Structures Stiffness

K. Youssefi and B. Furman Engineering 10, SJSU 1

Structures and Stiffness

B. FurmanK. Youssefi20SEP2007


Outline

• Newton’s 3rd Law• Hooke’s Law• Stiffness• Area moment of Inertia• Orientation of cross section and stiffness• Comparison of cross sections• Materials and stiffness


Newton’s 3rd Law

• Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporumduorum actiones in se mutuo semper esseæquales 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.


Newton’s 3rd Law - example

M M

Free body diagram T, tension

T, tension

M*g

•Isolate the body of interest

•Put back the forces that are acting


Hooke’s Law

• Robert Hooke (1635-1702)– Materials resist loads (push or pull back) in

response to applied loads• This ‘resistance’ is accomplished by deformation of

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


Hooke’s Law, cont.

• Hooke found that deflection was proportional to load

Load, N

Deflection, mm

slope, kSlope of Load-Deflection curve:

deflection

loadk =

The “Stiffness”


Stiffness

• Stiffness in tension and compression– Forces F applied, length L, cross-sectional area, A,

and material property, E (Young’s modulus)

AE

FL=δF

δF

k =

AE

FLF=

L

AEk =

F

L

A


Stiffness, cont.

• Stiffness in bending

• 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 (radius of curvature, Ri)• Outer “fibers” (B) are in tension (radius of curvature, Ro)

A

Ri

B

Ro

F


Review Question 1

• Stiffness is defined as:A. Force/Area

B. Deflection/ForceC. Force/Deflection

D. Force x DeflectionE. Mass/area


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 an 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.

The higher the area moment of inertia, the less a structure deflects (higher stiffness)

Mathematically, the area moment of inertia appears in the denominator of the deflection equation, therefore;


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

Load

Maximum distance of 4 inch to the centroid I2

2

2 x 8 beam


Moment of Inertia Equations for Selected Profiles

π (d)4

64I =

Round solid section

Rectangular solid section

b

hbh31I =

12

b

h

1I =

12hb3

d Round hollow section

π

64I = [(do)

4 – (di)4]

do

di

BH3 -1I =

12bh31

12

Rectangular hollow section

H

B

h

b


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)/(.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 high by .125 thick.

H

B

h

b

B = 2.25, H = 1.25

b = 2.0, h = 1.0


Review Question 2

• Which cross section has the larger I?A.

B.

Rectangular Horizontal

Rectangular Vertical


Stiffness Comparisons for Different sections

Square Box Rectangular Horizontal

Rectangular Vertical


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 structure deflects (higher stiffness)

Deflection of a Cantilever Beam

Fixed end

Support


Modulus 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.


Wind Turbine Structure

The support structure should be optimized for weight and stiffness.

Support Structure


Review Question 3

• Which material has the higher stiffness?A. Steel

B. Aluminum C. Alumina ceramic

D. NylonE. Unobtanium


Examples of Achieving Structural Stiffness

‘Ribbing’‘Gusset’

‘Boss’


Examples of Achieving Structural Stiffness, cont.

http://en.wikipedia.org/wiki/Image:FT_Rail.jpg



Welded ‘box’construction





‘Flange’



Structures Stiffness

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