Elasticity

Tutorial 7

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what is the Modulus of Elasticity, E?

Question 1

a measure of the strength of a materialb

a measure of the stiffness of a materialc

a material’s tendency to be deformed elastically

i.e. how a material responds to stress d

the slope of the stress-strain curve a

a and de

what are the units for the Modulus of Elasticity, E?

Question 2

MPab

MNc

no units a

what is the stress in a rod?

Question 3a

81.5 MPab

163 MPac

80 MPa a

a footbridge carrying a load of 80 kN is supported by two 25 mm dia. aluminium rods 3 m long. Neglecting the self-weight of the rod and given that theModulus of Elasticity of aluminium is 70,000 MPA

3m

80 kN

what is the strain in a rod?

Question 3b

0.00116 (1.16 x 10-3)b

0.02 (2 x 10-2)c

3 mm a

a footbridge carrying a load of 80 kN is supported by two 25 mm dia. aluminium rods 3 m long. Neglecting the self-weight of the rod and given that theModulus of Elasticity of aluminium is 70,000 MPA

3m

80 kN

how much does the rod lengthen?

Question 3c

4.5 mmb

3.5 mmc

35 mm a

a footbridge carrying a load of 80 kN is supported by two 25 mm dia. aluminium rods 3 m long. Neglecting the self-weight of the rod and given that theModulus of Elasticity of aluminium is 70,000 MPA

3m

80 kN

are the rods strong enough ?

Question 3d

yesa

nob

a footbridge carrying a load of 80 kN is supported by two 25 mm dia. aluminium rods 3 m long. Neglecting the self-weight of the rod and given that theModulus of Elasticity of aluminium is 70,000 MPA

3m

80 kN

given that the maximum allowable tensile stress for aluminium is 120 MPA

what does elastic behaviour mean ?

Question 4

the material stretches under tensiona

the material responds to stress in a linear wayb

the strain is linearly proportional to the to stressc

the deformations are irreversibled

the deformations are reversiblee

b, c and df

b, c and eg

what does plastic behaviour mean ?

Question 5

the material can be bent and reshapeda

the material responds to stress in a linear wayb

the deformations are largec

the deformations are irreversible / permanentd

the deformations are reversiblee

a, c and df

a, c and eg

what does brittle behaviour mean ?

Question 6

the material responds to stress in a linear waya

the material fails suddenly soon after the yield stressb

the material is strong in tensionc

the material is strong in compression and weak in tensiond

a, b and ce

a, b and df

b and dg

is steel ?

Question 7

plastica

brittleb

elasticc

both elastic and plasticd

draw the stress / strain curve for an elasto-plastic material

Question 8

show mea

next questionb

draw the stress / strain curve for a brittle material

Question 9

show mea

next questionb

what is the difference between the curves ?

Question 10

nonea

the elasto-plastic one has an elastic range whereas the brittle one hasn’t

b

the brittle curve has almost no plastic rangec

what are the advantages of elasto-plastic materials ?

Question 11

nonea

they are strongerb

large deformations after the yield stress is reachedc

give warning of dangerd

c and de

b, c and df

which of these lists contains all brittle materials ?

Question 12

concrete, timber, bricka

masonry, cast iron, glass, cement, high-strength carbon steel

b

concrete, glass, brick, timber c

steel, concrete, glass, brickd

why do we use brittle materials ?

Question 13

because we like thema

they are, in the main, cheap and good in compressionb

they are, in the main, cheap and good in tensionc

how do we cure brittleness ?

Question 14

by not using brittle materialsa

by introducing elastic material to take care of tensile stresses

b

you don’tc

what does having a high value ofthe Modulus of Elasticity, E, mean ?

Question 15

the material is strongera

the material deforms lessb

a and bc

what dimensions should the column be (nearest 25mm) ?

Question 16a

350 x 350 mm square or 375 mm dia

a

275 x 275 mm square or 325 mm diab

325 x 325 mm square or 350 mm diac

a ground-floor reinforced concrete column in a multi-storey building is3m high and carries a load of 3.2 MN. Given that the max. allowable stress for concrete is 30MPa and the Modulus of Elasticity, E, of concrete is 25,000MPa

what is the actual stress in the column?

Question 16b

2.6 MPaa

32.0 MPab

26.1 MPac

a ground-floor reinforced concrete column in a multi-storey building is3m high and carries a load of 3.2 MN. Given that the max. allowable stress for concrete is 30MPa and the Modulus of Elasticity, E, of concrete is 25,000MPa

what is the strain in the column?

Question 16c

1.0a

0.001 (10-3)b

0.01 (10-2)c

a ground-floor reinforced concrete column in a multi-storey building is3m high and carries a load of 3.2 MN. Given that the max. allowable stress for concrete is 30MPa and the Modulus of Elasticity, E, of concrete is 25,000MPa

by how much does the column shorten?

Question 16d

3.0 mma

1.0 mmb

30 mmc

a ground-floor reinforced concrete column in a multi-storey building is3m high and carries a load of 3.2 MN. Given that the max. allowable stress for concrete is 30MPa and the Modulus of Elasticity, E, of concrete is 25,000MPa

what are safety factors?

Question 17a

warning signsa

margins of safetyb

the amount by which we over-design a structurec

why do we use safety factors?

Question 17b

to ensure that the structure doesn’t collapse a

to provide a margin of safety so that failure is extremely unlikely

b

to prevent over-designingc

what value of safety factors do we use in buildings?

Question 17c

1.0 – 3.0a

1.0 - 2.5b

1.5 – 2.5c

2.0 – 3.0d

which would require a lower safety factor ?

Question 17d

concretea

steelb

timberc

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the modulus of elasticity is a substance's tendency to be deformed elastically (i.e. non-permanently) when a force is applied to it.

The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region:

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the modulus of elasticity has nothing to do with strength

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the modulus of elasticity indicates how stiff a material is, but that’s not what it is

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E = stress / strainsince strain has no units, E has the same units as stress

i.e. MPa

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E = stress / strainWhat are the units for stress? What are the units for strain?

think again

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f = F / AForce, F = 40 kN (per rod),

A = πR2 = 3.142 x 12.5 x 12.5 =491 mm2

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not correct

f = F / AWhat is the force? What is the area?

(try keeping everything to Newtons and mms)

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not correct

f = F / AWhat is the force? Don’t forget that 80 kN is carried by 2 rods

What is the area?(try keeping everything to Newtons and mms)

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E = f / eso, e = f/ E

E = 81.5 / 70,000

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a rather large strain, don’t you think?What are the units of strain?

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check your calculations

E = f / eSo, e = f / E

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e = ΔL / L so, ΔL = e x L

change in length = strain x original length

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a rather large deformation, don’t you think?

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don’t guess! check your calculations

e = ΔL / Lstrain = change in length / original length

so, ΔL = e x L

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actual stress is 81.5 MPamaximum allowable stress is 120 MPa

actual stress < maximum allowable stress

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what is the actual stress?what is the maximum allowable stress?

is the actual stress greater or less than the max. allowable stress?

so?

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elastic behaviour means all of these.the material strains in a linear relationship to stress and

the deformations are reversible.Also the deformations are very small

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all materials stretch under tension

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means other things too

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we’re talking aboutelastic behaviour

think of a spring

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plastic behaviour means all of these.The material can be bent and reshaped,

the material has large deformations,the deformations are irreversible.

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means other things too

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we’re talking aboutplastic behaviour

this is elastic behaviour

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we’re talking aboutplastic behaviour

(e) is elastic behaviour

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brittle behaviour means all of these.The material behaves elastically up to the yield point,

Then snaps suddenly.Brittle materials are generally weak in tension.

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means other things too

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we’re talking aboutbrittle behaviour

brittle materials are not good in tension

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steels are generally elasto-plastic.i.e. they are elastic up to their yield stress and then plastic High-strength-carbon steels can be brittle

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what else?

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generally not

some high-strength carbon steels are brittle

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strainplasticrange

elasticrange

stre

ss ultimatefailure

yield stress

yield point

Elasto-Plastic Behaviour Graph

At first, the material behaves elastically, up to the yield stress. After the yield stress is reached, large deformations occur for very little increase in stress. Note the large plastic range. The material still has strength after the yield point.

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strainelasticrange

stre

ss ultimatefailure

yield stress

yield point

Brittle Behaviour Graph

At first, the material behaves elastically, up to the yield stress. After the yield stress is reached, the material snaps suddenly with no warning

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brittle materials fail almost immediately the yield point is reached

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you’re not trying

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brittle materials DO have an elastic range

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elasto-plastic materials because they undergo large, visible deformations give a good warning of impending failure.

They are also strong in tension

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you’re not trying

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how many times must it be said

elasticity has NOTHING to do with strength

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and …. ???

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timber is not brittle

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steel is not brittle

some high-strength carbon steels are brittle but not generally.

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for example, concrete is a relatively cheap materialand it is a good sound and heat insulator and also fireproof

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what’s ‘like’ got to do with it?

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no!, no!, no!

brittle materials are not strong in tension.Think of what happens if you pull on chalk. That’s why stone beams crack easily on the underside.

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that’s why we have reinforced concrete

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we just said how good they are for many purposes

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c’mon – a little bit of effort

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the Modulus of Elasticity has to do with stiffness and not strength.The higher the value of E, the more the material resists deformation

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strength has nothing to do with elasticity

you should really know by now

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f = F / A, A = F / f, A = 3.2 x 106 / 30 (keep everything to Newtons and mms)A = 106,667 mm2

√106,667 = 326.7πD2/4 = 106,667, D = 368.5

so, 327x327 mm needs to be upsized to 350x350mm and 368.5 mm to 375 mm

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f = F / A, A = F / f, A = 3.2 x 106 / 30 (keep everything to Newtons and mms)A = 106,667 mm2

√106,667 = ?πD2/4 = 106,667, D = ?

check your calculations

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f = F / A, A = F / f, A = 3.2 x 106 / 30 (keep everything to Newtons and mms)A = 106,667 mm2

√106,667 = ?πD2/4 = 106,667, D = ?

dimensions need to be upsized not downsized to nearest 25 mm

check your calculations

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f = F / A, f = 3.2 x 106 / (350 x 350) (keep everything to Newtons and mms)f = 26.1 N/mm2

Remember 1 N/mm2 = 1 MPa

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check your calculations

f = F / A, f = 3.2 x 106 / (350 x 350) (keep everything to Newtons and mms)

Remember 1 N/mm2 = 1 MPa

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E = f / e, e = f / E, e = 26.1 / 25,000e = 0.001 = 10-3

strains are very small quantities

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this is a very large value

E = f / e, e = f / E

strains are very small quantities

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check your calculations

E = f / e e = f / E strain = stress / Modulus of Elasticity

So what is f? what is E?

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e = ΔL / L, ΔL = e x L = 0.001 x 3000ΔL = 3.0 mm

elastic deformations are also very small

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check your calculations

e = ΔL / L, ΔL = e x LΔL = strain x original length

So what is e? what is L?

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this is a very large value

e = ΔL / L, ΔL = e x LΔL = strain x original length

So what is e? what is L?

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safety factors are factors by which we over-design a structureto allow a margin of safety

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safety factors are not warning signs.Think of the words

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they provide margins of safetybut that’s not what they are.

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safety factors are factors by which we over-design a structureto allow a certain margin of safety.

They ensure that failure is extremely unlikelywithout totally over-designing the structure

They vary depending on the structure and the material

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You can’t fully ensure that failure won’t occur under all circumstances

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partly correct, but what do they do with regards to the safety of a structure?

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depends on the building and on the material. The more serious the result of collapse or the more likely,

The greater the factor of safety.Of course the more we over-design the greater the cost

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what’s the use of a safety factor of 1?

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finished !

you’ve graduated as an elasticity master

since steel is fairly homogeneous and its properties are well knownwithin small tolerances, it is much more predictable.

Therefore one can use a lower safety factor

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concrete varies a lot

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timber varies a lot