Journal of Structural Engineering, Vol. 133, No. 6, June 1, 2007.

1.Introduction

The tensile capacity of concrete is usually neglected when calculatingthe strength of a reinforced concrete beam or slab, eventhough concrete continues to carry tensile stress between thecracks due to the transfer of forces from the tensile reinforcementto the concrete through bond. This contribution of the tensileconcrete is known as tension stiffening, and it affects the member’sstiffness after cracking and hence its deflection and thewidth of the cracks.

With the advent of high-strength steel reinforcement, reinforcedconcrete slabs usually contain relatively small quantities oftensile reinforcement, often close to the minimum amount permittedby the relevant building code. For such members, the flexuralstiffness of a fully cracked cross section is many times smallerthan that of an uncracked cross section, and tension stiffeningcontributes greatly to the stiffness after cracking. In design, deflectionand crack control at service-load levels are usually thegoverning considerations, and accurate modeling of the stiffnessafter cracking is required.

The most commonly used approach in deflection calculationsinvolves determining an average effective moment of inertia [Ie]for a cracked member. Several different empirical equations areavailable for Ie, including the well-known equation developed byBranson [1965] and recommended in ACI 318 [ACI 2005]. Othermodels for tension stiffening are included in Eurocode 2 [CEN1992] and the [British Standard BS 8110 1985]. Recently,Bischoff [2005] demonstrated that Branson’s equation grossly overestimates thtie average sffness of reinforced concrete memberscontaining small quantities of steel reinforcement, and heproposed an alternative equation for Ie, which is essentially compatiblewith the Eurocode 2 approach.

In this paper, the various approaches for including tensionstiffening in the design of concrete structures, including the ACI318, Eurocode 2, and BS8110 models, are evaluated critically andempirical predictions are compared with measured deflections.Finally, recommendations for modeling tension stiffening instructural design are included.

2.Flexural Response after Cracking

Consider the load-deflection response of a simply supported, reinforcedconcrete slab shown in Fig. 1. At loads less than thecracking load, Pcr, the member is uncracked and behaves homogeneouslyand elastically, and the slope of the load deflection plotis proportional to the moment of inertia of the uncracked transformedsection, Iuncr. The member first cracks at Pcr when theextreme fiber tensile stress in the concrete at the section of maximum moment reaches the flexural tensile strength of the concrete or modulus of rupture, fr. There is a sudden change in the local stiffness at and immediately adjacent to this first crack. On the section containing the crack, the flexural stiffness drops significantly, but much of the beam remains uncracked. As load increases, more cracks form and the average flexural stiffness of the entire member decreases.

If the tensile concrete in the cracked regions of the beam carried no stress, the load-deflection relationship would follow the dashed line ACD in Fig. 1. If the average extreme fiber tensile stress in the concrete remained at fr after cracking, the loaddeflection relationship would follow the dashed  the actual response lies between these two extremes and is shown in Fig. 1 as the solid line AB. The difference between the actual response and the zero tension response is the tension stiffening effect （ in Fig. 1）.

As the load increases, the average tensile stress in the concrete reduces as more cracks develop and the actual response tends toward the zero tension response, at least until the crack pattern is fully developed and the number of cracks has stabilized. For slabscontaining small quantities of tensile reinforcement [typicallytension stiffening may be responsible for morethan 50% of the stiffness of the cracked member at service loads and  remains significant up to and beyond the point where the steel yields and the ultimate load is approached]. The tension stiffening effect decreases with time under sustained loads, probably due to the combined effects of tensile creep, creep rupture, and shrinkage cracking, and this must be accounted for in long-term deflection calculations.

3.Models for Tension Stiffening

The instantaneous deflection of beam or slab at service loads may be calculated from elastic theory using the elastic modulus of concrete Ec and an effective moment of inertia, Ie. The value of Ie for the member is the value calculated using Eq. [1] at midspan for a simply supported member and a weighted average value calculated in the positive and negative moment regions of a continuous span

               （1）

where Icr=moment of inertia of the cracked transformed section;Ig=moment of inertia of the gross cross section about the centroidal axis [but more correctly should be the moment of inertia of the uncracked transformed section, Iuncr]; Ma=maximum moment in the member at the stage deflection is computed; Mcr=cracking moment =(frIg / yt); fr=modulus of rupture of concrete (=7.5 fc in psi and 0.6 fc in Mpa); and yt=distance from the centroidal axis of the gross section to the extreme fiber in tension.

A modification of the ACI approach is included in the Australian Standard AS3600-2001 (AS 2001)to account for the fact that shrinkage-induced tension in the concrete may reduce the cracking moment significantly. The cracking moment is given by Mcr=(fr− fcs)Ig / yt, where fcs is maximum shrinkage-induced tensile stress in the uncracked section at the extreme fibre at which cracking occurs（Gilbert 2003）.

                        （2）

where distribution coefficient accounting for moment level and degree of cracking and is given by

                            （3）

and 1=1.0 for deformed bars and 0.5 for plain bars; 2=1.0 for a single, short-term load and 0.5 for repeated or sustained loading; sr=stress in the tensile reinforcement at the loading causing first cracking (i.e., when the moment equals Mcr), calculated while ignoring concrete in tension; s is reinforcement stress at loading under consideration (i.e., when the in-service moment Ms is acting), calculated while ignoring concrete in tension; cr=curvature at the section while ignoring concrete in tension; and uncr=curvature on the uncracked transformed section.

For slabs in pure flexure, if the compressive concrete and the reinforcement are both linear and elastic, the ratio sr /s in Eq.(3) is equal to the ratio Mcr /Ms. Using the notation of Eq.（1）, Eq.(2) can be reexpressed as

                    （4）

For a flexural member containing deformed bars under shortterm loading, Eq. (3) becomes =1−（Mcr /Ms）2 and Eq.（4）can be rearranged to give the following alternative expression for Ie for short-term deflection calculations [recently proposed by Bischoff (2005)]:                       (5)

This approach, which has now been superseded in the U.K. by the Eurocode 2 approach, also involves the calculation of the curvature at particular cross sections and then integrating to obtain the deflection. The curvature of a section after cracking is calculated by assuming that (1) plane sections remain plane; (2) the concrete in compression and the reinforcement are assumed to be linear elastic; and（3）the stress distribution for concrete in tension is triangular, having a value of zero at the neutral axis and a value at the centroid of the tensile steel of 1.0 MPa instantaneously, reducing to 0.55 MPa in the long term.

4.Comparison with Experimental Data

To test the applicability of the ACI 318, Eurocode 2, and BS 8110 approaches for lightly reinforced concrete members, the measured moment versus deflection response for 11 simply supported, singly reinforced one-way slabs containing tensile steel quantities in the range 0.0018<<0.01 are compared with the calculated responses. The slabs (designated S1 to S3, S8, SS2 to SS4, and Z1 to Z4) were all prismatic, of rectangular section, 850 mm wide, and contained a single layer of longitudinal tensile steel reinforcement at an effective depth d (with Es=200,000 MPa and the nominal yield stress fsy=500 Mpa). Details of each slab are given in Table 1, including relevant geometric and material properties.

The predicted and measured deflections at midspan for each slab when the moment at midspan equals 1.1, 1.2, and 1.3 Mcr are presented in Table 2. The measured moment versus instantaneousdeflection response at midspan of two of the slabs (SS2 and Z3) are compared with the calculated responses obtained using the three code approaches in Fig. 2. Also shown are the responses if cracking did not occur and if tension stiffening was ignored.

5.Discussion of Results

It is evident that for these lightly reinforced slabs, tension stiffening is very significant, providing a large proportion of the postcracking stiffness. From Table 2, the ratio of the midspan deflection obtained by ignoring tension stiffening to the measured midspan deflection (over the moment range Mcr to 1.3 Mcr)is in the range 1.38–3.69 with a mean value of 2.12. That is, on average, tension stiffening contributes more than 50% of the instantaneous stiffness of a lightly reinforced slab after cracking at service load.

For every slab, the ACI 318 approach underestimates the instantaneous deflection after cracking, particularly so for lightly reinforced slabs. In addition, ACI 318 does not model the abrupt change in direction of the moment-deflection response at first cracking, nor does it predict the correct shape of the postcracking moment-deflection curve.

The underestimation of short-term deflection using the ACI318 model is considerably greater in practice than that indicated by the laboratory tests reported here. Unlike the Eurocode 2 and BS 8110 approaches, the ACI 318 model does not recognize or account for the reduction in the cracking moment that will inevitably occur in practice due to tension induced in the concrete by drying shrinkage or thermal deformations. For many slabs, cracking will occur within weeks of casting due to early drying or temperature changes, often well before the slab is exposed to its full service loads.

By limiting the concrete tensile stress at the level of the tensile reinforcement to just 1.0 MPa, the BS 8110 approach overestimates the deflection of the test slabs both below and immediately above the cracking moment. This is not unreasonable and accounts for the loss of stiffness that occurs in practice due to restraint to early shrinkage and thermal deformations. Nevertheless, the BS 8110 approach provides a relatively poor model of the

postcracking stiffness and incorrectly suggests that the average tensile force carried by the cracked concrete actually increases as M increases and the neutral axis rises. As a result, the slope of the BS 8110 postcracking moment-deflection plot is steeper than the measured slope for all slabs. The approach is also more tedious to use than either the ACI or Eurocode 2 approaches.

In all cases, deflections calculated using Eurocode 2[ Eqs.(3)–(5)] are in much closer agreement with the measured deflection over the entire postcracking load range. As can be seen in Fig. 2, the shape of the load-deflection curve obtained using Eurocode 2 is a far better representation of the actual curve than that obtained using Eq. (1). Considering the variability of the concrete material properties that affect the in-service behavior of slabs and the random nature of cracking, the agreement between the Eurocode 2 predictions and the test results over such a wide range of tensile reinforcement ratios is quite remarkable. With the ratio of () in Table 2 varying between 0.80 and 1.39 with a mean value of 1.07, the Eurocode 2 approach certainly provides a better estimate of short-term behavior than either ACI 318 or BS8110.

6.Conclusions

Although tension stiffening has only a relatively minor effect on the deflection of heavily reinforced beams, it is very significant in lightly reinforced members where the ratio Iuncr / Icr is high, such as most practical reinforced concrete floor slabs. The models for tension stiffening incorporated in ACI （2005）, Eurocode 2 (CEN 1992), and BS 8110 （1985） have been presented and their applicability has been assessed for lightly reinforced concrete slabs.Instantaneous deflections calculated using the three code models have been compared with measured deflections from 11 laboratory tests on slabs containing varying quantities of steel reinforcement. The Eurocode 2 approach （Eq.（5） has been shown to more accurately model the shape of the instantaneous load-deformation response for lightly reinforced members and be far more reliable than the ACI 318 approach （Eq.（1）.