Tapered beam Deflection
Tapered beam Deflection
(OP)
Hello,
I'm hoping someone here can help. I have a question concerning my dinghy mast that have been puzzling me for some time now and I have been unable to find any reference book or website that covers this topic. I am a Chartered Engineer, but not a structural one, so know the basics of structural theories but confess that this is beyond me.
If a sailboat mast of uniform section is unstayed then it can be considered as a cantilever beam and I am able to calculate the deflection using simple beam theory (assume for now that the sail applies a UDL to the mast). How do I calculate the deflection, but when the mast section is uniformly tapered (so that the 'I' value decreases linearly from root to tip)? Clearly the deflection will be greater - can I replicate this by simply applying a uniformly increasing load to a non-tapered section? If not, how else can I calculate the deflection? It must be possible to do this using hand calcs rather than FEA, surely. For the sake of detail, I should add that the mast section is a hollow cylinder and that the taper applies in all axes iso that the section remains round. I guess this makes it a high aspect ratio frustum of a cone.
Many thanks in advance for any suggestions, but be warned - if I get a good response I may pose another question!
I'm hoping someone here can help. I have a question concerning my dinghy mast that have been puzzling me for some time now and I have been unable to find any reference book or website that covers this topic. I am a Chartered Engineer, but not a structural one, so know the basics of structural theories but confess that this is beyond me.
If a sailboat mast of uniform section is unstayed then it can be considered as a cantilever beam and I am able to calculate the deflection using simple beam theory (assume for now that the sail applies a UDL to the mast). How do I calculate the deflection, but when the mast section is uniformly tapered (so that the 'I' value decreases linearly from root to tip)? Clearly the deflection will be greater - can I replicate this by simply applying a uniformly increasing load to a non-tapered section? If not, how else can I calculate the deflection? It must be possible to do this using hand calcs rather than FEA, surely. For the sake of detail, I should add that the mast section is a hollow cylinder and that the taper applies in all axes iso that the section remains round. I guess this makes it a high aspect ratio frustum of a cone.
Many thanks in advance for any suggestions, but be warned - if I get a good response I may pose another question!






RE: Tapered beam Deflection
RE: Tapered beam Deflection
y = w * L^4 / (7.872EI * (db/da)^3.282)
where:
I is the moment of inertia at the free end
db is the diameter at the fixed end
da is the diameter at the free end
Hope this helps. Not sure if I'll be able to handle another question, but give it a try.
RE: Tapered beam Deflection
RE: Tapered beam Deflection
_____________________________________
I have been called "A storehouse of worthless information" many times.
RE: Tapered beam Deflection
y'=-M/EJ.
For a cantilever with UDL, if you take x=0 at the beam tip, you have M=wx2/2
For a round thin hollow section J=πtR3. So J doesn't vary linearly: if the thickness is constant then the variation will be with the cube of x, if the section area is constant, it changes with the square of x.
Let's take the first case: you may write
R=Rt+(Rr-Rt)x/L
Now
y'=-wx2/2Eπt(Rt+(Rr-Rt)x/L)3
This can be integrated analytically on http://integrals.wolfram.com/index.jsp but the result is too long to be reported here (contains the inverse of a linear form and a log times a linear form), or can be integrated numerically by using a spreadsheet.
I hope this gives at least a picture of the problem you are facing, that is really solvable by elementary means, though some patience is needed.
prex
http://www.xcalcs.com
Online tools for structural design
RE: Tapered beam Deflection
RE: Tapered beam Deflection
v(x), being the slope of the beam, is the integral of [M(x)/EI] for a constant I, but in your case I is a function of x, so v(x) = integral[M(x)/EI(x)]
now M(x) is going to be proportional to x^2 (for a UDL)
maybe you can linearise I(x) ... i'd think this would be close enough given your assumption of loading, maybe piece-wise linear.
btw, i think a UDL is very simplistic assumption. the load from the sail is probably more proportional to the sail chord and leech tension (from the vang). and how are you accounting for the shrouds (which restrict deflection of the mast, creating a node in it's displaced shape)
good luck, i think you can solve this by hand methods
RE: Tapered beam Deflection
broekie's formula is sure a lot better.
www.SlideRuleEra.net
RE: Tapered beam Deflection
Anyway, with such good answers its time for the next question...
I was considering stiffening the mast to prevent it bending too much in gusty winds. I know I could use shrouds (stays) to hold the mast rigid or just increase the sectional inertia but I want to do it differently.
Do you know what diamonds wires are? They are just tensioned wires that run from the base of the mast to a point higher up the mast, say 70% of the length. At the midpoint of the wire (so 35% of the mast height) a spreader bar, perpendicular to the mast, holds the wire out so that you have what looks a bit like a suspension bridge on its side. There is one either side of the mast, so that they form a diamond. Simple.
What I can't work out is how to calculate how much stiffer the mast is with these attached. Obviously, wire tension and spreader length will increase the siffness, but how? Is it just a case of the spreader bar applying a point load equal to the tension in the wire (resolved through the diamond angle), which opposes the applied sail force?
RE: Tapered beam Deflection
diamond wires work (i think) because of their tension. the diamond connects to two points on the mast (i know you know, but it helps explain my subsequent waffle). these two points form the major diagonal of the diamond. consider one point remains fixed, and the other point deflects. this'll strain the two diamond wires connected to it, increasing the tension in one, reducing it in the other (hence the preload in the wires, you wouldn't want one of the wires going slack. also the spreader (the minor diameter of the diamond) is going to rotate with the mast's deflection, which'll strain the wires some.
maybe you can solve this iteratively, tho' i think FEA would be much easier (and non-linear at that). consider the mast without the diamonds, get the deflected shape, the deflection at the apex of the diamond (where it attaches to the mast) and the slope where the spreaders attach. now look at the diamond with these enforced displacements; one corner is fixed, the two spreader points move (oppositely) and the thrid corner deflects (to the side). the lengths of the diamond wires will all change, causing changes in the internal loads in the wires (increasing the tension in some, reducing it in others). i think this'll produce, effectively, a restorative couple to the mast (a moment opposite the the deflection from the loading). apply this couple to the mast, recalculate the displacements and slopes, and iterate !
an interesting problem, good luck
RE: Tapered beam Deflection
It's not that I want to turn the discussion towards analysis software (that's been talked about often enough) it's just that I don't want anyone to suffer needlessly.
Just an idea.
Regards,
-Mike
RE: Tapered beam Deflection
the "uniform load" w is not uniform since the mast is tapered, the wind is on the diameter which is tapered if the wind on the mast is the thing that causes the moment and deflection. I'm not a sailor, but isn't the sail connected to the mast with a series of ropes which would put discreet load points on the length of the mast?
_____________________________________
I have been called "A storehouse of worthless information" many times.
RE: Tapered beam Deflection
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: Tapered beam Deflection
Many thanks for all the words of wisdom, it seems these aren't such trivial problems after all...
RE: Tapered beam Deflection
The AASHTO formula only works for relatively thick wall thicknesses. When the wall thickness:diameter ratio is less than about 1:8 two things happen. Firstly, the taper has very little effect on deflection when compared to an untapered section (for same fixed end section size). Secondly, reducing the free end daimeter/MOI actually decreases the deflection, giving very curious results. I suppose that teaches me a lesson about understanding the source and limitations of empirical data! Extrapolation is always dodgy...
RE: Tapered beam Deflection
It is not perfect because there is no horizontal shear connection between the points of connection.
RE: Tapered beam Deflection
the loading (UDL or varying DL) leads to a simple enough moment curve, and the MoI expression is straight-forward too, but integrating them probably isn't pretty.
still, given the UDL assumption, i think linearising MoI, at least peicewise, should be accurate enough. for the diamond wires, i think drawing the deformed diamond will illuminate the wire strains/stresses/loads.
RE: Tapered beam Deflection
y = w*L^4 / (2E*pi*t*(Rb-Ra)^4) * (3*Ra*(-ln(Rb/Ra) - Ra/Rb + Ra^2/6/Rb^2 + 1/2) + Rb)
For a triangular load,
y = W*L^3 / (6E*pi*t*(Rb-Ra)^5) * (12*Ra^2*ln(Rb/Ra) + Ra*(8*Ra^2/Rb - Ra^3/Rb^2 - 8*Rb) + Rb^2)
where:
w = uniform load
W = total load
t = wall thickness
Ra = radius measured to wall centerline at free end
Rb = radius measured to wall centerline at fixed end
Sorry about the confusion my first answer may have caused. Probably should read the fine print a little more carefully.
RE: Tapered beam Deflection
RE: Tapered beam Deflection
_____________________________________
I have been called "A storehouse of worthless information" many times.
RE: Tapered beam Deflection
if you get less deflection in a tapered beam for the same inertia at the fixed end and the same load distribution, then you better double check your results: this would be a physically inconsistent situation violating the principle of energy conservation.
prex
http://www.xcalcs.com
Online tools for structural design
RE: Tapered beam Deflection