Simultaneous shape and topology optimization of prestressed concrete beams

This paper presents a new optimization approach for the design of prestressed concrete beams. The prestressing tendon is modeled as a chain of linear segments that transfer point forces to the concrete domain according to the tendon’s angles. The concrete beam is modeled as a discretized continuum following density-based approaches to topology optimization. The shape of the tendon and the topology of the surrounding concrete are optimized simultaneously within a single problem formulation. A special filtering technique is developed in order to ensure that the tendon is covered by concrete, thus shape and topological variables are tightly coupled. Several test cases demonstrate the applicability of the proposed optimization procedure. The deformation of the optimized designs due to external loads is counteracted by the deformation due to prestressing, hence by tuning the force in the tendon the total deformation can approach zero. Consequently, the beams exhibit a compression-only response meaning that the common goal of prestressed concrete design is achieved.     

Optimization of prestressed concrete solid and voided slabs

A methodology is presented for the minimum cost optimization of prestressed concrete solid and voided slabs. Apart from the rationale imparted to the design process, the method is shown to yield considerable savings in prestressing steel and therefore savings in energy. Computer programs developed for the implementation of the methodology have facilitated the speedy running of the procedure a myriad of times over a wide range of design parameters such as span, live load, material and erection cost ratios, concrete and strand strengths, prestress effectiveness etc. Qualitative and quantitative conclusions regarding the behavioural characteristics of the parameters and the manner of their interplay in the optimal picture are advanced.     

An approximate nonlinear design procedure for partially prestressed concrete beams

The theoretical basis and the main results of an approximate nonlinear design procedure which attempts to predict the behavior at ultimate of prestressed and partially prestressed concrete members is presented. Difficulties encountered in simulating the actual behavior of concrete and steel are discussed; particular emphasis is put on comparing the various available stress-strain relationships for prestressing steels, and observed variability in their actual properties. Recommended equations and corresponding coefficients are given. Typical results showing the influence on ultimate behavior of major variables such as the type and amount of prestressing steel, the amount of nonprestressed tensile and compressive reinforcement, and the ultimate concrete compressive strain are presented and compared with code derived results. They help point out where the code recommendations, if used, may lead to an unsafe design and what modifications may be needed to make them more representative of observed trends.     

Optimum design of prestressed concrete beams using constrained differential evolution algorithm

This paper is concerned with the cost minimization of prestressed concrete beams using a special differential evolution-based technique. The optimum design is posed as single-objective optimization problem in presence of constraints formulated in accordance with the current European building code. The design variables include geometrical dimensions that define the shape of the cross section and the amount of prestressing steel. A special (μ?+?λ)-constrained differential evolution method is performed in order to solve the optimization problem. Its search mechanism depends on several mutation strategies whereas an archiving-based adaptive tradeoff model is in charge of selecting a specific constraint-handling technique. Finally, numerical examples are included to illustrate the application of the presented approach.     

VTOP. An improved software for design optimization of prestressed concrete beams

Optimum tendon layout of prefabricated prestressed concrete beams was an early application of design optimization methodologies. But most of the approaches found in the literature were not suitable to be implemented in real life bridge engineering. To fill the existing gap a research has been conducted in the past years by the authors to optimize prefabricated concrete beams formulating the problem in such a way that could be applied directly by bridge designers in beam and slab bridge decks. The software produced, labelled VTOP, contains the necessary capabilities for daily applications and possesses a user friendly graphical interface. This paper describes the problem formulation carried out and includes several examples to show the efficiency of the computer code implemented. Some of them are examples of single prefabricated concrete beams and also an example showing the overall analysis of a bridge deck and the subsequent optimization of the prefabricated beam subjected to the internal forces produced by the external loading is included in the paper.     

Damage assessment of prestressed concrete beams using artificial neural network (ANN) approach

An artificial neural network (ANN) based approach is presented for the assessment of damage in prestressed concrete beams from natural frequency measurements. The details of an experimental programme suitably designed and carried out to induce the desired extents of damages in the prestressed concrete beams and generate the training and test data for the ANN are presented. The analysis of the static and dynamic behavior of perfect and damaged prestressed concrete beams reveal that there exists a close relationship among the natural frequency, deflection, crack width, first crack load, ultimate load and degree of damage. Therefore, these parameters were mainly used as input data for training and testing the ANN. A feed forward ANN learning by back propagation algorithm implemented using MATLAB has been employed in this study. The main focus of this work has been to study the feasibility of using an ANN trained with only natural frequency data to assess the damage in prestressed concrete beams. This is explored by comparing the performance of an ANN trained only with natural frequency data with other ANNs trained with a mix of static and dynamic data. It has been demonstrated that an ANN trained only with dynamic data can assess the damage with less than 10% error, when the error is the difference between the actual damage in percent and predicted damage in percent. The shortcomings of this study have also been presented.     

Interactive CAD of prestressed concrete using a mini-computer

An interactive approach to the design of prestressed concrete members and structures is described. This has been developed for use on an office-based mini-computer system and programs are suitable for engineers who have little experience of computing. Emphasis is placed on the use of visual display facilities to present the results of calculations necessary for design decisions, and data are stored in disk files which are automatically accessed as required. Individual program modules are produced for each separate stage of the design process, and these can either be used to undertake isolated calculations or can be combined to form suites for designing complete structures. Attention is paid to the simplification of analyses wherever possible both to minimize time delays and to accommodate limits of computer size. It is found that in this way complex designs can be produced with much greater flexibility than is possible with automatic approaches, and much more quickly than by manual methods.     

Modular interactive analysis-design of prestressed concrete structural elements

A computer system for interactive analysis-design of pretensioned prestressed concrete members has been developed adding another working tool to the designer's collection of design aids. The interactive concept allows the designer to interact with the computer, making decisions that might be impractical or impossible to include in a program but allowing the computer to carry out rigorous computations. The process of designing pretensioned prestressed concrete members is organized into distinct phases. In order to produce an optimum design, various phases of the design process may have to be carried out numerous times depending on the effect of the design variables that are changed during the process. The modular interactive design system allows each module to be examined in any order and in any frequency after initial data has been input. This flexibility has been obtained by treating each module as a subroutine called by a control module. The system is designed to communicate with the designer in a conversational mode. Instructions are given to the user before requiring a response and the user has before him sufficient information to make necessary decisions. The present system can be used for the analysis-design of simple span beams and one-way slabs, with or without cantilever end spans, and of structures where the dead load is carried by simple beam action and superimposed dead and live loads are carried by continuous beam action where mild steel is provided to develop continuity over interior supports. The system is designed to be used on a minicomputer.     

Computer aided design of prestressed concrete highway bridges

The paper presents a computer aided design system for prestressed concrete highway bridges which, starting from few geometrical data, provides the complete geometry, prestressing steel, reinforcing steel, amount of materials and cost of all the bridge elements: deck, bearings, piers, abutments and foundations. Different configurations are devised, from short and medium to long-span bridges, accounting for different deck super-structures and erection methods. All the results are displayed on the computer screen and can be printed. The system provides also DXF files containing the general layout, cross-sections and prestressing arrangement of box girder bridges. This system allows, in a short time, an accurate design and an economical estimation of a particular bridge, taking into account the most important technical requirements. It is a useful decision-making tool for both design and administration engineers.     

An interactive design algorithm for prestressed concrete girders

The primary purpose of this paper is to discuss an interactive design and analysis algorithm for prestressed concrete girders. Prestressed concrete highway bridge girder design is used for the prototype computer program to simplify the incorporation of design code requirements and loading conditions. The computer code can be extended to include other prestressed concrete girder applications. The second purpose arises from the search for the optimum prestressed concrete girder design. Linear programming is discussed as a possible method to arrive at the optimum girder cross-section and prestressing strand design. However, manufacturing standardization and techniques make selection of the optimum crosssection, prestressing force, and strand centroid eccentricity by mathematical methods rather academic. Therefore, design optimization, to be practical, must be based on standard cross-sections and prestressing strand position templates. The algorithm, guided by the engineer, selects, from tabulated standard crosssections and associated combinations of prestressing forces and eccentricities, the cross-section, prestressing force, and eccentricity to satisfy the problem constraints. The kern boundaries are calculated in the analysis portion of the algorithm. The engineer, using the kern boundaries, determines the path of the strands by specifying the strand hold-down points and associated strand centroid eccentricity. The algorithm also provides for shear reinforcement, dead load deflections at transfer and after placement of the slab, and auxiliary nonprestressed tension reinforcement at transfer.     

Optimal design of prestressed concrete pipes using linear programming

Non cylinder composite typ: prestressed concrete pipes extensively used in India for high pressure water supply lines are being designed at present by the conventional ‘analysis’ type structural design process. Geometric properties are assumed and then analysis is carried out, for checking accepted design behaviour or safe pressure carrying capacity. The authors here, have considered design behaviour at various loading stages and pressures, as an input in the design process called ‘synthesis’ and geometric properties are generated as an output. Optimal design solution is found using well known linear programming technique. Illustrative design example considering Veeranam Lake—Madras 230 km pipe line, shows substantial economy over conventionally designed pipes.     

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.     

Optimization of rigid-plastic shallow curved beams

The minimum weight problem of a shallow circular beam is studied in the case when the beam has a piece-wise constant thickness. The minimum of the weight is sought under the condition that the deflections of the beam of piece-wise constant thickness do not exceed those of the reference beam of constant thickness for given values of the external loading. The beam is subjected to uniformly distributed transverse pressure and to axial dead load. The material of the beam is assumed to be ideally rigid-plastic. Moderately large deflections are taken into account. Necessary optimality conditions are derived and used in order to establish the optimal values of the design parameters.     

Finite element analysis of reinforced and prestressed concrete structures including thermal loading

This paper describes the application of finite element techniques to the solution of nonlinear concrete problems. Reinforced concrete thick plates and shells are first considered for which both a perfect and strain-hardening plasticity approach are employed to model the compressive behaviour. A dual criterion for yielding and crushing in terms of stresses and strains is considered, which is complemented with a tension cut-off representation. Degenerate thick shell elements employing a layered discretisation through the thickness are adopted and both reduced and selectively integrated 8-node serendipity and heterosis elements are considered.Thermal loading of prestressed concrete structures is also considered which necessitates the inclusion of time effects in the analysis. The technique described in this paper involves concurrently solving an uncoupled set of equations within a time interval to provide both the displacement and temperature increments. A two-level time stepping scheme is employed to predict temperature changes within a time interval and elasto-viscoplastic material analysis is performed using an explicit forward-difference scheme incorporating an equilibrium iteration procedure. The constitutive model for the concrete is essentially identical to that employed for the plate and shell analysis.Numerical examples are presented for both types of analysis and comparison is made with experimental results whenever possible. Additionally, results for thermal loading are presented which indicate that a full transient thermal-mechanical analysis is sometimes essential in order to obtain a realistic structural response.     

Dynamic analysis of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load

The dynamic response of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load is studied by using Lagrange equations. In the study, for using the Lagrange equations, trial functions denoting the deflection of the beam and the rotation of the cross-sections are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The effects of the value of the eccentricity of the compressive load, the excitation frequency, the constant velocity of the transverse moving harmonic load and viscous damping of the material of beams are studied in detail. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with exact solutions based on the Euler–Bernoulli beam theory obtained for the special cases of the investigated problem.     

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.     

Minimum cost design of multispan partially prestressed concrete T-beams using DCOC

A previous study on the minimum cost design of multispan partially prestressed concrete (PPC) beams using DCOC is extended to multispan T-beams. The cost of construction which includes the costs of concrete, prestressing steel, nonprestressing steel and formwork is minimized. The design constraints include limits on the maximum deflection, flexural and shear strengths, in addition to ductility requirements, and upper and lower bounds on design variables as stipulated by the Australian Code AS 3600. Based on Kuhn-Tucker necessary conditions, the optimality criteria are explicitly derived in terms of the design variables-effective depth, eccentricity of prestressing steel and non-prestressing steel ratio. The prestressing profile is prescribed by parabolic functions. The self-weight of the structure is included in the equilibrium equation of the real system, as is the secondary effect resulting from the prestressing force. An iterative procedure for updating the design variables is outlined. Two numerical examples of multispan PPC beams with T crosssection are solved to show the applicability and efficiency of the DCOC-based technique.     

Design optimization of prestressed concrete spans for high speed ground transportation

The problem of designing stiff, lightweight, and least expensive elevated spans to support ground vehicles is investigated. A computer program using the direct search method was developed to calculate optimum geometric configurations of prestressed concrete girders with nonlinear constraint conditions involving stresses and deflections; with specified inputs on loading, unit costs and overall size; and with checks on buckling, shear and ultimate section strength. Parameters allow-for choices between stiff, expensive configurations or flexible, less expensive designs. Several numerical results for simple spans are included.     

Optimum topology and shape design of prestressed concrete bridge girders using a genetic algorithm

The purpose of this study is to optimize the topology and shape of prestressed concrete bridge girders. An optimum design approach that uses a genetic algorithm (GA) for this purpose is presented. The cost of girders is the optimum design criterion. The design variables are the cross-sectional dimensions of the prefabricated prestressed beams, the cross-sectional area of the prestressing steel and the number of beams in the bridge cross-section. Stress, displacement and geometrical constraints are considered in the optimum design. AASHTO Standard Specifications for Highway Bridges are taken into account when calculating the loads and designing the prestressed beams. A computer program is coded in Visual Basic for this optimization. Many design examples from various applications have been optimized using this program. Several of these examples are presented to demonstrate the efficiency of the algorithm coded in the study.     

Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending

A secant approach is illustrated for the ultimate limit state (ULS) analysis of prestressed reinforced concrete sections subjected to axial load and biaxial bending in presence of softening stress–strain laws. The stiffness matrix and the resultant loads are evaluated analytically by a novel methodology, termed fiber-free, which represents a computationally efficient alternative to fiber approaches. Extensive computations of the ULS domains of benchmark test cases show that the robustness of the proposed algorithmic strategy is substantially unaffected by the amount of reinforcement, prestressing and softening, though localized non-convex regions have been occasionally experienced in presence of softening.