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Problem 01 Invent Yourself

Problem 01 Invent Yourself

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Problem 01 Invent Yourself

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  1. Problem 01Invent Yourself reporter: Denise Sacramento Christovam

  2. Problem 1 Invent Yourself It is more difficult to bend a paper sheet, if it is folded “accordion style” or rolled into a tube. Using a single A4 sheet and a small amount of glue, if required, construct a bridge spanning a gap of 280 mm. Introduce parameters to describe the strength of your bridge, and optimise some or all of them.

  3. Contents

  4. Strength of Materials - Paper • Intermolecular forces Hydrogen Bonds Cellulose

  5. Strength of Materials - Paper Linear structure composed of cellulose fibers intertwined. For calculation purposes, the A4 sheet may be considered isotropic. http://mosmanibphilosophy.wordpress.com/2012/09/05/what-is-contained-in-a-blank-sheet-of-paper/

  6. Strength of Materials - Paper Cardboard is more resistant than bond paper for the same bridge structure. Cardboard- 250 g/m² Bond Paper- 75 g/m²

  7. Strength of Materials - Paper F • Hooke’s Law analogy: 1 x Greater grammage 2 F Lower grammage Increases with the resistance θ1 θ2 x

  8. Strength of Materials - Paper • Flexural rigidity: Tension from the torque required to bend the structure Thickness t Young Modulus Second moment of inertia

  9. Bridge Types

  10. Bridge Types - Accessories

  11. Bridge Types – Trussed Bridge • Structure of connected elements forming triangular units. • Typically straight, that may be stressed from tension and compression. • External forces and reactions are considered to act only on the nodes. Howe Truss Brown Truss Pratt Truss Warren Truss

  12. Force Distribution – Plane Bridge Tangent Tangent W Bridge falls due to its own weight.

  13. Force Distribution – Rectangular Bridge L Lateral View W Weight generates forces in the surface the paper Greater resistance (as already shown)

  14. Force Distribution – Tubular Bridge W Concentrated load

  15. Force Distribution – Fanfolded Bridge Until the bridge collapses (max. load) Series of triangular structures Every opening angle has an optimal number of folds, as will be shown in experiments.

  16. Force Distribution – Trussed Bridge • Considering the most stable truss bridges, Warren and Pratt: Warren Truss Pratt Truss Load Compression Our bridge (modified due to difficulty during making process) Tension Limitations: single A4 sheet. Offers greater stability than the fanfolded owing to the presence of diagonal beams throughout the structure Offers more resistance since it provides the compression cancelling tension, balancing the entire structure Being irregular (triangles with different angles and sides), there’s no total cancellation of the tensions Equilateral triangles are very stable and uniform structures; they balance the force distribution better than any other truss

  17. Force Distribution – Trussed Bridge • Euler’s equation for columns: • Optimization: small triangles • Glue used: scholar • No chemical reactions with paper • More efficient distribution of tension along the bridge’s joints • Represents ≈ 8% of the paper bridge’s mass Must be low for maximum load supported K=1,0 Slenderness ratio

  18. Torque on the supports • Minimizing the influence of torque and activation energy x N f N’ R = 2h Equilibrium Imminence of falling Falling τ High thickness = more force required to fall over = no loss of contact

  19. Material • Types of paper: • Bond A4 (75g/m²) • Cardboard A4 (120 g/m²) • Cardboard A4 (250 g/m²) • Corrugated fiberboard A4 (237 g/m²) • Newsprint A4 (48 g/m²) • Wrapping tissue A4 (20 g/m²) • Coins of different masses: • R$ 0.05: 4,10 g • R$ 0.10: 4,80 g • R$ 0.25: 7,55 g • R$ 0.50: 7,81 g • Weights of different masses • Sand • Filler • 2 supports for the bridge • Wooden plank • Glue • Ruler (millimetres) (±0,5 cm) • Scale (±0,05 g)

  20. Experimental Description • Experiment 1: Different weight arrangements. • Experiment 2: Type of Bridge. • Experiment 3: Variation of each type of bridge’s configuration. • Experiment 4: Accessories to increase the bridge’s strength. • Experiment 5: Variationof paper’s characteristics.

  21. Experiment 1: Load Distribution Punctual Distribution Bridge Supports 28 cm

  22. Experiment 1: Load Distribution Sand’s density: 2,00g/ml Total volume: 500 ml Falling time: 70,18 s Measuring

  23. Experiment 1: Load Distribution Uniform Distribution 1 2

  24. Experiment 1: Load Distribution 3 4

  25. Experiment 1: Load Distribution Standard bridge: fanfolded bridge (“accordion”) Note: number of folds= 13

  26. Experiment 1: Load Distribution Optimization: uniform distribution (4)

  27. Experiment 2: Bridge types Plane Rectangular Tubular Fanfolded Trussed Standard paper: bond (75 g/m²)

  28. Experiment 2: Bridge types Optimization: triangular bridge (5)

  29. Experiment 3: Bridge parameters 1. Plane bridge: number of folds 2 folds 1 coin: 4,1g 3 folds 3 coins: 23,4g

  30. Experiment 3: Bridge parameters 2. Tubular bridge: diameter Diameter Variation Load (10-1 g) Optimization: small diameter (more folds) Diameter (cm)

  31. Experiment 3: Bridge parameters Using the inclined plane method to determine the wood-paper static friction coefficient: Surface (wall) Wooden Block Surface (wall) β

  32. Experiment 3: Bridge parameters 3. Fanfolded bridge: internal angle Optimization:n α sin²(α) Each angle has an optimal number of folds

  33. Experiment 3: Bridge parameters Optimization: 16 folds

  34. Experiment 4: Accessories used to increase strength 1. Fanfolded bridge: holding bands Removed bands from the same sheet

  35. Experiment 4: Accessories used to increase strength 1. Fanfolded bridge: holding bands Supported nearly 386g

  36. Experiment 4: Accessories used to increase strength 1. Fanfolded bridge: holding bands 386g 225g

  37. Experiment 4: Accessories used to increase strength 2. Fanfolded bridge: holding bands + tubes Band Bridge Tube Support Supported nearly 1098g

  38. Experiment 5: Paper Variation 1. Rectangular bridge: Cardboard (120g/m²) Cardboard (250g/m²) Supported nearly 415g Supported nearly 690g

  39. Experiment 5: Paper Variation 2. Fanfolded bridge: Cardboard (120g/m²) Cardboard (120g/m²) Cardboard (250g/m²) Supported nearly 525g Supported nearly 447g

  40. Experiment 5: Paper Variation 2. Fanfolded bridge: Corrugated fiberboard (237g/m²) Newsprint (48 g/m²) Wrapping tissue (20 g/m²) Supported nearly 732g Supported nearly 30g Supported nearly 12g

  41. Experiment 5: Paper Variation Optimization: high grammage

  42. Weight Arrangement • Bridge • Accessories • Paper Conclusion

  43. References • BS EN 1993 Eurocode 3: Design of steel structures. Various parts, BSI • BS EN 1993-1-8:2005. Eurocode 3: Design of steel structures. Design of joints, BSI • Washizu, K.Variationalmethods in ElasticityandPlasticity, PergamonPress, 1974. ISBN 978-0-08-026723-4. • Murnaghan, F. D. (1937): "Finite deformations of an elastic solid", enAmericanJournalofMathematics, 59, pp. 235-260. • V. V. Novozhilov (1953):FoundationsofNon-linearTheoryofElasticity, GraylockPress, Rochester • Bushnell, David. “Buckling of Shells—pitfall for designers” AIAA Journal, Vol. 19, No. 9, 1981. • http://www-classes.usc.edu/architecture/structures/Arch213A/213A-lectures-print/19-Buckling-print.pdf • Teng, Jin Guang. “Buckling of thin shells: Recent advances and trends”