Validation of the model with experimental data: BSI® UHPFRC

∂ (4.79)

It is also possible to find the deformational capacity i.e.χ of a section in softening regime, for a desired bending moment, Mi, as:

( )

Mi Mi Mi

M χ = ⇒χ (4.80)

which is a necessary piece of information when a certain rotation of the element has to be assured, typically in statically indeterminate systems (Chapters 5 and 6).

Force-displacement relationship

With the defined M-χ relationship of the cracked section (Equations 4.77 and 4.78), the deflections of the beam can be obtained using Equations 4.71 for the pre-peak phase and Equation 4.73 for the post-peak phase.

4.4 Application of the developed model

4.4.1 Validation of the model with experimental data: BSI® UHPFRC

The results of the model are compared with experimental results obtained in the test programme on thin beams made of BSI® UHPFRC, performed at the EPFL (Appendix T1). Beams of heights varying between 25 - 75 mm are tested in three point bending, with a constant span L = 420 mm.

Some of the experimental results in terms of force - mid-span displacement are shown in Figure 4.49 a), together with results from the analytical model (black line) and numerical model (grey line). The results are obtained with material properties in tension based on results from uniaxial tensile test (§ 3.3.2.8). Experimentally achieved bending strengths are plotted in Figure 4.49 b) against element’s height.

A significant dispersion in measured bending strengths, and in the force-displacement response in general, was also noted considering the elements of constant height, Figure 4.50 b). More remarkable scatter of the results is noted in the region of higher solicitations, characterised by macrocrack opening, while the non-linear response due to microcracking shows less dispersed results.

β = 0.2

β = 0.4

model

a) b)

0 2.5 5 7.5

Δ mm

0 25 50

PkN

0 50 100

h mm

0 15 30

fequMPa

Figure 4.49: a) Measured and simulated force-mid-span deflection response for UHPFRC beams of different heights h =25-75 mm; b) measured bending strength for beams of different heights

For the material with a pseudo-plastic phase up to εu= 2.5 ‰, the fictitious crack is assumed to start propagating at σequ = nfct fct=2.4 fct = 21.6 MPa (Figure 4.50, dashed line). A similar dispersion in results of bending test, applying similar materials, is reported by other researchers (e.g. [Reineck, Greiner 2004], [Graybeal 2006]).

a) b)

0 5 10 15 20 25 30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

deflection [mm]

mean value characteristic v.

σ

[MPa]

equ

4.0

Figure 4.50: Beam in four-point bending: a) comparison of results of numerical model (black line) and the mean value of test results (grey line); b) test results on plates, h=50 mm, in four-point bending, with span L = 420 mm [Simon 2006]; tensile resistance fct = 9 MPa and pseudo-plastic plateau in tension

The scatter in the results can be explained principally by the non-uniformity of the material resistance influenced by the effective fibre volume, which varies along the element. For the same reason, it can be expected that the bending response, modelled with a single crack opening, and the tensile material law obtained from uniaxial test (characterising the weakest section along a tie¹), will not precisely follow the behaviour of every tested sampler. However, a mean value is very well represented with the model using mentioned material properties (Figure 4.50 a)). The results of the bending response of five beams of constant height h=50 mm made of BSI (Figure 4.50) are obtained from the test performed at the LCPC. The experimental programme performed at EPFL included

1 In uniaxial tension, there is a greater probability of localised deformation in the weakest section (due to element size), than in an element in bending where the maximal force concentrates in a smaller region.

0 2 4

Δ mm

0 10 20 30

ΣequMPa

model

test (mean) microcracking

elements of variable heights (Figure 4.49) but did not include more than three specimens of the same size, and the mean value given in Figure 4.50 is found more representative)

a) b)

0 5

Δ mm

0 5

PkN

P 25a: test and simulation

0 5

Δ mm

0 5

PkN

P 25a, simulations with fctvar.

c) d)

0 5

Δ mm

0 5

PkN

P 25a, simulations with GFvar.

0 5

Δ mm

0 5

PkN

P 25a, simulations with fct fctm

Figure 4.51: Simulated and measured response for a beam of h = 25 mm: a) simuation with tensile material properties obtained from uniaxial tensile test; b) variation of tensile strengths, corresponding to a constant stress of pseudo-plastic plateau; c) variation of strain softening slope; d) simulation with strain hardening (fctm=9 MPa, fct=10 MPa)

Several phenomena characterising UHPFRC elements enable the scatter in measured bending response once the macrocrack starts to propagate to be explained, and can be demonstrated using the developed models (Figure 4.51 to 4.53 ):

- statistically, more resistant sections are present in the region of maximal solicitation, due to locally increased fibre effectiveness; consequently a crack starts to propagate in the most loaded section for a higher load level, as if a positive tensile hardening slope had developed;

alternatively, due to the gradient of bending moment, a crack may start to propagate in an asymmetric section (Appendix T1);

- the crack pattern is irregular, dissipating more energy than the idealised linear crack; the tested beams had a relatively large width (200 mm), and the crack in some samples propagated along a very irregular line (Appendix T1); a longer crack line dissipates more energy, resulting in increased bending resistance;

- more than one macrocrack develops; similarly to the previous phenomenon, by dissipating more energy, more macrocracks enable a slight increase in resistance, and most of all a significant increase in deformations at peak-force; two or three macrocracks developing prior to peak-force were observed in some of the tested beams by means of photogrammetry analysis.

The simulations presented in Figures 4.51 to 4.53 demonstrate that a slight change in tensile material properties, respecting the range of possible tensile properties according to the tensile tests on notched ties for example (§ 3.3.2.8), enable the behaviour of some samples to be approached more accurately. In Figure 4.51 a), the modelled response is plotted against the experimental curve for a beam 25 mm thick. It can be seen that resistance is slightly underestimated, while deformations are

fct= 10 MPa

fct= 9.5 MPa

fct= 9 MPa

dσ / dw=

3.34 MPa/mm

5 MPa/mm

6.8 MPa/mm

less accurately predicted. The same Figure b) shows the influence of increase in the constant-stress pseudo-plastic plateau, while in c) the influence of the decrease in the strain-softening slope, dσ/dw, is shown. It can be noted that the increased pseudo-plastic stress, obviously increasing the strength, overestimates stiffness in the pre-peak part, while the decreased softening slope (increased initial part of GF) shows a tendency to increasing deformability. (According to uniaxial tensile test dσ/dw= 6.8 MPa/mm, 5 MPa/mm is a possible value for UHPFRC, and 3.34 MPa/mm is a value that corresponds to a slope that would be obtained using a well accepted strain-softening curve given by Equation 3.33, with γ = 4.) Finally, assuming a tensile strain hardening with a slope rising from fctm = 9 MPa to fct = 10 MPa, and maintaining the maximal strain prior to crack opening and the tensile softening slope, the measured curve is well simulated (Figure 4.51 d)).

0 2 4

Δ mm

0 5 10 15

PkN

P 4b: test and simulation

Figure 4.52: Simulated and measured response for a beam of h = 38 mm (nominal h = 40 mm) a) b)

0 2 4

Δ mm

0 10 20 30

PkN

P 5a: test and simulation

0 1.5 3

Δ mm

0 10 20 30

PkN

P 5a, simulations with fct fctm

c) d)

0 1.5 3

Δ mm

0 20 40

PkN

P 6 a and b: test and simulation

0 1.5 3

Δ mm

0 20 40

PkN

P 6a, simulations with fct fctm

Figure 4.53: Simulated (black lines) and measured behaviour (grey lines) of beam of nominal heigth h=50 mm

fct= 11 MPa

fct= 10 MPa

fctm=const= 9 MPa

fct= 11 MPa

fct= 10 MPa

fctm=const=9 MPa

The behaviour of 38-mm high beam is well modelled with tensile behaviour obtained from uniaxial tensile test (Figure 4.52).

The behaviour of beams of thicknesses 50 and 60 mm is presented in Figure 4.53. Two very closely spaced macrocracks were observed in the sections close to mid-span, before Pmax is attained, in the case of the beam of h=50 mm, explaining its higher strengths and deformability.

In document Structural implications of ultra-high performance fibre-reinforced concrete in bridge design (Page 108-112)