Strength of Pressure Equipment Components Taking into Account the Deterioration Concept

Relationships for critical and allowable normal stress and shear stress were proposed, as well as the strength conditions with respect to critical state and with respect to allowable state, respectively, in the case of static loading and cyclic loading, taking into account the concept of deterioration. Values for the deterioration were obtained against experimental data for tubular specimens, with only one load, with two loads acting simultaneously, as well as for samples with rectangular cross-section.

where k 1 α  and cr σ is the critical normal stress of the crackless sample. The deterioration due to crack in this case is, where,   where N is the number of accumulated cycles, kg is a coefficient that characterizes the crack growth rate, which is related to the geometry specimen material and load [39]. a is the crack length and ai is the initial crack length related to material defect or other microstructural features. Under cyclic loading with normal stresses of a cracked sample, the critical stresses for a sample cyclically loaded according to V.V. Jinescu and al. [4] are,

c. The use of critical stresses
As to use the critical stresses calculated with the above relationships ((2); (5); (9)), the deterioration ( (3) and (6)) must be calculated. These critical stresses are used in connection with the principle of critical energy [14; 18; 21; 37], applied here to pressure equipment design.
 For static loading: the strength conditions are [18; 19], (10) where the specific energy participations with respect to the static critical state due to loading with normal stress, σ, and with shear stress, τ, respectively, are: cr cr (11) The stresses σ and τ are the effective applied normal stress and effective applied shear stress, respectively; the allowable conditions are, as follows, (12) where the specific energy participations with respect to allowable state in the case of static loading, are: al al (13) The allowable stresses are defined as:  (14) where the specific energy participations with respect to critical state, in the case of cyclic loading are [14]: a a P P (16) where the specific energy participation with respect to allowable state are as follows: (17) In relationships (17)  The deterioration due to crack depends on crack shape, crack depth and crack length. As to use the relationships established in the paper, the deterioration due to crack must be obtained.

The deterioration of tubular samples
The deterioration due to only one load We further analyze the loads: under internal pressure, p, and separately under axial force, F. In all cases, the loading occurred down to the yield point, the latter being considered the limit stress   y cr σ σ  . a. In cylindrical shells under internal pressure, the maximum stress is the hoop or circumferential stress, (19) where     Figure 3, b, depending on the ratio s R c m  for four values of the reported depth a/s. b. For a tubular specimen with semi-elliptical circumferential crack on the inner surface (Fig. 4,a), under axial force F, by processing the experimental data in [2], we have obtained the curves in Figure 4 Fig. 4, b were represented in Figure 5, a, and b. 7. The correlation between the pcr(a;θ)/pcr ratio and the Mb,cr(a;θ)/Mb,cr ratio in the case of tubular specimens with circumferential semi-elliptical crack at the inner surface, characterized by a/s=0.75 and θ/π=0.40 (Processed according to [2]).

The deterioration due to effect superposition of two loads
In the case of specimens from materials considered ideal-plastic (α = 1) with an axial crack, the critical state under double load S1 and S2 results from the general relation where S1,cr and S2,cr are the critical loads of the crackless specimen under load S1 and load S2, respectively, whereas For specimens with circumferential cracks, c in relationship (25) is replaced by θ. This general relationship is further applied to two particular loading cases.
 Loading under internal pressure and bending moment (Fig. 6). For a tubular specimen with circumferential crack (Fig. 1, b), the relationship for the calculation of damage is obtained from equation (25) in the form of  (26) By using relation (26) on the basis of three pairs of values of the reported loads, represented in Figure 7, the values of the damages are listed in Table 1 Figure 7 shows the dependence of pcr(a;θ)/pcr şi Mb,cr(a;θ)/Mb,cr, processed according to [2]. The last column of the Table 1 features the average value of the damage.

The deterioration of samples with rectangular cross-section
The deterioration due to cracks in several rectangular cross-section sample ( Figure 8) were experimentally obtained. Specimens with fully penetrated cracks, perpendicular to the specimen axis, were tested.   The deterioration decreases with the size of 2c1 increase.

Conclusions
The objective of the paper was to introduce the concept of deterioration in the calculus of the ultimate (critical) stress of pressure equipment with cracks.
Relationships for critical normal stress and for critical shear stress of a cracked component have been proposed ((2) and (5)). Strength criterions were proposed for statically (10) and for cyclically (14) loaded components. At the same time relationship for the allowable strength criteria were proposed, for static loading (12) and for cyclic loading (16).
The critical stresses as well as the allowable stresses, depends on the deterioration produced by the crack. On this account for several samples the deterioration produced by cracks were calculated on the basic of experimental data in the case of only one load (tubular sample and sample with rectangular section), and in the case of two different loads (tubular sample).