Moment of Inertia – Bending Loading


Let’s take a look at one of the most basic parameters necessary in analyzing a structure’s ability to handle bending loads: the moment of inertia.
Remember that a moment is simply the effect a force has on a single point.  Consider the scenario of a person placing a wrench on a bolt and applying a force to the end of the wrench handle.  Such a force creates a moment about the center of the bolt head.
Now consider the cross-section of the actual wrench handle.  By applying a force to the end of the handle, the handle experiences a bending load.
Breaking Bricks with Low Moment of InertiaThe moment of inertia about an axis of bending indicates its ability to resist such bending.  This applies to many stuctures – trusses, beams, boards, etc.  Think about a martial artist breaking a wooden board or stack of bricks.  Have you ever seen anyone strike the narrow end of a structure standing on end?  Of course not – this is because the moment of inertia is far too high around that axis.  Rather, the board or bricks are always struck on the flat side where the moment of inertia – the structure’s ability to resist bending – is lowest.
The point of all this relates intimately to design.  Many structures have a consistent moment of inertia along their entire length simply because of material availability or ease of manufacturing.  For example, many structural beams have a consistent cross-sectional area even though, structurally, there is less material needed at the ends of the beam than at the center of the beam span.  The beam has a consistent cross-section because it is much easier and cheaper to make a structural beam this way.  This is an acceptable justification.
Other manufacturing technologies – such as injection molding or cast metal – should never have an inefficient design because there are very few limits to the geometry which can be make using these manufacturing techniques.  Take a look at these examples of well designed products that utilize a variable moment of inertia based on their designed loading:
Cantilever Material Racks:
Tapered Cantilever Racks
Chair Base Legs:
Chair Base with Tapered Cantilever Legs
Notice that the cantilevered parts in the above two images all use a tapered shape.  To understand the efficiency of the tapered cantilever, take a look at the  moment diagram of a simple cantilever beam (blue line in the image to the left).  Notice that just as the moment diagram is tapered, so should the structure be taped for efficient use of materials without sacrificing the structure’s ability to resist bending.
The next post will take a closer look at some basic design techniques of metal castings, as relates to bending and the analysis of moments of inertia.

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