# 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"

The Principle axis is inclined at an angle 19.16 degree