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Mechanics in Energy Resources Engineering - Chapter 7 Analysis of Stress and Strain

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Week 10, 3 May

Mechanics in Energy Resources Engineering - Chapter 7 Analysis of Stress and Strain

Ki B k Mi PhD Ki-Bok Min, PhD

Assistant Professor

E R E i i

Energy Resources Engineering Seoul National University

(2)

schedule

• Ch.7 Analysis of Stress and Strain

– 3 May, 10 May, 12 May3 May, 10 May, 12 May

• Ch.8 Application of Plane Stress

– 17 May, 19 May

• Ch 9 Deflection of BeamsCh.9 Deflection of Beams

– 24 May, 26 May, 31 May

• Ch.10 Statically Indeterminate Beams

– 2 June, 7 June2 June, 7 June

• Final Exam: 9 June

(3)

Outline

• Introduction

• Plane Stress

• Plane Stress

• Principal Stresses and Maximum Shear Stresses

• Mohr’s Circle for Plane Stress H k ’ L f Pl St

• Hooke’s Law for Plane Stress

• Triaxial Stress

• Plane Strain

(4)

Introduction

• Stresses in cross section

• Stresses in inclined section: larger stresses may occur

Finding the normal and shear stresses acting on inclined section is – Finding the normal and shear stresses acting on inclined section is

necessary

M i t t f Ch 5!

– Main content of Ch.5!

?

(5)

Introduction

• We have already learned this!

– Uniaxial Stress & Stresses in inclined sectionUniaxial Stress & Stresses in inclined section

2 1

cos 1 cos 2

x 2 x

sin cos sin 2 2

x

x

   

– Pure Shear & Streses in inclined section

cos 2

sin 2

cos 2

(6)

Introduction

• ONE instrinsic state of stress can be expressed in many many different ways depending on the reference axis (or orientation of element).

– Similarity to force: One intrinsic state of force (vector) can be y ( ) expressed similarly depending on the reference axis.

– Difference from force: we use different transformation equations q from those of vectors

– Stress is NOT a vector BUT a (2S ess s O a ec o U a ( nd order) tensor o de ) e so they do not ey do o combine according to the parallelogram law of addition

(7)

Plane Stress D fi iti

Definition

• Plane Stress: Stresses in 2D plane

• Normal stress σ : subscript identify the face on which the

• Normal stress, σ : subscript identify the face on which the stress act. Ex) σx

Sh t 1 t b i t d t th f hi h th

• Shear stress, τ : 1st subscript denotes the face on which the stress acts, and the 2nd gives the direction on that face. Ex) τxy

Positive x face Positive y face

(8)

Plane Stress D fi iti

Definition

• Sign convention

– Normal stress: tension (+), compression (-)Normal stress: tension ( ), compression ( ) – Shear stress:

acts on a positive face of an element in the positive direction of an axis (+) :

acts on a positive face of an element in the positive direction of an axis (+) : plus-plus or minus-minus

acts on a positive face of an element in the negative direction of an axis (-):

acts on a positive face of an element in the negative direction of an axis ( ):

plus-minus or minus-plus

Positive normal & shear stresses

(9)

Plane Stress D fi iti

Definition

• Shear stresses in perpendicular planes are equal in magnitude and directions shown in the below.

– Derived from the moment equilibrium

xy yx

• In 2D (plane stress) we need three components to describe a

• In 2D (plane stress), we need three components to describe a complete state of stress

x xy

x y xy y

yx y

(10)

Plane Stress

St i li d ti

Stresses on inclined sections

• Stresses acting on inclined sections assuming that σx, σy, τxy are known.

– x1y1 axes are rotated counterclockwise through an angle θ – Strategy??? 

– Strategy??? 

– wedge shaped stress element

(11)

Plane Stress

St i li d ti

Stresses on inclined sections

Free Body Diagram express in terms of “Force”

• Force Equilibrium Equations in x1 and y1 directions

i

F A A A

1 1 0 0 0

0 0

sec cos sin

tan sin tan cos 0

x x x xy

y yx

F A A A

A A

   

 

i

F A A A

1 1 1 0 0 0

0 0

sec sin cos

tan cos tan sin 0

y x y x xy

y yx

F A A A

A A

   

 

(12)

Plane Stress

St i li d ti

Stresses on inclined sections

• Using xy yx and simplifying

2 2

cos sin 2 sin cos

 

1 cos sin 2 sin cos

x x y xy

 

   

1 1

2 2

sin cos cos sin

x y x y xy

   

– When θ = 0,

   

1 1y y y

x1 x

x y1 1 xy

– When θ = 90,

x1 x x y1 1 xy

x1 y

x y1 1 xy

 

(13)

Plane Stress

T f ti E ti

Transformation Equations

• From half angle and double angle formulas

2 1

cos (1 cos 2 )

2 sin2 1(1 cos 2 )

2 1

sin cos sin 2

• Transformation equations for plane stress

2 2( ) s cos s

2

1 cos 2 sin 2

2 2

x y x y

x xy

 

1 1 sin 2 cos 2

2

x y

x y xy

   

– Intrinsic state of stress is the same but the reference axis are different

– Derived solely from equilibrium applicable to stresses in any kind of materials (linear or nonlinear or elastic or inelastic)

(14)

Plane Stress

T f ti E ti

Transformation Equations

With σy=0.2σx & τxy=0.8 σx With σy 0.2σx & τxy 0.8 σx

F θ  θ 90

• For σy1, θ  θ + 90,

– Making summations 1

cos 2 sin 2

2 2

x y x y

y xy

 

S f th l t ti di l f f l

1 1

x y x y

– Sum of the normal stresses acting on perpendicular faces of plane stress elements is constant and independent of θ

(15)

Plane Stress

S i l C f Pl St

Special Cases of Plane Stress

• Uniaxial stress

1 cos 2

x

x sin 2

• Pure Shear

1 1 cos 2

x 2

1 1 sin 2 2

x

x y  

Bi i l St

1 sin 2

x xy

1 1 cos 2

x y xy

• Biaxial Stress

1 cos 2

2 2

x y x y

x

1 2 2

x

1 1 sin 2

2

x y

x y

 

2

(16)

Plane Stress E l 7 1 Example 7-1

• Determine the stress acting on an element inclined at an angle θ = 45°

(17)

Plane Stress E l 7 1 Example 7-1

90°

(18)

Principal Stresses and Maximum Shear St

Stresses

(19)

Mohr’s Circle for Plane Stress

(20)

Hooke’s Law for Plane Stress

(21)

Triaxial Stress

(22)

Plane Strain

(23)

Outline

• Introduction

• Plane Stress

• Plane Stress

• Principal Stresses and Maximum Shear Stresses

• Mohr’s Circle for Plane Stress H k ’ L f Pl St

• Hooke’s Law for Plane Stress

• Triaxial Stress

• Plane Strain

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