PSI - Issue 24

Fabio Bruzzone et al. / Procedia Structural Integrity 24 (2019) 167–177 F. Bruzzone et al. / Structural Integrity Procedia 00 (2019) 000–000

171

5

In Nassar and Abdoud (2009), an analytical approach is developed and compared to FE analysis for better com puting the equivalent area A . The authors divided formulation in two possible solutions on the basis of the stress distribution. If the stress envelope is completely inside the joint thickness ( D A ≥ l K · tan ϕ + γ · d , with γ equal to the ratio between the contact area under the bolt head and bolt shank diameter) the clamped member sti ff ness is expressed as:

E 1 · E 2 · d · π · tan ϕ γ · d + 2 · l 2 · tan ϕ − d γ · d + 2 · l 2 · tan ϕ + 3 · d

K p =

(10)

γ + 3 γ − 1

γ · d + 2 · l 1 · tan ϕ − d γ · d + 2 · l 1 · tan ϕ + 3 · d

( E 1 + E 2 ) · ln

+ E 1 · ln

+ E 2 · ln

If the stress is only partially developed inside the joint ( γ · d < D A < l K · tan ϕ + γ · d ):

E 1 · E 2 · π · tan ϕ

K p =

ln

γ − 1

+ 4 · E 1 · (2 · l 2 · tan ϕ − D A + γ · d ) + 4 · E 2 · (2 · l 1 · tan ϕ − D A + γ · d ) ( D a + 3 d ) · ( D a − d ) (11)

γ + 3 2 · D A + 3 · d

D A − d

E 1 + E 2 d

·

Similar approach of Nassar and Abdoud (2009) is proposed in Haidar et al. (2011), where the di ff erence is related to the pressure distribution assumed inside the clamped members. In such a case a third order polynomial distribution is assumed, and the obtained equations are closed to that of Nassar and Abdoud (2009):

0 . 5 · E · π · tan ϕ

(12)

K p =

(3 · γ + 7) ( D A − d ) (3 · D A + 7 d ) · ( γ − 1)

+ 10 · ( γ · d − D A + l K · tan ϕ ) (3 · D A + 7 · d ) · ( D A − d )

l K

ln

d ·

If the stress is only partially developed inside the joint ( γ · d < D A < l K · tan ϕ + γ · d ):

0 . 5 · E · π · tan ϕ

(13)

K p =

(3 · γ + 7) · ( d − D A + l K · tan ϕ ) ( γ − 1) · (3 γ · d + 7 · d + 3 · l K · tan ϕ )

ln

In Yildirim (1988) the equation for computing the clamped member sti ff ness was derived not considering the support of FEA but analysing experimental data. Without direct access to the publication Yildirim (1988), data to be reported here are collected in Canyurt and Sekercioglu (2015). The sti ff ness of the clamped members can be computed using:

l 1 l 2

d · E ·

d

·

l 1 l 2

0 . 045 ·

− 0 . 0075 · l K

l K

π 4 ·

K p = 0 . 86 ·

(14)

All the previous models require identification of a validity range or they are not complete in terms of ratio between bolt and clamped member action diameter, or they do not distinguish between ESV or DSV. According to those criticisms, the authors developed the following analysis procedure to settle down a unique formula able to easily compute the sti ff ness of the clamped members.