Chapter 12: Review of Centroids and Moments of Inertia

Example 12.2, page no. 833

In [1]:
import math 

#initialisation
A1 = 6*0.5                      # Partial Area in in2
A2 = 20.8                       # from table E1 and E3
A3 = 8.82                       # from table E1 and E3
y1 = (18.47/2.0) + (0.5/2.0)    # Distance between centroid C1 and C2
y2 = 0                          # Distance between centroid C2 and C2
y3 = (18.47/2.0) + 0.649        # Distance between centroid C3 and C2

#calculation
A = A1 + A2 + A3                        # Area of entire cross section
Qx = (y1*A1) + (y2*A2) - (y3*A3)        # First moment of entire cross section
y_bar = Qx/A                            # Distance between x-axis and centroid of the cross section
print "The distance between x-axis and centroid of the cross section is ", round(-y_bar,2), "inch"

The distance between x-axis and centroid of the cross section is  1.8 inch

Example 12.5, page no. 840

In [2]:
import math 

#initialisation
A1 = 6*0.5                          # Partial Area in in2
A2 = 20.8                           # from table E1 and E3
A3 = 8.82                           # from table E1 and E3
y1 = (18.47/2.0) + (0.5/2.0)        # Distance between centroid C1 and C2
y2 = 0                              # Distance between centroid C2 and C2
y3 = (18.47/2.0) + 0.649            # Distance between centroid C3 and C2

#calculation
A = A1 + A2 + A3                    # Area of entire cross section
Qx = (y1*A1) + (y2*A2) - (y3*A3)    # First moment of entire cross section
y_bar = Qx/A                        # Distance between x-axis and centroid of the cross section
c_bar = -(y_bar)

I1 = (6*0.5**3)/12.0                # Moment of inertia of A1 
I2 = 1170                           # Moment of inertia of A2 from table E1
I3 = 3.94                           # Moment of inertia of A3 from table E3
Ic1 = I1 + (A1*(y1+c_bar)**2)       # Moment of inertia about C-C axis of area C1
Ic2 = I2 + (A2*(y2+c_bar)**2)       # Moment of inertia about C-C axis of area C2
Ic3 = I3 + (A3*(y3-c_bar)**2)       # Moment of inertia about C-C axis of area C3
Ic = Ic1 + Ic2 + Ic3                # Moment of inertia about C-C axis of whole area
print "The moment of inertia of entire cross section area about its centroidal axis C-C", round(Ic), "in^4"

The moment of inertia of entire cross section area about its centroidal axis C-C 2200.0 in^4

Example 12.7, page no. 851

In [3]:
import math 
import numpy

#initialisation
Ix = 29.29e06                   # Moment of inertia of crosssection about x-axis
Iy = 5.667e06                   # Moment of inertia of crosssection about y-axis
Ixy = -9.336e06                 # Moment of inertia of crosssection 

#calculation
tp1 = (numpy.degrees(numpy.arctan((-(2*Ixy)/(Ix-Iy)))))/2.0  # Angle definig a Principle axix
tp2 = 90 + tp1  
print "The Principle axis is inclined at an angle", round(tp1,2), "degree"
print "Second angle of inclination of Principle axis is", round(tp2,2), "degree"
Ix1 = (Ix+Iy)/2.0 + ((Ix-Iy)/2.0)*math.cos(math.radians(tp1)) - Ixy*math.sin(math.radians(tp1))
Ix2 = (Ix+Iy)/2.0 + ((Ix-Iy)/2.0)*math.cos(math.radians(tp2)) - Ixy*math.sin(math.radians(tp2))
print "Principle Moment of inertia corresponding to tp1", round(Ix1), "mm^4"
print "Principle Moment of inertia corresponding to tp2", round(Ix2), "mm^4"

The Principle axis is inclined at an angle 19.16 degree
Second angle of inclination of Principle axis is 109.16 degree
Principle Moment of inertia corresponding to tp1 31700001.0 mm^4
Principle Moment of inertia corresponding to tp2 22420295.0 mm^4