Structures Stiffness

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

    Structures and Stiffness

    B. Furman

    K. Youssefi20SEP2007

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    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

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    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.

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    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

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    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

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    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

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    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

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    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

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    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

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    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;

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    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

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    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

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    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

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    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

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    K. Youssefi and B. Furman Engineering 10, SJSU 15

    Review Question 2 Which cross section has the larger I?

    A.

    B.

    RectangularHorizontal

    RectangularVertical

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    K. Youssefi and B. Furman Engineering 10, SJSU 16

    Stiffness Comparisons for Different sections

    Square Box Rectangular

    Horizontal

    Rectangular

    Vertical

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    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

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    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.

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    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.

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    K. Youssefi and B. Furman Engineering 10, SJSU 20

    Wind Turbine Structure

    The support structure should be optimized for weight and stiffness.

    SupportStructure

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    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

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    K. Youssefi and B. Furman Engineering 10, SJSU 22

    Examples of Achieving Structural

    Stiffness

    Ribbing

    Gusset

    Boss

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    K. Youssefi and B. Furman Engineering 10, SJSU 23

    Examples of Achieving Structural

    Stiffness, cont.

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

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    Examples of Achieving Structural

    Stiffness, cont.

    Welded box

    construction

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    K. Youssefi and B. Furman Engineering 10, SJSU 25

    Examples of Achieving Structural

    Stiffness, cont.

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    K. Youssefi and B. Furman Engineering 10, SJSU 26

    Examples of Achieving Structural

    Stiffness, cont.

    Flange

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    Examples of Achieving Structural

    Stiffness, cont.