- Open Access
Computer Aided Design of RC Structures
© The Author(s) 2013
- Received: 5 August 2012
- Accepted: 31 December 2012
- Published: 22 May 2013
After reviewing the background and motivations for using modern computational methods for the design of reinforced concrete structures, an algorithm making use of the object oriented programming language Python and professionally developed finite element software is presented for the sizing and placement of the reinforcement in RC structures. The developed method is then used to design the reinforcement of a deep beam. To validate the design, two identical deep beam specimens were manufactured with the obtained steel, and then tested in the laboratory. It was found that the experimental results corroborated those predicted with the finite element design method.
- automated design
- non-linear finite element analysis
- object oriented programming
- deep beams
Since the first application of the finite element to the analysis of reinforced beams by Ngo and Scordelis (1967), a large number of approaches for modeling the behavior of concrete as a material or the behavior of reinforced concrete structures have been developed. While it is not the aim of the present study to provide a detailed review of the very large body of literature on this subject, as there are many good quality reviews published in the literature (ASCE 1982; de Borst 2002), it is still worthwhile to briefly describe some of the major developments that have occurred in this area.
Ngo and Scordelis (1967) analyzed simple beams using a triangular element. The cracking of concrete was modeled using the discrete approach. This approach, which is physically and intuitively appealing, was implemented by letting a crack grow when the nodal force at the node ahead of the crack tip exceeded a tensile strength criterion. Then, the node is split into two nodes and the tip of the crack is assumed to propagate to the next node. This process, of course, requires the redefinition of the mesh every time a crack propagates. This proved computationally intensive even in two dimensions, let alone in three dimensions. This prompted Rashid (1968) to develop the smeared crack approach, where a crack is numerically, rather than physically, modeled. In this approach, cracking of the concrete occurs when the principal tensile stress exceeds the ultimate tensile strength. The elastic modulus of the material is then assumed to be zero in the direction parallel to the principal tensile stress direction. Since these pioneering works, many other developments have taken place; in particular the experimental work of Kupfer et al. (1969), Palaniswamy and Shah (1974), and Kotsovos and Newman (1977), which revealed many aspects of the peculiar behavior of concrete such as exhibiting nonlinearity in both tension and compression, and its failure envelope depended on all the stress invariants. This prompted the development of failure criteria for concrete (Willam and Warnke 1975; Ottosen 1977), and plasticity based models (Chen 1976). Up to the middle of the 80s, the classical plasticity based models formed the majority of the constitutive models for concrete. Yet, they still could not capture the progressive degradation of the mechanical properties caused by the initiation and coalescence of micro-cracks. Notable developments in this area were the recognition that concrete is not a completely brittle material and crack orientation changes with loading history (Vecchio and Collins 1982). This led to the development of strain softening models. Hillerborg (1976) developed the fictitious crack model, which considers a tension softening fracture zone to avoid stress concentration at the tip of the crack. Recognizing that micro-cracking in the fracture process zone is not continuous, Bazant and Oh (1983) introduced the crack band model. Another significant contribution was the development of the rotating crack model (Cope et al. 1980; Gupta and Akbar 1983; de Borst and Nauta 1985; Rots 1988). The 80s have also seen the development of continuum damage mechanics as a framework for modeling degradation of the mechanical properties of concrete (Mazars 1984; Mazars and Pijaudier-Cabot 1989). When combined with plasticity, damage models; such as the ones proposed by Lubliner et al. (1989) and Lee and Fenves (1998), form a powerful class of models capable of describing the macroscopic behavior of concrete. The later have been implemented in the commercial finite element software ABAQUS (Simulia 2011).
Yet despite more than four decades of research and a large body of literature on the application of finite elements in the analysis of reinforced concrete, very few of these achievements have reached the design office; the finite element is still used as a verification tool rather than as a design tool. Hu and Lin (2006) analysed a PWR prestressed concrete containment vessel to verify its structural integrity under internal pressure. More recently, Syroka et al. (2011) and Mercan et al. (2010) analyzed reinforced concrete corbels and prestressed spandrel beams respectively, and on both occasions the finite element predictions were compared with experimental results. There have been few attempts at using the finite element method as a design tool for reinforced concrete structures (Tabatai and Mosalam 2001; An and Maekawa 2004; Khennane 2005). Unlike in other fields of engineering; such as metal forming, where numerical simulations are being conducted on a routine basis to design industrial parts (Khelifa et al. 2007), or the automobile industry, which simulates crash tests extensively even though it is possible to develop a product solely through prototyping, automated design of concrete structures has not attracted a lot of attention. This lack of interest can be explained by the difficulties associated with modeling the complex behavior of reinforced concrete, and by the fact that civil engineering structures are unique. The other likely reason is the impossibility to assess the validity of finite element designed structures with actual proof testing because of their size.
The aim of the present study therefore is to use the finite element method to design reinforced structural elements, which can be manufactured and subsequently tested to validate the designs. A deep beam is chosen because its behavior encompasses all the difficulties associated with modeling reinforced concrete structures under a state of generalized stress, and also for the ease of manufacture and testing. The technique, initially developed by Khennane (2005), which consists in using professional software because of user friendliness and proven reliability as opposed to “in-house” written software was adapted. The general purpose finite element code ABAQUS was selected. ABAQUS not only offers robust concrete models such as the one based on the concept of damaged plasticity theory developed by Lubliner et al. (1989) and Lee and Fenves (1998)., but it also comes with a scripting interface, which is an extension of the object oriented programming language Python (2011). For instance in ABAQUS, it is possible to write a Python script that automates the following tasks: creating and modifying the components of a model, such as parts, materials, loads, and steps; creating, modifying, and submitting analysis jobs; reading from and writing to the output databases; and, viewing the results of an analysis.
2.1 Finite Element Modelling
Before optimizing the reinforcement, it is necessary to develop a finite element model for the reinforced concrete deep beam. Two dimensional four-node continuum plane stress (CPS4) elements were used to model the concrete while two nodded truss elements were used for the reinforcement. Perfect bond was assumed between the reinforcement and the surrounding concrete. This was achieved by embedding the truss elements in the continuum elements representing the concrete. This type of representation allows the reinforcement to be treated as an integral part of the basic element, and its stiffness contribution can be evaluated using the principle of superposition.
The dilatation angle was assumed to be 50°. A detailed description of the CDPM model and its implementation can be found in the ABAQUS 6.10 documentation (Simulia 2011). It is worth to note that the CDPM is primarily intended for the analysis of concrete under cyclic/dynamic loading. And as such, it includes material softening and stiffness degradation, which at times, can lead to convergence difficulties. These can be avoided by using a viscoplastic regularization of the constitutive equations.
2.2 Design Methodology
The background and motivations for using modern computational methods for the design of reinforced concrete structures are presented. An algorithm making use of professionally developed finite element software is presented for the design of the reinforcement. It was used to design the reinforcement for a deep beam whose behavior encompasses the complexities of reinforced concrete structures, and which can be easily cast and tested in the laboratory to check the design. The rationale for the design is that the steel bars carrying the loads once the concrete is cracked should be strained as close as possible to the steel yield strain. Other criteria such a limiting deflection or a crack opening could be used separately or in conjunction with steel yielding, but that is more a case of generalization rather than one of principle.
To validate the design, two deep beam specimens with the obtained steel were cast and tested in the laboratory. It was found that the experimental results corroborated those predicted with the finite element design method. The failure load was found to be within 10 % of the target load. It was also found that this variation was consistent with experimental scatter. The measured mid-span displacements at failure were also consistent with those predicted by the method. It was found that the design resulted in a more ductile behavior. The measured steel strains were also found to be in the vicinity of the yield strains as anticipated by the method. Most importantly, it was experimentally proven that the method uses the reinforcing steel more efficiently.
Based on this work, it can be concluded that the current state of the art of the constitutive modeling of concrete is sufficient as to warrant the use of the finite element method as a design tool for reinforced concrete structures. Yet, one can still argue that the new design may not be economical after all as it involves a lot of cutting of steel bars to comply with the design. This may be true for a one-off job, but it is definitely not the case for the pre-cast industry that manufactures thousands of panels, where automated cutting and welding are used and the savings on steel could be substantial.
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