 - Axle Configurations for Truck Types Considered in Study Table 6.4.2.2.1-1 - Effect of Load Configuration on Distribution Factor DISTRIBUTION FACTOR (g)
*
PERCENT DIFFERENCE WITH HS-20
HS-20
HTL-57
4A-66
B-141
NCHRP 12-26
HTL-57
4A-66
B-141
NCHR P 1226
Average*
1.293
1.261
1.285
1.268
1.304
-2.4
-0.6
-1.9
+0.9
Max. S 5000 mm
2.220
2.162
2.205
2.178
2.308
-2.6
-0.7
-1.9
+4.0
Min. S 1000 mm
0.713
0.717
0.713
0.715
0.755
+0.6
0.0
+0.3
+5.9
Max. L 60 000 mm
0.982
0.958
0.983
0.952
1.033
-2.4
+0.1
-3.1
+5.2
Min. L 6000 mm
1.630
1.625
1.624
1.623
1.807
-0.3
-0.3
-0.4
+10.9
S L ts Kg
= = = =
2200 mm 20 000 mm 185 mm 2.33 x 1011 mm4
Lecture - 6-12
printed on June 24, 2003 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 previous studies, and many simplified methods and formulas were developed by previous researchers for multi-lane loading moment distribution factors. The AASHTO formula, the formulas presented by other researchers, and the formulas developed in the study are discussed in the following according to their application. Table 6.4.2.2.3a-1 is taken from the specifications and summarized criteria for moment in interior beams or elements for various types of cross-sections. Similar tables exist for moment in exterior griders, for shear in interior girders and shear in exterior girders. Table 6.4.2.2.3a-2 describes how the term L (length) may be determined for use in the live load distribution factor equations given in Table 6.4.2.2.3a-1. In the rare occasion when the continuous span arrangement is such that an interior span does not have any positive uniform load moment (i.e., no uniform load points of contraflexure), the region of negative moment near the interior supports would be increased to the centerline of the span, and the L used in determining the live load distribution factors would be the average of the two adjacent spans.
Lecture - 6-13
printed on June 24, 2003 Table 6.4.2.2.3a-1 - Distribution of Live Loads Per Lane for Moment in Interior Beams Type of Beams
Wood Deck on Wood or Steel Beams Concrete Deck on Wood Beams Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, Tand Double T-Sections
Applicable Cross-Section from Table 4.6.2.2.1-1
Distribution Factors
a, l l a, e, k and also i, j if sufficiently connected to act as a unit
See Table S4.6.2.2.2a-1 One Design Lane Loaded: S/3700 Two or More Design Lanes Loaded: S/3000 One Design Lane Loaded: S 4300
0.06
0.4
S L
0.3
0.1
K g 3
Lt s
S 2900
1800
1100 S 4900 110 ts 300 6000 L 73 000 Nb 4
d
0.6
0.2
S L
0.1
K g 3
Lt s
use lesser of the values obtained from the equation above with Nb = 3 or the lever rule
Nb = 3
One Design Lane Loaded:
2100 S 4000 18 000 L 73 000 Nc 3
S 1100
1.75
300 L
0.35
1 N c
0.45
Two or More Design Lanes Loaded: 13 N c Concrete Deck on Concrete Spread Box Beams
S
Two or More Design Lanes Loaded: 0.075
Multicell Concrete Box Beam
Range of Applicability
b, c
0.3
S 430
1 L
0.25
One Design Lane Loaded: S 910
0.35
Sd
If Nc > 8 use Nc = 8
0.25
L2
1800 S 3500 6000 L 43 000 450 d 1700 Nb 3
Two or More Design Lanes Loaded: S 1900
0.6
Sd
0.125
L2
Use Lever Rule
Lecture - 6-14
S > 3500
printed on June 24, 2003 Type of Beams
Applicable Cross-Section from Table 4.6.2.2.1-1
Concrete Beams used in Multibeam Decks
f
Distribution Factors
One Design Lane Loaded: k
h
0.5
b 2.8L
where: g if sufficiently connected to act as a unit
Range of Applicability
I J
900 b 1500 6000 L 37 000 5 Nb 20
0.25
k 2.5(N b )
0.2
1.5
Two or More Design Lanes Loaded: b k 7600
0.6
b L
0.2
I J
0.06
Regardless of Number of Loaded Lanes: S/D where: C = K (W/L)
K
Skew
D = 300 [11.5 - NL + 1.4NL (1 - 0.2C)2] when C 5
NL
45°
6
D = 300 [11.5 - NL] when C > 5 g, i, j if connected only enough to prevent relative vertical displacement at the interface
Steel Grids on Steel Beams
Concrete Deck on Multiple Steel Box Girders
a
b, c
(1
K=
µ) I J
for preliminary design, the following values of K may be used: Beam Type Nonvoided rectangular beams Rectangular beams with circular voids: Box section beams Channel beams T-beam Double T-beam
K 0.7 0.8 1.0 2.2 2.0 2.0
One Design Lane Loaded: S/2300 If tg< 100 mm S/3050 If tg 100 mm Two or More Design Lanes Loaded: S/2400 If tg< 100 mm S/3050 If tg 100 mm
S
1800 mm
S
3200 mm
Regardless of Number of Loaded Lanes: 0.05
0.85
N L N b
0.425 N L
Lecture - 6-15
0.5
N L N b
1.5
printed on June 24, 2003 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 which moment is being calculated.
Negative Moment - End spans of continuous spans, from end to point of contraflexure under a uniform load on all spans
The length of the span for which moment is being calculated.
Negative Moment - Near interior supports of continuous spans, from point of contraflexure to point of contraflexure under a uniform load on all spans
The average length of the two adjacent spans.
Positive Moment - Interior spans of continuous spans, from point of contraflexure to point of contraflexure under a uniform load on all spans
The length of the span for which moment is being calculated.
Shear
The length of the span for which shear is being calculated.
Exterior Reaction
The length of the exterior span.
Interior Reaction of Continuous Span
The average length of the two adjacent spans.
Moment Distribution to Interior Girders, Multi-Lane Loading The AASHTO formula for moment distribution for multi-lane loading is given as S/1800 for reinforced concrete T-beam bridges with girder spacing up to 3000 mm, and as S/1700 for steel girder bridges and prestressed concrete girder bridges with girder spacing up to 4300 mm, where S is the girder spacing. When the girder spacing is larger than the specified limit, s imple beam distribution is to be used to calculate the load distribution factors. Marx, et al, at the University o f Illinois, developed a formula for wheel load distribution for moment which included multiple lane reduction factors and is applicable to all beam-and-slab bridges. The formula is based on girder spacing, span length, slab thickness and bridge girder stiffness. A formula which does not consider a reduction for multi-lane loading was developed at Lehigh University. The Lehigh formula includes terms for the number of traffic lanes, number of girders, girder spacing, span length and total curb-to-curb deck width. Sanders and Elleby (NCHRP Report 83) developed a simple formula based on orthotropic plate theory for moment distribution on beam-and-slab bridges. Their formula includes terms for girder
Lecture - 6-16
printed on June 24, 2003 spacing, number of traffic lanes and a stiffness parameter based on bridge type, bridge and beam geometry and material properties. A full-width design approach, known as Henry's Method, is used by the State of Tennessee. Henry's Method includes factors for number of girders, total curb-to-c urb bridge deck width and a reduction factor based on number of lanes. A formula developed as part of NCHRP Project 12-26 includes the effect of girder spacing, span length, girder inertia and slab thickness. The multiple lane reduction factor is built into the formula. This formula, applicable to cross-sections with four or more beams, is given by:
g 0.075
S 2900
0.6
S L
0.2
K g
0.1
(6.4.2.2.3a-1)
3
Lt s
where: S
=
girder spacing (1100 mm
L
=
span length (6000 mm
L
Kg
=
n(I+Aeg2) (4 x 108
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 of girder to mid-point of slab)
ts
=
slab thickness (110 mm
Kg
S
ts
4900 mm)
73 000 mm)
300 mm)
This formula is dependent on the inertia of the girder and, thus, a value for Kg must be assumed for initial design. For this purpose, Kg/LtS3 may be taken as unity. All of the above formulas were evaluated using direct finite element analysis with the GENDEK-5 program and a database of 30 bridges; subsequently, they were evaluated using the MSI method and database of more than 300 bridges. It was found that Equation 6 and the Illinois formulas are accurate and produce results that are as accurate as the Level 2 methods. Moment Distribution to Exterior Girders, Multi-Lane Loading Previous AASHTO Specifications recommend a simple beam distribution of wheel loads in the transverse direction for calculating wheel load distribution factors in edge girders. Any load that falls
Lecture - 6-17
printed on June 24, 2003 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 interior girder is distributed to these girders by assuming t hat the slab acts as a simple beam in that region. Any wheel load that falls inside of the first interior girder is assumed to have no effect on the edge girder. Marx, et al, at the University o f Illinois, developed a formula for the exterior girder based on certain assumptions in the placement of loads and may not be applicable to all bridges. This formula includes terms similar to those used in their formula for moment distribution to interior girders. A formula, depending on wheel position, alone was developed as part of this study which results in a correction factor for the edge girder. The factor must be applied to the distribution factor for the interior girder to obtain a distribution factor for the edge girder. This formula is given by:
e
0.77
d e
(6.4.2.2.3a-2)
2800
where: de
=
distance from edge of the roadway, usually the face of curb, to the center of the exterior web of the exterior cell, in mm
If the edge of the lane is outside of the exterior girder, the distance is positive; if the edge of the lane is to the interior sid e of the girder, the distance is negative. It was found that the formula developed in Project 12-26 resulted in accurate correction factors and was simpler than the previous AASHTO procedure. Moment Distribution to Interior Girders, Single-Lane Loading The literature search performed in this study did not reveal any simplified formula for single-lane loading of beam-and-slab bridges. The formula developed as part of the study is as follows:
g 0.06
S 4300
0.4
S L
0.3
K g 3
Lt s
0.1
(6.4.2.2.3a-3)
where the parameters are the same as those given for Equation 6.4.2.2.3a-1. This formula is applicable to interior girders only.
Lecture - 6-18
printed on June 24, 2003 Moment Distribution to Exterior Girders Simple beam distribution in the transverse direction should be used for single-lane loading of edge girders. One other investigation, applicable to load distribut ion for both shear and moment, is required for exterior beams of beam-slab bridges with diaphragms or cross-frames. This addition was not developed as part of NCHRP 12-26, but was added by the NCHRP 12-33 Editorial Committee. This distribution is based on treating the cross-section as a transversely rigid unit which deflects and rotates as a straight line. The live load is positioned for maximum effect on an exterior beam (one lane, two lane, three lane, etc., each with its appropriate multiple presence factor). The total vertical force and moment about the centroid of the cross-section is applied to the area of the cross-section, i.e., the number of beams, and the section modulus, i.e., the sum of the square of the distances of each beam from the centroid of the beams divided by the distance to the exterior beam. The specification puts this in equation form as: N L
R
N L N b
X ext
ev (6.4.2.2.3a-4)
N b
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 lane load from the center of gravity of the pattern of girders (mm)
x
=
horizontal distance from the center of gravity of the pattern of girders to each girder (mm)
Xext
=
horizontal distance from the center of gravity of the pattern of girders to the exterior girder (mm)
Nb
=
number of beams or girders
Shear Distribution No formula was found from previous research for the calculation of wheel load distribution factors for shear. Therefore, the formulas developed as part of the 12-26 study are reported for different cases as follows. The formula for multi-lane loading of interior girders is:
Lecture - 6-19
printed on June 24, 2003
g 0.2
S 3600
S 10 700
2
(6.4.2.2.3a-5)
The correction formula for multi-lane loading edge girder shear is: e
0.6
d e 3000
(6.4.2.2.3a-6)
The formula for shear distribution factor due to single-lane loading is:
g 0.36
S 7600
(6.4.2.2.3a-7)
Equation 6.4.2.2.3a-7 is applicable to interior girders only. Simple beam distribution in the transv erse direction should be used for single-lane loading of edge girders. Correction for Skew Effects Previous AASHTO Specifications did not include approximate formulas to account for the effect of skewed supports. However, some researchers have developed correction factors for such effects on moments in interior girders. Marx, et al, at the University of Illinois, developed four correction formulas for skew, one each for skew angles of 0, 30, 45 and 60 degrees. Corrections for other values of skew are obtained by straight-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 concrete I-girders was developed as part of the research performed at Lehigh University. This formula is based on the number of traffic lanes, number of girders, girder spacing, span length and total curb-to-curb deck width, and includes a variable term for skew angle. A correction factor for moment in skewed supports was also developed as part of Project 12-26. This formula is: r 1
c 1 (tan θ)1.5
(6.4.2.2.3a-8)
where, for θ > 30 ,:
Lecture - 6-20
printed on June 24, 2003
c 1
0.25
0.25
K g 3
Lt s
S L
0.5
(6.4.2.2.3a-9)
If θ 30 , c1 is taken as zero. In calculating c1, θ should not be taken as greater than 60 . The other parameters are as defined previously. From the literature review, no correction formulas were obtained for shear effects due to skewed supports. In Project 12-26, it was found that shear in interior girders need not be corrected for skew effects; that is, the shear distribution to interior girders is similar to that of a straight bridge. A correction formula for shear at the obtuse corner of the exterior girder of two girder systems and all girders of a multi-girder bridge was developed as part of this study and is given as: 3
r 1
0.2
L t s
k g
0.3
tan θ
(6.4.2.2.3a-10)
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 dis tribution factor in the exterior girder of non-skewed bridges. Therefore, the product of factors g, e and r must be obtained to find the obtus e corner shear distribution factor in a beam-and-slab bridge. The distribution factors calculated for moments are plotted as a function of girder spacing for Spans 9, 18, 27, 36 and 60 m in Figure 6.4.2.2.3a-1. For comparisons, AASHTO (1989) distribution factors are also shown. Girder distribution factors, specified by AASHTO (1989), are conservative for larger girder spacing. For shorter spans and girder spacings, AASHTO (1989) produces smaller distribution factors than calculated values.
Lecture - 6-21
printed on June 24, 2003
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 by various researchers in the past. Bridge deck behavior has been well studied and many recommendations have been made for detailed analysis of these bridges. However, there is a limited amount of information on simplified wheel load distribution formulas in the literature. In this context, a “girder” is a notional I-shape consisting of one web of a multi-cell box and the associated half-flanges on each side of the web. Moment Distribution to Interior Girders Scordelis, at the University of California, Berkeley, presented a formula for prediction of wheel load distribution for moment distribution in prestressed and reinforced concrete box girder bridges. The formula is based on modification of distribution factors obtained for a rigid cross-section. The formula predicts load distribution factors in reinforced concrete box girders with high accuracy and for prestressed concrete box girders with acceptable accuracy. Sanders and Elleby also presented a simple formula for moment distribution factors which is similar to their formula for beamand-slab bridges. The following formulas, developed as part of NCHRP 12-26, may be used to predict the moment load distribution factors in the interior girders of concrete box girder bridges due to single-lane and multi-lane loadings. These formulas are applicable to both reinforced
Lecture - 6-22
printed on June 24, 2003 and prestressed concrete bridges, and the multiple presence factor is accounted for. For single-lane loading:
g
S 1100
1.75
300 L
0.35
1 N c
0.45
(6.4.2.2.3b-1)
For multi-lane loading:
g
13 N c
0.3
S 430
1 L
0.25
(6.4.2.2.3b-2)
where: S
=
girder spacing, in mm
L
=
span length, in mm
Nc
=
number of cells
Moment Distribution to Exterior Girders The factor for load distribution for exterior girders shall be We/4300 mm, where We is the width of the exterior girder, taken as the top slab width measured from the mid-point between girders to the edge of the slab. Shear Distribution No formula for shear load distribution was obtained from previous research for box girder bridges, but the following were developed as part of NCHRP 12-26. The shear distribution factor for interior girder multi-lane loading of reinforced and prestressed concrete box girder bridges is: g
S 2200
0.9
d L
0.1
(6.4.2.2.3b-3)
where: S = girder spacing, in mm d = girder depth, in mm L = span length, in mm
Lecture - 6-23
printed on June 24, 2003 The distribution factor for shear in the interior girders due to single-lane loading may be obtained from: S 2900
g
0.6
d L
0.1
(6.4.2.2.3b-4)
where the parameters are as defined in Equation 6.4.2.2.3b-3. A correction formula for shear in the exterior girder for multilane loading is:
e
0.64
d e
(6.4.2.2.3b-5)
3800
where: de
=
distance from edge of the roadway, usually the face of curb, to the center of the exterior web of the exterior cell, in mm
Correction for Skew Effects The following formula was developed for correction of moment due to skewed supports for values of θ from 0 to 60 : r 1.05
0.25 (tan θ)
1.0
(6.4.2.2.3b-6)
If θ > 60 , use 60 in Equation 6.4.2.2.3b-6. Another formula was developed in Project 12-26 for correction of shear at the obtuse corner of an edge girder. It must be applied to the shear distribution factor for the edge girder of a non-skew ed bridge and must, therefore, be used in conjunction with the edge girder correction factor of Equation 6.4.2.2.3b-5. This formula, applicable for values of θ up to 60 , is: r 1
c 1 (tan θ)
(6.4.2.2.3b-7)
where: c1
=
0.25 + L/(70d)
d
=
bridge depth, in mm
L
=
span length, in mm
Lecture - 6-24
printed on June 24, 2003 6.4.2.2.3c Simplified Formulas for Slab Bridges The literature search did not reveal any simplified formulas for wheel load distribution in slab bridges other than those recommended by AASHTO. Therefore, the following are formulas that were developed as part of NCHRP 12-26. Moment Distribution, Multi-Lane Loading Equation 6.4.2.2.3c-1 was developed to predict wheel load distribution (distribution design width) for mom ent in slab bridges due to multi-lane loading. Multiple presence factors are already accounted for in the formula:
E 2100
0.12 L1 W 1
0.5
W N L
(6.4.2.2.3c-1)
where: E
=
the transverse distance over which a wheel line is distributed
L1
=
L
W1
=
W
L
=
span length, in mm
W
=
bridge width, in mm, edge-to-edge
18 000 mm 18 000 mm
Moment Distribution, Single-Lane Loading The equation below predicts wheel load distribution for moment due to single-lane loading. E 250
0.42 (L1 W 1 )0.5
(6.4.2.2.3c-2)
where the parameters are as defined in Equation 6.4.2.2.3c-1. Correction for Skew Effects Equation 6.4.2.2.3b-6 may be used to account for the reduction of moment in skewed bridges. According to previous AASHTO Specifications, slab bridges are adequate for shear if they are designed for moment. A quick check of this assumption was made and it was concluded that it is a valid assumption. Therefore, no formula or method is presented for calculation of shear in slab bridges.
Lecture - 6-25
printed on June 24, 2003 6.4.2.2.3d Simplified Formulas for Multi-Beam Decks which are Sufficiently Interconnected to Act as a Unit Only one formula, other than those presented in the previous AASHTO Specifications, was obtained for load distribution in multibeam decks. This formula, developed by Arya at the University of Illinois, is applicable to both box and open section multi-beam bridges and predicts interior beam moment responses due to single-lane and multi-lane loading. However, a number of simplified formulas developed in the study are valid only for multi-box beam decks and do not apply to open sections. Therefore, the response of multi-beam decks made of open members, such as channels, may or may not be accurately predicted by the formulas developed in that study. Moment Distribution to Interior Girders, Multi-Lane Loading The formula developed by Arya for interior girder load distribution in multi-beam decks includes terms for the maximum number of wheels that can be placed on a transverse section of the bridge, number of beams, beam width and span length. A variation of the formula was also proposed for multi-beam decks made of channels, which includes consideration of the overall depth of the channel section and its average thickness, defined as its area divided by its length along the centerline of the thickness. The following formula was developed in Project 12-26 to predict load distribution factors for interior beam moment due to multilane loading. The multiple presence reduction factor is already accounted for in the formula.
g
b k 7600
0.6
b L
0.2
I J
0.06
(6.4.2.2.3d-1)
where: 0.2
k
=
2.5(Nb)
1.5
b
=
beam width, in mm
L
=
span length, 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 a beam; an estimated value for these properties must, therefore, be used in preliminary design. The term I/J may be taken as unity for this case.
Lecture - 6-26
printed on June 24, 2003 Moment Distribution to Interior Girders, Single-Lane Loading Arya also presented a load distribution formula for multi-beam decks designed for one traffic lane. The formulation and parameters were similar to those presented for multi-lane loading. A variation of that equation was also presented for calculation of the interior beam moment distribution factor for a single-lane, channel section, multibeam deck. It should be noted that Arya's equations are not applicable to cases of only one-lane loading with more than one traff ic lane. A formula for wheel load distribution for moment in the interior girders due to single-lane loading was also developed in NCHRP 1226. This formula is as follows:
g
b k 2.8L
0.5
I J
0.25
(6.4.2.2.3d-2)
All parameters are defined in Equation 6.4.2.2.3d-1. Equation 6.4.2.2.3c-2 is also dependent on inertia and torsional constants, and a value of 1.0 may be used as an approxim ation for the term I/J during preliminary design. Moment Distribution to Exterior Girders The moment in the edge girder due to multi-lane loading in multi-beam decks comprised of box units is obtained by using a correction factor applied to the interior girder distribution factors for multi-lane loading. This correction factor may be found from the following formula:
g 1.04
d e 7600
(6.4.2.2.3d-3)
where: de =
distance from edge of the lane to the center of the exterior web of the exterior girder, in mm
For exterior beams of sufficiently interconnected multi-beam bridge decks comprised of T-shaped units subjected to multi-lane loading, Equation 6.4.2.2.3b-2 applies. For single-lane loading and for multi-beam decks comprised of either box units or units other than box units, the lever rule is used.
Lecture - 6-27
printed on June 24, 2003 Shear Distribution Distribution factors for shear in interior girders of multi-beam decks in "Bridge Decks Comprised of Box Units" due to multi-lane loading may be calculated from the following formula: 0.4
b 4000
g
b L
0.1
I J
0.05
(6.4.2.2.3d-4)
where the parameters are as defined in Equation 6.4.2.2.3d-1. Distribution factors for shear in the interior girders of multibeam decks in "Bridge Decks Comprised of Box Units" due to singlelane loading are obtained from the following formula: b g 0.70 L
0.15
I J
0.05
(6.4.2.2.3d-5)
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 are dependent on inertia and torsional constants, and a value of 1.0 may be used as an approximation for the term I/J during preliminary design. The shear in the edge girder of multi-beam deck in "Bridge Decks Comprised of Box Units" due to multi-lane loading can be found using a correction factor applied to interior girder distribution factors. This correction factor is obtained from the formula:
e
1.02
d e 15 000
(6.4.2.2.3d-6)
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder, in mm
For shear in exterior beams of sufficiently interconnected mult ibeam bridge decks comprised of T-shaped units, Equations 6.4.2.2.3b-4 through 6.4.2.2.3b-5 and the lever rule should be used, where appropriate. Correction for Skew Effects The moment in any beam in a skewed bridge may be obtained by using a skew reduction factor given by Equation 6.4.2.2.3b-6.
Lecture - 6-28
printed on June 24, 2003 The shear in the interior beams of a skewed multi-beam bridge comprised of box beams is usually of the same order as that of the shear in the obtuse corner and must be obtained by applying a correction factor to the respons e of the edge girder in a straight bridge. This correction factor may be calculated from the formula: c 1 (tan θ)0.5
r 1
(6.4.2.2.3d-7)
where:
c 1
L 90d
(6.4.2.2.3d-8)
6.4.2.2.3e Simplified Formulas for Multi-Beam Decks which are not Sufficiently Interconnected to Act as a Unit The LRFD Specification contains the same provis ions for load distribution in this type of bridge superstructure as appeared in recent editions of the Standard Specification, and as repeated below for completeness. The key difference between bridges treated herein, as compared to Section 6.4.2.2.3d, is the degree of transverse interconnection of units. If box, T, channel or other precast units are interconnected through a structural slab, or sufficiently transversely post-tensioned to produce a similar level of continuity, then the discussion of Section 6.4.2.2.3d applies. If the interconnection between the units is expected to transmit shear, but relatively little moment over the bridge service life, then the provisions herein apply. The Specification provides for the computation of a bending moment distribution factor, regardless of the number of lanes, given by:
g
S 300D
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)I J
Lecture - 6-29
printed on June 24, 2003 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 by AASHTO, was obtained from previous research for determining load distribution factors in spread box beam bridges. This formula was developed at Lehigh University for predicting the response of interior beams due to multi-lane loading and was later adopted by AASHTO. A correction factor for skewed bridges was also presented. In addition, a number of simple formulas were developed as part of NCHRP Project 12-26. Moment Distribution to Interior Beams, Multi-Lane Loading A formula developed in Project 12-26 for moment in interior spread box beams due to multi-lane loading is as follows:
g
S 1900
0.6
Sd
0.125
L2
(6.4.2.2.3f-1)
where S
=
girder spacing (mm)
L
=
span length (mm)
d
=
beam depth (mm)
Moment Distribution to Interior Beams, Single-Lane Loading A similar formula was developed for distribution to interior beams due to single-lane loading:
Lecture - 6-30
printed on June 24, 2003 S 910
g
0.35
Sd
0.25
(6.4.2.2.3f-2)
L2
where the parameters are as defined in Equation 6.4.2.2.3f-1. Moment Distribution to Exterior Girders The moment in edge girders due to multi-lane loading may be calculated by applying a correction factor to the interior girder distribution factor:
e
0.97
d e
(6.4.2.2.3f-3)
8700
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder (mm)
The distribution factor for moment in the edge girder due to single-lane loading may be obtained by simple-beam distribution, i.e., the lever rule, in the same manner as was described for beam-andslab bridges. Shear Distribution The distribution factor for shear in the interior girders due to multi-lane loading may be calculated from the following:
g
S 2250
0.8
d L
0.1
(6.4.2.2.3f-4)
where the parameters are as defined previously. The distribution factor for shear in the interior girders due to single-lane loading may be obtained from:
g
S 3050
0.6
d L
0.1
(6.4.2.2.3f-5)
where the parameters are again as defined previously. The shear in the edge girder due to multi-lane loading can be found by applying a correction factor to the interior girder equation. This correction factor is:
Lecture - 6-31
printed on June 24, 2003
e
0.8
d e 3050
(6.4.2.2.3f-6)
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder (mm)
The wheel load distribution factor for shear in the edge girder due to single-lane loading may be obtained by simple-beam distribution in the same manner as was described for beam-and-slab bridges, i.e., the lever rule. Correction for Skew Effects Research at Lehigh University also resulted in a formula for correction of wheel load distribution for m oment in interior girders due to multi-lane loading in skewed bridges. NCHRP 12-26 concludes that Equation 6.4.2.2.3b-6 was also applicable to this case. The shear in the interior beams of a skewed bridge is the same as that of a straight bridge. However, the shear in the obtuse corner must be obtained by applying a correction factor to the distribution factor for the edge girder in a straight bridge, given by Equation 6.4.2.2.3b-7, which C1 is taken as:
c 1
(Ld )0.5 6S
(6.4.2.2.3f-7)
6.4.2.2.3g Response of Continuous Bridges The response of continuous bridges was stud ied by modeling a number of two-span continuous bridges where each span is similar to the average bridge. The wheel load distribution factor for each case was compared to that of a simple bridge and correction factors for continuity were obtained. In the case of beam-and-slab bridges, a complete parameter study was performed, and it was found that the correction factor is generally independent of bridge geometry. These factors are given in the table below. When the NCHRP 12-26 factors were incorporated into the LRFD Specification, it was decided that 5% corrections were unwarranted given that the distribution factors are an approximation of actual behavior and are, therefore, subject to some variability. The continuity correction for negative moment, a 10% increase, was also neglected on the basis that experimentally observed "fanning" of the reaction tends to reduce the negative moment as compared to a typical beam calculation.
Lecture - 6-32
printed on June 24, 2003 Table 6.4.2.2.3f-1 - Continuity Correction Factors Beam-and-Slab Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent
c c c c
= = = =
1.05 1.10 1.00 1.05
c c c c
= = = =
1.00 1.10 1.00 1.00
Box Girder Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent Slab Bridges Positive moment Negative moment
c = 1.00 c = 1.10
Multi-Beam Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent
c c c c
= = = =
1.00 1.10 1.00 1.05
c c c c
= = = =
1.00 1.10 1.00 1.05
Spread box beam bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent 6.4.2.3 TRUSS AND ARCH BRIDGES 6.4.2.3.1 General The approximate method for load distribution to lines of trusses and arches is a so-called "lever rule", which is simply a matter of summing moments about one line of trusses or arches to find the reaction on the other line. This approach is illustrated by calculations in Lecture 7.
Lecture - 6-33
printed on June 24, 2003 6.5 REFINED METHODS 6.5.1 Deck Slabs Where refined analysis of deck slabs is desirable, finite element analysis is recommended. Elements should be chosen to simulate both bending and in-plane or membrane effects. If the analysis utilizes only plate/membrane or shell elements and has only one or two elements through the thickness of the deck, then the refined analysis will report an essentially bending-type respons e in the deck. There has been much experimental and analytic work that suggests that bending is not the primary source of strength in bridge decks, but that the development of membrane action, analogous to a shallow arch or dome load path within the deck is the primary source of strength. This type of action will only be determined through a very rigorous modeling of the deck. 6.5.2 Beam Slab Bridges Relatively rigorous models of beam slab bridges can be developed using general purpose commercial finite element programs, finite strip programs or special purpose greater finite element-based computer programs which have been specifically developed to simplify the analysis of bridge-type structures. These more custom-oriented programs often contain mesh generating capabilities, automatic load placement capabilities and code checking. Detailed bridge deck analysis using a finite element computer program may be used to produce accurate results. However, extreme care must be taken in preparation of the model, or inaccurate results will be obtained. Important points to consider are selection of a program capable of accurately modeling responses being investigated, calculation of element properties, mesh density and support conditions. Every model should be thoroughly checked to ensure that nodes and elements are generated correctly. Another important point is the loading. Truck loads should be placed at positions that produce the maximum response in the components being investigated. In many cases, the truck location is not known before preliminary analysis is performed and, therefore, many loadings should be investigated. This problem is more pronounced in skewed bridges. Many computer programs have algorithms that allow loads to be placed at any point on the elements. If this feature is not present, equivalent nodal loads must be calculated. Distribution of wheel loads to various nodes must also be performed with care, and the mesh should be fine enough to minimize errors that can arise because of load approximations. Many computer programs, especially the general purpose finite element analysis programs, report stresses and strains, not shear and moment values. Calculation of shear and moment values from the
Lecture - 6-34
printed on June 24, 2003 stresses must be carefully performed, usually requiring an integration over the beam cross-section. Some programs report stresses at node points rather than Gaussian integration points. Integration of stresses at node points is normally less accurate and may lead to inaccurate results. Many graphical and computer-based methods are available for calculating wheel load distribution. One popular method consists of design charts based on the orthotropic plate analogy, similar to that presented in the Ontario Highway Bridge Design Code. As computers become readily available to designers, simple computer-based methods, such as SALOD, become more attractive than nomographs and design charts. Also, grillage analysis presents a good alternative to other simplified bridge deck analysis methods and will generally produce more accurate results. The grillage analogy may be used to model any one of the five bridge types studied in this research. Each bridge type requires special modeling techniques. A major advantage of plane grid analysis is that shear and moment values for girders are directly obtained and integration of stresses is not needed. Loads normally need to be applied at nodal points, and it is recommended that sim plebeam distribution be used to distribute wheel loads to individual nodes. If the loads are placed in their correct locations, the results will be close to those of detailed finite element analysis. As indicated previously, the designer has to be responsible for constructing a suitable model and determining that the results are accurate. It is possible to make seemingly small errors in computer models which can have dramatic effects on the results which are obtained. 6.5.3 Example of Modeling Errors The modeling of diaphragms and boundary conditions at supports and bearings is vital to obtaining the proper results when using these sophisticated programs. The burden of correctly handling these factors rests with the designer. Consider the following example which shows how a very small modeling error produced very erroneous results. The framing plan shown on Figure 6.5.3-1 represents an actual bridge that was designed using a grid-type approach. The designer had a good model for this structure, except that the rotational degree of freedom corresponding to the global "x" axis was fixed at all of the bearings. This did not allow the diaphragms at the piers and abutments to respond correctly to the imposed loadings and deformations, and also had the effect of producing artificially stiff ends on the girders by virtue of vector resolution between global and local systems. The effect of this condition on the reactions obtained at the abutments and piers was dramatic. Modest uplift was reported at the acute corner along the near abutment show n in Figure 6.5.3-1, and a very substantial uplift was reported at the acute angle at the far
Lecture - 6-35
printed on June 24, 2003 abutment. This is shown in the table in Figure 6.5.3-1, as is a moment diagram for non-composite dead loads which reflects the incorrect reactions. Also shown in the table of reactions on Figure 6.5.3-1 are the correct reactions determined when the structure was modeled using the generic STRESS Computer Program with proper boundary conditions at the supports. In this case, a positive reaction is found at all bearings, and a significantly different moment diagram for noncomposite dead load also resulted. The correct reactions and moment diagram are also shown on Figure 6.5.3-1. The modeling of the degrees of freedom at the lines of support on this structure was also investigated utilizing a relatively complete three-dimensional finite element analysis and the SAPIV Computer Program. The model used is illustrated in Figure 6.5.3-2, which shows how the deck slab, girders and cross-frames were modeled in their proper relative positions in the cross-section whic h extended along the bridge from end-to-end. Also shown on this figure is a comparison of the reactions obtained from STRESS and from S APIV by applying all of the non-composite loads in a single loading. The comparison between these reactions is excellent. In order to verify that the order of pouring the deck slab units would not contribute to an uplift situation, the pouring sequence was replicated in a three-dimensional SAPIV analysis. The results of the analysis of the three stages of the pouring sequence are also shown in Figure 6.5.3-2, as well as the total accumulated load at the end of the pour. Comparison of the sequential loading with the application of a single loading of non-composite dead load also showed relatively good agreement in this case. The important point demonstrated in the example of Figures 6.5.3-1 and 6.5.3-2 is that seemingly small errors in modeling of the structure can result in very substantial changes in the reactions, shears and moments. The designer must be aware of this potential when utilizing the more refined analysis techniques. Incidentally, there are cases in which an uplift reaction due to skew and/or curvature is possible. The simple span bridge shown in Figure 6.5.3-3 and reported on in the November 1, 1984, issue of Engineering News-Record, was analyzed at the request of t he owner. In this case, the uplift reactions computed by the designer were verified. Sometimes modeling problems occur because User's Manuals are not clear, or a "bug" exists, of which the program's author/vendor is not aware. Such a case is illustrated for the simply-supported, partially-curved and skewed bridge in Figure 6.5.3-4. Initially, this bridge was modeled with extra joints at locations other than diaphragms in an effort to improve live load determination. As a result, the number of points along each girder were not equal, but there was no indication in the program's descriptive literature that this type of arrangement of program input could potentially cause a problem. The resulting moment envelopes for the middle and two
Lecture - 6-36
printed on June 24, 2003 exterior girders are shown in Figure 6.5.3-4, and are obviously unusual 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 responding properly to the unequal number of nodes per girder, that nodes should be essentially "radial", and that it was not certain that nodes to which diaphragms were not connected were legitimate. The revised model, shown in Figure 6.5.3-4, produced clearly better results, as shown in the indicated moment envelopes.
Lecture - 6-37
printed on June 24, 2003
Figure 6.5.3-1 - Framing plan, comparative live load reactions and moment envelopes showing effect of proper and improper rotational boundary condition, as reflected in grid analysis.
Lecture - 6-38
printed on June 24, 2003
Figure 6.5.3-2 - Finite element idealization and reactions obtained for structure shown in Figure 6.
Lecture - 6-39
printed on June 24, 2003
Figure 6.5.3-3 - Framing Plan of Curved Span with Skewed Piers
Figure 6.5.3-4 - Comparative moment envelopes for the middle and two outside girders of curved skewed system showing the results of apparent "bug" in algorithm for applying live load. 6.5.4 Other Types of Bridges The Specification contains additional requirements for the rigorous analysis of cellular-type structures, truss bridges, arch bridges, cable-stayed bridges and suspension bridges. Generally speaking, refined analysis will involve a computer model which accurately affects the geometry, relative component stiffnesses,
Lecture - 6-40
printed on June 24, 2003 boundary conditions and load supply to the structure. Suspension bridges will almost always be analyzed us ing a large deflection theory. The deflection theory may also be applied to arches and cable-stayed bridges. In the case of the cable-stayed bridge of moderate span, it may be sufficiently accurate to evaluate the second order effects on the deck system of the tower by supplementary calculations in providing a correction factor, developed on a bridge-specific basis. The change in stiffness of the cables caused by change in sag as the cable load changes can be accounted for using the so-called "Ernst" equations, given in the Specification, for modified modulus of elasticity.
Lecture - 6-41
printed on June 24, 2003 REFERENCES Jones, 1976, "A Simple Algorithm for Computing Load Distribution in Multi-Beam Bridge Decks", Proceedings, 8th ARRB Conference, 1976
Lecture - 6-42
printed on June 24, 2003
APPENDIX A The Load Distribution Problem and its Solution in NCHRP 12-26
printed on June 24, 2003 Older editions of the AASHTO Specifications allow for simplified analysis of bridge superstructures using the concept of a load distribution factor for bending moment in interior girders of most types of bridges, i.e., beam-and-slab, box girder, slab, multi-box beam and spread-box beam. This distribution factor is given by:
g
S 300 D
(A-1)
where: g
=
a factor used to multiply the total longitudinal response of the bridge due to a single longitudinal line of wheel loads in order to determine the maximum response of a single 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 have taken place over the last 55 years have led to inconsistencies in the load distribution criteria including: 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-supported bridges. Although it was stated that these procedures apply to the design of normal (i.e., supports oriented perpendicular to the longitudinal girders) highway bridges, there are no other guidelines for determining when the procedures are applicable. Because modern highway and bridge design practice requires a large number of bridges to be constructed with skewed supports, on curved alignments, or continuous over interior supports, it is increasingly important that the limitations of load distribution criteria be fully understood by designers. Advanced computer technology has become available in recent years which allows detailed finite element analysis of bridge decks. However, many computer programs exist which employ different formulations and techniques. It is important that the computer methodology and formulation that produces the most accurat e results be used to predict the behavior of bridge decks. In order to identify the most accurate computer programs, data from full-sc ale and prototype bridge load tests were compiled. The bridge tests were then modeled by different computer programs and the experimental and computer results were compared. The programs that produced the most accurate 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 of applicability. 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 selected in order to achieve national representation. This resulted in a database of 365 beam-and-slab bridges, 112 prestressed concrete and 121 reinforced concrete box girder bridges, 130 slab bridges, 67 multi-box beam bridges and 55 spread-box beam bridges. This bridge database was studied 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. A hypothetical bridge that has all the average properties obtained from the database, referred to as Lecture - 6-A1
printed on June 24, 2003 the "average bridge" was created for each of the beam-and-slab, box girder, slab, multi-box beam and 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 also prepared. In evaluating simplified formulas, it is important to understand the effect of various bridge parameters on load distribution. Bridge parameters were varied one at a time in the average bridge for the bridge type under consideration. Load distribution factors for both shear and moment were obtained for all such bridges. Variation of load distribution factors with each parameter shows the importance of each parameter. Simplified formulas can then be developed to capture the variation of load distribution factors with each of the important parameters. A brief description of the method used 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 function of the form axb, where x is the value of the given parameter, and a and b are coefficients to be determined based on the variation of x. Second, it is assumed that the effects of different parameters are independent of each other, which allows each parameter to be considered separately. 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 paramet ers included in 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 for S, then: g1 = (a)(S1b1)(Lb2)(tb3)(...)
(A-2)
g2 = (a)(S2b1)(Lb2)(tb3)(...)
(A-3)
therefore: g 1
S1
g 2
S2
b1
(A-4)
or:
ln b1 ln
g 1 g 2
(A-5)
S1 S2
If n different values of S are examined and successive pairs are used to determine the value of b1, n 1 different values for b1 can be obtained. If these b1 values are close to each other, an exponential 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 power factors, i.e., b1, b2, and so on, are determined, the value of "a" can be obtained from the average bridge, i.e.,
Lecture - 6-A2
printed on June 24, 2003
a
g o b1
So
b2
Lo
b3
t o
(A-6)
(...)
This procedure was followed during the entire course of the NC HRP 12-26 study to develop new formulas as needed. In certain cases where an exponential function was not suitable to model the effect of a parameter, slight variation from this procedure was used to achieve the required accuracy. However, this procedure worked quite well in most cases and the developed formulas demonstrate high accuracy. Because certain assumptions were made in the derivation of simplified formulas and some bridge parameters were ignored altogether, it is important to verif y the accuracy of these formulas when applied to real bridges. The database of actual bridges was used for this purpose. Bridges to which the formula can be applied were identified and analyzed by an accurate method. The distribution factors obtained from the accurate analysis were compared to the results of the simplified methods. The ratio of the approximate to accurate distribution factors was calculated and examined to assess the accuracy of the approximate method. Average, standard deviation, and minimum and maximum ratio values were obtained for each formula or simplified method. The method 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 slightly conservative results. The minimum and maximum values show the extreme predictions that each method or formula produced when a specific database was used. Although these values may change slightly if a different set of bridg es is used for evaluation, the minimum and maximum values allow identification of where shortcom ings in the formula may exist that are not readily identified by the average or standard deviation values. It was previously mentioned that different subsets of the database of bridges were used to evaluate different formulas. When a subset included a large number of bridges (100 or more), a Level 2 method was used as the basis of comparison. When it included a smaller number of bridges (less than 100), a Level 3 method was used. As a result, LANELL (an influence surface method) was used for verification of formulas for moment distribution in box girder bridges, and a Multi-dimensional Space Interpolation (MSI) method was used for verification of formulas for straight 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 the development of a number of computer programs for structural analysis. Programs that can be applicable to bridge deck analysis can be divided into two categories. One includes general purpose structural analysis programs such as SAP, STRUDL and FINITE. The other category is specialized programs for analysis of specific bridge types, such as GENDEK, CURVBRG and MUPDI. In the search for the best available computer program for analysis of each bridge type, all suitable computer programs (general and specific) that were available at the time of the 12-26 research were evaluated. In order to achieve meaningful comparisons and assess the level of accuracy 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-A3
printed on June 24, 2003 •
by comparison of the averages and standard deviations of the ratios of analytical to experimental results, and
•
by comparison of statistical differences of analytical and experimental results. Five bridge types were considered: beam-and-slab, box girder, slab, multi-box beam, and spread-box beam.
For analysis of beam-and-slab bridges, the following comput er programs and models were evaluated: GENDEK-PLATE, GENDEK-3, GENDEK-5, CURVBRG, SAP and MUPDI. It was found that, in general, GENDEK-5 analysis using plate elements for the deck slab and eccentric beam elements for the girders is very accurate. This program is also general enough to cover all typical cases, i.e., straight, skew, moment and shear. However, for analysis of curved open girder steel bridges, CURVBRG was the most accurate program. MUPDI was also found to be a very accurate and fast program; however, skewed bridges cannot be analyzed wit h this program and shear values near 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 were evaluated. MUPDI was the fastest and most practical program for analysis of straight bridges for moment, but FINITE was found to be the most practical program for skewed bridges and for obtaining accurate shear results. Therefore, MUPDI was selected for the evaluation of LANELL (a Level 2 method for moment in straight bridges which was, in turn, used for evaluation of Level 1 methods) and FINITE was selected for other cases. For the analysis of slab bridges, computer programs MUPDI, FINITE, SAP and GENDEK were evaluated. Shear results cannot be obtained accurately in slab bridges and, therefore, were not 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-skewed prismatic 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 of California, Berkeley, for analysis of multi-beam bridges (referred to as the POWELL program herein). Various modeling techniques were studied using different grillage models and different plate 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 later used in evaluation of more simplified methods. POWELL is also very accurate in reporting moments in straight bridges, but it uses a finite strip formulation, s imilar to MUPDI, and, therefore, is incapable of modeling skewed supports , and shear results near supports and load loc ations cannot be accurately obtained. This program was used to evaluate simplified methods for straight bridges. For analysis of spread-box beam bridges, computer programs SAP, MUPDI, FINITE and NIKE-3D were evaluated. FINITE produced the most accurate results, especially when shear was considered. MUPDI was selected to evaluate simplified methods for calculation of moments in straight 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 computer methods have been unavailable. The Ontario Highway Bridge Design Code uses one such method based on orthotropic plate theory. Other graphical methods have also been developed and reported. A good example of the influence surface method is the computer program SALOD Lecture - 6-A4
printed on June 24, 2003 developed by the University of Florida for the Florida Department of Transportation. This program uses influence surfaces, obtained by detailed finite element analysis, which are stored in a database accessed by SALOD. One advantage of influence surface methods is that the response of the bridge deck to different truck types can be readily computed. A grillage analysis using plane grid models can also be used with minimal computer resources to calculate the response of bridge decks in most bridge types. However, the properties for grid members must be calculated with care to assure accuracy. Level 2 methods used to analyze the five bridge types (beam-and-slab, box girder, slab, multi-beam and spread-box beam bridges) are discussed below. The following methods were evaluated for analysis of beam-and-slab bridges: plane grid analysis, the nomograph-based method included in the Ontario Highway Bridge Design Code (OHBDC), SALOD and Multi-dimensional Space Interpolation (MSI). All of these methods are applicable for single- and multi-lane loading for moment. The OHBDC curves were developed for a truck other than HS-20, and using the HS-20 truck in the ev aluation process may have introduced some discrepancies. The method presented in OHBDC was also found to be time consuming, and inaccurate interpolation between curves was probably a common source of error. SALOD can be used with any truck and, therefore, the "HS" truck was used in its evaluation. The MSI method was developed based on HS-20 truck loading for single- and multiple-lane loading. MSI was found to be the fastest and most accurate method and was, therefore, selected for the evaluation of Level 1 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 were evaluated. The comments made about OHBDC for beam-and-slab bridges are valid for box girder bridges 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 found to be the most accurate method and, thus, was used in the evaluation of Level 1 methods. SALOD also 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 accurate evaluation. In the analysis of multi-beam bridges, a met hod presented in Jones, 1976, was evaluated. The method is capable of calculating distribut ion factors due to a single concentrated load and was modified for this study to allow wheel line loadings. The results were found to be in very good agreement with POWELL. However, because this method was only applicable for moment distribution 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 a Level 2 method. In general, Level 2 graphical and influence surface methods generated accurate and dependable results. While these methods are sometimes difficult to apply, a major advantage of some of them is that different trucks, lane widths, and multiple presence live load reduction fact ors may be considered. Therefore, if a Level 2 procedure does not provide needed flexibility, its use is not warranted because the accuracy of it is on the same order as a simplified formula. MSI is an example 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 some of the methods mentioned above. In addition, a general purpose plane grid analysis program is Lecture - 6-A5
printed on June 24, 2003 available to most bridge designers. Therefore, this method of analysis is considered a Level 2 method. However, the user has the burden of producing a grid model that will produce sufficiently accurate 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 distribution factors for each of the bridge types studied. Level 1 Methods: Simplified Formulas The current AASHTO Specifications recommend use of simplified formulas for determining load distribution factors. Many of these formulas have not been updated in years and do not provide optimum accuracy. A number of other formulas have been developed by researchers in recent years. Most of these formulas are for moment distribution for beam-and-slab bridges subjected to multi-lane truck loading. While some have considered correction factors for edge girders and skewed supports, very little has been reported on shear distribution factors or distribution factors for bridges other than beam-and-slab. The sensitivity of load distribut ion factors to various bridge parameters was also determined as part of the study. In general, beam spacing is the most significant parameter. However, span length, 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 various bridge parameters for each bridge type. Ignoring the effect of bridge parameters, other than beam spacing, can result in highly inaccurate (either conservative or unconservative) solutions. A major objective of the research in Project 12-26 was to evaluate older AASHTO Specifications and other researchers' published work to assess their accuracy and develop alternate formulas whenever a more accurate method could be obtained. The formulas that were evaluated 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-A6
printed on June 24, 2003
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-A7
printed on June 24, 2003
Figure A-5 - Effect of Parameter Variation on Spread-Box Beam Bridges
Lecture - 6-A8
printed on June 24, 2003 LECTURE 6 - ANALYSIS I
6.1 OBJECTIVE OF THE LESSON The objectives of this lesson are to acquaint the student with: •
the various analysis techniques required and/or recommended for determining the force effects and components of bridges, and
•
the use of approximate and refined methods for the determination for force effects in conventional girder-type structures.
The background on the development of new, improved, distribution factors which were developed under NCHRP Project 12-26 has been included for reference in an Appendix. The use of grid and finite element types of analysis for multibeam bridges is also recommended in the LRFD Specification. These methods require considerable care in structural modeling, and several examples of the large effects of seemingly small errors in structural models will be presented.
6.2 ACCEPTABLE METHODS OF STRUCTURAL ANALYSIS Article S4.4 contains a list of methods of analysis that are considered suitable for analysis of bridges. These include the classical 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-type methods, analysis based on series expansions and the yield-line method for the non-linear analysis of plates and railings. Some of these methods of analysis are suitable for hand calculations, but for any problem of large size, some sort of computer solution will almost always be required for practical design purposes. This is because almost all of these methods, with the possible exception of the series methods and the yield-line methods, will eventually require the solution of large sets of simultaneous equations. The series method, while elegant from a mathematical point of view, will typically require a computer program to expand the series sufficiently to yield good results in a practical time frame. Yield-line methods, which could be considered the extension of plastic design to two-dimensional surfaces, are typically a hand calculation procedure. The use of computer programs in bridge design brings up a philosophical problem as to the responsibility for error. Almost all vendors of commercial computer programs disavow any responsibilit y for error. A release from liability is usually implicit in their use and may even be an explicit part of obtaining a license. This means that an organization using a computer program must be relatively certain of
Lecture - 6-1
printed on June 24, 2003 the results that it obtains. It is not necessary for every engineer in a large design section to have personally confirmed every computer program, but it is necessary that some verification testing be done or that the results of previous verification testing be obtained in order to produce the required level of confidence. Computer programs can be verified against universally accepted closed- form solutions, other computer programs which have been previously verified, or the results of testing. Many computer programs for design use also contain code checking capabilities. Others have portions of the applicable design specification embedded in the coding of the program. In order to identify the specification edition which may have been tied to a given release of a program and also to provide a means for determining which structures may have been designed with a version of a program later found to contain errors, the specification requires that a name, version and a release date of software be identified in the contract drawing.
6.3 PRINCIPLES OF MATHEMATICAL MODELING 6.3.1 Structural Material Behavior The LRFD Specification recognizes both elastic and inelastic behavior of materials for analysis purposes. Inelastic material behavior is implicit in many of the equations and procedures specified for the calculation of cross-sectional resistance, such as calculating the nominal resistance of a concrete beam or column, the nominal plastic moment resistance of a compact and adequately braced steel cross-section, or the bearing capacity of a spread footing. Often, the force effects to which this resistance will be compared will be calculated on a basis of a linear structural analysis with elastic material properties having been assumed. This dichotomy has existed in the bridge specification since the introduction of load factor design in the early 1970's. It continues through the LRFD Specification. On the other hand, there are certain assumed failure modes at extreme events and the use of mechanism and unified autostress design procedures for steel girders, where permitted, which require analysis based on non-linear behavior. Many times, this analysis will take a form analogous to plastic design of steel frames. For example, seismic design may be based on the formation of plastic hinges at the top and bottom of the columns of a bent. Ship collision forces may be absorbed in a comparable inelastic manner. Furthermore, it is anticipated that future seismic design provisions will be based on extensive research currently underway to develop a step-by-step nonlinear force displacement relationship for the lateral displacement of piers. Where inelastic analysis is used, the Designer must be cert ain that a ductile failure mode is obtained through proper detailing. Rules
Lecture - 6-2
printed on June 24, 2003 for achieving this are presented in the sections for steel and conc rete design. 6.3.2 Geometry 6.3.2.1 GENERAL Most analyses done for the purpose of designing bridges are based on the assumption that the displacements caus ed by the loads are relatively small and, therefore, it is suitably accurate to base the calculations on the undeformed shape. This is typically referred to as the small deflection theory, and it is routinely used in the design of beam-type structures and bridges which resis t loads through a couple whose tensile and compressive forces remain essentially in fixed positions relative to each other while the bridge deflects. This will be the case for a truss and for tied-arches. For other types of structures and components and for certain types of analysis, the effect of the deflections must be considered in the development of the force equilibrium equations, i.e., the equations of equilibrium are written for the displaced shape. Almost all engineers are aware that the study of structural stability requires consideration of the displaced shape, in fact, if the displaced shape is not part of the original formulation of the problem, one would never be able to determine that a column, shell or plate can buckle. Consider for a moment the simple pin-ended column. Unless the deflected shape of the column is taken into account, the moment caused by the axial load acting on the displaced shape would not be accounted for. It is this moment which causes the column to move laterally, i.e., to buckle. Almost a century ago, it was found that the only reasonably accurate way to calculate force effects in suspension bridges of any size was to include the deflection of the cable in the formulation of the problem and, therefore, the displacement of a stiffening truss or stiffening girder. As conventional, i.e., not tied, arches became longer and more slender, an effect directly analogous to that observed in suspension bridges can become significant enough that it must be accounted for in the design of the arch rib. In fact, because the arch rib is in compression and can buckle, the effect of large deflections can be especially important. Finally, the compression members of frames in bents can also be susceptible to this phenomenon. Where non-linear effects arriving either out of material nonlinearity or large deflections become significant, then super position of forces does not apply. This means that each load case under investigation must be studied separately under the full effect of all of the 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 a variety of individual loads and then combines, or superimposes, the
Lecture - 6-3
printed on June 24, 2003 force effects calculated for each individual load to make up whatever group 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 analyze each load combination separately, as identified above, certain approximate methods have been developed to allow the designer to add a correction to a set of force effects calc ulated in a linear manner. These are sometimes called single-step adjustment methods, the most commonly used of which is moment magnification for beam columns, which has been part of the AASHTO Specifications sinc e the early 1970's. For beam columns, the moment magnification proces s is given by the equations below. Mc = δb M2b + δs M2s
(6.3.2.2-1)
f c = δb f 2b + δs f 2s
(6.3.2.2-2)
for which: C m
δb
1
δs
1
1.0
P u
(6.3.2.2-3)
φP e
1 ΣP u
(6.3.2.2-4)
φΣP e
where: Pu
=
factored axial load (N)
Pe
=
Euler buckling load (N)
φ
=
resistance factor for axial compression as specified in Specification Sections 5, 6 and 7, as applicable
M2b
=
moment on compression member due to factored gravity loads that result in no appreciable sidesway calculated by conventional first order elastic frame analysis, always positive (N mm)
f 2b
=
stress corresponding to M2b (MPa)
M2s
=
moment on compression member due to factored lateral or gravity loads that result in sidesway, ∆,
Lecture - 6-4
printed on June 24, 2003 greater than u/1500, calculated by conventional first order elastic frame analysis, always positive (N mm) f 2s
=
stress corresponding to M2s (MPa)
It may appear that the moment magnification f actor contains the Euler buckling load, Pe. However, Pe is only a convenient substitution for a group of terms related to the displacement of the beam column. A derivation of the moment magnification equation can be found in many textbooks on steel or concrete design. For cases where the shape of the beam column is expected to be radically different from that given by the simply-supported case, or the loads significantly different from those indicated above, then it is possible to make an adjustment to account for a different init ial elastic shape through the factor cm. The moment magnification procedure has also been extended to arches, and this has been available in the AASHTO Specifications for many years and is reproduced as Article S4.5.3.2.2c with no further refinement. 6.3.2.3 REFINED METHODS The effect of large deflections can also be rigorously accounted for through iterative solutions of equilibrium equations, taking into account updated positions of the structure, or by using geometric stiffness terms. In some cases, e.g., the suspension bridge, solutions are available to the differential equations of equilibrium which can be solved in a trial and error fashion, or through series expan sion. 6.3.3 Modeling Boundary Conditions Points of expansion or other forms of articulation in the structure are commonly idealized as frictionless units. Where past practice indicates that this has been a reasonable conservative approach, continued use is warranted. There are other instances where the potential for nonfunctional expansion devices and/or the possibility that joints may close should also be investigated. This might be the case, for example, in a seismic analysis where analysis may be made, assuming that expansion joints are operable and open, and then another analysis might be made, assuming that they are closed and nonfunctional in order to simulate, or bound, t he effects of joints reaching the limits of travel during the earthquake. The possibility of reaching the limit of expansion travel should also be investigated when evaluating non-linear effects on substructure elements. It may be possible that the amount of moment magnification may be reduced because expansion dams will close, jamming the structure against the abutments before the full movement implicit in the moment magnification factor can be reached. This will reduce the moment magnification factor and, hence, the design moment.
Lecture - 6-5
printed on June 24, 2003 Similarly, the effect of boundary conditions at foundatio n units should also be evaluated. Foundation units are seldom fully fixed or fully pinned, and an evaluation of the potential movement of a foundation unit may be necessary in order to properly assess response, as well as secondary moments caused by change in geometry. Here again, bounding of the range of probable movement may 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 made based on the plan geometry of the superstructure. The first simplification involves the possibility of replacing the superstructure for analysis purposes wit h a single-line element called a spine beam. This may be done when the transverse distortion of the superstructure is small in comparison with the longitudinal deformation. Generally, if the superstructure is a torsionally stiff closed section or sections whose length exceeds 2.5 times their width, it may be idealized as a line element whose dimensions may be determined as given in the Specification. This can be used to significantly simplify analysis models. The second simplification deals with when it is possible to consider curved superstructures as straight for the purpose of analysis. If the superstructure is a torsionally stiff closed section and the central angle of a segment between piers is less than 12 , then the segment may be considered straight. If the superstructure is made of torsionally weak open sections, then the effects of curvature may be neglected when the subtended angle is less than that given in Table 6.4.1-1. Table 6.4.1-1 - Limiting Central Angle for Neglecting Curvature in Determining Primary Bending Moments
No. of Beams
Angle for One Span
Angle for Multiple Spans
2
2
3
3 or 4
3
4
5 or more
4
5
Lecture - 6-6
printed on June 24, 2003 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 deck slabs by analyzing a strip of deck as a continuous beam. Provisions are 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 secondary direction exceeds 1.5 times the spacing in the primary direction, then all of the wheel loads applied to the deck are considered to be applied to the primary strip. The secondary strip is designed on a basis of percentage of reinforcement in the primary strip. If the spacing of the supporting components on the secondary direction is less than 1.5 times that in the primary direction, then a crossed sticks analogy is used. The width of the equivalent strips in each direction is provided by Table S4.6.2.1.3-1 and the wheel load is distributed between two idealized intersecting strips according to the relative stiffness of each strip. Once the wheel loads have been assigned to the strips, for either case identified above, then the force effects are calculated based on a continuous beam. For the purpose of analyzing the continuous beam, the span length of each span is taken as a centerto-center of supporting components. For the purpose of calculating moment and shear at a design section, some offset from the theoretical center of support is permitted as given in the Specification. Decks which form an integral part of a cellular cross-section are supported on webs which are monolithic with the deck. Therefore, when the deck rotates, the web of the box girder rotates giving rise to bending stresses throughout the cross-section. For the purpose of analyzing this effect, a cross-sectional frame action procedure is identified in the Specification. In the case of fully filled and partially filled grids, the results of recent research are incorporated in LRFD Article S4.6.2.1.8 to given bending 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 for assigning portions of a design lane to a supporting component. These are commonly called distribution factors. It is important to remember that the approximate distribution factors, specified in the LRFD Specification, are on a lane, i.e., axle basis, not a wheel basis. The distribution factors are given for t he various kinds of bridges shown in Figure 6.4.2.2.1-1.
Lecture - 6-7
printed on June 24, 2003 SUPPORTING COMPONENTS
TYPE OF DECK
Steel Beam - Revised Factors
Cast-in-place concrete slab, precast concrete slab, steel grid, glued/spiked panels, stressed wood
Closed Steel or Precast Concrete Boxes - Revised Factors
Cast-in-place concrete slab
Open Steel or Precast Concrete Boxes - Revised Factors
Cast-in-place concrete slab, precast concrete deck slab
Cast-in-Place Concrete Multicell Box - Revised Factors
Monolithic concrete
Cast-in-Place Concrete Tee Beam - Revised Factors
Monolithic concrete
Precast Solid, Voided or Cellular Concrete Boxes with Shear Keys - Revised Factors
Cast-in-place concrete overlay
Precast Solid, Voided or Cellular Concrete Box with Shear Keys and with or without Transverse Post-Tensioning - Revised Factors (in some cases)
Integral concrete
Precast Concrete Channel Sections with Shear Keys
Cast-in-place concrete overlay
Precast Concrete Double Tee Section with Shear Keys and with or without Transverse Post-Tensioning
Integral concrete
Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning
Integral concrete
Concrete I or Bulb-Tee Sections - Revised Factors
Cast-in-place concrete, precast concrete
Wood Beams - Revised Factors
Cast-in-place concrete or plank, glued/spiked panels or stressed wood
TYPICAL CROSSSECTION
Figure 6.4.2.2.1-1 - Common Deck Superstructures Covered in LRFD Specification Articles 4.6.2.2.2 and 4.6.2.2.3
Lecture - 6-8
printed on June 24, 2003 Some of the distribution factors are new to the Specification as a result of an extensive project on load distribution known as NCHRP Project 12-26. Where the distribution factors for a given type of cross-section have been developed under that project and are new to the Specification, the words "revised factors" appear in the column identified as "supporting components". Where those words do not appear, the distribution factors have been retained from earlier editions of the AASHTO Standard Specifications. Some simplifications have been made in utilizing the distribution factors from NCHRP 12-26. In particular, correction factors for various aspects of structural action, typically involving continuity, which were less than 5%, were omitted from the LRFD Specifications. Similarly, an increase in moments over piers, thought to be on the order of 10%, was not included because stresses at or near internal bearings have been shown to be reduced below that calculated 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 the multiple presence factors, given in Lecture 3, are built into the distribution factors, whereas, the multiple presence factors in earlier editions of the AASHTO Standard Specifications are built into the NCHRP 12-26 factors. Additionally, the factors appropriate for the LRFD Specification are based on a lane of live load, rather than a "line of wheels". Finally, when the SI version of the LRFD Specification is used, conversion to that system of units has also been accou nted for. Various limits on span, spacing and other characteristics are provided in the Specifications for each of the distribution factors. These parameters identify the range for which the factors were developed. They were not evaluated for factors beyond the ranges indicated. Therefore, for structures which do not comply with these limitations, a rigorous analysis by grid or finite elements should be used. Furthermore, the distribution factors usually apply for structures which are: •
essentially constant in deck width,
•
have four or more beams, unless noted,
•
have beams which are parallel and approximately of the same stiffness,
•
have overhangs that do not exceed 0.9 m, unless specifically noted,
•
have in-plan curvatures less than those specified above, and
•
have a cross-section consistent with one of the cross-sections identified 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 - 6-9
printed on June 24, 2003 and development of the new distribution factors. The discussion below was taken from the NCHRP Research Resul ts Digest No. 187, a summary of the Final Report of Project 12-26 as summarized by Ian M. Friedland, NCHRP Project Coordinator. Live load distribution on highway bridges is a key response quantity in determining member size and, consequently, strength and serviceability. It is of critical importance both in the design of new bridges and in the evaluation of the load-carrying capacity of existing bridges. Using live load distribution factors, engineers can predict bridge response by treating the longitudinal and transverse effects of live loads as uncoupled phenomena. Empirical live load distribution factors for stringers and longitudinal beams have appeared in the AASHTO Standard Specifications for Highway Bridges with only minor changes since 1931. Findings of recent studies suggest a need to update these specifications in order to provide improved predictions of live load distribution. Live load distribution is a function of the magnitude and location of truck live loads and the response of the bridge to these loads. The NCHRP 12-26 study focused on the second factor mentioned above: the response of the bridge to a predefined set of loads, namely, the HS family of trucks. In Project 12-26, three levels of analysis were considered for each bridge type. The most accurate level, Level 3, involves detailed modeling of the bridge deck. Level 2 includes either graphical methods, nomographs and influence surfaces, or simplified computer programs. Level 1 methods provide simple formulas to predict lateral load distribution, using a wheel load distribution factor applied to a truck wheel line to obtain the longitudinal response of a single girder. The major part of the research in Project 12-26 was devoted to the Level 1 analysis methods because of its ease of application, established use, and the surprisingly good correlation with the higher levels of analysis in their application to a majority of bridges. The formulas presented in the current AASHTO specifications were evaluated, and alternative formulas were developed that offer improved accuracy, wider range of applicability, and in some cases, easier application than the current AASHTO formulas. These formulas were developed for interior and exterior girder moment and shear load distribution for single or multiple lane loadings. In addition, correction factors for continuous superstructures and skewed bridges were developed. The formulas presented in previous AASHTO Specifications, although simpler, do not present the degree of acc uracy demanded by today's Bridge Engineers. In some cases, these formulas can result in highly unconservative results (more than 40%), in other cases they may be highly conservative (more than 50%). In general, the formulas developed in Project 12-26 are within 5% of the result s of an accurate
Lecture - 6-10
printed on June 24, 2003 analysis. Table 6.4.2.2.1-1 shows comparisons with moment distribution factors obtained from AASH TO, Level 1, Level 2 and Level 3 methods for simple span bridges. Table 6.4.2.2.1-1 - Comparison of interior girder moment distribution factors by varying levels of accuracy using the "average bridge" for each bridge type
NCHRP 12-26 (Level 1)
Grillage (Level 2)
Finite Element (Level 3)
1.413 (S/1700)
1.458
1.368
1.378
Box girder a
1.144
1.143
0.970
1.005
Slabb
1820
1710
1900
1890
Multi-box beama
0.646
0.597
0.540
0.552
Spread box beama
1.564
1.282
1.248
1.241
Bridge Type Beam-and-slaba
AASHTO
a
Number of wheel lines per girder
b
Wheel line distribution width, in mm
In addition, the study resulted in recommendations for use of computer programs to achieve more accurate results. The recommendations focus on the use of plane grid analysis, as well as detailed finite element analysis, where different truck types and their combinations may be considered. 6.4.2.2.2 Influence of Truck Configuration The formulas developed in Project 12-26 for the Level 1 analysis were based on the standard AASHTO "HS" trucks. A limited parametric study conducted as part of the research showed that variations in the truck axle configuration or truck weight do not significantly affect the wheel load distribution factors. The group of axle trains used for this study are shown in Figure 6.4.2.2.2-1. It is anticipated that smaller gage widths would result in larger distribution factors, and larger gage widths would result in smaller distribution factors. Table 6.4.2.2.1-1 gives the variation of wheel load distribution factors with different axle configurations applied to a number of beamand-slab bridges. The differences were below 1% in many cases and, in all cases, the formulas resulted in good predictions. Therefore, with some caution, these formulas may be applied to other truck types. Obviously, Levels 2 and 3 analyses may also be applied for trucks significantly different from the AASHTO family of trucks.
Lecture - 6-11
printed on June 24, 2003
Figure 6.4.2.2.2-1 - Axle Configurations for Truck Types Considered in Study Table 6.4.2.2.1-1 - Effect of Load Configuration on Distribution Factor DISTRIBUTION FACTOR (g)
*
PERCENT DIFFERENCE WITH HS-20
HS-20
HTL-57
4A-66
B-141
NCHRP 12-26
HTL-57
4A-66
B-141
NCHR P 1226
Average*
1.293
1.261
1.285
1.268
1.304
-2.4
-0.6
-1.9
+0.9
Max. S 5000 mm
2.220
2.162
2.205
2.178
2.308
-2.6
-0.7
-1.9
+4.0
Min. S 1000 mm
0.713
0.717
0.713
0.715
0.755
+0.6
0.0
+0.3
+5.9
Max. L 60 000 mm
0.982
0.958
0.983
0.952
1.033
-2.4
+0.1
-3.1
+5.2
Min. L 6000 mm
1.630
1.625
1.624
1.623
1.807
-0.3
-0.3
-0.4
+10.9
S L ts Kg
= = = =
2200 mm 20 000 mm 185 mm 2.33 x 1011 mm4
Lecture - 6-12
printed on June 24, 2003 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 previous studies, and many simplified methods and formulas were developed by previous researchers for multi-lane loading moment distribution factors. The AASHTO formula, the formulas presented by other researchers, and the formulas developed in the study are discussed in the following according to their application. Table 6.4.2.2.3a-1 is taken from the specifications and summarized criteria for moment in interior beams or elements for various types of cross-sections. Similar tables exist for moment in exterior griders, for shear in interior girders and shear in exterior girders. Table 6.4.2.2.3a-2 describes how the term L (length) may be determined for use in the live load distribution factor equations given in Table 6.4.2.2.3a-1. In the rare occasion when the continuous span arrangement is such that an interior span does not have any positive uniform load moment (i.e., no uniform load points of contraflexure), the region of negative moment near the interior supports would be increased to the centerline of the span, and the L used in determining the live load distribution factors would be the average of the two adjacent spans.
Lecture - 6-13
printed on June 24, 2003 Table 6.4.2.2.3a-1 - Distribution of Live Loads Per Lane for Moment in Interior Beams Type of Beams
Wood Deck on Wood or Steel Beams Concrete Deck on Wood Beams Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, Tand Double T-Sections
Applicable Cross-Section from Table 4.6.2.2.1-1
Distribution Factors
a, l l a, e, k and also i, j if sufficiently connected to act as a unit
See Table S4.6.2.2.2a-1 One Design Lane Loaded: S/3700 Two or More Design Lanes Loaded: S/3000 One Design Lane Loaded: S 4300
0.06
0.4
S L
0.3
0.1
K g 3
Lt s
S 2900
1800
1100 S 4900 110 ts 300 6000 L 73 000 Nb 4
d
0.6
0.2
S L
0.1
K g 3
Lt s
use lesser of the values obtained from the equation above with Nb = 3 or the lever rule
Nb = 3
One Design Lane Loaded:
2100 S 4000 18 000 L 73 000 Nc 3
S 1100
1.75
300 L
0.35
1 N c
0.45
Two or More Design Lanes Loaded: 13 N c Concrete Deck on Concrete Spread Box Beams
S
Two or More Design Lanes Loaded: 0.075
Multicell Concrete Box Beam
Range of Applicability
b, c
0.3
S 430
1 L
0.25
One Design Lane Loaded: S 910
0.35
Sd
If Nc > 8 use Nc = 8
0.25
L2
1800 S 3500 6000 L 43 000 450 d 1700 Nb 3
Two or More Design Lanes Loaded: S 1900
0.6
Sd
0.125
L2
Use Lever Rule
Lecture - 6-14
S > 3500
printed on June 24, 2003 Type of Beams
Applicable Cross-Section from Table 4.6.2.2.1-1
Concrete Beams used in Multibeam Decks
f
Distribution Factors
One Design Lane Loaded: k
h
0.5
b 2.8L
where: g if sufficiently connected to act as a unit
Range of Applicability
I J
900 b 1500 6000 L 37 000 5 Nb 20
0.25
k 2.5(N b )
0.2
1.5
Two or More Design Lanes Loaded: b k 7600
0.6
b L
0.2
I J
0.06
Regardless of Number of Loaded Lanes: S/D where: C = K (W/L)
K
Skew
D = 300 [11.5 - NL + 1.4NL (1 - 0.2C)2] when C 5
NL
45°
6
D = 300 [11.5 - NL] when C > 5 g, i, j if connected only enough to prevent relative vertical displacement at the interface
Steel Grids on Steel Beams
Concrete Deck on Multiple Steel Box Girders
a
b, c
(1
K=
µ) I J
for preliminary design, the following values of K may be used: Beam Type Nonvoided rectangular beams Rectangular beams with circular voids: Box section beams Channel beams T-beam Double T-beam
K 0.7 0.8 1.0 2.2 2.0 2.0
One Design Lane Loaded: S/2300 If tg< 100 mm S/3050 If tg 100 mm Two or More Design Lanes Loaded: S/2400 If tg< 100 mm S/3050 If tg 100 mm
S
1800 mm
S
3200 mm
Regardless of Number of Loaded Lanes: 0.05
0.85
N L N b
0.425 N L
Lecture - 6-15
0.5
N L N b
1.5
printed on June 24, 2003 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 which moment is being calculated.
Negative Moment - End spans of continuous spans, from end to point of contraflexure under a uniform load on all spans
The length of the span for which moment is being calculated.
Negative Moment - Near interior supports of continuous spans, from point of contraflexure to point of contraflexure under a uniform load on all spans
The average length of the two adjacent spans.
Positive Moment - Interior spans of continuous spans, from point of contraflexure to point of contraflexure under a uniform load on all spans
The length of the span for which moment is being calculated.
Shear
The length of the span for which shear is being calculated.
Exterior Reaction
The length of the exterior span.
Interior Reaction of Continuous Span
The average length of the two adjacent spans.
Moment Distribution to Interior Girders, Multi-Lane Loading The AASHTO formula for moment distribution for multi-lane loading is given as S/1800 for reinforced concrete T-beam bridges with girder spacing up to 3000 mm, and as S/1700 for steel girder bridges and prestressed concrete girder bridges with girder spacing up to 4300 mm, where S is the girder spacing. When the girder spacing is larger than the specified limit, s imple beam distribution is to be used to calculate the load distribution factors. Marx, et al, at the University o f Illinois, developed a formula for wheel load distribution for moment which included multiple lane reduction factors and is applicable to all beam-and-slab bridges. The formula is based on girder spacing, span length, slab thickness and bridge girder stiffness. A formula which does not consider a reduction for multi-lane loading was developed at Lehigh University. The Lehigh formula includes terms for the number of traffic lanes, number of girders, girder spacing, span length and total curb-to-curb deck width. Sanders and Elleby (NCHRP Report 83) developed a simple formula based on orthotropic plate theory for moment distribution on beam-and-slab bridges. Their formula includes terms for girder
Lecture - 6-16
printed on June 24, 2003 spacing, number of traffic lanes and a stiffness parameter based on bridge type, bridge and beam geometry and material properties. A full-width design approach, known as Henry's Method, is used by the State of Tennessee. Henry's Method includes factors for number of girders, total curb-to-c urb bridge deck width and a reduction factor based on number of lanes. A formula developed as part of NCHRP Project 12-26 includes the effect of girder spacing, span length, girder inertia and slab thickness. The multiple lane reduction factor is built into the formula. This formula, applicable to cross-sections with four or more beams, is given by:
g 0.075
S 2900
0.6
S L
0.2
K g
0.1
(6.4.2.2.3a-1)
3
Lt s
where: S
=
girder spacing (1100 mm
L
=
span length (6000 mm
L
Kg
=
n(I+Aeg2) (4 x 108
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 of girder to mid-point of slab)
ts
=
slab thickness (110 mm
Kg
S
ts
4900 mm)
73 000 mm)
300 mm)
This formula is dependent on the inertia of the girder and, thus, a value for Kg must be assumed for initial design. For this purpose, Kg/LtS3 may be taken as unity. All of the above formulas were evaluated using direct finite element analysis with the GENDEK-5 program and a database of 30 bridges; subsequently, they were evaluated using the MSI method and database of more than 300 bridges. It was found that Equation 6 and the Illinois formulas are accurate and produce results that are as accurate as the Level 2 methods. Moment Distribution to Exterior Girders, Multi-Lane Loading Previous AASHTO Specifications recommend a simple beam distribution of wheel loads in the transverse direction for calculating wheel load distribution factors in edge girders. Any load that falls
Lecture - 6-17
printed on June 24, 2003 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 interior girder is distributed to these girders by assuming t hat the slab acts as a simple beam in that region. Any wheel load that falls inside of the first interior girder is assumed to have no effect on the edge girder. Marx, et al, at the University o f Illinois, developed a formula for the exterior girder based on certain assumptions in the placement of loads and may not be applicable to all bridges. This formula includes terms similar to those used in their formula for moment distribution to interior girders. A formula, depending on wheel position, alone was developed as part of this study which results in a correction factor for the edge girder. The factor must be applied to the distribution factor for the interior girder to obtain a distribution factor for the edge girder. This formula is given by:
e
0.77
d e
(6.4.2.2.3a-2)
2800
where: de
=
distance from edge of the roadway, usually the face of curb, to the center of the exterior web of the exterior cell, in mm
If the edge of the lane is outside of the exterior girder, the distance is positive; if the edge of the lane is to the interior sid e of the girder, the distance is negative. It was found that the formula developed in Project 12-26 resulted in accurate correction factors and was simpler than the previous AASHTO procedure. Moment Distribution to Interior Girders, Single-Lane Loading The literature search performed in this study did not reveal any simplified formula for single-lane loading of beam-and-slab bridges. The formula developed as part of the study is as follows:
g 0.06
S 4300
0.4
S L
0.3
K g 3
Lt s
0.1
(6.4.2.2.3a-3)
where the parameters are the same as those given for Equation 6.4.2.2.3a-1. This formula is applicable to interior girders only.
Lecture - 6-18
printed on June 24, 2003 Moment Distribution to Exterior Girders Simple beam distribution in the transverse direction should be used for single-lane loading of edge girders. One other investigation, applicable to load distribut ion for both shear and moment, is required for exterior beams of beam-slab bridges with diaphragms or cross-frames. This addition was not developed as part of NCHRP 12-26, but was added by the NCHRP 12-33 Editorial Committee. This distribution is based on treating the cross-section as a transversely rigid unit which deflects and rotates as a straight line. The live load is positioned for maximum effect on an exterior beam (one lane, two lane, three lane, etc., each with its appropriate multiple presence factor). The total vertical force and moment about the centroid of the cross-section is applied to the area of the cross-section, i.e., the number of beams, and the section modulus, i.e., the sum of the square of the distances of each beam from the centroid of the beams divided by the distance to the exterior beam. The specification puts this in equation form as: N L
R
N L N b
X ext
ev (6.4.2.2.3a-4)
N b
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 lane load from the center of gravity of the pattern of girders (mm)
x
=
horizontal distance from the center of gravity of the pattern of girders to each girder (mm)
Xext
=
horizontal distance from the center of gravity of the pattern of girders to the exterior girder (mm)
Nb
=
number of beams or girders
Shear Distribution No formula was found from previous research for the calculation of wheel load distribution factors for shear. Therefore, the formulas developed as part of the 12-26 study are reported for different cases as follows. The formula for multi-lane loading of interior girders is:
Lecture - 6-19
printed on June 24, 2003
g 0.2
S 3600
S 10 700
2
(6.4.2.2.3a-5)
The correction formula for multi-lane loading edge girder shear is: e
0.6
d e 3000
(6.4.2.2.3a-6)
The formula for shear distribution factor due to single-lane loading is:
g 0.36
S 7600
(6.4.2.2.3a-7)
Equation 6.4.2.2.3a-7 is applicable to interior girders only. Simple beam distribution in the transv erse direction should be used for single-lane loading of edge girders. Correction for Skew Effects Previous AASHTO Specifications did not include approximate formulas to account for the effect of skewed supports. However, some researchers have developed correction factors for such effects on moments in interior girders. Marx, et al, at the University of Illinois, developed four correction formulas for skew, one each for skew angles of 0, 30, 45 and 60 degrees. Corrections for other values of skew are obtained by straight-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 concrete I-girders was developed as part of the research performed at Lehigh University. This formula is based on the number of traffic lanes, number of girders, girder spacing, span length and total curb-to-curb deck width, and includes a variable term for skew angle. A correction factor for moment in skewed supports was also developed as part of Project 12-26. This formula is: r 1
c 1 (tan θ)1.5
(6.4.2.2.3a-8)
where, for θ > 30 ,:
Lecture - 6-20
printed on June 24, 2003
c 1
0.25
0.25
K g 3
Lt s
S L
0.5
(6.4.2.2.3a-9)
If θ 30 , c1 is taken as zero. In calculating c1, θ should not be taken as greater than 60 . The other parameters are as defined previously. From the literature review, no correction formulas were obtained for shear effects due to skewed supports. In Project 12-26, it was found that shear in interior girders need not be corrected for skew effects; that is, the shear distribution to interior girders is similar to that of a straight bridge. A correction formula for shear at the obtuse corner of the exterior girder of two girder systems and all girders of a multi-girder bridge was developed as part of this study and is given as: 3
r 1
0.2
L t s
k g
0.3
tan θ
(6.4.2.2.3a-10)
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 dis tribution factor in the exterior girder of non-skewed bridges. Therefore, the product of factors g, e and r must be obtained to find the obtus e corner shear distribution factor in a beam-and-slab bridge. The distribution factors calculated for moments are plotted as a function of girder spacing for Spans 9, 18, 27, 36 and 60 m in Figure 6.4.2.2.3a-1. For comparisons, AASHTO (1989) distribution factors are also shown. Girder distribution factors, specified by AASHTO (1989), are conservative for larger girder spacing. For shorter spans and girder spacings, AASHTO (1989) produces smaller distribution factors than calculated values.
Lecture - 6-21
printed on June 24, 2003
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 by various researchers in the past. Bridge deck behavior has been well studied and many recommendations have been made for detailed analysis of these bridges. However, there is a limited amount of information on simplified wheel load distribution formulas in the literature. In this context, a “girder” is a notional I-shape consisting of one web of a multi-cell box and the associated half-flanges on each side of the web. Moment Distribution to Interior Girders Scordelis, at the University of California, Berkeley, presented a formula for prediction of wheel load distribution for moment distribution in prestressed and reinforced concrete box girder bridges. The formula is based on modification of distribution factors obtained for a rigid cross-section. The formula predicts load distribution factors in reinforced concrete box girders with high accuracy and for prestressed concrete box girders with acceptable accuracy. Sanders and Elleby also presented a simple formula for moment distribution factors which is similar to their formula for beamand-slab bridges. The following formulas, developed as part of NCHRP 12-26, may be used to predict the moment load distribution factors in the interior girders of concrete box girder bridges due to single-lane and multi-lane loadings. These formulas are applicable to both reinforced
Lecture - 6-22
printed on June 24, 2003 and prestressed concrete bridges, and the multiple presence factor is accounted for. For single-lane loading:
g
S 1100
1.75
300 L
0.35
1 N c
0.45
(6.4.2.2.3b-1)
For multi-lane loading:
g
13 N c
0.3
S 430
1 L
0.25
(6.4.2.2.3b-2)
where: S
=
girder spacing, in mm
L
=
span length, in mm
Nc
=
number of cells
Moment Distribution to Exterior Girders The factor for load distribution for exterior girders shall be We/4300 mm, where We is the width of the exterior girder, taken as the top slab width measured from the mid-point between girders to the edge of the slab. Shear Distribution No formula for shear load distribution was obtained from previous research for box girder bridges, but the following were developed as part of NCHRP 12-26. The shear distribution factor for interior girder multi-lane loading of reinforced and prestressed concrete box girder bridges is: g
S 2200
0.9
d L
0.1
(6.4.2.2.3b-3)
where: S = girder spacing, in mm d = girder depth, in mm L = span length, in mm
Lecture - 6-23
printed on June 24, 2003 The distribution factor for shear in the interior girders due to single-lane loading may be obtained from: S 2900
g
0.6
d L
0.1
(6.4.2.2.3b-4)
where the parameters are as defined in Equation 6.4.2.2.3b-3. A correction formula for shear in the exterior girder for multilane loading is:
e
0.64
d e
(6.4.2.2.3b-5)
3800
where: de
=
distance from edge of the roadway, usually the face of curb, to the center of the exterior web of the exterior cell, in mm
Correction for Skew Effects The following formula was developed for correction of moment due to skewed supports for values of θ from 0 to 60 : r 1.05
0.25 (tan θ)
1.0
(6.4.2.2.3b-6)
If θ > 60 , use 60 in Equation 6.4.2.2.3b-6. Another formula was developed in Project 12-26 for correction of shear at the obtuse corner of an edge girder. It must be applied to the shear distribution factor for the edge girder of a non-skew ed bridge and must, therefore, be used in conjunction with the edge girder correction factor of Equation 6.4.2.2.3b-5. This formula, applicable for values of θ up to 60 , is: r 1
c 1 (tan θ)
(6.4.2.2.3b-7)
where: c1
=
0.25 + L/(70d)
d
=
bridge depth, in mm
L
=
span length, in mm
Lecture - 6-24
printed on June 24, 2003 6.4.2.2.3c Simplified Formulas for Slab Bridges The literature search did not reveal any simplified formulas for wheel load distribution in slab bridges other than those recommended by AASHTO. Therefore, the following are formulas that were developed as part of NCHRP 12-26. Moment Distribution, Multi-Lane Loading Equation 6.4.2.2.3c-1 was developed to predict wheel load distribution (distribution design width) for mom ent in slab bridges due to multi-lane loading. Multiple presence factors are already accounted for in the formula:
E 2100
0.12 L1 W 1
0.5
W N L
(6.4.2.2.3c-1)
where: E
=
the transverse distance over which a wheel line is distributed
L1
=
L
W1
=
W
L
=
span length, in mm
W
=
bridge width, in mm, edge-to-edge
18 000 mm 18 000 mm
Moment Distribution, Single-Lane Loading The equation below predicts wheel load distribution for moment due to single-lane loading. E 250
0.42 (L1 W 1 )0.5
(6.4.2.2.3c-2)
where the parameters are as defined in Equation 6.4.2.2.3c-1. Correction for Skew Effects Equation 6.4.2.2.3b-6 may be used to account for the reduction of moment in skewed bridges. According to previous AASHTO Specifications, slab bridges are adequate for shear if they are designed for moment. A quick check of this assumption was made and it was concluded that it is a valid assumption. Therefore, no formula or method is presented for calculation of shear in slab bridges.
Lecture - 6-25
printed on June 24, 2003 6.4.2.2.3d Simplified Formulas for Multi-Beam Decks which are Sufficiently Interconnected to Act as a Unit Only one formula, other than those presented in the previous AASHTO Specifications, was obtained for load distribution in multibeam decks. This formula, developed by Arya at the University of Illinois, is applicable to both box and open section multi-beam bridges and predicts interior beam moment responses due to single-lane and multi-lane loading. However, a number of simplified formulas developed in the study are valid only for multi-box beam decks and do not apply to open sections. Therefore, the response of multi-beam decks made of open members, such as channels, may or may not be accurately predicted by the formulas developed in that study. Moment Distribution to Interior Girders, Multi-Lane Loading The formula developed by Arya for interior girder load distribution in multi-beam decks includes terms for the maximum number of wheels that can be placed on a transverse section of the bridge, number of beams, beam width and span length. A variation of the formula was also proposed for multi-beam decks made of channels, which includes consideration of the overall depth of the channel section and its average thickness, defined as its area divided by its length along the centerline of the thickness. The following formula was developed in Project 12-26 to predict load distribution factors for interior beam moment due to multilane loading. The multiple presence reduction factor is already accounted for in the formula.
g
b k 7600
0.6
b L
0.2
I J
0.06
(6.4.2.2.3d-1)
where: 0.2
k
=
2.5(Nb)
1.5
b
=
beam width, in mm
L
=
span length, 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 a beam; an estimated value for these properties must, therefore, be used in preliminary design. The term I/J may be taken as unity for this case.
Lecture - 6-26
printed on June 24, 2003 Moment Distribution to Interior Girders, Single-Lane Loading Arya also presented a load distribution formula for multi-beam decks designed for one traffic lane. The formulation and parameters were similar to those presented for multi-lane loading. A variation of that equation was also presented for calculation of the interior beam moment distribution factor for a single-lane, channel section, multibeam deck. It should be noted that Arya's equations are not applicable to cases of only one-lane loading with more than one traff ic lane. A formula for wheel load distribution for moment in the interior girders due to single-lane loading was also developed in NCHRP 1226. This formula is as follows:
g
b k 2.8L
0.5
I J
0.25
(6.4.2.2.3d-2)
All parameters are defined in Equation 6.4.2.2.3d-1. Equation 6.4.2.2.3c-2 is also dependent on inertia and torsional constants, and a value of 1.0 may be used as an approxim ation for the term I/J during preliminary design. Moment Distribution to Exterior Girders The moment in the edge girder due to multi-lane loading in multi-beam decks comprised of box units is obtained by using a correction factor applied to the interior girder distribution factors for multi-lane loading. This correction factor may be found from the following formula:
g 1.04
d e 7600
(6.4.2.2.3d-3)
where: de =
distance from edge of the lane to the center of the exterior web of the exterior girder, in mm
For exterior beams of sufficiently interconnected multi-beam bridge decks comprised of T-shaped units subjected to multi-lane loading, Equation 6.4.2.2.3b-2 applies. For single-lane loading and for multi-beam decks comprised of either box units or units other than box units, the lever rule is used.
Lecture - 6-27
printed on June 24, 2003 Shear Distribution Distribution factors for shear in interior girders of multi-beam decks in "Bridge Decks Comprised of Box Units" due to multi-lane loading may be calculated from the following formula: 0.4
b 4000
g
b L
0.1
I J
0.05
(6.4.2.2.3d-4)
where the parameters are as defined in Equation 6.4.2.2.3d-1. Distribution factors for shear in the interior girders of multibeam decks in "Bridge Decks Comprised of Box Units" due to singlelane loading are obtained from the following formula: b g 0.70 L
0.15
I J
0.05
(6.4.2.2.3d-5)
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 are dependent on inertia and torsional constants, and a value of 1.0 may be used as an approximation for the term I/J during preliminary design. The shear in the edge girder of multi-beam deck in "Bridge Decks Comprised of Box Units" due to multi-lane loading can be found using a correction factor applied to interior girder distribution factors. This correction factor is obtained from the formula:
e
1.02
d e 15 000
(6.4.2.2.3d-6)
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder, in mm
For shear in exterior beams of sufficiently interconnected mult ibeam bridge decks comprised of T-shaped units, Equations 6.4.2.2.3b-4 through 6.4.2.2.3b-5 and the lever rule should be used, where appropriate. Correction for Skew Effects The moment in any beam in a skewed bridge may be obtained by using a skew reduction factor given by Equation 6.4.2.2.3b-6.
Lecture - 6-28
printed on June 24, 2003 The shear in the interior beams of a skewed multi-beam bridge comprised of box beams is usually of the same order as that of the shear in the obtuse corner and must be obtained by applying a correction factor to the respons e of the edge girder in a straight bridge. This correction factor may be calculated from the formula: c 1 (tan θ)0.5
r 1
(6.4.2.2.3d-7)
where:
c 1
L 90d
(6.4.2.2.3d-8)
6.4.2.2.3e Simplified Formulas for Multi-Beam Decks which are not Sufficiently Interconnected to Act as a Unit The LRFD Specification contains the same provis ions for load distribution in this type of bridge superstructure as appeared in recent editions of the Standard Specification, and as repeated below for completeness. The key difference between bridges treated herein, as compared to Section 6.4.2.2.3d, is the degree of transverse interconnection of units. If box, T, channel or other precast units are interconnected through a structural slab, or sufficiently transversely post-tensioned to produce a similar level of continuity, then the discussion of Section 6.4.2.2.3d applies. If the interconnection between the units is expected to transmit shear, but relatively little moment over the bridge service life, then the provisions herein apply. The Specification provides for the computation of a bending moment distribution factor, regardless of the number of lanes, given by:
g
S 300D
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)I J
Lecture - 6-29
printed on June 24, 2003 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 by AASHTO, was obtained from previous research for determining load distribution factors in spread box beam bridges. This formula was developed at Lehigh University for predicting the response of interior beams due to multi-lane loading and was later adopted by AASHTO. A correction factor for skewed bridges was also presented. In addition, a number of simple formulas were developed as part of NCHRP Project 12-26. Moment Distribution to Interior Beams, Multi-Lane Loading A formula developed in Project 12-26 for moment in interior spread box beams due to multi-lane loading is as follows:
g
S 1900
0.6
Sd
0.125
L2
(6.4.2.2.3f-1)
where S
=
girder spacing (mm)
L
=
span length (mm)
d
=
beam depth (mm)
Moment Distribution to Interior Beams, Single-Lane Loading A similar formula was developed for distribution to interior beams due to single-lane loading:
Lecture - 6-30
printed on June 24, 2003 S 910
g
0.35
Sd
0.25
(6.4.2.2.3f-2)
L2
where the parameters are as defined in Equation 6.4.2.2.3f-1. Moment Distribution to Exterior Girders The moment in edge girders due to multi-lane loading may be calculated by applying a correction factor to the interior girder distribution factor:
e
0.97
d e
(6.4.2.2.3f-3)
8700
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder (mm)
The distribution factor for moment in the edge girder due to single-lane loading may be obtained by simple-beam distribution, i.e., the lever rule, in the same manner as was described for beam-andslab bridges. Shear Distribution The distribution factor for shear in the interior girders due to multi-lane loading may be calculated from the following:
g
S 2250
0.8
d L
0.1
(6.4.2.2.3f-4)
where the parameters are as defined previously. The distribution factor for shear in the interior girders due to single-lane loading may be obtained from:
g
S 3050
0.6
d L
0.1
(6.4.2.2.3f-5)
where the parameters are again as defined previously. The shear in the edge girder due to multi-lane loading can be found by applying a correction factor to the interior girder equation. This correction factor is:
Lecture - 6-31
printed on June 24, 2003
e
0.8
d e 3050
(6.4.2.2.3f-6)
where: de
=
distance from edge of lane to the center of exterior web of the exterior girder (mm)
The wheel load distribution factor for shear in the edge girder due to single-lane loading may be obtained by simple-beam distribution in the same manner as was described for beam-and-slab bridges, i.e., the lever rule. Correction for Skew Effects Research at Lehigh University also resulted in a formula for correction of wheel load distribution for m oment in interior girders due to multi-lane loading in skewed bridges. NCHRP 12-26 concludes that Equation 6.4.2.2.3b-6 was also applicable to this case. The shear in the interior beams of a skewed bridge is the same as that of a straight bridge. However, the shear in the obtuse corner must be obtained by applying a correction factor to the distribution factor for the edge girder in a straight bridge, given by Equation 6.4.2.2.3b-7, which C1 is taken as:
c 1
(Ld )0.5 6S
(6.4.2.2.3f-7)
6.4.2.2.3g Response of Continuous Bridges The response of continuous bridges was stud ied by modeling a number of two-span continuous bridges where each span is similar to the average bridge. The wheel load distribution factor for each case was compared to that of a simple bridge and correction factors for continuity were obtained. In the case of beam-and-slab bridges, a complete parameter study was performed, and it was found that the correction factor is generally independent of bridge geometry. These factors are given in the table below. When the NCHRP 12-26 factors were incorporated into the LRFD Specification, it was decided that 5% corrections were unwarranted given that the distribution factors are an approximation of actual behavior and are, therefore, subject to some variability. The continuity correction for negative moment, a 10% increase, was also neglected on the basis that experimentally observed "fanning" of the reaction tends to reduce the negative moment as compared to a typical beam calculation.
Lecture - 6-32
printed on June 24, 2003 Table 6.4.2.2.3f-1 - Continuity Correction Factors Beam-and-Slab Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent
c c c c
= = = =
1.05 1.10 1.00 1.05
c c c c
= = = =
1.00 1.10 1.00 1.00
Box Girder Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent Slab Bridges Positive moment Negative moment
c = 1.00 c = 1.10
Multi-Beam Bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent
c c c c
= = = =
1.00 1.10 1.00 1.05
c c c c
= = = =
1.00 1.10 1.00 1.05
Spread box beam bridges Positive moment Negative moment Shear at simply-supported end Shear at continuous bent 6.4.2.3 TRUSS AND ARCH BRIDGES 6.4.2.3.1 General The approximate method for load distribution to lines of trusses and arches is a so-called "lever rule", which is simply a matter of summing moments about one line of trusses or arches to find the reaction on the other line. This approach is illustrated by calculations in Lecture 7.
Lecture - 6-33
printed on June 24, 2003 6.5 REFINED METHODS 6.5.1 Deck Slabs Where refined analysis of deck slabs is desirable, finite element analysis is recommended. Elements should be chosen to simulate both bending and in-plane or membrane effects. If the analysis utilizes only plate/membrane or shell elements and has only one or two elements through the thickness of the deck, then the refined analysis will report an essentially bending-type respons e in the deck. There has been much experimental and analytic work that suggests that bending is not the primary source of strength in bridge decks, but that the development of membrane action, analogous to a shallow arch or dome load path within the deck is the primary source of strength. This type of action will only be determined through a very rigorous modeling of the deck. 6.5.2 Beam Slab Bridges Relatively rigorous models of beam slab bridges can be developed using general purpose commercial finite element programs, finite strip programs or special purpose greater finite element-based computer programs which have been specifically developed to simplify the analysis of bridge-type structures. These more custom-oriented programs often contain mesh generating capabilities, automatic load placement capabilities and code checking. Detailed bridge deck analysis using a finite element computer program may be used to produce accurate results. However, extreme care must be taken in preparation of the model, or inaccurate results will be obtained. Important points to consider are selection of a program capable of accurately modeling responses being investigated, calculation of element properties, mesh density and support conditions. Every model should be thoroughly checked to ensure that nodes and elements are generated correctly. Another important point is the loading. Truck loads should be placed at positions that produce the maximum response in the components being investigated. In many cases, the truck location is not known before preliminary analysis is performed and, therefore, many loadings should be investigated. This problem is more pronounced in skewed bridges. Many computer programs have algorithms that allow loads to be placed at any point on the elements. If this feature is not present, equivalent nodal loads must be calculated. Distribution of wheel loads to various nodes must also be performed with care, and the mesh should be fine enough to minimize errors that can arise because of load approximations. Many computer programs, especially the general purpose finite element analysis programs, report stresses and strains, not shear and moment values. Calculation of shear and moment values from the
Lecture - 6-34
printed on June 24, 2003 stresses must be carefully performed, usually requiring an integration over the beam cross-section. Some programs report stresses at node points rather than Gaussian integration points. Integration of stresses at node points is normally less accurate and may lead to inaccurate results. Many graphical and computer-based methods are available for calculating wheel load distribution. One popular method consists of design charts based on the orthotropic plate analogy, similar to that presented in the Ontario Highway Bridge Design Code. As computers become readily available to designers, simple computer-based methods, such as SALOD, become more attractive than nomographs and design charts. Also, grillage analysis presents a good alternative to other simplified bridge deck analysis methods and will generally produce more accurate results. The grillage analogy may be used to model any one of the five bridge types studied in this research. Each bridge type requires special modeling techniques. A major advantage of plane grid analysis is that shear and moment values for girders are directly obtained and integration of stresses is not needed. Loads normally need to be applied at nodal points, and it is recommended that sim plebeam distribution be used to distribute wheel loads to individual nodes. If the loads are placed in their correct locations, the results will be close to those of detailed finite element analysis. As indicated previously, the designer has to be responsible for constructing a suitable model and determining that the results are accurate. It is possible to make seemingly small errors in computer models which can have dramatic effects on the results which are obtained. 6.5.3 Example of Modeling Errors The modeling of diaphragms and boundary conditions at supports and bearings is vital to obtaining the proper results when using these sophisticated programs. The burden of correctly handling these factors rests with the designer. Consider the following example which shows how a very small modeling error produced very erroneous results. The framing plan shown on Figure 6.5.3-1 represents an actual bridge that was designed using a grid-type approach. The designer had a good model for this structure, except that the rotational degree of freedom corresponding to the global "x" axis was fixed at all of the bearings. This did not allow the diaphragms at the piers and abutments to respond correctly to the imposed loadings and deformations, and also had the effect of producing artificially stiff ends on the girders by virtue of vector resolution between global and local systems. The effect of this condition on the reactions obtained at the abutments and piers was dramatic. Modest uplift was reported at the acute corner along the near abutment show n in Figure 6.5.3-1, and a very substantial uplift was reported at the acute angle at the far
Lecture - 6-35
printed on June 24, 2003 abutment. This is shown in the table in Figure 6.5.3-1, as is a moment diagram for non-composite dead loads which reflects the incorrect reactions. Also shown in the table of reactions on Figure 6.5.3-1 are the correct reactions determined when the structure was modeled using the generic STRESS Computer Program with proper boundary conditions at the supports. In this case, a positive reaction is found at all bearings, and a significantly different moment diagram for noncomposite dead load also resulted. The correct reactions and moment diagram are also shown on Figure 6.5.3-1. The modeling of the degrees of freedom at the lines of support on this structure was also investigated utilizing a relatively complete three-dimensional finite element analysis and the SAPIV Computer Program. The model used is illustrated in Figure 6.5.3-2, which shows how the deck slab, girders and cross-frames were modeled in their proper relative positions in the cross-section whic h extended along the bridge from end-to-end. Also shown on this figure is a comparison of the reactions obtained from STRESS and from S APIV by applying all of the non-composite loads in a single loading. The comparison between these reactions is excellent. In order to verify that the order of pouring the deck slab units would not contribute to an uplift situation, the pouring sequence was replicated in a three-dimensional SAPIV analysis. The results of the analysis of the three stages of the pouring sequence are also shown in Figure 6.5.3-2, as well as the total accumulated load at the end of the pour. Comparison of the sequential loading with the application of a single loading of non-composite dead load also showed relatively good agreement in this case. The important point demonstrated in the example of Figures 6.5.3-1 and 6.5.3-2 is that seemingly small errors in modeling of the structure can result in very substantial changes in the reactions, shears and moments. The designer must be aware of this potential when utilizing the more refined analysis techniques. Incidentally, there are cases in which an uplift reaction due to skew and/or curvature is possible. The simple span bridge shown in Figure 6.5.3-3 and reported on in the November 1, 1984, issue of Engineering News-Record, was analyzed at the request of t he owner. In this case, the uplift reactions computed by the designer were verified. Sometimes modeling problems occur because User's Manuals are not clear, or a "bug" exists, of which the program's author/vendor is not aware. Such a case is illustrated for the simply-supported, partially-curved and skewed bridge in Figure 6.5.3-4. Initially, this bridge was modeled with extra joints at locations other than diaphragms in an effort to improve live load determination. As a result, the number of points along each girder were not equal, but there was no indication in the program's descriptive literature that this type of arrangement of program input could potentially cause a problem. The resulting moment envelopes for the middle and two
Lecture - 6-36
printed on June 24, 2003 exterior girders are shown in Figure 6.5.3-4, and are obviously unusual 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 responding properly to the unequal number of nodes per girder, that nodes should be essentially "radial", and that it was not certain that nodes to which diaphragms were not connected were legitimate. The revised model, shown in Figure 6.5.3-4, produced clearly better results, as shown in the indicated moment envelopes.
Lecture - 6-37
printed on June 24, 2003
Figure 6.5.3-1 - Framing plan, comparative live load reactions and moment envelopes showing effect of proper and improper rotational boundary condition, as reflected in grid analysis.
Lecture - 6-38
printed on June 24, 2003
Figure 6.5.3-2 - Finite element idealization and reactions obtained for structure shown in Figure 6.
Lecture - 6-39
printed on June 24, 2003
Figure 6.5.3-3 - Framing Plan of Curved Span with Skewed Piers
Figure 6.5.3-4 - Comparative moment envelopes for the middle and two outside girders of curved skewed system showing the results of apparent "bug" in algorithm for applying live load. 6.5.4 Other Types of Bridges The Specification contains additional requirements for the rigorous analysis of cellular-type structures, truss bridges, arch bridges, cable-stayed bridges and suspension bridges. Generally speaking, refined analysis will involve a computer model which accurately affects the geometry, relative component stiffnesses,
Lecture - 6-40
printed on June 24, 2003 boundary conditions and load supply to the structure. Suspension bridges will almost always be analyzed us ing a large deflection theory. The deflection theory may also be applied to arches and cable-stayed bridges. In the case of the cable-stayed bridge of moderate span, it may be sufficiently accurate to evaluate the second order effects on the deck system of the tower by supplementary calculations in providing a correction factor, developed on a bridge-specific basis. The change in stiffness of the cables caused by change in sag as the cable load changes can be accounted for using the so-called "Ernst" equations, given in the Specification, for modified modulus of elasticity.
Lecture - 6-41
printed on June 24, 2003 REFERENCES Jones, 1976, "A Simple Algorithm for Computing Load Distribution in Multi-Beam Bridge Decks", Proceedings, 8th ARRB Conference, 1976
Lecture - 6-42
printed on June 24, 2003
APPENDIX A The Load Distribution Problem and its Solution in NCHRP 12-26
printed on June 24, 2003 Older editions of the AASHTO Specifications allow for simplified analysis of bridge superstructures using the concept of a load distribution factor for bending moment in interior girders of most types of bridges, i.e., beam-and-slab, box girder, slab, multi-box beam and spread-box beam. This distribution factor is given by:
g
S 300 D
(A-1)
where: g
=
a factor used to multiply the total longitudinal response of the bridge due to a single longitudinal line of wheel loads in order to determine the maximum response of a single 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 have taken place over the last 55 years have led to inconsistencies in the load distribution criteria including: 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-supported bridges. Although it was stated that these procedures apply to the design of normal (i.e., supports oriented perpendicular to the longitudinal girders) highway bridges, there are no other guidelines for determining when the procedures are applicable. Because modern highway and bridge design practice requires a large number of bridges to be constructed with skewed supports, on curved alignments, or continuous over interior supports, it is increasingly important that the limitations of load distribution criteria be fully understood by designers. Advanced computer technology has become available in recent years which allows detailed finite element analysis of bridge decks. However, many computer programs exist which employ different formulations and techniques. It is important that the computer methodology and formulation that produces the most accurat e results be used to predict the behavior of bridge decks. In order to identify the most accurate computer programs, data from full-sc ale and prototype bridge load tests were compiled. The bridge tests were then modeled by different computer programs and the experimental and computer results were compared. The programs that produced the most accurate 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 of applicability. 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 selected in order to achieve national representation. This resulted in a database of 365 beam-and-slab bridges, 112 prestressed concrete and 121 reinforced concrete box girder bridges, 130 slab bridges, 67 multi-box beam bridges and 55 spread-box beam bridges. This bridge database was studied 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. A hypothetical bridge that has all the average properties obtained from the database, referred to as Lecture - 6-A1
printed on June 24, 2003 the "average bridge" was created for each of the beam-and-slab, box girder, slab, multi-box beam and 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 also prepared. In evaluating simplified formulas, it is important to understand the effect of various bridge parameters on load distribution. Bridge parameters were varied one at a time in the average bridge for the bridge type under consideration. Load distribution factors for both shear and moment were obtained for all such bridges. Variation of load distribution factors with each parameter shows the importance of each parameter. Simplified formulas can then be developed to capture the variation of load distribution factors with each of the important parameters. A brief description of the method used 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 function of the form axb, where x is the value of the given parameter, and a and b are coefficients to be determined based on the variation of x. Second, it is assumed that the effects of different parameters are independent of each other, which allows each parameter to be considered separately. 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 paramet ers included in 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 for S, then: g1 = (a)(S1b1)(Lb2)(tb3)(...)
(A-2)
g2 = (a)(S2b1)(Lb2)(tb3)(...)
(A-3)
therefore: g 1
S1
g 2
S2
b1
(A-4)
or:
ln b1 ln
g 1 g 2
(A-5)
S1 S2
If n different values of S are examined and successive pairs are used to determine the value of b1, n 1 different values for b1 can be obtained. If these b1 values are close to each other, an exponential 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 power factors, i.e., b1, b2, and so on, are determined, the value of "a" can be obtained from the average bridge, i.e.,
Lecture - 6-A2
printed on June 24, 2003
a
g o b1
So
b2
Lo
b3
t o
(A-6)
(...)
This procedure was followed during the entire course of the NC HRP 12-26 study to develop new formulas as needed. In certain cases where an exponential function was not suitable to model the effect of a parameter, slight variation from this procedure was used to achieve the required accuracy. However, this procedure worked quite well in most cases and the developed formulas demonstrate high accuracy. Because certain assumptions were made in the derivation of simplified formulas and some bridge parameters were ignored altogether, it is important to verif y the accuracy of these formulas when applied to real bridges. The database of actual bridges was used for this purpose. Bridges to which the formula can be applied were identified and analyzed by an accurate method. The distribution factors obtained from the accurate analysis were compared to the results of the simplified methods. The ratio of the approximate to accurate distribution factors was calculated and examined to assess the accuracy of the approximate method. Average, standard deviation, and minimum and maximum ratio values were obtained for each formula or simplified method. The method 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 slightly conservative results. The minimum and maximum values show the extreme predictions that each method or formula produced when a specific database was used. Although these values may change slightly if a different set of bridg es is used for evaluation, the minimum and maximum values allow identification of where shortcom ings in the formula may exist that are not readily identified by the average or standard deviation values. It was previously mentioned that different subsets of the database of bridges were used to evaluate different formulas. When a subset included a large number of bridges (100 or more), a Level 2 method was used as the basis of comparison. When it included a smaller number of bridges (less than 100), a Level 3 method was used. As a result, LANELL (an influence surface method) was used for verification of formulas for moment distribution in box girder bridges, and a Multi-dimensional Space Interpolation (MSI) method was used for verification of formulas for straight 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 the development of a number of computer programs for structural analysis. Programs that can be applicable to bridge deck analysis can be divided into two categories. One includes general purpose structural analysis programs such as SAP, STRUDL and FINITE. The other category is specialized programs for analysis of specific bridge types, such as GENDEK, CURVBRG and MUPDI. In the search for the best available computer program for analysis of each bridge type, all suitable computer programs (general and specific) that were available at the time of the 12-26 research were evaluated. In order to achieve meaningful comparisons and assess the level of accuracy 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-A3
printed on June 24, 2003 •
by comparison of the averages and standard deviations of the ratios of analytical to experimental results, and
•
by comparison of statistical differences of analytical and experimental results. Five bridge types were considered: beam-and-slab, box girder, slab, multi-box beam, and spread-box beam.
For analysis of beam-and-slab bridges, the following comput er programs and models were evaluated: GENDEK-PLATE, GENDEK-3, GENDEK-5, CURVBRG, SAP and MUPDI. It was found that, in general, GENDEK-5 analysis using plate elements for the deck slab and eccentric beam elements for the girders is very accurate. This program is also general enough to cover all typical cases, i.e., straight, skew, moment and shear. However, for analysis of curved open girder steel bridges, CURVBRG was the most accurate program. MUPDI was also found to be a very accurate and fast program; however, skewed bridges cannot be analyzed wit h this program and shear values near 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 were evaluated. MUPDI was the fastest and most practical program for analysis of straight bridges for moment, but FINITE was found to be the most practical program for skewed bridges and for obtaining accurate shear results. Therefore, MUPDI was selected for the evaluation of LANELL (a Level 2 method for moment in straight bridges which was, in turn, used for evaluation of Level 1 methods) and FINITE was selected for other cases. For the analysis of slab bridges, computer programs MUPDI, FINITE, SAP and GENDEK were evaluated. Shear results cannot be obtained accurately in slab bridges and, therefore, were not 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-skewed prismatic 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 of California, Berkeley, for analysis of multi-beam bridges (referred to as the POWELL program herein). Various modeling techniques were studied using different grillage models and different plate 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 later used in evaluation of more simplified methods. POWELL is also very accurate in reporting moments in straight bridges, but it uses a finite strip formulation, s imilar to MUPDI, and, therefore, is incapable of modeling skewed supports , and shear results near supports and load loc ations cannot be accurately obtained. This program was used to evaluate simplified methods for straight bridges. For analysis of spread-box beam bridges, computer programs SAP, MUPDI, FINITE and NIKE-3D were evaluated. FINITE produced the most accurate results, especially when shear was considered. MUPDI was selected to evaluate simplified methods for calculation of moments in straight 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 computer methods have been unavailable. The Ontario Highway Bridge Design Code uses one such method based on orthotropic plate theory. Other graphical methods have also been developed and reported. A good example of the influence surface method is the computer program SALOD Lecture - 6-A4
printed on June 24, 2003 developed by the University of Florida for the Florida Department of Transportation. This program uses influence surfaces, obtained by detailed finite element analysis, which are stored in a database accessed by SALOD. One advantage of influence surface methods is that the response of the bridge deck to different truck types can be readily computed. A grillage analysis using plane grid models can also be used with minimal computer resources to calculate the response of bridge decks in most bridge types. However, the properties for grid members must be calculated with care to assure accuracy. Level 2 methods used to analyze the five bridge types (beam-and-slab, box girder, slab, multi-beam and spread-box beam bridges) are discussed below. The following methods were evaluated for analysis of beam-and-slab bridges: plane grid analysis, the nomograph-based method included in the Ontario Highway Bridge Design Code (OHBDC), SALOD and Multi-dimensional Space Interpolation (MSI). All of these methods are applicable for single- and multi-lane loading for moment. The OHBDC curves were developed for a truck other than HS-20, and using the HS-20 truck in the ev aluation process may have introduced some discrepancies. The method presented in OHBDC was also found to be time consuming, and inaccurate interpolation between curves was probably a common source of error. SALOD can be used with any truck and, therefore, the "HS" truck was used in its evaluation. The MSI method was developed based on HS-20 truck loading for single- and multiple-lane loading. MSI was found to be the fastest and most accurate method and was, therefore, selected for the evaluation of Level 1 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 were evaluated. The comments made about OHBDC for beam-and-slab bridges are valid for box girder bridges 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 found to be the most accurate method and, thus, was used in the evaluation of Level 1 methods. SALOD also 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 accurate evaluation. In the analysis of multi-beam bridges, a met hod presented in Jones, 1976, was evaluated. The method is capable of calculating distribut ion factors due to a single concentrated load and was modified for this study to allow wheel line loadings. The results were found to be in very good agreement with POWELL. However, because this method was only applicable for moment distribution 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 a Level 2 method. In general, Level 2 graphical and influence surface methods generated accurate and dependable results. While these methods are sometimes difficult to apply, a major advantage of some of them is that different trucks, lane widths, and multiple presence live load reduction fact ors may be considered. Therefore, if a Level 2 procedure does not provide needed flexibility, its use is not warranted because the accuracy of it is on the same order as a simplified formula. MSI is an example 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 some of the methods mentioned above. In addition, a general purpose plane grid analysis program is Lecture - 6-A5
printed on June 24, 2003 available to most bridge designers. Therefore, this method of analysis is considered a Level 2 method. However, the user has the burden of producing a grid model that will produce sufficiently accurate 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 distribution factors for each of the bridge types studied. Level 1 Methods: Simplified Formulas The current AASHTO Specifications recommend use of simplified formulas for determining load distribution factors. Many of these formulas have not been updated in years and do not provide optimum accuracy. A number of other formulas have been developed by researchers in recent years. Most of these formulas are for moment distribution for beam-and-slab bridges subjected to multi-lane truck loading. While some have considered correction factors for edge girders and skewed supports, very little has been reported on shear distribution factors or distribution factors for bridges other than beam-and-slab. The sensitivity of load distribut ion factors to various bridge parameters was also determined as part of the study. In general, beam spacing is the most significant parameter. However, span length, 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 various bridge parameters for each bridge type. Ignoring the effect of bridge parameters, other than beam spacing, can result in highly inaccurate (either conservative or unconservative) solutions. A major objective of the research in Project 12-26 was to evaluate older AASHTO Specifications and other researchers' published work to assess their accuracy and develop alternate formulas whenever a more accurate method could be obtained. The formulas that were evaluated 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-A6
printed on June 24, 2003
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-A7
printed on June 24, 2003
Figure A-5 - Effect of Parameter Variation on Spread-Box Beam Bridges
Lecture - 6-A8
