• Full Attendance 10 Announcement

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(1)

Announcement

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

• Homework due April 10th

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

• Full Attendance 10

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

• Participation = max 5

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

(2)

Thermal stress-strain

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

Internal equilibrium Node equilibrium

Global equilibrium

Compr ession

(3)

Thermal stress-strain

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

𝛿 ≠ 0 𝜀 ≠ 0 𝜎int = 0 𝑅𝑟𝑒𝑐 = 0

𝛿 = 0 𝜀 = 0 𝜎int ≠ 0 𝑅 𝑟𝑒𝑐 ≠ 0

𝛿 ≠ 0 𝜀 ≠ 0 𝜎int ≠ 0 𝑅 𝑟𝑒𝑐 ≠ 0

Compr ession

= 𝐸𝛼 𝛥𝑇

= 𝛼 𝛥𝑇

(4)

Stresses on inclined sections

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

𝜎

𝑥

= 𝑃

𝐴

(5)

Stresses on inclined sections

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

𝜎

𝑥

= 𝑃

𝐴 𝜏

𝑚𝑎𝑥

= 𝜎

𝑥

2 = 𝑃

2𝐴

(6)

Stresses on inclined sections

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

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

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

(8)

Homework

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

(9)

Homework

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

(10)

Chapter 3 Torsion

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

• 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

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

• Torque (twisting moment)

• A moment of couple

• Notation

(12)

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

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

Angle of twist (Angle of rotation)

𝝓 = 𝟎 𝝓 = 𝝓𝐦𝐚𝐱

It is a pure shear!

(13)

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

Torsion analysis

Torsion, T

Shear stress, Compatibility Shear strain, 𝜸

𝜏 = 𝐺𝛾

𝝉

(14)

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

• Angle of twist (angle of rotation)

Torsional deformation of a circular bar

𝜙(𝑥)

(15)

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

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

Shear strains at the outer surface

Shear strain under pure shear

(16)

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

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

Shear strains at the outer surface

Maximum shear strain on surface

Rate of twist

=Angle of twist per unit length

(17)

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

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

Shear strains at the outer surface

Maximum shear strain on surface

Rate of twist

(18)

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

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

Shear strains within the bar

(19)

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

• Hooke’s law in shear

Link with shear stress

(20)

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

• Shear stress with Torsion force

The Torsion Formula

Polar moment of inertia

Torsion formula

(21)

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

• Generalized torsion formula

The Torsion Formula

Torsion formula

Generalized torsion formula

(22)

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

• Angle of twist

The Torsion Formula

Torsion formula

Generalized torsion formula Angle of twist

Rate of twist

(Angle of twist per unit length)

(23)

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

• Shear stress with Torsion force

The Torsion Formula

Polar moment of inertia

(24)

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

• Net torsion calculation

The Non-uniform Torsion

(25)

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

• Compatibility relation

Statically indeterminate torsional member

Equation of equilibrium Two unknowns

Equation of compatibility

(26)

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

• Compatibility relation

Statically indeterminate torsional member

Equation of equilibrium Equation of compatibility

Solved!

(27)

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

• Stresses on inclined planes

Stress and strain in pure shear

(28)

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

• Stresses on inclined planes

Stress and strain in pure shear

(29)

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

• Strains in pure shear

Stress and strain in pure shear

(30)

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

• Stresses on inclined planes

Relationship btw E and G

(31)

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

Relationship btw E and G

(=)

(32)

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

Relationship btw E and G

(=)

𝜀

max2

≈ 0 𝑠𝑖𝑛𝛾 ≈ 𝛾

for very small strains

(33)

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

Relationship btw E and G

(=)

(34)

Chalk failure pattern prediction

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

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)

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

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

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