BITS Pilani K K Birla Goa Campus VIKAS CHAUDHARI
MECHANICS OF SOLIDS
(ME F211)
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Mechanics of Solids
Chapter-4
Stress and Strain
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Stress and Strain
Objectives
Stress
To know state of stress at a point
To solve for plane stress condition applications
Strain
To know state of strain at a point
To solve for plane strain condition applications
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Stress and Strain
Stress
Stress vector can be defined as
A
FT
A
n
0
)(
limT is force intensity or stress acting on a plane
whose normal is ‘n’ at the point O.
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Stress and Strain
Characteristics of stress
The physical dimensions of stress are force per unit area.
Stress is defined at a point upon an imaginary plane, which
divides the element or material into two parts.
Stress is a vector equivalent to the action of one part of the
material upon another.
The direction of the stress vector is not restricted
Stress vector may be written in terms of its components with
respect to the coordinate axes in the form
kTjTiTT zn
y
n
x
nn )()()()(
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Stress and Strain
Body cut by a plane ‘mm’ passing through point
‘O’ and parallel to y-z plane and
Consider the free body of the left part of the
plane ‘mm’
Divide plane mm in large number of small areas,
i.e. y z
A force F is acting on the small area A (A is
centered on the point O).
F is inclined to the surface mm at some
arbitrary angle.
Figure shows the rectangular components of
the force vector F acting on the small area
centered on point O.
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Stress and Strain
Definition of positive and negative faces
Positive face of given section
If the outward normal points in a positive coordinate direction then that face is called as positive face
Negative face of given section
If the outward normal points in a negative coordinate direction then that face is called as Negative face
Face Positive Negative
x face 1-2-3-4 5-6-7-8
y face 3-4-5-6 1-2-7-8
z face 1-4-5-8 2-3-6-7
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Stress and Strain
Stress components on positive x face
Normal Stress
Shear Stress
Similarly on y face y, yx & yz and on z face z, zx & zy
stress components will exist.
0lim
x
xx
Ax
F
A
0
0
lim
lim
x
x
y
xyA
x
zxz
Ax
F
A
F
A
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Stress and Strain
3-Dimentional State of Stress OR Triaxial State of Stress
x xy xz
yx y yz
zx zy z
A knowledge of the nine stress components is
necessary in order to determine the components
of the stress vector T acting on an arbitrary plane
with normal n.
Stress components acting on the six sides of a parallelepiped.
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Stress and Strain
Plane stress condition
Stress components in the z direction has a very small value compared to the other two directions and moreover they do not vary throughout the thickness.
For example: Thin sheet which is being pulled by forces in the plane of the sheet.
The state of stress at a given point will only depend upon the four stress components.
x xy
yx y
x
y
z
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Stress and Strain
Plane stress condition (Stress components in z direction are zero)
If take the xy plane to be the plane of the sheet, then x , ’x , y ,
’y , xy , ’xy , yx and ’yx will be the only stress components acting
on the element, which is under observation.
x ’x xy
’xy
yx
’yx ’y
y
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Stress and Strain
Equilibrium of a Differential Element in Plane Stress
Stress components in plane stress expressed in terms of partial derivatives.
Following figure must satisfy equilibrium conditions i.e. M = 0 and F = 0
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Stress and Strain
Equilibrium of a Differential Element in Plane Stress
M = 0 is satisfied by taking moments about the center of the element After simplification, we obtain In the limit as x and y go to zero
02 2 2 2
xy yx
xy xy yx yx
x x y yM y z x y z x z y x z k
x x
02 2
xy yx
xy yx
x y
x x
xy yx
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Stress and Strain
Equality of Cross Shears
This Equation says that in a body in plane stress the shear-stress
components on perpendicular faces must be equal in magnitude.
It can also be shown : yz = zy and xz = zx
Definition of positive and negative xy
xy
xy
Positive Shear Negative Shear
xy
xy
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Stress and Strain
Equilibrium of a Differential Element in Plane Stress
F = 0 is satisfied by following two conditions After simplification, we obtain
0
0
yxxx x yx x yx
y xy
y y xy y xy
F x y z y x z y z x zx y
F y x z x y z x z y zy x
0
0
xyx
xy y
x y
x y
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Stress and Strain
Stress Components Associated with Arbitrarily Oriented Faces in Plane Stress
x’ x’y’
x
xy y
M
P N
MP = MN cos
NP = MN sin
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Stress and Strain
Resolve forces in normal and along the oblique plane i.e. along x’ and y’
' ' cos sin sin cos 0x x x xy y xyF MN MP MP NP NP
2 2
'cos sin 2 sin cos
x x y xy
2
cos2-1sin and
2
12coscos 22
' cos2 sin 22 2
x y x y
x xy
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Stress and Strain
Resolve forces in normal and along the oblique plane i.e. along x’ and y’
' ' ' sin cos sin sin 0y x y x xy y xyF MN MP MP NP NP
2 2
' ' ( )sin cos (cos sin )x y y x xy
' ' sin 2 cos22
y x
x y xy
' cos2 sin 22 2
x y x y
y xy
Similarly
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Stress and Strain
Mohr's Circle Representation of Plane Stress
122
2
2
x y
xyR
2
x yOC
x
y
O C
x
y
xy
xy
,
,
x xy
y xy
x
y
xy
xy y
y
x x y
x
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Stress and Strain
Mohr's Circle Representation of Plane Stress
x
y
O C
x
y
xy
xy
,
,
x xy
y xy
x
y
xy
xy y
y
x x y
x
x
x’
y’
x’y’
x’y’
y’
x’
' ' '
' ' '
' ,
' ,
x x y
y x y
x
y
2
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Stress and Strain
Mohr's Circle Representation of Plane Stress
90
22tan
12
1
yx
xy
22
2,1 )2
(2
xy
yxyx
22
max )2
( xyyx
x
y
O C
x
y
xy
xy 1 2
max
max
21
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Stress and Strain
Problem:
Draw the mohr’s circle for following state of stresses
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Problem:
Draw the mohr’s circle for following state of stresses
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Problem:
Draw the mohr’s circle for following state of stresses
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Problem:
Draw the mohr’s circle for following state of stresses
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Problem:
Draw the mohr’s circle for following state of stresses
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Stress and Strain
Problem:
Addition of Two States of stress
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Stress and Strain
Problem:
Addition of Two States of stress
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Stress and Strain
Problem:
Find the principal stress directions if the stress at a point is sum of the two states of stresses as illustrated
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Problem:
Find the principal stress directions if the stress at a point is sum of the two states of stresses as illustrated
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Stress and Strain
Problem:
Find the principal stress and the orientation of the principal axes of stress for the following cases of plane stress.
a. b. c.
x = 40MPa x = 140MPa x = -120MPa
y = 0 y = 20MPa y = 50MPa
xy = 80MPa xy = -60MPa xy = 100MPa
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Stress and Strain
Problem:
For given plane stress state find out normal stress and shear stress at a plane 450 to x- plane. Also find position of principal planes, principal stresses and maximum shear stress. Draw the mohr’s circle and represent all the stresses.
x x
xy
xy y
y
y= 50 MN/m2
x=110 MN/m2
xy= 40 MN/m2
= 450
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Stress and Strain
Problem:
If the minimum principal stress is -7MPa, find x and the angle which the principal axes make with the xy axes for the case of plane stress illustrated
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Stress and Strain
Problem: A rectangular plate is under a uniform state of plane stress in the xy plane. It is known that the maximum tensile stress acting on any face (whose normal lies in the xy plane) is 75MPa. It is also known that on a face perpendicular to the x axis there is acting a compressive stress of 15MPa and no shear stress. No explicit information is available as to the values of the normal stress y, and shear stress xy acting on the face perpendicular to the y axis. Find the stress components acting on the face perpendicular to the a and b axes which are located as shown in the lower sketch. Report your results in an unambiguous sketch.
x
y 15MPa 15MPa
y
y
yx
yx
x
b
a
30o
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Stress and Strain
Analysis of Deformation
n n n nu u i u j u k
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Displacement of Continuous Body
a. Rigid-body translation
b. Rigid-body rotation about c
c. Deformation without rigid-
body motion
d. Sum of all the displacements
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Stress and Strain
Definition of Strain Components
(b) Uniform Strain (c) Non-uniform Strain
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Stress and Strain
Strain
Normal Strain
Shear Strain
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Stress and Strain
Plane Strain (stain components in z direction are zero)
0
' 'limxx
O C OC
OC
0
' 'limyy
O E OE
OE
0 00 0
lim ' ' ' lim ' ' '2
xyx xy y
COE C O E C O E
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Stress and Strain
Relation between Strain and Displacement in Plane Strain
The x and y components of the displacement of point O are indicated by u and v.
u and v must be continuous functions of x and y to ensure that the displacement be geometrically compatible i.e. no hole or void are created by the displacement.
Using the concept of partial derivatives, the displacement of point E and C can be expressed as shown in figure.
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Stress and Strain
Relation between Strain and Displacement in Plane Strain
; ; x y xyu v v u
x y x y
x xy
yx y
xy yx
0 0
' 'lim limxx x
x u x x xO C OC u
OC x x
0 0
' 'lim limyy y
y v y y yO E OE v
OE y y
0 00 0
lim ' ' ' lim2 2 2
xyx xy y
v x x u y y v uC O E
x y x y
Plain strain condition where
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Strain Components Associated with Arbitrary Sets of Axes
' ' ' '
' ' ' '; ;
' ' ' 'x y x y
u v v u
x y x y
It could be possible to express these displacement components either as functions of x' and y' or as functions of x and y.
Vikas Chaudhari BITS Pilani, K K Birla Goa Campus
Stress and Strain
Strain Components Associated with Arbitrary Sets of Axes
'
'
' '
' ' '
' ' '
' ' '
' ' '
' ' ' ' ' '
' ' ' ' ' '
x
y
x y
u u x u y
x x x y x
v v x v y
y x y y y
v u v x v y u x u y
x y x x y x x y y y
'cos 'sin
'sin 'cos
' cos sin
' sin cos
x x y
y x y
u u v
v u v
By Chain Rule of Partial Derivatives
From geometry, following relations are obtained
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Stress and Strain
Strain Components Associated with Arbitrary Sets of Axes
'
2 2
'
2 2
'
cos sin cos cos sin sin
cos sin sin cos
cos sin sin cos
x
x
x x y xy
u v u v
x x y y
u v v u
x y x y
'
'
' '
cos 2 sin 22 2 2
cos 2 sin 22 2 2
sin 2 cos 22 2 2
x y x y xy
x
x y x y xy
y
x y y x xy
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Stress and Strain
Mohr's Circle Representation of Plane strain
x
y
O C
x
y
,2
,2
xy
x
xy
y
x
y
x xy
xy y
2
2
xy
2
xy
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Stress and Strain
Mohr's Circle Representation for any arbitrary Plane
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Stress and Strain
Problem:
A sheet of metal is deformed uniformly in its own plane so that the
strain components related to a set of axes xy are
x = -200 X 10-6
y = 1000X 10-6
xy = 900 X 10-6
find the strain components associated with a set of axes x'y‘ inclined
at an angle of 30° clockwise to the xy set. Also to find the principal
strains and the direction of the axes on which they exist.
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Stress and Strain
References
1. Introduction to Mechanics of Solids by S. H. Crandall et al
(In SI units), McGraw-Hill
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