Issue 38

S. Bennati et alii, Frattura ed Integrità Strutturale, 38 (2016) 377-391; DOI: 10.3221/IGF-ESIS.38.47

Stage 3 – Application of imposed loads – Non-linear response and failure of the system In stage 3, non-linear response is expected due to either plasticity of the steel beam or softening of the adhesive. However, as the numerical examples below will demonstrate, for current material properties and geometry, the steel beam turns out to be the weakest element of the system, while the adhesive behaves elastically up to very high values of the imposed load. A detailed description of the structural response in stage 3 would require considering the progressive plasticisation of the beam at the mid-span and neighbouring cross sections. At the same time, large deformations and displacements are expected to occur, bringing further non-linearity into the problem. For the sake of simplicity, here we do not explicitly analyse the above sketched non-linear response and limit ourselves to consider the ultimate limit state corresponding to the complete plasticisation of the mid-span cross section of the beam. or illustration purposes, we apply the model to standard IPE steel beams [15]. For each cross section of the series from IPE 120 to IPE 600, we first carry out a preliminary design to determine the span and permanent load of the “existing” unstrengthened beam. Then, we imagine to apply the Sika ® CarboDur ® FRP strengthening system [16] and calculate the loads corresponding to the elastic limit states in the beam, adhesive, and laminate. Lastly, we take into account the ultimate limit state corresponding to the plasticisation of the mid-span steel cross section and evaluate the increased load bearing capacity of the strengthened beam. Material properties and geometry of structural elements The material properties, factored according to the Eurocodes [20, 21] and Italian regulations on FRP strengthening [22, 23], are the following:  steel (grade S235): Young’s modulus, E s = 210 GPa; characteristic yield stress, f yk = 235 MPa; partial factor for material,  s = 1.05; design yield stress, f yd = f yk /  s = 223.81 MPa;  adhesive (Sikadur ® -30): shear modulus, G a = 4.923 GPa; characteristic strength,  k = 15 MPa; environmental conversion factor,  a = 0.85; partial factor for material,  a = 1.2; design strength,  0 =  a  k /  a = 10.63 MPa;  laminate (CarboDur ® S): longitudinal Young’s modulus, E f = 165 GPa; characteristic tensile strength, f fk = 3100 MPa; partial factor for material,  f = 1.1; design tensile strength, f fd =  a f fk /  f = 2395.45 MPa. Concerning the geometry of structural elements, the geometric properties of the steel cross sections are listed in Tab. 1. The thickness of the adhesive layer is assumed t a = 1 mm. The laminates have width b f = 60 mm and thickness t f = 1.3 (type S613) or 2.6 mm (type S626), depending on the steel cross section. The distance between the beam supports and laminate anchor points is a = 500 mm for all cross sections. F A PPLICATION

Flange thickness h f (mm)

Moment of inertia I b (mm 4 ) 3178000 5412000 8693000 13170000 19430000 27720000 38920000 57900000 83560000 117700000 162700000 231300000 337400000 482000000 671200000 920800000

Elastic section modulus W b (mm 3 )

Plastic section modulus Z b (mm 3 )

Width

Height h b (mm)

Area

Cross section

b b

A b

(mm)

(mm 2 )

IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600

64 73 82 91

120 140 160 180 200 220 240 270 300 330 360 400 450 500 550 600

6.3 6.9 7.4 8.0 8.5 9.2 9.8

1321 1643 2009 2395 2848 3337 3912 4595 5381 6261 7273 8446 9882

52960 77320 108700 146300 194300 252000 324300 428900 557100 713100 903600 1156000 1500000 1928000 2441000 3069000

60730 88340 123900 166400 220600 285400 366600 484000 628400 804300 1019000 1307000 1702000 2194000 2787000 3512000

100 110 120 135 150 160 170 180 190 200 210 220

10.2 10.7 11.5 12.7 13.5 14.6 16.0 17.2 19.0

11550 13440 15600

Table 1 : Geometric properties of steel cross sections.

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