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### Transcript of Week 4 - 1 slides

• Mechanics of Solids

• Thermal Strain

Thermal Deformation

Thermal Strain

Thermal Stress

thermal = E thermal A B

• 25 mm 20 mm

A B C

300 mm 450 mm

D D=25 mm

a

a

Section a-a

0.2 mm

E

Segment AB: EAB = 44.7 GPa; AB = 26 x10-6/oC

Segment CD: ECD = 68.9 GPa; CD = 24 x 10-6/oC

A B C D D=25 mm

E FAB FDC

FAB = FCD = F

Y-AB = 152 MPa

Y-CD = 255 MPa

= 152 x (252-202) = 107.5 kN

= 255 x 12.52 = 125.2 kN

107.5 kN

4-85 The AM1004-T61 magnesium alloy tube AB is capped with a rigid plate. The gap between E and end C of the 6061-T6 aluminum alloy solid circular rod CD is 0.2 mm when the temperature is 30oC . Determine the highest temperature to which it can be raised without causing yielding either in the tube or the rod. Neglect the thickness of the rigid cap. (p. 157)

Thermal Strain

• 25 mm 20 mm

A B C

300 mm 450 mm

D D=25 mm

a

a

Section a-a

0.2 mm

E

A B C D D=25 mm

E FAB FDC

FAB = FCD = F 107.5 kN

D=25 mm FDC FDC

AD= 0.2 mm = AB + CD (Compatibility)

T F = 0.2 mm

4-85 The AM1004-T61 magnesium alloy tube AB is capped with a rigid plate. The gap between E and end C of the 6061-T6 aluminum alloy solid circular rod CD is 0.2 mm when the temperature is 30oC . Determine the highest temperature to which it can be raised without causing yielding either in the tube or the rod. Neglect the thickness of the rigid cap. (p. 157)

Thermal Strain

• 4-85 The AM1004-T61 magnesium alloy tube AB is capped with a rigid plate. The gap between E and end C of the 6061-T6 aluminum alloy solid circular rod CD is 0.2 mm when the temperature is 30oC . Determine the highest temperature to which it can be raised without causing yielding either in the tube or the rod. Neglect the thickness of the rigid cap. (p. 157)

25 mm 20 mm

A B C

300 mm 450 mm

D D=25 mm

a

a

Section a-a

0.2 mm

E

A B C D D=25 mm

E FAB FDC

FAB = FCD = F 107.5 kN

AD = 0.2 mm = AB + CD (Compatibility)

= 0.2 mm

=- 1.02 mm = -1.43 mm

=2.65 mm

T = 142 oC Segment AB: EAB = 44.7 GPa; AB = 26 x10-6/oC

Segment CD: ECD = 68.9 GPa; CD = 24 x 10-6/oC Y-AB = 152 MPa

Y-CD = 255 MPa Th = 30 + T = 172 oC

Thermal Strain

= E Saint-Venant Principle Hookes Law

15 kN

A D

20 kN

B C

a

a

N [kN]

L

NAB

A B C D 0

NBC

NCD

NBC - NAB = - 20 kN

NCD - NBC= + 15 kN

• Poissons Ratio

When a deformable body is stretched by a tensile force, not only does it elongate but it also contract laterally, i.e. it would contract in other two dimensions. Likewise, a compressive force acting on a deformable body cause it to contract in the direction of force and yet its sides expand laterally

Lateral strain is the same in all lateral direction;

Poissons ratio is a constant.

Usually 0 v 0.5. For most linearly elastic material v 0.3;

• Lateral strain is the same in all lateral direction;

Poissons ratio is a constant.

Usually 0 v 0.5. For most linearly elastic material v 0.3;

z

y

x

o

zz

yy

xx

z

= + + zz

'yy

'xx

'zz

=

'z =zz/E ''z z = - (yy/E) = - (xx/E)

z + z + z = [zz - (yy + xx ) ]/E x = [xx - (yy + zz ) ]/E

y = [yy - (xx + zz ) ]/E

Poissons Ratio

• Lateral strain is the same in all lateral direction;

Poissons ratio is a constant.

Usually 0 v 0.5. For most linearly elastic material v 0.3;

z

y

x

o

zz

yy

xx

= [zz - (yy + xx ) ]/E

x = [xx - (yy + zz ) ]/E

y = [yy - (xx + zz ) ]/E

dx dz

dy

dV=dx dy dz

(1+x)dx z (1+z)dz

(1+y)dy

V

V = (1+x) (1+y) (1+z) dx dy dz

= (1+x) (1+y) (1+z) dx dy dz - dx dy dz

= (x+y+z) dx dy dz

e= V dV = x+y+z Volume Strain: =

1 - 2 E

(zz + yy + xx)

Poissons Ratio

• Lateral strain is the same in all lateral direction;

Poissons ratio is a constant.

Usually 0 v 0.5. For most linearly elastic material v 0.3;

z

y

x

o

zz

yy

xx

dx dz

dy

dV=dx dy dz

(1+x)dx

(1+z)dz

(1+y)dy

V = (1+x) (1+y) (1+z) dx dy dz

e= V dV

Volume Strain:

= 1 - 2 E (zz + yy + xx)

zz = yy = xx = p

= 3(1 - 2)

E p e k = Volume Modulus of elasticity:

(Bulk modulus)

Poissons Ratio

• 3-25 The plastic rod is 200 mm long and 15 mm in diameter. If an axial load of 300 N is applied to it, determine the change in its length and the change in its diameter. E=2.70GPa, v=0.4. (p. 111)

200 mm

300 N 300 N

A B

= 0.126/200=0.00063

= - 0.00063 x 0.4 = - 0.00025

= - 0.00025 x 15 = - 0.00377 mm

Ans.

Ans.

Poissons Ratio

• Lateral strain is the same in all lateral direction;

Poissons ratio is a constant.

Usually 0 v 0.5. For most linearly elastic material v 0.3;

z

y

x

o

zz

yy

xx

dx dz

dy

dV=dx dy dz

(1+x)dx

(1+z)dz

(1+y)dy

V = (1+x) (1+y) (1+z) dx dy dz

e= V dV

Volume Strain:

= 1 - 2 E (zz + yy + xx)

zz = yy = xx = p

= 3(1 - 2)

E p e k = Volume Modulus of elasticity:

(Bulk modulus)

Poissons Ratio

• Strain Energy

Work of a Force:

F W = F x

P

x

F = f(x)

x 0

F = k x

P

(k = P/=EA/L )

W

W

External Work

• P

x 0

F = k x

P

(k = P/ )

External Work

Internal Work Strain Energy y

y

dx dz dy

dFy = ydA = ydxdz

dz = dz

We = Ui

W

Strain Energy

• Internal Work Strain Energy

0

z

Strain Energy Density - Strain-energy per unit volume

i

Modulus of resilience - When stress reaches the proportional limit, strain energy density is referred to Modulus of resilience

pl

pl

x fr

y

y

dx dz dy

Modulus of toughness -The higher the modulus of toughness, the more strain energy can be stored (absorbed) inside the material before fracturing.

z

Strain Energy

• Internal Work Strain Energy y

y

dx dz dy

F

Total Strain Energy inside deformable body The above equation is the elemental strain energy, by integrating it, the total volume and substituting Hooke's law we have that:

External Work

If the deflection of a structure or member under a single concentrated load P is known, the corresponding Work W may be obtained by writing

W

Strain Energy

• 14-73 Determine the horizontal displacement of join B. Each A-36 member has a cross-sectional area of 1250 mm2. (p. 760)

A

B

C 1.8 m

2.4 m

1 kN

1.8 m

Strain Energy

Mechanics of SolidsSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17