slides chapter 3 stress & equilibrium
TRANSCRIPT
-
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
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
2/14
Traction Vector
P1
P2
P3
p
(Externally Loaded
Body
F
n
A
(Sectioned
Body
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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==
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
3/14
Stress Tensor
Traction on an!blique "lane
x
z
y
n T n
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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=
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
4/14
Stress Transformation
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
=
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
nl l nmnnml mml nnmml l
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=τ′
+τ++τ++τ+σ+σ+σ=τ′
τ+τ+τ+σ+σ+σ=σ′
τ+τ+τ+σ+σ+σ=σ′
τ+τ+τ+σ+σ+σ=σ′
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
5/14
T#o$%imensionalStress Transformation
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
θθ−
θθ
=
100
0cossin
0sincos
ijQ
)sin(coscossincossin
cossin2cossin
cossin2sincos
22
22
22
θ−θτ+θθσ+θθσ−=τ′
θθτ−θσ+θσ=σ′
θθτ+θσ+θσ=σ′
xy y x xy
xy y x y
xy y x x
θτ+θσ−σ
=τ′
θτ−θσ−σ−σ+σ=σ′
θτ+θσ−σ
+σ+σ
=σ′
2cos2sin2
2sin2cos22
2sin2cos22
xy
x y
xy
xy y x y x
y
xy
y x y x
x
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
6/14
"rincipal Stresses & %irections
('eneral CoordinateSystem
("rincipal CoordinateSystem
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
0]det[ 322
1
3=+σ−σ+σ−=σδ−σ I I I ijij 321 ,, σσσ
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
7/14
Traction Vector Components
Mohr’s Circles of Stress
Admissible N and S valueslie in t!e s!aded area
T nn A
S
N
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
2/12 )|(| N S
N
−=
⋅=n
n
T
nT
0))((
0))((
0))((
21
2
13
2
32
2
≥σ−σ−+
≤σ−σ−+
≥σ−σ−+
N N S
N N S
N N S
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
8/14
Example 3$ Stress Transformation
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
9/14
Spherical) %e*iatoric) !ctahedraland *on +ises Stresses
" " " S#!erical Stress $ensor
" " " %eviatoric Stress $ensor
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
" " " &cta!edral 'ormaland S!ear Stresses
" " " von ises Stress
ijkk ij δσ=σ3
1~
ijkk ijij δσ−σ=σ3
1ˆ
ijijij σ+σ=σ ˆ~ {
( ) 2/12212/1213232221
1321
623
1])()()[(
3
1
31
31)(
31
I I
I
oct
kk oct
−=σ−σ+σ−σ+σ−σ=τ
=σ=σ+σ+σ=σ
2/12
13
2
32
2
21
2/1222222
])()()[(2
1
)](6)()()[(
2
1ˆˆ
2
3
σ−σ+σ−σ+σ−σ=
τ+τ+τ+σ−σ+σ−σ+σ−σ=σσ=σ=σ zx yz xy x z z y y xijijvonMisese
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
10/14
Stress %istribution Visuali,ation -sin./$% 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)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
11/14
%istribution "lots
%is0 -nder %iametrical
Compression
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
12/14
Equilibrium Equations
F
T n
V
S
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
⇒=+⇒= ∫∫∫ ∫∫ ∑ 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
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
13/14
Stress & Traction Components
in Cylindrical Coordinates
Equilibrium Equations
σθ
x "
x
x *
r
θ
#
dr
σ #
σr
τr θ τr# τ
θ #
dθ
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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
-
8/20/2019 slides Chapter 3 Stress & Equilibrium
14/14
Stress & Traction Components in
Spherical Coordinates
Equilibrium Equations
σ$
x "
x
x *
$
θ
φ
τ$θ
σφ
σθ
τ$φ
τφθ
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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