### Stiffness

```Structures and Stiffness
ENGR 10
Introduction to Engineering
Ken Youssefi/Thalia Anagnos
Engineering 10, SJSU
1
Wind Turbine Structure
The Goal
The support structure should be optimized for
weight and stiffness (deflection)
Support
Structure
Ken Youssefi
Engineering 10, SJSU
2
Wind Turbine Structure
Hollow tapered tube
Lattice structure
Hollow tube with guy wire
Ken Youssefi
Engineering 10, SJSU
3
Wind Turbine Structure
Structural support
Tripod support
Tube with guy
wire and winch
Ken Youssefi
Engineering 10, SJSU
4
Wind Turbine Structure
Three giant wind
turbine provides
15% of the
power needed.
in Bahrain
Ken Youssefi
Engineering 10, SJSU
5
Support structure failure,
New York. Stress at the
base of the support
tower exceeding the
strength of the material
Ken Youssefi
Engineering 10, SJSU
6
Support structure failure,
Denmark. Caused by
high wind
Ken Youssefi
Engineering 10, SJSU
7
Failure at the thin
Support structure
failure, UK
Lightning strike,
Germany
Ken Youssefi
Engineering 10, SJSU
8
Many different forms
Ken Youssefi
Engineering 10, SJSU
9
Balsa wood
PVC Pipe
Cardboard
Engineering 10, SJSU
10
Recycled
Materials
Foam Board
Ken Youssefi
Engineering 10, SJSU
11
Metal Rods
Old Toys
Engineering 10, SJSU
12
Spring Stiffness
Δx
Compression
spring
Tension
spring
F
F
F = k (Δx)
where
k = spring constant
Δ x = spring stretch
F = applied force
Ken Youssefi
Engineering 10, SJSU
13
Stiffness (Spring)
• Deflection is proportional to load, F = k (∆x)
slope, k
Deflection (mm or in.)
k 
deflection
Ken Youssefi
The “Stiffness”
Engineering 10, SJSU
14
Stiffness (Solid Bar)
• Stiffness in tension and compression
– Applied Forces F, length L, cross-sectional area, A,
and material property, E (Young’s modulus)
F
F
End view
L
F
F
δ
L
k 
A
E is constant for a given material
F

k 
E (steel) = 30 x 106 psi
AE
E (Al) = 10 x 106 psi
L
 
FL
AE
Ken Youssefi
E (concrete) = 3.4 x 103 psi
Stiffness for components
in tension-compression
Engineering 10, SJSU
E (Kevlar, plastic) = 19 x 103 psi
E (rubber) = 100 psi
15
Stiffness
• Stiffness in bending
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
Ken Youssefi
Engineering 10, SJSU
16
Stiffness of a Cantilever Beam
Wind
Deflection of a Cantilever Beam
L = length
Fixed end
Support
F = force
Y = deflection = FL3 / 3EI
Fixed end
Ken Youssefi
Engineering 10, SJSU
17
Concept of Area Moment of Inertia
Wind
Deflection of a Cantilever Beam
L = length
Fixed end
Support
F = force
Y = deflection = FL3 / 3EI
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)
Ken Youssefi
Engineering 10, SJSU
18
Clicker Question
kg is a unit of force
A) True
B) False
Ken Youssefi
Engineering 10, SJSU
19
Clicker Question
All 3 springs have the same
initial length. Three springs
same force F. Which spring
has the greatest stiffness?
A.
B.
C.
D.
E.
Ken Youssefi
K1
K2
K3
They are all the same
I don’t know
Engineering 10, SJSU
K2
K3
K1
F
F
F
20
Note:
Intercept = 0
Default is:
• first column plots on
x axis
• second column plots
on y axis
Ken Youssefi
Engineering 10, SJSU
21
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
22
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
A2
A1
y1
y2
X
Ken Youssefi
Area, A
X
Engineering 10, SJSU
23
Moment of Inertia – Comparison
1
4”
Maximum distance of
4 inch to the centroid
I2
2 x 8 beam
and location
2
I1
2”
1”
2 x 8 beam
Maximum distance of 1 inch
to the centroid
I2 > I1 , orientation 2 deflects less
Ken Youssefi
Engineering 10, SJSU
24
Moment of Inertia Equations for Selected Profiles
d
Round solid section
I=
do
Round hollow section
 (d)4
di
64
I=
Rectangular solid section
 [(d )4 – (d )4]
o
i
64
Rectangular hollow section
b
I=
1
bh3
12
h
h
b
I=
B
I=
1
hb3
12
H
b
1
1
BH3 bh3
12
12
h
Ken Youssefi
Engineering 10, SJSU
25
Show of Hands
1.0 inch
•
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
hollow rectangular section 2.25”
wide X 1.25” high X .125” thick
b
H
h
B
B = 2.25”, H = 1.25”
b = 2.0”, h = 1.0”
Ken Youssefi
Engineering 10, SJSU
26
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 = .1667 , Area = 2
2.0
Now, consider a hollow rectangular section 2.25 inch wide by 1.25 high
by .125 thick.
b
B = 2.25, H = 1.25
h
b = 2.0, h = 1.0
H
B
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
(.1995 - .1667)/(.1667) x 100= .20 = 20% less deflection
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
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
27
Clicker Question
versus deflection for
three structures.
Which is stiffest?
C
B
A. A
B. B
C. C
A
D. I don’t know
Deflection
(inch)
Engineering 10, SJSU
28
Stiffness Comparisons for Different sections
Stiffness = slope
Square
Ken Youssefi
Box
Rectangular
Horizontal
Engineering 10, SJSU
Rectangular
Vertical
29
Material and Stiffness
E = Elasticity Module, a measure of material deformation under a load.
Deflection of a Cantilever Beam
Support
L = length
F = force
Y = deflection = FL3 / 3EI
Fixed end
The higher the value of E, the less a structure
deflects (higher stiffness)
Ken Youssefi
Engineering 10, SJSU
30
Material Strength
Standard Tensile Test
Ductile Steel (low carbon)
Standard Specimen
Sy – yield strength
Su – fracture strength
σ (stress) = Load / Area
ε (strain) = (change in length) / (original length)
Ken Youssefi
Engineering 10 - SJSU
31
Common Mechanical Properties
•
Yield Strength (Sy) – the
highest stress a material
can withstand and still
return exactly to its original
•
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)
Ken Youssefi
Engineering 10 - SJSU
32
Modules of Elasticity (E) of Materials
Steel is 3 times
stiffer than
Aluminum and
100 times stiffer
than Plastics.
Ken Youssefi
Engineering 10, SJSU
33
Density of Materials
Plastic is 7 times
lighter than steel
and 3 times lighter
than aluminum.
Ken Youssefi
Engineering 10, SJSU
34
Impact of Structural Elements on
Overall Stiffness
P
Rectangle deforms
P
Triangle rigid
Ken Youssefi / Thalia Anagnos
Engineering 10, SJSU
35
Clicker Question
The higher the Modulus of Elasticity (E),
the lower the stiffness
A. True
B. False
Ken Youssefi
Engineering 10, SJSU
36
Clicker Question
Which of the following materials is
the stiffest?
A.
B.
C.
D.
E.
Ken Youssefi
Cast Iron
Aluminum
Polycarbonate
Steel
Fiberglass
Engineering 10, SJSU
37
Clicker Question
The applied load affects the stiffness
of a structure.
A. True
B. False
Ken Youssefi
Engineering 10, SJSU
38
Stiffness Testing
Ken Youssefi
Engineering 10, SJSU
39
Stiffness Testing Apparatus
pulling on
tower
Dial gage
to measure
deflection
weights
Successful
testers
Ken Youssefi
Engineering 10, SJSU
40
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