# Structures and Stiffness

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

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

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

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

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

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 6

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

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

9 Many different forms Ken Youssefi Engineering 10, SJSU 9

10 Balsa wood PVC Pipe Cardboard Engineering 10, SJSU 10

11 Foam Board Recycled Materials Ken Youssefi Engineering 10, SJSU 11

12 Metal Rods Old Toys Engineering 10, SJSU 12

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

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

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) F k F F FL AE L L k δ F AE L A End view F 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 Ken Youssefi Engineering 10, SJSU 15

16 Stiffness in bending Stiffness 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 Ken Youssefi Engineering 10, SJSU 16 A B

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

18 Fixed end Concept of Area Moment of Inertia Deflection of a Cantilever Beam Wind Support L = length F = force Y = deflection = FL 3 / 3EI 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) Ken Youssefi Engineering 10, SJSU 18

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

20 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? K 1 K 2 K 3 A. K 1 B. K 2 C. K 3 D. They are all the same E. I don t know F F F Ken Youssefi Engineering 10, SJSU 20

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

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 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. Ken Youssefi Engineering 10, SJSU 22

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

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

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

26 Show of Hands 1.0 inch A designer is considering two cross sections as shown. Which will produce a stiffer structure? A. Solid section B. Hollow section C. I don t know 2.0 inch hollow rectangular section 2.25 wide X 1.25 high X.125 thick h b B H B = 2.25, H = 1.25 b = 2.0, h = 1.0 Ken Youssefi Engineering 10, SJSU 26

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

28 Clicker Question Load (lbs) The plot shows load versus deflection for three structures. Which is stiffest? A. A B. B C. C D. I don t know C B A Deflection (inch) Engineering 10, SJSU 28

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

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

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

32 Ductility Common Mechanical Properties Yield Strength (S y ) the highest stress a material can withstand and still return exactly to its original size when unloaded. Ultimate Strength (S u ) - the greatest stress a material can withstand, fracture stress. Modulus of elasticity (E) - the slope of the straight portion of the stress-strain curve. - 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) Ken Youssefi Engineering 10 - SJSU 32

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 33

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

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

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

37 Clicker Question Which of the following materials is the stiffest? A. Cast Iron B. Aluminum C. Polycarbonate D. Steel E. Fiberglass Ken Youssefi Engineering 10, SJSU 37

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

39 Stiffness Testing Ken Youssefi Engineering 10, SJSU 39

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

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