An uncoupled higher-order beam theory and its finite element implementation

• Published in: International journal of mechanical sciences Vol. 134; pp. 525 - 531
• Main Authors: Geng, P.S., Duan, T.C., Li, L.X.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2017
• Subjects: Finite element method; Generalized displacement; Generalized stress (strain); The principle of virtual work; Uncoupled constitutive relations; The principle of virtual work; Finite element method; Generalized displacement; Generalized stress (strain); Uncoupled constitutive relations
• ISSN: 0020-7403; 1879-2162
• DOI: 10.1016/j.ijmecsci.2017.10.041

•The general mode of axial displacement is proposed in an orthogonal expansion form, and the uncoupled constitutive relations are then derived for beam problems.•The principle of virtual work is proposed and the variationally consistent decoupled beam theory is then established.•The finite element method is formulated as simple as for a three-dimensional elastic problem.•The higher-order beam element can capture the effect of the clamped end and the load jump via smoothly modeling the warping of cross section. A beam problem is though classical but not theoretically settled down at present. Different from the previous work, the current study starts with defining the generalized displacements. Together with the assumptions and the shear stress free condition, the axial displacement is first mathematically expanded in two terms and then decomposed into an orthogonal form in terms of the generalized displacements. The generalized stresses are accordingly defined, and the uncoupled constitutive relations are derived for beam problems after the generalized strains are properly measured. The principle of virtual work is proposed and the variationally consistent higher-order beam theory is eventually established, which can reduce to the variationally asymptotic lower-order beam theory. With these fundamentals, the finite element method is finally formulated as easy as for a three-dimensional elastic problem, and applied to typical problems. The results show that the higher-order beam element can capture the effect of the clamped end and the load jump via smoothly modeling the warping of cross section by using a locally refined mesh while accurately modeling the deflection. With the current framework, modern beam structures accounting for the effect of nonlocal elasticity, small scales and material heterogeneities can be readily solved. [Display omitted]