Modeling the tensile steel reinforcement strain in RC-beams subjected to cycles of loading and unloading
Introduction
To ensure the serviceability of reinforced concrete structures, deflection control is an important design objective. Excessive concrete cracking and excessive deformation are one of the most common causes of damage and result in large annual cost to the construction industry.
Reinforced concrete beams in service are usually cracked, as the tensile strength of concrete is low [1]. Under normal service conditions, the concrete between the primary cracks is able to continue to carry tensile stress, due to the transfer of forces from the tensile reinforcement to the concrete through bond. This phenomenon is known as tension stiffening, which affects the beam’s stiffness and hence its deflection, especially for lightly reinforced concrete beams [2]. As a result, the tension stiffening effect must be accurately modeled to simulate the in-service behavior of reinforced concrete structures, particularly under repeated loading [3].
Most of existing models for the flexural reinforced concrete beams deal with monotonic loading. The smeared-crack model is a popular way to simulate the tension stiffening effect. In this approach, an average stress-strain relation is considered for the whole tension area to account for the “average” deformation response of a cracked member [4]. A modified constitutive relationship for the steel reinforcement [5], [6] or an updated descending branch of the tensile stress-strain curve for concrete have been developed and implemented in finite element analyses [5], [7], [8], [9], [10], [11], [12]. In addition, the so-called microscopic models based on the bond-slip mechanism and discrete cracking have been proposed by Floegl and Mang [13], Gupta and Maestrini [14], and Choi and Cheung [15].
Alternatively, several empirical models have been widely accepted by engineers in design for the control of deflections, involving determination of the effective moment of inertia () for a cracked member under monotonic loading. Branson developed a well-known model [16], which was adopted by the ACI Building Code [17]. Branson’s equation gives a weighted average of the uncracked and cracked stiffness of the reinforced concrete cross-section at any load level, but it has been shown to overestimate the effective stiffness of lightly reinforced concrete beams and slabs [2]. In comparison with Branson’s model, Bischoff suggested a weighted average of the uncracked and cracked flexibility of reinforced concrete cross-sections [18], [19]. Experiments carried out on reinforced slabs having reinforcement ratios ranging between 0.18% and 0.84% demonstrated that Bischoff’s model is more accurate than Branson’s model for lightly reinforced concrete members [2]. A statistical study that employed data from nine experimental programs involving a total of 80 specimens showed a similar conclusion that the Branson’s model overestimated the stiffness significantly for reinforcement ratios ranging between 0.4% and 0.8% [20]. In fact, the accuracy of deflection predictions made using all the techniques decreased significantly for beams with small reinforcement ratios [20]. In this paper, an alternative empirical model to Bischoff and Branson’s models is proposed. Contrary to Bischoff and Branson’s models, which are both based on weighted averages of uncracked and cracked moment of inertia, the model proposed is based on the local measurement and modeling of the steel reinforcement strains in the tensile zone.
Due to the tension stiffening effect, the distribution of the tensile strain along the reinforcement bars between the cracks is not uniform, but rises to a maximum at each crack and drops to a minimum mid-way between adjacent cracks. In some experimental studies, strain gauges glued to the bar surface have been used to measure steel strain [21]. However, this is not entirely satisfactory, as the bond properties of the reinforcement bars are affected by the strain gauges [22]. Alternatively, magnetic methods and fiber-optical strain measurements have been used to measure the distribution of strain in the reinforcing steel [22], [23]. In this paper, instead of measuring the distribution of reinforcing steel strain, a simple apparatus was adopted to measure the overall steel reinforcement strain over a length of the cracked tension zone containing two or more primary cracks. The mid-span deflection of the test beam was also measured. In total, eleven reinforced concrete beams were tested. Using the experimental results, a model for the overall steel reinforcement strain, termed the effective steel strain, has been developed for monotonic loading. The moment of inertia can then be deduced by homogenization across the height of the reinforced concrete beam, thereby providing an alternative way to calculate the effective moment of inertia (). The performance of the new model is compared to results from Bischoff’s model results for different reinforcement ratio.
Unlike existing models of tension stiffening, the current model considers the residual or permanent deformation of reinforced concrete beams subjected to cycles of loading and unloading. After cracking, the deflection of a beam does not return to zero when it is unloaded due to a permanent residual deformation [24]. As a result, the bending stiffness during cycles of loading and unloading is significantly higher than that predicted by any of the models mentioned previously, which were all developed to assess the stiffness under monotonic loading. A framework to calculate the over stiffness and the irrecoverable deflection of cracked reinforced concrete beams subject to cycles of loading and unloading was developed by Castel et al. [24]. In this model, the reduction in stiffness, that occurs when cracking is stabilized, is governed by the steel-concrete bond degradation between the so-called cover-control cracks. The reduction in tension stiffening due to cover-control cracking was considered by implementing a damage variable to account for the loss of bond [24], [25]. The damage variable was calibrated to predict the overall stiffness during cycles of loading and unloading. However, the calibration was carried out using a limited amount of data. In this paper, further validation of the calibration of is reported using a larger number of beams with different reinforcement ratios and based on both overall steel reinforcement strain and deflection. Using the overall steel reinforcement strain measured during the cycles of loading and unloading permits a much more accurate and convincing validation than by simply comparing measured and predicted deflection.
Section snippets
Experiments
The aim of the experiments was to measure the overall strains in the tensile reinforcement and the displacement of beams subjected to both monotonic loading and cycles of loading and unloading, and then to use this information to determine the overall stiffness of the members. Experimental results from eleven reinforced concrete beams are reported in this paper.
Two different concrete mixes were used, with average compressive strengths of 38 MPa and 46 MPa after 28 days, respectively. The flexural
Modeling steel reinforcement effective strain under monotonic loading
Before cracking, the effective strain of the tensile reinforcement can be calculated as:where is the moment of inertia of the uncracked section; is the depths of neutral axis at the uncracked section; and is the cracking moment.
When the applied bending moment causes the steel reinforcement to first yield , it is assumed that tension stiffening can be neglected, leading to effective strain of reinforcement at first yield to be given by
Modeling steel reinforcement average strain under cycles of loading and unloading
The steel strain between cracks can be written as [3]where is the steel strain at the cracked section (where ); is the steel strain at uncracked section (where ) and the function is taken as,where x is the distance from the cracks.
Accordingly, the concrete strain between the cracks is given by:where is the concrete strain at the level of the tensile reinforcement at .
The relationship
Experimental validation based on overall stiffness
The typical response of a reinforced concrete member subjected to cycles of loading and unloading is illustrated in Fig. 12. The path OABC is a monotonic load-deflection envelop for a typical reinforced concrete beam. The stiffness of reinforced concrete beams at point B can be calculated in terms of the effective moment of inertia () as shown in Fig. 12. For the unloading phase, following the path BD, the deflection does not return to zero, leading to an irreversible deflection labeled
Conclusions
In this paper, a model of the overall stiffness of in-service reinforced concrete beams has been developed. For monotonic loading, based on experimental results obtained from RC beams with reinforcement ratios between 0.56% and 0.88%, a new model of effective moment of inertia is suggested. The new model provides a more accurate and generally more conservative estimate of the effective moment of inertia than Bishcoff’s model, when the reinforcement ratio is close to the minimum ratio. For
Acknowledgments
The authors are very grateful for the funding from the projects 51308468, 51308471 and 51378432 supported by National Natural Science Foundation of China, and the projects DP110103028 and DP140100529 supported by the Australian Research Council. Their thanks also go to the Fundamental Research Funds for the Central Universities with Grant No. 2682014CX072 and 2682015ZD13. The first author wishes to thank the Key Laboratory of High-speed Railway Engineering, Ministry of Education, Southwest
References (31)
- et al.
Deriving stressstrain relationships for steel fibre concrete in tension from tests of beams with ordinary reinforcement
Eng Struct
(2012) - et al.
A simplified model for predicting the shear response of reinforced concrete membranes
Thin-Wall Struct
(1994) - et al.
An efficient tension-stiffening model for nonlinear analysis of reinforced concrete members
Eng Struct
(2008) - et al.
Tension stiffening model for planar reinforced concrete members
Comput Struct
(1996) - et al.
Deflection prediction of reinforced concrete beams by design codes and computer simulation
Eng Struct
(2013) - et al.
Modeling of steel and concrete strains between primary cracks for the prediction of cover-controlled cracking in RC-beams
Eng Struct
(2011) - et al.
Effect of arrangement of tensile reinforcement on flexural stiffness and cracking
Eng Struct
(2016) - et al.
Modeling short-term tension stiffening in reinforced concrete prisms using a continuum-based finite element model
Eng Struct
(2009) Tension stiffening in lightly reinforced concrete slabs
J Struct Eng
(2007)- et al.
Finite-element modeling to calculate the overall stiffness of cracked reinforced concrete beams
J Struct Eng
(2012)