PSI - Issue 37

5

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

710

[ 11 12 13 21 22 23 31 32 33 41 42 43 51 52 53 61 62 63 0 0 0 0 0 0 0 0 0 − 0 0 0 − 0 0 0 −

14 15 16 24 25 26 34 35 36 44 + 45 46 54 55 + 56 64 65 66 + −

0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 − 0 0 0 0 0 0

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ]

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ]

=

] [

[

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ]

= [ 1 2 3 4 5 6 7 8 9 ] [

(4)

The external load is null at the internal Node j when the MT joint is in working condition, then the relationship between forces and displacements at nodes i and J can be calculated as

∆ ∆ ∆ ∆ ∆ ∆ ]

∆ ∆ ∆ ∆ ∆ ∆ ] = [

1 − 2 −5 1 4 − 2 −5 1 6 −

8 −5 1 4 9 − 8 −5 1 6 ] [

(5)

[

Therefore, the stiffness matrix of the hybrid finite element is expressed as = [ 1 − 2 −5 1 4 − 2 −5 1 6 − 8 −5 1 4 9 − 8 −5 1 6 ]

(6)

Based the similar derivation process, the stiffness matrix of the element in Fig. 6(a) can be also obtained. These hybrid finite elements with MT joints would be used for further modeling timber frame. a b

Beam element MT joint

Beam element MT joint

K PY K NX

Node i

Node j

Node I

Node J

Node j

Node i

K M θ

P I

P i

P j

P i

P j

P J

N

N

N

N

N

N

I

i

j

M I

M i

M j

i

j

J

M i

M j

M J

Figure 6 - (a) hybrid finite element with MT joint on the left; (b) hybrid finite element with MT joint on the right