**Journal of Theoretical**

and Applied Mechanics

and Applied Mechanics

**56**, 4, pp. 1153-1162, Warsaw 2018

DOI: 10.15632/jtam-pl.56.4.1153

### Mechanical nature of a single walled carbon nanotube using Legendre’s polynomials

Euler-Bernoulli’s beam theory and Hamilton’s principle are employed to derive the set of

governing differential equations. An efficient variational method is used to determine the

solution of the problem and Legendre’s polynomials are used to define basis functions. Significance

of using these polynomials is their orthonormal property as these shape functions

convert mass and stiffness matrices either to zero or one. The impact of various parameters

such as length, temperature and elastic medium on the buckling load is observed and the

results are furnished in a uniform manner. The degree of accuracy of the obtained results

is verified with the available literature, hence illustrates the validity of the applied method.

Current findings show the usage of nanostructures in vast range of engineering applications.

It is worth mentioning that completely new results are obtained that are in validation with

the existing results reported in literature.

*Keywords*: Euler-Bernoulli’s beam theory, nonlocal elasticity theory, Legendre’s polynomial, aspect ratio, critical buckling load, Winkler and Pasternak elastic constants

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