Issue 59
M. Shariyat, Frattura ed Integrità Strutturale, 59 (2022) 423-443; DOI: 10.3221/IGF-ESIS.59.28
*
, N R
, N R
Σ
/ Σ
a
a
1
2
1
1
2
2
(26)
*
, N R
, N R
Σ
/ Τ
a
a
1
1
1
Therefore, according to Eqns. (22) and (26), one may deduce that:
2
2
, , m u m u 1 1 2 2
, m u
1
1
a
a
1
1
1 1
R
R
1
1
2
t
2 t
t
1
1
1
, m u
a
a
2
R
R
1
1
t
2
12
(27)
2
eq
, , m u m u 1 1 2 2
1
a
1
R
1
t
( ) ( ) t
1
1
2
a
2
R
1
2
2
The time history of eq t stress has to be plotted and the contents of the cycle histograms may be determined based on the Rainflow procedure; so that Miner’s damage accumulation rule can be applied to the stochastic time history sample to evaluate the resulting fatigue damage:
( ) n
1 i
i eq
D
(28)
N
eq
where
m
m
1
* 1 Σ Γ
u
R
,
, eq
1
eq
N
(29)
eq
a eq
eq N for the i th Rainflow cycle counting stage of the histogram of the equivalent
( ) i eq n is the number of cycles associated
stress. To check the matrix failure, one may start from Eqn. (23), expressing all fatigue strengths in terms of Y instead of X . Using a procedure similar to that led to Eqn. (27), the following equivalent stress expression may be proposed for checking the matrix cracking:
2
2
, , m u m u 1 1 2 2
, m u
a
a
2
2
2 2
R
R
1
1
2
t
1 t
2
2
( ) t
2
2
2
, m u
1
a
a
1
R
R
1
1
t
12
(30)
1
eq
, , m u m u 1 1 2 2
a
2
R
1
2
t
( ) ( ) t
2
1
2
1
a
1
R
1
1
Similarly, starting from Eqn. (24), shear failure may be checked by using the following equivalent stress expression:
2
2
, m u
, m u
a
a
1 t
2 t
R
R
1
1
2 ( ) t
12
12
, m u
, m u
a
a
1
1 1
2
2 2
R
R
1
1
1
2
t
1
2
(31)
eq
2
, m u
a
R
1
1 2 t t
12
, m u
, m u
a
a
1
2
1 1
1 1
R
R
1
1
1
1
1
2
So that, Eqn. (28) may be replaced by:
430
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