Comparison Of The Deflection In Two Cantilever Box ... - Size
was made to the mathematical derivation of the relation between deflections in two cantilever box beams of identical design but of .... B.I. and P.O.. Premium.
In Two Cantilever Box Beams By Frederic K. Howard In Part 2 of my article "Load Tests on a Scale Spar Structure" in last month's SPORT AVIATION reference was made to the mathematical derivation of the relation between deflections in two cantilever box beams of identical
design but of different sizes. It can be shown that the deflection on the actual spar will vary as the square of the scale factor. Consider two cantilever beams, A and B, shown in accompanying sketches, which F.re related to each other in size as indicated. Each beam is mounted inverted and horizontal, pinned rigidly at one end and so loaded that the
load varies directly with the depth of the spar. In spars of
As the size of any beam is changed — that is, as the design is scaled up cr scaled down, 1 and I will of course change. If the pattern of loading remains constant, then c will also be constant. If identical materials are used and the construction of the beam is not altered, E likewise will remain unaffected by the scaling up or scaling down. The relation between the maximum deflections under load of the two beams A and B below (B having dimensions n times those of A) can most clearly be brought out by the following question: If, under a total load W, beam A deflects at its free end an amount D, what value will r have when beam B is loaded to a total amount n3W and as a result
deflects at its free end rD? (Assuming of course both
A
beams have the same pattern of load distribution along their spans.) To compare the deflection of the two beams, it is necessary first to determine the relation between their moments of inertia. Although it seems likely that the relation itself is independent of the actual cross section design of the beam, it is probably a sounder proof to consider the type of cross section actur.lly used in the box beam design. (This cross section type is shown in the sketch.) The value of I for beam A and for beam B will in each case be the sum of the separate values of I for the two vertical webs, the two horizontal webs, and the four flanges.
w; -V)
— j~
B
Considering first beam A: I \ = 2I\ + 2I'n + 4 I ' K where I\ is the moment of inertia of a vertical web; I ' M is the moment of inertia of a
n
03*.
"f"
D
horizontal web about an axis through the centroid of the spar; I'|. is the moment of inertia of a flange also about an axis through the centroid of the spar.
bad..3 , . , „ n about an axis through its own 12 b.,d3 centroid and IK = about an axis through its centroid. 12 lv =
constant section this would be a uniformly distributed load; in tapered cantilever spars it is one which increases uniformly to the fixed end. In both instances the ...
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The M-th dimensional p-th order polynomial chaos consists in a set of multidimensional Hermite polynomials in { }M ii 1. = ξ. , whose degree does not exceed p.
The M-th dimensional p-th order polynomial chaos consists in a set of multidimensional Hermite polynomials in { }M ii 1. = ξ. , whose degree does not exceed p.
