Nonlinear analysis of reinforced concrete hyperboloid cooling towers—II. Parametric study and results

In this second and final part of this series of papers the details of parametric studies conducted to assess the influence of various geometric and material parameters on the load-displacement response of three reinforced concrete hyperboloid cooling towers are presented. The material model adopted for the nonlinear finite element analysis is described in part I of this paper.     

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.     

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 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 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 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 finite element analysis of concrete structures using new constitutive models

Nonlinear finite element analysis was applied to various types of reinforced concrete structures using a new set of constitutive models established in the fixed-angle softened-truss model (FA-STM). A computer code FEAPRC was developed specifically for application to reinforced concrete structures by modifying the general-purpose program FEAP. FEAPRC can take care of the four important characteristics of cracked reinforced concrete: (1) the softening effect of concrete in compression, (2) the tension-stiffening effect by concrete in tension, (3) the average (or smeared) stress–strain curve of steel bars embedded in concrete, and (4) the new, rational shear modulus of concrete. The predictions made by FEAPRC are in good agreement with the experimental results of beams, panels, and framed shear walls.     

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 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.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

Nonlinear analysis of reinforced concrete cylindrical shells

In this paper, analysis of reinforced concrete cylindrical shells is performed using a strain-based finite element. The shell element employed is bidimensional, cylindrical circular and has four-nodes and five nodal degrees of freedom. The nonlinearities due to concrete cracking and yielding of the steel are taken into account. The constitutive models for the materials employ the smeared cracking concept and a finite element layered approach. Concrete is modeled by a strain-induced orthotropic-elastic model under plane state of stress. A bilinear steel model is used and the stress/reversal with Baushinger effect is included. Examples show the good accuracy provided by this analysis.     

Efficient finite element modelling of reinforced concrete beams retrofitted with fibre reinforced polymers

This paper presents a new simple and efficient two-dimensional frame finite element (FE) able to accurately estimate the load-carrying capacity of reinforced concrete (RC) beams flexurally strengthened with externally bonded fibre reinforced polymer (FRP) strips and plates. The proposed FE, denoted as FRP–FB-beam, considers distributed plasticity with layer-discretization of the cross-sections in the context of a force-based (FB) formulation. The FRP–FB-beam element is able to model collapse due to concrete crushing, reinforcing steel yielding, FRP rupture and FRP debonding.The FRP–FB-beam is used to predict the load-carrying capacity and the applied load-midspan deflection response of RC beams subjected to three- and four-point bending loading. Numerical simulations and experimental measurements are compared based on numerous tests available in the literature and published by different authors. The numerically simulated responses agree remarkably well with the corresponding experimental results. The major features of this frame FE are its simplicity, computational efficiency and weak requirements in terms of FE mesh refinement. These useful features are obtained together with accuracy in the response simulation comparable to more complex, advanced and computationally expensive FEs. Thus, the FRP–FB-beam is suitable for efficient and accurate modelling and analysis of flexural strengthening of RC frame structures with externally bonded FRP sheets/plates and for practical use in design-oriented parametric studies.     

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.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed 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.     

Nonlinear finite element analysis of shells-part II. two-dimensional shells

A general nonlinear finite element formulation is given for two-dimensional problems. The formulation applies to the practically important cases of shells of revolution, tubes, rings, beams and frames. The approach is deduced from a corresponding three-dimensional formulation [4] and this enables a simplified implementation, especially with respect to constitutive software. Uniform reduced-integration Lagrange elements are employed and shown to be very effective for the class of problems considered.     

A finite element model for the nonlinear analysis of reinforced and prestressed masonry walls

A materially nonlinear layered finite element model is proposed for the analysis of reinforced and/or prestressed masonry wall panels under monotonie loadings in the plane and/or out of the plane, capable of evaluating both the serviceability load and the ultimate load. An orthotropic incrementally linear relationship and equivalent uniaxial concept are used to represent the behaviour of masonry under biaxial stresses while a uniaxial bilinear elasto-plastic model with hardening is employed for rebar and the so-called ‘power-formula’ is adopted to describe the stress-strain relationship of prestressing steel.

After cracking, the smeared coaxial rotating crack model is adopted and tension stiffening, reduction in compressive strength and stiffness after cracking, and strain softening in compression are accounted for. The modified Newton-Raphson iteration method is employed to ensure convergency of non linear solution.

The proposed finite element model has been tested by a comparison with experimental data available in literature, both for reinforced and prestressed wall panels. The analysis of results shows good agreement between the values obtained by the proposed model and those obtained experimentally.     

Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part I. Quasistatic problems

The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part.The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation.The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].     

汽车钢板弹簧非线性有限元分析

为在汽车动力学特性设计中准确计算汽车悬架的作用力,用HyperMesh建立钢板弹簧的有限元模型,并在钢板弹簧的接触面中添加接触单元模拟板间摩擦.利用Abaqus计算钢板弹簧刚度,考虑中心螺栓和U形螺栓的预紧力,分析不同摩擦因数对钢板弹簧迟滞特性的影响.计算结果表明该模型可用于分析考虑钢板之间摩擦作用时钢板弹簧的受力情况,探讨非线性有限元分析过程中钢板弹簧的刚度和迟滞特性.