Title: The size and shape of the cross-section of the piece of material used
1What 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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2Why 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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3Which 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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4Where 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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5The 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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6A 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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7A 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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8A 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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9A Measure of Strength - Z (cont.)
- The bigger the Section Modulus, the stronger the
section - Contains d2, so depth is important
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10Stiffness and Strength
- Strength --gt Failure of Element
- Stiffness --gt Amount of Deflection
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11Other 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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12What about Non-Rectangles?
- Can be calculated, with a little extra work
- Manufacturers publish tables of properties
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1312/28
14How 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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15How 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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16Using 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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17Strength 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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18Using 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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19Strength 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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20What 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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21Deflection Formulas - Simple Beams
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22Deflection Formulas - Cantilevers
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23Deflection 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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24Using 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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25Deflection 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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26Deflection Checking Example (cont.)
- 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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27Using 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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28What 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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29What Else can go Wrong?
- Deep beams are economical but subject to lateral
buckling
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