Validation of the model with experimental data: other materials

The model is validated for other materials both with and without tensile strain hardening. Two cases of strain hardening materials are studied:

- a material with a more pronounced tensile strength than that of the tested UHPFRC, and with a similar ultimate hardening strain;

- a material with a lower tensile strength and a more pronounced ultimate hardening strain.

An SFRC is studied as a material without tensile strain hardening but with a pronounced bending hardening, due to significant GF

4.4.2.1 Application to materials with strain-hardening Multi-scale fibre-reinforced concrete

A UHPFRC known as CEMTECmultiscale©), mentioned in Section 3.2.3, [Rossi et al. 2005], is used here as an example of a strain hardening material with a high tensile strengths, in the range of 20 MPa, (Figure 4.54 a)), which is approximately two times greater than the tensile strength of the majority of UHPFRCs. The strain hardening slope is low, as is usual for UHPFRCs, with ultimate tensile strain εu= 2.5 ‰. The compressive strength is fc=220 MPa and modulus of elasticity Ec = 55 GPa. More information on the material, data on tested elements and the numerical FEM modelling can be found in [Tailhan et al. 2003], [Parant 2003].

a) b) c)

Figure 4.54: Measured data for multi-scale fibre-reinforced concrete CEMTECmultiscale©: a) uniaxial stress-strain relationship in tension, b) bending stress-mid-span displacement curve of beams h = 40 mm in four-point bending test; c) bending stress-mid-span displacement curve of beams h = 200 mm in four-point bending test; Data from [Tailhan et al. 2003]

The simulation of element behaviour using the non-linear beam analysis developed in this thesis is shown in comparison to measured data in Figure 4.55. The analysed element is a beam of 40 x 100 x 600 mm, subjected to four-point bending, with 3 x 140 = 420 mm of the span. Analysis is performed with the above-mentioned material properties, assuming various constant levels of pseudo-plastic tensile stress (18 -22 MPa).

a) b)

Figure 4.55: Simulated force-displacement curve (black line) for 40-mm thick beam with different levels of pseudo-plastic tensile stress, fct, and average measured curve (grey line):

a) zoom in the region before localisation of deformation in the cracked section;

b) position of the proposed design level at ULS (without safety factor) for different fct

It can be seen that both force and deformations are satisfactorily simulated with the proposed modelled. Maximal equivalent bending stress prior to crack localisation can be estimated using Equation 4.16: measured bending strength, achieved with crack opening, is only slightly higher than the value σequ(Mpl, max), considering elements of heights of 40 – 200 mm (Figure 4.54 d) and c)). Moreover, the pseudo-plastic behaviour was considered as the representative material behaviour according to authors [Tailhan et al. 2003], and it was used to model numerically the complete pre-peak behaviour, allowing an increased length of the quasi-plastic plateau.

Engineered cementitious composite

Engineered cementitious composites (ECC) are cementitious fibre-reinforced materials characterised by a very pronounced tensile strain before strain softening occures. In this paragraph, the behaviour of elements made of an ECC presented in [Kunieda et al. 2002] is simulated. Material characteristics in tension are given in Figure 4.56 a) and b), with tensile strength equal to 4.8 MPa,εu= 3.5‰ and Ec = 23.3 GPa.

a) b) c)

Figure 4.56: Data for ECC: a) uniaxial stress-strain relationship in tension, elastic and hardening region; b) stress-crack opening relationship; c) Force-mid-span displacement of tested beams and plates, h=100 mm; from [Kunieda et al. 2002]

macrocrack opening

The model is applied to elements of various sizes corresponding to the geometry of tested samples, tabulated in the mentioned literature source. Examples of the modelling of element behaviour in four-point bending are plotted in Figure 4.57. The beam of dimensions 100 x 100 x 400 mm, height, width and length, subjected to forces introduced with a span of 3 x 100=300 mm, is simulated using the analytical model (Figure 4.57 a)), assuming various levels of constant pseudo-plastic tensile plateau. In the same Figure b) the same element is modelled using strain hardening slope as defined in Figure 4.56 a), and the predicted response of a triple-height element (300 mm) is also shown.

a) b)

0 0.5

Δ mm

0 5 10 15

ΣflexMPa

Figure 4.57: Simulated force-displacement curves (black lines) and lower and upper limits of measured response: a) beam in four-point bending, h=100 mm, modelled assuming different levels of constant pseudo-plastic stress, fct ; b) beams in four-point bending, h=100 and 300 mm, modelled using strain hardening

Even in the case of more pronounced strain hardening deformations, the model can be seen to predict well the response, and strain hardening dominates in achieving bending strengths, with only a slight additional increase in bending stress with crack opening. With a tensile strength of 4.8 MPa and an elastic tensile stress 0.8·4.8 = 3.8 MPa, for an average pseudo-plastic strain fct = 4.3 MPa, the macrocrack can be assumed to start propagating at σequ(Mpl, max) = 2.43 fct = 10.5 MPa, according to Equation 4.16 (dashed line in Figure 4.57a)). The bending stress is increased for a lower percentage with the macrocrack opening in comparison to materials with lower εu.

4.4.2.2 Application to materials without strain hardening

Results of the developed model are compared with the results of other models for bending in the presence of a fictitious crack, published in [RILEM 2002], and some of them presented in §4.2. The studied element is a beam subjected to three-point bending, with a constant cross-section (150x150 mm), with a span L = 500 mm. Tensile behaviour is modelled as linear elastic, with ft=3MPa and the initial strain softening curve has a slope of 30 MPa/mm up to to w= 0.05 mm and zero stress at wc= 10 mm: The modulus of elasticity is Ec=35 GPa.

The comparison of predictions of various models is presented in Figure 4.58 a), and the comparison of an average prediction and the model developed in the present study are ploted in Figure 4.58 b, as gray and black line respectively. It can be concluded that the model enables also good prediction of bending response of quasi-brittle materials withouth hardening plateau.

crack opening

Test Simulations:

fct=4.8 MPa fct=4.3 MPa fct=3.8 MPa

a) b)

0 0.2 0.4

Δ mm

10 20 30

PkN

h 150.`mm, Pmax 23.97` , ΔPmax 0.06`mm

Figure 4.58: Bending response of SFRC: a) prediction of various models rapported in [RILEM 2002]; b) comparison of the average prediction by the models presented in a), grey line, and the model developed in the present study (black line)