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blodgett - table 5 - welds as a line
table 5 in blodgett gives various properties for calculating loads on a weld per unit length. i have just been following the calculation through from first principles and keep ending up with a term in the equation that shouldn't be there. heres my working:
box section width b, depth d,
i=2(d3/12) + 2(b/12 + bd2/4)
s=2i/d
s=d2/3 + b/3d + bd
with units in terms of length squared. but the middle term in the equation is unitless so it doesn't work. if i neglect it, the equation for s is as blodgett presented.
what am i missing?
typically when calculating the "moment of inertia" for welds perpendicular to the line of axis being considered, only the parallel axis theorem portion (the ad^2 term or specifically in this case the 2*bd^2/4 term) are considered as contributing because the contribution of the moment of intertia about the thickness of the weld is miniscule. the moment of inertia of the weld about its own axis would be (1/12)(b)(t)^3 in lieu of the b/12 term in your equation (where t is the weld thickness). as the t is small, the t^3 is very small and thus typically ignored. the erronious middle term of the final equation thus drops out completely.
the equation for the i of a box section when you include the thickness, t, is:
i=2(td^3/12) + 2(bt^3/12 + tbd^2/4)
note that this is the same as the i equation you have with t = 1.
following the actual equation through:
s=td^2/3 + bt^3/3d + tbd
which again is your equation with t = 1.
oddly enough, if you just multiply the equation you have by the thickness you have, you will get very nearly the correct answer for i. if you take blodgett's version of s (without the middle term) and multiply it by the thickness you have, you get very nearly the correct answer for s. they are not linear, but for small thicknesses, it is very close. set up both equations on excel and work through a number of thicknesses to prove it to yourself.
why does it work? if the thickness is small compared to the depth and width, the t^3 term in the middle becomes inconsequential compared to the d^3, bd^2, d^2, and bd terms. it's an approximation, but a very close one if t stays less than d/5 and b/5 (it's about a 1% difference at that level).
btw, which blodgett book and in what chapter is the table 5 you're looking at? i looked through all of my blodgett books and couldn't find the one you're talking about...
thanks folks. i think my quandry came about as a result of 'over analysis'. i was looking for an exact derivation, it didn't occur to me that they formulae were approximations albeit very close. it just goes to show why you really shouldn't use equations in text books unless you know where they come from.
my copy of blodgetts is design of welded structures, eighth printing july 1976 with table 5 on page 7.4-7.

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