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 2.4 Suspension Bridge 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 ⁄ ⁄ � 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!.