4 stress transformation 2015 bach khoa

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CHAPTER 4: STRESS TRANSFORMATION

4.1 Introduction

4.2 Plane Stress State

4.3 Transformation of plane stress

4.4 Morh’s circle for plane stress

4.5 Hooke’s Laws

4.6 Transformation of plane strain

4.7 Morh’s circle for plane strain

Stress state at a point is the set of all stresses acting on all faces passing through this point

• The most general state of stress at a point may be represented by 6 components,

) ,

,

: (Note

stresses shearing

, ,

stresses normal

, ,

xz zx

zy yz

yx xy

zx yz xy

z y x

• Same state of stress is represented by a different set

of components if axes are rotated

• The first part of the chapter is concerned with how the components of stress are transformed under a rotation of the coordinate axes The second part of the chapter is devoted to a similar analysis of the transformation of the components of strain

Shear stress: two subscripts

+ First subscript denotes the face on which

the stress acts

+ Second gives the direction on the stress

vector

Positive face (+): normal axis follows the

positive direction of the original axis

Negative face (-): normal axis follows the

negative direction of the original axis

SIGN CONVENTION:

Positive direction (+): stress vector follows

positive direction of the axis

Negative direction (-): stress vector follows

negative direction of the axis

positive direction - positive face = positive stress

negative direction-negative face = positive stress

positive direction-negative face = negative stress

negative direction-negative face = negative stress

4.2 PLANE STRESS STATE

• Plane Stress - state of stress in which two faces of

the cubic element are free of stress For the illustrated example, the state of stress is defined by

0 ,

, y xy and z  zx  zy 



• State of plane stress also occurs on the free surface

of a structural element or machine component, i.e.,

at any point of the surface not subjected to an

external force

4.2 PLANE STRESS STATE

• State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate

cos sin

cos cos

sin cos

0

cos sin

sin sin

sin cos

cos cos

0

A A

A A

A F

A A

A A

A F

xy y

xy x

y x y

xy y

xy x

x x

4.3 TRANSFORMATION OF PLANE STRESS

• The equations may be rewritten to yield

θ is positive if the rotation is counter

clockwise from x to x’

• The previous equations are combined to yield parametric equations for a circle,

2 2

2 2

2

2 2

where

xy

y x

y x

ave

y x ave

max,

90

by separated angles

two defines

: Note

2 2

tan

2 2

y x

xy p

xy y

x y

4.3 TRANSFORMATION OF PLANE STRESS

Maximum Shearing Stresses

2

45

by from

offset

and 90

by separated angles

two defines :

Note

2 2

y x

ave

p

xy

y x

s

xy y

x R

For the state of plane stress shown,

determine (a) the principal panes,

(b) the principal stresses, (c) the

maximum shearing stress and the

corresponding normal stress

SOLUTION:

• Find the element orientation for the principal stresses from

y x

• Determine the principal stresses from

2

2 min

max,

2

y x

333

1 10 50

40 2

2 2

tan

p

y x

xy p

max,

40 30

20

2 2

MPa 70

MPa 40

MPa 50







4.3 TRANSFORMATION OF PLANE STRESS

EXAMPLE 4.01

MPa 10

MPa 40

MPa 50

40 30

• With the physical significance of Mohr’s circle for plane stress established, it may be applied with simple geometric considerations Critical values are estimated graphically or calculated

• The principal stresses are obtained at A and B

y x

xy p

tan

min max,

The direction of rotation of Ox to Oa is the same as CX to CA

• For a known state of plane stress

plot the points X and Y and construct the circle centered at C

xy y

x  

 , ,

2 2

2

y x

y x

4.4 MORH’S CIRCLE FOR PLANE STRESS

• With Mohr’s circle uniquely defined, the state

of stress at other axes orientations may be depicted

• For the state of stress at an angle  with

respect to the xy axes, construct a new diameter X’Y’ at an angle 2 with respect to XY.

• Normal and shear stresses are obtained

from the coordinates X’Y’

• Mohr’s circle for centric axial loading:

0

x A

P  



A

P xy

y x

4.4 MORH’S CIRCLE FOR PLANE STRESS

For the state of plane stress shown,

(a) construct Mohr’s circle, determine

(b) the principal planes, (c) the

principal stresses, (d) the maximum

shearing stress and the corresponding

MPa 30

20 50

MPa

20 2

10 50

FX CF

y x

EXAMPLE 4.02

• Principal planes and stresses

50 20



MPa 70

max 



50 20



MPa 30

30

40 2

4.4 MORH’S CIRCLE FOR PLANE STRESS







EXAMPLE 4.03

For the state of stress shown,

determine (a) the principal planes

and the principal stresses, (b) the

stress components exerted on the

element obtained by rotating the

given element counterclockwise

60 100

2

2 2

R

y x

ave







4.4 MORH’S CIRCLE FOR PLANE STRESS

4

2 20

48 2

max  

6 52 cos 52 80

6 52 cos 52 80

6 52 4

67 60

180

X K

CL OC

OL

KC OC

OK

y x y x









• Stress components after rotation by 30o

Points X’ and Y’ on Mohr’s circle that

correspond to stress components on the

rotated element are obtained by rotating

XY counterclockwise through 2   60 

MPa 3

41

MPa 6

111

MPa 4

48







4.4 MORH’S CIRCLE FOR PLANE STRESS

• Transformation of stress for an element

rotated around a principal axis may be

represented by Mohr’s circle

• The three circles represent the normal and shearing stresses for rotation around each principal axis

• Points A, B, and C represent the

principal stresses on the principal planes

(shearing stress is zero)

min max

Application of Morh’s circle to the Three-Dimensional Analysis of Stress

• In the case of plane stress, the axis perpendicular to the plane of stress is a principal axis (shearing stress equal zero)

c) planes of maximum shearing stress are at 45o to the principal planes

b) the maximum shearing stress for the element is equal to the maximum “in-plane” shearing stress

a) the corresponding principal stresses are the maximum and minimum normal stresses for the element

• If the points A and B (representing the

principal planes) are on opposite sides of the origin, then

4.4 MORH’S CIRCLE FOR PLANE STRESS

Application of Morh’s circle to the Three-Dimensional Analysis of Stress

• If A and B are on the same side of the origin (i.e., have the same sign), then

c) planes of maximum shearing stress are

at 45 degrees to the plane of stress

b) maximum shearing stress for the element is equal to half of the maximum stress

a) the circle defining max, min, and

max for the element is not the circle corresponding to transformations within the plane of stress

• For an element subjected to multi-axial loading, the normal strain components resulting from the stress components may be determined from the

principle of superposition This requires:

1) strain is linearly related to stress 2) deformations are small

E E

E

E E

E

E E

E

z y

x z

z y

x y

z y

x x

• With these restrictions:

GENERALISED HOOKE’S LAW

4.5 HOOKE’S LAW

DILATATION: BULK MODULUS

• Relative to the unstressed state, the change in volume is

     

e) unit volum per

in volume (change

dilatation

2 1

1 1 1

1 1

x

z y x

z y x z

y x

k

p E

p e

• Subjected to uniform pressure, dilatation must be negative, therefore

2 1

0   

SHEARING STRAIN

• A cubic element subjected to a shear stress will

deform into a rhomboid The corresponding shear

strain is quantified in terms of the change in angle between the sides,

zx zx

yz yz

4.5 HOOKE’S LAW

Relation Among E, , and G

• An axially loaded slender bar will elongate in the axial direction and contract in the transverse directions

• Plane strain - deformations of the material

take place in parallel planes and are the same in each of those planes

• Example: Consider a long bar subjected

to uniformly distributed transverse loads State of plane stress exists in any

transverse section not located too close to the ends of the bar

• Plane strain occurs in a plate subjected along its edges to a uniformly distributed load and restrained from expanding or contracting laterally by smooth, rigid and fixed supports

: strain of

components

x y xy z  zx  zy 



4.6 TRANSFORMATION FOR PLANE STRAIN

• State of strain at the point Q results in

different strain components with respect

to the xy and x’y’ reference frames

 

 x y

OB xy

xy y

x OB

xy y

cos sin

sin cos

2 1

2 2

2

sin 2

2

2

sin 2

2

cos 2

2

2

sin 2

2

cos 2

2

xy y

x y

x

xy y

x y

x y

xy y

x y

x x

• The equations for the transformation of plane strain are of the same form as the equations for the transformation of plane

stress - Mohr’s circle techniques apply

• Abscissa for the center C and radius R ,

2 2

2 2

xy p

2 tan

4.7 MORH’S CIRCLE FOR PLANE STRAIN

Three-Dimensional Analysis of Strain

• Previously demonstrated that three principal axes exist such that the perpendicular

element faces are free of shearing stresses

• By Hooke’s Law, it follows that the shearing strains are zero as well and that the principal planes of stress are also the principal planes of strain

• Rotation about the principal axes may be represented by Mohr’s circles

Three-Dimensional Analysis of Strain

• For the case of plane strain where the x and y

axes are in the plane of strain,

- the z axis is also a principal axis

- the corresponding principal normal strain

is represented by the point Z = 0 or the

origin

• If the points A and B lie on opposite sides

of the origin, the maximum shearing strain

is the maximum in-plane shearing strain, D and E

• If the points A and B lie on the same side of

the origin, the maximum shearing strain is out of the plane of strain and is represented

by the points D’ and E’

4.7 MORH’S CIRCLE FOR PLANE STRAIN

Three-Dimensional Analysis of Strain

• Consider the case of plane stress,

• If B is located between A and C on the

Mohr-circle diagram, the maximum

shearing strain is equal to the diameter CA

• Strain perpendicular to the plane of stress

b

b a

a

E

E E

E E

Three-Dimensional Analysis of Strain

• Strain gages indicate normal strain through changes in resistance

3

2 3

2 3

2 2

2

2 2

2 2

1 1

1

2 1

2 1

cos sin

sin cos

cos sin

sin cos

cos sin

sin cos

x

xy y

x

xy y