chapter 7 combined stresses and strains.(som-201)
TRANSCRIPT
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
1/40
Vijay
MEC201Mechanics of
Solids
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
2/40
Vijay
Combined
Stresses and
Strains
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
3/40
Vijay
Combined stresses
P
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
4/40
Vijay
Wire over pulley
Section atAA
100 N
500
Dia 1A A
Bending stresses
+
Direct stresses=
Resultant stresses
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
5/40
Vijay
Spirally-woundcylindrical vessel
z
r
zz
nn
nt
Consider a spirally-wound cylindrical vessel of diameter 40 mm. Thevessel is made from a thin copper strip of width 30 mm wound spirally.The edges of the strip are brazed together. The diameter of thecylinder is 40 mm and the wall thickness is 2 mm. What is the shear
strength of the brazed joint if the internal pressure that this cylindercan withstand is 3MPa.
30 mm
(
)
40
mm
= 0.234 rad= 42.2o
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
6/40
Vijay
Spirally-woundcylindrical vessel
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
7/40Vijay
Spirally-woundcylindrical vessel
zz
nn
nt
nn and nt are obtained as 23.2 MPa and = 7.5
MPa, respectively
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
8/40Vijay
Plain stresses
x
z
y
xxxx
yy
yy
yxxy
xyyx
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
9/40Vijay
Two-Dimensional Stress
x
yxxxx
yy
yy
xy
xy
y
x
y
x
x
y
First convert stresses in toforces since m is of forcesnot stresses.
Take depth = 1
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
10/40Vijay
Stresses on Plane
xx
yy
xy
x
y
n
nnnt
+
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
11/40Vijay
Stresses on Plane...
xx
yy
xy
xy
nnnt+
+xy
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
12/40Vijay
Stresses on Plane...
xx
yy
xy
x
y
nnnt
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
13/40Vijay
Stresses on Plane...
xx
yy
xyx
y
nnnt
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
14/40Vijay
Stresses on Plane...
Ifxy= 0
2
n
n'
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
15/40Vijay
Stresses on Plane...
Ifxy 0
X
Y
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
16/40Vijay
( nn , nt )
Stresses on Plane...
X
Y
N
x
y
x
yn
t
2
2
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
17/40
Vijay
Stresses on Plane...
( nn , nt )
X
Y
N2
2x
y
x
yn
t
Plot the stress components on the -plane using the convention thata positive shear stress on an x-like face is plotted downward while apositive stress on a y-like face is plotted upwards. Reverse for thenegative shear stresses
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
18/40
Vijay
X- and Y- Faces
A
B
B
A
A
B
A
B
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
19/40
Vijay
Principal Planes
x
y
X
Y
2
AB a
a,b =
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
20/40
Vijay
Uni-axial Tension
xx
XY
C
D
Ductile Failure
Brittle Failure
C
D
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
21/40
Vijay
Uni-axial Compression
yy
Y X
S1
S2
S1
S2
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
22/40
Vijay
Hydrostatic Loading
1
1
X,Y No shearCompressive stressindependent of direction
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
23/40
Vijay
Equal Tensile Loading
1
1
X,Y
No shear
Tensile stressindependent ofdirection
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
24/40
Vijay
Shear Loading
Z
2 1 Principal Directions
o
o
z
1
2
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
25/40
Vijay
Torsion Failure
ox
y 12
Principal Directions
Torsion failure of aductile material
Torsion failure of abrittle material
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
26/40
Vijay
An Example
x
y
40 kPa
110 kPa
50 kPa
X
Y
CP2
P1
C = (( + )/ , ) = ( , ),110 50 2 0 80 0 R = [{( )/ }110 50 2 2 + 402)]= 50P1= (( + ), ) = ( , ),80 50 0 130 0 P2= (( ), ) = ( , )80 50 0 30 0
P1
P2
12
(c)
2
63 7 MPa1
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
27/40
Vijay
Torsional Pendulum
.63 7 MPa
.41 8 MPa
z
50 N60o
=1 mm
L= 1 m
z
S2
1
Z
S1
S2
2
1
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
28/40
Vijay
Torsional Pendulum
z
50 N60o
z
S2
1
Z
S1
S2
2
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
29/40
Vijay
Displacement and strain
A
D
C
B
D
B
u
u+(u/x)x
v
v+(v/y)y
v+(v/x)x
x,u
y,v
u+(u/y)y
A
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
30/40
Vijay
Transformation of planestrains
y,v
x,uA
B
B
C
ED
X
y
B
A
The coordinates of point B arelcos and lsin
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
31/40
Vijay
Transformation of planestrains
y,v
x,uA
B
B
C
ED
X
y
B
A
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
32/40
Vijay
y
xC
( xx +yy)/2
Mohr circle of strain
X
Y
/2
O
2
Y
X
y'y
x'x
x'y /2
x'
y'
yy
xx
yx
yy
xx
xy
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
33/40
Vijay
An Example:
In a condition of plane stress in a steelplate, xx = 130 MPa, yy = -70 MPa, xy=
80 MPa, Find the principal strains.
y
x
Y
Radius = sqrt(1002
+ 802
)= 128
X
1 = 30 + 128 = 158 MPa
2 = 30 - 128 = -98 MPa
30
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
34/40
Vijay
Example, continued:
In the last example:1 = 158 MPa
2 = -98 MPa
1 = (1 2)/E = (158 0.25(-98))/200103 = 0.910-3
2 = (21)/E = (-98 0.25(158))/200103 = -
0.6910-3
Find principal strains
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
35/40
Vijay
Distortion in computergraphics
x
y1% stretch
5%
com
pression
/2
y
x
S2
S1
Radius = [0.01 (0.05)]/2 = 0.03.
Maximum angular strain S1S2equal to two
times the radius of the circle, i.e., 0.06
e a ons p e ween
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
36/40
Vijay
e a ons p e weenelastic properties E, G
and
12
x
y
y
x
o
x
y 12
X
Y
12
X
Y
12
=o/G 1 = 2 = o
e a ons p e ween
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
37/40
Vijay
e a ons p e weenelastic properties E, G
and yx
x
y
o
=o/G
d d i S i
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
38/40
Vijay
Bonded Resistance StrainGauges
Soldertabs
Alignmentmarks
A
ctive
gridlength
Backing sheet
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
39/40
Vijay
An application
Air Gauge
Cantilevered reed
Modelvocalchord
easur ng s ear
-
8/8/2019 Chapter 7 Combined Stresses and Strains.(SOM-201)
40/40
Vij
easur ng s earstresses with strain
gauges o
z
/2
O
Z
o/2
o/2
P1P2
z
P1P2
= o/2
o/2