A reinforced concrete (RC) is a composite structure that consists of steel reinforcement and concrete with different material properties. A basic cha- racteristic of RC is that concrete, weak in tension, is reinforced by steel reinforcement, which is strong in tension. RC behaves as a composite structure under load, but when cracks are generated in concrete, it shows complicated nonlinear behavior in which the superposition is not generally applicable as in case of linear behavior. The main phenomena after cracking are bond action between reinforcement and concrete, aggregate interlock along the crack inter- face, dowel action by the local bending of reinforcement crossing cracks and compressive deterioration of cracked concrete.
Since the FEM has been developed initially for isotropic continuous material, its application to RC structures was extremely difficult, as it
becomes discontinuous after cracking. The first application of the FEM to RC was a crack analysis of RC beams by Scordelis and Ngo in 1967 (Ref. 5.5). They represented concrete and reinforcement separately using different sets of finite elements. Their models for cracking and bond slip could simulate physical phenomena splendidly. They investigated the propagation of shear cracking and the subsequent role of shear reinforcement in detail. It was unique that they could trace in detail the change of internal stress condition with load that had been difficult to observe in the physical experiment. This research gave a significant effect on the subsequent researches on RC.
Isohata and Takiguchi published papers on the application of FEM to RC shear problems in 1971. The research on the FEM analysis of RC structures in 1970's and 1980's was oriented, first, towards the formulation of constitutive laws for the modeling of material behavior of RC, and second, the application of FEM to clarify nonlinear behavior of RC members. The IABSE Colloquiums held in Delft, the Netherlands in 1981 (Ref. 5.6) was the first international conference in this area. Modeling of the material behavior of RC was discussed, and it was concluded that a future problem was to fill up the gap between FEM researchers and experimental researchers.
A committee on the shear strength of RC structures was established under the chairmanship of Okamura, the University of Tokyo in Japan Concrete Institute (JCI), from 1981 to 1984. The shear problem, an important pro- blem in the earthquake-resistant design, was discussed from the viewpoints of macroscopic models and microscopic FEM models. Publication of test data of selected test specimens for the verification of analytical models was a significant activity of the committee. It can be said that the research in this field was drastically advanced by the systematic research activities mainly by young committee members in only four years (Ref. 5.7).
In 1983, an international blind competition for the analytical prediction of behavior of RC panels was managed by Collins at the University of Toronto.
The experimental result was suppressed from disclosure during the analysis, and FEM researchers were asked to attend the competition and to submit analysis corresponding to the experiment. But many FEM researchers failed to predict the behavior with sufficient accuracy. Applicants with better predic- tion had confirmed the concrete compressive deterioration characteristics by their own biaxial tests before the analysis. Many analytical researchers of RC learned from this international competition that it is important to carry out basic experiments for the modeling and to evaluate the reliability of analytical
models. This experience was succeeded by the integrated research supported by the Ministry of Education Grant-in-Aids for Scientific Research, "Basic experiment on accuracy improvement of FEM analysis of RC structures and development of analytical models", from 1986 to 1989, represented by Morita of Kyoto University. A cooperative research group mainly composed of young researchers made lively discussions overcoming academic clique. The research fruits were presented in one session of the ASCE Structures Congress and also in the Tokyo seminar in 1989. The Tokyo seminar was very successful with more than 200 participants (Ref. 5.8).
The first US-Japan seminar on the FEM analysis of RC structures was held in Tokyo in 1985, and analytical models for applying FEM to RC structures (RCFEM) were discussed. It was characteristic for the US side to introduce the concept of fracture mechanics in their research reports. Aoyama and Noguchi reported future prospects of RCFEM. They indicated the necessity of the direct application of FEM to the practical design and the application of FEM to the development of macroscopic models and design equations as future research goals (Refs. 5.9 and 5.10).
In JCI committee on "FEM analysis and design method of RC structures"
under the chairmanship of Noguchi of Chiba University from 1986 to 1988, future problems indicated in the above US-Japan seminar were made to be activity goals. A design practitioner group published "Guideline on the appli- cation of FEM analysis to RC design" (Ref. 5.11). A researcher group verified the validity of previously proposed shear strength equations and macroscopic models of RC members by FEM analysis, aiming at the development of ratio- nal macroscopic models and design equations. A calculation method of shear strength derived from a macroscopic model was adopted in the Architectural Institute of Japan ultimate strength design guidelines, based on the activities of the above-mentioned JCI committee (Ref. 5.12).
The second US-Japan seminar on the FEM analysis of the RC structure was held in Columbia University, New York in 1991, and Japanese basic and systematic research on the application of RCFEM to development and design of new structures was introduced. A gap between FEM analytical researchers, experimental researchers and practical designers was discussed. Shirai reported on a detailed questionnaire results on the application of nonlinear FEM analysis to practical design, collected from design practitioners of thirteen construction companies in Japan. His report represented the characteristics of the Japanese research (Ref. 5.13).
Over three years from 1992 to 1995, the integrated research on "Reconstruc- tion of the shear design method of reinforced concrete structures by extremely precise FEM analysis" was carried out, represented by Noguchi of Chiba University and supported by the Ministry of Education Grant-in-Aids for Scientific Research. This was a cooperative research based on previous re- searches and by young generation researchers standing aloof from academic clique. The emphasis was placed on the application of the FEM analysis on the shear design of RC structures by making full use of basic researches.
5.2.2. Modeling of RC
When the FEM is applied to RC structures, it is necessary to consider the form that is easy to express characteristics of reinforced concrete structures with FEM (Refs. 5.14 and 5.15).
5.2.2.1. Two-Dimensional Analysis and Three-Dimensional Analysis In previous FEM analysis, two-dimensional analysis that assumes plane stress state or plane strain state is widely used except for special structures like nuclear pressure vessels. It has been applied not only to structures such as shear walls with explicit plane stress condition but also to beams, columns and beam-column joints, which do not necessarily exhibit plane stress or plane strain conditions. By progress of research on the constitutive laws and advance in computer hardware such as workstations, three-dimensional analysis has come to be gradually used. Three-dimensional stress flow is generated in a RC member subjected to two-directional input load, beam-column joints with lateral beams, concrete column confined with steel plates or lateral reinforcing bars, and footings. In these members, three-dimensional analysis is desirable for representing more realistic state of stress and deformation.
5.2.2.2. Modeling of Concrete
When Scordelis and Ngo applied the FEM to RC beams in 1967 for the first time, the model of a beam shown in Figs. 5.3-5.5 was used. It was two- dimensional analysis, and the plane stress condition was assumed. The con- crete was made to have a unit thickness except for reinforcement position.
Reinforcing steel was idealized into plane elements, and concrete elements over- lapping steel elements were modified to have reduced thickness.
UNIT . , WIDTH - 4 /
^ ^ " , - i f "
-AV&&0
Fig. 5.3. Analytical model for RC simple beam (Ref. 5.5).
Normal direction to crack surfaces
t Node Parallel direction
to crack surfaces
The same coordinate before cracking Fig. 5.4. Crack linkage element (Ref. 5.14).
Zero stiffness normal to cracks (a) Discrete Crack Model (b) Smeared Crack Model
Fig. 5.5. Crack models (Ref. 5.14).
Though concrete is a composite material composed of aggregate, sand and cement, it is usually handled as a uniform material like steel in the FEM analysis. In the two-dimensional analysis, triangle and quadrilateral elements are usually used. In the three-dimensional analysis, a layered shell element is often used, dividing the concrete into the thickness direction. This ele- ment can represent reinforcement layers. It is also possible to consider crack
propagation and concrete compressive failure by stiffness evaluation for each layer. However, out-of-plane shear deformation cannot be considered.
5.2.2.3. Modeling of Reinforcement
In the analytical example of Fig. 5.3, the reinforcement was expressed like a long column of a plane material. It was overlaid with a concrete layer, and connected by a link element that expressed bond behavior. According to the type of analysis, reinforcing bars can be represented by one of the following elements: a bar element like a truss or a beam, a layer in a shell element, a plane and a hexahedron solid. A reinforcement layer or a truss element is usually used, as the effect of bending stiffness and dowel action of a reinforcing bar is not so large.
5.2.2.4. Modeling of Cracks
In the analytical example of Fig. 5.3, concrete cracks were closely set in ad- vance to actual locations between elements. This expression method is called a discrete crack model. Concrete nodes on both sides of crack surfaces are connected by a crack link element that consists of two orthogonal springs, as shown in Fig. 5.4. A large value is given to the spring stiffness before the crack opens. After cracking, the spring stiffness in the orthogonal direction is set to zero, and the spring in the parallel direction is used to express shear transfer across the crack plane. The unique feature of the discrete crack model is that the crack width can be evaluated. It is effective when a small number of cracks will open like in case of shear tension failure of a beam with small amount of lateral reinforcement.
On the other hand, a smeared crack model handles the concrete as an orthogonal material with zero stiffness normal to crack directions in an ele- ment as shown in Fig. 5.4. In the smeared crack model, cracks are distributed uniformly in one direction in the element. It is not necessary to set a crack path before the analysis like discrete crack model. It is easy to divide an RC element into finite elements by using this model, and it is suitable for elements with many cracks widely distributed, such as a shear wall. However, spacing and width of cracks cannot be evaluated.
5.2.2.5. Modeling of Bond between Reinforcement and Concrete
Unless we can assume a perfect bond between reinforcement and concrete, it is necessary to express bond slip in the FEM model. For the expression of
bond, there are two ways. In the analytical example of Fig. 5.3, the bond slip is
represented by a bond link element with two orthogonal springs between nodes of reinforcement and concrete, as shown in Fig. 5.3. The spring stiffness along the longitudinal direction of reinforcement represents the bond characteristics, determined from bond stress and slip relationship. Characteristics of dowel action of reinforcement are represented by the spring stiffness normal to the longitudinal direction.
Another method is the tension-stiffening model. This model assumes that concrete can carry some tensile stress caused by bond after cracking. This method is used for members like shear walls in which reinforcing bars are arranged uniformly and bond slip is relatively small.