# Assessment of the Durability of Threaded Joints
## 4 Shakedown of the Threaded Joints  
Due to the considerable mechanical and thermal effect, the element of the structure may show low-cycle fatigue cracks, and residual strain may accumulate. In this case, the parameters of the variable nonelastic strain process limit the designed durability. If the residual strain resulting from the plastic flow at the start of loading no longer accumulates after a certain number of cycles, this means that the structure has adapted to the load. The sufficient shakedown conditions are based on the [PERSON] [42; 43] principle (static shakedown theorem) and [PERSON] [44] kinematic theorem. The experimental investigation conducted on threaded joints showed that in certain cases, the total stress caused by the tightening and cyclic bending reaches the plastic area and penetrates to a certain depth. Further repetitive variable loading affects the strength and reliability of the joint. It was noticed that the cyclic plastic strain no longer accumulates, i.e., the joint shakedown takes place as soon as a favorable field of residual stress has developed after a small number of cycles. In case of violation of the shakedown conditions, plastic strain with a variable sign (usually local) may occur, or single-sign strain may grow in each cycle of the structure or a part thereof. This results in a progressive shape change that covers the entire structural element or its part. Theoretical papers on the topic of shakedown provide a revision of the cyclic strength margin of threaded joints.  
The alternating plastic strain appears where, at any moment in the cycle, there are no constant stresses, the total amount of which together with the stresses relatively proportionate (according to [PERSON]'s law) to the external action would be less than the threshold stresses \(\sigma_{s}\), at least at one point in the structure. If the stresses in the structure are proportionate to only one parameter, plastic strains with variable sign start developing, where the proportional strain interval exceeds \(2\sigma_{s}\).  
In case of a change of conditions, the alternating plastic strain may not develop. Instead, the progressive shape change may appear in cases where constant stresses (necessary for the total stresses at each point of the body at any moment of the cycle, without variation of the loads) do not exceed the threshold stresses as they do not satisfy the equilibrium conditions (with zero loads in individual cases, where no constant external forces are present).  
This is the static assessment of the bottom threshold of the progressive shape change. The upper threshold may be analyzed from the kinematic approach; the progressive shape change will take place in the case of the distribution of the increment of irreversible strain, at which the action of the minimum threshold stresses of the cycle and tensile stresses caused by the difference of the external action is not positive and calculated for the entire volume of the structural element.  
The threshold values of the external parameters can be determined by analyzing the shaking conditions. In case the values are slightly exceeded, this leads to the increase in the interval of nonelastic strains and their incremental growth per cycle. The normal action of the structure occurs where the plastic strain does not vary during stationary loading. The shaking conditions of elements whose durability is determined by a fairly small number of load variation cycles may be used as a failure criterion. Experimental and theoretical investigations of threaded joints subjected to tension and bending may be used to complete the assessment of the possibility of shakedown.