### 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!**

### Thermal stress-strain

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

**Internal equilibrium**
**Node equilibrium**

**Global equilibrium**

Compr ession

### Thermal stress-strain

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

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

### Stresses on inclined sections

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

### 𝜎

_{𝑥}

### = 𝑃

### 𝐴

### Stresses on inclined sections

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

### 𝜎

_{𝑥}

### = 𝑃

### 𝐴 𝜏

_{𝑚𝑎𝑥}

### = 𝜎

_{𝑥}

### 2 = 𝑃

### 2𝐴

### 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. **

### Homework

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

### Homework

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

### Homework

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

### 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

### Notation

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

• Torque (twisting moment)

• A moment of couple

• Notation

@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 = 𝝓_{𝐦𝐚𝐱}

It is a pure shear!

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

### Torsion analysis

**Torsion, T**

**Shear stress, ** **Compatibility ** **Shear strain, ** 𝜸

### 𝜏 = 𝐺𝛾

### 𝝉

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

• Angle of twist (angle of rotation)

### Torsional deformation of a circular bar

### 𝜙(𝑥)

@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@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

@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 surfaceRate of twist

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

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

### Shear strains within the bar

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

• Hooke’s law in shear

### Link with shear stress

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

• Shear stress with Torsion force

### The Torsion Formula

Polar moment of inertia

Torsion formula

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

• Generalized torsion formula

### The Torsion Formula

Torsion formula

Generalized torsion formula

@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)

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

• Shear stress with Torsion force

### The Torsion Formula

Polar moment of inertia

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

• Net torsion calculation

### The Non-uniform Torsion

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

• Compatibility relation

### Statically indeterminate torsional member

Equation of equilibrium Two unknowns

Equation of compatibility

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

• Compatibility relation

### Statically indeterminate torsional member

Equation of equilibrium Equation of compatibility

Solved!

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

• Stresses on inclined planes

### Stress and strain in pure shear

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

• Stresses on inclined planes

### Stress and strain in pure shear

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

• Strains in pure shear

### Stress and strain in pure shear

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

• Stresses on inclined planes

### Relationship btw E and G

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

### Relationship btw E and G

**(=)**

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

### Relationship btw E and G

**(=)**

### 𝜀

_{max}

^{2}

### ≈ 0 𝑠𝑖𝑛𝛾 ≈ 𝛾

for very small strains

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

### Relationship btw E and G

**(=)**

## 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**

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