Load Path and Equilibrium of Bridge Structures
Zhao LIU, Ph.D., Prof. School of Civil Eng., Southeast University, China Email: [email protected] February 18, 2020 OUTLINE
1. Introduction 2. Graphical method for bridge structures 3. Load path visualization in continuum bodies 4. Conceptual design from perspective of load paths 5. Conclusions
2 1. Introduction
Newton’s three laws of motion:
Mathematical Principles of Natural Philosophy (1687)
First law v =const. Kinetics
Second law F = ma Dynamics
Third law F = R Statics
Force equilibrium is also a cornerstone for bridge design
3 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 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
6 2.Graphical method for bridge structures
2.1 Girder Bridge
RL RR
RL=∑ F ii x/ L = 257.5 kN
RR=∑ F ii( L −= x ) / L 142.5 kN
7 2.Graphical method for bridge structures
2.1 Girder Bridge
Choose for your convenience Force scale factor, and Length scale factor
Hands-on exercise? 8 2.Graphical method for bridge structures
2.1 Girder Bridge
pole
9 2.Graphical method for bridge structures
2.1 Girder Bridge
10 2.Graphical method for bridge structures
2.1 Girder Bridge
parallel projection Funicular polygon Force polygon
11 2.Graphical method for bridge structures
2.1 Girder Bridge 2.Graphical method for bridge structures
2.1 Girder Bridge
Bending moment diagram
13 2.Graphical method for bridge structures
2.1 Girder Bridge
Moment diagram
M= (Length * Lscale )( H f * F scale )
Shear diagram
V*= Force Fscale 2.Graphical method for bridge structures
2.2 Arch Bridge
Force polygon
15 Funicular polygon 2.Graphical method for bridge structures
2.2 Arch Bridge
Funicular polygon Force polygon
16 2.Graphical method for bridge structures
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
21 2.Graphical method for bridge structures
2.3 Cable-stayed Bridge
22 2.Graphical method for bridge structures
Who is Steinman? 1922 2.Graphical method for bridge structures
2.4 Suspension Bridge
24 2.Graphical method for bridge structures
2.4 Suspension Bridge
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. 3. Load path visualization in continuum bodies
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
For a beam continuum subjected to a point load, it is hard to find clear load path. Various visualizations and interpretations have been defined.
Stress Trajectory Principal tension, and compression
Stress contours: Tension
Compression 3. Load path visualization in continuum bodies
Various visualizations and interpretations have been defined.
Funicular polygon Force polygon
29 3. Load path visualization in continuum bodies
-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.
30 30 3. Load path visualization in continuum bodies
-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
CASE 1: Sydney Harbor Bridge, Australia (1932)
125m
504m
32 4. Conceptual design: from perspective of load path
H H
V V 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
= 2 = ( 8 2 𝑉𝑉 𝑞𝑞 𝑙𝑙⁄ 𝐻𝐻 𝑞𝑞 𝑙𝑙 ⁄ 𝑙𝑙� 4. Conceptual design: from perspective of load path
CASE 2: Xiegang Bridge, Suzhou, China (2015)
A through-type trussed arch bridge
72m 220m 72m
34 4. Conceptual design: from perspective of load path
Load path features: tied arch + continuous beam
Count er- wei ght 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
Although balanced, less robust
Expansion joints
37 4. Conceptual design: from perspective of load path
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
Expansion joints 4. Conceptual design: from perspective of load path
Case 4: Shunyi Bridge, Beijing, China (2006)
Span Length=120m, Width=5m, Height of tower=30m Design load: q=5kN/m2 Two lattice-type steel towers, hinged at their base
39 4. Conceptual design: from perspective of load path
Which rotational direction shall be released for the bottom pivot pins?
transverse
longitudinal 4. Conceptual design: from perspective of load path
Free rotation in transverse Restricted in longitudinal
41 On Dec 6, 2006, the bridge collapsed when loading for acceptance test.
42 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 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
h
Drop-in
悬臂梁 h 挂梁 悬臂梁 Cantilever span Cantilever
h 4. Conceptual design: from perspective of load path
Case 7: Corner cracking in ledge beam
Honrizontal tension
Vertical tension
47 4. Conceptual design: from perspective of load path
Case 8: Built-in bolt under tension
Skin rebar is helpful
48 4. Conceptual design: from perspective of load path
Case 9: Force transfer from gusset plate to chord
Welded on side plates is more effective
49 4. Conceptual design: from perspective of load path
Case 9: Force transfer from gusset plate to chord
Welded on mid-sides is more effective
50 5. Conclusions
(1) For the integrity of a bridge structure, layout of structural system and construction details should be equally emphasized.
We i g h t We i g h t
L2 L1 L2
Proportioning, dimensioning, detailing
- L1/L2, h/L, f/L - Well defined load paths: vertical, transverse, longitudinal - Counterweights - Member sizes - Joints 51 - … … 51 5. Conclusions
(2) The form and geometry of a structure is the expression of force in itself.
Load path – Short and direct, asap – Smooth transition at nodes – Less-concentrated – Redundant Force equilibrium – Robust – Reliable – Resilience, at extreme events
52 5. Conclusions
(3) The family of graphic statics can be illuminating and thought-provoking, since it reveals things otherwise difficult to see when using purely numerical or matrix method.
However, graphical statics is not all-powerful, which can be cumbersome to deal with indeterminate structures, or to find displacement solution.
53 5. Conclusions
(4) Strut-and-tie models are developed to capture load path in structural concrete, so that to provide good details of reinforcement and to determine load carrying. The STM is based on lower bound theorem of plasticity, so it can be assured to deliver safe designed structure.
However, uniqueness problem in STM haunt its users in many circumstances.
54 Thank you for your attention!