Week 6_ Principal Stresses and Strains

7/23/2019 Week 6_ Principal Stresses and Strains

1/26

ENEM 14012 Principal stresses andstrains

Ref Beer 2012


2/26

Introduction

In 3D case, stressat a point has sico!ponents

x y z

xy yz zx


3/26

Introduction

"o# theco!ponents of

stresses aretransfor!ed undera rotation of aes

$a!e is the case#ith strains


4/26

%ransfor!ation of stress plane

stress

2D casex

y xy

xy

xy

xy

y

x

0z zx yz = = =

, ,x y xy


5/26

Plane stress

& thin plate

%hic'ness is s!allco!pared to other

di!ension Plane stress(

independent of )and unifor! o*erthic'ness

, ,x y xy


6/26

Plane stress


7/26

%ransfor!ation of Planestress


8/26


E+uiliriu! e+uations(2 2

2 2

2 2

2

2

0 : cos sin 2 sin cos

0 : ( ) sin cos (cos sin )

sin 2 2 sin cos

cos 2 cos sin

1 cos 2cos2

1 cos 2sin

2

x x x y xy

y x y x y xy

F

F

= = + +

= = +

=

=

+=

=


9/26


Puttin- tri-ono!etric relations

cos 2 sin 22 2

sin 2 cos 22

cos 2 sin 2 , 902 2

x y x y

x xy

x y

x y xy

x y x y o

y xy

x y x y C

+

= + +

= +

+ = = +

+ = + =


10/26

Principal stresses and !a shearin-

stress

It is a circle

2 2 2 2

2 2

2 2 2

( ) ( )2 2

( )2 2

( )

x y x y

x x y xy

x y x y

ave xy

x ave x y

if and R

R

+

+ = +

+ = = +

+ =


11/26

Principal stresses


12/26

Principal stresses

Ma *alue of nor!alstress

sin 2 cos 2 02

2tan2

x y

x y xy

xy

P

x y

= + =

=


13/26

Principal stresses

%he e+uation de.nest#o an-les that areapart

/or , apart

Principal stresses

Principal planes, no

shear stress in thisplane

180o

P 90o


14/26

Principal stresses

Ma and !innor!al stresses

max min

2 2

max,min ( )2 2

ave ave

x y x y

xy

R and R

= + =

+ = +


15/26

Ma shearin- stress

Point D and E

rdinate is !ashearin- stress

&scissa is

2

x y

ave

+=


16/26

Ma shearin- stress

2

cos 2 sin 22 2

x yx

x y x y

x xy

+=

+ = + +

2

x y

ave

+

=


17/26

Ma shearin- stress

2 2

max

cos 2 sin 2 02

tan2

2

( )2

,

2

x y

S xy S

x y

S

xy

x y

xy

x y

ave

R

normal stress

+ =

=

= = +

+ = =


18/26

rientation of ele!ent

/or !a shear and!a nor!al stress

tan 2

2

x y

Sxy

=

2tan 2

xy

P

x y

=


19/26

rientation of ele!ent

Relati*e orientation

1tan 2

tan 2

sin 2 cos20

cos 2 sin 2

sin 2 sin 2 cos 2 cos 2 0

cos(2 2 ) 0

2 2 90

45

S

P

s P

s P

s P s P

s P

o

s P

o

S P

=

+ =

+ =

=

=

=


20/26

Mohrs circle( plane stress

%he circle #e sa# is Mohr circle

er!an en-ineer tto Mohr 13561718

%he circle can e used as analternati*e solution of prole!

Is ased on -eo!etric consideration,not re+uired the use of !an9for!ulas


21/26

Mohrs circle( plane stress


22/26

Principal stress and !ai!u! shear

stress


23/26

%hin6#alled pressure *essels

&pplication of planestress

:all o;ers little

resistance to endin- Internal forces on the

#all are tan-ent tothe #all surface

%9pes( c9lindrical andspherical pressure*essels


24/26

%hin6#alled pressure *essels


25/26

$tresses on the #all

Hoop stress Longitudinal stress

1

1

0 : (2 ) (2 ) 0ZF t x p r x

pr

t

= =

=

22

2

1 2

0 : (2 ) 0

2

2

x

F rt p r

pr

t

= =

=

=

r10


26/26

$phere

1 2

1

max

2

2 4

pr

tpr

t

= =

= =

Week 6_ Principal Stresses and Strains

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