RC shear critical beams tested by Pogdorniak-Stanik

Further comparisons between numerical and experimental results are also provided in terms of cracking development and crack widths, as depicted in Figure 2.18, which shows the crack pattern at failure for the three specimens tested by Vecchio and Shim [74]. It should be noticed that the reported numerical crack pattern corresponds to numerical failure load, which is very close to the experimental one for all the considered beams, as observed from previous comparisons. As can be seen, the model exhibits a fine capability for reproducing the diagonal-tension crack experimentally observed for these beams, so correctly describing the very brittle and sudden failure, typical of beams containing no shear reinforcement. Furthermore, maximum crack widths, which represent one of the most difficult parameters to predict in numerical analyses, are substantially comparable to experimental ones.

Figure 2.18 Experimental (left side, [74]) vs numerical (right side) crack patterns and crack widths at failure

Table 2.6 Geometrical details of the analyzed beams

Figure 2.19 Geometric dimensions and reinforcement arrangement

Table 2.7 Properties of the reinforcing bars

Properties of reinforcing steel are listed in Table 2.7; the elastic modulus Es

is evaluated according to Eurocode 2 [115], since the experimental value was not measured. As regards concrete properties, only the cylindrical compressive strength - fc = 37 MPa - is provided in [75]; as a consequence, the other parameters required for the definition of the adopted constitutive relation are

Sample b [mm]

h [mm]

d [mm]

L_Span [mm]

L_tot [mm]

Longitudinal Reinforcement

A_s/bd (%)

BN25 300 250 225 1352 1502 3 M15 0.89

BN25D 300 250 225 1352 1502 3 M15, 10 #3 1.31

BN50 300 500 450 2700 3000 2 M20, 1 M25 0.81

BN50D 300 500 450 2700 3000 2 M20, 1 M25, 10M10 1.11

508585858585

300

500

M25 M20

M25 M20 M10

50 M15

300

250

BN50 BN50D

BN25 BN25D

25 40 40 40 40 40

M15

#3

Bar size d_b [mm]

A_b [mm²]

f_y [MPa]

f_u [MPa]

E_s [MPa]

#3 9.5 71.3 508 778 200000

M10 11.3 100 458 692 200000

M15 16 200 437 643 200000

M20 19.5 300 438 667 200000

M25 25.2 500 490 689 200000

evaluated through the correlations suggested in [115]. Even in this case, the compressive post-peak nonlinearity parameter B is assumed equal to 0.1.

All the tests were performed under loading control, by applying a central point load in several steps. At the end of each step, the load was lowered to approximately 90% of its current peak value and then increased again. During all test execution, crack pattern evolution and crack width were monitored in detail by noting them at each load stage.

Numerical analyses are performed by following the same modeling choices already described in §2.4.2.1 for the shear critical beams tested by Vecchio and Shim [74].

Figure 2.20 Comparison between numerical and experimental [75] results in terms of applied load P vs. midspan deflection δ

0 50 100 150 200 250

0 1 2 3 4 5 6

Experimental NLFEA BN25

BN25D

P [kN]

δ[mm]

(a)

0 50 100 150 200 250 300 350

0 1 2 3 4 5 6 7 8 9

Experimental NLFEA

P [kN]

BN50

δ[mm]

BN50D

(b)

Figure 2.20 shows a comparison between numerical and experimental results in terms of applied load P vs. midspan deflection δ: Experimental evidence is well met by numerical predictions both as regards stiffness and failure load and mode. Numerical analyses are also able to predict the variation of bearing capacity related to a different amount and arrangement of longitudinal reinforcement in the cross-section: beams BN25D and BN50D which contain additional distributed reinforcements on element sides fail at a higher shear stress than the corresponding specimens BN25 and BN50, as also evidenced by experimental tests. Therefore, the satisfactory agreement between experimental and numerical results for both the experimental program carried out by Vecchio and Shim [74] (see §2.4.2.1) and by Podgorniak-Stanik [75] proves that the proposed model is able to reliably describe the behavior up to failure (in terms of strength, stiffness, ductility), for different specimen geometries, as well as different reinforcement arrangements.

Comparisons with Code provisions are then provided. The ultimate load is computed as already described in §2.4.2.1, by applying the relations suggested in MC2010 [57] and EC2 [115]; see respectively Equations (2.25)-(2.26) and (2.27)-(2.28).

Table 2.8 Comparison between experimental [75], numerical and Design Code results in terms of failure load and their relative ratio

The results presented in Table 2.8 indicate that the failure load is predicted with accuracy by numerical simulations for all the examined specimens. As regards beams BN25 and BN50, also EC2 relation provides precise peak values, while the approach suggested in MC2010 [57] tends to underestimates them. On the contrary the Design Codes fail in providing the correct value for the beam with the designation D, both EC2 and MC2010 relations indeed do not account for the small longitudinal bars distributed along the web of these elements, so providing the same value for the two series of beams.

Moreover, the model provides accurate results not only in terms of global load-deformation response but also in terms of crack widths and crack pattern evolution up to failure, as depicted in Figure 2.21 and Figure 2.22.

Sample P_u,exp [kN]

P_u,num [kN]

P_u,MC2010 [kN]

P_u,EC2 [kN]

P_u,num P_u,exp

P_u,MC2010 P_u,exp

P_u,EC2 P_u,exp

BN25 145 157 115 148 1.08 0.80 1.02

BN25D 244 215 115 148 0.88 0.47 0.61

BN50 260 266 193 241 1.02 0.74 0.93

BN50D 322 327 193 241 1.02 0.60 0.75

Figure 2.21 Numerical vs. experimental [75] crack pattern and crack width at failure

Figure 2.21 shows the numerical and experimental [75] crack pattern at failure for the four investigated specimens. As previously described for the beams tested by Vecchio and Shim [74], the numerical crack pattern refers to the failure load obtained by simulations, which is almost superimposed to the experimental one for all the considered specimens, as shown in Table 2.8. The model is able to reproduce the brittle experimental failure characterized by the presence of a critical diagonal-tension crack.

BN25 wf-max =0.25 mm

wf-max =0.43 mm33

BN25D

wf-max =0.25 mm wf-max =0.21 mm

BN50

wf-max =0.20 mm wwf-max f-max ==0.43 mm 0.20 mm

w_f-max =0.43 mm BN50D

^{wf-max =0.25 mm BN50D wf-max =0.35 mm}

Failure

BN25 → Pu,exp = 145 kN BN25D → Pu,exp = 225 kN BN50 → Pu,exp = 260 kN BN50D → Pu,exp = 322 kN

Failure

BN25 → Pu,exp = 145 kN BN25D → Pu,exp = 225 kN BN50 → Pu,exp = 260 kN BN50D → Pu,exp = 322 kN

Failure

Figure 2.22 Numerical (left side) vs. experimental (right side, [75]) crack patterns and crack widths at different loading stages for specimen BN25

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.