The size and shape of the cross-section of the piece o

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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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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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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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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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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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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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What about Non-Rectangles

• Can be calculated with a little extra work
• Manufacturers publish tables of properties

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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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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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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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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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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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Deflection Formulas - Simple Beams
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Deflection Formulas - Cantilevers
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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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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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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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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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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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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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What Else can go Wrong

• Deep beams are economical but subject to lateral
  buckling

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