Issue 55
M. M. Konieczny et alii, Frattura ed Integrità Strutturale, 55 (2021) 241-257; DOI: 10.3221/IGF-ESIS.55.18
2
** B d 1
t r
C
C C
d
1 1
(5)
2
*
2
*
r dr C
dr
B
r
C B
r
r
r
where the function of the transverse force intensity t(r) for the most common methods of plate support was assumed to take the form [16]:
2 3 k t r k r k r r 1
2
(6)
where: k 1 , k 2 , k 3 – lateral force function coefficients. The solution to Eqn. (5) is given by the function of the plate deflection angle ( r ):
4
3
3 k r
1 k r
2 k r
1
r
1 λ D r
λ
2 D r
(7)
2
3 λ 3 λ
1 λ 1 λ
4 λ 4 λ
1
*
C B
1
2
1
2
1
2
r
in which 1 λ and 2 λ are roots of the characteristic equation:
**
1 λ r r C C C C
B
2
λ
0
(8)
*
B
In function (7), which is the general integral of the differential Eqn. (5), the constant coefficients k 1 , k 2 , k 3 depend on the load method, while D 1 and D 2 are integration constants whose numerical values are determined on the basis of the boundary conditions given in this study. After determining the function φ (r) from compounds (2) and (3), the intensities of radial and circumferential moments in the base material (steel) m r1 , m θ 1 and in the plating layer (titanium) m r2 , m θ 2 are determined. The deflection of the bimetallic perforated plate with the plating layer is determined by the relationship: 3 w r r dr D (9) where D 3 – integration constant. The stress in the cross-sections of the bimetalic perforated plate (Fig. 3) is derived on the basis of the compounds: 1) Radial stress r : a) in the base material of the plate (steel) 1 r :
3 r m
, dla 1 2 h z h
z
(10)
1
r
3
3
C h h
1
r
1
2
b) in the applied layer of the plate (titanium) 2 r :
3 r m
2 2 h z h a
z
, dla
(11)
2
r
3 3
2
C h a h
r
1
2
2) Circumferential stress : a) in the base material of the plate (steel)
1 :
3 m
1
z
, dla 1 2 h z h
(12)
3
3
C h h
1
1
2
245
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