457.201 재료역학 및 실험 Mechanics of Materials and Lab.

(1)

Announcement

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

• To be updated

(2)

Progress

• Chapter 1 Material properties

• Chapter 2 Axially loaded member

• Chapter 3 Torsion

• Chapter 4 Shear and Bending moment

• Chapter 5 Stress in beams

• Chapter 7 Stress and Strain

• Chapter 9 Deflection of beams

• Chapter 11 Columns

(3)

Chapter 7 Analysis of Stress and Strain

KEYWORDS:

Plane stress Mohr’s Circle

Uniaxial, Biaxial, Triaxial stress Principal stress

Flexure formula

Shear formula

Torsion formula

(4)

Introduction

(Sign convention) All we need to know..

𝜎 x 𝜎 y

=applied normal stress

𝜃 = angle rotation

=applied shear stress

𝜏 𝑥𝑦

=new normal stress

=new shear stress

𝜏 𝑥

₁

𝑦

₁

𝜎 𝑥

₁

𝜎 𝑦

₁

(5)

Introduction

(Sign convention)

At certain rotation angle, there will be no shear stress

𝜎₁ 𝜎₂ 𝜃_𝑝1

=Principal stress

=Principal angle

𝜏₁₂ =?

(6)

Introduction

(Sign convention)

Because x-y or x1-y1 axis is always perpendicular, 𝜎₁ 𝜎₂ 𝜃_𝑝1

=Principal stress

=Principal angle

𝜎₂ 𝜎₁ 𝜃_𝑝2

=Principal stress

=Principal angle +𝟗𝟎 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛, 𝜃_𝑝2= 𝜃_𝑝1 + 90

(7)

Mohr’s Circle

Derivation can be tedious but final result is very useful!

Point on Circle

(8)

Mohr’s Circle

Principal stress can be also found easily

Point on Circle

(9)

Transformation

Wedge-shape stress element

(10)

Transformation

Force equilibrium condition

<x1 direction>

(11)

Transformation

Force equilibrium condition

<y1 direction>

(12)

Transformation equations for plane stress

<x1 direction> - normal <y1 direction> - shear

(13)

Transformation equations for plane stress

<x1 direction> - normal

<y1 direction> - normal

𝜃 = 𝜃 + 90°

(14)

Observation

<x1 direction> - normal

<y1 direction> - normal

(15)

Observation

<uniaxial stress> <x1 direction> - normal

<y1 direction> - shear

(16)

Observation

<pure shear> <x1 direction> - normal

(17)

Observation

<biaxial stress> <x1 direction> - normal

(18)

Principal stress and Maximum shear stress

𝜃_𝑝1

𝜃_𝑝2=𝜃_𝑝1+90

𝜃_𝑠 𝜃_𝑝1

𝜃_𝑠

𝜎₁ = 𝑚𝑎𝑥 𝜎₂ = 𝑚𝑖𝑛

𝜏₁₂ = 0 𝜏₁₂ = 0

𝜎₁ ≠0 𝜏₁₂ = 𝑚𝑎𝑥 𝜏₂₁ = 𝑚𝑎𝑥 𝜎₂ ≠0

Principal stress

Max. shear stress 𝜏₁₂

𝜏₂₁ 𝜏₂₁

𝜏₁₂

(19)

Principal stress

𝑎𝑡 𝜃_𝑝 𝜎₁ = 𝑚𝑎𝑥 𝜎₂ = 𝑚𝑖𝑛 Principal stress

Differentiation for max or min of 𝜎_𝑥1

(20)

Principal stress

Your homework

(21)

𝜎

_𝑥1

+ 𝜎

_𝑦1

= 𝜎

_𝑥

+ 𝜎

_𝑦

𝜎

₁

+ 𝜎

₂

= 𝜎

_𝑥

+ 𝜎

_𝑦

Finally…

using

Principal angle = Principal stress =

`

Principal stress

(22)

Maximum Shear Stress

Differentiation for max of

𝑎𝑡 𝜃_𝑠 𝜏₁₂ = 𝑚𝑎𝑥 𝜏₂₁ = 𝑚𝑎𝑥 Max. shear stress

(23)

Maximum Shear Stress

𝑎𝑡 𝜃_𝑠 𝜏₁₂ = 𝑚𝑎𝑥 𝜏₂₁ = 𝑚𝑎𝑥 Max. shear stress

Maximum shear stress =

Max shear angle =

(where )

(24)

Max. shear stress angle vs Principal stress angle

𝑎𝑡 𝜃_𝑠 𝜏₁₂ = 𝑚𝑎𝑥

𝜏₂₁ = 𝑚𝑎𝑥

Max. shear stress 𝑎𝑡 𝜃_𝑝

𝜎₁ = 𝑚𝑎𝑥 𝜎₂ = 𝑚𝑖𝑛 Principal stress

(25)

Max. shear stress angle vs Principal stress angle

𝜏₂₁ = 𝑚𝑎𝑥 Max. shear stress

𝑎𝑡 𝜃_𝑝

(26)

Max. shear stress angle vs Principal stress angle

𝜏₂₁ = 𝑚𝑎𝑥 Max. shear stress

𝑎𝑡 𝜃_𝑝

at arbitrary angle 𝜃

(27)

Mohr’s circle

𝜎

𝜏

0

(28)

Mohr’s circle

𝜎

𝜏 0

Point on Circle

𝝉

_𝒙𝒚

𝜎

_𝒙

(29)

Mohr’s circle

𝜎

𝜏 0

Point on Circle

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

𝒚 𝜎_𝑥 + 𝜎_𝑦

2

(30)

Mohr’s circle

𝜎

𝜏 0

Point on Circle

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

2

𝑅

(31)

Mohr’s circle application I

𝜎

𝜏 0

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

2

𝑅

Point on Circle

2𝜃

(32)

Mohr’s circle application I

𝜎

𝜏 0

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

2

𝑅

Point on Circle

2𝜃

𝜎

_𝒙𝟏

𝜎

_𝒚𝟏

𝝉

_{𝒙𝟏𝒚𝟏}

(33)

Point on Circle

Mohr’s circle application II

𝜎

𝜏 0

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

2

𝑅

2𝜃

_𝑝

principal angle 𝜃_𝑝

𝜎

_𝟏

𝜎

𝟐

(34)

Point on Circle

Mohr’s circle application III

𝜎

𝜏 0

𝝉

_𝒙𝒚

𝜎

_𝒙

𝜎

_𝒚

𝜎_𝑥 + 𝜎_𝑦 2

𝑅

2𝜃

_𝑠

max. shear angle 𝜃_𝑠

𝝉

_𝒎𝒂𝒙

𝝉

_𝒎𝒂𝒙

= 2(𝜃

_𝑝+45

)

𝜏_𝑚𝑎𝑥

θ_s

(35)

Mohr’s circle Example

(36)

Mohr’s circle Example