PSI - Issue 37

Pan Yu et al. / Procedia Structural Integrity 37 (2022) 706–713 Pan Yu / Structural Integrity Procedia 00 (2021) 000 – 000

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2.2. CF joint The timber column of heritage frame is placed freely on the stone base. When the rotational angle of CF is small, the full cross section of CF contacts with the stone base, the distribution of the compressive stress at CF is shown in Fig. 5(a). When the rotational angle of CF is relatively large, the partial cross section of CF contacts with the stone base, the compressive stress at one side of the CF is large, but compressive stress at opposite side of the CF is null due to the uplifting of CF, and the distribution of the compressive stress at CF is presented in Fig. 5(b). The relationship between the restoring moment and angle at the CF (Yu, 2021) in the whole process of rotation can be obtained through the integration as a b

N c

N c

Column

Column

h r

h l

h r

θ

θ

Stone base

σ l Stone base

σ σ 2 ( x )

σ σ 1 ( x )

σ r

σ r

x

L θ

x

h c

h c

Figure 5 - compressive stress at CF (a) Stress state 1; (b) Stress state 2 { M = h 3 ∙b 12 tan( ) 0 ≤ θ ≤ θ 1 M = N ℎ 2 − 1 3 √ 2N 3 ∙cot( ) 3 θ 1 < θ (2) where N is the vertical load at CF, is the elastic module parallel to the grain, is the height of the column, ℎ and b is the height and width of the cross section of the column, θ is the rotational angle of the CF, and the critical rotational angle θ 1 of CF can be calculated as (Yu, 2021) θ 1 = arctan ( N h ∙2 2 ∙b ) (3) 3. Hybrid finite elements 3.1. Hybrid finite element with MT joint The semi-rigid property of bolted connection was modelled by the virtual springs, then the hybrid finite element with bolted connection was proposed for further studying on the model of steel structure (Law, 2003; 2005). The similar approach is adopted for modeling the heritage timber frame, two hybrid finite elements with MT joints at the one end of the beam element are shown in Fig. 6. The stiffness matrix of the element with MT joint at the right end of the beam in Fig. 6(b) is derived here. The MT joint is simulated by three springs whose stiffnesses are , and . Nodes i and J are external nodes, Node j is the internal node, the forces and displacements at Node i are , , and , , , and the corresponding incremental forms of these variables are ∆ , ∆ , ∆ and ∆ , ∆ , ∆ . The notations of the forces and displacement at Node J and j are similar defined. The relationship of incremental forces and displacements at the nodes i , j and J of the element in Fig. 6(b) is expressed as