Worn Condition Ship Set Stability Estimation

The work studies the influence of wearing and corrosion of vessel bridging elements on the rigidity of the longitudinal framing reinforced plates. An assessment method of a potential plates stability loss between the reinforcing plates until the moment of reinforced plate general stability loss.


Introduction
Stability is one of the most important qualities of any engineering structure including a vessel hull. In the structure of a vessel hull many elements and constructions -beams, plates, bridging -work in the conditions of axial compression or plane stress condition, in which stability loss is possible and it poses a great threat to integrity and durability of the hull.
In this connection the problem of plates stability reinforced with stiffening ribs has a significant practical interest. For this purpose it is important to determine ribs stiffness value so that they could serve as the plates rigid support. Then to determine the necessary distance between the stiffening ribs of vessel bridging and to assess the ability of vessel bridging elements to resist strains, which appear during vessel exploitation, taking into account a possible deterioration of connections. This problem is solved by the methods of vessel construction mechanics. And the solutions known in study and research works [1 -3] basically take into account elastic deformations and do not reflect the possibility of losing plates stability between the ribs until the moment of losing the general stability of a reinforced plate. A rigorous solution of this problem, taking into account the influence of Hook's law deviation and floor plates or skin stability loss, is mathematically complicated and demands using the methods of finite elements analysis.
It is worth noting that despite a great number of works devoted to reinforced plates stability the question of critical assessment of deterioration and corrosion influence on such plates stability has not been paid enough attention to. Uneven character of reinforced plates elements deterioration makes this influence quite substantial, which is reflected on the quantitative results as well as on the qualitative picture of stability loss.
Due to problems complexity in stability assessment in order to solve them approximate methods are used in practice, which enable to reduce the problem of bridging stability solution to the assessment of isolated beams stability, however, an assessment of these solutions correctness with due regard to connections deterioration has not been performed. Taking into account the abovementioned, the deterioration and corrosion influence analysis for vessel bridging stability as well as the assessment of approximate solutions correctness of reinforced plates stability assessment can be considered quite timely.

Problem Statement
The demand of hull structures stability is an integral part of strength standards and vessel construction standards of the leading registries of shipping. Stability of hull constructions longitudinal elements under compression stresses from general bend must be checked by comparing critical (corrected Eulerean) stresses σ кр , which are determined taking into account the decrease of normal elasticity and connections deterioration module by the middle of predicted durability with predicted compression stresses.
The basic supporting macroelement of a vessel hull is bridging. When the stability of longitudinals is provided the bridging stability depends on transverse beams rigidity (deep beams and ground timbers). The required inertia moment of these connections under the longitudinal framing system is determined according to P.F. Papkovich's formula [4] (1) where -coefficient depending on the degree of beam supporting section restraint; -beam case bay; l -distance between deep beams; b -distance between longitudinal beams; i -inertia moment of longitudinal beam cross section; Coefficient taking into account Hook's law deviation; -function depending on the number of bridging case bay, the number of stability loss half-waves and parameter where -Eulerean stress of longitudinal beam. Applied to the conditions of deep beams fastening to the board and on girder taking into account the formula for critical stresses and approximal dependence of function the initial function (1) will take the form of (2) As a result of a decreasing of vessel hull resistance moment due to connections corrosion deterioration general transverse stresses in the process of a vessel exploitation can increase by 10-30% for the vessels of 200-100 metres correspondingly.
Taking into account that the parameter with the course of time always exceeds the initial value characteristic for a new vessel, because an even deterioration is more noticeable on the smaller beams with inertia moment of cross section in comparison to a framing anchor having inertia moment , then beam rigidity in rundown condition exceeds the rigidity necessary for providing bridging stiffness on the level of a longitudinal beam stiffness. If the initial rigidity of deep beams provides a desired bridging stability, then in the rundown condition their inertia moments will guarantee the bridging stability at the level of longitudinal beams stability.
Thus, bridging stability in rundown condition is determined by the stability of panels reinforced with longitudinal ribs (longitudinal framing system). In order to assess the stability of such reinforced panels the following formula is used when using longitudinal framing system (3) where -a number of half-waves developing at the moment of stability loss within the limits of bay length a; -function depending on critical stresses level, geometrical and physical parameters as well as on stiffening ribs number and the number of half-waves both in longitudinal and transverse directions; F -surface area of transverse section of stiffening rib; S -plate thickness; -stress. There is a critical value of stiffening ribs transverse section inertia moment where ribs are rigid supports for the plate. The increase of inertia moment in excess of the critical value does not lead to the increase of reinforced panel stability. In order to determine the critical value of inertia moment in the formula (3) it is necessary to accept stresses as equal to Eulerian stresses for the plate with sizes a and b, simply supported by stiff contour, that is to consider the ribs to be absolutely stiff.
In the presence of sufficiently great number of ribs function tends to one. Taking this into account, in the practice of vessel constructions engineering it is recommended to report stability condition (3) in terms of (4) However, using the formula (4) it should be noted that while increasing the number of stiffening ribs the value of minimal frame space is limited by the technology specifications.
Corrosion deterioration influences Eulerean stresses due to decreasing of beam geometrical characteristics as well as a significant reduction of an attached belt. However, using formula (4) as a particular variant of formula (3) can lead to logical misbeliefs and incorrect quantitative evaluations. The analysis of the connections deterioration influence on their stability taken in this work proves the possibility of incorrect conclusions if formula (4) is used to assess vessel bridging beams stability.
Beams elements deterioration can lead to the increase of Eulerean stresses under the general decrease of Eulerean forces value. Taking into consideration that while calculating the imaginary hull girder the present moment stresses in the points of hull section are determined, an incorrect statement can appear about the growth of connections stability while their deterioration. From here it follows the conclusion that the assessment of framing wide-flange beams by the value of Eulerean stresses is not always eligible.
Further analysis of reinforced plates taking into account deterioration and corrosion is taken on the basis of the correspondence (3).
In the general case σ кр = k c σ с (5) where σ кр -critical stresses of construction and its elements; -compression stresses acting in the constructions elements; k c -stability coefficient.
According to Sea vessels strength standards as of 1991 k c = 1 for beams of longitudinal framing and bridging in general; k c = 0.8 -for sheerstrake steel plate elements, upper-deck stringer, flat keel and bilge strake; k c = 0.6 -for other steel plate elements of the hull included into the composition of imaginary hull girder.
Thus, proceeding from Sea vessels strength standards the criterion of deck grillage plates stability is written as σ кр ≤σ экспл. where -the greatest stresses acting in operation conditions and a criterion of stiffening ribs stability is written as σ кр ≤ where -steel fluidity limit. In other words, the plates stability is checked for the action of service condition and stiffening ribs stability for load appearing in vessel hull limit state. Taking into account the possibility of the bridging plates stability loss before the moment of the general loss of stiffened panel stability, there is a necessity to assess the plates transverse deformations influence (bridging antiplane) for stiffened panels stability taking into account the deterioration of the plates and stiffening ribs section elements.

Study results
A rigorous solution of this problem with respect for deviations of the Hook's law and losses in stability of floor or shell plates is extremely complex. However, due to a certain conditionality of