#### Concrete under compressive stress

This section briefly discusses the key relationships concerning the behaviour of concrete under compressive stress.

The behaviour of concrete in actual structures can differ slightly from the behaviour that lab tests seem to indicate. There are some tests specially recommended for the evaluation of concrete properties in real structures.[12-15] However, to anticipate the behaviour of a future structure, conclusions by previous investigations, most of them from laboratory works, need to be taken into account. There are some stress–strain relationships available to model the concrete material. In this work, the authors have used the proposed equation by Hognestad,[16] that is:

- (1)

where is the maximum compression resistance of the concrete (measured in uniaxial compression tests on cylindrical specimens) and ɛ_{0} is the strain corresponding to .

When establishing stress versus strain relationships for concrete cracked due to shear forces, it is important to remember that the deformation state it experiences is very different from what occurs in test membranes under uniaxial compression, where stress causes some transverse strain as a result of the *Poisson* effect. When cracked reinforced concrete is subjected to shear, concrete suffers significant stress because of transverse strain, as a result of which it becomes less resistant to compression and less rigid (the *softening effect*) than test specimens in uniaxial compression tests.

In Model A, where a reduction in the maximum compression stress of the concrete and its corresponding strain is paramount, the reduction factor for stress (β_{σ}) and for strain (β_{ɛ}) is given by Eqs. (7), (8), and (9) (Fig. 2).

- (7)

- (8)

- (9)

The *K*_{c} parameter accounts for strains in concrete and the *K*_{f} parameter takes into account that in high strength concrete, the softening effect is more pronounced as a consequence of smoothed fracture planes, resulting in the earlier onset of local compression stability failure or in earlier crack-slip failure.[19]

Vecchio and Collins also proposed a simpler Model B that only uses a reduction factor for compression stress, given by Eqs. (10) and (11).

- (10)

- (11)

Later, Vecchio et al.[23] extended Model B to include high resistance concretes, by using Model A and incorporating a factor *K*_{f} expressed by Eq. (12).

- (12)

These models were based on the results of proportional stress tests on concrete panels with equal percentages of reinforcement in the two orthogonal directions, in which the test pieces failed because the concrete was crushed before the rebars gave way and in which a lesser *softening effect* was observed than that predicted by earlier theoretical models. Therefore, Vecchio[24] proposed using the parameter β from Eq. (13) for both models.

- (13)

In the previous equation *C*_{s} = 0.55 and *C*_{d} assume the values of *K*_{c} (Eqs. (8) and (11)). Coefficients *K*_{σ} and *K*_{ɛ} are defined in Table 1.

Table 1. Coefficients *K*_{σ} and *K*_{ɛ} | Proportional action | Sequential action |
---|

| α =45° | α =90° | α =45° | α =90° |
---|

*K*_{σ} | 400 | 400 | 400 | 250 |

*K*_{ɛ} | 160 | 550 | 160 | 0 |

Hsu[27] later proposed Eq. (17) to conservatively describe the *softening effect*, for use in conjunction with Eq. (14). According to Hsu, this is a conservative expression that provides a lower limit for the tests carried out during the investigation.

- (17)

Afterwards, based on tests in which the principal direction of stresses was at an oblique angle relative to the rebars, Mikame et al.[28] proposed Eq. (18) as the coefficient to describe the reduction in resistance to compression.

- (18)

Figure 3 represents the reduction factors calculated from previous models as well as from some less common models not discussed here. The deviations between the curves from the several proposed models are notable. Therefore, more reliable models need to be considered. None of the studied models considers the thickness of the panels as a parameter. These aspects justify further investigation in this area, especially with a view to incorporating variables that have not been considered relevant before, such as the above-mentioned thickness of the panels.

#### Reinforcements under tension

In the elastic phase, the average stress versus strain of rebars surrounded by concrete, when subject to tension, can be mathematically described by the curve obtained from uniaxial tensile tests on isolated rebars (free from concrete).

However, when concrete cracks, there are large differences in the distribution of tensile stresses and strains occurring in the rebars. These differences occur because throughout the length of a crack the entire traction stress is absorbed by the rebars, while between the cracks, the tensile stress is absorbed by concrete, by adherence in the interface. This is the so-called *stiffening effect*.

While some models for rebars embedded in concrete (CFT, MCFT) incorporate simplified average elastic stress versus strain relationships with a perfectly plastic yield plateau, others (Softened Truss Model–STM, Rotating Angle Softened Truss Model–RASTM) incorporate more elaborate mathematical expressions which take into account the spatial differences in tensions and strains mentioned above.

To more easily enable these equations to be applied, Belarbi and Hsu[26] take certain values as given: *E*_{p} = 0.025 *E*_{s}, *f*_{0} = 0.89 *f*_{y} e *f*_{n} = 0.091 *f*_{y}, resulting in Eq. (24).

- (24)