Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T06:45:18.267Z Has data issue: false hasContentIssue false
  • English
  • Français

A Cantilever Subjected to Axial Force, Bending Moment, and Shear Force

Published online by Cambridge University Press:  12 August 2014

W.-D. Tseng ,
J.-Q. Tarn  and
C.-C. Chang
W.-D. Tseng*
Affiliation:
Department of Construction Engineering, Nan Jeon University of Science and Technology, Tainan, Taiwan
J.-Q. Tarn
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan
C.-C. Chang
Affiliation:
Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan
*
*[email protected]
Get access

Abstract

We present an exact analysis of the displacement and stress fields in an elastic 2-D cantilever subjected to axial force, shear force and moment, in which the end conditions are exactly satisfied. The problem is formulated on the basis of the state space formalism for 2-D deformation of an orthotropic body. Upon delineating the Hamiltonian characteristics of the formulation and by using eigenfunction expansion, a rigorous solution which satisfies the end conditions is determined. The results show that the end condition alters the stress significantly only near the end, and elementary solutions in the form of polynomials can give sufficiently accurate results except near the ends. Such a system would give rise to localized stresses and displacements in the immediate neighborhood of the ends, and the effect may be expected to diminish with distance on account of geometrical divergence.

Keywords

2-D cantileverEnd effectsHamiltonianState spaceSymplectic orthogonalityOrthotropic
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 5 , October 2014 , pp. 549 - 559
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd Ed., McGraw-Hill, New York (1970).Google Scholar
2.Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body, Mir, Moscow (1981).Google Scholar
3.Tarn, J. Q., “A State Space Formalism for Anisotropic Elasticity, Part I: Rectilinear Anisotropy,” International Journal of Solids and Structures, 39, pp. 5143–5155 (2002).CrossRefGoogle Scholar
4.Hildebrand, F. B., Advanced Calculus for Applications, 2nd Ed., Prentice-Hall, Englewood Cliffs, New Jersey (1976).Google Scholar
5.Tarn, J. Q., Tseng, W. D. and Chang, H. H., “A Circular Elastic Cylinder Under its Own Weight,” International Journal of Solids and Structures, 46, pp. 2886–2896 (2009).CrossRefGoogle Scholar
6.Zhong, W. X., A New Systematic Methodology for Theory of Elasticity (in Chinese), Dalian University of Technology Press, Dalian, China (1995). [Text in Chinese]Google Scholar
7.Chen, J. T. and Chen, P. Y., “A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending,” Journal of Mechanics, 23, pp. 211–222 (2007).CrossRefGoogle Scholar
8.Alzaharnah, I. T., “Flexural Characteristics of a Cantilever Plate Subjected to Heating at Fixed End,” Journal of Mechanics, 25, pp. 1–8 (2009).CrossRefGoogle Scholar
9.Liu, C. C., Yang, S. C. and Chen, C. K., “Nonlinear Dynamic Analysis of Micro Cantilever Beam Under Electrostatic Loading,” Journal of Mechanics, 28, pp. 63–70 (2012).CrossRefGoogle Scholar