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1

ENCE717 – Bridge EngineeringBridge Analysis

Chung C. Fu, Ph.D., P.E.

(http: www.best.umd.edu)

Introduction – Bridge Analysis

7. Numerical Methods (3.0)

Numerical Methods – 3D FEM

 Advantages and Disadvantages of Grid Analysis

 Advantages and Disadvantages of 3D FEM Analysis

3D FEM Example

Numerical Methods – Elastic Stabili ty (3.2.8 & 14.0)

Numerical Methods – Creep and Shrinkage Analysis (3.3

& 5.2)

Numerical Methods – Influence Surface (3.4)

Numerical Methods – 3D FEM

Numerical Methods –

 Applications in Bridge Analysis

Issues include:

1) what types of element a bridge model should be

used;

2) when a 2D model is sufficient and when a 3D

model is necessary; and

3) how to correctly interpret FEM results from bridge

engineering perspectives, especially when a

bridge is modeled as plate or shell elements.

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Numerical Methods –

Elements used in Bridge Analysis

In general, truss, frame(/beam) and shell(/plate) elements can

cover most bridge analyses; Truss element is also called link element. Bridge bearings,

hangers, prestress tendons, cables, and etc. can be modeled

as truss elements.

In line models, girders, stringers, diaphragms, pylons,

columns, piers, and etc. are usually modeled as frame

elements.

Shell element combines in-plane stress/strain behaviortogether with bending of a plate, either as a thin plate or a

thick plate.

Numerical Methods – Advantages and

Disadvantages of Grid Analysis

Middle- and short-span girder bridges, an intermediate model,

or the so-called grid model, is widely used;

Grid model: each node of an element has only vertical

displacements, bending rotation and torsional displacements.

Element internal forces contain bending and torsional

moments plus shear, accordingly;

• Some behaviors of a wide thin walled box girder, such as

warping when torsion is restrained, distortion when insufficient

diaphragm is used and shear lagging due to longitudinal shear

deformations of flanges, cannot be represented in a gridmodel;

Numerical Methods – Advantages and

Disadvantages of 3D FEM Analysis (1)

For truss or frame elements, internal forces output from FEM

analyses can be used directly for engineering design and

code checks;

For shell elements, the original FEM results are not

meaningful and cannot be used in design or code because the

stresses in each element’s local coordinate system.

•Shell elements of  a web in a box girder and vertical shear stresses

Numerical Methods – Advantages and

Disadvantages of 3D FEM Analysis (2)

a) Curves – axial stresses distribution

from a shell element model

b) Straight lines – axial stresses

distribution re-computed from beam

bending theory by using equivalentinternal forces obtained from stress

integration

•Stresses along horizontal direction after unfolded

•Major principal stresses

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3D FEM Example - MD28 in Tuscarora 3D FEM Example - MD28 in Tuscarora

Pattern: Allcracks are

initiated from

skewed

abutment and

normal to the

abutment line.

Then, turnparallel to the

girder lines

3D FEM Analysis - 55’ Span, 15 Skew

• Little difference

4 Skewed Ties at 5’ and 20’

from Supports

2 Staggered Ties (Full-

Width)

3D FEM Analysis - 55’ Span, 30 Skew

• Skewed ties show significant improvement

4 Skewed Ties at 5’ and 20’

from Supports

2 Staggered Ties (Full-

Width)

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Numerical Methods – Elastic Stability Numerical Methods – Elastic Stability

Figure 14.12 – The first mode of a simple arch

bridge bulking, out-of-plane ( 408.516

Elastic Stabili ty Example

Figure 14.13 – The second mode of a

simple arch bridge bulking, out-of-plane

( 1046.208

Figure 14.14 – The third mode of a

simple arch bridge buckling, in-plane

( 1259.367

Numerical Methods –

Creep and Shrinkage Analysis

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Numerical Methods –

Creep and Shrinkage Analysis

Figure 3.13  –  Moment distribution of  a 3‐span continuous bridge built span by span, 

without consideration of  concrete creep considered (kN∙M  )

Figure 3.14  –  Moment distribution of  a 3‐span continuous bridge 8 years after built span 

by span, with consideration of  concrete creep considered (kN∙M  )

Creep and Shrinkage Typical Time Curve (1)

4.04.0

3.53.5

3.03.0

2.52.5

2.02.0

1.51.5

3.723.72

3.033.03

2.572.57

2.222.22

2.002.00

1.701.70

1.441.44

1.01.0

0.50.5

00 33 77 1414 21212828 42425656 33 44 55 66 99 11 1.51.5 22 33 55

DaysDays MonthsMonths YearsYears

     1     1 . .     2     2

     0     0

     1     1 . .     0     0

     7     7

     1     1 . .     0     0

     0     0

     0     0 . .     9     9

     6     6

     0     0 . .     9     9

     1     1

     0     0 . .     9     9

     4     4

     0     0 . .     9     9

     0     0

     0     0 . .     8     8

     8     8

tt

DURATION OF LOADINGDURATION OF LOADING

     T     T     O     O

     T     T     A     A     L     L

     E     E     L     L     A     A     S     S     T     T     I     I     C     C

     A     A     N     N     D     D

     C     C     R     R     E     E     E     E     P     P

     S     S     T     T     R     R     A     A     I     I     N     N

4.0

3.5

3.0

2.5

2.0

1.5

3.72

3.03

2.57

2.22

2.00

1.70

1.44

1.0

0.5

0 3 7 14 21 28 42 56 3 4 5 6 9 1 1.5 2 3 5

Days Months Years

     1 .     2

     0

     1 .     0

     7

     1 .     0

     0

     0 .     9

     6

     0 .     9

     1

     0 .     9

     4

     0 .     9

     0

     0 .     8

     8

t

DURATION OF LOADING

     T     O

     T     A     L

     E     L     A     S     T     I     C

     A     N     D

     C     R     E     E     P

     S     T     R     A     I     N

Creep and Shrinkage Typical Time Curve (2) Moment due to Creep

Free Cantilever Statical SystemFree Cantilever Statical System

Changed Statical System (Midspan Continuous)Changed Statical System (Midspan Continuous)

MMFinal (t)Final (t)

½L½L ½L½L

MMII M =M =II

FixedFixed FixedFixed

qq

qLqL22

88

MMIIIIM =M =IIII

qLqL22

1212qLqL22

2424

MMIIII

MMII

MMcr (t)cr (t)

Free Cantilever Statical System

Changed Statical System (Midspan Continuous)

MFinal (t)

½L ½L

MI M =I

Fixed Fixed

q

qL2

8

MIIM =II

qL2

12qL2

24

MII

MI

Mcr (t)

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Structural Concrete subjected to Creep

elel (t )(t )00

cr cr  (t )(t )

PP PP

PPef ef  PPef ef 

Cantilever BeamCantilever Beam

Simple BeamSimple Beam

elel ( )( )tt 00cr cr (t )(t )

el (t )0

cr  (t )

P P

Pef  Pef 

Cantilever Beam

Simple Beam

el ( )t 0cr (t )

Structural Concrete subjected to Creep

PP

Post-Tensioned BeamPost-Tensioned BeamPP

PP PP

PPef ef 

PPef ef 

elel (t )(t )00

elel (t )(t )00

elel (t )(t )

PT TendonPT Tendon

P

Post-Tensioned BeamP

P P

Pef 

Pef 

el (t )0

el (t )0

el (t )

PT Tendon

Collapse of Palau Bridge due to CreepNumerical Methods –

Influence Surface

Figure 3.19  –  Influence surface of  a tied arch bridge

For spatial bridge analyses, traditional lateral load distribution theories,

influence lines and simplified calculation methods are substituted by

spatial structural analyses and influence surface loading.