slides Chapter 3 Stress & Equilibrium

8/20/2019 slides Chapter 3 Stress & Equilibrium

1/14

Chapter 3 Stress and Equilibrium

Body and SurfaceForces

(b) Sectioned Axially Loaded Beam

Surface Forces: T ( x )

S

(a) Cantilever Beam Under Self-Wei!t

Loadin

Body Forces: F( x )

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island


2/14

Traction Vector

P1

P2

P3

p

(Externally Loaded

Body

F

n

A

(Sectioned

Body


A A ∆

∆

= →∆F

n x T

n

0lim),(

3213

321

3211

),(

),(

),(

eeeen x T

eeeen x T

eeeen x T

n

2

n

n

z zy zx

yz y yx

xz xy x

σ+τ+τ==τ+σ+τ==

τ+τ+σ==


3/14

Stress Tensor

Traction on an!blique "lane

x

z

y

n T n


σττ

τστ

ττσ

==

z zy zx

yz y yx

xz xy x

][

3

2

1

)(

)()(

e

e

eT n

z z y yz x xz

z zy y y x xy

z zx y yx x x

nnn

nnnnnn

σ+τ+τ+

τ+σ+τ+τ+τ+σ=

j ji

n

i nT σ=


4/14

Stress Transformation


=

333

222

111

nml

nml

nml

Qij pq jqipij QQ σ=σ′

)()()(

)()()(

)()()(

)(2

)(2

)(2

131313131313131313

323232323232323232

212121212121212121

333333

2

3

2

3

2

3

222222

2

2

2

2

2

2

111111

2

1

2

1

2

1

nl l nmnnml mml nnmml l



l nnmml nml

l nnmml nml

l nnmml nml

zx yz xy z y x zx

zx yz xy z y x yz

zx yz xy z y x xy

zx yz xy z y x z

zx yz xy z y x y

zx yz xy z y x x

+τ++τ++τ+σ+σ+σ=τ′



τ+τ+τ+σ+σ+σ=σ′




5/14

T#o$%imensionalStress Transformation


θθ−

θθ

=

100

0cossin

0sincos

ijQ

)sin(coscossincossin

cossin2cossin

cossin2sincos

22

22

22

θ−θτ+θθσ+θθσ−=τ′

θθτ−θσ+θσ=σ′

θθτ+θσ+θσ=σ′

xy y x xy

xy y x y

xy y x x

θτ+θσ−σ

=τ′

θτ−θσ−σ−σ+σ=σ′

θτ+θσ−σ

+σ+σ

=σ′

2cos2sin2

2sin2cos22

2sin2cos22

xy

x y

xy

xy y x y x

y

xy

y x y x

x


6/14

"rincipal Stresses & %irections

('eneral CoordinateSystem

("rincipal CoordinateSystem


0]det[ 322

1

3=+σ−σ+σ−=σδ−σ I I I ijij 321 ,, σσσ


7/14

Traction Vector Components

Mohr’s Circles of Stress

Admissible N and S valueslie in t!e s!aded area

T nn A

S

N


2/12 )|(| N S

N

−=

⋅=n

n

T

nT

0))((

0))((

0))((

21

2

13

2

32

2

≥σ−σ−+

≤σ−σ−+

≥σ−σ−+

N N S

N N S

N N S


8/14

Example 3$ Stress Transformation



9/14

Spherical) %e*iatoric) !ctahedraland *on +ises Stresses

" " " S#!erical Stress $ensor

" " " %eviatoric Stress $ensor


" " " &cta!edral 'ormaland S!ear Stresses

" " " von ises Stress

ijkk ij δσ=σ3

1~

ijkk ijij δσ−σ=σ3

1ˆ

ijijij σ+σ=σ ˆ~ {

( ) 2/12212/1213232221

1321

623

1])()()[(

3

1

31

31)(

31

I I

I

oct

kk oct

−=σ−σ+σ−σ+σ−σ=τ

=σ=σ+σ+σ=σ

2/12

13

2

32

2

21

2/1222222

])()()[(2

1

)](6)()()[(

2

1ˆˆ

2

3

σ−σ+σ−σ+σ−σ=

τ+τ+τ+σ−σ+σ−σ+σ−σ=σσ=σ=σ zx yz xy x z z y y xijijvonMisese


10/14

Stress %istribution Visuali,ation -sin./$% or 3$% "lots of "articular Contour Lines

• Particular Stress Components

• Principal Stress Components

• Maximum Shear Stress

• von Mises Stress

• Isochromatics (lines of principal stress dierence =constant; same as max shear stress)

• Isoclinics (lines along hich principal stresses haveconstant orientation)

• Isopachic lines (sum of principal stresses =

constant)• Isostatic lines (tangent oriented along a particular

principal stress; sometimes called stresstra!ectories)



11/14

%istribution "lots

%is0 -nder %iametrical

Compression


P

P

(a %is0

"roblem

I

(b +ax Shear Stress

Contours

(1sochromatic

Lines

(c +ax "rincipal

Stress

Contours

I

(d Sum of "rincipal

Stress Contours

(1sopachic

Lines

(e *on +ises

Stress Contours

(f Stress Tra2ectories

(1sostatic Lines


12/14

Equilibrium Equations

F

T n

V

S


⇒=+⇒= ∫∫∫ ∫∫ ∑ 00 V iS n

i dV F dS T F

0

0

0

=+

∂

σ∂+

∂

τ∂+

∂

τ∂

=+∂

τ∂+

∂

σ∂+

∂

τ∂

=+∂τ∂+

∂τ∂+

∂σ∂

z z yz xz

y

zy y xy

x zx yx x

F

z y x

F z y x

F z y x

xz zx

zy yz

yx xy

jiij

τ=τ

τ=τ

τ=τ

⇒σ=σ⇒=×∑ 0F r


13/14

Stress & Traction Components

in Cylindrical Coordinates


σθ

x "

x

x *

r

θ

#

dr

σ #

σr

τr θ τr# τ

θ #

dθ


σττ

τστ

ττσ

=

θ

θθθ

θ

z z rz

z r

rz r r

σ

z z z r rz z

z z r r

z rz r r r r

eeeT

eeeT

eeeT

σ+τ+τ=

τ+σ+τ=

τ+τ+σ=

θθ

θθθθθ

θθ

011

021

0)(

11

=+τ+∂

σ∂+

θ∂τ∂

+∂

τ∂

=+τ+∂τ∂

+θ∂σ∂

+∂τ∂

=+σ−σ+∂

τ∂

+θ∂

τ∂

+∂

σ∂

θ

θθθθθ

θ

θ

z rz z

r z

r r

r

F r z r r

F r z r r

F r z r r

z rz

r

rz r


14/14

Stress & Traction Components in

Spherical Coordinates


σ$

x "

x

x *

$

θ

φ

τ$θ

σφ

σθ

τ$φ

τφθ


σττ

τστττσ

=

θφθθ

φθφφ

θφ

R

R

R R R

σ

θθφφθθθ

θφθφφφφ

θθφφ

σ+τ+τ=

τ+σ+τ=

τ+τ+σ=

eeeT

eeeT

eeeT

R R

R R

R R R R R

0)3cot2(1

sin

11

0]3cot)[(1

sin

11

0)cot2(1

sin

11

=+τ+φτ+θ∂σ∂

φ+

φ∂

τ∂+

∂τ∂

=+τ+φσ−σ+θ∂

τ∂

φ+

φ∂

σ∂+

∂

τ∂

=+φτ+σ−σ−σ+θ∂

τ∂

φ+

φ∂

τ∂+

∂

σ∂

θθφθθφθθ

φφθφφθφφ

φθφ

θφ

F R R R R

F R R R R

F R R R R

R

R

R R R

R

r

r

R R

slides Chapter 3 Stress & Equilibrium

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