Why Does a Beam Bend Only About the Centroidal Axis?

[chandran]

in beam bending the resistance of a cross section is measure by
moment of inertia about centroidal why. How do we know that the bends
only about the centroidal axis . Any detailed discussion on this?

 

[jasc15]

Looking at the cross section of the beam in bending, you should be able to understand that some portion of the beam is in tension, and some portion is in compression. So is follows from logic that there must be a point in between that experiences no stress, called the centroidal axis. Maybe that is not rigorous enough explanation and I am sure someone around here can elaborate.

 

[charng]

The centroidal axis is an important concept in beam bending as it helps us understand the resistance of a cross section to bending. The moment of inertia about the centroidal axis is a measure of the distribution of the cross-sectional area around this axis. This means that the centroidal axis is the axis through which the cross section can resist bending most effectively.

We know that a beam bends only about the centroidal axis because of the principle of minimum potential energy. This principle states that for a given external load on a beam, the deformation of the beam will be such that the potential energy of the system is minimized. In other words, the beam will deform in a way that requires the least amount of energy. Since the centroidal axis is the axis of maximum resistance to bending, this is where the beam will deform the least and therefore require the least amount of energy.

To further elaborate, imagine a beam with a non-uniform cross section. If we try to bend this beam about an axis other than the centroidal axis, the distribution of the cross-sectional area will not be symmetrical and the beam will require more energy to deform. This is because the moment of inertia about the non-centroidal axis will be larger, leading to a higher resistance to bending. However, if we bend the same beam about the centroidal axis, the distribution of the cross-sectional area will be symmetrical and the beam will require the least amount of energy to deform.

In summary, the centroidal axis is the axis of maximum resistance to bending and the principle of minimum potential energy dictates that a beam will deform only about this axis to minimize the energy required for deformation. This is why we consider the moment of inertia about the centroidal axis as a measure of the resistance of a cross section to bending.