Dean Hackett

Structures

Week 2

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In previous sessions…

• Brief review of previous learning:– Types of motion– Classes of lever– Turning moments

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Types of structure

• Mass• Frame• Shell

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Forces

• Compression• Tension• Torsion• Shear• Bending

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Applying forces

• Distributed (UDL)• Concentrated (point)

• Static• Dynamic

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Reinforcing structures

• Create the following shapes from the modelling materials supplied. Ensure free-moving pin joints.

• Reinforce each shape internally using pin joints• Reinforce each shape internally using only string

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Combining external forces

Force A

Force B

What direction will the ball move in?

Resultant force C

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Using vectors

Force A10N

Force B10N

Any value that has both magnitude and direction can be drawn as a vector

Resultant force C

Draw forces accurately and to scale

Complete the parallelogram (in this case a square)

Draw in the resultant

Measure magnitude and angle

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Parallelogram of forces

Force A15N

Force B10N

Resultant force C

Draw forces accurately and to scale

Complete the parallelogram (in this case a rectangle)

Draw in the resultant

Measure magnitude and angle

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Parallelogram of forces

Force A15N

Force B10N

Resultant force C

Draw forces accurately and to scale

Complete the parallelogram (in this case a parallelogram)

Draw in the resultant

Measure magnitude and angle

60°

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Parallelogram of forces

Force A15N

Force B10N

Resultant force C

Draw forces accurately and to scale

Complete the parallelogram

Draw in the resultant

Measure magnitude and angle

60°

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Parallelogram of forces

Force A37N

Force B25N

Resultant force C

Draw forces accurately and to scale

Complete the parallelogram

Draw in the resultant

Measure magnitude and angle45°

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Triangle of forces

Force A260N

Force B180N

Resultant force D

Redraw forces accurately and to scale, with arrows nose to tail

Complete the triangle by drawing in the resultant

Note that the resultant runs from start point ‘a’ to end point ‘c’

60°The equilibrant completes the triangle with all arrows running nose to tail

a

bc

This closed shape with all vectors running in sequence means the forces are in equilibrium

Equilibrant force E

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Triangle of forces

Force A236N

Force B115N

Redraw forces accurately and to scale, with arrows nose to tail

Draw in equilibrant, completing the triangle20°

Calculate equilibrant for these forces using triangle of forces

Equilibrant

AA

BB

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Polygon of forces

Force A27N

Force B18N

Redraw forces accurately and to scale, with arrows nose to tail,

Draw in equilibrant ensuring flow of arrows is continued

50°

Force C20N

60°

45°

Force D6N

Note that the order in which the forces are drawn does not matter, as long as the flow is consistent

AABB

CC

DD

AA BB

CCDD

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Polygon of forces

Force A150kN

Force B70kN

Redraw forces accurately and to scale, with arrows nose to tail,

Draw in equilibrant ensuring flow of arrows is continued

60°

Force C180kN

30°

Force D160kN Note magnitude and

direction.

AA

BBCC

DD

Equilibrant

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Internal forces If a compressive force is applied to the top of the column, what force must the column be applying back, in order to remain in equilibrium?

What is happening at the base of the column?

The red arrows indicate that the column is under compression and is, therefore, a strut

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Internal forces If a load is applied to the top of the structure, what do the internal forces look like?

Are these struts or ties?

The red arrows indicate that the members are under compression (they are pushing back) and are, therefore, struts

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Internal forcesIf we do not know what the internal forces are

doing, we can still construct a triangle of forces:

150N

45° 45°

Label spaces as per Bow’s Notation

A B

C

Draw the force that you do know, ab

We don’t know if bc is in compression or tension, so draw a line across the end of ab at the correct anglea

b

Assuming the structure is in equilibrium, there is only one way to complete the triangle using the force ca

Measure the magnitude and note direction of the constructed vectors.

c

Transfer findings to original problem

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Internal forces

150N

60° 60°

Redo the calculations using a steeper angle

A B

C

What do you notice about the forces in individual members?

What are the problems in designing a structure in this way?

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Internal forcesRedo the calculations using a shallower angle

What do you notice about the forces in individual members?

What are the problems in designing a structure in this way?

150N

30° 30°

A B

C

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Hookes' Law

0

1

2

3

4

5

6

7

8

9

10

0 100 200 300 400 500

Extension

Lo

ad

• Gradually load up and measure extension of a spring or other materials

Hookes’ Law

• Complete at least two full sequences, completing table as you go

• Use Excel to plot graphs of load/extension

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Stress

• Nominal Stress σ = Load Р/Original area А (N/m2)

2.4kN5kN

0.15m2

0.08m2Which rod is under the most stress?

Bar A

5000/0.15 = 33333N/m2

= 33.3kN/m2

Bar B

2400/0.08 = 30000N/m2

= 30kN/m2

A B

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Strain

• Strain ε = extension e/original length L (no units!)

Which rod is under the most strain?

215m

m

180m

m

180m

m

150m

m

A B

Bar A

35/180 = 0.19

Bar B

30/150 = 0.2

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Young’s Modulus

• Modulus of elasticity E = stress σ /strain ε (N/m2)

Which rod is the most elastic?A B

Bar A

33300/0.19 =

175kN/m2 (175 kPa)

Bar B

30000/0.2 =

150kN/m2 (150 kPa)

33.3kN/m2 30.0kN/m2

0.19 0.2

Note that a lower modulus of elasticity means more flexibility

Pascals are a measure of load over an area or ‘pressure’.

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Young’s ModulusComplete the table on Excel to calculate Young’s modulus for your test pieces

Mass (g) Load (N)

Stress = load / area area =

1.57mm2

Length (mm)

Extension (mm)

Strain = Extension /

original length

Young’s modulus = stress /

strain

0 0 0 205 0 0  

100 0.981 0.624841 230 25 0.121951 5.12

200 1.962 1.249682 285 80 0.390244 3.20

300 2.943 1.874522 340 135 0.658537 2.85

400 3.924 2.499363 390 185 0.902439 2.77

500 4.905 3.124204 440 235 1.146341 2.73

600 5.886 3.749045 490 285 1.390244 2.70

700 6.867 4.373885 540 335 1.634146 2.68

800 7.848 4.998726 590 385 1.878049 2.66

900 8.829 5.623567 640 435 2.121951 2.65

Plot a graph of stress/strain

Youngs modulus = stress / strain

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5

Strain (extension / original length)

Str

ess

N/m

m2

(lo

ad /

are

a)

Compare elasticity of the springs with other groups

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Young’s ModulusWhat does the graph show?

The modulus of a material can be plotted against many other characteristics such as cost, thermal conductance, working temperature range, etc.

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Common Beam Sections

Closed Sections Open Sections

Which of these sections is the most efficient?

What problems might you expect to be associated with the different sections?

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Task: Avoiding stress

• Devise a method for testing the strongest practical way of producing a box corner in acrylic.

• Create test pieces and test your ideas.

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Weblinks

• www.greenhomebuilding.comBig resources for sustainable home design

• www.sustainableabc.comFantastic resources for sustainable design

• www.architect.org/links/sustainable_architecture.html Good set of eco links

• www.ecosustainable.com.au Huge set of eco links

• www.naturalspace.com Fantastic site, beautiful case studies

• www.lanxun.com/pce/index.htmRange of design programs to try

• www.architecturalresources.infoNice but problematical site with good tutorials