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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

1/28

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

2/28

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)

3/28

Where Elasticity Comes in

- Stress is proportional to strain
- Parts further from the centre strain more
- The outer layers receive greatest stress

4/28

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

5/28

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

6/28

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

7/28

A Measure of Strength - Z

- Simple solutions for rectangular sections

- Doing the maths (in the Notes)
- gives the Section Modulus

For a rectangular section

8/28

A Measure of Strength - Z (cont.)

- The bigger the Section Modulus, the stronger the

section - Contains d2, so depth is important

9/28

Stiffness and Strength

- Strength --gt Failure of Element
- Stiffness --gt Amount of Deflection

10/28

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

11/28

What about Non-Rectangles?

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

12/28

12/28

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

13/28

How do we Use these Properties? (cont.)

- Go back to the bending moment diagrams
- Maximum stress occurs where bending moment is a

maximum

14/28

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)

15/28

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

16/28

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

17/28

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)

18/28

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

19/28

Deflection Formulas - Simple Beams

20/28

Deflection Formulas - Cantilevers

21/28

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

22/28

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

23/28

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

24/28

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

25/28

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

26/28

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

27/28

What Else can go Wrong?

- Deep beams are economical but subject to lateral

buckling

28/28

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