Nonlinear analysis of reinforced concrete

This paper discusses the development of two different computer programs for nonlinear analysis of reinforced concrete structures.The first program handles plane stress problems. Flow theory of plasticity is used in the modelling of concrete and reinforcement. General four-noded quadrilateral elements with selective sampling of strain are used in the discretization.The second program is developed for analysis of plates and shells. Endochronic theory is used in the constitutive law for concrete whereas an overlay model is utilized for the reinforcement. Geometric nonlinearities are accounted for through updating of coordinates for the triangular shell elements.Several examples of applications of the two programs are given. The plane stress program is used for analysis of a beam and two different corbels, while the shell program has been applied to a square plate and a shell with geometric nonlinearities.     

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.     

Nonlinear analysis of reinforced concrete structures

For the prediction of yield and failure of concrete under combined stress, a generalization of the Mohr-Coulomb behavior is made in terms of the principal stress invariants. The generalized yield and failure criteria are developed to account for the two major sources of nonlinearity: the progressive cracking of concrete in tension, and the nonlinear response of concrete under multiaxial compression. Using these criteria, incremental stress-strain relationships are established in suitable form for the nonlinear finite element analysis.For the analysis of reinforced concrete members by finite elements, a method is introduced by which the effect of reinforcement is directly included. With this approach, the stress-strain laws for the constituent materials of reinforced concrete are uncoupled permitting efficient and convenient implementation of a finite element program. The applicability of the method is shown on sample reinforced concrete analysis problems.     

Edge reinforced circular elastic inclusions in cylindrical shells

The effect of having an edge reinforcement around a circular elastic inclusion in a cylindrical shell is studied. The influence of various parameters of the reinforcement such as area of cross section and moment of inertia on the stress concentrations around the inclusion is investigated. It is found that for certain inclusion parameters it is possible to get an optimum reinforcement, which gives minimum stress concentration around the inclusion. The effect of moment of inertia of the reinforcement of SCF is found to be negligible. The results are plotted in a non-dimensional form and a comparison with flat plate results is made which show the curvature effect. In the limiting case of a rigid reinforcement the results tend to those of a rigid circular inclusion. Results are also presented for different values of μ_e the ratio of extensional rigidity of shell to that of the inclusion.     

3-D limit load analysis of reinforced concrete shells of arbitrary shape

A limit analysis method for thick reinforced concrete shells of arbitrary shape is developed using a 3-D concrete model based on a Mohr-Coulomb fracture theory in a solid-like isoparametric element. The proposed approach is well suited to engineering requirements as is illustrated by a HP shell case study     

Nonlinear static analysis of reinforced concrete frames by network models

The simulation of reinforced concrete frames by networks, with bars obeying uniaxial stres-strain laws of concrete or steel, is proposed. Formulae for the determination of concrete bar sections are derived. Concrete σ-ε law, including cracking and plastic behavior, is described by 3 constitutive variables; steel σ-ε law, including plastic behavior, is described by 1 constitutive variable. A simple program is presented for the nonlinear static analysis of such network models based on the incremental loading technique. This program is used for the analysis of a plane, one story reinforced concrete frame under cyclic horizontal loading of its girder, for which experimental data are available. The computational results are found in good agreement with the experimental ones.     

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.     

Free vibration analysis of circular cylindrical shells

Determination of the vibrational characteristics of circular cylindrical shells often requires significant computational effort. This paper presents the results of a comprehensive, computer based, numerical investigation of the free vibration of circular cylindrical shells. An analytical procedure which accurately predicts the natural frequencies and radial mode shapes (corresponding to axial wave number and circumferential wave number both equal to one) for a wide range of circular cylindrical shells is developed. The procedure is applicable to shells either with or without a top closure. Several numerical examples are presented which illustrate application of the procedure and verify its accuracy.     

Buckling analysis of ring-stiffened oval cylindrical shells

An energy principle is employed to derive the equations governing the stability of a simply-supported, eccentrically ring-stiffened, oval, orthotropic cylindrical shell. The kinematic relations used are those of Love-type shell theory and the effect of reinforcing rings is accounted for by a distributed stiffness approach. The cylinder is subjected to a combination of uniform axial and lateral pressures.

It is determined that the domain of stability of such a stiffened cylinder is bounded by two distinct solutions, herein denoted as corresponding to ‘long’ and ‘short’ axial wavelengths, with the extent of the short wavelength solution being dependent upon the degree of stiffening afforded by the rings.

The analysis of the effects of ring eccentricity shows that ovals are affected in a similar manner to circular cylinders in that outside rings provide the greatest capacity for sustaining axial compression, while inside rings are capable of supporting the greatest lateral pressure.

Finally, it is found that the buckling load of an oval cylinder under uniform lateral pressure slightly exceeds the corresponding value for an equivalent circular cylinder. As a further verification of this phenomenon, a Rayleigh-Ritz procedure is employed to determine the buckling load of an oval ring under uniform radial load. The results of this analysis corroborate those obtained for the cylinder.     

High-precision finite element analysis of cylindrical shells

This paper presents the finite element formulation to study the free vibration of cylindrical shells. The displacement function for the high-precision shell element with 16 degrees of freedom is approximated by a Hermitian polynomial of beam function type. The explicit formulation for the high-precision element is extremely efficient. For the purpose of comparison, the subject element is used to study the sample case of free vibration of a shell structure. The results are in good agreement with those published. The study shows that solution accuracy with fewer elements is assured and that accurate solutions are obtainable in the high-frequency range.     

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 three-dimensional resonance analysis of shells

This study details geometrically and physically nonlinear analysis of thin shells in resonance regions of vibration. Generalized analysis of motion with implementation of the FETM-method and of updated Lagrangian formulation is also studied. Utilization of the multigrid spatial simulation mesh for geometric representation of the shell as well as of the anisotropy of material is put forth. Special numerical techniques for solving nonlinear resonance equations of motion are presented. Illustrative numerical solutions of nonlinear resonance response of thin shell structures are performed.     

Linear and nonlinear stability analysis of cylindrical shells

The paper describes the application of a curved isoparametric shell element to large displacement analyses including instability phenomena. A total Lagrangian formulation has been adopted using the standard incremental/iterative solution procedure. The linear stability analyses usually performed for the initial position were repeated at several advanced fundamental states on the nonlinear prebuckling path. Thus a current estimate of the final failure load is given. The method has been applied to several perfect and imperfect cylindrical shells under uniform pressure or wind load. Finally the example of a cylindrical panel under one concentrated transverse load is discussed.     

Discrete energy method for the analysis of cylindrical shells

This paper presents an investigation into the use of the closely associated finite difference technique for the analysis of shell structures as a feasible alternative to the finite element method. The method discretises the total energy of the structure into energy due to extension and bending and that due to shear and twisting, contributed by two separate sets of rectangular elements formed by a suitable finite difference network. The derivatives in the corresponding energy expressions are replaced by their finite difference forms and the nodal displacements then constitute the undetermined parameters in the variational formulation. The formulation is also extended to a cylindrical shell element of rectangular planform. The results obtained by DEM are compared with existing results and they show excellent agreement.     

Probabilistic analysis of reinforced concrete columns

The subject of this work is the probabilistic finite element analysis of reinforced concrete columns. Concrete properties are represented as homogeneous Gaussian random fields. The yield stress and position of steel reinforcement, dimensions of the column cross-section and axial load are considered as random variables. The Monte Carlo method is employed to obtain expected values and standard deviations of the rupture load. The partial safety factors method is used for columns design and structural safety is evaluated by means of the reliability index, which is obtained through simulations. The effects of main parameters on the reliability index are investigated. It is shown that the correlation length of random fields for concrete properties may have a significant effect on reliability. Therefore, simplified procedures, which do not consider spatial variations of concrete properties are inappropriate for safety analysis.     

Nonlinear analysis of reinforced concrete plate and shell structures using 20-noded isoparametric brick elements

This paper describes an efficient and accurate, 3-D finite element model which may be adopted in the nonlinear analysis of reinforced concerete plate and shell structures. Benchmark tests are undertaken to check the computational model.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.     

Nonlinear analysis of shells using the MITC formulation

Summary The formulation of general shell elements using the method of mixed interpolation of tensorial components (MITC) is reviewed. In particular three elements that were formulated using the MITC method are examined: the MITC4 and MITC8 that were developed for general nonlinear analysis under the restriction of small strains and the MITC4-TLH that was developed for finite strain elasto-plastic analysis of shells. In memoriam of Juan Carlos Simo.     

Static analysis of cylindrical shells by using B spline functions

This paper is concerned with the application of cubic B₃ spline, quintic B₅ spline functions and eigen-functions which satisfy the boundary conditions to obtaining the approximate solution for the static analysis of a flat cylindrical shell. Unified computational schemes suited for various types of boundary conditions of cylindrical shell are presented in the paper. In comparison with the conventional finite element method and finite strip method, the main features of the present method are higher accuracy, less unknowns, easier programming and more economy in computing storage and time requirement. The numerical results presented are in good agreement with the exact solution.     

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.