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Applied Structural Mechanics Fundamentals of Elasticity Part 2

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 16 2 Tensor algebra and analysis Example Application of the previous formulas to cylindrical coordinates Single-valued relations between Cartesian coordinates X1 and cylindrical coordinates 41 read as follows see Fig. 2.5 x cosí3 x siní3 x3 43. 2.42a Position vector r cos eL 41 sin ịỊ e2 t- e3 . 2.42b Covariant base vectors according to 2.25 1B1 - cos Í3 e1 sin f e2 K2 - sin f eL cos e2 2.43 Covariant metric components according to 2.4a gij Ei Sj For example g22 - Ẽ2 s2 i sinV t- i fcosV - 2- Covariant metric tensor Eij r 1 0 0 0 0 . 1 ê.44 According to 2.5c because of gịj gị g. 0 for i Ạ j Orthogonal base Contravariant components from 2.5c 2.5 Curvilinear coordinates 17 Contravariant metric tensor 1 0 gij 0 Ô-2 2.45 0 0 1 . Determinant g of the covariant metric tensor 1 0 0 0 0 1 2.46 0 0 CHRISTOFFEL symbols of the first kind according to 2.30 For example P221 y g21 2 g12 f2 - g22fl ị 0 0 - 2í . CHRISTOFFEL symbols in matrix notation M 0 0 0 0 - 0 0 0 0 e 0 0 o 0 0 . 0 3 0 CHRISTOFFEL symbols of the second kind according to 2.31 For example r glk r . 1- -f1 0 0 0 - 0 - f1 . JuJu Ju Jr n CHRISTOFFEL symbols in matrix notation 0 -Í1 0 0 0 ư r 0 v 0 0 0 0 2.48 1 a1 0 0 rj o. 18 3 State of stress LAPLACE operator according to ÍỈ.Ĩ9 ỉ r -7 4 g gj1 gj2 i 2 gJ3 3 J y s - 7 g gn i i A s22 2 2 i 8 3. ị1 M í y 22 í 33 A i l-ii j ly í 22 í 33. 2.49 3 State of stress 3.1 Stress vector The essential objective of structural analysis is the calculation of stresses and deformations of bodies. As shown in Fig. 3.1 we make a cut through the body which is in equilibrium under external loads in the form of volume forces fj surface tractions Pi and concentrated forces Fk. A resulting force A F is transmitted at every element A A of the cut. Fig. 3.1 Cut through a body Fig. 3.2 Resolution of the stress .