Exemplu de verificare

Bending stiffness of bolted joint of open sections

Steel

Connection

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A sample chapter from book Benchmark cases for advanced design of structural steel connections

10.2.1 Description

The prediction of rotational stiffness is verified on a bolted eaves moment joint. A bolted joint of open section column HEB and beam IPE is studied and the joint’s behaviour is described on moment-rotation diagram. The results of analytical model by the component-based finite element method (CBFEM) are compared with the component method (CM). The numerical results in form of a benchmark case are available.

10.2.2 Analytical model

The rotational stiffness of a joint should be determined from deformation of its basic components, which are represented by the stiffness coefficient k_i. The rotational stiffness of the joint S_j is obtained from:

\[ S_j = \frac{E z^2}{\mu \Sigma_i \frac{1}{k_i}} \]

where

\(k_i\) — the stiffness coefficient for the joint component i;

\(z\) — the lever arm, see 6.2.7;

\(μ\) — the stiffness ratio, see 6.3.1.

The joint components that are taken into account in this example are column web panel in shear k₁ which equals infinity for a stiffened column and a single equivalent stiffness coefficient k_eq for end-plate joint with two or more bolt-rows in tension.

\[k_{\mathit{1}} = 0.38 \, \frac{A_{\mathit{vc}}}{\beta \, z}\]

\[k_{eq} = \frac{(k_{eff,0}h_{r,0}) + (k_{eff,1}h_{r,1}) + (k_{eff,2}h_{r,2}) + (k_{eff,3}h_{r,3}) + (k_{eff,4}h_{r,4})}{z_{eq}}\]

\[k_{eff,i} = \frac{1}{\frac{1}{k_{5,i}} + \frac{1}{k_{10}} + \frac{1}{k_{4,i}}}\]

\[z_{eq} = \frac{(k_{eff,0}h_{r,0}^2) + (k_{eff,1}h_{r,1}^2) + (k_{eff,2}h_{r,2}^2) + (k_{eff,3}h_{r,3}^2) + (k_{eff,4}h_{r,4}^2)}{(k_{eff,0}h_{r,0}) + (k_{eff,1}h_{r,1}) + (k_{eff,2}h_{r,2}) + (k_{eff,3}h_{r,3}) + (k_{eff,4}h_{r,4})}\]

\[S_{\mathit{j,\,ini}} = \frac{E \, z_{\mathit{eq}}^{2}}{\mu \left( \frac{1}{k_{\mathit{eq}}} + \frac{1}{k_{\mathit{1}}} \right)}\]

where

\(h_{r,i}\) — distance from the bolt row to beam bottom flange see Drawing 10.2.1

\(k_i\) — the stiffness coefficient for the joint component i

\(z_{eq}\) — is the equivalent lever arm

\[ \textsf{\textit{\footnotesize{Drawing 10.2.1 }}}\]

An open section beam IPE 330 is connected with bolted end-plate to a column HEB 200 in the example. The end-plate thickness is 15 mm, the bolt type is M24 8.8 and the assembly is shown in Fig. 10.2.1. Other examples have different column cross-sections. The stiffeners are inside column opposite to beam flanges with thickness of 15 mm. Beam flanges are connected to the end-plate with welds throat thickness of 8 mm. The beam web is connected with weld throat thickness of 5 mm. Plasticity is applied in welds. The material of the beam, column and end-plate is S235. The joint is loaded in bending. The design resistance is limited by the component column web panel in shear. The calculated stiffness coefficients of the basic components, initial stiffness, stiffness by design resistance and rotation of the beam are summarised in Tab. 10.2.1. Joints with column height below 260 mm had the web panel in shear failure mode, other had beam flange in tension, so their bending resistances are equal.

Tab. 10.2.1 Results of the analytical model (Component method)

\[ \textsf{\textit{\footnotesize{Fig. 10.2.1 Joint geometry with dimensions}}}\]

10.2.3 Verification of stiffness

Detailed information about the prediction of stiffness in CBFEM may be found in chapter 3.9. The CBFEM analyses allows to calculate secant rotational stiffness in any stages of loading. The design resistance is reached by 5% plastic strain in a component column web panel in shear. The rotational stiffness calculated by CBFEM compared with CM. The comparison shows good agreement in initial stiffness and correspondence of joint’s behavior. The calculated stiffness from CBFEM and CM are summarized in Fig 10.2.2.

Tab. 10.2.2 Verification CBFEM to CM

\[ \textsf{\textit{\footnotesize{Fig. 10.2.2 Verification of the bending resistance CBFEM to CM}}}\]

\[ \textsf{\textit{\footnotesize{Fig. 10.2.3 Verification of the bending stiffness CBFEM to CM}}}\]

\[ \textsf{\textit{\footnotesize{Fig. 10.2.4 Sensitivity study for the beam height}}}\]

\[ \textsf{\textit{\footnotesize{Fig. 10.2.5 Sensitivity study for the beam height (initial stiffness)}}}\]

10.2.4 Global behavior and verification

A comparison of the global behavior of a bolted eaves moment joint described by the moment-rotation diagram is prepared. The joint is analyzed and the stiffness of the connected beam is calculated. The main characteristic is the initial stiffness calculated by 2/3 M_j,Rd, where M_j,Rd is the design moment resistance of the joint. M_c,Rd stands for a design moment resistance of the analysed beam. The moment-rotation diagrams are shown in Fig. 10.2.6-10.2.16