The centroid is the geometric center of a shape. If the material has a uniform density then the centroid will also be the center of mass as it is this point about which all the moments due to the weight of the material at each point net out to zero.
In our model we are only going to be dealing with cross-sections that are composed of uniform density material (the overall section we are dealing with may not be uniform but the component parts we use to build it up are uniform) and so the two concepts will coincide.
As our cross-sections are going to be symmetrical about the vertical axis then the centroid will lay on the y-axis ( the vertical axis) and can be calculated as:
y(av) =1/A* Int(y dA)
where ‘Int’ means integral with respect to area dA, y is the y-value of a point on the cross-section and A is the cross-section.
The centroid is the 'average y value' of the surface of the cross-section.
Intuitively, the top layer of a board (between the footpads) will typically be in tension and the bottom skin in compression (ie water pushing up between your stance). So, it follows that there must be a point somewhere between the two where the tension is zero. It turns out that for the types of materials and geometry we’ll be limiting ourselves to, this so called ‘neutral axis’ coincides with the centroid.
The neutral axis is important for our model because it is the point about which all moments due to the stresses in the cross section will attempt to rotate that cross section until these moments all net out to zero - that it will bend until the moments net out to zero as the moments due to the tension above the neutral axis net out with the opposing moments due to the compression of the material below the neutral axis. These internal moments arise as a reaction the externally applied moments . This is the main mechanism for bending that we shall model (pure bending).
We will see that the value of the moment of inertia (‘I’ in the Bernoulli Euler Beam question) depends on the location of the centroid and in fact that as we introduce concave the centroid of the cross-section moves and gives rise to increased stiffness.