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**Structures and Stiffness**

ENGR 10 Introduction to Engineering Ken Youssefi/Thalia Anagnos Engineering 10, SJSU

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**Wind Turbine Structure**

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

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**Wind Turbine Structure**

Hollow tapered tube Lattice structure Hollow tube with guy wire Ken Youssefi Engineering 10, SJSU

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**Wind Turbine Structure**

Tripod support Structural support Tube with guy wire and winch Ken Youssefi Engineering 10, SJSU

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**Wind Turbine Structure**

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

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**Support structure failure, New York**

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

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**Support structure failure, Denmark. Caused by high wind**

Ken Youssefi Engineering 10, SJSU

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**Support structure failure, UK **

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

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Many different forms Ken Youssefi Engineering 10, SJSU

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**Balsa wood PVC Pipe Cardboard Lots of different materials are possible**

Engineering 10, SJSU

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Recycled Materials Foam Board Ken Youssefi Engineering 10, SJSU

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Metal Rods Old Toys Engineering 10, SJSU

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**Spring Stiffness F = k (Δx) where k = spring constant**

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

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**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

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**Stiffness (Solid Bar) Stiffness in tension and compression F L F**

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

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**Stiffness Stiffness in bending**

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

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**Stiffness of a Cantilever Beam**

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

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**Concept of Area Moment of Inertia**

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

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**Clicker Question kg is a unit of force True False Engineering 10, SJSU**

Ken Youssefi Engineering 10, SJSU

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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? F K1 K2 K3 K1 K2 K3 They are all the same I don’t know Ken Youssefi Engineering 10, SJSU

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**first column plots on x axis second column plots on y axis**

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

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**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

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**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 Ken Youssefi Engineering 10, SJSU

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**Moment of Inertia – Comparison**

1 Maximum distance of 4 inch to the centroid I2 Same load and location 2 2 x 8 beam 4” Load 2 x 8 beam I1 2” 1” Maximum distance of 1 inch to the centroid I2 > I1 , orientation 2 deflects less Ken Youssefi Engineering 10, SJSU

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**Moment of Inertia Equations for Selected Profiles**

Round hollow section 64 I = [(do)4 – (di)4] do di Round solid section d (d)4 64 I = Rectangular solid section b h bh3 1 I = 12 BH3 - 1 I = 12 bh3 Rectangular hollow section H B h b b h 1 I = 12 hb3 Ken Youssefi Engineering 10, SJSU

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**Show of Hands Solid section Hollow section I don’t know**

2.0 inch 1.0 inch 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 H B h b B = 2.25”, H = 1.25” b = 2.0”, h = 1.0” Ken Youssefi Engineering 10, SJSU

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**Example – Optimization for Weight & Stiffness**

Consider a solid rectangular section 2.0 inch wide by 1.0 high. 1.0 I = (1/12)bh3 = (1/12)(2)(1)3 = , Area = 2 2.0 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 I = (1/12)bh3 = (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 ( )/(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. 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

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**Clicker Question Load (lbs) C**

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 Engineering 10, SJSU

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**Stiffness Comparisons for Different sections**

Stiffness = slope Square Box Rectangular Horizontal Rectangular Vertical Ken Youssefi Engineering 10, SJSU

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**Material and Stiffness**

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

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**Material Strength Standard Tensile Test σ (stress) = Load / Area**

Ductile Steel (low carbon) Sy – yield strength Su – fracture strength Standard Specimen σ (stress) = Load / Area ε (strain) = (change in length) / (original length) Ken Youssefi Engineering 10 - SJSU

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**Common Mechanical Properties**

– the highest stress a material can withstand and still return exactly to its original size when unloaded. Yield Strength (Sy) - the greatest stress a material can withstand, fracture stress. Ultimate Strength (Su) - the slope of the straight portion of the stress-strain curve. Modulus of elasticity (E) - 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 - 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 Ken Youssefi Engineering 10 - SJSU

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**Modules of Elasticity (E) of Materials**

Steel is 3 times stiffer than Aluminum and 100 times stiffer than Plastics. Ken Youssefi Engineering 10, SJSU

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

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**Impact of Structural Elements on Overall Stiffness**

Rectangle deforms P Triangle rigid Ken Youssefi / Thalia Anagnos Engineering 10, SJSU

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

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**Clicker Question Cast Iron Aluminum Polycarbonate Steel Fiberglass**

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

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**Clicker Question True False**

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

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Stiffness Testing Ken Youssefi Engineering 10, SJSU

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**Stiffness Testing Apparatus**

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

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