Some basics of engineering mechanics

[Gertsch <lgertsch@slate.Mines.EDU>]

On Sun, 4 Feb 1996, Rob Meyerson wrote:

 [much snipped] 
> Sorry, your wrong on this one.  Set this one up as a physics problem regarding
> the justification of stresses (I can't think of the real term, but this is 
> close
> enough).  With the feet under the body, the stress diagram looks like this:
> 
>          1
>          1 w
>          1
>          V
> 
> In this case, the force on the feet (w) equals the mass times the 
> acceleration due to gravity (simple Newtonian stuff).
> 
> With the forelimb held off to the side, the diagram looks like this:
> 
>         --------->1
>          \ A      1
>           \       1
>            \      1
>             \     1 x
>           H  \    1
>               \   1
>                \  1
>                 \ 1
>                  >V
> 
> Where:
>         H = Hypotenus (with a value equal to w).
> 
>         A = The angle between hypotenuse and horizontal.
> 
>         X = Total load applied to the feet.
> 
> Trigonometry says that X = H.cos A.  Since the cosine of any angle (provided
> A < 90) is always less than one, then X < H.  Therefore, the total load 
> applied
> to the feet is less with the feet held out to the side.

The total load applied to the feet is *fixed* by the portion of the 
animal's weight they support.  Your vector diagram is correct in itself, 
but you have misinterpreted it.  In the terms of your diagram, "X" is 
fixed, but as the angle A decreases, the un-named horizontal leg -- let's 
call it "Y" -- increases.  What this means is that as the sprawl 
increases, the stress in the horizontal portion of the arms also increases.

Remember that, for static equilibrium (i.e., so the critter can simply 
stand), the sum of forces and torques in all possible directions must 
equal zero.  Restricting our inquiry here to the sum of forces in the 
vertical direction, that means the resistive forces pushing up through 
the soles of our critter's feet must equal its weight.  However its 
legs are angled makes no difference to that fact.  It does, however, make 
differences in the stress distribution within its bones and at its joints.

Leslie Gertsch
research assistant professor
Earth Mechanics Institute
Colorado School of Mines