(
c u ct c u ct)
ct ct
pl
c =−f + f Eε + f Eε − f
σ , 2 2 . (4.18)
The minimal required ratio between compressive stress and tensile strength is
( )
(
c u ct c u ct)
ct ct
ct pl c
ct f E f E f
f
n = σf =− + ε + ε −
2 1 2
, 1
. (4.19) The plots of the expression 4.19, as a function of εu, for different levels of tensile strengths are given
in Figure 4.16. The values obtained for nct justify the interest in providing sufficient compressive strength in order to exploit the tensile capacities of UHPFRC, considering in addition that the deformation, and thus maximal compressive stress, will continue to increase in the following regime (regime C). For a typical UHPFRC, the compressive stress required at the end of the pseudo-plastic phase is 5-6 times tensile strength, which is largely satisfied. This also allows Equations 4.10 to 4.15 to be applied in further analysis. Plots in Figure 4.16 show an increasing requirement for compressive stress as a function of εu, and a decreasing requirement as a function of tensile strength.
0 5 10
0 0.01
εu [-]
σc / fct
fct = 3 MPa fct=10 MPa fct=30 MPa
Figure 4.16: Minimal ratio of compressive stress to tensile strengths required to enable exploitation of maximal tensile strain εu
4.3.4.2 Element behaviour in presence of pseudo-plastic yielding
For the given boundary conditions (Figure 4.11 c)) the function of curvature χ(M), Equation 4.11, can be expressed as a continuous function along the length of the element, with M = M(P, x) being the continuous function of the moment along x. Plots in Figure 4.17 a) represent the distribution of curvature along the element length (up to mid-span) for different load levels.
It can be noted that for load levels of up to approx. 70-80 % of maximal load in the pseudo-plastic phase Ppl,max curvatures deviate slightly from the elastic curvatures. The significant increase in curvature occurs when the load approaches the level of Ppl,max (see also Figure 4.13), causing a considerable local increase in deformations in the vicinity of the most loaded section. Thus, multi-microcracking, which in a uniaxial stress state prevents strain localisation, causes the concentration of deformations in the case of a non-constant tensile force along the element. Similar conclusions can be drawn regarding the deformations of a member subjected to a uniformly distributed load (Figure 4.17 b)).
UHPFRC
nct
a) b)
Figure 4.17: Distribution of curvature along length of beam at various load levels, for boundary conditions of : a) three-point bending; b) uniformly distributed load along a simple beam
The force-displacement relationship for the beam can finally be obtained e.g. by the integration of curvature. For simple symmetric load cases, closed form solutions can be established from the following
The solution for the mid-span deflection of a beam subjected to three-point bending is obtained in the following form:
⎥=
Otherwise, for an arbitrary monotonic function of the bending moment, performing a numerical integration applying any standard integration procedure gives satisfactory results.
The force-displacement relationships for UHPFRC beams of different heights are plotted in Figure 4.18. In Figure 4.18 b) the simulated curves are plotted against the measured data of beams of 25, 40 and 60 mm height, in constant span L = 420 mm, subjected to a three-point bending test (Appendix T1). The analytical results show very good agreement with experimental results. As previously stated, the bending strength attained prior to the development of macrocracking is size-independent, and it can be seen that a high proportion of bending strength (more than 85%) is achieved in this regime. Regarding the deformational capacity of the elements, it is evident that the thinner elements are more deformable which is further discussed in § 4.4.3.
a)
0 0.02 0.04
ΔL2 h 0
10 20 30
ΣequMPa
h 25, 40, 50, 60, 75 mm
b)
0 0.02 0.04
ΔL2 h 0
10 20 30
ΣequMPa
h 25, 40, 60 mm
Figure 4.18: Bending stress over mid-span displacement for beams of various heights, h, subjected to three-point bending, with 420 mm span: a) simulations with fct = 9 MPa, εu = 2.5 o/oo
and Ec=60 GPa, b) simulations and measured response of UHPFRC beams (Appendix T1)
In addition, longitudinal deformations along the length of the beam can also be predicted by applying the developed equations. Let us observe the elongation of two points on the tensile (lower) side of the element. For the sake of comparison with experimental data, let us consider the two points at the position x = L / 2 - lm /2 and L /2 + lm /2, L /2 being the point of introduction of the force (Figure 4.19).
lm
θ
θ
l + lm Δ m l + lm Δ m, device
hm
x v(x)
L P
Figure 4.19: Measurement of non-linear tensile deformations: initial position and rotation of measurement points caused by rotation of beam
The elongation of the two points, using symmetry with respect to mid-point, is determined by the following expression:
thick
thin
dx integration can be performed. The calculated elongation, Δlm, is plotted in Figure 4.20 against the measured data as a grey line. Owing to the geometry of the measurement device (Figure 4.19), the experimentally captured deformations are higher than the beam deformations, due to the rotation of the measurement base. The error is more pronounced for slender beams because of their higher deformability (Figure 4.20 a)).
a) b) c)
Figure 4.20: Calculated and measured elongations of the measurement base during the elastic- pseudo-plastic phase, for beams of various depths; continuous grey lines denote actual elongation, while the black lines denote calculated elongation of the same measurement base but captured at the position of the measurement device
The relationship between measured elongation, Δlm. device, and actual elongation, Δlm., can be obtained from the geometry of the device (Figure 4.19) if the rotation θ of the member at the position of the measurement device is known:
( )
The rotation θ can be obtained from integration of the curvature function, respecting the boundary conditions:
for each level of load P. The obtained measured elongation (Equations 4.24 and 4.27) shows good agreement with experimental data (Figure 4.20, black line).
Δlm
Δlm, m. device
test data
These results, together with results obtained from the photogrammetry analysis (Figure 4.28 showing no localisation of deformation in one crack prior to Ppl, max), confirm1 that the multi-microcracking observed in uniaxial tension also develops in bending and can be well modelled using pseudo-plastic material properties in tension.
4.3.4.3 Inelastic unloading and actual element stiffness in multi-microcracking regime