Zhao LIU, Ph.D., Prof. School of Civil Eng., Southeast University, China Email: [email protected] February 18, 2020 Load Path and Equilibrium of Bridge Structures
Zhao LIU, Ph.D., Prof.School of Civil Eng., Southeast University, China
Email: [email protected] 18, 2020
Load Path and Equilibrium of
Bridge Structures
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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?
2.Graphical method for bridge structures
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pole
2.1 Girder Bridge
2.Graphical method for bridge structures
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2.1 Girder Bridge
2.Graphical method for bridge structures
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Funicular polygon Force polygonparallel projection
2.1 Girder Bridge
2.Graphical method for bridge structures
2.1 Girder Bridge
2.Graphical method for bridge structures
Bending moment diagram
2.1 Girder Bridge
2.Graphical method for bridge structures
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Moment diagram
Shear diagram
M ( * )( * )scale f scaleLength L H F=
V * scaleForce F=
2.1 Girder Bridge
2.Graphical method for bridge structures
2.2 Arch Bridge
Funicular polygon
Force polygon
2.Graphical method for bridge structures
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2.2 Arch Bridge
Force polygonFunicular polygon
2.Graphical method for bridge structures
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2.2 Arch Bridge
Ideal arch axis
Horizontal thrust
2.Graphical method for bridge structures
2.2 Arch Bridge
2.Graphical method for bridge structures
2.2 Arch Bridge
Robert Maillart: Salginatobel bridge, Switzerland, 1930
2.Graphical method for bridge structures
2.2 Arch Bridge
Gustave Eiffel: Ponte de Dona Maria Pia, Porto, Portugal A railway bridge (1877)
2.Graphical method for bridge structures
2.2 Arch Bridge
2.Graphical method for bridge structures
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2.3 Cable-stayed Bridge
2.Graphical method for bridge structures
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2.4 Suspension Bridge
Who is Steinman? 1922
2.Graphical method for bridge structures
2.4 Suspension Bridge
2.Graphical method for bridge structures
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2.4 Suspension Bridge
2.Graphical method for bridge structures
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
3. Load path visualization in continuum bodies
Stress TrajectoryPrincipal tension, andcompression
Stress contours:Tension
Compression
3. Load path visualization in continuum bodies
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
3. Load path visualization in continuum bodies
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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.
3. Load path visualization in continuum bodies
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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.
3. Load path visualization in continuum bodies
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
4. Conceptual design: from perspective of load path
A through-type trussed arch bridge
220m72m 72m
CASE 2: Xiegang Bridge, Suzhou, China (2015)
4. Conceptual design: from perspective of load path
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Load path features: tied arch + continuous beam
Counter-weight
4. Conceptual design: from perspective of load path
Case 3:
Yingzhou Bridge Luoyan, China(2009)
3 drop-in spans
6 expansion joints
4. Conceptual design: from perspective of load path
Expansion joints
Although balanced, less robust
4. Conceptual design: from perspective of load path
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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
4. Conceptual design: from perspective of load path
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)
4. Conceptual design: from perspective of load path
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Which rotational direction shall be released for the bottom pivot pins?
longitudinal
transverse
4. Conceptual design: from perspective of load path
Restricted in longitudinalFree rotation in transverse
4. Conceptual design: from perspective of load path
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On Dec 6, 2006, the bridge collapsed when loading for acceptance test.
Case 5: Tension at hammer head bent cap
4. Conceptual design: from perspective of load path
Case 5: Tension at hammer head bent cap
4. Conceptual design: from perspective of load path
Case 6: Cracking at corner joint and cap beam of a straddle bent
4. Conceptual design: from perspective of load path
Case 7: Corner cracking in dapped-end beam
挂梁悬臂梁 悬臂梁h
h
h
CantileverDrop-in
span Cantilever
4. Conceptual design: from perspective of load path
Case 7: Corner cracking in ledge beam
Vertical tension
Honrizontal tension
4. Conceptual design: from perspective of load path
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Case 8: Built-in bolt under tension
Skin rebar is helpful
4. Conceptual design: from perspective of load path
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Welded on side plates
is more effective
Case 9: Force transfer from gusset plate to chord
4. Conceptual design: from perspective of load path
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Case 9: Force transfer from gusset plate to chord
Welded on mid-sides
is more effective
4. Conceptual design: from perspective of load path
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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!