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

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


Lattice structure


Hollow tube with guy wire

Hollow tapered tube



Tube with guy wire and winch

Tripod support

Structural support



World Trade Center in Bahrain

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


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


Support structure failure, Denmark. Caused by high wind


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

Lightning strike, Germany

Support structure failure, UK


Many different forms

Engineering 10, SJSU 10

Cardboard

Balsa wood

PVC Pipe


Foam Board

Recycled Materials


Metal Rods

Old Toys


Spring Stiffness

FF

Δx

where k = spring constantΔ x = spring stretchF = applied force

F = k (Δx)

Compression spring

Tension spring


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:deflection

loadk

The “Stiffness”


Stiffness (Solid Bar)• Stiffness in tension and compression

– Applied Forces F, length L, cross-sectional area, A, and material property, E (Young’s modulus)

AE

FL

F

k

L

AEk

Stiffness for components in tension-compression

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

FF

LEnd view

A

FF

L δ


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


Stiffness of a Cantilever Beam

Y = deflection = FL3 / 3EI

F = forceL = length

Deflection of a Cantilever Beam

Fixed end

Support

Fix

ed

end

Wind


Concept of Area Moment of Inertia


F = forceL = length


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;

Fix

ed

end

Wind

Clicker Question


kg is a unit of force

A)TrueB)False

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

F

F

K1

K2

K3

A. K1

B. K2

C. K3

D. They are all the sameE. I don’t know


Note: Intercept = 0

Default is:• first column plots

on x axis • second column

plots on y axis


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.


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

X X

Area, A

A1

A2

y1

y2


Moment of Inertia – Comparison

Load

2 x 8 beam

Maximum distance of 1 inch to the centroid

I1

I2 > I1 , orientation 2 deflects less

1

2” 1”

Maximum distance of 4 inch to the centroid I2

Same load and location

2

2 x 8 beam

4”


Moment of Inertia Equations for Selected Profiles

(d)4

64I =

Round solid section

Rectangular solid section

b

hbh31I =

12

b

h

1I =

12hb3

d Round hollow section

64

I = [(do)4 – (di)

4]

do

di

BH3 - 1

I =12

bh31

12

Rectangular hollow section

H

B

h

b


Show of Hands

• 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

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”


Example – Optimization for Weight & StiffnessConsider a solid rectangular section 2.0 inch wide by 1.0 high.

I = (1/12)bh3 = (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)bh3 = (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


Clicker Question

Deflection (inch)

Load (lbs)

C

B

A

The plot shows load versus deflection for three structures. Which is stiffest?

A. A

B. B

C. C

D. I don’t know


Stiffness Comparisons for Different sections

Square Box Rectangular Horizontal

Rectangular Vertical

Stiffness = slope


Material and Stiffness

E = Elasticity Module, a measure of material deformation under a load.


F = forceL = length

The higher the value of E, the less a structure deflects (higher stiffness)


Fixed end

Support

Ken Youssefi Engineering 10 - SJSU 31

Material StrengthStandard Tensile Test

Standard Specimen

Ductile Steel (low carbon)

Sy – yield strength

Su – 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 (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)


Modules of Elasticity (E) of Materials

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


Density of Materials

Plastic is 7 times lighter than steel and 3 times lighter

than aluminum.

Plast

ics

Alum

inum

sSte

el

Impact of Structural Elements on Overall Stiffness

Ken Youssefi / Thalia Anagnos Engineering 10, SJSU 35

Rectangle deforms

Triangle rigid

P

P

Clicker Question


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

A. True

B. False

Clicker Question


Which of the following materials is the stiffest?

A. Cast IronB. AluminumC. PolycarbonateD. SteelE. Fiberglass

Clicker Question


The applied load affects the stiffness of a structure.

A. TrueB. False


Stiffness Testing


weights

Load pulling on tower

Dial gage to measure deflection

Successful testers

Stiffness Testing Apparatus

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

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