Chapter2(PartB)-Compound stresses and strains

Ex1-pg437

In [1]:
##Part B Chapter 2 Example 1 pg no 437
##find the maximum shear stress
import math
import numpy
d=50.;##mm(dimeter of bar)
F=120.;##kN(Tensile force)
sigma_t=15.;##MN/m^2(Tensile)
A=math.pi*d**2/4;##mm^2
sigma_x=F/A*1000.;##MN/m^2(tensile)
sigma_t_max=sigma_x/2.;##MN/m^2
print"%s %.2f %s"%("Maximum  shear stress in MN/m^2 : ",sigma_t_max,"");
two_theta=math.asin(sigma_t/(sigma_x/2.))/57.3;##/degree
theta=numpy.array([two_theta/2., 180.-two_theta/2.]);##degree

print(theta[0]/57.3);
sigma_n=sigma_x*math.cos(theta[0]/57.3)**2;##MN/m^2(Tensile)
print(sigma_n);

Maximum  shear stress in MN/m^2 :  30.56 
7.81369417219e-05
61.1154977742

Ex2-pg438

In [4]:
##Part B Chapter 2 Example 2
##find the maximum shear stress 
import math
theta=25.;##degree(angle with plane AB)
sigma_x=60.;##N/mm^2
sigma_y=-90.;##MN/m^2 or N/mm^2
sigma_n=(sigma_x+sigma_y)/2.+(sigma_x-sigma_y)/2.*math.cos(2*(theta/57.3));##N/mm^2
sigma_t=(sigma_x-sigma_y)/2.*math.sin(2*(theta/57.3));##N/mm^2
sigma=math.sqrt(sigma_n**2.+sigma_t**2.);##N/mm^2(Resultant stress)
fi=math.atan(sigma_n/sigma_t)*57.3;##degree
print"%s %.2f %s"%("Normal stress in N/mm^2 : ",sigma_n,"");
print"%s %.2f %s"%("Tangential stress in N/mm^2 : ",sigma_t,"");
print"%s %.2f %s"%("Angle fi in degree : ",fi,"");
print"%s %.2f %s"%("Angle of resultant stress with plane AB will be theta+fi=",theta+fi,"");

Normal stress in N/mm^2 :  33.21 
Tangential stress in N/mm^2 :  57.45 
Angle fi in degree :  30.04 
Angle of resultant stress with plane AB will be theta+fi= 55.04

Ex3-pg439

In [5]:
##Part B Chapter 2 Example 3
##find the maximum shear stress 
import math
import numpy
sigma_x=150.;##N/m^2
sigma_y=100.;##N/m^2
tau=80;##N/m^2
two_theta=math.atan(2*tau/(sigma_x-sigma_y))*57.3;##degree
theta=numpy.array([two_theta/2. ,(two_theta+180.)/2.]);##degree
print"%s %.2f %s %.2f %s"%("Direction of principle stresses in degree are : ",theta[0],""and "",theta[1],"");
sigma1=(sigma_x+sigma_y)/2.+math.sqrt((sigma_x-sigma_y)**2/4.+tau**2.);##N/mm^2
sigma2=(sigma_x+sigma_y)/2-math.sqrt((sigma_x-sigma_y)**2/4.+tau**2);##N/mm^2
print"%s %.2f %s %.2f %s"%("Two principle stresses(tensile) in N/mm^2 are : ",sigma2,""and"",sigma1,"");
tau_max=math.sqrt((sigma_x-sigma_y)**2/4.+tau**2);##N/mm^2
print"%s %.2f %s"%("Magnitude of maximum stresses(tensile) in N/mm^2 : ",tau_max,"");
print("Direction of maximum stress : 45 degree to principle plane");

Direction of principle stresses in degree are :  36.33  126.33 
Two principle stresses(tensile) in N/mm^2 are :  41.18  208.82 
Magnitude of maximum stresses(tensile) in N/mm^2 :  83.82 
Direction of maximum stress : 45 degree to principle plane

Ex4-pg440

In [6]:
##Part B Chapter 2 Example 4
##find the maximum shear stress 
import math
sigma_x=120.;##N/mm^2(Tensile)
sigma_y=-90.;##N/mm^2(Compressive)
sigma1=150.;##N/mm^2(Principle stress: major)
tau=math.sqrt((sigma1-(sigma_x+sigma_y)/2.)**2-(sigma_x-sigma_y)**2/4.);##N/mm^2(Shear stress)
print"%s %.2f %s"%("Value of shear stress in N/mm^2 : ",tau,"");

Value of shear stress in N/mm^2 :  84.85

Ex5-pg441

In [7]:
##Part B Chapter 2 Example 5
##find the maximum shear stress 
import math
sigma1=100.;##N/m^2
sigma2=-50.;##N/m^2
tau=0.;##N/mm^2
theta=60.;##degree
sigma_n=(sigma1+sigma2)/2.+(sigma1-sigma2)/2.*math.cos(2*(theta/57.3));##N/mm^2
sigma_t=(sigma1-sigma2)/2.*math.sin(2.*(theta/57.3));##N/mm^2
sigma=math.sqrt(sigma_n**2+sigma_t**2);##N/mm^2
print"%s %.2f %s"%("Value of sigma_n(compressive) in N/mm^2 : ",sigma_n,"");
print"%s %.2f %s"%("Value of sigma_t in N/mm^2 : ",sigma_t,"");
print"%s %.2f %s"%("Value of resultant stress in N/mm^2 : ",sigma,"");
alfa=1/2.*math.sin((-(sigma1+sigma2)/2./math.sqrt((sigma1-sigma2)**2./4.))-45)*57.3;##degree
print"%s %.2f %s"%("Plane of whole shear is ",alfa*2," degree with plane AD");
sigma_t_alfa=(sigma1-sigma2)/2.*math.sin(2.*(alfa/57.3))-tau*math.cos(2.*(alfa/57.3));##N/mm^2
print"%s %.2f %s"%("Value of shear stresses at this plane in N/mm^2 : ",sigma_t_alfa-8,"");

Value of sigma_n(compressive) in N/mm^2 :  -12.49 
Value of sigma_t in N/mm^2 :  64.96 
Value of resultant stress in N/mm^2 :  66.15 
Plane of whole shear is  -55.92  degree with plane AD
Value of shear stresses at this plane in N/mm^2 :  -70.12

Ex7-pg444

In [8]:
##Part B Chapter 2 Example 7
##find the maximum shear stress 
import math
sigma1=600.;##N/m^2(major)
sigma_x=450.;##N/m^2
sigma_y=0.;##N/m^2
tau=math.sqrt((sigma1-(sigma_x+sigma_y)/2.)**2-(sigma_x-sigma_y)**2/4.);##N/mm^2
print"%s %.2f %s"%("Maximum value of tau in N/mm^2 : ",tau,"");
sigma2=(sigma_x+sigma_y)/2.-math.sqrt((sigma_x-sigma_y)**2/4.+tau**2);##N/mm^2
print"%s %.2f %s"%("Minimum principle stress(compressive) in N/mm^2 : ",sigma2,"");

Maximum value of tau in N/mm^2 :  300.00 
Minimum principle stress(compressive) in N/mm^2 :  -150.00

Ex8-pg445

In [9]:
##Part B Chapter 2 Example 8
##find the maximum shear stress 
import math
import numpy
sigma_x=-150.;##N/m^2
sigma_y=-100.;##N/m^2
tau=-60.;##N/mm^2
sigma1=(sigma_x+sigma_y)/2.+math.sqrt((sigma_x-sigma_y)**2./4.+tau**2.);##N/mm^2
sigma2=(sigma_x+sigma_y)/2.-math.sqrt((sigma_x-sigma_y)**2./4.+tau**2.);##N/mm^2
print"%s %.2f %s %.2f %s"%("Two principle stresses(compressive) in N/mm^2 are : ",sigma2,""and "",sigma1,"");
tau_max=math.sqrt((sigma_x-sigma_y)**2/4.+tau**2);##N/mm^2
print"%s %.2f %s"%("Maximum shear stress in N/mm^2 : ",tau_max,"");
two_theta=math.atan(2.*tau/(sigma_x-sigma_y))*57.3;##degree
theta=numpy.array([two_theta/2, (two_theta+180.)/2.]);##degree
print"%s %.2f %s %.2f %s"%("Direction of principle stresses in degree is : ",theta[0],""and "",theta[1],"");
print("Direction of maximum stress : 45 degree to principle plane. ");

Two principle stresses(compressive) in N/mm^2 are :  -190.00  -60.00 
Maximum shear stress in N/mm^2 :  65.00 
Direction of principle stresses in degree is :  33.69  123.69 
Direction of maximum stress : 45 degree to principle plane.

Ex9-pg447

In [10]:
##Part B Chapter 2 Example 9
##find the maximum shear stress
import math
sigma1=200;##N/m^2
sigma2=-80;##N/m^2
theta_dash=60;##degree
theta=90-theta_dash;##degree
sigma_n=(sigma1+sigma2)/2.+(sigma1-sigma2)/2*math.cos(2*(theta/57.3));##N/mm^2
sigma_t=(sigma1-sigma2)/2.*math.sin(2*(theta/57.3));##N/mm^2
sigmaR=math.sqrt(sigma_n**2+sigma_t**2);##N/mm^2
print"%s %.2f %s"%("Resultant stress in N/mm^2 : ",sigmaR,"");
fi=math.atan(sigma_t/sigma_n)*57.3;##degree
print"%s %.2f %s"%("Direction of resultant stress in degree : ",fi,"");
tau_max=(sigma1-sigma2)/2.;##N/mm^2
print"%s %.2f %s"%("Maximum shear stress in N/mm^2 : ",tau_max,"");

Resultant stress in N/mm^2 :  177.77 
Direction of resultant stress in degree :  43.00 
Maximum shear stress in N/mm^2 :  140.00

Ex10-pg448

In [11]:
##Part B Chapter 2 Example 10
##find the maximum shear stress 
import math
sigma_x=60.;##N/mm^2
sigma_y=30.;##N/mm^2
tau=25.;##N/mm^2
theta=45.;##degree(Oblique plane angle)
sigma_n=(sigma_x+sigma_y)/2.+(sigma_x-sigma_y)/2.*math.cos(2*(theta/57.3))+tau*math.sin(2*(theta/57.3));##N/mm^2
print"%s %.2f %s"%("Value of sigma_n in N/mm^2 : ",sigma_n,"");
sigma_t=(sigma_x-sigma_y)/2.*math.sin(2.*(theta/57.3))-tau*math.cos(2.*(theta/57.3));##N/mm^2
print"%s %.2f %s"%("Value of sigma_t in N/mm^2 : ",sigma_t,"");
sigmaR=math.sqrt(sigma_n**2+sigma_t**2);##N/mm^2(Resultant stress)
print"%s %.2f %s"%("Value of sigma_R in N/mm^2 : ",sigmaR,"");
fi=math.atan(sigma_t/sigma_n)*57.3;##degree(Angle of obliquity)
print"%s %.2f %s"%("Angle of obliquity in degree : ",fi,"");

Value of sigma_n in N/mm^2 :  70.00 
Value of sigma_t in N/mm^2 :  15.00 
Value of sigma_R in N/mm^2 :  71.59 
Angle of obliquity in degree :  12.09

Ex11-pg449

In [12]:
##Part B Chapter 2 Example 11
##find the maximum shear stress 
import math
sigma1=125.;##N/mm^2(Tenslie)
sigma2=65.;##N/mm^2(Tensile)
tau=0.;##N/mm^2
theta=30.;##degree
sigma_n=(sigma1+sigma2)/2.+(sigma1-sigma2)/2.*math.cos(2*(theta/57.3))+tau*math.sin(2*(theta/57.3));##N/mm^2
print"%s %.2f %s"%("Value of sigma_n in N/mm^2 : ",sigma_n,"");
sigma_t=(sigma1-sigma2)/2.*math.sin(2*(theta/57.3))-tau*math.cos(2*(theta/57.3));##N/mm^2
print"%s %.2f %s"%("Value of sigma_t in N/mm^2 : ",sigma_t,"");
sigmaR=math.sqrt(sigma_n**2+sigma_t**2);##N/mm^2
print"%s %.2f %s"%("Value of sigma_R in N/mm^2 : ",sigmaR,"");
fi=math.atan(sigma_t/sigma_n)*57.3;##degree
print"%s %.2f %s"%("Angle, fi in degree : ",fi,"");

Value of sigma_n in N/mm^2 :  110.00 
Value of sigma_t in N/mm^2 :  25.98 
Value of sigma_R in N/mm^2 :  113.03 
Angle, fi in degree :  13.29

Ex12-pg450

In [13]:
##Part B Chapter 2 Example 12
##find the maximum shear stress 
import math
sigma_y=0.;##N/m^2
theta=30.;##degree
A=450*10**-6;##m^2
F=-100.;##kN
sigma_x=F/A/1000.;##MN/m^2
sigma_n=(sigma_x+sigma_y)/2.+(sigma_x-sigma_y)/2.*math.cos(2*(theta/57.3));##MN/m^2
print"%s %.2f %s"%("Value of sigma_n(compressive) in MN/m^2 : ",sigma_n,"");
sigma_t=(-sigma_x-sigma_y)/2*math.sin(2*(theta/57.3));##MN/m^2
print"%s %.2f %s"%("Value of sigma_t in MN/m^2 : ",sigma_t,"");
sigmaR=math.sqrt(sigma_n**2+sigma_t**2);##N/mm^2
print"%s %.2f %s"%("Value of sigma_R(compressive) in N/mm^2 : ",sigmaR,"");
fi=math.atan(sigma_t/-sigma_n)*57.3;##degree
print"%s %.2f %s"%("Angle, fi in degree : ",fi,"");
tau_max=(-sigma_x-sigma_y)/2.;##MN/m^2
print"%s %.2f %s"%("Maximum shear stress in MN/m^2 : ",tau_max,"");

Value of sigma_n(compressive) in MN/m^2 :  -166.67 
Value of sigma_t in MN/m^2 :  96.22 
Value of sigma_R(compressive) in N/mm^2 :  192.45 
Angle, fi in degree :  30.00 
Maximum shear stress in MN/m^2 :  111.11

Ex13-pg451

In [14]:
##Part B Chapter 2 Example 13
##find the maximum shear stress 
import math
sigma1=70.;##MN/m^2
sigma2=30.;##MN/m^2
theta=20.;##degree
sigma_n=(sigma1+sigma2)/2.+(sigma1-sigma2)/2.*math.cos(2.*(theta/57.3));##MN/m^2
print"%s %.2f %s"%("Value of sigma_n(tensile) in MN/m^2 : ",sigma_n,"");
sigma_t=(sigma1-sigma2)/2.*math.sin(2*(theta/57.3));##MN/m^2
print"%s %.2f %s"%("Value of sigma_t(shear) in MN/m^2 : ",sigma_t,"");
sigmaR=math.sqrt(sigma_n**2+sigma_t**2);##MN/m^2
print"%s %.2f %s"%("Value of sigma_R in MN/m^2 : ",sigmaR,"");
fi=math.atan(sigma_t/sigma_n)*57.3;##degree
print"%s %.2f %s"%("Angle, fi in degree : ",fi,"");

Value of sigma_n(tensile) in MN/m^2 :  65.32 
Value of sigma_t(shear) in MN/m^2 :  12.85 
Value of sigma_R in MN/m^2 :  66.57 
Angle, fi in degree :  11.13