Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method

This paper deals with some aspects related to three-dimensional numerical modelling of reinforced concrete structures using the Finite Element Method (FEM). Some subjects such as the solution technique of the non-linear equilibrium equations and the constitutive model for concrete and reinforcement steel are emphasised and commented. A robust method for the evaluation of the intersecting points of the embedded reinforcement bars into the three-dimensional finite element mesh is also presented. The main advantages of the Generalised Displacement Control Method with the Generalised Displacement Parameter to improve the response of the concrete and reinforced concrete analyses are highlighted. Finally, a series of numerical examples related to the above-mentioned aspects are presented.     

Simulation of continuous fibre reinforced thermoplastic forming using a shell finite element with transverse stress

A shell finite element with transverse stress is presented in this paper in order to simulate the forming of thermoplastic composites reinforced with continuous fibres. It is shown by an experimental work that many porosities occurs through the thickness of the composite during the heating and the forming process. Consequently the reconsolidation i.e. the porosity removing by applying a compressive stress through the thickness is a main point of the process. The presented shell finite element keeps the five degrees of freedom of the standard shell elements and adds a sixth one which is the variation in thickness. A locking phenomenon is avoided by uncoupling bending and pinching in the material law. A set of classical validation tests will prove the efficiency of this approach. Finally a forming process is simulated. It shows that the computed transverse stresses are in good agreement with porosity removing in the experiments.     

Nonlinear finite element analysis of reinforced concrete stiffened and cellular slabs

A novel approach is presented in this paper for linear and nonlinear finite element analysis of reinforced and prestressed concrete cellular slabs based on a slab-beam model. Mindlin/Timoshenko assumptions are adopted in the slab-beam model which thus allows for transverse shear deformations. Several examples are presented to illustrate the accuracy and limitations of the method.     

Behavior of reinforced concrete structures predicted by the finite element method

With current nonlinear analysis computer capabilities, considerable strides are being made in developing procedures, the utilization of which, allows one to analyze reinforced concrete structures while taking into account cracking and other characteristics of the constituent materials. Objective of the analyses is the determination of displacements as well as concrete and steel forces at various stages of loading. Cracking and the consequent loss of tensile strength is a major characteristic that must be modeled in any program for the analysis of reinforced concrete members. The three basic approaches have been employed by various investigators to account for tension cracking are discussed. Advantages and problems associated with these approaches are discussed. The formulation used to delineate compressive stress characteristics of concrete is evaluated. Solution procedures are described. Peculiarities required of solution methods in order to be suitable for the analysis of reinforced concrete systems are noted.     

Design optimization of simply supported concrete slabs by finite element modelling

This paper outlines the finite element prediction process for the development of charts for accurate peak load determination of simply supported, reinforced concrete slabs under uniformly distributed loading. Through a series of parametric studies using a simple concrete model, the simulation of tests on four simply supported slabs was used as a basis for establishing a set of optimum parameter values and computational conditions, which guarantees acceptable solution. The reliability of the established parameter values for prediction purposes was verified by the direct simulation of 11 other slabs. Following the successful reliability check, the finite element model was used for analysing 270 “computer model” slabs, from which charts were developed. These charts serve for quick and reliable peak load determination of arbitrary simply supported slabs. A comparative study of the direct finite element and chart predictions, with values from analytical and design methods, reveals the superiority of the charts over the latter methods, with accuracy comparable to that of the optimised finite element model. The chart prediction is noted to be accurate to within 4% of test results. A strategy for displacement determination is also established, with the same degree of success and the paper discusses possible practical applications of the developed finite element system.     

Flexural analysis of reinforced fibrous concrete members using the finite element method

A numerical procedure based on the finite element method is developed for the geometric and material nonlinear analysis of reinforced concrete members containing steel fibres and subjected to monotonic loads. The proposed procedure is capable of tracing the displacements, strains, stresses, crack propagation, and member end actions of these structures up to their ultimate load ranges. A frame element with a composite layer system is used to model the structure. An iterative scheme based on Newton-Raphson's method is employed for the nonlinear solution algorithm. The constitutive models of the nonlinear material behaviour are presented to take into account the nonlinear stress-strain relationships, cracking, crushing of concrete, debonding and pull-out of the steel fibres, and yielding of the reinforcement. The geometric nonlinearity due to the geometrical change of both the structure and its elements are also represented. The numerical solution of a number of reinforced fibrous concrete members are compared with published experimental test results and showed good agreement.     

Nonlinear finite element for modeling reinforced concrete columns in three-dimensional dynamic analysis

A new finite element is proposed for slender, flexure-dominated reinforced concrete columns subjected to cyclic biaxial bending with axial load, and its implementation into a program for the nonlinear static or dynamic analysis of structures in three-dimensions, is described. The element belongs to the class of distributed inelasticity discrete models for the nonlinear dynamic response analysis of frame structures to earthquake ground motions. The element tangent flexibility matrix is constructed at each time step by Gauss-Lobatto integration of the section tangent flexibility matrix along the member length. The tangent flexibility matrix of the cross-section relates the increment of the vector of the three normal stress resultants N, M_y, M_z, to the vector increment of the section deformation measures. ε_o, _y, _z, and is constructed on the basis of the bounding surface of the cross-section, which is defined as the locus of points in the space of the normalized N, M_y, M_z, which correspond to ultimate strength. The bounding surface concept enables the model to produce realistic predictions for the nonlinear response of the cross-section to any arbitrary loading path in the space N-M_y-M_z.The bounding surface is introduced and utilized in a very flexible manner, enabling a variety of cross-sectional shapes to be treated in a unified way. As this flexibility is at the expense of computational simplicity and memory size requirements, emphasis is placed on algorithmic techniques to facilitate numerical implementation and to increase computational efficiency.     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

Non-linear behaviour of reinforced concrete beams: From 3D continuum to 1D member modelling

Various levels of modelling of reinforced concrete beams are presented and compared. Their complementary character is outlined in some simple cases. Further studies are proposed.     

Nonlinear finite element analysis of curved beams

Mixed curved-beam finite elements are developed for the geometrically nonlinear analysis of deep arches. The analytical formulation is based on a form of the nonlinear deep-arch theory with the effects of transverse shear deformation and bending-extensional coupling included. The fundamental unknowns consist of the six internal forces and generalized displacements of the arch. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Numerical studies are presented to demonstrate the high accuracy of the solutions obtained by the mixed models and to show that their performance is considerably less sensitive to variations in the arch geometry than that of the displacement models.     

Simulation of the dynamic response of concrete beams externally reinforced with carbon-fiber reinforced plastic

Laboratory-scale concrete beams 3 × 3 × 30 in (7.62 × 7.62 × 76.2 cm) in size were impulsively loaded to failure in a drop-weight impact machine. The beams had no internal steel reinforcement, but instead were externally reinforced on the bottom or tension side of the beams with one-, two- and three-ply unidirectional carbon-fiber reinforced plastic (CFRP) panels. In addition, several of the beams were also reinforced on the sides, as well as the bottom, with three-ply CFRP. The beams were simply supported and loaded at beam midspan, and sustained dynamic loads with amplitudes up to 10 kips (44.5 kN) and durations less than 1 ms. Experimental measurements included total load, midspan displacement and strains, and a high-speed framing camera (10 000 frames s⁻¹) which gave insight into the failure mechanisms. Dynamic beam behavior was also studied numerically using the finite-element method. The numerical simulations gave insight into the localized displacement behavior of the beams when subjected to an intense, impulsive load, as well as the phenomenon known as the ‘traveling plastic hinge.’ Results showed that the numerical simulations did very well in predicting the displacement-time behavior of two beam types studied in detail; a plain concrete beam and a beam reinforced with three-ply CFRP on the bottom.     

New approach to optimization of reinforced concrete beams

The present paper outlines an application of genetic algorithm based strategies to a class of optimization tasks associated with the design of steel reinforced concrete structures. In this particular case, the principal design objective is to minimize the total cost of a structure. The resulting structure, however, should not only be marked with a low price but also comply with all strength and serviceability requirements for a given level of the applied load. To solve such a complex optimization problem with a number constraints calls for an efficient and yet reliable optimization technique. Here, the problem is addressed with the help of the augmented simulated annealing method. As an example, a simple continuous steel reinforced beam is analyzed to assess applicability of the proposed approach.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

A finite element model for thick beams

A general method for obtaining higher order beam theories is reviewed and cast in a form for creating a finite element model. Reissner's principle and Legendre polynomial series expansions are key features in the development. A thick beam element is produced having capabilities of representing nonlinear distributions, through the thickness, of all stress and deformation variables. The model can be used to analyze most thick beams and localized stress conditions. Beam problems are solved and the performance of the thick beam element model is assessed.     

Optimal force transmission in reinforced concrete deep beams

In the literature on the theory of optimal structural design several theories have been developed and discussed in most general terms but the examples solved to illustrate them are usually very simple and with little practical significance. Solutions of many more problems of practical significance are required to exploit the available theories to the fullest extent and, thus, to narrow the gap between the theoretical development and its application.In this paper the well established theory of pin-jointed frameworks is applied to rationally design non homogeneous structures the material of which is weak in tension and rigid, ideal plastic in compression. Several optimal solutions are obtained and their validity is discussed in qualitative terms in light of the existing experimental evidence.Although the discussion in this paper is limited to the deep beams and the reinforced concrete, the method is equally well applicable to other materials such as reinforced plastics and, thus, can be used in other fields of research.     

A finite element model for poroelastic beams with axial diffusion

The finite element method is used for those fluid-saturated poroelastic rods in which diffusion is possible only in the axial direction as a result of the microgeometry of the solid skeleton material. Variational principles are developed first for this purpose. Two types of variables, the displacements and pore pressure, are involved in the time dependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy’s law) into the Euler–Lagrange equations of the functionals. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solutions for both types of variables are found to coincide well with the existing analytical solutions. Some interesting results are demonstrated which are not available by analytical methods.     

A finite element formulation for beams with thin walled cross-sections

The paper presents a finite element model for calculation of stresses and deformations of beams with thin walled cross-sections. The beam model takes into account deformations due to shear. Warping is accounted for by a modified sector coordinate formulation. As interpolation functions between the seven degrees of freedom at each node are used the analytical solutions for the special case of a double symmetric cross-section. Therefore, depending on the external loading, each prismatic beam can in most cases be treated as a single element. The assembly of the beam elements to the global model is performed by use of transition matrices which assures compatibility between the elements in the sence of least squares.     

Weight optimization for flexural reinforced concrete beams with static nonlinear response

The weight optimization of reinforced concrete (RC) beams with material nonlinear response is formulated as a general nonlinear optimization problem. Incremental finite element procedures are used to integrate the structural response analysis and design sensitivity analysis in a consistent manner. In the finite element discretization, the concrete is modelled by plane stress elements and steel reinforcement is modelled by discrete truss elements. The cross-sectional areas of the steel and the thickness of the concrete are chosen as design variables, and design constraints can include the displacement, stress and sizing constraints. The objective function is the weight of the RC beams. The optimal design is performed by using the sequential linear programming algorithm for the changing process of design variables, and the gradient projection method for the calculations of the search direction. Three example problems are considered. The first two are demonstrated to show the stability and accuracy of the approaches by comparing previous results for truss and plane stress elements, separately. The last one is an example of an RC beam. Comparative cost objective functions are presented to prove the validity of the approach.     

Geometrically nonlinear finite element analysis of tapered beams

A novel finite element methodology is developed capable of analyzing the geometrically nonlinear behavior of thin-walled framed structures composed of non-prismatic members. The pertinent element matrices are formulated on the basis of a modified version of the variational theorem of Hellinger and Reissner. Finite geometry changes are consistently described by using an updated Lagrangian (U.L.) formulation. Validity, accuracy and reliability of the proposed scheme are examined on the basis of several well-selected test examples.