Species of curve generated by a deflected uniform rod Best answer on the web
Posted in: darrelrussell.com edit
07 Jan 2009
The phrase "elastic curve" is an inclusive one and certainly applies to cases in which all the bending is attributed to discrete external loading (rather than the "body forces" acting along the length of the curve due the beam's own weight).
However as a way of emphasizing the lack of such influences, I would suggest the combined phrase "thin-beam elastic curve".
The phrase "cantilever beam" would denote the arrangement you describe of being rigidly mounted (not "hinged") on the side of a building. If the rod does not extend horizontally, you might clarify this point by elaborating this as an "inclined cantilever beam".
In the "alternate" terminology of splines (which are "thin" by definition), the usual prescription is just the length of the spline and the location of two endpoints (or possibly additional points through which it is made to pass). If you wish to use this kind of language to describe your curve, it would be proper to say "a (two-point) spline, clamped at one end". This implies that the slope as well as the location is being prescribed, but only at one (clamped spline would tend to be read as meaning clamped at both ends, which is not your case).
regards, mathtalk-ga
regards, mathtalk-ga
One term for such a rod, rigidly supported at one end only (the other end being "free" but subject to load), is "cantilever beam". For a general introduction to the fascinating topic of beams and their deflections, see here, esp. the subsection entitled "Deflections":
[The Remarkable Theory of Beams, by J. B. Calvert]
http://www.du.edu/~jcalvert/tech/beam.htm
"Galileo studied beams, and although he did not get it quite right, he showed how the subject should be approached. The theory of beams was only perfected in the late 17th century with the rise of the science of elasticity, and was shown to be a subject of great complexity for which a full and accurate solution was very difficult."
A fully general treatment would need to manage not only the beam's cross-section but also the possibility that although the rod itself is of uniform composition, the material properties are anisotropic (different in different directions). Some mention of these difficulties is made here:
[Elasticity, by J. B. Calvert]
http://www.du.edu/~jcalvert/tech/elastic.htm
"Fortunately, most materials possess some symmetry in their properties, and this symmetry can be used to reduce the number of independent elastic constants."
A plane figure formed by a "uniform elastic rod" subject to varying perpendicular load or "stress" along its (fixed) length is called an elastic curve. The equation for an elastic curve has no general solution in simple functions, although special cases (constant curvature = circle) do. Historically the investigation of elastic curves led to the topic of elliptic integrals:
[Physics to Mathematics: from Lintearia to Lemniscate]
http://www.ias.ac.in/resonance/Apr2004/pdf/Apr2004p21-29.pdf
[A Property of Euler's Elastic Curve]
http://math.tulane.edu/~vhm/papers_html/EU.pdf
However if as an approximation (accurate for small deflections) the length of the curve is replaced by Cartesian coordinate x (taking y as the deflection distance about the "neutral" x-axis), then a simplified version is:
d^4 y / dx^4 = - w(x)/EI
where w(x) represents the load and EI is a constant. E (or sometimes Y) is Young's modulus, and I is the moment of inertia about the "neutral axis", here the x-axis; the product EI is known as the "flexural rigidity".
If no weight is imputed to the rod itself, then w(x) would be zero along the length, except at the very end of the rod. In this way one can rationalize that the curve is a cubic polynomial:
y(x) = c_0 + c_1*x + c_2*x^2 + c_3*x^3
[If some modest uniform distribution of weight of the rod itself is introduced, then in this simplified model an easily determined fourth order coefficient is to be added.]
These coefficients can be determined from consideration of the "boundary conditions" of the rod. At the rigidly supported end (x=0 using the most convenient coordinates) we fix both the location and the slope of the rod, so that:
c_0 = c_1 = 0
At the other end of the rod we have a deflection -D (location) of the rod's tip. However the slope is not specified there, so we must look around for an additional boundary condition.
The simplest treatment would be to argue that if the rod were extended beyond this point, then (again assuming weightlessness) its extension would be straight. Continuity would then suggest the second derivative is zero at the loaded/deflected end of the rod.
Let x = L be this end of the rod (recalling our approximation of the length by Cartesian coordinate), so that our conditions are:
y(L) = -D
y"(L) = 0
and we have two equations to solve for c_2 and c_3:
c_2 + L * c_3 = -D/L^2
c_2 + 3L * c_3 = 0
One easily solves that:
c_3 = (1/2)D/L^3
c_2 = -3L * c_3 = -(3/2)D/L^2
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The approximation of the shape of a flexible (but incompressible) rod by a cubic polynomial, as we sketched above, is closely related to the topic of splines and their convenient numerical proxies, cubic splines. As in the discussion above, a true spline is parameterized with respect to arc length while the cubic splines use a simpler functional dependence on a Cartesian coordinate.
[The Curve of Least Energy]
http://people.csail.mit.edu/bkph/papers/Least_Energy.pdf
"In a thin beam, curvature at a point is proportional to the bending moment. The total elastic energy stored in a thin beam is therefore proportional to the integral of the square of the curvature. The shape taken on by a thin beam is the one which minimizes its internal strain energy. This is why we call the curve sought here the minimum energy curve. A thin metal or wooden strip used by a draftsman to smoothly connect a number of points is called a spline."
In fact we can relate the "true spline" problem to that of the cantilever beam in the following way. Where the cantilever beam is of length L and passes from horizontal at the origin downward smoothly through the "deflection" at some point (x,-D), consider the spline of length 2L which passes through the symmetric pair of points (x,-D) and (-x,-D). Such a spline is naturally symmetric with respect to the y-axis.
Given a method for determining the true spline between these two points of a given length L, we could vary the horizontal spacing (from -x to x) until the resulting curve passes exactly through the origin. The right-hand half of this curve would then be the "true" shape of the cantilever beam.
regards, mathtalk-ga
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Suggested Search Terms:
"cantilever beam"
"elastic curve"
"flexible rod"
"spline approximation"
Your response is comprehensive and obviously adequate for the question as asked. However, as this was the first question I have asked on Google Answers, and not being cognizant of the complexity of bending theory, I should have phrased my question more specifically.
You mention elastic curve, lintearia, and splines as possible curves depending on material composition, etc. I found some confusion in reading the attached articles as to where the distinction is made between a deflection curve due solely to an applied force, as compared to one due to the mass of the beam itself. The "elastic curve" seems to refer to the latter.
I am in need of a designation of the curve generated by deflected rod due only to a perpendicularly applied force, as would be exhibited by a deflected floor or ceiling mounted cantilever. I would now ask:
What kind of curve is generated by a deflected aluminum cantilever (strongly isotropic material)? Assume a very small value of deflection, Less than 1% of the length of the cantilever, and assume no other forces (gravity or mass of the bar itself) contribute to the bending. A simple answer as to the most appropriate name of curve should suffice. With the name of the curve, I can then explore the articles again, as it relates to the specific cantilever I just outlined. Thank you.