Use of Lagrange polynomials to build refined theories for laminated beams, plates and shells

Pagani, Alfonso; Carrera, Erasmo; Augello, R.; Scano, D.

doi:10.1016/j.compstruct.2021.114505

Cited by 19 publications

(4 citation statements)

References 48 publications

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“…To address those limitations, many advanced beam theories (Berdichevsky et al., 1992; El Fatmi and Zenzri, 2002; Silvestre and Camotim, 2002) have been put forward in the past years, including Carrera Unified Formulation (CUF) proposed by Carrera and Giunta (2010). This higher-order 1D model allows for the use of different cross-sectional expansion polynomials, such as Taylor polynomials (Carrera et al., 2013), Lagrange expansion (Pagani et al., 2021), and Jacobi polynomials (Carrera et al., 2023), to evaluate 3D displacement field. The CUF beam theory allows for the selection of the desired beam and expansion order, providing higher efficiency and lower computational costs.…”

Section: Introductionmentioning

confidence: 99%

A consistent crack bandwidth for higher-order beam theories: Application to concrete

Shen,

Tiago Arruda,

Pagani

2023

International Journal of Damage Mechanics

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Higher-order theories have a broad range of successful applications but also suffer from localization instability and mesh-size dependency when modeling quasi-brittle materials such as concrete with strain-softening behavior. To overcome the above difficulties, this paper proposes a fracture energy regularization method with a unified, consistent crack bandwidth specifically tailored for higher-order beam theories. The Carrera unified formulation (CUF) is applied to develop scalable structural theories and related finite elements. To evaluate the accuracy of the new crack bandwidth, three typical experimental quasi-static benchmarks of pure concrete structures are utilized. A modified Mazars damage model with tensile and compressive softening laws is implemented in these benchmarks. The comparison between numerical and experimental results demonstrates that the proposed method can accurately determine the correct crack bandwidth and preserve the dissipated energy per unit area of a fracture surface. Moreover, this robust estimation of crack bandwidth reduces the mesh dependency in general, ensuring the high efficiency of the CUF model.

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Section: Introductionmentioning

confidence: 99%

A consistent crack bandwidth for higher-order beam theories: Application to concrete

Shen,

Tiago Arruda,

Pagani

2023

International Journal of Damage Mechanics

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“…In the domain of CUF theories, Carrera and Giunta [4] used Higher Order Theories (HOT) derived from the Taylor polynomials. Furthermore, Carrera et al [5] used Lagrange-like expansions over the cross-section. Concerning the FE models, Carrera et al [1] used two-, three-and four-node Lagrange-like shape functions in the CUF framework.…”

Section: Introductionmentioning

confidence: 99%

“…The classical theories can be derived through penalization techniques from first-order Taylor. Furthermore, Carrera et al [5] used Lagrange-like expansions along the thickness direction. Finally, Carrera et al [1] used four-, eight-and nine-node FEs to study composite plates.…”

Section: Introductionmentioning

confidence: 99%

Analysis of composite beams, plates, and shells using Jacobi polynomials and NDK models

Scano

2023

Materials Research Proceedings

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Abstract. In this work, hierarchical Jacobi-based expansions are explored for the static analysis of multilayered beams, plates, and shells as structural theories as well as shape functions. Jacobi polynomials, denoted as P_p^((γ,θ) ), belong to the family of classical orthogonal polynomials and depend on two scalars parameters 𝛾 and 𝜃, with p being the polynomial order. Regarding the structural theories, layer wise and equivalent single-layer approaches can be used. It is demonstrated that the parameters 𝛾 and 𝜃 of the Jacobi polynomials are not influential for the calculations. These polynomials are employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate of finite element stiffness matrices straightforwardly. Furthermore, Node-dependent Kinematics is used in the CUF framework to build global-local models to save computational costs and obtain reliable results simultaneously.

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“…Higher-order beam finite element method based on Carrera unified formulation (CUF) [1] allows for one-dimensional analysis along the beam direction, with three-dimensional results obtained by expanding the cross-section using different polynomials such as Taylor [2] and Lagrange [3,4]. The Lagrange expansion is a popular choice for many engineering analyses due to its ability to fit well with arbitrary cross-sections.…”

Section: Introductionmentioning

confidence: 99%

A study of characteristic element length for higher-order finite elements

Shen

2023

Materials Research Proceedings

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Abstract. The utilization of a fracture energy regularization technique, based on the crack band model, can effectively resolve the issue of mesh-size dependency in the finite element modelling of quasi-brittle structures. However, achieving accurate results requires proper estimation of the characteristic element length in the finite element method. This study presents practical calculation methods for the characteristic element length, particularly for higher-order finite elements based on the Carrera Unified Formulation (CUF). Additionally, a modified Mazars damage model that incorporates fracture energy regularization is employed for damage analysis in quasi-brittle materials. An experimental benchmark is adopted then for validation, and the result shows that the proposed methods ensure accurate regularization of fracture energy and provide mesh-independent structural behaviors.

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Use of Lagrange polynomials to build refined theories for laminated beams, plates and shells

Cited by 19 publications

References 48 publications

A consistent crack bandwidth for higher-order beam theories: Application to concrete

A consistent crack bandwidth for higher-order beam theories: Application to concrete

Analysis of composite beams, plates, and shells using Jacobi polynomials and NDK models

A study of characteristic element length for higher-order finite elements

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