RC shear critical beams tested by Vecchio and Shim

Finally, Figure 2.14 shows panel stress-strain response for uniaxial and biaxial compressive loading. In this case, the adopted model is able to provide an accurate prediction of the experimental response both in pre- and post-peak regions, correctly catching crushing failure.

Moreover, the evolution of volumetric strains ΔV/V= 1c+2c+3c, reported in Figure 2.15, proves the effectiveness of the followed approach in representing the volume reduction and then the dilatation typical of concrete loaded in compression. These last graphs show two distinct numerical curves, which are respectively obtained by following a different procedure for the evaluation of Poisson coefficient  in the post-peak region. At first (dotted line, Figure 2.15),  is determined by keeping fixed the secant bulk modulus Kf for a given change of the secant elastic modulus Ec, as suggested in [71,99]. Subsequently (continuous line, Figure 2.15)  is calculated by following the approach proposed by in this thesis, by applying Equation (2.20). It can be observed that this latter approach provides more regular curves, which better fit the available experimental data.

The three beams were characterized by a progressively increasing concrete compressive strength fc. This value, together with the other concrete mechanical properties (the peak strain c0, the initial elastic value of the Young modulus Ec, the tensile strength fct) necessary to define the adopted constitutive relation are reported in Table 2.3.

Table 2.2 Geometrical details of the analyzed beams

As the initial value of Poisson coefficient  was not experimentally measured it is assumed equal to 0.2. The compressive post-peak nonlinearity parameter B is set equal to 0.1. Moreover, as only the concrete split cylinder strength fsp was available, the value of fct reported in Table 2.3 is calculated according to Eurocode 2 [115].

Table 2.3 Concrete mechanical properties applied in simulations

The main characteristics of the reinforcement, both in terms of geometrical details (bar dimeter db and area Ab) and steel properties (elastic modulus Es and yield fy and ultimate fu strength), can be found in Table 2.4.

Table 2.4 Geometrical details and mechanical properties of the reinforcement Sample b

[mm]

h [mm]

d [mm]

L_Span [mm]

L_tot [mm]

Longitudinal Reinforcement

As/bd (%)

OA1 305 552 457 3660 4100 2 M25^b, 2 M30 1.72

OA2 305 552 457 4570 5010 2 M25^a, 3 M30 2.22

OA3 305 552 457 6400 6840 2 M25^b, 4 M30 2.73

Sample f_c [MPa]

f_ct [MPa]

E_ci [MPa]

ε_{c 0} (–)

OA1 22.6 2.131 36500 0.0016

OA2 25.9 3.031 32600 0.0021

OA3 43.5 2.821 34300 0.0019

Bar size d_b [mm]

A_b [mm²]

f_y [MPa]

fu

[MPa]

E_s [MPa]

M25^a 25.2 500 440 615 210000

M25^b 25.2 500 445 680 220000

M30 29.9 700 436 700 200000

All the tests were performed under loading control, with a central point load, until the approaching of the ultimate stage, when the procedure was switched to displacement control so to allow the evaluation of the post-peak behavior. As already mentioned, the purpose of this experimental program was to recreate, as much as possible, the Bresler and Scordelis tests [114], in terms of geometrical dimensions, reinforcement details, material strengths and loading. Compared to these latter, the beams tested by Vecchio and Shim [74] exhibited indeed a very similar behavior, with only few minor differences; as a consequence, only the specimens described in [74] are considered in the FE analyses reported herein, but in the following graphs, also the results obtained by Bresler and Scordelis [114] on nominally identical beams are reported.

Taking advantage of the symmetry of the problem, only one half of each beam is simulated, by adopting a FE mesh constituted by quadratic, isoparametric 8-node membrane elements with reduced integration (4 Gauss integration points). Numerical analyses are performed under displacement control, by applying an increasingly displacement at the loading point, in order to achieve a better numerical convergence and evaluate also the post-peak behavior.

Numerical and experimental results are first compared by considering the global response, in terms of applied load P vs midspan deflection , as can be seen from Figure 2.17. On the same graphs, the experimental results obtained by Bresler and Scordelis [114] on nominally identical beams are reported too.

The graphs highlight the high accuracy of the proposed model in representing the global behavior both at serviceability (cracking load) and at ultimate limit state. The peak load is accurately predicted, as well as the brittle shear failure characterized by no ductility.

For comparison, design Code provisions are also considered. In more details, both the relations suggested in Eurocode 2 [115] and Model Code 2010 [57] (EC2 and MC2010 in the following) are analyzed. Table 2.5 compares the experimental ultimate load capacity Pu,exp to the numerical value Pu,num and to the maximum load to failure provided by Design Codes: Pu,MC2010 and Pu,EC2, by applying respectively MC2010 and EC2 relations. The maximum load is obtained as the double the ultimate resistant shear, depurated by the self-weight.

Table 2.5 Comparison between experimental [74] , numerical and Design Code results in terms of failure load and their relative ratio

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

OA1 331 297 212 265 0.90 0.64 0.80

OA2 320 359 225 290 1.12 0.70 0.91

OA3 385 370 254 341 0.96 0.66 0.88

Figure 2.17 Comparison between numerical and experimental [74] results in terms of applied load P vs. midspan deflection 

0 50 100 150 200 250 300 350 400

0 2 4 6 8 10

Vecchio-Shim beam Bresler-Scordelis beam NLFEA

P [kN]

 [mm]

OA1 (a)

0 50 100 150 200 250 300 350 400

0 3 6 9 12 15 18

Vecchio-Shim beam Bresler-Scordelis beam NLFEA

P [kN] OA2

 [mm]

(b)

0 50 100 150 200 250 300 350 400

0 5 10 15 20 25 30 35 40

Vecchio-Shim beam Bresler-Scordelis beam NLFEA

P [kN] OA3

 [mm]

(c)

With reference to MC2010 [57], the ultimate resistant shear force is computed for the Level II Approximation, by applying the following Equation:

w c ck dg x

c ,

Rd f zb

z k V .

γ

ε ⋅

⋅ +

= +

1000 1300 1500

1 4

0 , (2.25)

being z the internal lever arm, bw the minimum width of the cross-section and kdg

a parameter related to the maximum aggregate size (equal to 20 mm for these beams). εx represents the longitudinal strain calculated at the mid-depth of the effective shear depth and it is in turn calculated according to MC2010 [57]:







 



 

 +  +

= z

N e z V

M A

E ^{Ed Ed}

Ed s s x

ε m ∆

2 1 2

1 , (2.26)

where ∆e is the load eccentricity and Ned, Ved, Med are the sectional forces. The applied shear Ved in Equation (2.26) is posed equal to the ultimate resistant shear force of the element - Equation (2.25); thus an iterative procedure is required.

The ultimate resistant shear force according to EC2, [115], can be instead calculated as:

(

l ck

)

cp w Rd,cmin c

, Rd c

,

Rd C k f k b d V

V  ≥



+

= ρ 13 1σ

100 , (2.27)

where

d b k f

k .

V_{Rd,cmin ck cp} _w









 +

= 1σ

12 32

035

0 , (2.28)

being d the effective depth of the cross-section, ρ^l the geometrical ratio (defined as As/bwd), σ^cp the axial stress due to loading or pre-stressing. The parameters Crd,c, k and k1 are respectively set equal to 0.18/γ^{c , 1+√}(200/d) and 0.15. Further details can be found in EC2, [115], to which reference is made.

Referring to experimental tests, in Equations (2.25) and (2.27) the characteristic value of compressive strength fck is substituted by its mean value fc and the partial safety factor for concrete (γ^c) is posed equal to 1.

As can be observed in Table 2.5, the NLFEA ultimate load and the experimental value are very close to each other, also varying the span length, while MC2010 and EC2 relations provides values further from experimental observations, even if the latter provides better estimations. However, MC2010 and EC2 values can be considered quite satisfactory, considering the high scatter typical of shear tests and bearing in mind that they should provide results on the safe side.

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

Nel documento Development of a nonlinear model for RC/FRC elements applied to the analysis of tunnel linings under fire conditions (pagine 108-113)