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 

The objective of this project is to develop new recommended LRFD live-load distribution-factor design equations for shear and moment that are

simpler to apply and have a wider range of applicability than those in the current LRFD. The need for refined methods of analysis should be minimized.

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The Problem

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Basics Behavior Stiff Deck relative to Girders – better distribution, more uniform

All analysis and numerical approaches attempt to quantify this behavior Somewhere between Equal (Rigid body) and Lever Rule

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Simplicity Accuracy

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Simplicity Accuracy

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Literature Review  Current

Specifications & Simplified Approach  Modeling Techniques  Field Testing  Parametric Effects  Bridge Type  Nonlinear effects

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PI Bias for a Simple Method • Analytically based approach • Canadian Specification Orthotropic Plate Theory  space

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Type of Deck

Closed Steel or Precast Concrete Boxes

Cast in place concrete slab, precast concrete Cast in place concrete slab

Open Steel or Precast Concrete Boxes

Cast-in-place concrete slab, precast concrete slab

Steel Beam

Cast-in-Place Concrete Multicell Boxes Cast-in-Place Concrete Tee Beam Precast Solid, Voided, or Cellular Concrete Boxes with Shear Keys Precast Solid, Voided, or Cellular Concrete Boxes with Shear Keys and with or without Transverse Posttensioning Precast Concrete Channel Sections with Shear Keys Precast Concrete Double Tee Section with Shear Keys and with or without Transverse Posttensioning Precast Concrete Tee Section with Shear Keys and with or without Transverse Reinforcement Precast Concrete I or BulbTee Sections

Wood Beams

Slabs

AASHTO Letter (see Table 4.6.2.2.1-1)

Q Analytical Group Type

a

slab on girders

b

slab on girders

c

slab on girders

Number of Bridges NBI 1990 - most NBI Total Number Skewed recent Inventory 14275 66263 151398 30.0% 43.8% 1464 3075 4847 3.1% 63.4% 1811

Monolithic Concrete

d

slab on girders

Monolithic Concrete

e

slab on girders

Cast-in-place concrete overlay

f

monolithic slab and girders

3546

5718

3.8% 360 5633 0.8% 629 28106 1.3% 5329 17766 11.2% 933

Integral Concrete

g

monolithic slab and girders

Cast-in-place concrete overlay

h

slab on girders

2.0%

2848

49.0% 82 9.2%

895

208 Integral Concrete

i

slab on girders

53.6% 1396

40 0.1%

62.0% 3677 65.3% 11340 40.3% 9514

300

0.4%

35.3% 851

208 Integral Concrete

j

slab on girders

300

0.4%

35.3% 851

Cast-in-place concrete, precast concrete

k

slab on girders

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

l

slab on girders

Not Applicable

Not Applicable

Slabs

Total:

14168 29.8% 53285

26691 50.1%

1571 3.3% 26629

4213 15.8%

6799

20728

14.3% 47587 100.0%

55869

353845

37.1% 150825 42.6%

NBI Database

Supporting Components

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Summary Table (NBI Data) Type

Bridge Percentages by Type 1990-present Total Inventory

Steel Beam Concrete I Precast Concrete Boxes with Shear Keys Slabs

30.0%

42.8%

29.8%

15.1%

11.2%

5.0%

14.3% 85.3%

15.8% 78.7%

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Data Source

Reference

Total No. Bridges

NCHRP 1226

1

809

TN Tech Set 1

2

24

LRFR

3

653

Parametric Bridges

N/A

74

Bridge Types

Number of Bridges

Parameter Ranges Number of Span Length (ft) Girder Spacing (ft) Spans min. max min max n/a 12 93 2.42 16 n/a 12 205 2 15.5 n/a 18.75 136.2 3.21 10.5 n/a 43.3 243 6 20.75 n/a 35.2 147 6.58 10.67 n/a 14.2 68 n/a n/a n/a 21 112.7 n/a n/a n/a 29.3 136.5 6.42 11.75 n/a 58 281.7 8.67 24 1-6 44.38 81.49 5.67 13.75 2-6 115.49 159.00 8.33 10.29 3-5 67.42 74.33 9.00 10.58 4-5 66.00 88.50 8.17 12.58 2-3 98.75 140.00 9.00 10.33 2-4 140.00 182.00 9.33 11.50 1-3 170.67 252.00 9.00 9.38

Slab Thickness (in) min max 5 11 4.42 12 5 9 n/a n/a n/a n/a 9.8 36 0 11 6 8.5 5 9.5 7.75 8.75 8.25 8.27 8.25 8.75 7.00 9.00 8.00 9.25 8.00 9.00 8.50 8.50

Skew Angle (deg) min max 0 52.98 0 66.1 0 47.7 n/a n/a n/a n/a 0 70 0 55.8 0 52.8 0 60.5 0.00 48.49 0.00 26.70 0.00 33.50 0.00 31.56 0.00 26.23 0.00 50.16 4.50 31.95

Aspect Ratio (L/W) min max 0.32 3.26 0.4 4.53 0.31 3.12 0.52 8.13 0.53 5.5 0.21 2.56 0.22 5.96 0.54 3.11 0.75 8.02 1.68 2.03 1.43 4.97 1.45 1.53 1.91 2.74 2.24 3.05 1.60 5.11 3.28 7.00

Conc. T-Beam Steel I-Beam Prestressed I-Beam Prestressed Conc. Box R/C Box Slab Multi-Box Conc. Spread Box Steel Spread Box Precast Conc. Spread Box Precast Conc. Bulb-Tee Precast Conc. I-Beam CIP Conc. T-Beam CIP Conc. Multicell Steel I-Beam Steel Open Box

71 163 94 112 121 127 66 35 20 4 4 3 3 4 4 2

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

5.22

Spread Box Beams

27

1

100.00

190.00

5.00

20.00

6.00

12.00

N/A

N/A

1.40

8.00

Adjacent Box Beams

23

1

100.00

210.00

3.00

5.83

5.00

6.00

N/A

N/A

1.13

9.60

Slab 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

1560

1-7

12.00

300.00

2.00

24.00

0.00

36.00

0.00

70.00

0.21

9.60

Summary:

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Data Sources 1. NCHRP 12-26 Bridge Database 800 + Bridges can be used in an automated process to generate simplified and rigorous analyses.

2. Tenn. Tech. Database Detailed descriptions and rigorous analysis are available from a recent TT study for TN DOT. Results, structural models, etc., are readily available.

3. Virtis/Opis Database Bridges

4. Parametrically Generated Bridges

650+ bridges may be exported from Virtis/Opis to supply real bridges to both simplified and rigorous methods.

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

Condense to a Common Database

Common Database A Format NCHRP 12-50

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A

Common Database Format NCHRP 12-50

BRASS-Girder (LRFD)TM

Rigorous Analysis (Basis) SAP AASHTO FE Engine Ansys

Simplified Analysis Methods:       

Standard Specifications (S over D) LRFD Specifications Rigid Method Lever Rule Adjusted Equal Distribution Method Canadian Highway Bridge Design Code Sanders

B

Common Database Format NCHRP 12-50

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B

Common Database Format NCHRP 12-50

Studies Directed Toward:  Skew  Lane Position  Diaphragms

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.

Comparisons and Regression Testing (NCHRP 12- 50 Process) Tasks 6 & 9 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 BridgeTech, Inc.

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, reaction, translation) • Single and multiple lanes loaded • Critical longitudinal position • Accounts for barrier, etc. • 4-ft truck transverse truck spacing • POI at least tenth points BridgeTech, Inc.

Computation of Distribution Factor

Distribution Factor

Rigorous Action  / Number Lanes   g  Action from Beamline  for same Longitudinal Position   

M rigorous g M beam

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Using Distribution Factors M design ( rigorous estimate ) M beam g

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Example of Standard Specification Results Moment at 1.4 One-lane Loaded Exterior I-Girder Std. S/D vs. Rigorous

Std. Spec. (S/D) Distribution Factor

1.4

Unit slope = good R2 = poor

y = 0.9914x + 0.2962 2 R = 0.3834

1.2

1 1

1

0.8

Poor R2 = little hope 

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rigorous Distribution Factor

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

1.4

slope = poor

R2 = good Lever Rule Distribution Factor

1.2 1

y = 1.63x - 0.2644 R2 = 0.8889

1

1

0.8

0.6

Apply affine transformation

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rigorous Distribution Factor BridgeTech, Inc.

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

Calibrated Lever Rule Distribution Factor

1.4

slope = good

1.2 1

y = 0.978x + 0.0413 R2 = 0.8889

1

1

R2 = good and is the same

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rigorous Distribution Factor BridgeTech, Inc.

Simple Method

Affine Transformation Concept Rotation to Unity by multiplication

Original Simple Method Raise or lower by addition/substratio n

Rigorous Method

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Affine Transformation (example) y1 1.63 x 0.2644

y1 1.63x 0.2644 y2    x 0.1622 1.63 1.63 1.63

y1 y3 y 2 0.1622  0.1622 x 0.1622 0.1622 x 1.63 g (Calibrated lever rule )am g Lever rule bm

Unit slope

Unit slope

where 1 am  0.61 and bm 0.1622 1.63 and g (Calibrated lever rule ) is the calibrated distribution factor, and g ( Lever rule ) is the lever rule distribution factor computed with the typical manual approach. BridgeTech, Inc.

Simple Method

Affine Transformation Concept

Rigorous Method

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Moment Distribution Factor Computation Number of Loaded Lanes

Girder

One

Interior and Exterior

Distribution Factor N L   mg m m  a g  b  m   m m lever rule N g   





Multiple Presence Factor

m = 1.2 Use integer part of

Two or more Loaded Lanes

Interior and Exterior

W F  N  mg m m a  c st m  L  10 N g      N g  

Wc 12

to determine number of loaded lanes for multiple presence. m shall be greater than or equal to 0.85.

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Multiple Presences Number of Multiple Loaded Presence Factor Lanes "m" 1 1.20 2 1.00 3 0.85 4 or more 0.65

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Lever Rule Equations (aids) Number Girder of Loaded Location Lanes Distribution Factor 1 de  2 2S

Range of Application

Loading Diagram

Number of Wheels to Beam

6'

d e S 6 ft

1

de S

de

S

1 6'

d 3 1 e  S S

d e S 6 ft

2 de

S

6'

d 3 1 e  S S

Exterior

4'

d e S 10 ft

2 de

S

S

6'

2 or more

3 3d e 8   2 2S S

4'

6'

10  d e S  16 ft

3 de

S

6'

2 d 16 2 e  S S

6'

4'

6'

16  de S  20 ft

4 de

S

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Calibration Coefficients (Moment) Structure Type Steel I-Beam Precast Concrete I-Beam Precast Concrete Bulb-Tee Beam Precast Concrete Double Tee with Shear Keys with or without PostTensioning Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning Precast Concrete Channel with Shear Keys Cast-in-Place Concrete Tee Beam Cast-in-Place Concrete Multicell Box Beam Adjacent Box Beam with Cast-inPlace Concrete Overlay Adjacent Box Beam with Integral Concrete Precast Concrete Spread Box Beam Open Steel Box Beam

AASHTO LRFD Cross Section Type a k k

Moment One Loaded Lane Exterior Interior

am

bm

am

bm

0.61

0.16

1.20

-0.37

0.65

0.15

1.40

-0.41

0.41

-0.03

1.51

-0.69

i

j h e d f g b, c c

0.50

0.06 0.77 Use Article 4.6.2.2.3

-0.17 BridgeTech, Inc.

Calibration Coefficients (Moment) Structure Type Steel I-Beam Precast Concrete I-Beam Precast Concrete Bulb-Tee Beam Precast Concrete Double Tee with Shear Keys with or without PostTensioning Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning Precast Concrete Channel with Shear Keys Cast-in-Place Concrete Tee Beam Cast-in-Place Concrete Multicell Box Beam Adjacent Box Beam with Cast-inPlace Concrete Overlay Adjacent Box Beam with Integral Concrete Precast Concrete Spread Box Beam Open Steel Box Beam

AASHTO LRFD Cross Section Type a k k

Moment One Loaded Lane Exterior Interior

am

bm

am

bm

0.61

0.16

1.20

-0.37

0.65

0.15

1.40

-0.41

0.41

-0.03

1.51

-0.69

i

j h e d f g b, c c

0.50

0.06 0.77 Use Article 4.6.2.2.3

-0.17 BridgeTech, Inc.

Calibration Coefficients (Shear) Shear

Structure Type Steel I-Beam Precast Concrete I-Beam Precast Concrete Bulb-Tee Beam Precast Concrete Double Tee with Shear Keys with or without PostTensioning Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning Precast Concrete Channel with Shear Keys Cast-in-Place Concrete Tee Beam Cast-in-Place Concrete Multicell Box Beam Adjacent Box Beam with Cast-inPlace Concrete Overlay Adjacent Box Beam with Integral Concrete Precast Concrete Spread Box Beam Open Steel Box Beam

AASHTO LRFD Cross Section Type a k k

Exterior Two or More Lanes One Loaded Lane Loaded

Interior One Loaded Two or More Lane Lanes Loaded

av

bv

av

bv

av

bv

av

bv

0.81

0.09

0.94

0.07

1.09

-0.15

0.91

0.05

0.79

0.09

0.94

0.05

1.24

-0.22

1.21

-0.17

0.85

0.04

0.90

0.04

1.27

-0.22

1.04

-0.06

0.63

0.14

0.79 0.12 0.85 Use Article 4.6.2.2.3

-0.01

1.02

-0.10

i

j h e d f g b, c c

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Structural Factor (Moment) Multiple Lanes Loaded Two or more loaded lanes

Structure Type Steel I-Beam Precast Concrete I-Beam Precast Concrete Bulb-Tee Beam Precast Concrete Double Tee with Shear Keys with or without PostTensioning Precast Concrete Tee Section with Shear Keys and with or without Transverse Post-Tensioning Precast Concrete Channel with Shear Keys Cast-in-Place Concrete Tee Beam Cast-in-Place Concrete Multicell Box Beam Adjacent Box Beam with Cast-inPlace Concrete Overlay Adjacent Box Beam with Integral Concrete Precast Concrete Spread Box Beam Open Steel Box Beam

AASHTO LRFD Cross Section Type a k k

F st

Uniform Distribution

i 1.15 j h e

1.10

d f

1.10

Nlane / Ngirder

g b, c c

1.00 Use Existing Specification

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Example 2’

12  P  6  P  Rexterior      0.9 P 10  2  10  2   g Lever Rule 0.9

6’

P/2

P/2

gCalibrated am g Lever Rule bm 0.61 0.9  0.16 gCalibrated 0.71

4’

10’

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Calibration Results Slab-on-Girder Bridges Calibration Constants

Type of Bridge Girder Location

Shear

Exterior

Basic Method

Lanes Loaded

Interior

Exterior

Interior

Initial Trend Line - Lever Rule and Henry's Method (Henry's Method already calibrated)

Computed Calibration Factors (for Lever Rule Calibration)

Recommended Calibration Factors

Slope

Intercept

R2

Figures

a

b

a or F st

b

Regression Plot

R2

1.2386

-0.1112

0.9661

13a, N-125

0.8074

0.0898

0.81

0.09

13b, K-16

0.9656

1.0657

-0.0713

0.9674

14a, N-117

0.9384

0.0669

0.94

0.07

14b, K-15

0.9485

1 Lane

0.9209

0.1355

0.9198

15a, N-93

1.0859

-0.1471

1.09

-0.15

15b, K-12

0.9198

2 or More Lanes

1.1041

-0.0553

0.9343

16a, N-85

0.9057

0.0501

0.91

0.05

16b, K-11

0.9343

1.6396

-0.2679

0.8894

17a, N-61

0.6099

0.1634

0.61

0.16

17b, K-8

0.8894

1.0440

0.1098

0.8757

18a, N-51

n/a

n/a

1.15

n/a

18b, K-7

0.8757

0.8346

0.3076

0.4497

19a, N-29

1.1982

-0.3686

1.20

-0.37

19b, K-4

0.4508

1.0112

0.0658

0.9216

20a, N-19

n/a

n/a

1.15

n/a

20b, K-3

0.9216

1 Lane Calibrated 2 or More Lanes Lever

Moment

I-Girders (a, h, i, j, k)

Action

Calibrated Lever 1 Lane Henry's Method 2 or More Lanes Calibrated Lever 1 Lane Henry's Method 2 or More Lanes

Quite Good (typical) BridgeTech, Inc.

Statistical Comparison Conceptual Mean

Number of Samples

Standard deviation

1.00

g simplified g rigorous BridgeTech, Inc.

Shift Simple Upward by a factor Increase by a factor that is related to the COV

 a

1.00

Simple / Rigorous

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Analysis Factors Type of Bridge

Shear

Basic Method

Calibrated Lever

Moment

I-Girders (a, h, i, j, k)

Action

Calibrated Lever Henry's Method Calibrated Lever Henry's Method

Girder Location

Exterior Interior

Analysis Factor Computations No. of Std. Dev. Offset b = 1 No. of Std. Dev. Offset b = 0.5 No. of Std. Dev. Offset b = 0.0 No. of Computed Rounded Rounded Computed Rounded No. of Std. Computed No. of Std. Dev. Analysis Std. Dev. Analysis Analysis Factor Dev. Offset Analysis Factor Analysis Factor Analysis Offset Offset Factor Factor ( b = 1) (b = 0.5) Factor (b = β ga g a (rounded) β ga g a (rounded) β ga g a (rounded) 1.0 1.049 1.05 0.5 1.020 1.05 0.0 0.991 1.00 1.0 1.053 1.05 0.5 1.019 1.05 0.0 0.986 1.00 1.0 1.069 1.10 0.5 1.035 1.05 0.0 1.001 1.00 1.0 1.102 1.10 0.5 1.051 1.05 0.0 1.000 1.00

Lanes Loaded

Figures

Ratio of Means

Inverse

COV

1 Lane 2 or More Lanes 1 Lane 2 or More Lanes

13c 14c 15c 16c

S/R 1.010 1.014 0.999 1.000

(S/R) -1 0.991 0.986 1.001 1.000

V S/R 0.058 0.067 0.069 0.102

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

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

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

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

Exterior

Interior

High due to high COV

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Example Continued 12  P  6  P   Rexterior       0.9P 10  2  10  2    g Lever Rule 0.9 g Calibrated a m g Lever Rule bm 0.61 0.9  0.16

Previous Example

g Calibrated 0.71

gCalibrated g 0.71

a mg 1.05  1.2  0.71 a mg 0.89 BridgeTech, Inc.

All effects are now separated an understandable Analysis

a mg 1.05  1.2  0.71 Effect of Multiple Presence

a mg 0.89 Variability in analysis

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Skew  Adjustments

for shear

 No

iteration  Commentary M&M 20-07 Study  Neglect decrease for moment

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Curvature  No

change

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

Applicable Cross-Section from Table 4.6.2.2.1-1 a, e and also h, i, j

Range of Correction Factor

1.0 0.20 tan 

if sufficiently connected to act as a unit

Applicability 060 3.5 S 16.0 20 L 240 N b 4

Precast concrete I and bulb tee beams

K

1.0 0.09 tan 

060 3.5 S 16.0 20 L 240 N b 4

Cast-in-Place Concrete Multicell Box

D

12.0 L   1.0 0.25  tan  70d  

060 6.0 S 13.0 20 L 240 35 d 110 N c 3

Concrete Deck on Spread Concrete Box Beams

Concrete Box Beams Used in Multibeam Decks

B, c

f, g

Ld 1.0  12.0 tan  6S

12.0 L 1.0  tan  90 d

060 6.0 S 11.5 20 L 140 18 d 65 N b 3 060 20 L 120 17 d 60 35 b 60 5 Nb 20

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W 1'-9"

1'-9" ts

Push-the-limits bridges Bridge No. 1 2 3 4 5 6 7 8 9 10 11

Overhang

Recommended Slab Girder minimum slab Total Thickness, Spacing, thickness Span Bridge ts S (AASHTO STD Length, L Width, W (ft) Table 8.9.2) (in) (ft) (ft) 12 12 12 12 12 14 16 18 20 12 12

8.80 8.80 8.80 8.80 8.80 9.60 10.40 11.20 12.00 8.80 8.80

9.00 9.00 9.00 9.00 9.00 9.75 10.50 11.25 12.00 9.00 9.00

240 260 280 300 200 200 200 200 200 160 160

44 44 44 44 44 48 54 60 68 58 53

S

S

S

Overhang

No.of girders

Overhang (ft)

4 4 4 4 4 4 4 4 4 5 4

4 4 4 4 4 3 3 3 4 5 8.5

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Many Parameter Studies  Skew  Diaphragm

Cross-frame Stiffness  End Cross-frames  Intermediate Cross-frames  Typical

Example

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Span 1

Span 5 R = 43.82 kips DF = 0.621

R = 34.30 kips DF = 0.564

R = 34.30 kips DF = 0.564

R = 43.82 kips DF = 0.621 25

With Support Diaphragms

1

5 R = 43.82 kips DF = 0.621

R = 34.30 kips DF = 0.564 R = 52.15 kips DF = 0.739

R = 28.00 kips DF = 0.460

R = 34.30 kips DF = 0.564

R = 43.82 kips DF = 0.621 R = 46.62 kips DF = 0.660

R = 43.78 kips DF = 0.720 25

30°

With Support Diaphragms

1 R = 46.62 kips DF = 0.660

R = 43.78 kips DF = 0.720 R = 20.07 kips DF = 0.330

5

R = 61.87 kips DF = 0.876

R = 28.00 kips DF = 0.460

R = 52.15 kips DF = 0.739 R = 52.94 kips DF = 0.870

R = 54.89 kips DF = 0.778 25

60°

With Support Diaphragms

1 R = 52.94 kips DF = 0.870

5 R = 54.89 kips DF = 0.778

R = 61.87 kips DF = 0.876

R = 20.07 kips DF = 0.330 BridgeTech, Inc.

Regression Testing Complete database used to compare:  LRFD  S/D  Rigorous  Again,

used 12-50

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Is this simpler?  Consistent

approach for most bridge

types  Based upon lever rule (shear and onelane moment) – and adjusted  Based uniform distribution (multiplelanes loaded – and adjusted  Independently accounts for multiple presence BridgeTech, Inc.

Is this simpler?  Independently

accounts for variability of simple analysis wrt rigorous  Lever rule aids are provided in appendix  No iterative approach, i.e., independent of cross section and span lengths  Same for positive and negative moment areas  Skew corrections are based upon S/L (readily known) BridgeTech, Inc.

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  Recalibrate

uniform method parallel the calibrated lever  Improve one-lane loaded for moment  Review/revise tub girder systems  Develop

presentation materials to help explain this in a more understandable manner  Suggestions welcome! BridgeTech, Inc.

Questions, Discussion

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End of AASHTO Talk

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Extra slides

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Modeling Appendix Exterior Longitudinal Girder

le1

li /2

li

t1

h1 h

h2 le2

li /2

l e1

Centroid Axis of Box-Girder

tw

tw

hs li

Interior Longitudinal Girder

t2

li

h l e2 * e

l

Exterior Longitudinal Girder

Interior Longitudinal Girder

Closed Section For Torsional Rigidity

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Shear Distribution Factors for CIP Concrete Multicell Box Beam Bridges -- Validation

Multicell Box Beam

Bridge TTU BridgeID No. Lanes Span Type Bridge No. Ext Int Loaded 1 1 2 3 12 1007 1008 1 2 2 3 1 1 2 3 13 1009 1010 1 2 2 3 1 1 2 3 14 1011 1012 1 2 2 3 1 1 2 15 1013 1014 1 2 2

Beam End L L L L L L L L L L L L L L L L L L L L L L

SAP2000 BTLiveLoader Difference Exterior Interior Exterior Interior Exterior Interior 0.620 0.344 0.618 0.338 0.29% 1.84% 0.644 0.876 0.644 0.889 0.06% 1.45% 0.654 0.900 0.653 0.915 0.25% 1.69% 0.618 0.346 0.620 0.336 0.34% 3.03% 0.637 0.885 0.640 0.891 0.51% 0.68% 0.645 0.904 0.649 0.913 0.57% 0.93% 0.643 0.317 0.639 0.310 0.63% 2.08% 0.669 0.861 0.664 0.815 0.80% 5.36% 0.679 0.885 0.672 0.877 1.00% 0.95% 0.635 0.318 0.547 0.289 13.84% 9.10% 0.667 0.854 0.606 0.759 9.06% 11.18% 0.679 0.883 0.629 0.817 7.37% 7.50% 0.722 0.264 0.718 0.257 0.43% 2.52% 0.774 0.955 0.780 0.948 0.70% 0.65% 0.779 1.022 0.784 1.033 0.67% 1.02% 0.706 0.267 0.698 0.261 1.17% 2.26% 0.767 0.941 0.769 0.930 0.15% 1.18% 0.776 1.016 0.778 1.022 0.24% 0.63% 0.652 0.317 0.658 0.314 0.88% 0.95% 0.675 0.914 0.681 0.936 0.83% 2.40% 0.642 0.319 0.645 0.335 0.43% 5.12% 0.671 0.908 0.675 0.983 0.62% 8.30% BridgeTech, Inc.

2

Method Rating Based on the Value of the Correlation Coefficient (R ) between Each Simplified Method and Rigorous Analysis excellent ≥0.9

0.90 > good ≥0.80

Bridge Girder Lanes Action Set Locations Loaded Exterior Shear Interior 1 Exterior Moment Interior Exterior Shear Interior 2 Exterior Moment Interior Exterior Shear Interior 3 Exterior Moment Interior Exterior Shear Interior 4 Exterior Moment Interior

Lever Rule

0.80 > acceptable ≥0.70

Henry's Method 1 excellent good 2 or more excellent acceptable 1 excellent poor 2 or more excellent excellent 1 good good 2 or more good good 1 bad bad 2 or more acceptable excellent 1 excellent good 2 or more excellent excellent 1 excellent poor 2 or more good excellent 1 excellent excellent 2 or more excellent excellent 1 poor bad 2 or more poor excellent 1 excellent acceptable 2 or more excellent excellent 1 excellent acceptable 2 or more good excellent 1 poor poor 2 or more good excellent 1 acceptable bad 2 or more poor excellent 1 excellent poor 2 or more excellent excellent 1 excellent poor 2 or more excellent excellent 1 poor bad 2 or more acceptable good 1 bad excellent 2 or more poor good

LRFD good good good excellent good good bad acceptable good excellent excellent good good excellent excellent excellent excellent excellent excellent excellent poor good poor poor excellent excellent acceptable excellent bad poor bad poor

0.70 > poor ≥0.50 Method CHBDC STD bad bad good good bad poor bad acceptable poor poor excellent excellent poor poor acceptable good poor acceptable acceptable good poor poor bad bad acceptable good good good poor poor poor poor

bad < 0.5 Sanders

bad bad bad bad good good good good bad bad acceptable bad bad bad acceptable poor poor poor poor poor good good good excellent poor poor poor poor poor poor good good acceptable bad acceptable poor good poor excellent poor poor poor acceptable poor poor bad poor bad acceptable poor good acceptable excellent acceptable excellent acceptable bad bad poor bad bad bad poor bad

Best Method Lever Lever Lever Lever Lever Lever CHBDC Henry's Lever Lever Lever Henry's Henry's Henry's LRFD Henry's Lever Lever Lever Henry's CHBDC Henry's Lever Henry's Lever Lever STD Henry's CHBDC Henry's Henry's Henry's

Slab On I

CIP Tees vvcc

Spread

vvcccvc Boxes

Adjacent vvcc Boxes

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Basics Continued 

 





Deflection is the easiest state variable to predict analytically/numerically 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

d w EI 2 M ( x ) dx 3 d w EI 3 V ( x ) dx

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Prerequisites 

We are not proposing to take any one simplified method “as is”. (unless it really works well).



Analytically-based approaches can be implemented at different levels (i.e., compute stiffness parameters) – empirical methods cannot.



Analytically-based approaches can be more easily extended (in case of limits of application), than empirically-based methods.



Analytically-based approaches can be as simple as empirical approaches BridgeTech, Inc.

Task 1 -- Literature Review

Michael Patritch Graduate Student TN Tech

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Task 1 -- Literature Critical Findings  Simplified

methods

 Sanders

and Elleby  “Equal Distribution Method” – name is a misnomer  Canadian Standards  Juxtaposition of stiffness extremes  Stiffness

effects

 Testing  Analysis

and modeling BridgeTech, Inc.

Sanders and Elleby  

NCHRP study Limitations    



Considered   



Span to 120-ft Slab on Beam (Orthotropic Plate Theory) Multi beam (Articulated Orthotropic Plate) CIP Boxes (Folded Plate) Aspect ratio Relative long/trans flexural stiffness Relative torsonal stiffness

Field tests for some validation

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AASHTO LRFD 

NCHRP 12-26



Empirically based



Includes stiffness parameters in equations



“Ugly” equations



Embedded multiple presence factors



No rational analytical basis



Resort to lever rule when empiricism fails



Works reasonably well for interior girders



Limitations are of concern BridgeTech, Inc.

Sanders and Elleby (cont)

g ( wheel ) S / D

Double for LFRD Design Lane

g S / D

2

NL  2N L   C D 5   3 1    10  7   3 NL D 5  10

For C 3 For C 3

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Equal Distribution Method (TN DOT)  All

beams carry equal live load (interior/exterior)  g = NL/N g  Interpolate number of lanes  Adjust g by empirical factors from research  Research is on interior beams  Simple but purely empirical  Limited sample for rigorous comparison BridgeTech, Inc.

TN DOT Equal Distribution Method

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Sanders and Elleby (cont) W C L

1 2

E I1    2G J1 J t   but

C = K(W/L) Bridge Type

Beam and slab (includes concrete slab bridge)

Beam Type and Deck Material

K

Concrete deck: Noncomposite steel I-beams

3. 0

Composite steel I-beams

4. 8

Nonvoided concrete beams (prestressed or reinforced)

3. 5

Separated concrete box-beams

1. 8

Concrete slab bridge

0. 6

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Canadian Specification  Analytically

based upon orthotropic plate

theory  Very similar to Sanders and Elleby  Use either stiffness parameter approach or good estimation tables (easy)  Few limitations  More rational limits for skew and curvature

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Canadian Specification (cont)     

 



Slab Voided slab, including multi-cell box girders with sufficient diaphragms, Slab-on-girders Steel grid deck-on-girders Shear-connected beam bridges in which the interconnection of adjacent beams is such as to provide continuity of transverse flexural rigidity across the cross-section Box girder bridges in which the boxes are connected by only the deck slab and transverse diaphragms, if present Shear-connected beam bridges in which the interconnection of adjacent beams is such as not to provide continuity of transverse flexural rigidity across the cross-section Numerous wood systems ….

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Canadian Specification (cont)

M g avg

n M Lane m  Ng

M g Fm M g avg S N Fm   C f Ce  F 1     100 100  

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Canadian Specification (cont) S N Fm   C f C e  F 1     100 100  

Lane position effect

Lane width effect

F C K Span Length

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Canadian Specification (cont) Similar procedures for shear – different values

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Canadian Specification (cont) Skew limit Plan View

1     Slabs  Bridge Width tan( skew angle)  6    1 Span Length    Slab on Beams   18 

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Canadian Specification (cont)

2

Span Length 1.0 b Radius of Curvature

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Kennedy and Sennah  Steel

Boxes  Possibly concrete with modifications  Similar to LRFD approach (authors claim better accuracy)  Empirical  “Ugly” equations

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European Practice

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Examples

AASHTO LRFD Henry (old method) CSA Sanders and Elleby

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Summary AISI Example 2 Method

Interior Moment

AASHTO LRFD Canadian Bridge Design Code Sanders and Elleby Equal Distribution Factor Method

0.694 0.895 0.710 0.774

PCI Example 4

Exterior

Shear 0.955 0.895 0.710 0.774

Moment 0.840 1.034 0.710 0.774

Interior

Shear 0.840 1.034 0.710 0.774

Moment 0.732 0.783 0.946 0.655

Exterior

Shear 0.884 0.783 0.946 0.655

Moment 0.733 0.930 0.946 0.655

Shear 0.733 0.930 0.946 0.655

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Task 2 – Range of structural forms, materials and range of application  Range

of application

 There

is no reason at this point to limit range of application (we can include range outside of conventional practice and geometries)  “All” parameters will be included in the database  A large amount of data is available from several sources (see Task 6 tables)  Additional data can be added

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Task 3 -- Analytical method 

Mathematical model 

  

Equilibrium, compatibility, and constitutive relationships Beam theory Kirchhoff plate theory Results in governing ODE or PDF

x d w p   4 dx EI  x 4

p x , y   4 w D BridgeTech, Inc.

Task 3 -- Analytical method  Numerical

methods

 Finite

difference  Finite element method (plate or shell elements)  Grillage  Finite Strip Method  Harmonic analysis (Sanders and Elleby)

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Theorem Requirements Independent of the assumptions!  Calculated internal actions modeling and applied

forces are in equilibrium No instability or fracture  Materials

and section/member behavior must be ductile

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The lower bound theorem is one of the most important theorems/concepts in structural engineering.

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The lower bound theorem is one of the most important theorems/concepts in structural engineering. Offers wonderful assurance as the models are often simple approximations to the real world.

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Task 4 -- Process Detail and Chaining NCHRP 12-50

Database

TN Tech, Set 1

NCHRP 12-62 12-26

CSA

Sanders

LRFR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Process 12-50 -> BRASS -> Computation -> 12-50 12-50 -> BRASS -> Input File -> FEA -> 12-50 12-50 -> BRASS -> Input File -> FSM -> 12-50 TT Set 1 -> Process 1 TT Set 1 -> Process 2 TT Set 1 -> Process 3 TT Set 2 -> Process 1 TT Set 2 -> Process 2 TT Set 2 -> Process 3 CSA -> Process 1 CSA -> Process 2 CSA -> Process 3 Sanders -> Process 1 Sanders -> Process 2 Sanders -> Process 3 LRFR -> Process 1 LRFR -> Process 2 LRFR -> Process 3

Effort Small Large Medium Small-> Small Small -> Large Small -> Medium Small-> Small Small -> Large Small -> Medium Depends Depends Depends Small-> Small Small -> Large Small -> Medium Small-> Small Small -> Large Small -> Medium

Do it? yes yes yes yes yes yes yes yes yes maybe maybe maybe maybe maybe maybe yes yes yes

Sample

All All All Typical Typical Typical

Typical Typical Typical

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Task 4 -- Process Matrix AASHTO Std Spec. AASHTO LRFD Lever Rule Rigid method CSA Modified Henry JOSE Rigorous FEA Rigorous FSM

Method

Comparison of Methods and Available Databases of Bridges AASHTO Std AASHTO LRFD AASHTO LRFD Lever Rule AASHTO LRFD Rigid Method Sanders Canadian Specification (CSA) Henry (Modified) Rigorous FEA Rigorous FSM

Notes S/D equation, program into an automated approach LRFD Eqs, program into an automated approach Program into an automated approach Program into an automated approach Program into an automated approach Program into an automated approach Program into an automated approach FEA engine and/or a commercial FEA engine Available FSM program

NCHRP 12-50 process 1 process 1 process 1 process 1 process 1 process 1 process 1 process 2 process 3

TN Tech Set 1 process 4 process 4 process 4 process 4 process 4 process 4 process 4 process 5 process 6

Database NCHRP 12-26 process 7 process 7 process 7 process 7 process 7 process 7 process 7 process 8 process 9

CSA process 10 process 10 process 10 process 10 process 10 process 10 process 10 process 11 process 12

Sanders process 13 process 13 process 13 process 13 process 13 process 13 process 13 process 14 process 15

LRFR process 16 process 16 process 16 process 16 process 16 process 16 process 16 process 17 process 18

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Task 6 – Parametric Study  Slab

on Steel Girders

 321

Total Data Sets  AASHTO Type A

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Task 6 – Parametric Study  Slab

on Precast I and Bulb Tee Girders

 176

Total Data Sets  AASHTO Type K

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Task 6 – Parametric Study  Slab

on Concrete Tees

 74

Total Data Sets  AASHTO Types E and J

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Task 6 – Parametric Study  Slab

Bridges – 127 Slab Data Sets

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Task 6 – Parametric Study 

Spread Concrete Boxes  

94 Total Data Sets AASHTO Types B and C

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Task 6 – Parametric Study  Adjacent

Concrete Boxes

 307

Total Data Sets  AASHTO Types D, F, and G

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Task 6 -- Wood Bridges  Wood

bridges will not be addressed

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Additional Load Distribution Issues Additional Parameters Parameters Skew Barriers Diaphragms Skew Yes Maybe Yes Barriers Maybe Yes Maybe Diaphragms Yes Maybe Yes Location (e.g. Fatigue) Maybe Yes Maybe

Location Maybe Yes Maybe Yes

Perform separate parametric studies to focus exclusively on these effects

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Three primary questions for the panel  Lane

width to determine the number of

lanes  Live load position  Multiple presence factors

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Number of Lanes Loaded •Issue for our study – not total design load for girder systems •Integer number or decimal value •Should not be overly sensitive to the distribution factor

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Live Load Position – Interior Girder 6-4-6-6 (critical)

6-4-6-4-6 (critical)

6-6-6-6-6 (critical)

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Live Load Position (exterior girder) 2-6-4-6-6

2-6-4-6-4-6

2-6-6-6-6-6

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Multiple Presence – separate from live load distribution 

Task 6 distribution factors will be computed for one-, two-, three-, etc-lanes loaded. This could be combined, if necessary, later.



Research simplified methods will not include m.



Rearch simplified methods will permit one-, two-, three-, etc-lanes loaded to be computed and independently applied.



The specification can clearly indicate (apriori) how the number of controlling lanes, i.e., it can be explicit and simple. BridgeTech, Inc.

Example Number of lanes loaded 1

Multiple Presence Factor 1.2

Factor Required by Method x.X

2

1.0

y.Y

3

0.85

z.Z

4 or more

0.65

w.W Controls for Strength I

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