Refined beam elements with only displacement variables and plate/shell capabilities

By Erasmo Carrera, Marco Petrolo. 2012

245 citations recorded by the source corpus.

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.

Subfields: Civil and Structural Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/7f5d362f8047e5c47cb7da9a306810adee992e87

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In Civil and Structural Engineering

In Mechanics of Materials

Citations in the corpus

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials (2015, unknown): “Structural models included thin-walled sections, point loads and shell-like natural modes, whereas further studies have been addressed towards the analysis of open cross-sections, lateral edge enforced boundary conditions and layer-wise approaches [40].”
Unified formulation of geometrically nonlinear refined beam theories (2018, background): “For more details about LE beam theories, the readers are referred to the original paper by Carrera and Petrolo [52].”
Efficient CUF-based FEM analysis of thin-wall structures with Lagrange polynomial expansion (2020, background): “In general, the coordinate transformation of an arbitrary cross-section belonged to (x, z) coordinates to the natural square ðn, gÞ is trivial, for more details see [17, 18].”
Hierarchical theories of structures based on Legendre polynomial expansions with finite element applications (2017, methodology): “Three-node linear L3, four-node bilinear L4, nine-node cubic L9, and sixteen-node quartic L16 polynomials can be used to formulate refined beam theories (see Carrera and Petrolo [29]).”
Carrera Unified Formulation for Free-Vibration Analysis of Aircraft Structures (2016, methodology): “The present model uses the quadratic two-dimensional Lagrange functions to describe the solution over the cross section; this approach was presented in [40,41].”
Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies (2012, unknown): “A beam model with only displacement degrees of freedom has been developed (Carrera and Petrolo 2011, 2012); it has been concluded that full displacement models are required to accurately analyze beams with arbitrary sections subjected to combined bending and/or torsion loadings.”
Use of Lagrange multipliers to combine 1D variable kinematic finite elements (2013, background): “CUF for 1D theories was recently proposed by the first author and his co-workers [20, 21].”
Recent developments on refined theories for beams with applications (2015, unknown): “LE models were introduced in recent works by the ﬁrst author and his co-workers (Carrera and Petrolo, 2012a; Carrera et al., 2012a).”