Mathematics and Fly Fishing
From Quarterly of applied Math, V.3(3),pp 272-275, 1945.
Large Deflection of Cantilever Beams
By K. E. BISSHOPP AND D. C. DRUCKER (Armour Research Foundation)
Received April 6, 1945. This problem was considered by H. J. Barten, 'On the Deflection of a Cantilever Beam,' Quarterly of Applied Math., 2, 168-171 (1944). Previously an approximate solution had been obtained by Gross und Lehr in Die Federn, Berlin VDI Verlag, 1938.
The solution for large deflection of a cantilever beam cannot be obtained from elementary beam theory since basic assumptions are no longer valid. Specifically the elementary theory neglects the square of the first derivative in the curvature formula and provides no correction for the shortening of the moment arm as the loaded end of the beam deflects. For large finite loads, it gives deflections greater than the length of the beam!
The square of the first derivative and correction factors for the shortening of the moment arm become the major contribution to the solution of large deflection problems. The following theory, which utilizes these corrections, is in agreement with experimental observations.
The derivation is based on the fundamental Bernoulli-Euler theorem, which states that the curvature is proportional to the bending moment. It is assumed also that bending does not alter the length of the beam.
Considering a long, thin cantilever leaf spring, let L denote the length of beam, D the horizontal component of the displacement of the loaded end of the beam, d the Corresponding vertical displacement, P the concentrated vertical load at the free end, B the flexural rigidity, that is B =EI, when cross-sectional dimensions are of the same order of magnitude, and B =EI / (I - n2) for 'wide' beams, where n is the Poisson ratio. The exact expression for the curvature of the elastic line may be stated conveniently in terms of arc length and slope angle denoted by s and f, respectively, so that if x is the horizontal coordinate measured from the fixed end of the beam, the product of B and the curvature of the beam equals the bending moment M:
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(1) |
or
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(2) |
whence
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(3) |
The constant C can be evaluated directly by observing that the curvature at the loaded end is zero. Then if f0 is the corresponding slope
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(4) |
The value of f0 cannot be found directly from this equation but it is implied by the requirement that the beam be inextensible, so that
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(5) |
In order to evaluate this elliptic integral, denote PL2/B by a2 and let
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(6) |
Then
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(7) |
The the next step is to represent the deflection d in terms of a and an elliptic integral. Since
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and since we have df/ds from Eq. (4),
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Thus
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With the aid of Eq. (6) we obtain
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This equation can be split up into complete and incomplete elliptic integrals of the first and second kinds. In the notation of Jahnke and Emde,
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(8) |
so that
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(9) |
The horizontal displacement of the loaded end is calculated from Eqs. (1) and (4) with x=0 when f= 0. Thus
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or
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(10) |
From Eq.(6) we have sinf0=2k2-1.
Numerical results can be obtained by: (1) selecting values of k corresponding to tabulated values of the modular angle in the elliptic function tables and (2) determining q1 and a from Eq. (7). After this has been done, d/L and and (L-D)/L can be calculated from Eqs. (9) and (10) and plotted against a2 =PL2/B. The results of these calculations are shown in Fig. 1.
