Load Path and Equilibrium of Bridge Structures › ~ccfu › ref717 › Load path lecture-UMD.pdf · (3) The family of graphic statics can be illuminating and thought-provoking, since

Zhao LIU, Ph.D., Prof.School of Civil Eng., Southeast University, China

Email: [email protected] 18, 2020

Load Path and Equilibrium of

Bridge Structures

mailto:[email protected]

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OUTLINE

1. Introduction2. Graphical method for bridge structures3. Load path visualization in continuum bodies4. Conceptual design from perspective of load paths5. Conclusions

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Newton’s three laws of motion: Mathematical Principles of Natural Philosophy (1687)

1. Introduction

First law

Third law

Second law

v =const.

F = ma

F = R

Kinetics

Dynamics

Statics

Force equilibrium is also a cornerstone for bridge design

1. Introduction

Equilibrium: For a structure to stay put, all forces must cancel out

ΣFx = 0; ΣFy = 0; ΣM = 0

Load path: All forces or loads must go through bridge structure and eventually get to the ground.

1. Introduction

Equilibrium, reliable? Load path, efficient?

1. Introduction

Equilibrium conditions can be established by- Algebraical, or numerical methods - Geometrical, or graphical methods

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Nowadays-• More and more efforts have been placed on computational

and matrix methods; • less and less attention has been given to visual thinking

and hand drawing.

However, the graphical methods has - an aesthetic power, - a profound engineering insight

2.Graphical method for bridge structures

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RL RR

/ 257.5L i iR F x L kN= =∑( ) / 142.5R i iR F L x L kN= − =∑

2.1 Girder Bridge

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2.1 Girder Bridge

Choose for yourconvenience Force scale factor,

and Length scale factor

Hands-on exercise?


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pole

2.1 Girder Bridge


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2.1 Girder Bridge


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Funicular polygon Force polygonparallel projection

2.1 Girder Bridge


2.1 Girder Bridge


Bending moment diagram

2.1 Girder Bridge


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Moment diagram

Shear diagram

M ( * )( * )scale f scaleLength L H F=

V * scaleForce F=

2.1 Girder Bridge


2.2 Arch Bridge

Funicular polygon

Force polygon


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2.2 Arch Bridge

Force polygonFunicular polygon


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2.2 Arch Bridge

Ideal arch axis

Horizontal thrust


2.2 Arch Bridge


2.2 Arch Bridge

Robert Maillart: Salginatobel bridge, Switzerland, 1930


2.2 Arch Bridge

Gustave Eiffel: Ponte de Dona Maria Pia, Porto, Portugal A railway bridge (1877)


2.2 Arch Bridge


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2.3 Cable-stayed Bridge


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2.4 Suspension Bridge

Who is Steinman? 1922




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Kim, Namhee & Koh, Hyun-Moo. (2013). Preliminary Structural Form Planning for Suspension Bridge According to Force Flow. Journal of The Korean Society of Civil Engineers. 33. 10.12652/Ksce.2013.33.4.1315.

Load path, or force transfer- Different definitions for different investigators

Pictures in this slide after Malcolm Millais: Building Structures: From Concepts to Design 2nd Edition

3. Load path visualization in continuum bodies

Stress concentration around a hole

Stress contour by FEM

Force flow analogy Load path model by graphical method


Stress TrajectoryPrincipal tension, andcompression

Stress contours:Tension

Compression


For a beam continuum subjected to a point load, it is hard to find clear load path.

Various visualizations and interpretations have been defined.

Various visualizations and interpretations have been defined.

Force polygonFunicular polygon


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-For RC structures, the truss model or the strut-and-tie model serves practicing engineers to grasp load path in order to provide good details of reinforcement and to determine load carrying capacity of the members in very effective way.

-It demands clear understanding of load path.


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-The STM is based on lower bound theorem of plasticity, so it can be assured to deliver safe designed structure. - However, its uniqueness has been bothering practitioners.


4. Conceptual design: from perspective of load path

504m

125m

CASE 1: Sydney Harbor Bridge, Australia (1932)

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From the load path perspective, we can conclude Lower chord is the backbone of the arch bridge The towers at both side are vases, no structural function Its horizontal thrust and vertical reaction can be quickly estimated

�𝐻𝐻 = 𝑞𝑞 ⁄𝑙𝑙2 ( 8𝑙𝑙𝑉𝑉 = 𝑞𝑞 ⁄𝑙𝑙 2

VV

HH


A through-type trussed arch bridge

220m72m 72m

CASE 2: Xiegang Bridge, Suzhou, China (2015)


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Load path features: tied arch + continuous beam

Counter-weight


Case 3:

Yingzhou Bridge Luoyan, China(2009)

3 drop-in spans

6 expansion joints


Expansion joints

Although balanced, less robust


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Expansion joints

Unfavorable aspects of too many expansion joints Render the tension in ties more elusive Leave room for small dislocation or rotation at joints Reduce rideability of roadway Bring maintenance problem


Span Length=120m, Width=5m, Height of tower=30m Design load: q=5kN/m2

Two lattice-type steel towers, hinged at their base

Case 4: Shunyi Bridge, Beijing, China (2006)


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Which rotational direction shall be released for the bottom pivot pins?

longitudinal

transverse


Restricted in longitudinalFree rotation in transverse


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On Dec 6, 2006, the bridge collapsed when loading for acceptance test.

Case 5: Tension at hammer head bent cap


Case 5: Tension at hammer head bent cap


Case 6: Cracking at corner joint and cap beam of a straddle bent


Case 7: Corner cracking in dapped-end beam

挂梁悬臂梁悬臂梁h

h

h

CantileverDrop-in

span Cantilever


Case 7: Corner cracking in ledge beam

Vertical tension

Honrizontal tension


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Case 8: Built-in bolt under tension

Skin rebar is helpful


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Welded on side plates

is more effective

Case 9: Force transfer from gusset plate to chord


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Case 9: Force transfer from gusset plate to chord

Welded on mid-sides

is more effective


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(1) For the integrity of a bridge structure, layout of structural system and construction details should be equally emphasized.

5. Conclusions

L1L2 L2

Weight Weight

Proportioning, dimensioning, detailing- L1/L2, h/L, f/L- Well defined load paths: vertical, transverse, longitudinal- Counterweights- Member sizes- Joints- … …51

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(2) The form and geometry of a structure is the expression of force in itself.

5. Conclusions

Load path– Short and direct, asap– Smooth transition at nodes– Less-concentrated– Redundant

Force equilibrium– Robust– Reliable– Resilience, at extreme events

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(3) The family of graphic statics can be illuminatingand thought-provoking, since it reveals thingsotherwise difficult to see when using purelynumerical or matrix method.

However, graphical statics is not all-powerful, whichcan be cumbersome to deal with indeterminatestructures, or to find displacement solution.

5. Conclusions

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(4) Strut-and-tie models are developed to captureload path in structural concrete, so that to providegood details of reinforcement and to determine loadcarrying.

The STM is based on lower bound theorem ofplasticity, so it can be assured to deliver safedesigned structure.

5. Conclusions

However, uniqueness problem in STM haunt its users in many circumstances.

Thank you for your attention!

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Load Path and Equilibrium of Bridge Structures › ~ccfu › ref717 › Load path lecture-UMD.pdf · (3) The family of graphic statics can be illuminating and thought-provoking, since