TS according to Quast and modified Quast

Local Anisotropic Tension-Stiffening Augmentation

When using a concrete damage model, tension stiffening is represented by the local anisotropic tension-stiffening augmentation [4]. With an anisotropic damage model, this method replaces the previous, globally acting approach: instead of describing the tension zone of the cross-section once via a single stress-strain curve valid for the entire cross-section, it superimposes a reinforcement-induced tensile stress on the basic material response of the concrete at each concrete integration point. The correction is calculated locally (per integration point) and anisotropically (separately in both principal strain directions); the angle between the reinforcement direction and the respective principal tension direction is explicitly taken into account.

Activation Factor

At each concrete integration point, the principal strains ε₁, ε₂ and the associated principal strain angle θ_p are determined; only tension directions (ε_k > 0) are relevant for tension stiffening. Each reinforcement layer j contributes to the activation factor α_k of the principal direction k via three multiplicative partial factors:

• g_j (capacity factor): decreases with increasing reinforcement strain and becomes zero when the yield strain of the reinforcement is reached.
• a_j,k (angle factor): for the anisotropic damage model, 1 for parallel and 0 for perpendicular alignment of reinforcement and principal tension direction, and the complement for the respective other principal direction. For the isotropic damage model, a different calculation applies without complement formation, yielding the same value for both principal directions, which can still be non-zero at exactly 90° under biaxial tension; see the banner "Scope by material model" above.
• p_j (position factor): 1 within the concrete cover zone of the considered reinforcement layer, decreasing outside (strain compatibility between concrete and reinforcement).

Tension Stiffening | Activation Factor: Capacity, Angle, and Position

The contributions of all reinforcement layers are combined via a smoothed p-norm aggregator and limited to [0, 1]:

Target Stress Curve: Two Variants

The target stress in principal direction k is obtained as:

Two variants are available for the underlying target stress curve σ̂(ε), based on the two following subsections:

Notes for Existing Models

Quast

This model for capturing the contribution of the concrete in tension between the cracks is based on a defined stress-strain curve of the concrete in the tension zone (parabola-rectangle diagram).

The basic assumptions of the approach by Quast can be summarized as follows:

• Full contribution of the concrete in tension up to reaching the crack strain ε_cr or the calculated 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 yielding of the governing reinforcement bar.

In summary, this means that the calculated tensile strength f_ct,R 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 from the defined crack strain ε_cr until the yield strain of the reinforcement steel is reached in the governing steel fiber. This is achieved by the stress-strain curve shown in the image below in the tension zone of the concrete (parabola-rectangle diagram) and the determination of a reduction factor VMB (stiffening contribution of the concrete).

Stress-Strain Diagram of Concrete in Tension with Reduction Factor VMB = 0.4

The next image schematically shows the stress states for increasing loading with tension stiffening.

Stress States for Increasing Loads with Tension Stiffening

The stress-strain curve in the tension zone can be described by the following equations:

• The fullness of the parabola in the first section can be controlled by the exponent n_PR.
• The exponent should be adjusted such that the transition from the compression to the tension zone occurs with the same modulus of elasticity as much as possible.

To determine the reduction factor VMB, the strain at the most heavily tensioned steel fiber is used. The position of the reference point is illustrated in the following graphic.

Determination of Residual Tensile Strength for Tension Stiffening According to Quast

The reduction factor VMB decreases with increasing steel strain. In the diagram for the factor VMB (see image below), it can be seen that the factor VMB is reduced exactly to zero at the onset of yielding of the reinforcement.

Reduction Factor VMB – Tension Stiffening According to Quast

The curve of the reduction factor VMB in State II (ε > ε_cr) can be controlled via the exponent n_VMB.

• According to Pfeiffer [2], values of n_VMB = 1 (linear) to n_VMB = 2 (parabola) are empirical values for components subjected to bending stress.
• Quast [3] uses the exponent n_VMB = 1 (linear) in his model and thereby achieves good agreement in the recalculation of column tests.
• According to Pfeiffer [2], pure tension tests can be represented with n_VMB = 2 with acceptable agreement.

The assumption of a parabola-rectangle diagram for the cracked concrete tension zone is to be regarded as a computational aid. At first glance, there are large differences compared to experimentally determined stress-strain curves on the tension side of plain concrete.

Tension Stiffening According to Quast – Comparison of Model and Lab Test

When considering the existing stresses in the reinforced concrete cross-section subjected to bending stress, it becomes apparent that the parabola-rectangle diagram is indeed better able to describe the strains and stresses on average. In a bending beam, a concrete body forms between two cracks. This acts as a kind of wall into which tensile forces are gradually re-introduced by the reinforcement. As a result, a very irregular distribution of stress and strain develops. On average, however, a strain plane with a parabola-rectangle course can be constructed, by means of which the mean curvature can be captured.

Existing Stress State for Reinforced Concrete Under Bending

For the Quast model, the design values to be used were proposed as follows

• for the tensile strength f_ct,R

• for the crack strain ε_cr,R

The design value for the tensile strength f_ct,R is thus smaller than the specifications of the Eurocode. This is justified by the description of the stress-strain curve and the determination of the reduction factor VMB, in which the assumed tensile stress and the resulting tensile force are only gradually reduced after the tensile strain is exceeded. At a strain of 2 ⋅ ε_cr, an effective tensile stress of about 0.95 ⋅ f_ct,R still results. Thus, the reduction of the stiffness can be well predicted for bending stress. For pure tensile stress, the values given above for f_ct,R are too low. Here, according to Pfeiffer [2], the values from EC 2 should be used for the design value of the tensile strength. The values recommended by Quast [3] for f_ct,R = 1/20 ⋅ f_cm can be achieved by assuming 60% of the tensile strengths specified in EC 2. When assuming f_ct,R = 0.6 ⋅ fctm, on the one hand, cracking of the cross-section is predicted too early. On the other hand, however, a reduction of the tensile strength under permanent load (approx. 70%) or a temporarily higher load (e.g., the short-term application of the rare action combination), which leads to a damaged tension zone, is already taken into account. The individual design values for the tension zone of the concrete can be described as follows:

Modified Approach According to Quast

The modified approach according to Quast uses the same definition of the calculated tensile strength f_ct,R and the crack strain ε_cr,R as the approach according to Quast (see previous section). The difference lies in the manner of application to the concrete stress, not in the target curve itself:

• With the approach according to Quast, the concrete stress in the tension zone is directly set to the value of the target curve (σ_c = σ̂(ε), see target curve in section "Quast"); compared to the stress of the unmodified basic material model, this can mean both an increase and a reduction.
• With the modified approach, the target curve is instead purely additively superimposed on the concrete stress σ_c,Basis(ε) provided by the basic material model (e.g., damage model).

The concrete stress can thus only be increased by the tension stiffening, never reduced. This additive procedure is numerically more robust than the enforcing variant according to Quast, since it does not lead to an abrupt stiffness jump when the basic material stress already exceeds the target curve.