1-24. An element is subject to a normal stress S ~ N (260,202 )psi and a shear stress
T ~ N (900,2 )psi as shown in the figure. If the element is oriented 45 clockwise from its
current position, determine the distribution of the equivalent state of stress. Assume S and T are
independent.
S
T
Solution:
According to the stress diagram, we have
 x  0 ,  y  S ,  xy  T ,   45
 x' 
x  y
2
 
x'

 x  y
2
cos 2   xy sin 2 
 S
260
 T 
 900  1030psi
2
2
2
2
  
 20 
  S    T   
   70   70.7psi
 2 
 2 
2

x'
S S
S
 cos  90   T sin  90  
T
2 2
2
2
Thus,  x ' follows  x ' ~ N (1030,2 )psi .
  x  y
 2
 x' y'   

x' y'

S
2

S
S
sin  90   T cos  90  
 sin 2   xy cos 2 
2
2

  
  S   S  10psi
2
 2 
2
 130psi ,

x' y'
Thus,  x ' y ' follows  x ' y ' ~ N (130,2 )psi .
 y' 
x  y
2

 x  y
2
cos 2   xy sin 2 
S S
S
 cos  90   T sin  90  
T
2 2
2
1
 
y'
 S
260
 T 
 900  770psi
2
2
2
2
  
 20 
    S    T   
   70   70.7psi
 2 
 2 
2
2
y'
Thus,  y ' follows  y ' ~ N (770,2 )psi .
Ans.
2