Since the ultimate strength might be perceived as the most meaningful safety measure of the ship's hull girder structure, prediction of the ultimate bending moment becomes essential and unavoidable part of the ship structural concept design process. Methods employed should support multiple failure modes and their interactions, while giving precise prediction of collapse and post-collapse behavior of the structural members involved (particularly those under compression). On the other hand, multiple executions within design loop demand utilization of stable, robust and sufficiently fast algorithms.
2.1 Improved incremental-iterative method for
ultimate strength assessment of hull girder
Improved incremental-iterative method for longitudinal ultimate strength assessment is based on IACS prescribed incremental-iterative method [IACS, 2006 and Smith, 1977], see Figure 1a. Modifications of
Assessment of ultimate strength at the early design stage
Figure 1b. Flowchart of the improved incremental-iterative method for the longitudinal ultimate strength assessment.
the basic method are introduced in effort to enable inclusion of the effects disregarded by the basic method and thus improve the overall accuracy of the analysis. Incorporated method particularities include contemporary advances which improve the accuracy during multi-deck ship application, as well as the ability to consider vertical shear force influence on the ultimate hull girder strength. Influence of the shear stress and deck efficiency is incorporated into basic method in a manner illustrated by the figure 1b.
Figure 1a. Qualitative hull module moment to curvature (M- ) response curve, obtained by utilization of the Bernoulli- Euler beam idealization of the hull girder.
The effect of the vertical shear force on the hull girder ultimate strength is considered trough the influence of the warping induced shear stress distribution of the hull module on the collapse (buckling, yield) of the principal structural members, where energy based numerical method and
decomposition of a cross section into the line finite elements are used for the shear stress calculation.
An approximate procedure using linear-elastic 3D FEM analysis is used for prediction of the efficiency of each principal structural element in order to correct strains in case of multy deck ships cross-section. Although implementation of this modification has limitations regarding overall accuracy, relatively simple and not so time consuming nature of the procedure enables better structural response assessment and renders this modification of the basic method as convenient for the application within the optimization based concept design loop.
Incremental nature of the method enables prediction of the structural collapse dynamics and establishment of the collapse sequence of the principal structural members of the hull module. This enables subsequent redesign of the critical components resulting in a globally safer structure, especially if the methodology is employed within the optimization based concept design loop.
2.2 Non-linear coupled beam method
The CB-method is developed for global bending response of a ship with a long multi-deck superstructure. According to the idea the ship hull is dividing into longitudinal beams that have bending and axial stiffness. Each beam consists of part of deck or side structure and is connected to neighbor beam or beams, see Figure 2. The beams are connected by distributed springs, which transfer vertical forces and longitudinal shear forces between the beams. The stiffness of springs corresponds to the vertical elongation stiffness of the bulkhead or the side shell and to
Assessment of ultimate strength at the early design stage
the shear deformation stiffness of the structure connecting two decks. All stiffness parameters can have non-linear definitions corresponding to buckling or material yielding.
The longitudinally distributed line load can be applied on each beam separately or as a resultant load on the lower beam. Detailed description of the theory is presented by Naar et al. [Naar et al., 2004] and [Naar., 2006].
Figure 2. The basic concept to estimate the bending response of a passenger ship.
2.3 Validation
The intense validation of MS and CB-approaches against FE-approach is accomplished for prismatic chemical tanker structure and for various types of multi-deck ships. As finite element solver the explicit FE-code called LS-DYNA was used. All ships where modeled in full length as prismatic structures.
Validation confirmed good agreemnt of both methods with FE results for chemical tanker case. The accuracy of the MS-method compared to FE-approach is 3%. The accuracy for CB-method is smaller reaching to 10% compared to FE-results.
As an example of FEM simulation the tanker structure in sagging loading is presented in Figure 3.
Figure 3. Chemical tanker deck structure failure in sagging loading.
For the multi-deck ship cases agreement in results of both methods compared to FEM varies depending on the considered loading scenario. The accuracy of the CB- method compared to FE-approach in case of multi-deck ship depends whether the hogging or sagging loading is considered. In hogging the difference between the FE and CB results is between 2% and 6%, see Figure 4. In sagging the difference is more drastic by changing from 18-45%. For MS-method the difference between FE-results is between 2-21% for hogging between 0.1-4% for sagging.