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Structures and Stiffness

Structures and Stiffness. ENGR 10 Introduction to Engineering. Wind Turbine Structure. The Goal. The support structure should be optimized for weight and stiffness (deflection). Support Structure. Lattice structure. Hollow tube with guy wire. Wind Turbine Structure. Hollow tapered tube.

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Structures and Stiffness

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  1. Structures and Stiffness ENGR 10 Introduction to Engineering Engineering 10, SJSU

  2. Wind Turbine Structure The Goal The support structure should be optimized for weight and stiffness (deflection) Support Structure Engineering 10, SJSU

  3. Lattice structure Hollow tube with guy wire Wind Turbine Structure Hollow tapered tube Engineering 10, SJSU

  4. Structural support Tripod support Wind Turbine Structure Tube with guy wire and winch Engineering 10, SJSU

  5. Wind Turbine Structure Three giant wind turbine provides 15% of the power needed. World Trade Center in Bahrain Engineering 10, SJSU

  6. Support structure failure, New York. Stress at the base of the support tower exceeding the strength of the material Engineering 10, SJSU

  7. Support structure failure, Denmark. Caused by high wind Engineering 10, SJSU

  8. Blade failure, Illinois. Failure at the thin section of the blade Support structure failure, UK Lightning strike, Germany Engineering 10, SJSU

  9. Many different forms Engineering 10, SJSU

  10. Balsa wood PVC Pipe Cardboard Engineering 10, SJSU

  11. Recycled Materials Foam Board Engineering 10, SJSU

  12. Metal Rods Old Toys Engineering 10, SJSU

  13. Δx F F F = k (Δx) where k = spring constant Δx = spring stretch F = applied force Spring Stiffness Compression spring Tension spring Engineering 10, SJSU

  14. Deflection is proportional to load, F = k (∆x) Load (N or lb) slope, k Deflection (mm or in.) Slope of Load-Deflection curve: The “Stiffness” Stiffness (Spring) Engineering 10, SJSU

  15. F F L A F F End view δ L E is constant for a given material E (steel) =30 x 106 psi E (concrete) =3.4 x 103 psi E (Al) =10 x 106 psi E (Kevlar, plastic) =19 x 103 psi Stiffness for components in tension-compression E (rubber) =100 psi Stiffness (Solid Bar) • Stiffness in tension and compression • Applied Forces F, length L, cross-sectional area, A, and material property, E (Young’s modulus) Engineering 10, SJSU

  16. A B • How does the material resist the applied load? • Think about what happens to the material as the beam bends • Inner “fibers” (A) are in compression • Outer “fibers” (B) are in tension Stiffness • Stiffness in bending Engineering 10, SJSU

  17. Deflection of a Cantilever Beam Support F = force L = length Y = deflection = FL3 / 3EI Fixed end Fixed end Stiffness of a Cantilever Beam Wind Engineering 10, SJSU

  18. Deflection of a Cantilever Beam Support F = force L = length Y = deflection = FL3 / 3EI Fixed end Fixed end Mathematically, the area moment of inertia appears in the denominator of the deflection equation, therefore; The larger the area moment of inertia, the less a structure deflects (greater stiffness) Concept of Area Moment of Inertia Wind Engineering 10, SJSU

  19. Clicker Question kg is a unit of force • True • False Engineering 10, SJSU

  20. K3 K2 K1 F F F Clicker Question All 3 springs have the same initial length. Three springs are each loaded with the same force F. Which spring has the greatest stiffness? • K1 • K2 • K3 • They are all the same • I don’t know Engineering 10, SJSU

  21. Note: Intercept = 0 • Default is: • first column plots on x axis • second column plots on y axis Engineering 10, SJSU

  22. Concept of Area Moment of Inertia The Area Moment of Inertia is an important parameter in determine the state of stress in a part (component, structure), the resistance to buckling, and the amount of deflection in a beam. The area moment of inertia allows you to tell how stiff a structure is. The Area Moment of Inertia, I, is a term used to describe the capacity of a cross-section (profile) to resist bending. It is always considered with respect to a reference axis, in theXor Y direction. It is a mathematical property of a section concerned with a surface area and how that area is distributed about the reference axis. The reference axis is usually a centroidal axis. Engineering 10, SJSU

  23. Mathematical Equation for Area Moment of Inertia Ixx = ∑ (Ai) (yi)2 = A1(y1)2 + A2(y2)2 + …..An(yn)2 A (total area) = A1 + A2 + ……..An Area, A A2 A1 y2 y1 X X Engineering 10, SJSU

  24. Maximum distance of 4 inch to the centroid I2 Same load and location 2 2 x 8 beam Moment of Inertia – Comparison 1 Load 4” 2 x 8 beam I1 1” 2” Maximum distance of 1 inch to the centroid I2 > I1 , orientation 2 deflects less Engineering 10, SJSU

  25. Round hollow section do di  (d)4 [(do)4 – (di)4] I = I = 64 64 Rectangular solid section Rectangular hollow section 1 1 1 I = I = I = 12 12 12 b h bh3 1 h H bh3 BH3 - b 12 B b hb3 h Moment of Inertia Equations for Selected Profiles d Round solid section Engineering 10, SJSU

  26. 1.0 inch 2.0 inch b H h B Show of Hands • A designer is considering two cross sections as shown. Which will produce a stiffer structure? • Solid section • Hollow section • I don’t know hollow rectangular section 2.25” wide X 1.25” high X .125” thick B = 2.25”, H = 1.25” b = 2.0”, h = 1.0” Engineering 10, SJSU

  27. Now, consider a hollow rectangular section 2.25 inch wide by 1.25 high by .125 thick. B = 2.25, H = 1.25 b = 2.0, h = 1.0 I = (1/12)bh3 = (1/12)(2.25)(1.25)3 – (1/12)(2)(1)3= .3662 -.1667 = .1995 Area = 2.25x1.25 – 2x1 = .8125 b h H B Compare the weight of the two parts (same material and length), so only the cross sectional areas need to be compared. (2 - .8125)/(2) = .6 = 60% lighter Example – Optimization for Weight & Stiffness Consider a solid rectangular section 2.0 inch wide by 1.0 high. 1.0 2.0 I = (1/12)bh3 = (1/12)(2)(1)3 = .1667 , Area = 2 (.1995 - .1667)/(.1667) x 100= .20 = 20% less deflection So, for a slightly larger outside dimension section, 2.25x1.25 instead of 2 x 1, you can design a beam that is 20% stiffer and 60 % lighter Engineering 10, SJSU

  28. Load (lbs) C B A Deflection (inch) Clicker Question • The plot shows load versus deflection for three structures. Which is stiffest? • A • B • C • I don’t know Engineering 10, SJSU

  29. Stiffness Comparisons for Different sections Stiffness = slope Rectangular Horizontal Square Box Rectangular Vertical Engineering 10, SJSU

  30. Deflection of a Cantilever Beam Support F = force L = length Y = deflection = FL3 / 3EI Fixed end Material and Stiffness E = Elasticity Module, a measure of material deformation under a load. The higher the value of E, the less a structure deflects (higher stiffness) Engineering 10, SJSU

  31. Ductile Steel (low carbon) Sy – yield strength Su – fracture strength σ (stress) = Load / Area ε (strain) = (change in length) / (original length) Material Strength Standard Tensile Test Standard Specimen Engineering 10 - SJSU

  32. Yield Strength (Sy) • – the highest stress a material can withstand and still return exactly to its original size when unloaded. Ultimate Strength (Su) • - the greatest stress a material can withstand, fracture stress. Modulus of elasticity (E) • - the slope of the straight portion of the stress-strain curve. Ductility • - the extent of plastic deformation that a material undergoes before fracture, measured as a percent elongation of a material. % elongation = (final length, at fracture – original length) / original length Resilience • - the capacity of a material to absorb energy within the elastic zone (area under the stress-strain curve in the elastic zone) Toughness • - the total capacity of a material to absorb energy without fracture (total area under the stress-strain curve) Common Mechanical Properties Engineering 10 - SJSU

  33. Modules of Elasticity (E) of Materials Steel is 3 times stiffer than Aluminum and 100 times stiffer than Plastics. Engineering 10, SJSU

  34. Density of Materials Steel Plastic is 7 times lighter than steel and 3 times lighter than aluminum. Aluminums Plastics Engineering 10, SJSU

  35. Impact of Structural Elements on Overall Stiffness P Rectangle deforms P Triangle rigid Engineering 10, SJSU

  36. Clicker Question The higher the Modulus of Elasticity (E), the lower the stiffness • True • False Engineering 10, SJSU

  37. Clicker Question Which of the following materials is the stiffest? • Cast Iron • Aluminum • Polycarbonate • Steel • Fiberglass Engineering 10, SJSU

  38. Clicker Question The applied load affects the stiffness of a structure. • True • False Engineering 10, SJSU

  39. Stiffness Testing Engineering 10, SJSU

  40. Load pulling on tower Dial gage to measure deflection weights Stiffness Testing Apparatus Successful testers Engineering 10, SJSU

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