Appendix A: Review of Properties of Plane Area

Example A.1, Page No:486

In [4]:
import math

#Variable Decleration
A=2000 #Area of the plane in mm^2
Ix=40*10**6 #Momnet of Inertia in mm^4
d1=90 #Distance in mm
d2=70 #Distance in mm

#Calculations
Ix_bar=Ix-(A*d1**2) #Moment of Inertia along x_bar axis in mm^4
Iu=Ix_bar+A*d2**2 #Moment of Inertia along U-axis in mm^4

#Result
print Ix_bar
print "The moment of inertia along u-axis is",round(Iu,1),"mm^4"

23800000
The moment of inertia along u-axis is 33600000.0 mm^4

Example A.2, Page No:486

In [14]:
import math

#Variable Decleration
R=45 #Radius of the circle in mm
r=20 #Radius of the smaller circle in mm
h=100 #Depth of the straight section in mm

#Calculations
#Part 1

#Triangle
b=2*R #Breadth in mm
A_t=b*h*0.5 #Area in mm^2
Ix_bar_t=b*h**3*36**-1 #Moment of inertia in mm^4
y_bar1=2*3**-1*h #centroidal axis in mm
Ix_t=Ix_bar_t+A_t*y_bar1**2 #moment of inertia in mm^4

#Semi-circle
A_sc=pi*R**2*0.5 #Area of the semi-circle in mm^2
Ix_bar_sc=0.1098*R**4 #Moment of inertia in mm^4
y_bar2=h+(4*R*(3*pi)**-1) #Distance of centroid in mm
Ix_sc=Ix_bar_sc+A_sc*y_bar2**2 #Moment of inertia in mm^4

#Circle
A_c=pi*r**2 #Area of the circle in mm^2
Ix_bar_c=pi*r**4*4**-1 #Moment of inertia in mm^4
y_bar3=h #Distance of centroid in mm
Ix_c=Ix_bar_c+A_c*y_bar3**2 #Moment of inertia in mm^4

#Composite Area
A=A_t+A_sc-A_c #Total area in mm^2
Ix=Ix_t+Ix_sc-Ix_c #Moment of inertia in mm^4

#Part 2
y_bar=(A_t*y_bar1+A_sc*y_bar2-A_c*y_bar3)/(A) #Location of centroid in mm
Ix_bar=Ix-A*y_bar**2 #Moment of inertia in mm^4

#Result
print "Moment of inertia about x-axis is",round(Ix),"mm^4"
print "Moment of inertia about the centroidal axis is",round(Ix_bar),"mm^4"

Moment of inertia about x-axis is 55377079.0 mm^4
Moment of inertia about the centroidal axis is 7744899.0 mm^4

Example A.3, Page No:488

In [16]:
import math

#Variable Decleration
t=20 #Thickness in mm
h=140 #Depth in mm
w=180 #Width in mm

#Calculations
Ixy_1=0+(h*t*t*0.5*h*0.5) #product of inertia in mm^4
Ixy_2=0+((w-t)*t*(w+t)*0.5*t*0.5) #Product of inertia in mm^4
Ixy=Ixy_1+Ixy_2 #Product of inertia in mm^4

#Result
print "The Product of inertia is",round(Ixy),"mm^4"

The Product of inertia is 5160000.0 mm^4

Example A.4, Page No:495

In [51]:
import math

#Variable Decleration
t=30 #Thickness in mm
h=200 #Depth of the section in mm
w=160 #Width in mm
the=50 #Angle in degrees


#Calculations
A1=t*h #Area of the web portion in mm^2
A2=(w-t)*t #Area of the flange portion in mm^2
x_bar=(A1*t*0.5+A2*(t+(w-t)*0.5))/(A1+A2) #Location of x_bar in mm
y_bar=(A1*h*0.5+A2*t*0.5)/(A1+A2) #Location of y_bar in mm

#Simplfying the computation
a=t*h**3*12**-1
b=A1*(200*0.5-y_bar)**2
c=(w-t)*t**3*12**-1
d=A2*(t*0.5-y_bar)**2
Ix_bar=a+b+c+d #Moment of inertia about x-axis in mm^4

#Simplifying the computation
p=h*t**3*12**-1
q=A1*(t*0.5-x_bar)**2
r=t*(w-t)**3*12**-1
s=A2*((w-t)*0.5+t-x_bar)**2
Iy_bar=p+q+r+s #Moment of inertia about y-axis in mm^4

#Simplfying the computation
a1=(t*0.5-x_bar)*(h*0.5-y_bar)
a2=(t*0.5-y_bar)*((w-t)*0.5+t-x_bar)
Ixy_bar=A1*a1+A2*a2 #Moment of inertia in mm^4

#Part 1
#Simplfying the computation
a3=(Ix_bar+Iy_bar)*0.5
a4=(0.5*(Ix_bar-Iy_bar))**2
a5=Ixy_bar**2
I1=a3+np.sqrt(a4+a5) #Moment of inertia in mm^4
I2=a3-np.sqrt(a4+a5) #Moment of inertia in mm^4

ThetaRHS=-(2*Ixy_bar)/(Ix_bar-Iy_bar) #RHS of the tan term
theta1=arctan(ThetaRHS)*0.5*180*pi**-1 #Angle in degrees
theta2=theta1+90 #Angle in degrees

#Part 2
Iu=a3+np.sqrt(a4)*np.cos(2*the*pi*180**-1)-(Ixy_bar)\
      *np.sin(2*the*pi*180**-1) #Moment of inertia in mm^4
Iv=a3-np.sqrt(a4)*np.cos(2*the*pi*180**-1)+(Ixy_bar)\
      *np.sin(2*the*pi*180**-1) #Moment of inertia in mm^4
Iuv=np.sqrt(a4)*np.sin(2*the*pi*180**-1)+(Ixy_bar)\
      *np.cos(2*the*pi*180**-1) #Moment of inertia in mm^4
    
    
#Result
print "The Principal Moment of inertias are as follows"
print "I1=",round(I1),"mm^4 and I2=",round(I2),"mm^4"
print "Princial direction are theta1=",round(theta1,1), "degrees"\
       " theta2=",round(theta2,1),"degrees"
print "The moment of inertia along the uv-axis is",round(Iuv),"mm^4" 

The Principal Moment of inertias are as follows
I1= 47240734.0 mm^4 and I2= 11198811.0 mm^4
Princial direction are theta1= 31.6 degrees theta2= 121.6 degrees
The moment of inertia along the uv-axis is 10817183.0 mm^4

Example A.5, Page No:497

In [73]:
import math

#Variable Decleration
Ix_bar=37.37*10**6 #Moment of inertia in mm^4
Iy_bar=21.07*10**6 #Moment of inertia in mm^4
Ixy_bar=-16.073*10**6 #Moment of inertia in mm^4

#Calculations
b=(Ix_bar+Iy_bar)*0.5 #Parameter for the circle in mm^4
R=sqrt(((Ix_bar-Iy_bar)*0.5)**2+Ixy_bar**2) #Radius of the Mohr's Circle in mm^4

#Part 1
I1=b+R #MI in mm^4
I2=b-R #MI in mm^4
theta1=arcsin(abs(Ixy_bar)/R)*180*pi**-1*0.5 #Angle in degrees
theta2=theta1+90 #Angle in degrees

#Part 2
alpha=(100-theta1*2)*0.5 #Angle in degrees
Iu=round(b,2)+round(R,3)*round(np.cos(alpha*pi*180**-1),2) #MI in mm^4
Iv=round(b,2)-round(R,3)*round(np.cos(alpha*pi*180**-1),2) #MI in mm^4
Iuv=R*np.sin(2*alpha*pi*180**-1) #MI in mm^4

#Result
print "The Principal Moment of inertias are as follows"
print "I1=",round(I1),"mm^4 and I2=",round(I2),"mm^4"
print "Princial direction are theta1=",round(theta1,1), "degrees"\
       " theta2=",round(theta2,1),"degrees"
print "The moment of inertia along the uv-axis is",round(Iuv),"mm^4"  

The Principal Moment of inertias are as follows
I1= 47241205.0 mm^4 and I2= 11198795.0 mm^4
Princial direction are theta1= 31.6 degrees theta2= 121.6 degrees
The moment of inertia along the uv-axis is 10817230.0 mm^4