• Full Attendance 10 Announcement

(1)

Announcement

@Copyright Prof. Juhyuk Moon (문주혁), CEE, SNU

• Homework due April 10^th

• Lab session on April 5^th (Lab report due April 12^th)

• Full Attendance 10

(If you missed 2 classes of 20, mark will be 9) (Minimum absence is 25%)

• Participation = max 5

(If you are 10^th out of 50 students, you will get additional 4) (If you are 40^th out of 50 students, you will get additional 1) -> No one will take disadvantage!

(2)

Thermal stress-strain

Internal equilibrium Node equilibrium

Global equilibrium

Compr ession

(3)

Thermal stress-strain

𝛿 ≠ 0 𝜀 ≠ 0 𝜎_int = 0 𝑅_𝑟𝑒𝑐 = 0

𝛿 = 0 𝜀 = 0 𝜎_int ≠ 0 𝑅 _𝑟𝑒𝑐 ≠ 0

𝛿 ≠ 0 𝜀 ≠ 0 𝜎_int ≠ 0 𝑅 _𝑟𝑒𝑐 ≠ 0

Compr ession

= 𝐸𝛼 𝛥𝑇

= 𝛼 𝛥𝑇

(4)

Stresses on inclined sections

𝜎

_𝑥

= 𝑃

𝐴

(5)

Stresses on inclined sections

𝜎

_𝑥

= 𝑃

𝐴 𝜏

_𝑚𝑎𝑥

= 𝜎

_𝑥

2 = 𝑃

2𝐴

(6)

Stresses on inclined sections

𝜎

_𝑥

= 𝑃

𝐴 𝜏

_𝑚𝑎𝑥

= 𝜎

_𝑥

2 = 𝑃 2𝐴

E.g.,

If one material has:

Allowable shear stress = 40 MPa Allowable normal stress = 100 MPa

Then, the material under axially loaded will fail at 45 degree with axial stress of 80 MPa.

(7)

Homework

(8)

Homework

(9)

Homework

(10)

Chapter 3 Torsion

• Torsional deformation

• Circular bars of linearly elastic materials

• Non-uniform torsion

• Stress-strain in pure torsion

• Relationship between E and G

• Statically indeterminate torsional member

• Strain energy in torsion and pure shear

• And others

(11)

Notation

• Torque (twisting moment)

• A moment of couple

• Notation

(12)

• The angle of twist changes along the axis of the bar

Angle of twist (Angle of rotation)

𝝓 = 𝟎 𝝓 = 𝝓_𝐦𝐚𝐱

It is a pure shear!

(13)

Torsion analysis

Torsion, T

Shear stress, Compatibility Shear strain, 𝜸

𝜏 = 𝐺𝛾

𝝉

(14)

• Angle of twist (angle of rotation)

Torsional deformation of a circular bar

𝜙(𝑥)

(15)

• A small element abcd twists to ab’c’d

Shear strains at the outer surface

Shear strain under pure shear

(16)

Shear strains at the outer surface

Maximum shear strain on surface

Rate of twist

=Angle of twist per unit length

(17)

Shear strains at the outer surface

Maximum shear strain on surface

Rate of twist

(18)

• The shear strain is a function of radius of the bar

Shear strains within the bar

(19)

• Hooke’s law in shear

Link with shear stress

(20)

• Shear stress with Torsion force

The Torsion Formula

Polar moment of inertia

Torsion formula

(21)

• Generalized torsion formula

The Torsion Formula

Torsion formula

Generalized torsion formula

(22)

• Angle of twist

The Torsion Formula

Torsion formula

Generalized torsion formula Angle of twist

Rate of twist

(Angle of twist per unit length)

(23)

• Shear stress with Torsion force

The Torsion Formula

Polar moment of inertia

(24)

• Net torsion calculation

The Non-uniform Torsion

(25)

• Compatibility relation

Statically indeterminate torsional member

Equation of equilibrium Two unknowns

Equation of compatibility

(26)

• Compatibility relation

Statically indeterminate torsional member

Equation of equilibrium Equation of compatibility

Solved!

(27)

• Stresses on inclined planes

Stress and strain in pure shear

(28)

Stress and strain in pure shear

(29)

• Strains in pure shear

Stress and strain in pure shear

(30)

Relationship btw E and G

(31)

Relationship btw E and G

(=)

(32)

Relationship btw E and G

(=)

𝜀

_max²

≈ 0 𝑠𝑖𝑛𝛾 ≈ 𝛾

for very small strains

(33)

Relationship btw E and G

(=)

(34)

Chalk failure pattern prediction

A chalk has below material properties

Allowable normal stress (compression) = 8 MPa Allowable normal stress (tension) = 2 MPa

Allowable shear stress = 0.5 MPa

(35)

• Full Attendance 10 Announcement

Announcement

Thermal stress-strain

Thermal stress-strain

Stresses on inclined sections

𝜎

= 𝑃

𝐴

Stresses on inclined sections

𝜎

= 𝑃

𝐴 𝜏

= 𝜎

2 = 𝑃

2𝐴

Stresses on inclined sections

𝜎

= 𝑃

𝐴 𝜏

= 𝜎

2 = 𝑃 2𝐴

Homework

Homework

Homework

Chapter 3 Torsion

Notation

Angle of twist (Angle of rotation)

Torsion analysis

Torsion, T

Shear stress, Compatibility Shear strain, 𝜸

𝜏 = 𝐺𝛾

𝝉

Torsional deformation of a circular bar

𝜙(𝑥)

Shear strains at the outer surface

Shear strains at the outer surface

Shear strains at the outer surface

Shear strains within the bar

Link with shear stress

The Torsion Formula

The Torsion Formula

The Torsion Formula

The Torsion Formula

The Non-uniform Torsion

Statically indeterminate torsional member

Statically indeterminate torsional member

Stress and strain in pure shear

Stress and strain in pure shear

Stress and strain in pure shear

Relationship btw E and G

Relationship btw E and G

(=)

Relationship btw E and G

(=)

𝜀

≈ 0 𝑠𝑖𝑛𝛾 ≈ 𝛾

Relationship btw E and G

(=)

Chalk failure pattern prediction

A chalk has below material properties

Allowable normal stress (compression) = 8 MPa Allowable normal stress (tension) = 2 MPa

Allowable shear stress = 0.5 MPa

Vertical plane failure

Sharply inclined plane failure with 45 degree

Curved inclined plane with 45 degree crack line

A chalk has below material properties Allowable normal stress (compression) = 8 MPa Allowable normal stress (tension) = 2 MPa

Allowable shear stress = 0.5 MPa