Numerical model of the top bolted connection

A typical response of the bolted top connections, presented in Figure 5.1, was simulated by combining three material models: ElasticPP (EPP), ElasticPPGap (EPPGap) and Hysteretic, as shown in Figure 5.2 (a). In the first phase of the response (Figure 5.1, phase 1), the friction between the steel elements was activated due to the tightening torque in the bolt. The ElasticPP (Figure 5.2 b) model was used to simulate this friction. The properties of the model were defined using the common Coulomb friction model described in detail in the following subsections.

In the second phase of the connection response, the bolt washer reached the edge of the steel box, and the gap in the connection was depleted. At this moment, the stiffness of the top connection almost instantly increased (Figure 5.1, phase 2), which was simulated by the series combination of the ElasticPPGap (Figure 5.2 c) and the Hysteretic (Figure 5.2 d) material models.

To define the complete model of the top connection, the impact model (series combination of the ElasticPPGap and Hysteretic material models) was added in parallel to the friction model (ElasticPP), as shown in Figure 5.2 (a).

Figure 5.1: Typical hysteretic response of the top connection during the dynamic test on components Slika 5.1: Značilen histerezni odziv zgornjega stika med dinamičnim testom zgornjih stikov

Figure 5.2: Schematic presentation of the macro model: (a) combination of different hysteretic material models used for the numerical simulation of top and bottom connections, (b) ElasticPP, (c) ElasticPPGap and (d) Hysteretic material models

Slika 5.2: Shematski prikaz makro numeričnega modela: (a) kombinacija različnih histrereznih materialnih modelov za numerično simulacijo zgornjih in spodnjih stikov, (b) ElasticPP, (c) ElasticPPGap in (d) Hysteretic materialni modeli

The force–displacement relationship of the numerical model used for the simulation of top connections response is schematically presented in Figure 5.3. The model parameters are the size of the gap (dgap,top), the maximum displacement capacity (du), the friction force (Rfr,top), the

resistance of the top connection (Rmax,top) and stiffness (Kconn,top, Ki,top) as presented in following paragraphs. The recommended values are summarised in Table 5.

Figure 5.3: Schematic envelope of the numerical models of the top connection Slika 5.3: Shematski prikaz ovojnice numeričnega modela zgornjega stika

Table 5.1: Recommended values of the model parameters of the top connection Preglednica 5.1: Priporočene vrednosti modelnih parametrov zgornjega stika

Material

characteristic Value Material

characteristic Value Material

characteristic Value

cfr,top 0.4 Kconn,top 2·10⁴ kN/m KL 1·10⁴ kN/m

dgap,top* ±4.0 cm Ki,top 1.5·10³ kN/m Ry 0.01 kN

du* ±7.5 cm Rmax,top 58 kN px, py, d1, d2, b 0, 0, 0, 0, 0

Legend: cfr,top: friction coefficient between steel elements of the top connection, dgap,top: gap in the top connection, du: displacement capacity of the top connection, Rmax,top: resistance of the top connection, Kconn,top: initial stiffness of the top connection, Ki,top: bending stiffness of the top 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.

* 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 top connections are mounted centrally, then dgap,top

is half the width of available space in the steel box (cast in the panel) reduced by half of the thickness of the bolt washer. The position of connections has an important influence on the response of the panel, as will be demonstrated later.

Displacement capacity

The displacement capacity of the top connection consists of the variable gap in the top connections dgap,top, and the plastic deformation capacity of the bolt, which is about 3.5 cm. If the connections are installed in the middle of the gap, the total displacement capacity amounts to 7.5 cm.

Friction force

The friction force in the top connection was defined by using the common Coulomb friction model that assumes that the friction force is the product of the normal force on the surface and the constant coefficient of friction. Thus, the friction force in the top connection depends on the tightening torque in the bolt Tb and the coefficient of friction between the connection parts cfr,top (Zoubek, 2015):

𝑅_{𝑓𝑟,𝑡𝑜𝑝}= 𝑐_{𝑓𝑟,𝑡𝑜𝑝} 𝐹_𝑏 (5.1)

𝐹_𝑏 = ^𝑇𝑏

𝑐₀ 𝐷_𝑏 (5.2)

The friction coefficient in the threaded bolt c0 = 0.2 was considered. This is recommended friction coefficient for galvanised bolts without lubrication according to VDI 2230 standard (DIN VDI 2230 Part-1 cited by AmesWeb, 2020). The nominal diameter of the bolt Db was 16 mm.

It is recommended to use a friction coefficient cfr,top of 0.4 for this type of connection. The value is estimated from the maximum friction force observed during the tests of top connections Rfr,top = 8 kN and the tightening torque Tb = 65 Nm. The proposed value is in quite good agreement with the friction coefficients reported by Del Monte et al. (2019). They have evaluated the values of the static friction coefficient at about 0.45 and dynamic ones in the range 0.32–0.35, according to the tests on similar connection types.

It is not necessarily true that the tightening torque of the top connections in real precast structures will be 65 Nm, as prescribed by the producer. The tightening force also decreases during the seismic excitation due to loosening of the bolt. Generally, the friction forces activated in the connections are relatively small compared to the forces that occur in the main precast structure. The friction force observed during the shaking table tests was considerably smaller (Section 5.2.2). For the reasons listed above, using a friction force of 2 kN in the top connection is recommended. This force was also observed during the shake table tests.

Resistance of the top connection

According to the experimental results, the shear resistance of one top connection amounts to 58 kN for the connection with a stronger hot-rolled channel and around 34 kN if the cold-formed channel is used. Please see also the discussion provided in Section 3.3.5.

Stiffness

In general, the initial stiffness of the top connections (Kconn,top) is very large as long as the full friction force is not activated (see the recommended values in Table 5.1). After that, the stiffness is equal to zero as long as the gap is not depleted. Then the stiffness abruptly increases to Ki due to the activated bending stiffness of the bolt at the top.

The bending stiffness of the bolt at the top was estimated experimentally and analytically:

(1) Experimental estimation

The stiffness was experimentally estimated from the maximum force (58 kN) and displacement at the failure of the top connections. The calculation uses the displacement of the connection after the gap has been depleted (35 mm). The impact stiffness determined from experimental results is 1.7·10³ kN/m.

(2) Analytical estimation

The impact stiffness was analytically estimated with formulas proposed by Belleri et al. (2016), who tested very similar top connections. According to the static scheme presented in Figure 5.4, the following formula was proposed:

𝐾_{𝑖,𝑡𝑜𝑝}=^{12 𝐸 𝐼𝑏}

𝐿_𝑏²

𝐸 𝐼_𝑏 (𝑘_𝜃,𝑆1+𝑘_𝜃,𝑆2)+𝑘_𝜃,𝑆1 𝑘_𝜃,𝑆2𝐿_𝑏

(12 𝐸² 𝐼_𝑏²+4 𝐸 𝐼_𝑏 (𝑘_𝜃,𝑆1+𝑘_𝜃,𝑆2) 𝐿_𝑏+𝑘_𝜃,𝑆1 𝑘_𝜃,𝑆1 𝐿²_𝑏) (5.3) where EIb is flexural stiffness of the bolt, Lb is the length of the bolt, and kθ,S1 and kθ,S2 are elastic rotational stiffness of springs S1 and S2, respectively.

Figure 5.4: Static scheme of top connection (Belleri et al., 2016)

Slika 5.4: Shematski prikaz statičnega modela zgornjega stika (Belleri et al., 2016)

Calculated impact stiffness Ki is 1.4·10³ kN/m. Bolt length Lb = 60 mm, bolt diameter Db = 16 mm and the rotational stiffness kθ,S1 = 6·10³ kN/m were used. Because the channel lip failure was relevant, the rotational stiffness kθ,S2 was equal to zero.

The experimentally estimated stiffness agrees quite well with those evaluated as proposed by Belleri et al. (2016). Use Ki stiffness of 1.5·10³ kN/m in the numerical model is recommended.

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 large unloading stiffness KL of the Hysteretic model behaviour (see the envelope in Figure 5.3) were used to define the steep unloading branch.