What is the shape of an elastic band when suspended with a weight in the middle?

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When an elastic band is suspended from two points with a weight in the middle, it takes on a shape similar to a catenary due to the deformation caused by the weight. The weight of the elastic band itself is significant in determining this shape. The discussion suggests using the action principle to derive the second-order equation that describes this deformation. There is no widely recognized name for the specific shape of the elastic band under these conditions. The analysis indicates that both the weight of the band and the suspended weight influence its final form.
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What shape does an elastic band describe when hanging from two opposite points and suspending midway a weight W (causing deformation comparable to the droop without W)?
 
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Are we basically saying that the weight of the elastic band is significant?
 
Yes.
 
Use the action principle to derive the necessary 2nd order equation? I don't think there's a standard name for it.
 
Sounds right, as for a caternary.
 

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