CHAPTER 2 REVISION OF CONCRETE CONTRIBUTION
2.4 Comparison between numerical and experimental results
2.4.3 SFRC beams without shear reinforcement
Crack pattern evolution for increasing loads, in terms of crack distribution and width, is instead represented in Figure 2.22 for beam BN25. Similar results are obtained also for the other specimens, but herein omitted for sake of brevity.
As regards the first loading stage (LS1) depicted in Figure 2.22, the comparison highlights that the first numerical flexural crack occurs at approximately the same loading level registered during the experimental test. As loading increases, other flexural cracks develop, until the attainment of loading condition LS5, which is very close to failure load. At this point a significant shear diagonal crack starts to form as extension of existing cracks; this shear crack is well visible at failure, as depicted in Figure 2.21. Crack patterns and widths are reasonably well described during the entire loading history; hence, thanks to its fine capability for predicting crack pattern evolution, the proposed model can represent a valuable design tool also for serviceability verifications, where crack control represents one of the fundamental issues to be checked.
The attention is herein focused on four shear critical beams, tested by Cucchiara et al. [76] and subjected to four-point bending test. These beams were characterized by the same rectangular cross section - 150 mm wide and 250 mm deep – and by the same length (equal to 2500 mm), but had different shear spans a. All the specimens did not contain stirrups and were characterized by the same longitudinal reinforcement (2φ20), characterized by a Young modulus Es=232000 MPa and a yield strength fy=610 MPa. The main geometrical details and the amount of steel reinforcement of the analyzed beams are illustrated in Figure 2.23 e Table 2.9.
Figure 2.23 Geometric dimensions (in mm) and reinforcement arrangement of the examined beams
Two beams were reinforced by hooked-ended steel fibers, with length of 30 mm and equivalent diameter of 0.5 mm. For comparison, also two plain concrete beams having the same geometric characteristics as the SFRC ones are considered. All the examined specimens presented about the same compressive strength but, as expected, the fibrous concrete was characterized by more ductility and minor loss of strength in the post-peak phase. The fiber amount and the main concrete properties assumed in the simulations are provided in Table 2.10.
Table 2.9 Geometrical details of the analyzed beams A a=613
B a=438 φ20
150
250
a
P/2
1150 100
Sample b [mm]
h [mm]
d [mm]
a [mm]
Ltot
[mm]
Longitudinal Reinforcement
As/bd (%)
A00 150 250 219 613 2500 2 Φ20 1.91
A10 150 250 219 613 2500 2 Φ20 1.91
B00 150 250 219 438 2500 2 Φ20 1.91
B10 150 250 219 438 2500 2 Φ20 1.91
Table 2.10 Amount of fibers and concrete mechanical properties applied in simulations The adopted nomenclature is the same used by Cucchiara et al. [76] and is formed by a capital letter followed by two digits. Letters A and B respectively denote specimens with a shear span-to-depth ratio (a/d) equal to 2.8 and 2.0; the first digit refers to the amount of fibers (0 for plain concrete and 1 for Vf = 1%) and the second one to the amount shear reinforcement, which is equal to 0 for all the considered beams, since they did not contain stirrups.
All tests were carried out with monotonically increasing displacements.
Numerical analyses are then performed on the four selected beams: the same modeling choices already described in §2.4.2 for ordinary RC beams are applied.
Figure 2.24 Comparison between numerical and experimental [76] response of SFRC cylinder specimen subjected to compression
It is worth noting that the model proposed by Ottosen [70,72] and implemented with slight modifications in 2D-PARC constitutive law, was originally conceived only for plain concrete. To prove the consistence of the model also for fibrous concrete, a cylindrical specimen, realized with the same concrete mixture of SFRC beams and subjected to compression, is first modeled (Figure 2.24). As can be seen, numerical and experimental responses are almost superimposed
Sample Vf (%)
fc [MPa]
fct [MPa]
Eci [MPa]
εc 0 (–)
A00 0.0 41.20 1.84 26094 0.0025
A10 1.0 40.85 1.84 26236 0.0028
B00 0.0 41.20 1.84 26094 0.0025
B10 1.0 40.85 1.84 26236 0.0028
-50
-40
-30
-20
-10
0
-8 -6
-4 -2
0
Experimental NLFEA
σ2c [MPa]
ε
2c[‰]not only in the pre-peak branch, where the behavior of fibrous concrete is almost the same of concrete without fibers [76], but also in the descending part of the curves. This result can be easily obtained through a proper calibration of parameter B, which is set equal to 0.5 for the samples reinforced with fibers;
while it is maintained equal to 0.1 for the plain concrete ones.
Figure 2.25 Comparison between numerical and experimental [76] results for RC (A00, B00) and SFRC (A10, B10) beams in terms of applied load P vs. midspan deflection δ
The global response of the RC and SFRC beams is first analyzed. Figure 2.25 shows the comparison between numerical and experimental [76] results in terms of total load P vs midspan deflection δ for both the considered series of beams. For immediate comparison each graph reports the curves relative to the specimens characterized by the same shear span (A or B), with and without fiber addition (1 or 0). Experimental behavior for both the beams with greater (Figure
0 50 100 150 200 250
0 4 8 12 16 20
Experimental NLFEA
A00
A10
P [kN]
δ[mm]
(a)
0 50 100 150 200 250
0 4 8 12 16 20
Experimental NLFEA
B00
B10
P [kN]
δ[mm]
(b)
2.25a) and smaller (Figure 2.25b) span shear length is predicted closely.
Moreover, the improvement in terms of bearing load in the post-cracking stage due to the presence of fibers is correctly caught, so proving the accuracy of the proposed formulation also in case of SFRC elements.
Design Code provisions are then considered for the evaluation of the failure load. In the following only a theoretical explanation is provided, since the considered Code relations for SFRC specimens require the knowledge of the residual tensile strength fFtuk, which was not experimentally measured.
In more details EC2 [115] does not suggest any specific relation for specimens with fibrous reinforcement without stirrups; whereas MC2010 provides two formulas.
The first MC2010 relation, which has been validated and so it is considered reliable [57] is given by:
min F , Rd cp w
ck ctk Ftuk l
c F ,
Rd f . b d V
f . f . k
V ≥
+
+
= ρ σ
γ 100 1 75 015
18
0 3
1
, (2.29)
where
d b .
f k .
VRd,Fmin ck cp w
+
= 0035 2 015σ
2 1 3
, (2.30)
being d the effective depth of the cross-section, bw the minimum width of the cross-section, ρl the geometrical ratio (defined as As/bwd) and σcp the average axial stress due to loading or pre-stressing, while k represents a parameter that takes into account the size effect and it is set equal to 1+√(200/d). Further details can be found in MC2010 [57], to which reference is made. This approach is not consistent with the one adopted by MC2010 for RC beams reported herein in Equations (2.25)-(2.26), but it appears aligned to the one suggested by EC2, see Equations (2.27)-(2.28). The difference between EC2 relation for traditional RC elements - Equation (2.27) - and the above described formula - Equation (2.29) – lays indeed only in the addition of the term 7.5 fFtuk/ fctk in the round brackets, which is equal to zero for traditional RC beams.
The latter MC2010 relation is expressed as:
w Ftuk
f ck dg x
F F ,
Rd f kf cot zb
z k
V . ⋅
+
⋅ +
= + θ
ε
γ 1000
1300 1500
1 4 0
1 , (2.31)
being z the internal lever arm, θ the angle of the compressive stress field, kdg a parameter related to the maximum aggregate size. These parameters are in turn calculated according to MC2010, while kf is posed equal to 0.8. εx is the longitudinal strain calculated at the mid-depth of the effective shear depth by
Equation (2.26). This approach, more extensively explained in fib Bulletin 57
“Shear and punching shear in RC and FRC elements. Workshop proceedings.”
[116], even if not fully validated [57], is coherent with MC2010 formulation for RC elements, reported herein in Equations (2.25)-(2.26).
At last, the experimental crack distribution at failure is compared to the final numerical crack pattern in Figure 2.26: the brittle shear failure is well described in all cases, with the development of localized diagonal cracks running along an inclined line, from the support towards the loading point; thus proving once again the effectiveness of the proposed approach.
Figure 2.26 Comparison between numerical and experimental [76] crack patterns at failure for RC/SFRC specimens.