Chapter 3:Principal Stresses and Strains

Problem 3.8,page no.98

In [1]:
import math

#Given
#Variable declaration
sigma1=100           #Major principal stress in N/sq.mm
sigma2=-60           #Minor principal stress in N/sq.mm
theta=90-50          #Angle of inclination in degrees

#Calculation
sigman=round(((sigma1+sigma2)/2)+(((sigma1-sigma2)/2)*math.cos(math.radians(2*theta))),2)
sigmat=round((sigma1-sigma2)/2*(math.sin(math.radians(2*theta))),3)
sigmaR=round(math.sqrt(sigman**2+sigmat**2),3)
sigmat_max=int((sigma1-sigma2)/2)

#Result
print "Normal stress =",sigman,"N/mm^2"
print "Shear stress =",sigmat,"N/mm^2"
print "Resultant stress =",sigmaR,"N/mm^2"
print "Maximum shear stress =",sigmat_max,"N/mm^2"

Normal stress = 33.89 N/mm^2
Shear stress = 78.785 N/mm^2
Resultant stress = 85.765 N/mm^2
Maximum shear stress = 80 N/mm^2

Problem 3.9,page no.99

In [3]:
import math

#Given
#Variable declaration
sigma1=100        #Major principal stress in N/sq.mm
sigma2=-40        #Minor principal stress in N/sq.mm
theta=90-60       #Angle of inclination in degrees

#Calculation
sigman=((sigma1+sigma2)/2)+(((sigma1-sigma2)/2)*math.cos(math.radians(2*theta)))
sigmat=round((sigma1-sigma2)/2*(math.sin(math.radians(2*theta))),2)
sigmaR=round(math.sqrt(sigman**2+sigmat**2),1)
sigmat_max=int((sigma1-sigma2)/2)
phi=int(math.degrees(math.atan(sigmat/sigman)))

#Result
print "Resultant stress in magnitude =",sigmaR,"N/mm^2"
print "Direction of resultant stress =",phi,"degrees"
print "Maximum shear stress =",sigmat_max,"N/mm^2"

Resultant stress in magnitude = 88.9 N/mm^2
Direction of resultant stress = 43 degrees
Maximum shear stress = 70 N/mm^2

Problem 3.13,page no.111

In [2]:
import math

#Given
#Variable declaration
sigma1=120       #Major tensile stress in N/sq.mm
sigma2=-90       #Minor compressive stress in N/sq.mm
sigma_gp=150     #Greatest principal stress in N/sq.mm

#Calculation
 #case(a):Magnitude of the shearing stresses on the two planes
tau=round(math.sqrt(((sigma_gp-((sigma1+sigma2)/2))**2)-(((sigma1-sigma2)/2)**2)),3)
 #case(b):Maximum shear stress at the point
sigmat_max=int((math.sqrt((sigma1-sigma2)**2+(4*tau**2)))/2)

#Result
print "Shear stress on the two planes =",tau,"N/mm^2"
print "Maximum shear stress at the point =",sigmat_max,"N/mm^2"

Shear stress on the two planes = 84.853 N/mm^2
Maximum shear stress at the point = 135 N/mm^2

Problem 3.16,page no.115

In [4]:
import math

#Given
#Variable declaration
sigma1=600        #Major tensile stress in N/sq.mm
sigma2=300        #Minor tensile stress in N/sq.mm
tau=450           #Shear stress in N/sq.mm
theta1=45         #Angle of inclination in degrees
theta2=135        #Angle of inclination in degrees

#Calculation
sigman1=int(((sigma1+sigma2)/2)+(((sigma1-sigma2)/2)*math.cos(math.radians(2*theta1)))+(tau*math.sin(math.radians(2*theta1))))                                                    
sigman2=int(((sigma1+sigma2)/2)+(((sigma1-sigma2)/2)*math.cos(math.radians(2*theta2)))+(tau*math.sin(math.radians(2*theta2))))                                                         
sigmat1=int(round((sigma1-sigma2)/2*(math.sin(math.radians(2*theta1)))-(tau*math.cos(math.radians(2*theta1))),0))
sigmat2=int(round((sigma1-sigma2)/2*(math.sin(math.radians(2*theta2)))-(tau*math.cos(math.radians(2*theta2))),0))                                                         

#Result
print "Normal stress(when theta is 45 degrees)=",sigman1,"N/mm^2"
print "Normal stress(when theta is 135 degrees)=",sigman2,"N/mm^2"                                                         
print "Tangential stress(when theta is 45 degrees)=",sigmat1,"N/mm^2"
print "Tangential stress(when theta is 135 degrees)=",sigmat2,"N/mm^2"

Normal stress(when theta is 45 degrees)= 900 N/mm^2
Normal stress(when theta is 135 degrees)= 0 N/mm^2
Tangential stress(when theta is 45 degrees)= 150 N/mm^2
Tangential stress(when theta is 135 degrees)= -150 N/mm^2
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