Applied Mathematics, 2013, 4, 1568-1582 Published Online November 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.411212 Open Access AM Fracture Response of Reinforced Concrete Deep Beams Finite Element Investigation of Strength and Beam Size Guillermo A. Riveros 1 , Vellore Gopalaratnam 2 1 Information Technology Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS, USA 2 Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO, USA Email: Guillermo.A.Riveros@us.army.mil Received January 8, 2013; revised February 8, 2013; accepted February 15, 2013 Copyright © 2013 Guillermo A. Riveros, Vellore Gopalaratnam. This is an open access article distributed under the Creative Com- mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This article presents a finite element analysis of reinforced concrete deep beams using nonlinear fracture mechanics. The article describes the development of a numerical model that includes several nonlinear processes such as compres- sion and tension softening of concrete, bond slip between concrete and reinforcement, and the yielding of the longitu- dinal steel reinforcement. The development also incorporates the Delaunay refinement algorithm to create a triangular topology that is then transformed into a quadrilateral mesh by the quad-morphing algorithm. These two techniques al- low automatic remeshing using the discrete crack approach. Nonlinear fracture mechanics is incorporated using the fic- titious crack model and the principal tensile strength for crack initiation and propagation. The model has been success- ful in reproducing the load deflections, cracking patterns and size effects observed in experiments of normal and high-strength concrete deep beams with and without stirrup reinforcement. Keywords: Automatic Remeshing; Bond Slip; Concrete; Discrete Crack; Finite Element; Fracture Mechanics; Size Effects; Tensile Softening 1. Introduction Reinforced concrete (RC) deep beams have useful appli- cations in tall buildings, offshore structures, foundations, and military structures. A significant number of failures in RC structures initiate in tension regions caused by areas of high-stress concentrations or preexisting cracks. Stable growth of these tensile cracks, until peak loads, is associated with the development of large zones of frac- ture (fracture process zone (FPZ)). The growth of the FPZ, until peak load is reached, introduces the effect of structural size on the failure loads. Hence, if one was to design structures based on equations that were developed based on strength analysis, as in current American Con- crete Institute (ACI) code [1], the margin of safety pro- vided would depend upon the size of the structure. The margin of safety will be higher for smaller structures than for larger ones. It is also conceivable that this approach would lead to unconservative designs for some very large structures, e.g., deep slabs for underground storage tanks. Early attempts [2] to analyze failure in concrete struc- tures caused by crack growth were not successful, even though it was obvious that a fracture mechanics approach would be realistic to model brittle crack propagation type failures. The lack of success in the early attempts to ana- lyze crack propagation failures was due to the use of lin- ear elastic fracture mechanics (LEFM). LEFM assumes that the fracture process is small and can be replaced, and that the rest of the member volume remains elastic; however, research in the last four decades has resulted in modifications to LEFM to account for the distributed nature of pre-peak micro-cracking and the presence of a large FPZ in concrete [3-6]. These modifications have produced better results in the application of fracture me- chanics concepts to brittle failure in reinforced concrete. Theories that allow tensile softening and FPZ of rela- tively large sizes are classified as nonlinear fracture me- chanics models. A considerable effort has been committed to develop numerical models to simulate the fracture behavior of materials exhibiting tensile softening and FPZ, such as mortar, concrete, rock, or bricks used in civil engineering structures [4,7]. Two numerical methods to simulate frac- ture are available; the smeared crack approach and dis-