Explaining non-linear FE analyses
General
Ship hull strength assessments constitute an important design process with the purpose of deciding and quantifying the vessel’s available operational safety margins in harsh environmental conditions.Safety margins in this context mean in relation to two main types of strength limits and associated failure scenarios, i.e.
Fatigue failure. This is an integrated response effect over the lifetime of the vessel leading to cracks in structural details, brackets, hopper knuckles, hatch corners, etc. (FLS).
Buckling and collapse failure. This failure type may occur under extreme loads in overstressed parts of the hull structure and lead to local damage, buckling, etc. Structural collapse may also be more serious, involving a progressive scenario which ultimately leads to total hull girder collapse, excessive plate fracture with compartment flooding and total loss of the vessel (ULS).
Both phenomena threaten the ship’s overall integrity and it is therefore very important to be able to assess the “design” limits at which these types of failure scenarios may be a problem.
Fatigue
Fatigue failure (FLS) is a complex physical phenomenon and the analyses methods dealing with this are established at different levels of sophistication. Characteristic of them all are the non-linear and complex interactions between the ship’s movements and the dynamic hull response when the vessel travels through waves. Luckily, from a structural stress assessment point of view (FE), the methods required are normally sufficient when based on linear technology, i.e. the non-linear effects are mostly on the load and wave side.
Buckling and Collapse
When it comes to extreme loads and ship collapse (ULS), meaning an event that may occur once in a lifetime or with a similar frequency, the loads and stresses in the hull are so high that the structural response itself may deviate quite significantly from the linear hypothesis.
So what is meant by structurally non-linear behaviour? Obviously, it means that the highly practical and beloved hypothesis of proportionality between a “internal stress response” and the driving “external” load is violated. There are two main reasons for this deviation:
geometrical non-linearity; i.e. especially thin-walled plate and shell structures (a ship’s hull) will, when compressed beyond their buckling limit, change shape from their original “intact” shape into a buckled and deformed pattern. This change of shape can be significant and, if not corrected for in the computer analyses, the results can be very wrong.
physical non-linearity; the material (steel etc) will when strained beyond a certain level (yield strain) behave very differently than when strained at a lower load level. This sudden change in behaviour is due to the plastic and irreversible material response typical for metallic materials, i.e. a marked drop in stiffness will occur in the structural response when the material is loaded beyond the material yield plateau. For FRP and composite materials in general, the behaviour is somewhat different, characterized by a rather linear response up to first ply failure, beyond which the material stiffness gradually drops.
From an engineering perspective, departing from the linear hypothesis is a big step, as a new and complex world opens up with challenges, possibilities and problems. However, in order to assess the strength limits beyond which the ship may collapse and be severely damaged, there is no way around the real (non-linear) laws of physics.
The good thing though is that these non-linear laws of physics have been around for some time, and “smart” people have made computerized and modern analysis tools that solve these equations in a generic way. Especially during the past two-three decades, there has been a significant increase in the development of advanced structural analysis tools, and in particular the finite element method (FE) in its non-linear version has been successfully incorporated into several commercial software packages.
In the hands of skilled and trained persons such non-linear FE tools are indeed powerful, while without the necessary knowledge and understanding they are the opposite. This fact is important to note and it is clearly reflected in the range of applications seen around which are aimed at solving more tricky design and strength issues, such as failure and incident assessments, new design and concept evaluations, etc.
Range of applications
For ship strength in general, and container vessels’ assessments in particular, the non-linear FE tool can be used at different levels and for different structural problems. Some of the most obvious applications are
Longitudinal strength, hull girder ultimate capacity
Ship hulls bending and twisting under extreme weather conditions may trigger buckling and collapse mechanisms in the hull skin, frames, girders, etc.
These very complex failure mechanisms and the stress levels at which they take place and progress can be assessed with a high degree of accuracy and confidence. The analyses carried out by DNV Maritime’s Technical Consultancy department (360) were such an exercise and proved to be very useful.
Bow, stern or bottom slamming impact
Waves and slamming impacts on bow, stern or bottom structures may cause damage to the hull skin as well as to the internal supporting frame structures.
Collision and grounding damage assessment
Although they are rare events, ships may be subject to collision and grounding scenarios with dramatic results. The non-linear FE tool is again the most advanced and credible approach, both for predicting the extent of the damage during the incident itself and for assessing the residual strength of the damaged vessel after the incident.
Wave breaker design
Wave breaker structures on deck may be exposed to extreme loads from water on decks, breaking waves, etc and may be severely buckled and damaged.
Container lashing – sea fastening
For ships operating in heavy seas, large hull movements and rolling will strain the container lashing arrangements to the limit, with the risk of straps breaking and the containers falling overboard. Non-linear FE tools can be used to assess such problems, taking into account large ship roll motions, etc.
Ice loading – cold climate
Ships sailing through waters with ice may experience local and global extreme loads on the hull. For assessing such loads and possible hull damage and coping with the crushing properties of the ice itself in interaction with the ship’s structural response, non-linear FE tools are a must.
Lightweight structures
The use of new constructional material like FRP Composites, sandwich constructions, etc also challenges the linear technology. For normal design assessment, the linear approach will do, but for exploring the real strength limits accounting for all types of failure modes, delaminations, buckling, material rupture, wrinkling, etc, non-linear FE tools are invaluable (super-structures, hatch covers, lashing bridges, wave breakers, rudder).
Novel design concepts, special structures
The non-linear FE tool is also very useful in connection with the development of new and novel design concepts for which limited experience is available. Examples are LNG tanks with support and contact problems, lifting from supports, etc.
In such a perspective, new design challenges for the upcoming generation of ultra large container vessels may well identify structural problems for which non-linear FE analyses will be very useful (e.g. slamming and whipping).
Reflections for the future
It can be seen that the development of advanced computerized analysis tools is steadily progressing to the benefit of design and research engineers. Today, the main catch when using such tools is that the internal brain (equation solver) may break down and result in convergence problems for some types of complex analyses. This is a challenge that requires insight and a critical evaluation of the results.
The other catch is the computer time consumption being of the order of 1000 times or more that for the “same” (FE) model analysed using linear methods. With such computer time consumption, analyses of large problems and models become a real challenge. However, according to Moore’s law, computer hardware developments will lead to a doubling of the available calculation capacity every second year, so if this law holds in the future, analyses that take weeks today will only take minutes in the future.
With this in mind, we expect non-linear FE software and analysis tools to become more and more common and used by researchers, rule developers and ship designers in the future. However, they will be specialist tools as far as we can see and thus the linear approach will still constitute the basic technology in the design office for some decades to come.