Keywords: concrete, rheology, ageing, shrinkage, creep, non-linear analysis, time-dependent response, numerical modelling, changeable structural system.

This paper deals with the mechanical and rheological properties of concrete and with numerical modelling of the constitutive law of concrete considering ageing, decomposition and non-linear creep. The method developed for the general non-linear analysis of the time-dependent response of concrete structures is also presented. It is limited to reinforced, prestressed or composite plane frames. In addition to the influences of rheology, including the non-linear creep of concrete, the analyses also take into account the time-variable structural system.

The numerical modelling of the influence of long-term changeable high stresses on the compressive strength of concrete (Lopatic [1]) is based on the generalisation of the CEB-FIP model code 1990 [2] recommendations for concrete structures, which basically apply for constant stress level. Here, two opposite time effects are included at the same time. At low stress level the concrete strength increases due to the favourable influence of ageing, and on the other side concrete strength decreases due to its decomposition resulting from permanent action of high stresses. For the computational consideration of the influence of sustainable changeable high stress levels on the compressive strength of concrete, this paper gives a relatively simple model, where the whole history of previous loadings is included with the parameter of the already exploited limit time. With the help of this parameter it is possible to determine the compressive strength of concrete in an arbitrary time. As a basis for the development of the model and for the evaluation of the introduced model constant, the provisions of CEB-FIP MC 1990 [2] were used, appertaining to the decreasing compressive strength of concrete at constant high stress.

To include all types of non-linearity, the constitutive equations of concrete are written in incremental form. The increment of constitutive strain is obtained with the sum of the increment of the mechanical part of strain , originating from non-linear elastic behaviour of concrete under short-term loading, the increment of concrete strain due to concrete creep and the increment of concrete strain due to concrete shrinkage . For modelling of the viscoelastic properties of ageing concrete the generalised Kelvin model consisting of non-linear time-dependent spring and a chain of Kelvin bodies has been used which can, in addition to experimentally acquired functions of concrete creep, also approximate the creep functions of concrete according to the CEB-FIP 1990 model code [2]. The non-linearity of creep was included by functions which, depending on the momentary stress level, increase the speed of creep (Bazant & Prasannan [3], Bazant & Kim [4]). The main advantage of the selected incremental constitutive law of concrete is that only the stress, the situation indicator (possible tensile or compressive failure) and the hidden variables of all Kelvin units [5] in the previous time period have to be saved for each integration point. This enables an analysis of large structures with an almost unlimited number of time steps to be performed by using a PC.

The finite element method (FEM) has been applied in the presented analysis of time-dependent response of structures. It is limited to plane frames that can be reinforced, prestressed or composite. The level of stress can be arbitrary. In addition to the geometrical non-linearity of the structure, the non-linearity of the constitutive laws of material at short term loading and basic rheological influences, as well as the influence of high stress levels to the increase of concrete creep and the influence of sustained high stress levels to the reduction of compressive strength of concrete are also considered. By taking into account the rheological influences according to the non-linear theory, the constitutive law of concrete is carried out in the incremental form. Based on the given theoretical derivations, the NONREO computer program for the non-linear analysis of time-dependent response of reinforced concrete plane frames was prepared. Comparing the results of the computer simulations of structure behaviour with foreign experiments, the developed analysis method and the software have proven to be adequate and useful in practice.

1

J. Lopatic, "Time-dependent response of reinforced concrete structures at arbitrary stress level", P.Hd. thesis, University of Ljubljana, Slovenia, 1997.

2

CEB, "CEB-FIP Model code 1990", Buletin D'Information No 213/214, Lausane, 1993.

3

Z.P. Bazant, S. Prasannan, "Solidification theory for concrete creep", Journal of Engineering Mechanics, 115:8, "Part 1 - Formulation", pp 1691-1703, doi:10.1061/(ASCE)0733-9399(1989)115:8(1691); "Part 2 -Verification and application", pp 1704-1725, doi:10.1061/(ASCE)0733-9399(1989)115:8(1704), 1989.

4

Z.P. Bazant, J.K. Kim, "Improved prediction model for time-dependent deformations of concrete", Materials and structures, 24, "Part 2- Basic creep", pp 409-421; 25, doi:10.1007/BF02472014; "Part 3- Creep at drying", pp 21-28. doi:10.1007/BF02472209

5

Z.P. Bazant, "Dirichlet Series Creep Function for Aging Concrete", Journal of the Engineering Mechanics Division 99:EM2 , pp 367-387, 1973.

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