Upload
febri-al-farouk
View
236
Download
0
Embed Size (px)
Citation preview
8/2/2019 Structures Stiffness
1/27
K. Youssefi and B. Furman Engineering 10, SJSU 1
Structures and Stiffness
B. Furman
K. Youssefi20SEP2007
8/2/2019 Structures Stiffness
2/27
K. Youssefi and B. Furman Engineering 10, SJSU 2
Outline Newtons 3rd Law
Hookes Law
Stiffness
Area moment of Inertia Orientation of cross section and stiffness
Comparison of cross sections Materials and stiffness
8/2/2019 Structures Stiffness
3/27
K. Youssefi and B. Furman Engineering 10, SJSU 3
Newtons 3rd
Law Lex III: Actioni contrariam semper et
qualem esse reactionem: sive corporumduorum actiones in se mutuo semper essequales et in partes contrarias dirigi.
To every action there is always opposed an
equal reaction: or the mutual actions of two
bodies upon each other are always equal,and directed to contrary parts.
8/2/2019 Structures Stiffness
4/27
K. Youssefi and B. Furman Engineering 10, SJSU 4
Newtons 3rd
Law - example
M M
Free body diagram T, tension
T, tension
M*g
Isolate the body ofinterest
Put back the forcesthat are acting
8/2/2019 Structures Stiffness
5/27
K. Youssefi and B. Furman Engineering 10, SJSU 5
Hookes Law Robert Hooke (1635-1702)
Materials resist loads (push or pull back) inresponse to applied loads
This resistance is accomplished by deformationof
the material (changing its shape) Tension (stretching)
Compression (shortening)
Stretching or shortening of chemical bonds in atoms The science of Elasticity concerns forces and
deformations in materials
8/2/2019 Structures Stiffness
6/27
K. Youssefi and B. Furman Engineering 10, SJSU 6
Hookes Law, cont. Hooke found that deflection
was proportional to load
Load, N
Deflection, mm
slope, kSlope of Load-Deflection curve:
deflection
load
k=
The Stiffness
8/2/2019 Structures Stiffness
7/27
K. Youssefi and B. Furman Engineering 10, SJSU 7
Stiffness Stiffness in tension and compression
Forces F applied, length L, cross-sectional area, A,and material property, E (Youngs modulus)
AE
FL=
F
Fk=
E
FLF=
L
AEk=
F
L
A
8/2/2019 Structures Stiffness
8/27
K. Youssefi and B. Furman Engineering 10, SJSU 8
Stiffness, cont. Stiffness in bending
How does the material resist the applied load?
Think about what happens to the material as the beambends
Inner fibers (A) are in compression (radius of curvature, Ri)
Outer fibers (B) are in tension (radius of curvature, Ro)
A
Ri
B
Ro
F
8/2/2019 Structures Stiffness
9/27
K. Youssefi and B. Furman Engineering 10, SJSU 9
Review Question 1 Stiffness is defined as:
A. Force/Area
B. Deflection/Force
C. Force/DeflectionD. Force x Deflection
E. Mass/area
8/2/2019 Structures Stiffness
10/27
K. Youssefi and B. Furman Engineering 10, SJSU 10
Concept of Area Moment of Inertia
The Area Moment of Inertia, I, is a term used to describe thecapacity of a cross-section (profile) to resist bending. It is always
considered with respect to a reference axis, in the X or Ydirection. It is a mathematical property of a section concerned withan area and how that area is distributed about the reference axis.The reference axis is usually a centroidal axis.
The Area Moment of Inertiais an important parameter in determinethe state of stress in a part (component, structure), the resistance tobuckling, and the amount of deflection in a beam.
The area moment of inertia allows you to tell how stiffa structure is.
The higher the area moment of inertia, the less a
structure deflects (higher stiffness)
Mathematically, the area moment of inertia appears in the denominatorof the deflection equation, therefore;
8/2/2019 Structures Stiffness
11/27
K. Youssefi and B. Furman Engineering 10, SJSU 11
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
y1y2
8/2/2019 Structures Stiffness
12/27
K. Youssefi and B. Furman Engineering 10, SJSU 12
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
Load
Maximum distance of4 inch to the centroid I2
2
2 x 8 beam
8/2/2019 Structures Stiffness
13/27
K. Youssefi and B. Furman Engineering 10, SJSU 13
Moment of Inertia Equations for Selected Profiles
(d)4
64I =
Round solid section
Rectangular solid section
b
hbh31
I =12
b
h
1I =
12hb3
d Round hollow section
64I = [(do)
4 (di)4]
do
di
BH3 -1I =12
bh3112
Rectangular hollow section
H
B
h
b
8/2/2019 Structures Stiffness
14/27
K. Youssefi and B. Furman Engineering 10, SJSU 14
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)/(.1167) = .20 = 20% less deflection
(2 - .8125)/(2) = .6 = 60% lighter
Compare the weight of the two parts (same material and length),
compare areas. Material and length is the same for both profiles.
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 highby .125 thick.
H
B
h
b
B = 2.25, H = 1.25
b = 2.0, h = 1.0
8/2/2019 Structures Stiffness
15/27
K. Youssefi and B. Furman Engineering 10, SJSU 15
Review Question 2 Which cross section has the larger I?
A.
B.
RectangularHorizontal
RectangularVertical
8/2/2019 Structures Stiffness
16/27
K. Youssefi and B. Furman Engineering 10, SJSU 16
Stiffness Comparisons for Different sections
Square Box Rectangular
Horizontal
Rectangular
Vertical
8/2/2019 Structures Stiffness
17/27
K. Youssefi and B. Furman Engineering 10, SJSU 17
Material and Stiffness
E = Elasticity Modulus, a measure of material deformation under a load.
Y= deflection = FL3/ 3EI
F = forceL = length
The higher the value of E, the less a structuredeflects (higher stiffness)
Deflection of a Cantilever Beam
Fixed end
Support
8/2/2019 Structures Stiffness
18/27
K. Youssefi and B. Furman Engineering 10, SJSU 18
Modulus of Elasticity (E) of Materials
Steel is 3 times
stiffer than
Aluminum and
100 times stiffer
than Plastics.
8/2/2019 Structures Stiffness
19/27
K. Youssefi and B. Furman Engineering 10, SJSU 19
Density of Materials
Plastic is 7 times
lighter than steel
and 3 times lighter
than aluminum.
8/2/2019 Structures Stiffness
20/27
K. Youssefi and B. Furman Engineering 10, SJSU 20
Wind Turbine Structure
The support structure should be optimized for weight and stiffness.
SupportStructure
8/2/2019 Structures Stiffness
21/27
K. Youssefi and B. Furman Engineering 10, SJSU 21
Review Question 3 Which material has the higher stiffness?
A. Steel
B. Aluminum
C. Alumina ceramic
D. Nylon
E. Unobtanium
8/2/2019 Structures Stiffness
22/27
K. Youssefi and B. Furman Engineering 10, SJSU 22
Examples of Achieving Structural
Stiffness
Ribbing
Gusset
Boss
8/2/2019 Structures Stiffness
23/27
K. Youssefi and B. Furman Engineering 10, SJSU 23
Examples of Achieving Structural
Stiffness, cont.
http://en.wikipedia.org/wiki/Image:FT_Rail.jpg
8/2/2019 Structures Stiffness
24/27
K. Youssefi and B. Furman Engineering 10, SJSU 24
Examples of Achieving Structural
Stiffness, cont.
Welded box
construction
8/2/2019 Structures Stiffness
25/27
K. Youssefi and B. Furman Engineering 10, SJSU 25
Examples of Achieving Structural
Stiffness, cont.
8/2/2019 Structures Stiffness
26/27
K. Youssefi and B. Furman Engineering 10, SJSU 26
Examples of Achieving Structural
Stiffness, cont.
Flange
8/2/2019 Structures Stiffness
27/27
K. Youssefi and B. Furman Engineering 10, SJSU 27
Examples of Achieving Structural
Stiffness, cont.