Transformation of Stress | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


ANALYSIS OF STRESS 
• For this topic, the stresses to be considered 
are not on the perpendicular and parallel 
planes only but also on other inclined planes. 
x 
z 
y 
b 
A 
a 
b a 
P 
P 
(A body subjected to load P) 
Page 2


ANALYSIS OF STRESS 
• For this topic, the stresses to be considered 
are not on the perpendicular and parallel 
planes only but also on other inclined planes. 
x 
z 
y 
b 
A 
a 
b a 
P 
P 
(A body subjected to load P) 
• On plane a-a, normal force, N  produces normal 
stress. 
• On plane b-b, normal force, N produces normal 
stress and shear force, V produces shearing stress. 
ANALYSIS OF STRESS 
P 
P 
N = P cos ? 
V = P sin ? 
? 
? 
Area = A/cos ? 
?? =
?? ?? =
?? ?? 
Page 3


ANALYSIS OF STRESS 
• For this topic, the stresses to be considered 
are not on the perpendicular and parallel 
planes only but also on other inclined planes. 
x 
z 
y 
b 
A 
a 
b a 
P 
P 
(A body subjected to load P) 
• On plane a-a, normal force, N  produces normal 
stress. 
• On plane b-b, normal force, N produces normal 
stress and shear force, V produces shearing stress. 
ANALYSIS OF STRESS 
P 
P 
N = P cos ? 
V = P sin ? 
? 
? 
Area = A/cos ? 
?? =
?? ?? =
?? ?? 
ANALYSIS OF STRESS 
?? ?? =
?? ???????? ?? ???????? =
?? ?? ??????
2
?? = ?? ?? ??????
2
?? 
?? =
?? ???????? ?? ???????? = -
?? ?? ???????? ???????? = -?? ?? ???????? ???????? 
Page 4


ANALYSIS OF STRESS 
• For this topic, the stresses to be considered 
are not on the perpendicular and parallel 
planes only but also on other inclined planes. 
x 
z 
y 
b 
A 
a 
b a 
P 
P 
(A body subjected to load P) 
• On plane a-a, normal force, N  produces normal 
stress. 
• On plane b-b, normal force, N produces normal 
stress and shear force, V produces shearing stress. 
ANALYSIS OF STRESS 
P 
P 
N = P cos ? 
V = P sin ? 
? 
? 
Area = A/cos ? 
?? =
?? ?? =
?? ?? 
ANALYSIS OF STRESS 
?? ?? =
?? ???????? ?? ???????? =
?? ?? ??????
2
?? = ?? ?? ??????
2
?? 
?? =
?? ???????? ?? ???????? = -
?? ?? ???????? ???????? = -?? ?? ???????? ???????? 
• On an element, there are 6 components of stress: 
?
x
 , ?
y 
, ?
z
 , ?
xy
 , ?
yz
 , and ?
zx
  
• If the z axis is not considered or the stresses are 
independent with the z axis, then there exist only 
stresses in the x and y directions.  
x 
z 
y 
ANALYSIS OF STRESS 
Page 5


ANALYSIS OF STRESS 
• For this topic, the stresses to be considered 
are not on the perpendicular and parallel 
planes only but also on other inclined planes. 
x 
z 
y 
b 
A 
a 
b a 
P 
P 
(A body subjected to load P) 
• On plane a-a, normal force, N  produces normal 
stress. 
• On plane b-b, normal force, N produces normal 
stress and shear force, V produces shearing stress. 
ANALYSIS OF STRESS 
P 
P 
N = P cos ? 
V = P sin ? 
? 
? 
Area = A/cos ? 
?? =
?? ?? =
?? ?? 
ANALYSIS OF STRESS 
?? ?? =
?? ???????? ?? ???????? =
?? ?? ??????
2
?? = ?? ?? ??????
2
?? 
?? =
?? ???????? ?? ???????? = -
?? ?? ???????? ???????? = -?? ?? ???????? ???????? 
• On an element, there are 6 components of stress: 
?
x
 , ?
y 
, ?
z
 , ?
xy
 , ?
yz
 , and ?
zx
  
• If the z axis is not considered or the stresses are 
independent with the z axis, then there exist only 
stresses in the x and y directions.  
x 
z 
y 
ANALYSIS OF STRESS 
• This state of 2-dimensional stress is known as 
plane stress. 
   ?
z
 = ?
xz
 = ?
yz
 = 0  
All directions shown (axes and on 
element) are taken as positive. 
x 
y’ 
y 
x’ 
? 
?
x
 
? 
a 
a 
?
y
 
?
x
 
?
y
 
?
xy
 
ANALYSIS OF STRESS UNDER 2D