The effects of different strain contributions on the response of RC beams in fire
Introduction
Fire resistance presents an important aspect of safety of structures. It is well known that the temperature increase in fire conditions decreases load-carrying capacity of concrete, and increases its deformability. Due to structural and chemical changes in material caused by the elevated temperature, due to the internal stresses enforced by the temperature gradient, and due to high pore pressures caused by the evaporation of pore water, internal microcracks or damages appear in concrete. Further on, at an elevated temperature, the decomposition process of the cement stone in concrete begins, which is the consequence of the dehydration of the cement binder. Physical–chemical changes appear also in the stoneware, which leads to the decomposition of aggregate grains. For this reason the decrease of compressive strength of concrete at an elevated temperature depends also on the type of aggregate used. The elastic and shear moduli of concrete decrease nearly linearly with the increase of temperature [8], in contrast to the thermal extension coefficient of concrete, which increases non-linearly [26]. Due to stresses in concrete caused by the temperature gradient, due to the increase of pore pressures [11], [19], and due to the fact that the thermal extension coefficient of steel reinforcement increases with temperature much faster than that of concrete, concrete splitting may also appear.
The magnitude of concrete creep at elevated temperatures is much bigger than at room temperature. Cruz [9] measured the creep of concrete under a constant load and several different stabilized temperatures up to 650 ∘C and Anderberg and Thelandersson [1] up to 790 ∘C. They found out that temperatures only above 400 ∘C are somewhat influential. In contrast, the effect of creep of a steel reinforcement on stress and strain state in reinforced concrete frames is remarkable when temperature in reinforcement bars exceeds 400 ∘C [41].
A particularity of concrete is the so called transient strain. This has been found to have an important effect on the mechanical behaviour of concrete during the first heating [1], [29], [32]. It is irrecoverable and emerges as the result of the physico-chemical changes that take place only under the first heating. The transient strain aims to represent the inelastic deformations due to moisture effects such as the diffusion and the evaporation [20] and mismatch between thermal deformations of aggregate and mortar [31], [38]. Formally, it may be defined as that part of the total strain obtained in stressed concrete under heating, which cannot be accounted for otherwise (for further details, see [1], [32]).
Shrinkage of concrete becomes somewhat more intensive at elevated temperatures, yet the related strains are small compared to overall strains and can be disregarded in the analysis [16], [36]. In contrast, the bond strength between concrete and steel may decrease substantially with increasing temperature [31], [35]. Yet the decrease of the bond strength and the related increased bond slip seem to affect the bearing capacity and its ductility only when the structure is made from pre-stressed concrete [31].
Spalling of fire-exposed concrete is another important phenomenon, in particular, if concrete is densified by particles smaller than the cement grains such as micro-silica, and if the moisture content is more than 3% by weight. If the moisture content in traditional concrete is between 3% and 4%, the risk of spalling is small [15], [31]. Spalling may either progress slowly (‘progressive spalling’) as a consequence of deteriorating strength of concrete, steel and bonding, or appears instantaneously in the form of an explosion (‘explosive spalling’). While the effect of the former is only minor, the effect of explosive spalling can be disastrous. The explosive spalling is now well understood, being the result of a combined effect of high pore pressure and constrained deformations between aggregate and cement stone [19], [37], [39], [40]. Yet, and as pointed out by Hertz [15], further research is needed in order to develop a coherent theory and to establish the design methods for engineers.
It is clear that high temperature and its rapid variation in fire conditions trigger a number of complex and interrelated physical, mechanical and thermo-hydral phenomena in a reinforced concrete structure [22]. Comprehensive 3D numerical modelling is hence both theoretically and numerically a very difficult task. These analyses are typically used in the comprehensive study of behaviour of unique engineering structures, such as atomic power plants or highway tunnels [18], [27], where the fire response is modelled as a fully coupled thermo-hydro-mechanical problem [12], [13], [28], [38]. Much less sophisticated are the formulations, where the mechanical and the thermo-hydral processes are uncoupled, which makes it possible to study the fire response of a structure in two separate steps. In the first step, the heat and mass transfer during fire is determined, which is then, in the second step, employed as the time-dependent thermal load in the mechanical analysis [3], [10]. This is a reasonable assumption, because the contribution of the mechanical work to the change of temperature is small compared to the heat input during fire, and because water is much more incompressible than concrete [3]. For reinforced concrete frame structures made from traditional concretes, where explosive spalling is uncommon, we believe that the mathematical model employed in the design methods for engineers can further be simplified, i.e., the water and vapour transfer and phase changes in water can be neglected (as suggested in Eurocode 2 [10]) or indirectly accounted for by an artificially increased specific heat of concrete [10], or we do consider these effects, yet in an uncoupled sense [24]. The simplified mechanical model of the reinforced concrete framed structure is typically a framework of 1D beam elements, see [4], [7], [24], [25].
In the present paper we follow the above simplifying logic. The novelty of our approach is the introduction of the original strain-based planar beam finite element [4]. There are several advantages of this new finite element: (i) outstanding accuracy of both displacements and internal forces; (ii) insensitivity to locking; (iii) numerical robustness; (iv) good radius of attraction in Newton’s iteration. Furthermore, it enables us to introduce the strain-softening driven localization into the formulation in a natural way [4], [33]. This is an important issue, since the collapse of reinforced concrete structures in fire is typically a consequence of a series of strain localizations in the structure [3] due to the concrete softening in its post-peak stress–strain behaviour.
An aspect of utmost importance of the present fire analysis is that such a numerical analysis makes it possible to quantify the particular contributions of plastic, thermal, creep and transient strains to the total deformation of a structure. In fact, the main objective of the present paper is to show and discuss the effect of particular types of strains on behaviour, collapse load, resistance time and ductility of a reinforced concrete beam subject to fire.
The present computational model and the related computer program for the non-linear analysis of the response of planar reinforced concrete frames simultaneously exposed to fire and external mechanical loads up to failure have already been described to some extent in [4], [5], [6], [30]. That is why we here present only the details that are relevant to the present discussion.
Section snippets
Mechanical properties of concrete and reinforcing steel
In a time increment, , the geometric (or total) extensional strain increment, , of a generic material fibre of a beam is assumed to be the sum of increments of elastic, , plastic, , thermal, , creep, , and transient strain, , the latter being non-zero only in concrete. These increments of strain are assumed to be given functions of the stress and temperature, as described in the following.
Simply supported reinforced concrete beam with overhangs
Our first numerical example is a simply supported concrete beam with overhangs. This beam has been extensively tested by Lin et al. [26] and their results will be used to validate the present numerical model. In what follows three variants of this beam will be analysed and marked as , and [26]. Geometric, material and loading data are given in Fig. 2 and in Table 1, where and denote the compressive strength of concrete and the ultimate strength of steel at room temperature,
Conclusions
Modelling the behaviour of reinforced concrete frame structures in fire is a difficult task. In this paper we employ a two-step solution strategy and introduce a number of further simplifications in order to make the analysis practically feasible. The first step consists of determining the temperature distribution over the structure at each time during fire, which is, in the second step, used as the thermal load. The frame structure is modelled by strain-based, kinematically exact beam finite
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