Elastic Cantilever with Tip Load

Problem Statement

Note

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Analytical Solution

The tip deflection at the center of the end-loaded cross section \((\delta_C)\) is obtained from an approximate solution derived from 2D elasticity (based on a polynomial Airy stress function):

\[\delta_C = \frac{P L^3}{3 E I} + \frac{ \left( 4 + 5\nu \right) P L}{2 E H t}\]

where \(L\), \(H\) and \(t\) are the beam length, height and thickness, respectively, and \(I = H^3/12\) (Bergan and Felippa, 1985). The first term arises from bending deformation, and the second term arises from shear deformation. This equation yields 0.34183 + 0.014 = 0.35533 inches; however, Bergan and Felippa provide a value of 0.35583 inches — apparently, this is an error in their paper. That error has been propagated into many subsequent papers in which the value of 0.35583 inches is used. An alternative value is given by Cook (1987), who states that the shear term has a value of 0.016 inches, thereby giving a tip deflection of 0.35733 inches. Cook does not provide information about how his value is obtained. It is interesting to note that if the coefficient \((4 + 5\nu)\) in the shear term is replaced with \((5 + 4\nu)\), then one obtains Cook’s value. For the purposes of the present problem, we take the analytical tip deflection to be 0.35533 inches, because it corresponds with the above equation.[1]

References

Allman, D. J. “A Compatible Triangular Element Including Vertex Rotations For Plane Elasticity Analysis,” Computers and Structures, 19(1–2), 1–8 (1984).

Endnotes

Data File

CantileverElastic.dat

This is CantileverElastic.dat (empty for now).