Page 130 - Marine Structural Design

106 Part I Structural Design Principles

Taking a irregularly shaped plate, for example, we may estimate the displacements and
consequently the stresses within the plate under a given load for a specified material and
boundary conditions. The field variable of interest here is the displacement. Instead of
determining the displacement at every point in the plate, the finite element method divides the
plate into a finite number of elements, and provides the displacements at the nodal points of
each element. Interpolation functions are used within each element to describe the variations
of the field variable (e.g. displacement in this example) as a function of the local coordinates.
Using nodal displacements and interpolation function, the designer can compute the stress
variation within any given region.
Computation Based on FEM
Commercial software has been developed based on finite element theory. As input data for the
software, the designer define relevant coordinates of each node, element definitions, material
properties, boundary conditions, etc. Generally, the accuracy of the solution improves as the
number of elements increase, but the computational time and cost also increase. A high-speed
computer is required to perform and solve the large number of element assembly involved.
Different element types (rod, beam, membrane, solid, bending with 3-node, 4-node, 6-node, 8-
node, etc) are applied to various types of structures, which yield different accuracy and CPU
time. However, there is no substitute of experience when trying to determine the element
density and element type in order to achieve the required level of accuracy for the finite
element analysis of a particular structure.
The computer program determines the displacements at each node and then the stresses acting
through each element. One of the essential tasks in FE analysis is to analyze the results, which
is known as post-processing. The designer may view the results in tabular or graphical form. A
graphical view may be used initially to identify the regions and nodes of interest and
subsequently tabulate the output specified for the chosen areas of interest. If this were not the
case, the physical data of the whole structure may otherwise be too large to be structurally
assessed.
Marine Applications of FEM
The analyst may then use the results from the finite element analysis to strengthen the
structure via an increase in the material strength, via additional reinforcement, or by changing
the load path or the boundary conditions.
The critical areas where loads or stresses are concentrated, or where there are complex joint
details, will generally need to have a more detailed finite element model or finer element
mesh. The finite element analysis output will only be as good as the input data specified.
Again, it is particularly important for the designer to consider the limits of the model and
consequently the accuracy of analysis results.
Probably the most serious problem affecting ocean-going vessels in recent years has been
brittle fracture near bulkheads on very large bulk carriers. Such an effect could be easily
missed in a finite element model for such a vessel. Local flexibilityhigidity and material
behavior could be overlooked since the design emphasis is placed on increasing the stiffness
of local details to meet the requirements of the relevant codes.
In the following stiffness matrices are derived for 2D and 3D beam elements in order to
illustrate the finite element methods for offshore structural analysis and to prepare a theoretical
basis for Chapter 12 - 15.