Exact solutions for size-dependent bending of Timoshenko curved beams based on a modified nonlocal strain gradient model
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O R I G I NA L PA P E R
Pei Zhang
· Hai Qing
Exact solutions for size-dependent bending of Timoshenko curved beams based on a modified nonlocal strain gradient model
Received: 7 April 2020 / Revised: 26 August 2020 / Accepted: 5 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Size-dependent bending analysis of Timoshenko curved beams is performed with a modified nonlocal strain gradient integral model, in which the integral constitutive equation is transformed into an equivalent differential form equipped with two constitutive boundary equations. The governing equations and boundary conditions are derived via the minimum total potential energy principle and solved analytically using the Laplace transformation technique and its inverse version. In numerical examples, the inconsistency of the nonlocal strain gradient model is examined extendedly under different boundary and loading conditions, while consistent softening and stiffening responses can be observed via the modified nonlocal strain gradient integral model. In addition, within the modified nonlocal strain gradient model, numerical examples also show that the increase of the opening angle can affect the total size effects of the combination of the two scale parameters (i.e., nonlocal and gradient), and these effects are inconsistent for different beam boundaries. Finally, by comparing with the results of the Euler–Bernoulli theory, an interesting finding is that as the nonlocal (or gradient length-scale) parameter increases, the shear deformations of simply supported-simply supported beams (or clamped-clamped/simply supported beams) become more significant.
List of symbols u w φ L κ1 , κ2 b h σi j σi0j σi1j εi j ∇εi j A I
Circumferential displacement Radial deflection Rotation angle of cross section Length of curved beam Nonlocal length-scale parameters (introduced to describe the influence of the classical and higher-order nonlocal stress fields, see “Appendix A.” In this article, assuming κ1 = κ2 = κ) Width of curved beam Thickness of curved beam Components of total stress tensor (including the nonlocal and strain gradient effects see “Appendix A”) Components of classical nonlocal stress tensor (see “Appendix A”) Components of higher-order nonlocal stress tensor (see “Appendix A”) Components of strain tensor Components of strain gradient tensor Sectional area of curved beam Second moment of inertia
P. Zhang · H. Qing (B) State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China H. Qing E-mail: [email protected]
P. Zhang, H. Qing
ks N , M, Q N 0, N 0, M 0 N 1, N 1, M 1 R E G q Qˆ SS CS CC CF
Shear correction coefficient Total circumferential force, bending moment, and shear force (including the nonlocal and strain gradient effects, see Eq. [(4)] Classical nonlocal circumferential force, bending moment, and shear force [see Eq. (4)] Higher-order nonlocal circumferential force, bending moment, and shear force [see Eq. (4
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