Short Notes: Principal Stress & Strain | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Principal Stress & Strain  
• Consider a rectangular cross-section and we have to calculate the stress on 
an inclined section as shown in the figure.
Normal stress :
= < 7 COS* d
Stress on an inclined section 
Tangential stress
Resultant stress
o , = — s in Id
a , = P c o s d
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as 
Normal stress:
Page 2


Principal Stress & Strain  
• Consider a rectangular cross-section and we have to calculate the stress on 
an inclined section as shown in the figure.
Normal stress :
= < 7 COS* d
Stress on an inclined section 
Tangential stress
Resultant stress
o , = — s in Id
a , = P c o s d
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as 
Normal stress:
Tangential stress:
a r = r cos 26
Stress simple shear
Stress Induced by Axial Stress and Simple Shear
Normal stress
a = a, c o s ' 6 + < 7, s in " d + r s in 2 #
K l .
Tangential stress
Induced stress body diagram
s in 2 0 - r c o s 2 0
Principal Stresses and Principal Planes
The plane carrying the maximum normal stress is called the major principal plane 
and normal stress is called major principal stress. The plane carrying the minimum 
normal stress is known as minor principal stress.
Principal stress and planes
Major principal stress
Page 3


Principal Stress & Strain  
• Consider a rectangular cross-section and we have to calculate the stress on 
an inclined section as shown in the figure.
Normal stress :
= < 7 COS* d
Stress on an inclined section 
Tangential stress
Resultant stress
o , = — s in Id
a , = P c o s d
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as 
Normal stress:
Tangential stress:
a r = r cos 26
Stress simple shear
Stress Induced by Axial Stress and Simple Shear
Normal stress
a = a, c o s ' 6 + < 7, s in " d + r s in 2 #
K l .
Tangential stress
Induced stress body diagram
s in 2 0 - r c o s 2 0
Principal Stresses and Principal Planes
The plane carrying the maximum normal stress is called the major principal plane 
and normal stress is called major principal stress. The plane carrying the minimum 
normal stress is known as minor principal stress.
Principal stress and planes
Major principal stress
Minor principal stress
(Vl)
c r. — a.
<7, =
a x + <7, J i
~ r " \
(CTj - <7, )*
+ r
2 r
ta n 2 0 = -------------
<7j — <7,
(Jj + (7, = 0” j + O ’,
when 2 dp = 0
=> a , = <7, and <7, = < 7 :
t
- q2 
2
2 6P triangle
Across maximum normal stresses acting in plane shear stresses are zero. 
Computation of Principal Stress from Principal Strain
The three stresses normal to shear principal planes are called principal stress, 
while a plane at which shear strain is zero is called principal strain.
For two dimensional stress system, a3 = 0
„ _ El(£l+M£2) _ _E(M*l+£2)
, : - a z . :
l- /v \ - n
Maximum Shear Stress
The maximum shear stress or maximum principal stress is equal of one half the 
difference between the largest and smallest principal stresses and acts on the 
plane that bisects the angle between the directions of the largest and smallest 
principal stress, i.e., the plane of the maximum shear stress is oriented 45° from 
the principal stress planes.
Page 4


Principal Stress & Strain  
• Consider a rectangular cross-section and we have to calculate the stress on 
an inclined section as shown in the figure.
Normal stress :
= < 7 COS* d
Stress on an inclined section 
Tangential stress
Resultant stress
o , = — s in Id
a , = P c o s d
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as 
Normal stress:
Tangential stress:
a r = r cos 26
Stress simple shear
Stress Induced by Axial Stress and Simple Shear
Normal stress
a = a, c o s ' 6 + < 7, s in " d + r s in 2 #
K l .
Tangential stress
Induced stress body diagram
s in 2 0 - r c o s 2 0
Principal Stresses and Principal Planes
The plane carrying the maximum normal stress is called the major principal plane 
and normal stress is called major principal stress. The plane carrying the minimum 
normal stress is known as minor principal stress.
Principal stress and planes
Major principal stress
Minor principal stress
(Vl)
c r. — a.
<7, =
a x + <7, J i
~ r " \
(CTj - <7, )*
+ r
2 r
ta n 2 0 = -------------
<7j — <7,
(Jj + (7, = 0” j + O ’,
when 2 dp = 0
=> a , = <7, and <7, = < 7 :
t
- q2 
2
2 6P triangle
Across maximum normal stresses acting in plane shear stresses are zero. 
Computation of Principal Stress from Principal Strain
The three stresses normal to shear principal planes are called principal stress, 
while a plane at which shear strain is zero is called principal strain.
For two dimensional stress system, a3 = 0
„ _ El(£l+M£2) _ _E(M*l+£2)
, : - a z . :
l- /v \ - n
Maximum Shear Stress
The maximum shear stress or maximum principal stress is equal of one half the 
difference between the largest and smallest principal stresses and acts on the 
plane that bisects the angle between the directions of the largest and smallest 
principal stress, i.e., the plane of the maximum shear stress is oriented 45° from 
the principal stress planes.
r
, _ o v - a ^
ta n 26, = - a ‘ ^
1
i r \anldB
2 & = 2<9P ± 9 0 ° . & = 0 P- 45°
Principal Strain
For two dimensional strain system,
Where, ex = Strain in x-direction 
ey = Strain in y-direction 
Y xy = Shearing strain relative to OX and OY
20« triangle
O C o n d — — k A
dT
dx
iW)
Maximum Shear Strain:
The maximum shear strain also contains normal strain which is given as
45° Strain Rosette or Rectangular Strain Rosette
Rectangular strains Rosette are inclined 45° to each other
45° strain Rosette
Page 5


Principal Stress & Strain  
• Consider a rectangular cross-section and we have to calculate the stress on 
an inclined section as shown in the figure.
Normal stress :
= < 7 COS* d
Stress on an inclined section 
Tangential stress
Resultant stress
o , = — s in Id
a , = P c o s d
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as 
Normal stress:
Tangential stress:
a r = r cos 26
Stress simple shear
Stress Induced by Axial Stress and Simple Shear
Normal stress
a = a, c o s ' 6 + < 7, s in " d + r s in 2 #
K l .
Tangential stress
Induced stress body diagram
s in 2 0 - r c o s 2 0
Principal Stresses and Principal Planes
The plane carrying the maximum normal stress is called the major principal plane 
and normal stress is called major principal stress. The plane carrying the minimum 
normal stress is known as minor principal stress.
Principal stress and planes
Major principal stress
Minor principal stress
(Vl)
c r. — a.
<7, =
a x + <7, J i
~ r " \
(CTj - <7, )*
+ r
2 r
ta n 2 0 = -------------
<7j — <7,
(Jj + (7, = 0” j + O ’,
when 2 dp = 0
=> a , = <7, and <7, = < 7 :
t
- q2 
2
2 6P triangle
Across maximum normal stresses acting in plane shear stresses are zero. 
Computation of Principal Stress from Principal Strain
The three stresses normal to shear principal planes are called principal stress, 
while a plane at which shear strain is zero is called principal strain.
For two dimensional stress system, a3 = 0
„ _ El(£l+M£2) _ _E(M*l+£2)
, : - a z . :
l- /v \ - n
Maximum Shear Stress
The maximum shear stress or maximum principal stress is equal of one half the 
difference between the largest and smallest principal stresses and acts on the 
plane that bisects the angle between the directions of the largest and smallest 
principal stress, i.e., the plane of the maximum shear stress is oriented 45° from 
the principal stress planes.
r
, _ o v - a ^
ta n 26, = - a ‘ ^
1
i r \anldB
2 & = 2<9P ± 9 0 ° . & = 0 P- 45°
Principal Strain
For two dimensional strain system,
Where, ex = Strain in x-direction 
ey = Strain in y-direction 
Y xy = Shearing strain relative to OX and OY
20« triangle
O C o n d — — k A
dT
dx
iW)
Maximum Shear Strain:
The maximum shear strain also contains normal strain which is given as
45° Strain Rosette or Rectangular Strain Rosette
Rectangular strains Rosette are inclined 45° to each other
45° strain Rosette
Principal strains:
ea • + s i) + \ («i ~ £: )c o s20
< ? » = - (£\ + £ i ) - ^ ( £i - £i ) sin 2 0
ec - £ : ) c o s 2 (9
= \j(ea- ei) 2 + (eb-e ( y-