The main purpose of this article is to study the behaviour of thin RC membranes under compression in one direction and tension in the perpendicular direction. The influence of the thickness on the behaviour of the concrete panel is investigated. To study the behaviour of the panels, an experimental programme was carried out. Six RC panels under pure shear were tested up to failure. As a result, new experimental stress–strain relationships of thin plates under pure shear are presented. Such relationships take into account the reduction of the compressive strength due to perpendicular tensile stresses and the influence of the thickness of the RC panels. The good experimental values obtained from this investigation validate the effectiveness of the low cost testing equipment developed for this study. This was a good achievement, because previous tests on this topic were performed using expensive equipment, which very few laboratories could afford. In general, the values obtained for the stress–strain relationships are approximately within the range proposed by other investigators. It is also shown that the strength capacity predicted by European codes is higher than that obtained from the experimental tests. The values predicted by ACI code are very conservative.

The authors previously presented an article on state-of-the-art about compressive behaviour of concrete under perpendicular tensile stresses.[1] It is an important matter in many aspects regarding the behaviour of concrete elements, such as in hollow beams under torsion, where the section walls work as membranes mainly subjected to shear, with compressive stresses in one direction and tensile stresses in a perpendicular direction. These studies also show that the walls thickness may affect the behaviour of the compressive concrete in the struts differently from what the theoretical models had evaluated.[2-6] Thus, it is important to study this phenomenon, as well as its evolution with the panels/walls thickness.

As previously discussed[1] Compression FieldApproaches or Smeared Approaches as defined by ASCE-ACI Committee 445 on Shear and Torsion,[7] which are also used in the research work presented in this article, are particularly important.

There are only a few studies on this subject, possibly due to considerable costs of the tests. Some recent studies in this area[8-10] have analysed cases of walls or of strut and tie mechanisms but do not include experimental verification of the strength of concrete struts under perpendicular tension stresses.

The work developed for this article aims at studying the behaviour of concrete in compression in the main diagonal strut of a membrane, especially through stress versus strain diagrams. Particular attention was given to studying the influence of the plate thickness, comparing the experimental results with existing theoretical models, and to reduction factor of compressive strength of concrete through the action of transverse tensile strains obtained experimentally with the factors proposed in normative documents.

The design and validation of a simple and less onerous testing apparatus is also the objective of this study.

This article is the second of a series of two articles dealing with RC membranes under shear. In the first article of the series,[1] a state-of-the-art with regard to concrete struts under perpendicular tensile stress was presented. In the same article, the test specimens and the testing procedure of the specimens under shear were described, and the results of the strain and stress with respect to the principal direction were also presented and discussed.

Experimental Programme

Figure 1 illustrates the geometry adopted for tested membranes. Two series of similar membranes were tested (Series I and II), each containing three (membranes). The details of the testing models of Fig. 1 are presented in Tables 1 and 2. The adopted cover for the reinforcements is 2 cm.

The authors understand that some differences between lab tests and in situ actual strength of concrete is possible. Some tests are developed for in situ evaluation of concrete.[11-15] Nevertheless, the use of cubes or cylinders to evaluate the concrete strength is a common procedure and it was used in this investigation work also. Some cubes were tested and the values of the maximum stress varied between 24.9 and 26.5 MPa. Tensile tests to steel bars specimens were carried out in order to determine the medium yield stress, f_{ym}, and the yield strain corresponding to the beginning of yielding ɛ_{ym} (Table 3). The normative value usually indicated for ordinary reinforcements, for the elasticity model, was assumed: E_{s} = 200 GPa. The test apparatus is presented in Fig. 2. Test instrumentation and procedures were explained in the previous article.[1]

Table 3. Results from tension tests to steel bars

ϕ (mm)

f_{ym} (MPa)

ɛ_{ym} (×10^{−6})

6

595.2

2976

8

576.3

2881

10

570.18

2851

12

469.10

2346

Analysing the Experimental Results

The state of strain and stress of the elements tested were described in the previous article, and other aspects of the elements behaviour will now be described.

Stress–strain diagrams: concrete in compression

Experimental curves translate the stress versus strain behaviour of concrete carried by compression, as well as maximum stress and strain compared to the estimates based on Hognestad's parabola[16] and Thorenfeldt's curve,[17] which translate the behaviour of stress versus strain of concrete cylindrical specimens (ϕ 15× 30cm) when subjected to uniaxial compression. The aforementioned concrete behaviour aspects were also compared with the theoretical estimates proposed by other authors which were presented in the previous article by the authors.[1]

Experimental results should be represented in a three-dimensional form representing compressive stress in the concrete as a function of the principal strain 1 and 2 of the concrete strain state. To facilitate the comparative analysis, an orthogonal projection of these strains was represented in the plan defined by the axis corresponding to the principal compressive stress in concrete and to the principal compressive strain in concrete .

Figure 3 represents experimental diagrams for the behaviour of compressive concrete which were obtained combining the measured strains with a mechanical extensometer in the direction that corresponds to an angle of 45° from the axis x with the stress in the concrete in the same direction which was obtained through the procedure explained before.[1]

A qualitative analysis of Fig. 3 evidences that the experimentally obtained curves, as well as the state of stress and strain corresponding to compressive concrete strength are within these estimates based on the models proposed by other authors. This observation validates the testing procedure, as well as the results obtained. Figure 3 also suggests that the curves corresponding to the 4-cm thick membranes (especially SIE4 curve) present lower values compared with most curves proposed by other authors for the same state of strain. The curve corresponding to the 10-cm thick membrane, however, presents higher values of than the majority of curves proposed by other authors for the same state of strain. The curves corresponding to the 7-cm thick membrane present an intermediate behaviour. These observations suggest a possible influence of the membrane thickness in the behaviour of compressive concrete.

The non-existence of a drop-down curve after a peak in the experimental stress versus strain curves for the concrete under compression is observed, despite the fact that the tests were performed in deformation control. In fact, the membranes failure generally occurs in a sudden and explosive way. For the principal tensile strains mode below 0.015 (1.5%) with membranes loaded with deformation control, Zhang and Hsu[18] also obtained an explosive failure by concrete crushing without having observed a drop-down curve.

The quantification of the adequacy degree of the curves proposed by other authors to the experimental results was carried out resorting to an adjusting parameter C (Eq. (1)), adapted from Ferreira.[19]

Considering two curves, y_{1} and y_{2} defined by the same number of points (0 to imax), parameter C, which varies between 0 and 1, provides a comparison measure between the curves. This parameter becomes 1 when both curves are exactly the same and 0 when, for any x_{i} point, one of the functions is always higher then another, and both have opposing signals.

(1)

Parameters A_{1} and A are defined by Eqs. (2) and (3), respectively, and are represented by Fig. 4.

The geometric correspondence of the previous parameters is illustrated in the figure.

(2)

(3)

Table 4 presents the results of the application of the expressions previous to the experimental results.[16-25] As expected, it shows that the curves which translate concrete's uniaxial compressive behaviour present a lower adjusting degree regarding the experimental data. Table 5 presents an arithmetic average of parameter C for membranes with the same thickness.

By looking at the values of the previous tables and at the graphs, the curves proposed by Mikame et al.[21] and Vecchio et al. Model B[24] are better adjusted to the behaviour of the _{10-cm}thick membrane, presenting parameters C = 93.8% and C = 91.8%, respectively. This is due to the fact that they suggest higher values of for a given state of strain. The curve proposed by Belarbi and Hsu[22] better adjusts to the shape of the curves experimentally obtained for the 4-cm thick membranes, because it presents a medium parameter of C = 88.1%. This is due to the fact that the equation proposed by the authors suggests lower values of for the same values of ɛ_{c2}/ɛ_{0}.

The 7-cm thick membrane has an intermediate compressive concrete behaviour curve regarding what had been verified for the 4- and 10-cm thick curves. The expressions proposed by Vecchio and Collins,[20] Belarbi and Hsu,[22] and Mikame et al.[21] are the ones that better adjust to the progress of experimental curves obtained for these membranes with parameters C = 91.8%, C = 91.5% and C = 90.9%, respectively.

Failure load

Table 6 presents, for each of the tested models, the maximum load (Q), the load corresponding to the last reading of average strains on the concrete's surface (Q_{d}) and the normalised applied shear stress regarding compressive strength of concrete(ν_{n}). Medium strains are measured over a length that would cross some few cracks.

Table 6. Maximum loads

SIE4

SIE7

SIE10

SIIE4

SIIE7

Maximum load: Q (tf)

M21.3

–36.0

–49.2

–23.0

–35.5

Load for the last reading: Q_{d} (demec's) (tf)

–20.9

–36.0

–48.2

–23.0

–34.8

(Q_{d} -Q)/Q (%)

–1.7%

0.0%

–2.0%

0.0%

–2.2%

ν = Q/(eh) (MPa)

–6.1

–5.9

–5.6

–6.6

–5.8

ν_{n} = ν /f_{c}^{'} (%)

–30.8%

–28.8%

–29.4%

–31.5%

–29.8%

The maximum load presents reduced deviations regarding the load corresponding to the last reading using a strain meter before the failure occurred. Therefore, despite the readings regarding the values of the second line of Table 6, belonging to the last level of deformation to which the membranes were submitted, the results represent the state of stress immediately before the failure of that membrane. Also, according to Table 6 the normalised shear stress regarding compressive strength of concrete (ν_{n}), which corresponds to the failure of the membranes, presents a much reduced dispersion.

Stress and strain in concrete before failure

Table 7 presents the values of compressive stress in concrete f_{c45}, as well as medium strain in concrete along direction that corresponds to an angle of 45° and −45° from the axis x, corresponding to the last reading using a mechanical strain gauge for each membrane.

Table 7. Stresses and strains in concrete before failure

SIE4

SIE7

SIE10

SIIE4

SIIE7

f_{c45} (MPa)

–10.0

–10.7

–10.4

–11.2

–10.9

f_{c45} /f_{c}^{'}

50.7%

52.3%

54.5%

53.9%

56.3%

ɛ_{c45}

–0.001596

–0.001162

–0.001227

–0.001492

–0.001704

ɛ_{c45}/ɛ_{0}

72.6%

52.8%

55.8%

67.8%

77.4%

ɛ_{c-45}

0.005580

0.006471

0.007204

0.006349

0.007029

ɛ_{c-45}/ɛ_{c45}

–3.5

–5.6

–5.9

–4.3

–4.1

According to Table 7, the compressive strain in concrete (f_{c45}) normalised in relation to strength to uniaxial compression presents a much reduced dispersion. Also in Table 7, and as expected, the experimentally obtained compressive strength in concrete is considerably inferior to the uniaxial compressive strength which would be obtained in uniaxial compressive tests with concrete cylinders. When failure occurs, it is possible to see that the strain is considerably inferior to the value obtained in uniaxial compressive tests ɛ_{0}.

Maximum stress and corresponding strain

By analysing the membranes tests results, the readings presented a quotient of with reduced variations. Figure 5 presents the relationship experimentally obtained for the strain quotient ɛ_{c-45} /ɛ_{c45} in the test membranes. Considering that the quotient was approximately constant, strain versus stress diagrams for compressive concrete were drawn for each membrane by some authors (cited in Fig. 6), matching the quotient to the value corresponding to the last reading as obtained through mechanical strain gauges (Fig. 6). Stress and strain corresponding to maximum stress based on the curves proposed by other authors were also compared with the experimental results (Fig. 6).

Table 8 shows the percentage deviations of ɛ_{0} of the values for compressive strain (ɛ_{c2}^{max}) which correspond to the maximum compressive stress in concrete values estimated by the curves proposed by some authors (for ɛ_{c-45}/ɛ_{c45} obtained through the reading immediately before the membrane failure), as well as experimental values. Table 9 presents the percentage deviations of the values obtained through experiment for the compressive strain which corresponds to the maximum compressive stress supported by concrete regarding the values based on expressions proposed by other authors. From Fig. 6 and Tables 9 and 10, it appears that the strain corresponding to maximum compressive stress which concrete can support, in the presence of tensile stresses strains, is inferior to the value usually presumed for uniaxial compressive tests (−0.002 or −0.0022).

According to Vecchio and Collins,[20] the process to experimentally evaluate whether the softening effect is a phenomenon only associated to compressive stress in concrete, or also occurring in the strain corresponding to the maximum compressive stress, implies that the variable ɛ_{c1}, that is, which ensures that reinforced concrete elements present a tensile principal strain constant throughout the entire test or with a very reduced variation. The way these tests (to which this study refers) were carried out does not allow to control the tensile principal strain. Consequently, it is not possible to explicitly evaluate this aspect. In view of this, an indirect process was chosen to evaluate the softening effect, evaluating the curves adjustment proposed by other authors to the curves obtained through experiment (section Stress–Strain Diagrams: Concrete in Compression), as well as the deviation of stresses and strains regarding the estimated values based on the curves proposed by other authors for stress and strain corresponding to the concrete compressive strength.

Table 9 demonstrates that the compressive strain in concrete corresponding to membrane failure, for the quotients ɛ_{c1}/ɛ_{c2} experimentally registered and associated to expressions proposed by Vecchio and Collins,[20] Belarbi and Hsu,[22] and Vecchio Model B,[25] presents lower medium deviations regarding experimental values. However, it should be noted that, except for estimates based on the expression proposed by Mikame et al.,[21] which suggests higher values, or even the estimates based on the expressions proposed by Hsu[23] and Vecchio et al.[24] suggesting inferior values, there was a considerable dispersion of results and trends. Table 10 presents percentual deviations regarding uniaxial compressive strength (f_{c}^{'}), the maximum experimental compressive stress, and the estimates of the constitutive laws proposed by some authors for the quotient of principal strains (ɛ_{c1}/ɛ_{c2}) when the membranes failure was experimentally registered. Table 11 presents the percentual deviations of the experimental values of the compressive strength of concrete in the presence of transverse tensile stresses from the values obtained through the equations proposed by various authors. Figure 6 and Tables 10 and 11 show, as expected, that the value of compressive strength of concrete in the presence of high tensile strains presents considerable deviations regarding the results obtained through the uniaxial strain tests.

The estimates for concrete compression strength in the presence of transverse tensile stresses based on the equation proposed by Vecchio et al.[24] were better adjusted to the experimental results. As expected, the equation proposed by Hsu[23] provided inferior estimates to the ones which were observed (in proposing this expression, the author presented it as an inferior limit), and the same happened to the estimates obtained through the equations proposed by Belarbi and Hsu,[22] and by Vecchio and Collins.[20] The estimates based on the equations proposed by Mikame et al.[21] and Vecchio et al. Model B[24] were always higher than those obtained experimentally.

Influence of the thickness

The most correct way of evaluating the influence of thickness of each plate in compressive strength of concrete does not solely consist of normalising this parameter in relation to uniaxial compressive strength. It is also necessary to consider the transverse tensile strain and the quotient ɛ_{c1}/ɛ_{c2} regarding to concrete. As a consequence, the influence of the thickness must be studied by comparing experimental results with mathematical expressions which consider the aforementioned aspects.

Table 12 can be obtained by calculating the medium values of f_{c45}^{max}/f_{c2}^{max} in Table 11, for each thickness and in relation to each of the studied mathematical expressions. The values on the right column of Table 12, which are the arithmetic average of f_{c45}^{max}/f_{c2}^{max} for each plate's thickness, suggest that effective compressive strength of concrete in the presence of transverse tensile strains tends to increase with the thickness of the membranes, that is, it suggests that by increasing the thickness of the membranes the curves, which are better adjusted to the behaviour experimentally obtained, recommend higher compressive stress for concrete for the same strength ɛ_{c2}. This trend had already been suggested by the curves presented by Vecchio and Collins[20] and by parameter C.

This way, for 4-cm thick membranes, the best estimates for compressive strength correspond to those based on the equations proposed by Vecchio and Collins,[20] Belarbi and Hsu,[22] and Hsu[23]. The equation proposed by Vecchio et al. Model B[24] provides better estimates for the _{7-} and 10-cm thick membranes.

Comparing with normative provisions

This section compares reduction factor of compressive strength in concretes in the presence of transverse tensile strains (according to the presented experimental procedure) with the strength capacity reduction factors recommended in normative documents.

Table 13 and Fig. 7 compare the results obtained through experiment for the reduction factor of compressive strength in concrete, as well as the predictions carried out according to the provisions regarding CEN EN 1992-1-1. In reading Table 13 and Fig. 7, it is possible to conclude that the normative values for the reduction factor of compressive strength in concrete are very close to those experimentally verified.

Table 14 and Fig. 8 compare the results obtained through experiment for the reduction factor of compressive strength in concrete, as well as the predictions carried out according to the provisions regarding CEB-FIP MC 90.

In reading Table 14 and Fig. 8, it is possible to conclude that the normative values for the reduction factor of compressive strength in concrete are very close to those experimentally verified. The values of the reduction factor provided by CEB-FIP MC 90 are less restrictive than those experimentally verified and the values provided by FIB, CEB-FIP[28] present a higher correspondence to the values experimentally obtained.

Table 15 and Fig. 9 compare the results obtained through experiment for the reduction factor of compressive strength in concrete, as well as the predictions carried out according to the provisions regarding ACI 318–05, with α =90° and θ = 45°. By reading Table 15 and Fig. 9 it is possible to conclude that the strength capacity reduction factor suggested by the document in question provides conservative values regarding experimental values.

Table 16 and Fig. 10 compare the results obtained through experiment for the reduction factor of compressive strength in concrete, as well as the predictions carried out according to the provisions regarding CSA 23.3–04. In reading Table 16 and Fig. 10 it is possible to conclude that, for the values of the observed coefficient ɛ_{c1}/ɛ_{c2}, the equation in question generally suggests a greater capacity reduction of the compressive strength for concrete than that obtained through experiment.

The analysis of experimental results show the softening effect associated to compressive stress for concrete.

The analysis of the figures representing the stress versus strain compressive concrete behaviour shows that the form of the curve, as well as the stress and strain corresponding to the compressive strength for concrete experimentally obtained, are within the estimates based on the models proposed by other authors. This validates the test procedures that were implemented in this work, as well as the results that were obtained.

The results of this study suggest that the thickness of the membranes influences the behaviour of the concrete in the struts. The increase in thickness corresponds to an increase in the effective strength in concrete struts (in terms of stresses). This trend must, however, be cautiously considered for the following reasons:

the reduced number of validated tests (five membranes);

the normalised shear supported by the membranes suggests an opposing trend which, according to the authors, may be related to the dispersion which medium experimental tensile strains may have caused by the location of the cracks.

In any case, the work shows that the influence of the membrane's thickness must be subjected to specific studies geared towards the analysis of this variable. Furthermore, it is known that tests on larger concrete cubes with oriented damage (without friction between the concrete specimen and the machine platens) will lead to higher values of compressive strength when compared to the values obtained for smaller cubes.[31] The confinement effect of the outer concrete mass may have some influence on the overall strength of the specimens both in cube crushing and in the testing of panels reported here. Comparing the values of the reduction factor of compressive strength in concrete with the normative provisions shows that:

(i) CEN EN 1992-1-1[26] presents very similar reduction factors (shear + torsion) for the strength capacity, although slightly less restrictive, to those obtained;

(ii) CEB-FIP MC 90[27] shows reduction factors of compressive strength in concrete, in the simplified provisions regarding shear and torsion, similar to CEN EN 1992-1-1; in the most rigorous provisions, CEB-FIP MC 90[27] suggests reduction factors much less restrictive than those experimentally verified. FIB, CEB-FIP[28], in its turn, suggests reduction factors matching the experimental ones;

(iii) ACI 318–05[29] suggests limitations to strength capacity of elements carried by very restrictive shear or torsion provisions when compared to values which were experimentally obtained for the reduction factor of compressive strength in concrete;

(iv) CSA 23.3–04[30] suggests reduction factors of compressive strength in concretes more restrictive than those experimentally obtained.