The  further research  on the interaction between the vertical and horizontal bending was undertaken by the Hu, Y. et al, (2001). It  was found that the interaction curve is asymmetrical because the  hull cross-section is  not symmetrical about the horizontal axis and  the behavior of the structural members under compression is different from that under tension due to the non-linearity caused by buckling. An interaction equation  suitable for  bulk carriers is proposed based on the results of the analyzed ship.  As it is  well  know the  basic equation that relates the applied vertical and horizontal bending moments to the longitudinal stress are very simple and may be resumed as followed: yi yxi xiIx MIy M . .− = σ              (1) or it may be expressed as a function of the total moment by: yixi iIxIyMϕ ϕ σ sin . cos .− =           (2) Where  ϕ   is the angle that the bending moment vector makes with the baseline and xi and yi are the coordinates of the point in the referential located in any point of the neutral axis. For a given points of the cross section this relation is constant until the yield stress of  the material is reached in any  point of the section  or the local structure of hull  section  is damaged during  operation. Once the cases mentioned above occur the neutral axis moves away  from its original position and thus the constancy of the relation may be broken. Due to the same reason  the relation  between the angle of the moment vector  ϕ   and the angle of the neutral axis  θ  is constant in the linear elastic range  but is changed when  some damage of section is already  present. This relation may be expressed by: θ ϕ tgIItgxy=              (3) The Calculation Steps The assessment  of moment-curvature relationship is obtained  by imposing a sequence of increasing curvature to the  hull  girder. For each curvature, the average strain of element is determined assuming that plane  section remains plane after the  curvature is applied. The values of strain are introduced in the model that represents the load-shortening behavior  of each element. The  bending moment  resisted by the cross section is obtained from the summation  of the contributions from the inpidual elements. The calculated set of values defines the desired moment-curvature relation. The most general case corresponds to that in which the ship is subjected to curvature in the x and y directions respectively denoted as Cx , Cy. The overall curvature C is related to these two components by: 2 2y x C C C + =               (4) or         θ cos . C x C =  and  θ sin . C y = C     (5) adopting the right-hand  rule, where   θ   is the angle between the neutral axis and the x axis and is related to the components of the curvature by: xyCCtg = θ                (6) The strain at the centroid of an element i is iε  which depends  on its position and on the  hull curvature, as given by: y gi x gi iC x C y . . − = ε           (7) Where (xgi , ygi) are the coordinates  of centroid  of the element i referred to the central point at each curvature. Once the state of strain in  each element is determined, the corresponding average  stress may be calculated according to the method  described above, and consequently the components of the bending moment for a curvature C are given by: i i gi x A y M σ ∑ = ..           (8) ∑ = i i gi y A x M . .σ           (9) Where iσ   represents the stress of element i at (xgi , ygi). Ai represent the cross sectional area of element i. This is the  bending moment on the cross  section after calculating properly the instantaneous  position  of the intersection of the  neutral axis associated with each curvature and  the centerline (it is be called as center of force.). The condition to determine the correct position of neutral axis is: 0 = ∑ i iA σ              (10) The a trial-and error process has to be used to estimate correctly its position. The total net load in the section, NL, or the error in the shift estimate  ΔG should be less than  or equal  to sufficiently low  value  (Gord, J. M. , 1996). In this paper, the following equation is used. ∑ ∑ −≤ = i i i iA A NL 0610 σ σ         (11) Where i 0 σ   is the material yield stress of element i . The relationship of element strain—stress An applied curvature to the section induces compression in one side of  the axis of the section and tension in the other side. According to the algorithm described earlier, for a particular magnitude  of applied curvature, the induced strains are calculated  for all elements in the section. The corresponding stresses are calculated by using the stress-strain relationship for inpidual panels. 船体梁极限强度英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_63528.html