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Advances in Nano Research Volume 12, Number 2, February 2022 , pages 139-150 DOI: https://doi.org/10.12989/anr.2022.12.2.139 |
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Static deflection of nonlocal Euler Bernoulli and Timoshenko beams by Castigliano's theorem |
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Indronil Devnath, Mohammad Nazmul Islam, Minhaj Uddin Mahmood Siddique and Abdelouahed Tounsi
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| Abstract | ||
| This paper presents sets of explicit analytical equations that compute the static displacements of nanobeams by adopting the nonlocal elasticity theory of Eringen within the framework of Euler Bernoulli and Timoshenko beam theories. Castigliano's theorem is applied to an equivalent Virtual Local Beam (VLB) made up of linear elastic material to compute the displacements. The first derivative of the complementary energy of the VLB with respect to a virtual point load provides displacements. The displacements of the VLB are assumed equal to those of the nonlocal beam if nonlocal effects are superposed as additional stress resultants on the VLB. The illustrative equations of displacements are relevant to a few types of loadings combined with a few common boundary conditions. Several equations of displacements, thus derived, matched precisely in similar cases with the equations obtained by other analytical methods found in the literature. Furthermore, magnitudes of maximum displacements are also in excellent agreement with those computed by other numerical methods. These validated the superposition of nonlocal effects on the VLB and the accuracy of the derived equations. | ||
| Key Words | ||
| analytical solution; Castigliano's theorem; Eringen's nonlocal elasticity theory; Euler-Bernoulli beam theory; static displacements; Timoshenko beam theory | ||
| Address | ||
| Indronil Devnath and Mohammad Nazmul Islam: Department of Civil and Environmental Engineering, North South University, Dhaka, Bangladesh Minhaj Uddin Mahmood Siddique: Nippon Koei Bangladesh, Dhaka, Bangladesh Abdelouahed Tounsi: YFL (Yonsei Frontier Lab), Yonsei University, Seoul, Korea/ Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, 31261 Dhahran, Eastern Province, Saudi Arabia/ Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Algeria | ||