Transcript Document
Analysis of Stress and Strain
Review:
- Torsional shaft
- Axially loaded Bar
ππ
p
ο±
P
ππ
h
q
F P cos2 ο±
ο³ο± ο½ =
Aο±
A
ο΄ο± ο½
V
P cos ο± sin ο±
ο½
Aο±
A
P
ο΄xy
ο‘
ο΄nt
ο³n
ο΄yx
ο³ n ο½ ο΄ xy sin 2ο‘
ο΄ nt ο½ ο΄ xy cos 2ο‘
Questions:
(1) Is there any general method to determine stresses on any arbitrary plane
at one point if the stresses at this point along some planes are known?
(2) For an arbitrary loaded member, how many planes on which stresses are
known are required to determine the stresses at any plane at one point?
Analysis of Stress and Strain
ο³y
State of stress at one point:
Stress element:
y
ο³z
ο΄zy
ο΄yz
ο΄yx
ο΄yx
ο΄xy
ο΄zx ο΄xz
ο³x
ο³x
ο³y
ο΄xy
ο΄xy
ο΄yx
ο³y
z
- Use a cube to represent stress element. It is infinitesimal in size.
- (x,y,z) axes are parallel to the edges of the element
- faces of the element are designated by the directions of their
outward normals.
x
Sign Convention:
- Normal stresses: β+β tension; β-β compression.
- Shear stresses: β+β the directions associated with its subscripts are
plus-plus or minus-minus
β-β the directions associated with its subscripts are
plus-minus or minus-plus
ο³x
Plane Stress
Definition: Only x and y faces are subject to stresses, and all
stresses are parallel to the x and y axes.
ο³ z ο½ ο΄ xz ο½ ο΄ yz ο½ 0
ο³y
1
ο΄x y ο³
x
1 1
1
Stresses on inclined planes
ο΄xy
ο΄x y
1 1
ο±
ο³x
ο΄yx ο³
y
ο³x
1
ο₯F
ο₯F
x
ο½0
y
ο½0
Transformation equations for
plane stress
ο³ x ο½ ο³ x cos2 ο± ο« ο³ y sin 2 ο± ο« 2ο΄ xy sin ο± cos ο±
1
ο΄ x y ο½ ο ο¨ο³ x ο ο³ y ο© sin ο± cos ο± ο« ο΄ xy ο¨ cos2 ο± ο sin 2 ο± ο©
1 1
ο³x ο½
ο³x ο«ο³ y
1
ο΄x y ο½ ο
1 1
2
ο«
ο³ x οο³ y
2
ο³ x οο³ y
2
cos 2ο± ο« ο΄ xy sin 2ο±
sin 2ο± ο« ο΄ xy cos 2ο±
Transformation Equations
ο³x ο½
1
ο³y ο½
ο³ x ο«ο³ y
2
ο³ x ο«ο³ y
1
ο΄x y ο½ ο
1 1
2
ο«
ο
ο³ x οο³ y
2
ο³ x οο³ y
2
ο³ x οο³ y
2
cos 2ο± ο« ο΄ xy sin 2ο±
cos 2ο± ο ο΄ xy sin 2ο±
ο³ x ο«ο³ y ο½ ο³ x ο«ο³ y
1
1
sin 2ο± ο« ο΄ xy cos 2ο±
ο± angle between x1 and x axes, measured counterclockwise
Plane Stress β Special Cases
ο³x
Uniaxial Stress: ο³x
ο΄y
Pure Shear:
ο΄xy
x
ο΄xy
Biaxial Stress:
ο΄yx
ο³y
ο³x
ο³x
ο³y
Plane Stress
Example 1: A plane-stress condition exists at a point on the surface of
a loaded structure, where the stresses have the magnitudes and directions
shown on the stress element of the following figure. Determine the stresses
acting on an element that is oriented at a clockwise angle of 15o with
respect to the original element.
Principal Stresses
Principal stresses: maximum and minimum normal stresses.
Principal planes: the planes on which the principal stresses act
ο³x ο½
ο³x ο«ο³y
ο«
2
1
ο³x οο³ y
2
cos 2ο± ο« ο΄ xy sin 2ο±
ο
dο³ x1
dο±
ο½ο
ο³x οο³ y
2
2sin 2ο± ο« 2ο΄ xy cos 2ο± ο½ 0
ο
tan 2ο± p ο½
2ο΄ xy
ο³x οο³ y
ο± p : The angle defines the orientation of the principal planes.
Principal Stresses
tan 2ππ =
2ππ₯π¦
ππ₯ βππ¦
β cos 2ππ =
ππ₯1
OR
tan 2ππ =
ππ₯ βππ¦
2π
ππ₯ +ππ¦
2
β cos 2ππ = β
ππ₯1
, sin 2ππ =
ππ₯π¦
π
,π
=
ππ₯ βππ¦ 2
2
+ ππ₯π¦ 2
ππ₯ + ππ¦ ππ₯ β ππ¦ ππ₯ β ππ¦
ππ₯π¦
=
+
β
+ ππ₯π¦ β
2
2
2π
π
π1 = ππ₯1 =
2ππ₯π¦
ππ₯ βππ¦
+
ππ₯ βππ¦
2π
ππ₯ βππ¦ 2
2
+ ππ₯π¦ 2
, sin 2ππ = β
ππ₯π¦
π
,π
=
ππ₯ βππ¦ 2
2
ππ₯ + ππ¦ ππ₯ β ππ¦ βππ₯ + ππ¦
βππ₯π¦
=
+
β
+ ππ₯π¦ β
2
2
2π
π
π2 = ππ₯1 =
+ ππ₯π¦ 2
ο³1 ο³ ο³ 2
ππ₯ +ππ¦
2
β
ππ₯ βππ¦ 2
2
+ ππ₯π¦ 2
Shear Stress
Shear stresses on the principal planes:
ο΄x y ο½ ο
ο³ x οο³ y
1 1
2
sin 2ο± p ο« ο΄ xy cos 2ο± p ο½ 0
Example 2: Principal stresses in pure shear case:
ο΄y
ο΄xy
x
ο΄xy
ο΄yx
Maximum Shear Stresses
ο΄x y ο½ ο
1 1
ο³ x οο³ y
2
sin 2ο± ο« ο΄ xy cos 2ο±
dο΄ x1 y1
dο±
ο½ ο ο¨ο³ x ο ο³ y ο© cos 2ο± ο 2ο΄ xy sin 2ο± ο½ 0
ο³ x οο³ y
1
tan 2ο±s ο½ ο
ο tan 2ο±s ο½ ο
2ο΄ xy
tan 2ο± p
tan 2ππ = βtanΞ±
π
2
Let πΌ = β 2ππ
2ππ = β Ξ±
or
ο¦ ο³ οο³ y οΆ
ο³ οο³ 2
ο·ο· ο« ο΄ xy 2 ο½ 1
ο΄ max ο½ ο§ο§ x
2
ο¨ 2 οΈ
2
2ππ = Ο β Ξ±
ο±s ο½ ο± p ο
1
1
ο±s ο½ ο± p ο«
2
1
ο°
4
ο°
4
Plane Stress
Example 3: Find the principal stresses and maximum shear stresses and
show them on a sketch of a properly oriented element.
Mohrβs Circle For Plane Stress β
Equations of Mohrβs Circle
Transformation equations:
ο³x ο½
ο³x ο«ο³ y
1
ο΄x y ο½ ο
1 1
2
ο«
ο³ x οο³ y
2
ο³ x οο³ y
2
cos 2ο± ο« ο΄ xy sin 2ο±
(1)
sin 2ο± ο« ο΄ xy cos 2ο±
ο¦
ο§ ο³ x1 ο
ο¨
(1)2 + (2)2
ο¨ο³
ο³x ο«ο³ y οΆ
2
ο¦ ο³ x οο³ y οΆ
2
2
ο· ο« ο΄ x1 y1 ο½ ο§
οΈ
ο¨
ο©
2
2
2
ο· ο« ο΄ xy
οΈ
ο ο³ ave ο« ο΄ x1 y1 ο½ R 2
2
x1
(2)
ο³ ave ο½
ο³ x ο«ο³ y
2
2
,
ο¦ ο³ x οο³ y οΆ
R ο½ ο§ο§
ο¨
2
2
ο·ο· ο« ο΄ xy 2
οΈ
Two Forms of Mohrβs Circle
ο΄x y
1 1
ο³x
1
ο³x
1
ο΄x y
1 1
Construction of Mohrβs Circle
Approach 1: For the given state of stresses, calculate ο³ ave and R. The center
Of the circle is ( ο³ ave , 0) and the radius is R.
ο³x
1
ο΄x y
1 1
Construction of Mohrβs Circle
Approach 2: Find points corresponding to ο± = 0 and ο± = 90o and then draw a line.
The intersection is the origin of the circle.
ο³x
1
ο΄x y
1 1
Applications of Mohrβs Circle
Example 4: An element in plane stress at the surface of a large machine
is subject to stresses ππ₯ = 15000psi; ππ¦ = 5000psi; ππ₯π¦ = 4000psi
Using Mohrβs circle, determine the following quantities: (a) the stresses
acting on an element inclined at an angle of 40o, (b) the principal stresses
and (c) the maximum shear stress.
Plane Strain
Definition: Only x and y faces are subject to strains, and all
strains are parallel to the x and y axes.
ο₯ z ο½ ο§ xz ο½ ο§ yz ο½ 0
Note: Plane stress and plane strain do not occur simultaneously.
Plane Strain
Transformation Equations:
ο₯x ο½
1
ο₯y ο½
ο₯x ο«ο₯y
2
ο₯x ο«ο₯y
2
1
ο§x y
1 1
2
ο«
ο½ο
ο
ο₯x οο₯y
2
ο₯x οο₯y
ο₯x οο₯y
2
2
cos 2ο± ο«
cos 2ο± ο
sin 2ο± ο«
Principal Strains:
ο§ xy
2
ο§ xy
sin 2ο±
2
ο§ xy
sin 2ο±
2
ο₯x ο«ο₯y ο½ ο₯x ο«ο₯y
1
1
cos 2ο±
ο₯1 ο½
ο₯2 ο½
ο₯x ο«ο₯y
2
ο₯x ο«ο₯y
2
ο¦ ο₯ x ο ο₯ y οΆ ο¦ ο§ xy οΆ
ο·ο· ο« ο§ο§
ο·ο·
ο« ο§ο§
ο¨ 2 οΈ ο¨ 2 οΈ
2
2
ο¦ ο₯ x ο ο₯ y οΆ ο¦ ο§ xy οΆ
ο·ο· ο« ο§ο§
ο·ο·
ο ο§ο§
ο¨ 2 οΈ ο¨ 2 οΈ
2
2