STRESSES - STRENGTH OF MATERIAL BY IIT MADRAS GATE Notes | EduRev

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GATE : STRESSES - STRENGTH OF MATERIAL BY IIT MADRAS GATE Notes | EduRev

 Page 1


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Stresses 
 
Stress at a point 
Stress Tensor 
Equations of Equilibrium 
Different states of stress 
Transformation of plane stress 
Principal stresses and maximum shear stress 
Mohr's circle for plane stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Stresses 
 
Stress at a point 
Stress Tensor 
Equations of Equilibrium 
Different states of stress 
Transformation of plane stress 
Principal stresses and maximum shear stress 
Mohr's circle for plane stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction 
2.1 stress at a point  
 
 
Figure 2.1 
 
Consider a body in equilibrium under point and traction loads as shown in figure 2.1. 
After cutting the body along section AA, take an infinitesimal area ?A lying on the surface 
consisting a point C. 
The interaction force between the cut sections 1 & 2, through ?A is ?F. Stress at the point 
C can be defined, 
A0
F
lim
A ??
?
s=
?
 2.1 
?F is resolved into ?F
n
 and ?F
s
 that are acting normal and tangent to ?A.  
Normal stress, 
n
n
A0
F
lim 
A ??
?
s=
?
 
2.2 
Shear Stress, 
??
?
s=
?
s
s
A0
F
lim 
A
 2.3 
 
 
 
 
 
 
 
 
Top 
 
 
 
Page 3


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Stresses 
 
Stress at a point 
Stress Tensor 
Equations of Equilibrium 
Different states of stress 
Transformation of plane stress 
Principal stresses and maximum shear stress 
Mohr's circle for plane stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction 
2.1 stress at a point  
 
 
Figure 2.1 
 
Consider a body in equilibrium under point and traction loads as shown in figure 2.1. 
After cutting the body along section AA, take an infinitesimal area ?A lying on the surface 
consisting a point C. 
The interaction force between the cut sections 1 & 2, through ?A is ?F. Stress at the point 
C can be defined, 
A0
F
lim
A ??
?
s=
?
 2.1 
?F is resolved into ?F
n
 and ?F
s
 that are acting normal and tangent to ?A.  
Normal stress, 
n
n
A0
F
lim 
A ??
?
s=
?
 
2.2 
Shear Stress, 
??
?
s=
?
s
s
A0
F
lim 
A
 2.3 
 
 
 
 
 
 
 
 
Top 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
2.2 stress Tensor 
 
 
 
Figure 2.2 
Consider the free body diagram of an infinitesimally small cube inside the continuum as 
shown in figure 2.2. 
Stress on an arbitrary plane can be resolved into two shear stress components parallel to 
the plane and one normal stress component perpendicular to the plane. 
Thus, stresses acting on the cube can be represented as a second order tensor with nine 
components. 
xx xy xz
yx yy yz
zx zy zz
? ?
ss s
? ?
s= s s s
? ?
? ?
ss s
? ?
? ?
 
2.4 
 
Is stress tensor symmetric?  
 
 
Figure 2.3 
Page 4


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Stresses 
 
Stress at a point 
Stress Tensor 
Equations of Equilibrium 
Different states of stress 
Transformation of plane stress 
Principal stresses and maximum shear stress 
Mohr's circle for plane stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction 
2.1 stress at a point  
 
 
Figure 2.1 
 
Consider a body in equilibrium under point and traction loads as shown in figure 2.1. 
After cutting the body along section AA, take an infinitesimal area ?A lying on the surface 
consisting a point C. 
The interaction force between the cut sections 1 & 2, through ?A is ?F. Stress at the point 
C can be defined, 
A0
F
lim
A ??
?
s=
?
 2.1 
?F is resolved into ?F
n
 and ?F
s
 that are acting normal and tangent to ?A.  
Normal stress, 
n
n
A0
F
lim 
A ??
?
s=
?
 
2.2 
Shear Stress, 
??
?
s=
?
s
s
A0
F
lim 
A
 2.3 
 
 
 
 
 
 
 
 
Top 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
2.2 stress Tensor 
 
 
 
Figure 2.2 
Consider the free body diagram of an infinitesimally small cube inside the continuum as 
shown in figure 2.2. 
Stress on an arbitrary plane can be resolved into two shear stress components parallel to 
the plane and one normal stress component perpendicular to the plane. 
Thus, stresses acting on the cube can be represented as a second order tensor with nine 
components. 
xx xy xz
yx yy yz
zx zy zz
? ?
ss s
? ?
s= s s s
? ?
? ?
ss s
? ?
? ?
 
2.4 
 
Is stress tensor symmetric?  
 
 
Figure 2.3 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Consider a body under equilibrium with simple shear as shown in figure 2.3.  
Taking moment about z axis, 
() ( )
zyxxz y xyyz x
Mddd ddd =t - t =0 
 
xy yx
t =t 
 
Similarly, . 
xz zx yz zy
and t=t t =t
Hence, the stress tensor is symmetric and it can be represented with six 
components, , instead of nine components.  
xx yy zz xy xz yz
, , , , and ss s t t t
xx xy xz xx xy xz
yx yy yz xy yy yz
zx zy zz xz yz zz
???
ss s s t t
???
s= s s s = t s t
???
???
ss s t t s
???
???
?
?
?
?
?
?
 
2.5 
 
 
 
 
 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
Page 5


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Stresses 
 
Stress at a point 
Stress Tensor 
Equations of Equilibrium 
Different states of stress 
Transformation of plane stress 
Principal stresses and maximum shear stress 
Mohr's circle for plane stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction 
2.1 stress at a point  
 
 
Figure 2.1 
 
Consider a body in equilibrium under point and traction loads as shown in figure 2.1. 
After cutting the body along section AA, take an infinitesimal area ?A lying on the surface 
consisting a point C. 
The interaction force between the cut sections 1 & 2, through ?A is ?F. Stress at the point 
C can be defined, 
A0
F
lim
A ??
?
s=
?
 2.1 
?F is resolved into ?F
n
 and ?F
s
 that are acting normal and tangent to ?A.  
Normal stress, 
n
n
A0
F
lim 
A ??
?
s=
?
 
2.2 
Shear Stress, 
??
?
s=
?
s
s
A0
F
lim 
A
 2.3 
 
 
 
 
 
 
 
 
Top 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
2.2 stress Tensor 
 
 
 
Figure 2.2 
Consider the free body diagram of an infinitesimally small cube inside the continuum as 
shown in figure 2.2. 
Stress on an arbitrary plane can be resolved into two shear stress components parallel to 
the plane and one normal stress component perpendicular to the plane. 
Thus, stresses acting on the cube can be represented as a second order tensor with nine 
components. 
xx xy xz
yx yy yz
zx zy zz
? ?
ss s
? ?
s= s s s
? ?
? ?
ss s
? ?
? ?
 
2.4 
 
Is stress tensor symmetric?  
 
 
Figure 2.3 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Consider a body under equilibrium with simple shear as shown in figure 2.3.  
Taking moment about z axis, 
() ( )
zyxxz y xyyz x
Mddd ddd =t - t =0 
 
xy yx
t =t 
 
Similarly, . 
xz zx yz zy
and t=t t =t
Hence, the stress tensor is symmetric and it can be represented with six 
components, , instead of nine components.  
xx yy zz xy xz yz
, , , , and ss s t t t
xx xy xz xx xy xz
yx yy yz xy yy yz
zx zy zz xz yz zz
???
ss s s t t
???
s= s s s = t s t
???
???
ss s t t s
???
???
?
?
?
?
?
?
 
2.5 
 
 
 
 
 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
2.3 Equations of Equilibrium 
 
 
 
Figure 2.4 
 
Consider an infinitesimal element of a body under equilibrium with sides dx as 
shown in figure 2.4.  
  dy  1 × ×
B
x
, B
y
 are the body forces like gravitational, inertia, magnetic, etc., acting on the element 
through its centre of gravity.  
x
F0 =
?
, 
()
() () () ( )
yx
xx
xx y xx yx yx x
dx d 1 dy 1 dy dx 1 dx 1 B dx dy 1 0
xdy
?t ??
?s ??
s+ × -s × +t + × -t × + × × =
??
??
?
??
??
 
Similarly taking  and simplifying, equilibrium equations of the element in 
differential form are obtained as, 
y
F 0 =
?
yx
xx
x
xy yy
y
B0
xy
B0
xy
?t
?s
+ +=
??
?t ?s
+ +=
??
 
2.6 
 
Extending this derivation to a three dimensional case, the differential equations of 
equilibrium become, 
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