Structures and Stiffness - engineering.sjsu.edu · buckling, and the amount of deflection in a...
Transcript of Structures and Stiffness - engineering.sjsu.edu · buckling, and the amount of deflection in a...
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Ken Youssefi/Thalia Anagnos Engineering 10, SJSU 1
Structures and Stiffness
ENGR 10
Introduction to Engineering
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Ken Youssefi Engineering 10, SJSU 2
Wind Turbine Structure
The support structure should be optimized for
weight and stiffness (deflection)
Support
Structure
The Goal
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Ken Youssefi Engineering 10, SJSU 3
Lattice structure
Wind Turbine Structure
Hollow tube with guy wire
Hollow tapered tube
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Ken Youssefi Engineering 10, SJSU 4
Wind Turbine Structure
Tube with guy
wire and winch
Tripod support
Structural support
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Ken Youssefi Engineering 10, SJSU 5
Wind Turbine Structure
World Trade Center
in Bahrain
Three giant wind
turbine provides
15% of the
power needed.
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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
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Ken Youssefi Engineering 10, SJSU 7
Support structure failure,
Denmark. Caused by
high wind
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Ken Youssefi Engineering 10, SJSU 8
Blade failure, Illinois.
Failure at the thin
section of the blade
Lightning strike,
Germany
Support structure
failure, UK
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Ken Youssefi Engineering 10, SJSU 9
Many different forms
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Engineering 10, SJSU 10
Cardboard
Balsa wood
PVC Pipe
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Ken Youssefi Engineering 10, SJSU 11
Foam Board
Recycled
Materials
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Engineering 10, SJSU 12
Metal Rods
Old Toys
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Ken Youssefi Engineering 10, SJSU 13
Spring Stiffness
F F
Δx
where
k = spring constant
Δ x = spring stretch
F = applied force
F = k (Δx)
Compression
spring Tension
spring
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Ken Youssefi Engineering 10, SJSU 14
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”
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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
Fk
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
F F
L End view
A
F F
L δ
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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
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Ken Youssefi Engineering 10, SJSU 17
Stiffness of a Cantilever Beam
Y = deflection = FL3 / 3EI
F = force L = length
Deflection of a Cantilever Beam
Fixed end
Support
Fix
ed
en
d
Wind
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Ken Youssefi Engineering 10, SJSU 18
Concept of Area Moment of Inertia
Y = deflection = FL3 / 3EI
F = force L = 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
en
d
Wind
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Clicker Question
Ken Youssefi Engineering 10, SJSU 19
kg is a unit of force
A)True
B)False
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Clicker Question
Ken Youssefi Engineering 10, SJSU 20
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 same
E. I don’t know
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Ken Youssefi Engineering 10, SJSU 21
Note:
Intercept = 0
Default is:
• first column plots on
x axis
• second column plots
on y axis
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Ken Youssefi Engineering 10, SJSU 22
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.
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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
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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”
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Ken Youssefi Engineering 10, SJSU 25
Moment of Inertia Equations for Selected Profiles
(d)4
64 I =
Round solid section
Rectangular solid section
b
h bh3 1 I =
12
b
h
1 I =
12 hb3
d Round hollow section
64 I = [(do)
4 – (di)4]
do
di
BH3 - 1
I =
12 bh3 1
12
Rectangular hollow section
H
B
h
b
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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”
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Ken Youssefi Engineering 10, SJSU 27
Example – Optimization for Weight & Stiffness
Consider 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
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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
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Ken Youssefi Engineering 10, SJSU 29
Stiffness Comparisons for Different sections
Square Box Rectangular
Horizontal Rectangular
Vertical
Stiffness = slope
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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 = force L = length
The higher the value of E, the less a structure
deflects (higher stiffness)
Deflection of a Cantilever Beam
Fixed end
Support
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Ken Youssefi Engineering 10 - SJSU 31
Material Strength Standard 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)
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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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Ken Youssefi Engineering 10, SJSU 33
Modules of Elasticity (E) of Materials
Steel is 3 times
stiffer than
Aluminum and
100 times stiffer
than Plastics.
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Ken Youssefi Engineering 10, SJSU 34
Density of Materials
Plastic is 7 times
lighter than steel
and 3 times lighter
than aluminum.
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Impact of Structural Elements on
Overall Stiffness
Ken Youssefi / Thalia Anagnos Engineering 10, SJSU 35
Rectangle deforms
Triangle rigid
P
P
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Clicker Question
Ken Youssefi Engineering 10, SJSU 36
The higher the Modulus of Elasticity (E),
the lower the stiffness
A. True
B. False
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Clicker Question
Ken Youssefi Engineering 10, SJSU 37
Which of the following materials is the stiffest?
A. Cast Iron B. Aluminum C. Polycarbonate D. Steel E. Fiberglass
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Clicker Question
Ken Youssefi Engineering 10, SJSU 38
The applied load affects the stiffness of a structure.
A. True B. False
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Ken Youssefi Engineering 10, SJSU 39
Stiffness Testing
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Ken Youssefi Engineering 10, SJSU 40
weights
Load
pulling on
tower
Dial gage
to measure
deflection
Successful
testers
Stiffness Testing Apparatus