One-Dimensional Governing Equations and Finite Element Formulation

Chapter 1 suggests that the three-dimensional displacements on the midline of a cross-sectional wall can be expressed as products of the generalized 1D displacements and sectional shape functions. The displacements at a general point on a cross-section are calculated using those on the midline based on the kinematic assumption Kirchhoff’s thin plate theory, i.e., the assumption of no shear and normal strains in the thickness direction (ε_nn= γ_ns= γ_zn= 0 ). The local coordinates (z, n, s) defined on the midline of a cross-sectional wall will be used to facilitate the expression of the strain field from the three-dimensional displacements. A two-dimensional stress field can be derived assuming a plane-stress state (i.e.,σnn=τns=τzn=0) for thin walls.