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
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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
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Foam Board
Recycled Materials
Engineering 10, SJSU 12
Metal Rods
Old Toys
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Spring Stiffness
FF
Δx
where k = spring constantΔ x = spring stretchF = applied force
F = k (Δx)
Compression spring
Tension spring
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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:deflection
loadk
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)
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
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 = FL3 / 3EI
F = forceL = length
Deflection of a Cantilever Beam
Fixed end
Support
Fix
ed
end
Wind
Ken Youssefi Engineering 10, SJSU 18
Concept of Area Moment of Inertia
Y = deflection = FL3 / 3EI
F = forceL = 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;
Fix
ed
end
Wind
Clicker Question
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kg is a unit of force
A)TrueB)False
Clicker Question
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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
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Note: Intercept = 0
Default is:• first column plots
on x axis • second column
plots on y axis
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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
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
Ken Youssefi Engineering 10, SJSU 24
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”
Ken Youssefi Engineering 10, SJSU 25
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
Ken Youssefi Engineering 10, SJSU 26
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”
Ken Youssefi Engineering 10, SJSU 27
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
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. A
B. B
C. C
D. 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 = FL3 / 3EI
F = forceL = 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 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)
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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
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
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The higher the Modulus of Elasticity (E), the lower the stiffness
A. True
B. False
Clicker Question
Ken Youssefi Engineering 10, SJSU 37
Which of the following materials is the stiffest?
A. Cast IronB. AluminumC. PolycarbonateD. SteelE. Fiberglass
Clicker Question
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The applied load affects the stiffness of a structure.
A. TrueB. False
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Stiffness Testing
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weights
Load pulling on tower
Dial gage to measure deflection
Successful testers
Stiffness Testing Apparatus
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