A Centroidal Axis Perpendicular To Its Base

The centroid, or centre of gravity, of any object is the point within that object from which the force of gravity appears to act. Centroidal axis is any axis that passes through the centroid of the cross section. There can be an infinite number of centroidal axes. Two of these are the principal axes. Major principal axis is the centroidal axis about which the second moment of area is the largest (out of all the possible centroidal axes) and the minor principal axis is the one about which the second moment of area is the least. Centroidal axis is the CG of area.

Centroidal Axis

Centroidal Axis is any axis that passes through the centroid of the cross section. There can be an infinite number of centroidal axes. Two of these are the principal axes. Major principal axis is the centroidal axis about which the second moment of area is the largest (out of all the possible centroidal axes) and the minor principal axis is the one about which the second moment of area is the least. Centroidal axis is the CG of area.

If stresses are greater than the yield stress, and if all fibres across the depth of the section have yielded (fully plastic section), the neutral axis is the equal area axis (an axis perpendicular to the plane of loads that divides the cross section area into two equal halves). Equal area axis may be a centroidal axis, but depending on the shape, the equal area axis may not pass through the centroid.

Neutral Axis is the axis at which strain (and consequently stress) is zero when the beam is subjected to bending. Neutral axis is perpendicular to the plane of the loads. If stresses are linear and within the yield stress, the neutral axis passes through the centroid (that is, the neutral axis is one of the centroidal axes). Neutral axis is one where flexural stresses change sign.

When the beam is subjected to clockwise bending stress on the left face and anti clockwise on the right one the beam will deflect. Under the action of bending stress the upper portion will be in compression and lower in tension. However, along one axis in the middle there will be no stress nor compression or tension. That axis is the neutral axis. No Strain in this case.

Moment Of Inertia About Centroidal Axis

The Moment of Inertia (I) is a term used to describe the capacity of a cross-section to resist bending. It is always considered with respect to a reference axis such as X-X or Y-Y. It is a mathematical property of a section concerned with a surface area and how that area is distributed about the reference axis (axis of interest).

The reference axis is usually a centroidal axis. The moment of inertia is also known as the Second Moment of the Area and is expressed mathematically as:

                                           Ix = ∫Ay2 dA 

                                           Iy = ∫Ax2 dA 

where,          

             y = distance from the x axis to area dA 

             x = distance from the y axis to area dA  

The moment of inertia of an area with respect to any given axis is equal to the moment of inertia with respect to the centroidal axis plus the product of the area and the square of the distance between the 2 axes.

The parallel axis theorem is used to determine the moment of inertia of composite sections. 

                                          Ix = Ixc + Ad2 

                                          Iy = Iyc + Ad2

The radius of gyration of an area with respect to a particular axis is the square root of the

quotient of the moment of inertia divided by the area. It is the distance at which the entire area must be assumed to be concentrated in order that the product of the area and the square of this distance will equal the moment of inertia of the actual area about the given axis. In other words, the radius of gyration describes the way in which the total cross-sectional area is distributed around its centroidal axis. If more area is distributed further from the axis, it will have greater resistance to buckling. The most efficient column section to resist buckling is a circular pipe, because it has its area distributed as far away as possible from the centroid. 

Conclusion

Centroidal axis is a line which is passing through a point about which the whole weight is considered to be concentrated and the Neutral axis is a line about which total force is considered zero in any body which is being acted upon by some forces.