International Journal for Numerical Methods in Engineering, Dec 21, 2021
The extended/generalized finite element method has proven significant efficiency for handling cra... more The extended/generalized finite element method has proven significant efficiency for handling crack propagation and internal boundaries. In certain conditions, however, one of the major drawbacks relates to the representation of unrealistic traction oscillations, particularly in stiff interfaces. To the authors' best knowledge, the few remedies found in the literature depend on the type of underlying finite element, which in some aspects limits general applications. Since one of the major sources of oscillations is created by couplings within standard shape functions for certain crack arrangements, it is herein proposed a novel approach based on enrichment Laplace shape functions directly adapted to the underlying geometry of split subdomains. By doing so, all sources of oscillations are effectively removed, while enriched degrees of freedom are defined exclusively on one side of the domain. The performance is studied using both element and structural examples with highly stiff cracks. More importantly, further assessment in more complex crack propagation problems, including mixed‐mode fracture of concrete beams and a peel test, shows excellent agreement with experimental/numerical data in terms of load‐displacement curves and traction profiles. Results are shown to be objective with respect to the mesh for stiffness values virtually representing infinitely stiff interfaces.
Conventional solutions for the equations of equilibrium based on the well known Vlasov thin-walle... more Conventional solutions for the equations of equilibrium based on the well known Vlasov thin-walled beam theory uncouple the equations by adopting orthogonal coordinate systems. This technique introduces several modeling complications and limitations where eccentric supports or abrupt cross-sectional changes exist (in elements with rectangular holes, coped flanges, or longitudinal stiffened members, etc.). In this study, a general solution of the Vlasov thin-walled beam theory based on a non-orthogonal coordinate system is developed. A finite element formulation, which yields nodal values in exact agreement with those based on the closed form solution of the Vlasov field equations and boundary conditions, is derived. The advantages and modeling capabilities of the formulation are discussed in detail. General expressions for normal and shearing stresses under non-orthogonal coordinate systems are developed. For design purposes, an elastic interaction equation for general open thin-walled section is derived. The shear deformation effect due to warping torsion is included in the torsional analysis of open thin-walled beams of general cross-section. The principle of stationary complementary energy is adopted to formulate the governing field compatibility condition. The variational principle is then extended to formulate a finite element, which captures shear deformation effects and allows the use of a minimal number of elements. For squat beams, shear deformation effects are shown to gain significance. Field equations and boundary conditions are obtained for the buckling analysis of thin-walled members by using the principle of stationary complementary energy. Subsequently a finite element is derived which incorporates shear deformation effects, a feature that is neglected in most available buckling solutions. It is shown that conventional solutions which neglect shear deformation effects can overestimate the predicted buckling load in some cases. The proposed finite element formulation can be used for the problems of [...]
International Journal for Numerical Methods in Engineering, Dec 21, 2021
The extended/generalized finite element method has proven significant efficiency for handling cra... more The extended/generalized finite element method has proven significant efficiency for handling crack propagation and internal boundaries. In certain conditions, however, one of the major drawbacks relates to the representation of unrealistic traction oscillations, particularly in stiff interfaces. To the authors' best knowledge, the few remedies found in the literature depend on the type of underlying finite element, which in some aspects limits general applications. Since one of the major sources of oscillations is created by couplings within standard shape functions for certain crack arrangements, it is herein proposed a novel approach based on enrichment Laplace shape functions directly adapted to the underlying geometry of split subdomains. By doing so, all sources of oscillations are effectively removed, while enriched degrees of freedom are defined exclusively on one side of the domain. The performance is studied using both element and structural examples with highly stiff cracks. More importantly, further assessment in more complex crack propagation problems, including mixed‐mode fracture of concrete beams and a peel test, shows excellent agreement with experimental/numerical data in terms of load‐displacement curves and traction profiles. Results are shown to be objective with respect to the mesh for stiffness values virtually representing infinitely stiff interfaces.
Conventional solutions for the equations of equilibrium based on the well known Vlasov thin-walle... more Conventional solutions for the equations of equilibrium based on the well known Vlasov thin-walled beam theory uncouple the equations by adopting orthogonal coordinate systems. This technique introduces several modeling complications and limitations where eccentric supports or abrupt cross-sectional changes exist (in elements with rectangular holes, coped flanges, or longitudinal stiffened members, etc.). In this study, a general solution of the Vlasov thin-walled beam theory based on a non-orthogonal coordinate system is developed. A finite element formulation, which yields nodal values in exact agreement with those based on the closed form solution of the Vlasov field equations and boundary conditions, is derived. The advantages and modeling capabilities of the formulation are discussed in detail. General expressions for normal and shearing stresses under non-orthogonal coordinate systems are developed. For design purposes, an elastic interaction equation for general open thin-walled section is derived. The shear deformation effect due to warping torsion is included in the torsional analysis of open thin-walled beams of general cross-section. The principle of stationary complementary energy is adopted to formulate the governing field compatibility condition. The variational principle is then extended to formulate a finite element, which captures shear deformation effects and allows the use of a minimal number of elements. For squat beams, shear deformation effects are shown to gain significance. Field equations and boundary conditions are obtained for the buckling analysis of thin-walled members by using the principle of stationary complementary energy. Subsequently a finite element is derived which incorporates shear deformation effects, a feature that is neglected in most available buckling solutions. It is shown that conventional solutions which neglect shear deformation effects can overestimate the predicted buckling load in some cases. The proposed finite element formulation can be used for the problems of [...]
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