Ken Youssefi/Thalia Anagnos Engineering 10, SJSU 1 Structures and Stiffness - PowerPoint Presentation

Ken Youssefi/Thalia Anagnos Engineering 10, SJSU 1 Structures and Stiffness ENGR 10 Introduction to Engineering

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

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

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

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

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

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

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

Ken Youssefi Engineering 10, SJSU 9 Many different forms

Engineering 10, SJSU 10 Cardboard Balsa wood PVC Pipe

Ken Youssefi Engineering 10, SJSU 11 Foam Board Recycled Materials

Engineering 10, SJSU 12 Metal Rods Old Toys

Ken Youssefi Engineering 10, SJSU 13 Spring Stiffness F F Δ x where k = spring constant Δ x = spring stretch F = applied force F = k ( Δ x ) Compression spring Tension spring

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

Ken Youssefi Engineering 10, SJSU 15 Stiffness ( Solid Bar ) Stiffness in tension and compression Applied Forces F , length L, cross-sectional area, A, and material property, E (Young’s modulus) Stiffness for components in tension-compression E is constant for a given material E (steel) = 30 x 10 6 psi E (Al) = 10 x 10 6 psi E (concrete) = 3.4 x 10 3 psi E (Kevlar, plastic) = 19 x 10 3 psi E (rubber) = 100 psi F F L End view A F F L δ

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

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

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

Clicker Question Ken Youssefi Engineering 10, SJSU 19 kg is a unit of force True False

Clicker Question Ken Youssefi Engineering 10, SJSU 20 All 3 springs have the same initial length. Three springs are each loaded with the same force F . Which spring has the greatest stiffness? F F F K 1 K 2 K 3 K 1 K 2 K 3 They are all the same I don’t know

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

Ken Youssefi Engineering 10, SJSU 22 Concept of Area Moment of Inertia 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 the X or 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. 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.

Ken Youssefi Engineering 10, SJSU 23 Mathematical Equation for Area Moment of Inertia I xx = ∑ (A i ) (y i ) 2 = A 1(y1)2 + A2(y2 )2 + …..An(y n ) 2 A (total area) = A 1 + A 2 + ……..A n X X Area, A A 1 A 2 y 1 y 2

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

Ken Youssefi Engineering 10, SJSU 25 Moment of Inertia Equations for Selected Profiles  (d) 4 64 I = Round solid section Rectangular solid section b h bh 3 1 I = 12 b h 1 I = 12 hb 3 d Round hollow section  64 I = [( d o ) 4 – ( d i ) 4 ] d o d i BH 3 - 1 I = 12 bh 3 1 12 Rectangular hollow section H B h b

Ken Youssefi Engineering 10, SJSU 26 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 2.0 inch 1.0 inch hollow rectangular section 2.25” wide X 1.25” high X .125” thick H B h b B = 2.25”, H = 1.25” b = 2.0”, h = 1.0”

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

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

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

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

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

Ken Youssefi Engineering 10 - SJSU 32 - 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 Ductility Common Mechanical Properties - the capacity of a material to absorb energy within the elastic zone (area under the stress-strain curve in the elastic zone) Resilience - the total capacity of a material to absorb energy without fracture (total area under the stress-strain curve) Toughness – the highest stress a material can withstand and still return exactly to its original size when unloaded. Yield Strength (S y ) - the greatest stress a material can withstand, fracture stress. Ultimate Strength (S u ) - the slope of the straight portion of the stress-strain curve. Modulus of elasticity (E)

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

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

Impact of Structural Elements on Overall Stiffness Ken Youssefi / Thalia Anagnos Engineering 10, SJSU 35 Rectangle deforms Triangle rigid P P

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

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

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

Ken Youssefi Engineering 10, SJSU 39 Stiffness Testing

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