Numerical model of the bottom cantilever connection

The typical hysteretic response of the bottom connection is presented in Figure 5.5. Initially, the friction force was activated, followed by the sliding of the panel (Figure 5.5, phase 1). When the available gap in the connection was exhausted, the stiffness of the connection increased considerably due to the bending of the cantilever bracket (Figure 5.5, phase 2).

Figure 5.5: Typical hysteretic response of the bottom connection during the shake table test Slika 5.5: Značilen histerezni odziv spodnjega stika med testom na potresni mizi

The analysis showed that the friction response of the bottom connection under the dynamic loading had somewhat different characteristics. Thus, the friction model of the bottom connection was different. In the presented tests (and the real buildings subjected to the seismic excitations), the panels were subjected to the dynamic load. The friction force in the bottom connection was considerably affected by the velocity of connections’ excitations and damping, as observed in Section 3.3.5. Thus, the viscous friction model was used that assumes that the friction force is a linear function of the sliding speed (see Figure 5.6) to model friction in the bottom connection.

Figure 5.6: Schematic presentation of the macro model: (a) a combination of different hysteretic behaviours used for the numerical simulation of the bottom connections under dynamic loading, (b) Viscous, (c) ElasticPPGap and (d) Hysteretic material models

Slika 5.6: Shematski prikaz makro numeričnega modela: (a) kombinacija različnih histrereznih materialnih modelov za numerično simulacijo spodnjih stikov med dinamično obtežbo, (b) Viscous, (c) ElasticPPGap in (d) Hysteretic materialni modeli

Different friction models are available in the literature (Andersson et al., 2007; Liu et al., 2015).

Commonly, the friction force is physically explained by the Coulomb friction behaviour as the product of normal force on the surface and the coefficient of friction that is generally acknowledged to be constant. However, the friction force may depend on the sliding speed, and the coefficient of friction between two objects may vary according to the relative speed of motion (Rabinowicz, 1956;

Kragelskii, 1965).

The analysis of single component tests showed that the response of the bottom connections was rather viscoelastic, which implied that the parallel combination of the Viscous and Elastic models would be appropriate for simulating friction in the bottom connection. This model was used to simulate single component tests and was included in the original paper Modelling in-plane dynamic response of a fastening system for horizontal concrete facade panels in RC precast buildings (Starešinič et al., 2020).

However, it is difficult to explain the physical importance of the elastic spring in the bottom connections because there is no obvious source of stiffness during the sliding phase. Experimentally defined elastic stiffness was relatively small, and in principle, the viscous friction is usually modelled using only the Viscous material model. For this reason, the viscous friction model presented in Figure 5.6 (a, b) was used for the following numerical analyses.

As was the case for top connection, the significant increase of the connections’ stiffness in the second phase of the response was simulated by the series combination of the ElasticPPGap (Figure 5.6 c) and the Hysteretic (Figure 5.6 d) material models. The complete model of the bottom connection was defined by the parallel combination of friction and impact models (Figure 5.6 a).

The common Coulomb model was used to model the friction in the bottom connection during the quasi-static cyclic tests because there were no dynamic effects. Thus, this model was similar to that used for modelling the top connections response (Figure 5.2).

The force–displacement relationship of the numerical models used to simulate bottom connection responses is schematically presented in Figure 5.7. The model parameters are the size of the gap (dgap,bottom), the displacement capacity (du), friction force (Rfr,bottom), resistance of the cantilever (Rmax,bottom), damping (cvisc) and stiffness (Kconn,bottom, Ki,bottom) as presented in following paragraphs.

The recommended values are summarised in Table 5.2.

Figure 5.7: Schematic envelopes of numerical models of bottom connections: (a) during the cyclic test and (c) during the dynamic test

Slika 5.7: Shematski prikaz ovojnic numeričnega modela spodnjih stikov: (a) med ciklično obtežbo in (c) med dinamično obtežbo

Table 5.2: Recommended values of the model parameters of the bottom connection Preglednica 5.2: Priporočene vrednosti modelnih parametrov spodnjega stika

Material characteristic Value Material characteristic Value

dgap,bottom* ±4.5 cm Rfr,bottom 2 kN

cvisc,bottom 50 t/s Kconn,bottom 2·10³ kN/m

Rmax,bottom 176 kN Ki,bottom 1.5·10⁴ kN/m

px, py, d1, d2, b 0, 0, 0, 0, 0 KL 1·10⁴ kN/m

Ry 0.01 kN

Legend: dgap,bottom: gap in the bottom connection, cvisc,bottom: viscous damping coefficient, Rmax,bottom: resistance of cantilever, Rfr,bottom: friction force in the bottom connection, Kconn,bottom: initial stiffness of the bottom connection, Ki,bottom: bending stiffness of the bottom connection, KL: large unloading stiffness after the gap is depleted, px, py, d1, d2, b, Ry: specific parameters pinchx, pinchy, damage1, damage2, beta and Ry of the Hysteretic material model.

* Note that the value corresponds to the centrally positioned connection.

Size of the gap

The initial position of the connections depends on the actual construction and the possible residual displacements after the earlier excitations. If the bottom connections are mounted centrally, then dgap,bottom is half the width of the available space in the panel reduced by half of the thickness of the cantilever bracket. The position of connections has an important influence on the response of the panel, as will be demonstrated later.

Displacement capacity

The bottom connections did not fail during the tests. Thus it was not possible to define the displacement capacity of the bottom connection itself. However, the bottom connections always occur in pairs with the top connections, and top connections always fail before the bottom ones.

Therefore, the displacement capacity of the top connection can be considered as the displacement capacity of the complete connection assembly (please see the discussion about the failure provided in Section 3.3.4).

Friction force and damping

The friction in the bottom connection was considerably smaller than in the top connections. It was estimated from the results of the single component tests. The friction force of the top connections was subtracted from friction of the complete fastening system. The friction force in the bottom connections was thus estimated to be 2 kN.

The recommended value of the damping coefficient cvisc,bottom for the Viscous model was estimated based on the velocity and friction force measured in the tests. The value of 50 t/s was defined, which corresponds to a force of 2 kN at a velocity of 0.04 m/s.

Stiffness

The stiffness of the bottom connection was assumed to be zero during the sliding and until for as long as the gap is not depleted. Then the stiffness abruptly increases to Ki due to the activated bending stiffness of the steel cantilever.

The impact stiffness was experimentally estimated from the maximum force and displacement at the failure of the complete fastening system. The maximum force was estimated to 300 kN, which corresponds to two top and bottom connection pairs. Therefore, the force taken over by one bottom connection is 92 kN. The displacement of the connection after the gap has been depleted was 30 mm.

The impact stiffness of the bottom connection determined from experimental results amounts to 3.1·10³ kN/m. However, during the calibration of the dynamic test on the complete fastening system, the impact stiffness of the bottom connection was found to be much larger. To simulate the response of the connections accurately, the impact stiffness of the bottom connection was ten times larger than the impact stiffness of the top connections. Thus, a stiffness of 1.5·10⁴ kN/m was used for the simulation of dynamic tests. It is also proposed for further numerical analyses.

As in the case of top connections, the Hysteretic material model was used in series with EPPGap to model the response after the gap was closed. All the following specific parameters should be set to zero for this purpose: pinchx, pinchy, damage1, damage2 and beta. A relatively small parameter Ry and a large unloading stiffness KL of the Hysteretic model behaviour (see the envelopes in Figure 5.7 and Table 5.2) were used to define the steep unloading branch.

The initial stiffness of the bottom connections (Kconn,bottom) used for modelling the common friction behaviour during the quasi-static cyclic tests is, in general, very large as long as the friction force is not activated (see the recommended value in Table 5.2). After that, the stiffness is equal to zero as long as the gap is not depleted.

Resistance of the cantilever

The resistance of the bearing cantilever was analytically estimated. The steel bracket is made out of steel grade S355J0, with mean yield and ultimate strength 414 N/mm² and 546 N/mm², respectively (Braconi et al., 2013). The failure in shear and bending is considered for estimating shear resistance, as presented in Figure 5.8 and the following equations.

Shear

The critical element is bending resistance with a corresponding shear force of 176 kN. It was not reached during the tests or numerical analysis, even with extremely eccentrically positioned connections (see the parametric analysis in Chapter 6).

Figure 5.8: The scheme (a) of assumed critical cross sections and (b) scheme of a static model of bearing cantilever

Slika 5.8: Shematski prikaz (a) kritičnih prerezov in (b) statičnega modela jeklene konzole