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
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By combining the two e ff ects, the equation that can interpolate the surface obtained by the simulation is:
z = ( c · x d + e ) · y f · x g
(16)
By using the regression procedure, the coe ffi cient c , d , e , f and g can be estimated. This process was performed for each model and for each material, by interpolating the matrix of results.
5. Results and discussion
By applying the above illustrated procedure, five equations are estimated both for tapped thread joint (ESV) and bolted joint (DSV). In this section the intermediate results are reported for ESV, just to illustrate the procedure for obtaining the final equation. The five equations for the di ff erent materials are summed up in Table 3.
Table 3. Equations for clamped member sti ff ness evaluation in tapped thread joint (ESV). Material of the clamped member
Equation [N / mm]
6 · 1 . 927 − 6 · 0 . 836 − 0 . 501 · 6 · 1 . 077 − 0 . 631 · 6 · 1 . 39 − 0 . 8 · 6 · 0 . 558 − 0 . 351 · d w D A
4 . 5 · d
l K
d w / D A
Steel
K p = 10
3 . 5 · d 3 . 7 · d 3 . 05 · d l K
d w D A d w D A
l K l K
d w / D A
Aluminium
K p = 10
d w / D A
Brass
K p = 10
3 . 8 · d
d w D A
d w / D A
Cast iron
K p = 10
d w D A
l K
d w / D A
Magnesium
K p = 10
It is evident that the five equations have the same form, and the only di ff erence lies in the coe ffi cients. So the next step was identifying a possible unique expression for all the coe ffi cients, just depending on the Young modulus of the material. The generic formula can be expressed as:
C · d
6 · A − B ·
d w D A
l K
d w / D A
(17)
K p = 10
By plotting the values A , B and C assumed by the five di ff erent equations with respect to the Young modulus of the material, it is possible to highlight a relationship. In particular a linear relationship can be recognised for A and B coe ffi cient, whereas a polynomial relationship can be stated for the C coe ffi cient. In Figure 5 the fitting process is highlighted and in Table 4 the results are listed.
Table 4. Coe ffi cients for generalised equation of tapped thread joint (ESV). Coe ffi cient Expression
R 2
A = 8 . 664 e − 6 · E + 0 . 1786 B = 4 . 147 e − 6 · E + 0 . 188
A B C
0.995 0.990 0.999
C = 1 . 241 e − 15 · E 3 − 4 . 505 e − 10 · E 2 + 5 . 64 e − 5 · E + 1 . 318
By considering Equation (17) with the coe ffi cient of Table 4 for the ESV joint and making an evaluation of the errors with respect to the interpolated equation of Table 3, the maximum error is about 6%.
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