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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

Structures and Stiffness

ENGR 10

Introduction to Engineering

Engineering 10, SJSU

slide2

Wind Turbine Structure

The Goal

The support structure should be optimized for weight and stiffness (deflection)

Support Structure

Engineering 10, SJSU

wind turbine structure

Lattice structure

Hollow tube with guy wire

Wind Turbine Structure

Hollow tapered tube

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wind turbine structure1

Structural support

Tripod support

Wind Turbine Structure

Tube with guy wire and winch

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wind turbine structure2
Wind Turbine Structure

Three giant wind turbine provides 15% of the power needed.

World Trade Center in Bahrain

Engineering 10, SJSU

slide6

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

Engineering 10, SJSU

slide8

Blade failure, Illinois. Failure at the thin section of the blade

Support structure failure, UK

Lightning strike, Germany

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slide9

Many different forms

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slide10

Balsa wood

PVC Pipe

Cardboard

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slide11

Recycled Materials

Foam Board

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slide12

Metal Rods

Old Toys

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spring stiffness

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

stiffness spring
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

stiffness solid bar

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

stiffness

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

stiffness of a cantilever beam

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

concept of area moment of inertia

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

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clicker question
Clicker Question

kg is a unit of force

  • True
  • False

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clicker question1

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

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slide21

Note: Intercept = 0

  • Default is:
  • first column plots on x axis
  • second column plots on y axis

Engineering 10, SJSU

concept of area moment of inertia1
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

mathematical equation for area moment of inertia
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

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moment of inertia comparison

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

moment of inertia equations for selected profiles

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

show of hands

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

example optimization for weight stiffness

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

clicker question2

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

stiffness comparisons for different sections
Stiffness Comparisons for Different sections

Stiffness = slope

Rectangular Horizontal

Square

Box

Rectangular Vertical

Engineering 10, SJSU

material and stiffness

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

material strength

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

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common mechanical properties

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

modules of elasticity e of materials
Modules of Elasticity (E) of Materials

Steel is 3 times stiffer than Aluminum and 100 times stiffer than Plastics.

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density of materials
Density of Materials

Steel

Plastic is 7 times lighter than steel and 3 times lighter than aluminum.

Aluminums

Plastics

Engineering 10, SJSU

impact of structural elements on overall stiffness
Impact of Structural Elements on Overall Stiffness

P

Rectangle deforms

P

Triangle rigid

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clicker question3
Clicker Question

The higher the Modulus of Elasticity (E), the lower the stiffness

  • True
  • False

Engineering 10, SJSU

clicker question4
Clicker Question

Which of the following materials is the stiffest?

  • Cast Iron
  • Aluminum
  • Polycarbonate
  • Steel
  • Fiberglass

Engineering 10, SJSU

clicker question5
Clicker Question

The applied load affects the stiffness of a structure.

  • True
  • False

Engineering 10, SJSU

slide39

Stiffness Testing

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slide40

Load pulling on tower

Dial gage to measure deflection

weights

Stiffness Testing Apparatus

Successful testers

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