2.8.3.3 Stiffening effect of concrete in tension area

Stiffening effect of concrete in tension area

In cracked parts of the reinforced concrete, the tensile forces in the crack are resisted by the reinforcement alone. Between two cracks, however, tension stresses are transferred to the concrete through the (displaceable) bond. Thus, the concrete contributes to the resistance of internal tension forces, which leads to an increased stiffness of the component. This effect refers to the stiffening contribution of the concrete in tension between the cracks and is also called tension stiffening.

The increase of the structural component stiffness due to tension stiffening can be considered in two ways:

• A residual, constant tension stress, which remains after the crack formation, can be involved in the concrete's stress-strain diagram.

The residual tension stress is clearly smaller than the tensile strength of the concrete. Alternatively, it is also possible to introduce modified stress-strain relations for the tension zone that consider the contribution of the concrete in tension between the cracks in the form of a decreasing branch in the graph after the tensile strength is reached.

• Another approach is to modify the "pure" stress-strain diagram of the reinforcing steel.

In this case, a reduced steel strain ε_sm is applied in the relevant cross-section, with the strain resulting from ε_s2 and a reduction term due to the tension stiffening.

To consider tension stiffening, RF-CONCRETE NL uses the approach of modeling the concrete tensile strength according to Quast [4]. This model is based on a defined stress-strain relation of the concrete in the tension area (parabola-rectangle diagram). The basic assumptions of Quast's approach can be summarized as follows:

• full contribution of the concrete to tension until reaching the crack strain ε_cr or the calculational concrete tensile strength f_ct,R
• reduced stiffening contribution of the concrete in the tension zone according to the existing concrete strain
• no application of tension stiffening after the governing rebar starts yielding

To sum up, this means that the tensile strength f_ct,R used for the calculation is not a fixed value but relates to the existing strain in the governing steel (tension) fiber. The maximum tensile strength f_ct,R decreases linearly to zero, starting at the defined crack strain ε_cr until reaching the yield strain of the reinforcing steel in the governing steel fiber. This is achieved by means of the stress-strain relation in the tension area of the concrete shown in Figure 2.137 (parabola-rectangle diagram) and the determination of a reduction factor VMB (German: Versteifende Mitwirkung des Betons).

The stress-strain relation in the tension zone can be described with the following equations:

$$\begin{gathered} {\sigma }_c =\hspace{0.17em}\text{VMB} · \left[f_{ct,R} · \left(1 - {\left(1 - \frac{\epsilon }{{\epsilon }_{cr}}\right)}^{n_{PR}}\right)\right] \text{for} 0 < \epsilon < {\epsilon }_{cr} \\ {\sigma }_c =\hspace{0.17em}\text{VMB} · f_{ct,R} \text{for} \epsilon > {\epsilon }_{cr} (\text{constant}\text{ distribution}) \end{gathered}$$

The curvature of the parabola in the first section can be controlled by the exponent n_PR. The exponent should be adjusted in such a way that the transition from the compression zone to the tension zone is preferably achieved with the same modulus of elasticity.

To determine the reduction factor VMB, the strain at the most tensioned steel fiber is used. The position of the reference point is shown in Figure 2.139.

The reduction parameter VMB decreases with increasing steel strain. In the diagram for the factor VMB (see Figure 2.140), it is evident that the factor VMB is reduced to zero exactly at the point when the yielding of the reinforcement starts.

The distribution for the reduction factor VMB in state II (ε > ε_cr) can be controlled by means of the exponent n_VMB. According to Pfeiffer [5], the values n_VMB = 1 (linear) to n_VMB = 2 (parabola) are experiential values for structural elements subjected to bending. In his model, Quast [6] uses the exponent n_VMB = 1(linear), thus achieving good concordance when recalculating column tests. According to Pfeiffer [5], it is possible to describe pure tension tests with acceptable concordance by using n_VMB = 2.

The assumption of a parabola-rectangle diagram for the cracked concrete tension zone can be regarded as a calculation aid. At first glance, there are great differences compared to the experimentally determined stress-strain diagrams on the tension side of the pure concrete.

The given stresses in the reinforced concrete cross-section in bending show that the parabola-rectangle diagram is indeed better suited to describe the mean of the strains and stresses.

In a bending beam, a concrete body forms between two cracks. It acts as a sort of wall into which tension forces are gradually reintroduced by the reinforcement. This results in a very irregular distribution of stress and strain. On average, however, we can create a plane of strain with a parabola-rectangle distribution with which it is possible to consider the mean curvature.

Quast suggests the following calculation value for the tension strength f_ct,R and the crack strain ε_cr,R for his model.

$f_{ct,R} = \left|\frac{1}{20} · f_{cm}\right| {\epsilon }_{cr,R} = \left|\frac{1}{20} · {\epsilon }_{c1}\right|$

The calculational value for the tensile strength f_ct,R is thus smaller than specified by the Eurocode. This is due to the description of the stress-strain relation and the determination of the reduction parameter VMB, in which the assumed tension stress and the resulting tension force are only slowly reduced after exceeding the tension strain. For a strain of 2 ⋅ ε_cr, there is also an acting tension stress of about 0.95 ⋅ f_ct,R. Thus, in case of bending, the reduction of the stiffness can be predicted well. In case of pure tension, the values for f_ct,R mentioned above are too low. According to Pfeiffer [5], the values from EC 2 should be applied for the calculation value of the tensile strength.

The values for f_ct,R = 1/20 ⋅ f_cm recommended by Quast [6] can be reached by applying 60 % of the tensile strengths given in EC 2. On the one hand, the cracking of the cross-section is predicted too early when applying f_ct,R = 0.6 ⋅ f_ctm. On the other hand, this already takes into account a reduction of the tensile strength under permanent load (about 70 %) or a temporarily higher load (e.g. the short-term application of the rare action combination) that results in a damaged tension zone.

The individual calculation values for the concrete's tension zone can be described as follows: