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The size and shape of the cross-section of the piece of material used

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Title: The size and shape of the cross-section of the piece of material used


1
What is the 'Section' ?
  • The size and shape of the cross-section of the
    piece of material used
  • For timber, usually a rectangle
  • For steel, various formed sections are more
    efficient
  • For concrete, either rectangular, or often a Tee

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2
Why Different Shapes and Sizes
  • What shapes are possible in the material?
  • What shapes are efficient for the purpose?
  • Obviously, bigger is stronger, but less economical

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3
Which Way Around?
  • Beams are oriented one way
  • Depth around the X-axis is the strong way
  • Some lateral stiffness is also needed
  • Columns need to be stiff both ways (X and Y)

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4
Where Elasticity Comes in
  • Stress is proportional to strain
  • Parts further from the centre strain more
  • The outer layers receive greatest stress

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5
The Section Fights Back
  • The stresses developed resist bending
  • Equilibrium happens when the resistance equals
    the applied bending moment

All the compressive stresses add up to form a
compressive force C
C
a
T
All the tensile stresses add up to form a tensile
force T
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6
A Measure of Stiffness - I
  • Simple solutions for rectangular sections
  • Doing the maths (in the Notes)
  • gives the Moment of Inertia

For a rectangular section
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7
A Measure of Stiffness - I (cont.)
  • The bigger the Moment of Inertia, the stiffer the
    section
  • It is also called Second Moment of Area
  • Contains d3, so depth is important
  • The bigger the Modulus of Elasticity of the
    material, the stiffer the section
  • A stiffer section develops its Moment of
    Resistance with less curvature

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8
A Measure of Strength - Z
  • Simple solutions for rectangular sections
  • Doing the maths (in the Notes)
  • gives the Section Modulus

For a rectangular section
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9
A Measure of Strength - Z (cont.)
  • The bigger the Section Modulus, the stronger the
    section
  • Contains d2, so depth is important

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10
Stiffness and Strength
  • Strength --gt Failure of Element
  • Stiffness --gt Amount of Deflection

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11
Other Properties
  • The area tells how much stuff there is
  • used for columns and ties
  • directly affects weight and
  • cost
  • The radius of gyration is a derivative of I
  • used in slenderness ratio

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12
What about Non-Rectangles?
  • Can be calculated, with a little extra work
  • Manufacturers publish tables of properties

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13
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14
How do we Use these Properties?
  • Checking Beams
  • Designing Beams
  • given the beam section
  • check that the stresses deflection are
    within the allowable limits
  • find the Bending Moment and Shear Force
  • select a suitable section

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15
How do we Use these Properties? (cont.)
  • Go back to the bending moment diagrams
  • Maximum stress occurs where bending moment is a
    maximum

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16
Using Z to Check the strength of a Beam
  • Given the beam size and material
  • Find the maximum Bending Moment
  • Use Stress Moment/Section Modulus
  • Compare this stress to the Code allowable stress

M max BM
Z bd2 / 6
Actual Stress M / Z
Allowable Stress (from Code)
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17
Strength Checking Example
  • Given a softwood timber beam 250 x 50mm
  • Given maximum Bending Moment 4kNm
  • Given Code allowable stress 8MPa

Section Modulus Z bd2 / 6
50 x 2502 / 6
0.52 x 106 mm3
Actual Stress f M / Z
4 x 103 x 103
/ 0.52 x 106
7.69 MPa lt
8MPa
Actual Stress lt Allowable Stress
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18
Using Z to design a Beam for Strength
  • Given the maximum Bending Moment
  • Given the Code allowable stress for the material
  • Use Section Modulus Moment / Stress
  • Look up a table to find a suitable section

M max BM Allowable Stress (from Code)
required Z M / Allowable Stress
a) choose b and d to give Z gt than
required Z or
b) look up Tables of Properties
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19
Strength Designing Example
  • Given the maximum Bending Moment 4 kNm
  • Given the Code allowable stress for
  • structural steel 165 MPa

required Z 4 x 106 / 165 24 x 103 mm3
(steel handbooks give Z values in 103 mm3)
looking up a catalogue of steel purlins we find
C15020 - C-section 150 deep, 2.0mm thickness has
a Z 27.89 x 103 mm3
(smallest section Z gt reqd Z)
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20
What Controls Deflection?
  • Both E and I come into the deflection formula
  • (Material and Section properties)
  • The load, W, and span, L3
  • Note that I has a d3 factor
  • Span-to-depth ratios (L/d) are often used
  • as a guide

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21
Deflection Formulas - Simple Beams
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22
Deflection Formulas - Cantilevers
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23
Deflection Formula - Built-in Beams
  • The deflection is only one-fifth of a
  • simply supported beam
  • Continuous beams are generally stiffer than
    simply supported beam

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24
Using I to Check the Stiffness of a Beam
  • Given the beam size and material
  • Given the loading conditions
  • Use formula for maximum deflection
  • Compare this deflection to the Code allowable
    deflection

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25
Deflection Checking Example
  • Check the deflection of the steel channel
  • previously designed for strength
  • The maximum deflection lt L / 500

I 2.119 x 106 mm4
Section C15020
E 200 000 MPa
d (5/384) x WL3/EI mm
( Let us work in N and mm )
d (5/384) x 8000 x 40003 / (200000 x 2.119 x
106)
16 mm
8 mm
Maximum allowable deflection 4000 / 500
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26
Deflection Checking Example (cont.)
  • Need twice as much I
  • Could use same section back to back
  • 100 more material
  • A channel C20020 (200 deep 2mm thick)
  • has twice the I but only 27 more material

strategy for heavily loaded beams
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27
Using I to Design a Beam for Stiffness
  • Given the loading conditions
  • Given the Code allowable deflection
  • Use deflection formula to find I
  • Look up a table to find a suitable section

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28
What Section to Use?
better sections for beams
  • Beams need large I and Z in direction of
    bending
  • Need stiffness in other direction to resist
  • lateral buckling
  • Columns usually need large value of r
  • in both directions
  • Some sections useful for both

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29
What Else can go Wrong?
  • Deep beams are economical but subject to lateral
    buckling

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