By M. L. Ge, C. N. Yang

ISBN-10: 9971508281

ISBN-13: 9789971508289

ISBN-10: 9971508338

ISBN-13: 9789971508333

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**Sample text**

This famous Saint-Venant solution [103] of the problems of tension, pure bending, force bending and (in the general three-dimensional case) torsion plays an important role in the theory of rods. Thus, Love [100] tries to unify known exact solutions of the problem of bending for particular cases of the loading (moment, terminal transverse force, distributed force) into a general theory. However, in the general case with curvature, material non-homogeneity and anisotropy, as well as distributed external loads one needs to seek approximate solutions, which should get more accurate as the size of the cross section decreases (for plates and shells, under “cross section” we will understand a through-the-thickness element).

1007/978-3-7091-1777-4_2, © Springer-Verlag Wien 2014 37 38 2 Plane Bending of a Curved Rod Fig. 1 Geometry of a curved strip with a given middle line x 0 (s) and thickness H (Adapted from Vetyukov [160] with kind permission from Springer Science and Business Media) curvature. 2) is defined by the arc coordinate s and by the thickness coordinate n. 3) the gradient of the position vector is the two-dimensional identity tensor I2 . 133). The material may be generally anisotropic, and the fourth-rank tensor of elastic properties at plane stress 4 C as well as the density ρ may vary over the thickness of the strip in the case of a composite or a functionally graded structure [92].

We aim at obtaining the classical variant of the structural theory and introduce a formal small parameter in the problem to seek “beam” solutions, which vary in the axial direction much slower, than over the height: y is a “fast” variable, and x is a “slow” one. In the equations, we replace the derivatives ∂x by λ∂x , and the formal small parameter indicates smallness of the corresponding terms. It is equivalent to introducing the small parameter in the invariant differential operator of the plane elasticity problem: ∇ = λi∂x + j ∂y .

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