PSI - Issue 66

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ScienceDirect

Procedia Structural Integrity 66 (2024) 195–204 Structural Integrity Procedia 00 (2025) 000–000 Structural Integrity Procedia 00 (2025) 000–000

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

8th International Conference on Crack Paths 8th International Conference on Crack Paths

Assessment of multi-cracked non-uniform FGM beams using an equilibrium-based finite element formulation H.A.F.A. Santos a, ∗ , V.V. Silberschmidt b a A´rea Departamental de Engenharia Mecaˆnica, CIMOSM, UnIRE, Instituto Superior de Engenharia de Lisboa, Portugal b Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, UK Assessment of multi-cracked non-uniform FGM beams using an equilibrium-based finite element formulation H.A.F.A. Santos a, ∗ , V.V. Silberschmidt b a A´rea Departamental de Engenharia Mecaˆnica, CIMOSM, UnIRE, Instituto Superior de Engenharia de Lisboa, Portugal b Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, UK

Abstract Abstract

A novel, simple and effective finite element formulation is introduced for the quasi-static analysis of functionally graded non uniform Timoshenko beams with multiple and open concentrated cracks. This formulation relies on a complementary variational approach based on a set of approximations that satisfy all equilibrium conditions of the boundary-value problem in a strong form. As a result, this formulation is naturally free from the well-known shear-locking phenomenon. The effectiveness of the formulation is numerically demonstrated through its application to various problems, with the obtained results analysed and discussed. A novel, simple and effective finite element formulation is introduced for the quasi-static analysis of functionally graded non uniform Timoshenko beams with multiple and open concentrated cracks. This formulation relies on a complementary variational approach based on a set of approximations that satisfy all equilibrium conditions of the boundary-value problem in a strong form. As a result, this formulation is naturally free from the well-known shear-locking phenomenon. The effectiveness of the formulation is numerically demonstrated through its application to various problems, with the obtained results analysed and discussed. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of CP 2024 Organizers © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of CP 2024 Organizers. © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of CP 2024 Organizers. Keywords: Beams ; Non-uniform cross-sections; Functionally graded materials; Cracks; Finite element method; Equilibrium-based formulation. Keywords: Beams ; Non-uniform cross-sections; Functionally graded materials; Cracks; Finite element method; Equilibrium-based formulation.

1. Introduction 1. Introduction

The performance of structures is highly affected by the presence of damage, as it can lead to different types and levels of failure, ranging from a degradation of serviceability to a complete loss of functionality. For this reason, structural damage effects have been intensively investigated in several fields, such as Aeronautical, Mechanical and Civil Engineering, particularly with the aim to devise more efficient and accurate models. The one-dimensional straight beam is a very important model for structural analysis, widely applied to study the mechanical response of cracked beams and to understand the involved phenomena [12]. It demands for a low computational cost, while it is still able to represent the basic characteristics of a cracked member. The standard Euler-Bernoulli and Timoshenko beam theories have been enriched to include local variations in material properties or geometric characteristics. The different approaches employed to account for the presence of cracks can be grouped into three basic categories [7]: local stiffness reduction models [11]; continuous models [6]; and discrete spring (or lumped flexibility) models [2]. The performance of structures is highly affected by the presence of damage, as it can lead to different types and levels of failure, ranging from a degradation of serviceability to a complete loss of functionality. For this reason, structural damage effects have been intensively investigated in several fields, such as Aeronautical, Mechanical and Civil Engineering, particularly with the aim to devise more efficient and accurate models. The one-dimensional straight beam is a very important model for structural analysis, widely applied to study the mechanical response of cracked beams and to understand the involved phenomena [12]. It demands for a low computational cost, while it is still able to represent the basic characteristics of a cracked member. The standard Euler-Bernoulli and Timoshenko beam theories have been enriched to include local variations in material properties or geometric characteristics. The different approaches employed to account for the presence of cracks can be grouped into three basic categories [7]: local stiffness reduction models [11]; continuous models [6]; and discrete spring (or lumped flexibility) models [2].

∗ Corresponding author. E-mail address: hugo.santos@isel.pt ∗ Corresponding author. E-mail address: hugo.santos@isel.pt

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of CP 2024 Organizers 10.1016/j.prostr.2024.11.070 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of CP 2024 Organizers. 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of CP 2024 Organizers.