Learning how to calculate secondary moment using Bruhn's

[oriolbetriu]

Hi!

I am trying to learn how to calculate secondary moment, so that I can analize properly any strutted wing. I am following Bruhn's example, please see pages below. I think that I understand, the way the formulas are deduced from the force diagram, which ends up in a secondary order differential equation;

d^2M/dx^2 + 1/J^2*M = W

Which can be rewritten as;
d^2M/dx^2 + 1/J^2*M - W = 0
or;
y´´ + 1/J^2 + 1/J^2*y - W = 0

Solving this equation is sort of trivial. Because the roots of the differential equation can be solved, as if the equation was a second degree polinomy;
(-b +-(b^2 - 4*a*c)^(1/2))/(2*a) = (-(1/J^2) +-((1/J^2)^2 - 4*1*W))^(1/2)/(2*1)

If we look inside the root the value is positive, because (-4)*(-W) is positive, therefore the solution ought to be in the form of;
M = C1*e^(r1*alfa) + C2*e^(r2*alfa)

However Bruhn gives as the solution of the differential equation, the form that is used for imaginary numbers;
M = C1*sin(x/j) + C2*cos(x/j)

How is that so?! I can not tell what I am missing? I am stuck with this, and unable to advance. I never finished my aero studies, and I do not know anyone in my close circle, with whom I can discuss such things.

Any advice will be greatly appreciated!

Oriol

 

[Stress_Eng]

I still have the book recommended back when I was at uni. For a pin ended column, the 2nd order diff is
uyy + k^2.u = 0
The general solution is
u = A.cos(k.y) + B.sin(k.y) where k = (P/EI)^1/2
For beam column type problems, I’ve always seen this given as the general solution, followed by solving for the particular integral (additional terms as functions of the independent variable, in this example y).

 

[Sparweb]

Bruhn does not usually show all of his work. The point usually isn't to demonstrate a proof but only to illustrate principles. At a guess, Bruhn is using a trig identity in the differentiation. As a frequent user of Roark and Bruhn, I've noticed these derivations are normally solved with trigonometric functions, not exponential ones. School was too long ago for me to remember why. Someone more familiar with the methods to derive these solutions such as those in Roark can answer in detail (you might not find them here).

Also, not exactly sure why you referred to "imaginary numbers" but just in case you were taught to use "j" for your imaginary unit, in this context it doesn't mean that. You can find "j" as the square root of EI/P, none of which are negative. In Figure A5.66, the P vector is inward, making compression a positive value. No negatives under the square root, therefore no imaginary numbers.

 

[GregLocock]

Bruhn is a great resource but it is not a textbook, it is a reference work. In my ancient copy of this book https://www.sciencedirect.com/book/9780340719206/strength-of-materials-and-structures it is chapter 20, but in the new one chapter 18.
If you look in google books for

Strength of Materials and Structures​

By Carl T. F. Ross, The late John Case, A. Chilver

it is section 18.9

 

[rb1957]

Blasphemer !

I'm used to the trig solution to the diff equation.

I'm not 100% sure I'd use that to solve a strutted wing, but there could be application for it.

For myself, the wing is pinned to the fuselage, and the strut reacts the root bending moment and the rest is pretty easy calcs. Sure I'm making assumptions but they seem pretty defensible to me. I don't recall OEMs doing it much different (albeit from 70+ years ago).

 

[WKTaylor]

May be relevant... shed light on...this topic...
AFFDL-TR-69-42 Stress Analysis Manual

NOTE. 'Texts' teach... 'Manuals' are for quick/consistent working reference/calculations.

 

[Ng2020]

Rb. The inboard portion of the spar works as a beam column. However the error relative to a pure linear analysis is probably not that large if the spar is sufficiently stiff in bending.

 

[rb1957]

Sure, there are many interesting things happening, a nice shear lag problem ... the strut adds it's compression load on (or below) the lower surface, the fuselage attachment is usually on the upper surface