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Welcome to Monkey See!!!
Bending:
The first of my tasks is to understand what bending is. Bending is when one surface of an object is in tension while the other is in compression.
Fun Fact: This is a negative banana! By convention a positive moment is when there is compression on top not the bottom.
Bending theory provides me with this equation.
Bending Theory Equation[1]
By integrating this equation twice I can figure out the Bending Stiffness (S) of the rectangular beam.
Bending Stiffness Equation[2]
However, to understand this equation, I must understand the Second Moment of Area.
Basically it involves integrating the cross-sectional area dimensions of a shape like this:
Second Moment of Area of a Rectangular Beam[3]
There are also equations for all different kinds of cross-sectional areas!
For a circle:
For a semi-circle:
For a circular tube:
These equations look scary, but I only need to know what the equation means, and how to shove numbers into it.
There are also polar second moment of area equations, and this is like I have just done, but I need to do the integration twice.
I can use these equations with a further equation shown below to link how an applied moment can generate a maximum stress.
Where M = bending moment, y = distance from perimeter to the central axis, I = the second moment of area, and the equation gives the maximum stress.
Twisting:
The next concept I need to learn is torsion, which is twisting of a beam due to a rotational force. It is easier to show how this applies to rectangular beams, because rotation is easier to observe, but I only need to know how the numbers apply to circular beams.
As we can see, the cross sectional area of the circular beam remains constant while the rectangular one changes.
So it's time that I learnt about the equations that apply to torsional loads! Here is a diagram to help explain an example problem:
Twisting Equation[4]
This doesn't look so bad! I really like how this can be re-written to solve so many different problems. The Shear Modulus is a material property, sort of like Young's Modulus.
Whats Polar Second Moment of Area about though?!
It turns out that the polar second moment of area is very similar to the second moment of area I looked at earlier. The only difference is that it is calculated with respect to the polar axis (perpendicular to the plane). I suppose for circular beams I only need to worry about whether the beam is hollow or not as they all will have a circular cross sectional area.
Solid Beam: Hollow Beam:
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