Variation of stress resultants in concrete structures due to time-dependent creep and shrinkage strains of concrete

A general procedure for calculating the variation with time of the internal stress resultants and hence the stresses, in concrete structures is discussed. In particular, a study is made of the changes in the stress resultants due to time-dependent creep and shrinkage strains of concrete.A general procedure of calculating the variation in the stress resultants due to differential creep strains in concrete structures has been proposed by the author[1]. A similar procedure is followed in this paper to study these variations when creep and shrinkage strains take place simultaneously.The method leads to a system of n-linear differential equations of the form: X = AXt + B the solution of which is performed by a computer using Runge-Kutta numerical procedures.A reinforced concrete portal frame exhibiting creep and shrinkage strains is solved by the proposed method and the results are given in tabular and graphical form.     

Limit load analysis of thick-walled concrete structures - a finite element approach to fracture

The paper illustrates the interaction of constitutive modelling and finite element solution techniques for limit load prediction of concrete structures.On the constitutive side, an engineering model of concrete fracture is developed in which the Mohr-Coulomb criterion is augmented by tension cut-off to describe incipient failure. Upon intersection with the stress path the failure surface collapses for brittle behaviour according to one of three softening rules — no-tension, no-cohesion, and no-friction. The stress transfer accompanying the energy dissipation during local failure is modeled by several fracture rules which are examined with regard to ultimate load prediction.On the numerical side the effect of finite element idealization is studied first as far as ultimate load convergence is concerned. Subsequently, incremental tangential and initial load techniques are compared together with the effect of step size.Limit load analyses of a thick-walled concrete ring and a lined concrete reactor closure conclude the paper along with engineering examples.     

Multiscale hydro-thermo-mechanical model for early-age and mature concrete structures

Temperature and early-age mechanical properties in hydrating concrete structures present a significant risk for cracking, having a major impact on concrete durability. In order to tackle these phenomena, a multiscale analysis is formulated. It accounts for a high variety of cement properties, concrete composition, structure geometry and boundary conditions.The analysis consists of two steps. The first step focuses on the evolution of moisture and temperature fields. An affinity hydration model accompanied with non-stationary heat and moisture balance equations are employed.The second step contains quasi-static creep, plasticity and damage models. It imports the previously calculated moisture and temperature fields into the mechanical problem in the form of a staggered solution. The whole model has been implemented in the ATENA software, including also the effect of early-age creep, autogenous and drying shrinkage.Validation on selected structures shows a good prediction of temperature fields during concrete hardening and a reasonable performance of the mechanical part.     

混凝土内应力测量的应变砖传感器设计与应用

针对混凝土内应力传感器匹配误差问题,设计制作埋入式应变砖传感器,将混凝土作为传感器一部分整体进行标定,从而减少实际测量中匹配误差的影响。传感器设计中考虑温度和干扰补偿,并对应变计进行良好防护以确保20MPa压力下传感器绝缘性能良好。进行50kN静态加载试验,得到加载力同传感器输出的函数关系和非线性误差。经过激波管动态加载试验,得出应变砖固有频率大于30kHz,可用于动态加载的混凝土内应力测量。     

A simplified method of solution for the short cycle creep-plasticity problem

This paper describes a method for estimating the long-term effects on structures under cyclic changes of loading. For loads less than a certain critical amplitude (shakedown limit), the stress in the structure will a symptote to a cyclic stationary state consisting of an elastic part in response to the cyclic loading, plus a system of self-equilibrating residuals constant in time. It is shown that corresponding to this cyclic stationary state, the creep energy dissipation per cycle of loading is a maximum. Instead of following the exact time history to reach this state, in this paper it is found by a procedure of successive approximations. It corrects the admissible residual stress distribution at the beginning of a cycle by the creep and plastic strains accumulated over an entire cycle, which are in general not compatible, and requires additional self-equilibrating stresses to give an elastic strain distribution such that the total strain satisfies compatibility. The steady state is reached when no further correction is necessary. Convergence may be accelerated by a suitable choice of initial starting value, and by an artificial choice of the cycle time for the best computational convenience, upon which the steady-state solution can be proved to be independent. The procedure is a powerful device to obtain the cyclic steady-state solution, which will give an upper bound to the creep deformation per cycle and may also be used to find the shakedown limit. The formulation of the procedure in conjunction with the finite element method is given in detail and results of a few examples of the analysis are shown.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

Modelling of elastoplastic behaviour with non-local damage in concrete under compression

In the first part, an elastoplastic damage model is proposed for concrete under compression-dominated stresses. The model is applied to typical concrete in various loading conditions. The model’s parameters are determined from conventional triaxial tests. The comparison between experimental data and numerical simulations shows the performance of the model in predicting basic mechanical behaviour of concrete for a large range of stress. In the second part, an extension of the model is proposed by including a non-local damage formulation. The extended model is applied to the analysis of failure process zone in concrete related to material softening behaviour.     

Artificial neural network for predicting creep of concrete

The concrete is today the building material by excellence. Drying accompanies the hardening of concrete and leads to significant dimensional changes that appear as cracks. These cracks influence the durability of the concrete works. Deforming a concrete element subjected to long-term loading is the sum of said instantaneous and delayed deformation due to creep deformation. Concrete creep is the continuous process of deformation of an element, which exerts a constant or variable load. It depends in particular on the characteristics of concrete, age during loading, the thickness of the element of the environmental humidity, and time. Creep is a complex phenomenon, recognized but poorly understood. It is related to the effects of migration of water into the pores and capillaries of the matrix and to a process of reorganization of the structure of hydrated binder crystals. Applying a nonparametric approach called artificial neural network (ANN) to effectively predict the dimensional changes due to creep drying is the subject of this research. Using this approach allows to develop models for predicting creep. These models use a multilayer backpropagation. They depend on a very large database of experimental results issued from the literature (RILEM Data Bank) and on appropriate choice of architectures and learning processes. These models take into account the different parameters of concrete preservation and making, which affect drying creep of concrete as relative humidity, cure period, water-to-cement ratio (W/C), volume-to-surface area ratio (V/S), and fine aggregate-to-total aggregate ratio, or fine aggregate-to-total aggregate ratio. To validate these models, they are compared with parametric models as B3, ACI 209, CEB, and GL2000. In these comparisons, it appears that ANN approach describes correctly the evolution with time of drying creep. A parametric study is also conducted to quantify the degree of influence of some of the different parameters used in the developed neural network model.     

Explicit integration method with time step control for viscoplasticity and creep

Constitutive equations for viscoplasticity and creep are expressed in terms of stress components, temperature and internal variables. The resulting set of ordinary differential equations of the first order under the guise of the finite element method is integrated by the Euler forward scheme with automatic subincrementation. The time step length is set on the basis of a posteriori error estimate. Testing examples were computed, including benchmark problems introduced by Zienkiewicz and Cormeau. In every case a perfect match between the converged step length and the Cormeau formula for the critical time step was observed.     

Combined elastoplastic and limit analysis via restricted basis linear programming

In many engineering situations rigid-perfectly plastic limit analysis turns out to be insufficient and unreliable. On the other hand, elastic-plastic “historical” analysis is often laborious. For trusslike frames and suitably discretized structures the former analysis is traditionally cast into the mathematical model of linear programming; the latter has led to various formulations and solution methods explicitly or implicitly related to quadratic programming.In this paper the complete stress and strain response of elastic-perfectly plastic structures subjected to proportional loading up to collapse is determined by a method essentially consisting of a linear programming procedure supplemented by an additional rule which enforces a complementarity relation among variables at each pivotal step (“restricted basis entry”). Possible local unloadings are allowed for by means of suitable modifications of the current tableau. The method proposed provides a whole “exact” deformation history of the structure, the safety factor and the collapse mechanism. Lack of uniqueness in the deformation history and “limited mechanisms” prior to collapse are discussed. Nonholonomic (irreversible) analysis and holonomic (reversible nonlinear-elastic) analysis are compared.The computational effort is assessed and checked by examples and turns out to be roughly equivalent to that required by the familiar rigid-plastic limit analysis by the static approach via standard linear programming. Extensions of the method to strain-hardening behaviour, to nonproportional loading processes and to broader categories of structures are envisaged.     

Calculation of turbine rotors in secondary creep range

Turbine rotors in power plants are exposed to triaxial stresses by centrifugal forces at high temperatures which induce long time creep effects. In the first step, we implemented an algorithm for centrifugal volume forces in ADINA. In the next step, we tested the numerical behaviour of the modified Newton-algorithm in ADINA within the long time secondary creep range for simplified examples. Isotropie strain hardening was assumed in most cases of creep calculations.

The estimate of creep properties is based on the minimum least square method. The numerical stability in creep calculations is dependent on the magnitude of time step size and on a good fit between creep law properties and the real experimental material data.

In the examples of turbine rotor models with rotational symmetry we were able to estimate the state of stress and strain under creep conditions for life time periods up to 2 × 10⁵ hr.     

Optimum integer number and position of several groups of prestressing tendons for given concrete dimensions

The following design problem is solved: Given are the concrete dimensions and the loadings (e.g. dead load plus traffic loading). The engineer can choose the number of groups of tendons he wishes to use. For each group the following data are given: the location of the cross-sections at which the tendons are anchored, the prestressing force of one tendon, which can also be variable along the length of the beam to take friction losses approximately into consideration, the maximum and minimum number of tendons, the lower and upper bounds of the zone permissible for guaranteeing sufficient concrete coverage (these bounds can also vary along the length of the beam), the smallest permissible radius of curvature, as well as a relative price. Several load combinations, under service conditions with specified allowable maximum and minimum concrete stresses are selected by the engineer. A reduction factor for creep and shrinkage is also included in the input.Determined are (a) the overall most favorable integer number of tendons in each group, and (b) the position along the length of the beams, such that the stress margin that results in every section and under the stress conditions formulated under the most unfavorable load positions shall be a maximum.The optimization algorithm employs in part a linear program. Methods to eliminate redundant constraints on the level of the cross-sections and the elements are discussed.The procedure is illustrated with a highway bridge (deck on raking legs), using three groups of tendons.     

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.     

Neural network modeling of time-dependent creep deformations in masonry structures

Stresses and deformations in concrete and masonry structures can be significantly altered by creep. Thus, neglecting creep could result in un-conservative design of new structures and/or underestimation of the level of its effect on stress redistribution in existing structures. Brickwork has substantial creep strain that is difficult to predict because of its dependence on many uncontrolled variables. Reliable and accurate prediction models for the long-term, time-dependent creep deformation of brickwork structures are needed. Artificial intelligence techniques are suitable for such applications. A model based on radial basis function neural networks (RBFNN) is proposed for predicting creep and is compared to a multi-layer perceptron neural network (MLPNN) model recently developed for the same purpose. Accurate prediction of creep was achieved due to the simple architecture and fast training procedure of RBFNN model especially when compared to MLPNN model. The RBFNN model shows good agreement with experimental creep data from brickwork assemblages collected over the last 15 years.     

Stress,buckling and vibration of hybrid bodies of revolution

The analysis is applicable to bodies of revolution composed of thin shell segments, thick segments and discrete rings. The thin shell segments are discretized by the finite difference energy method and the thick or solid segments are treated as assemblages of 8-node isoparametric quadrilateral finite elements of revolution. Suitable compatibility conditions are formulated through which these dissimilar segments are joined without introduction of large spurious discontinuity stresses. Plasticity and primary or secondary creep are included. Axisymmetric prebuckling displacements may be moderately large. The nonlinear axisymmetric problem is solved in two nested iteration loops at each load level or time step. In the inner loop the simultaneous nonlinear equations corresponding to a given tangent stiffness are solved by the Newton-Raphson method. In the outer loop the plastic and creep strains and tangent stiffness are calculated by a subincremental procedure. The linear response to nonaxisymmetric loading is obtained by superposition of Fourier harmonics. Many examples are given to demonstrate the scope of the computer program, BOSOR6, derived from the analysis and to illustrate certain stress concentration effects in shell-type structures which cannot adequately be treated with use of thin shell theory.     

Numerical method for analysing open thin-walled structures under interaction of bending and torsion

A computer program is developed to analyse concrete beams of open thin-walled sections, at different stages of loading from zero load to failure. The program is divided into two parts; the first part deals with the beam from zero load to cracking. Of course, loading is combined bending, shear and torsion (warping torsion and St Venant's torsion). In this part of the program, Vlassov's theory has been used. The cracking load is defined as that load which causes principal tensile stresses equal to the tensile strength of concrete. The second part of the program deals with all post-cracking stages of loading from cracking point to failure. An iteration procedure is used until full convergence occurs at a particular cross-section. The geometrical properties are calculated; these include the contribution of steel in the cross-section and that of concrete in the compressive zones. The mathematical model is given. The computer results are compared with earlier experimental results, and the two sets of results show reasonable agreement. The program is written in FORTRAN.     

Semi-analytical solution of time-dependent electro-thermo-mechanical creep for radially polarized piezoelectric cylinder

History of strains, stresses, deformations and electric potentials of hollow cylinders made from PZT_5 have been investigated using successive elastic solution method. A differential equation containing creep strains for displacement is obtained. Since creep strains are time, temperature and stress dependent, the closed form solution cannot be found for this constitutive differential equation. Hence, a semi-analytical method is proposed. Electric potentials increase with time (similar to the radial stress histories), since it is induced by the radial stress histories during creep deformation of the cylinder, justifying industrial application of such a material as efficient actuators and sensors.     

Form finding of shells by structural optimization

Shell structures are known to be extremely parameter sensitive; even small changes of the initial design, e.g., to the shape of the shell, may drastically change the internal stress state. The ideal case for concrete shells is a pure membrane stress state in compression for all loading conditions. Since in many realistic situations the solution for an optimal shape is not obvious, the need for form finding methods is evident. This paper presents computational methods of structural optimization as a general tool for the form finding of shells. The procedure as a synthesis of design modelling, structural analysis and mathematical optimization is discussed with special emphasis on the modelling stage. Several examples show the power of the approach and the similarities to experimental solutions.