Giải thích hệ số PBN

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BridgeTech, Inc.

Simplified Live Load Distribution

Formula NCHRP 12-62

Research Team Jay A Puckett, Ph.D., P.E.

Dennis Mertz, Ph.D., P.E.

X Sharon Huo, Ph.D., P.E.

Mark Jablin, P.E.

Michael Patrick, Graduate Student

Matthew Peavy, P.E.

NCHRP Manager: David Beal, P.E.

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Objective

recommended LRFD live-load distribution-factor

design equations for shear and moment that are

The need for refined methods of analysis should

be minimized.

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The Problem

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Accuracy

Simplicity

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Accuracy

Simplicity

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PI Bias for a Simple Method

• Analytically based approach

• Canadian Specification

Orthotropic Plate Theory

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NBI 1990 - most recent

NBI Total Inventory Number Skewed

Precast Solid, Voided, or

Cellular Concrete Boxes with

Shear Keys and with or

Precast Solid, Voided, or

Cellular Concrete Boxes with

Shear Keys

Steel Beam Cast in place concrete

slab, precast concreteClosed Steel or Precast

Concrete Boxes

Cast in place concreteslab

Cast-in-place concrete orplank, glued/spiked panels

or stressed wood

Precast Concrete Channel

Sections with Shear Keys

Precast Concrete Double

Tee Section with Shear Keys

and with or without

Transverse Posttensioning

Precast Concrete I or

Bulb-Tee Sections

Precast Concrete Tee

Section with Shear Keys and

with or without Transverse

Reinforcement

Integral Concrete

Cast-in-place concreteoverlay

Integral Concrete

Cast-in-place concrete,precast concrete

Cast-in-place concreteslab, precast concrete slabMonolithic ConcreteMonolithic ConcreteCast-in-place concreteoverlay

j

slab on girders

slab on girdersslab on girders

monolithic slab and girders

slab on girders

ihg

slab on girders

slab on girdersslab on girders

Supporting Components Type of Deck

AASHTO Letter (see Table 4.6.2.2.1-1)

2848

a

dcb

fe

Number of Bridges Analytical Group Type

2810617766

5718

5633

151398Q

4847slab on girders

53285

Slabs Not Applicable Not Applicable Slabs

26629l

k

slab on girdersWood Beams

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Summary Table (NBI Data)

Type 1990-present Total Inventory

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min max min max min max min max min maxConc T-Beam 71 n/a 12 93 2.42 16 5 11 0 52.98 0.32 3.26Steel I-Beam 163 n/a 12 205 2 15.5 4.42 12 0 66.1 0.4 4.53Prestressed I-Beam 94 n/a 18.75 136.2 3.21 10.5 5 9 0 47.7 0.31 3.12Prestressed Conc Box 112 n/a 43.3 243 6 20.75 n/a n/a n/a n/a 0.52 8.13R/C Box 121 n/a 35.2 147 6.58 10.67 n/a n/a n/a n/a 0.53 5.5Slab 127 n/a 14.2 68 n/a n/a 9.8 36 0 70 0.21 2.56Multi-Box 66 n/a 21 112.7 n/a n/a 0 11 0 55.8 0.22 5.96Conc Spread Box 35 n/a 29.3 136.5 6.42 11.75 6 8.5 0 52.8 0.54 3.11Steel Spread Box 20 n/a 58 281.7 8.67 24 5 9.5 0 60.5 0.75 8.02Precast Conc Spread Box 4 1 - 6 44.38 81.49 5.67 13.75 7.75 8.75 0.00 48.49 1.68 2.03Precast Conc Bulb-Tee 4 2 - 6 115.49 159.00 8.33 10.29 8.25 8.27 0.00 26.70 1.43 4.97Precast Conc I-Beam 3 3 - 5 67.42 74.33 9.00 10.58 8.25 8.75 0.00 33.50 1.45 1.53CIP Conc T-Beam 3 4 - 5 66.00 88.50 8.17 12.58 7.00 9.00 0.00 31.56 1.91 2.74CIP Conc Multicell 4 2 - 3 98.75 140.00 9.00 10.33 8.00 9.25 0.00 26.23 2.24 3.05Steel I-Beam 4 2 - 4 140.00 182.00 9.33 11.50 8.00 9.00 0.00 50.16 1.60 5.11Steel Open Box 2 1 - 3 170.67 252.00 9.00 9.38 8.50 8.50 4.50 31.95 3.28 7.00

LRFR 3 653 Slab on RC, Prest., and

Steel Girders 653 1 - 7 18.00 243.00 2.33 18.00 0.00 8.00 N/A N/A 0.38 5.22Spread Box Beams 27 1 100.00 190.00 5.00 20.00 6.00 12.00 N/A N/A 1.40 8.00Adjacent Box Beams 23 1 100.00 210.00 3.00 5.83 5.00 6.00 N/A N/A 1.13 9.60Slab on Steel I-Beam 24 1 160.00 300.00 12.00 20.00 9.00 12.00 N/A N/A 2.76 6.82

Bridges Bridge Types

Total No.

24809

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Common Database Format NCHRP 12-50

1 NCHRP 12-26 Bridge

Database

800 + Bridges can be used in an

automated process to generate

simplified and rigorous analyses

3 Virtis/Opis Database Bridges

650+ bridges may be exported fromVirtis/Opis to supply real bridges toboth simplified and rigorous methods

2 Tenn Tech Database

Detailed descriptions and rigorousanalysis are available from a recent

TT study for TN DOT Results,structural models, etc., are readilyavailable

Data Sources

Condense to a Common Database

A

4 Parametrically Generated Bridges

74 Bridges were developed to test the limits of applicability of the proposed method

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Common Database Format

NCHRP 12-50

A

B

BRASS-Girder (LRFD) TM

Simplified Analysis Methods:

 Standard Specifications (S over D)

 LRFD Specifications

 Rigid Method

 Lever Rule

 Adjusted Equal Distribution Method

 Canadian Highway Bridge Design Code

 Sanders

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Simplified Moment and Shear Distribution Factor Equations

 Specification and Commentary Language

 Design Examples

 Final Report

Iterative Process Involving Tasks 7,8, and 9 through 12.

Common Database Format

Regression testing on “real” bridges (Virtis/Opis database, NCHRP 12-26 database)

(compare proposed method to current LRFD method)

Comparisons from parametric bridges and rigorous analysis

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Grillage Method (structural model)

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Influence Surfaces (structural model)

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Automated Live Load Positioning

• Critical live load placement

• Actions (shear, moment,

• Accounts for barrier, etc.

• 4-ft truck transverse truck

spacing

• POI at least tenth points

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Example of Standard Specification

Results

Moment at 1.4 One-lane Loaded Exterior I-Girder

Std S/D vs Rigorous

y = 0.9914x + 0.2962

R2= 0.3834

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Unit slope = good

R 2 = poor

hope 

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Lever Rule Results

Moment at 1.4 One-lane Loaded Exterior I-Girder Lever Rule vs Rigorous

y = 1.63x - 0.2644

R2= 0.8889

0 0.2 0.4 0.6 0.8 1 1.2 1.4

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Moment at 1.4 One-lane Loaded Exterior I-Girder Calibrated Lever Rule vs Rigorous

y = 0.978x + 0.0413

R2= 0.8889

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Calibrated Lever Rule Results

R 2 = good and is the

same

slope = good

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Rotation to Unity

by multiplication

Raise or lower by addition/substratio

n

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is the calibrated distribution factor, and

is the lever rule distribution factor computed with the typical man

Calibrated lever rule m Lever rule m

slope

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Number of

Loaded

Lanes

Use integer part of

m shall be greater than or equal to 0.85.

to determine number of loaded lanes for

multiple presence.

Interior and Exterior

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"m"

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e

d S6'

S

de6'

2 16

2 de

 

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Calibration Coefficients (Moment)

Precast Concrete I-Beam k

Precast Concrete Bulb-Tee Beam k

Precast Concrete Double Tee with

Shear Keys with or without

Post-Tensioning

i

Precast Concrete Tee Section with

Shear Keys and with or without

Transverse Post-Tensioning

j

Precast Concrete Channel with Shear

Cast-in-Place Concrete Tee Beam e 0.65 0.15 1.40 -0.41

Cast-in-Place Concrete Multicell Box

Adjacent Box Beam with

Cast-in-Place Concrete Overlay f

Adjacent Box Beam with Integral

Precast Concrete Spread Box Beam b, c 0.50 0.06 0.77 -0.17

Open Steel Box Beam c Use Article 4.6.2.2.3

Moment

1.51 -0.69 0.41 -0.03

1.20 -0.37 0.61 0.16

Structure Type

AASHTO LRFD Cross Section Type

One Loaded Lane Exterior Interior

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Calibration Coefficients (Moment)

Post-Tensioning

i

j

Cast-in-Place Concrete Tee Beam e 0.65 0.15 1.40 -0.41

Precast Concrete Spread Box Beam b, c 0.50 0.06 0.77 -0.17

Open Steel Box Beam c Use Article 4.6.2.2.3

Moment

1.51 -0.69 0.41 -0.03

1.20 -0.37 0.61 0.16

Structure Type

One Loaded Lane Exterior Interior

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Calibration Coefficients (Shear)

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

Steel I-Beam a

Post-Tensioning

i

Shear Exterior Interior One Loaded Lane

Two or More Lanes Loaded

One Loaded Lane

Two or More Lanes Loaded

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Structural Factor (Moment) Multiple

Lanes Loaded

Post-Tensioning

i

j

Cast-in-Place Concrete Tee Beam e 1.10

Precast Concrete Spread Box Beam b, c 1.00

Open Steel Box Beam c Use Existing

Specification

Structure Type

Two or more loaded lanes

1.15

1.10

F st

Uniform Distribution

N lane / N girder

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0.61 0.9 0.16 0.71

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Calibration Constants

Lanes Loaded Initial Trend Line - Lever Rule and Henry's

Method (Henry's Method already calibrated)

Computed Calibration Factors (for Lever Rule Calibration)

Recommended Calibration Factors

Quite Good (typical)

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Statistical Comparison Conceptual

1.00

Standard deviation

simplified rigorous

g g

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Analysis Factors

Type of Bridge

No of Std Dev.

Offset

Computed Analysis Factor

Rounded Analysis Factor (b = 1)

No of Std.

Dev Offset

Rounded Analysis Factor (b = 0.5)

No of Std Dev.

Offset

Rounded Analysis Factor (b =

S/R (S/R) -1 V S/R β g a g a (rounded) β g a g a (rounded) β g a g a (rounded)

1 Lane 13c 1.010 0.991 0.058 1.0 1.049 1.05 0.5 1.020 1.05 0.0 0.991 1.00

2 or More Lanes 14c 1.014 0.986 0.067 1.0 1.053 1.05 0.5 1.019 1.05 0.0 0.986 1.00

1 Lane 15c 0.999 1.001 0.069 1.0 1.069 1.10 0.5 1.035 1.05 0.0 1.001 1.00

2 or More Lanes 16c 1.000 1.000 0.102 1.0 1.102 1.10 0.5 1.051 1.05 0.0 1.000 1.00 Calibrated

Lever 1 Lane 17c 0.993 1.007 0.092 1.0 1.099 1.10 0.5 1.053 1.05 0.0 1.007 1.00 Henry's

Method 2 or More Lanes 18c 1.285 0.778 0.110 1.0 0.888 1.00 0.5 0.833 0.85 0.0 0.778 0.80 Calibrated

Lever 1 Lane 19c 0.996 1.004 0.244 1.0 1.248 1.25 0.5 1.126 1.15 0.0 1.004 1.00 Henry's

Method 2 or More Lanes 20c 1.139 0.878 0.068 1.0 0.945 1.00 0.5 0.912 0.95 0.0 0.878 0.90

Figures Action

Interior Exterior

Girder Location Basic

Method

Interior Exterior

Calibrated Lever

Analysis Factor Computations

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0.61 0.9 0.16 0.71

Calibrated a

a

mg mg

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Effect of Multiple

Presence Analysis

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Skew

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Curvature

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Type of Superstructure

Applicable Cross-Section from Table 4.6.2.2.1-1 Correction Factor

Range of Applicability Concrete Deck, Filled Grid,

Partially Filled Grid, or Unfilled

Grid Deck Composite with

Reinforced Concrete Slab on Steel

or Concrete Beams; Concrete

T-Beams, T- and Double T-Section

b

S L N

b

S L N

    

0 60 6.0 13.0

20 240

35 110 3

c

S L d N

20 140

18 65 3

b

S L d N

L d

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1'-9"

Overhang S

s 1'-9"

Bridge

No.

Girder Spacing, S (ft)

Recommended minimum slab thickness (AASHTO STD Table 8.9.2)

Slab Thickness,

t s (in)

Span Length, L (ft)

Total Bridge Width, W (ft)

No.of girders

Overhang (ft)

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Is this simpler?

types

one-lane moment) – and adjusted

(multiple-lanes loaded – and adjusted

presence

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Is this simpler?

simple analysis wrt rigorous

cross section and span lengths

areas

(readily known)

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Is it simpler?

 Many pages shorter

 Many variables eliminated from

notation and section

 Once affine transformations are

understood the adjustments from

lever are readily seen

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Go to report

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Additional work

calibrated lever

explain this in a more understandable manner

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Questions, Discussion

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End of AASHTO Talk

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Extra slides

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Exterior Longitudinal Girder

Closed Section For Torsional Rigidity

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Shear Distribution Factors for CIP Concrete Multicell

Box Beam Bridges Validation

Ext Int Exterior Interior Exterior Interior Exterior Interior

1

2

1014 1013

2

1

14 1011 1012

2 13

1

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1 excellent good good bad bad bad Lever

2 or more excellent acceptable good bad bad bad Lever

1 excellent poor good good good good Lever

2 or more excellent excellent excellent good good good Lever

2 or more good good good poor acceptable bad Lever

2 or more acceptable excellent acceptable acceptable acceptable poor Henry's

1 excellent good good poor poor poor Lever

2 or more excellent excellent excellent poor poor poor Lever

1 excellent poor excellent excellent good good Lever

2 or more good excellent good excellent good excellent Henry's

1 excellent excellent good poor poor poor Henry's

2 or more excellent excellent excellent poor poor poor Henry's

1 poor bad excellent acceptable poor poor LRFD

2 or more poor excellent excellent good good good Henry's

1 excellent acceptable excellent poor acceptable bad Lever

2 or more excellent excellent excellent acceptable acceptable poor Lever

1 excellent acceptable excellent acceptable good poor Lever

2 or more good excellent excellent good excellent poor Henry's

2 or more good excellent good poor acceptable poor Henry's

1 acceptable bad poor bad poor bad Lever

2 or more poor excellent poor bad poor bad Henry's

1 excellent poor excellent acceptable acceptable poor Lever

2 or more excellent excellent excellent good good acceptable Lever

1 excellent poor acceptable good excellent acceptable STD

2 or more excellent excellent excellent good excellent acceptable Henry's

2 or more acceptable good poor poor poor bad Henry's

1 bad excellent bad poor bad bad Henry's

2 or more poor good poor poor poor bad Henry's

Method Rating Based on the Value of the Correlation Coefficient (R 2 ) between Each Simplified

Method and Rigorous Analysis

Lanes Loaded

Girder Locations Action

Exterior Interior Exterior Interior Exterior

Interior Exterior Interior

Exterior Interior Exterior Interior

Exterior Interior

Slab On I

CIP Tees

Spread Boxes

Adjacent Boxes

vvcc

vvcccvc

vvcc

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 Interior girder load effects are

easier to predict than exterior

 Loads near midspan distribute

more uniformly than load

applied near supports.

 Relative stiffness is primary

and flexure is more important

than is torsion

 Most important parameter is

the girder spacing (or

cantilever span)

2 2 3 3

( ) ( )

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Prerequisites

method “as is” (unless it really works well).

implemented at different levels (i.e., compute

stiffness parameters) – empirical methods cannot.

extended (in case of limits of application), than

empirically-based methods.

empirical approaches

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Task 1 Literature Review

Michael Patritch Graduate Student

TN Tech

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Task 1 Literature Critical Findings

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 Considered

 Relative long/trans flexural stiffness

 Relative torsonal stiffness

 Field tests for some validation

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Sanders and Elleby (cont)

3

For 10

5

3

For 3

1 7

2 3

10 5

C

C N

N D

D S

g

L

L L

D S

wheel

Double for LFRD Design Lane

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