Effect of loading types and reinforcement ratio on an effective moment of inertia and deflection of a reinforced concrete beam

In the design of reinforced concrete structures, a designer must satisfy not only the strength requirements but also the serviceability requirements, and therefore the control of the deformation becomes more important. To ensure serviceability criterion, it is necessary to accurately predict the cracking and deflection of reinforced concrete structures under service loads. For accurate determination of the member deflections, cracked members in the reinforced concrete structures need to be identified and their effective flexural and shear rigidities determined. The effect of concrete cracking on the stiffness of a flexural member is largely dependent on both the magnitude and shape of the moment diagram, which is related to the type of applied loading. In the present study, the effects of the loading types and the reinforcement ratio on the flexural stiffness of beams has been investigated by using the computer program developed for the analysis of reinforced concrete frames with members in cracked state. In the program, the variation of the flexural stiffness of a cracked member has been obtained by using ACI, CEB and probability-based effective stiffness model. Shear deformation effect is also taken into account in the analysis and the variation of shear stiffness in the cracked regions of members has been considered by employing reduced shear stiffness model available in the literature. Comparisons of the different models for the effective moment of inertia have been made with the reinforced concrete test beams. The effect of shear deformation on the total deflection of reinforced concrete beams has also been investigated, and the contribution of shear deformation to the total deflection of beam have been theoretically obtained in the case of various loading case by using the developed computer program. The applicability of the proposed analytical procedure to the beams under different loading conditions has been tested by a comparison of the analytical and experimental results, and the analytical results have been found in good agreement with the test results.     

Nonlinear analysis of reinforced concrete hyperboloid cooling towers—I. Material model, finite element model and validation

This is the first part of a two-part series of papers in which the constitutive material modelling of reinforced concrete, in shell structures, which resist applied loads predominantly through membrane action, is presented. The material model includes the effects of tensile cracking, tension stiffening, compression softening, interface shear transfer, and change in material stiffness due to crack rotation. A four-noded isoparametric curved shell element has been used in the nonlinear finite element analysis. The results obtained by using the model for analysis of a shear wall panel subjected to in-plane loading have been compared with those from experimental investigation.     

A numerical model for the flexural analysis of short reinforced masonry columns including bond–slip

A numerical model for the flexural analysis of short reinforced masonry (RM) columns subjected to vertical and lateral loading is presented. A layered line element formulated for the analysis of reinforced concrete beam columns by the authors has been suitably modified for this purpose by incorporating the material nonlinearities associated with the RM. The modelling of the RM column accounts for the effect of the early cracking in the mortar joint and the bond–slip characteristics between the steel reinforcement and the masonry grout in addition to the material nonlinearity of the masonry and the steel. The model has been validated by comparing its prediction with the load–deflection response of four RM columns tested under varying levels of the axial load. The importance of including bond–slip characteristics in the modelling is demonstrated through the examples.     

Time-dependent modelling of RC structures using the cracked membrane model and solidification theory

A non-linear finite element model is presented for the time-dependent analysis of reinforced concrete structures under service loads. For the analysis of members in plane stress, the model is based on the cracked membrane model using a rotating crack approach combined with solidification theory for modelling creep. The numerical results are compared with a variety of long-term laboratory measurements, including development of deflections and cracking with time in a reinforced concrete beam, time-dependent change in support reactions of a continuous beam subject to support settlement and creep buckling of columns. The numerical results are in good agreement with the test data.     

Application of ADINA to nonlinear analysis of reinforced concrete shear walls with openings

This paper is concerned with application of ADINA to elasto-plastic analysis of the shear walls with openings. The authors analyzed the types of structures. One is the shear wall with many openings (the model of a secondary shield wall in nuclear power plant), on which scale model experiments were made. The other is the shear wall with openings in concrete rigid frame (the model of a shear wall in a building), on which parametric study was made.

In both cases, concrete is modeled using 8 nodes isoparametric 2 dimensional plane stress elements, reinforcing steels are modeled as truss elements. Concrete and elasto-plastic models are adopted for non-linear material model of concrete and reinforcing steel, respectively. The total numbers of nodes are 248–308, and that of 2D elements are 66–80.

Both analytical results are satisfactory from the view point of structural design. Close agreement to experimental results in the cracking load, crack extension, elasto-plastic stiffness and total strength was verified.     

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 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.     

Numerical analysis of sandwich beams

This paper describes the development of a numerical model for the physical nonlinear analysis of simply supported sandwich beams, specifically with foamed-concrete cores and concrete faces. The long-term behaviour is included in view of creep and shrinkage of both faces and core. The structural behaviour of sandwich beams is described by a fourth-order differential equation in the deformation w and a second-order differential equation in the shear deformation of the core γ_k. The flexural stiffness of the core is taken into account. The general-solution procedure is based on the finite-difference method, together with a successive-substitution algorithm using the secant flexural moduli of the core and faces and the secant shear modulus of the core. The option of tension stiffening is incorporated to represent the nonlinear behaviour of reinforced concrete in tension. The tension stiffening is numerically calculated from a distributed tensile load instead of a load acting on both ends of the reinforced bar. Creep and shrinkage are calculated separately from the differential equations with an algorithm based on increments of time. With the presented model, the time-dependent deflection along the axis of the beam and the state of stress in every fibre can be calculated.     

Minimum cost design of reinforced concrete beams using continuum-type optimality criteria

This paper outlines a general procedure for obtaining, on the basis of continuum-type optimality criteria (COC), economic designs for reinforced concrete beams under various design constraints. The costs to be minimized include those of concrete, reinforcing steel and formwork. The constraints consist of limits on the maximum deflection, and on the bending and shear strengths. However, the formulation can easily cater for other types of constraints such as those on axial strength. Conditions of cost minimality are derived using calculus of variation on an augmented Lagrangian. An iterative procedure based on optimality criteria is applied to a test example involving a reinforced concrete propped cantilever beam whose cross-section varies continuously. Numerical examples are presented in which the design variables are both the width and the depth or the depth alone, and the optimal costs are compared. The solution of the test example with depth alone as the design variable is confirmed by an alternative approach using discretized continuum-type optimality criteria (DCOC).     

Design optimization of 3D reinforced concrete structures having shear walls

This paper presents a computer-based method for the optimal design of three-dimensional Reinforced Concrete (RC) structures having beams subjected to shear force and bending moment, columns subjected to biaxial moments, biaxial shears and axial loads, and shear walls subjected to pure shear. Regarding the beams and columns, the design variables are the width, depth and area of longitudinal reinforcement of member sections. The design variables for the shear walls are the thickness of the wall, the area of vertical reinforcement, horizontal distance between the vertical stirrups, the area of horizontal reinforcement, vertical space between the horizontal stirrups, and the area of vertical flexural reinforcement. The Optimality Criteria (OC) method is applied to minimize the cost of the concrete, steel and formwork for the structure. ACI code [1] provisions concerning the strength and ductility of beams, columns and shear walls are taken as constraints. The constraints also impose upper and lower bounds on the dimensions of beams and columns, and on shear wall thickness, reinforcement area and the maximum and minimum vertical and horizontal spaces between the stirrups of the shear walls. Sensitivity analysis is conducted for both internal forces and the capacities of the sections of the beams, columns and shear walls. The features of the design method are illustrated by a solved example.     

Minimum cost design of RC beams using DCOC Part I: Beams with freely-varying cross-sections

A procedure for the economic design of reinforced concrete beams under several design constraints is outlined on the basis of discretized continuum-type optimality criteria (DCOC). The costs to be minimized involve those of concrete, reinforcing steel and formwork. The design constraints include limits on the maximum deflection in a given span, on bending and shear strengths, in addition to upper and lower bounds on design variables. An explicit mathematical derivation of optimality criteria is given based on the well known Kuhn-Tucker necessary conditions, followed by an iterative procedure for designs when the design variables are the depth and the steel ratio, or the depth alone. The computer code developed in Part I can handle freely-varying design variables along the members of any multispan beam. In Part II the DCOC and computer code are developed for designs when the member cross-section is assumed to be uniform along its entire length. Several test examples have been solved to prove the accuracy and efficiency of the DCOC-based techniques.     

Numerical modeling of masonry-infilled RC frames subjected to seismic loads

The behavior of masonry-infilled reinforced concrete frames under cyclic lateral loading is complicated because a number of different failure mechanisms can be induced by the frame-infill interaction, including brittle shear failures of the concrete columns and damage of the infill walls. In this study, nonlinear finite element models have been used to simulate the behavior of these structures. Diffused cracking and crushing in concrete and masonry are described by a smeared-crack continuum model, while dominant cracks as well as masonry mortar joints are modeled with a cohesive crack interface model. The interface model adopts an elasto-plastic formulation to describe the mixed-mode fracture of concrete and masonry. The model accounts for cyclic crack opening and closing, reversible shear dilatation, and joint compaction due to damage. The constitutive models have been validated with experimental data and successfully applied to the dynamic analysis of a three-story, two-bay, masonry-infilled, non-ductile, reinforced concrete frame tested on a shake table. The results have demonstrated the capabilities of the finite element method in capturing the nonlinear cyclic load–displacement response and failure mechanisms of the structure, and indicated the important contribution of infill walls to the seismic resistance of a non-ductile reinforced concrete frame.     

Minimum cost design of RC beams with segmentation using continuum-type optimality criteria

Economic designs of reinforced concrete beams with segmentation involving costs of concrete, reinforcing steel and formwork are considered. For practical reasons, the design variables are prescribed by means of an unknown constant and given shape function, whereas the constraints in addition to lower and upper bounds on design variables include maximum deflection in the span, flexural and shear strengths. Conditions of minimality are rigorously derived using calculus of variation on an augmented Lagrangian. An iterative procedure describing the computational aspects of the design is presented using a segmented propped cantilever beam as an example. Numerical examples are presented when the design variables involved are the depth alone or both the depth and steel ratio. A practical application of a simply supported beam is also considered and results compared with those obtained using nonlinear programming techniques (Karihaloo 1993; Kanagasundaram and Karihaloo 1990). An examination of the rate of convergence shows that the present technique is very promising.     

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.     

Design of framed foundation subjected to vibration

The potential effects of repeated loads on the stiffness and the natural frequencies of framed foundations which support vibrating equipment such as compressors or turbine generators are described. The reason for the reduction of stiffness of concrete under repeated loads is explained. A two-dimensional element and a bar element are utilized to idealize reinforced concrete structures. Principal stresses are checked to determine the status of cracking of concrete. If the principal stress is greater than the tensile capacity of concrete, a stress crack is assumed to develop in the perpendicular direction to the corresponding stress. The element stiffness is modified to account for the crack. The modulus of elasticity of concrete is modified under repeated loads according to the stress state. The modulus of elasticity of concrete is assumed to be constant in compression, but varying in tension. Example beams are analyzed to study the effect of the reduction of modulus in tension zone. The reduction of modulus is found to give significant effect on the overall stiffness and the natural frequency of the structure.     

Analysis of cyclic loading of plane RC structures

A numerical procedure for cyclic loading response of planar reinforced concrete structures is presented. A nonlinear orthotropic stress strain law for biaxially loaded plain concrete is developed and compared with experimental results for monotonic biaxial loading and uniaxial cyclic loading. The stress-strain law recognizes strength and ductility changes due to biaxial stress, and strength and stiffness degradation with cycles of loading. The stress strain law is incorporated into a finite element computer program which utilizes isoparametric quadrilaterals with extra non-conforming deformation modes. Numerical and experimental results are presented for a monotonically loaded shear wall-frame system and a cyclically loaded shear wall.     

Embedded fiber optic sensor for characterization of interface strains in FRP composite

Fiber reinforced polymer composites (FRP) have found widespread usage in repair and strengthening of concrete structures. FRP composites exhibit high strength to weight ratio, corrosion resistance, and convenient to use in repair applications. Several methods have been devised for repair of concrete beams with FRP fabrics, including wrapping of the cracked members, or adhesion of the fabric to the tension face of the members. A common cause of failure in such members is associated with debonding of the FRP substrate from the concrete in an abrupt manner. The mechanism of debonding is investigated through embedment of a distributed optical fiber sensor at the interface between the cracked concrete and the FRP fabric during repair of reinforced concrete beams under load. The fiber optic system consists of segmented long gauge length sensors along the length of an embedded optical fiber. This arrangement allows for complete interrogation of the interface deformations and debonding phenomenon. Experimental results pertaining to the load testing of FRP-repaired reinforced concrete beams are reported.     

Design optimization of 3D reinforced concrete structures

This paper presents a computer-based method for the optimal design of three-dimensional reinforced concrete (RC) skeletal structures having members subjected to biaxial moments, biaxial shears and axial loads. The width, depth and area of longitudinal reinforcement of member sections are taken as the design variables. The optimality criteria (OC) method is applied to minimize the cost of the concrete, steel and formwork for the structure. The primary focus of the paper concerns fundamental issues related to the formulation of design performance constraints on combined axial load, biaxial moments and biaxial shears. An example problem is solved with and without account for biaxial shear constraints to illustrate their influence on the design.     

Artificial intelligence-enhanced seismic response prediction of reinforced concrete frames

Existing physics-based modeling approaches do not have a good compromise between performance and computational efficiency in predicting the seismic response of reinforced concrete (RC) frames, where high-fidelity models (e.g., fiber-based modeling method) have reasonable predictive performance but are computationally demanding, while more simplified models (e.g., shear building model) are the opposite. This paper proposes a novel artificial intelligence (AI)-enhanced computational method for seismic response prediction of RC frames which can remedy these problems. The proposed AI-enhanced method incorporates an AI technique with a shear building model, where the AI technique can directly utilize the real-world experimental data of RC columns to determine the lateral stiffness of each column in the target RC frame while the structural stiffness matrix is efficiently formulated via the shear building model. Therefore, this scheme can enhance prediction accuracy due to the use of real-world data while maintaining high computational efficiency due to the incorporation of the shear building model. Two data-driven seismic response solvers are developed to implement the proposed approach based on a database including 272 RC column specimens. Numerical results demonstrate that compared to the experimental data, the proposed method outperforms the fiber-based modeling approach in both prediction capability and computational efficiency and is a promising tool for accurate and efficient seismic response prediction of structural systems.