materials Article Nonlinear ABAQUS Simulations for Notched Concrete Beams Ahmed Bahgat Tawfik 1, * , Sameh Youssef Mahfouz 1 and Salah El-Din Fahmy Taher 2   Citation: Tawfik, A.B.; Mahfouz, S.Y.; Taher, S.E.-D.F. Nonlinear ABAQUS Simulations for Notched Concrete Beams. Materials 2021, 14, 7349. https://doi.org/10.3390/ma14237349 Academic Editors: Dario De Domenico and Luís Filipe Almeida Bernardo Received: 29 October 2021 Accepted: 28 November 2021 Published: 30 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Construction and Building Engineering Department, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport (AASTMT), B 2401 Smart Village, Giza 12577, Egypt; symahfouz@aast.edu 2 Professor of Concrete Structures, Structural Engineering Department, Tanta University, Tanta 31527, Egypt; salah.taher@f-eng.tanta.edu.eg * Correspondence: ahmedbahagt@aast.edu Abstract: The numerical simulation of concrete fracture is difficult because of the brittle, inelastic- nonlinear nature of concrete. In this study, notched plain and reinforced concrete beams were investigated numerically to study their flexural response using different crack simulation techniques in ABAQUS. The flexural response was expressed by hardening and softening regime, flexural capacity, failure ductility, damage initiation and propagation, fracture energy, crack path, and crack mouth opening displacement. The employed techniques were the contour integral technique (CIT), the extended finite element method (XFEM), and the virtual crack closure technique (VCCT). A parametric study regarding the initial notch-to-depth ratio (a o /D), the shear span-to-depth ratio (S.S/D), and external post-tensioning (EPT) were investigated. It was found that both XFEM and VCCT produced better results, but XFEM had better flexural simulation. Contrarily, the CIT models failed to express the softening behavior and to capture the crack path. Furthermore, the flexural capacity was increased after reducing the (a o /D) and after decreasing the S.S/D. Additionally, using EPT increased the flexural capacity, showed the ductile flexural response, and reduced the flexural softening. Moreover, using reinforcement led to more ductile behavior, controlled damage propagation, and a dramatic increase in the flexural capacity. Furthermore, CIT showed reliable results for reinforced concrete beams, unlike plain concrete beams. Keywords: ABAQUS; finite element analysis (FEA); concrete damage plasticity (CDP); extended finite element method (XFEM); external post-tensioning (EPT) 1. Introduction Employing numerical finite element (FE) simulations for different concrete elements reduces the need for further physical testing and helps researchers undertake complex parametric studies precisely. The complex nonlinear behavior of concrete in both tension and compression makes it challenging to simulate. The complexity stems from the brittle response of concrete, leading to failure due to cracking or crushing when subjected to tension or compression, respectively. Also, it is difficult to simulate crack initiation and propagation due to the tensile damage. Various constitutive numerical models were presented to simulate the concrete behav- ior in tension and compression [1–5]. Y. Nikaido [6] improved a constitutive numerical model to simulate concrete behavior by considering compression stiffness recovery. Further- more, different studies were conducted to simulate the nonlinear behavior of different plain and reinforced concrete elements [7–11]. Zhang et al. [12] compared different numerical crack simulation techniques to simulate a notched concrete beam using the unified FE software package ABAQUS [13] without experimental result validation. These techniques included the virtual crack closure technique (VCCT) and the extended finite element method (XFEM). It was deduced that both VCCT and XFEM can capture the softening regime for concrete fracture well. An experimental work achieved by Yin et al. [14] was conducted on Materials 2021, 14, 7349. https://doi.org/10.3390/ma14237349 https://www.mdpi.com/journal/materials