Page 312 - Marine Structural Design

288 Part II Ultimate Strength

Dynamic loads applied as initial velocities of a colliding structure initiate the calculation with
the simulation of the dynamic motion of the colliding structure. The impact loads between the
structures are obtained as the results of the simulation. Once an impact force in a certain
direction is detected to be in tension during the simulation the contact is released. Thereafter,
the striker is assumed to move as a free body in that direction with a given constant velocity.
The criterion for re-establishing a contact is that the displacement of the striker will exceed the
displacement of the corresponding point on the affected structure. This type of loading is used
in Examples 14.5-14.7.
Dynamic loads applied as initial velocities of the strucklaffected structure are usually used
only in high-speed impacts. In such cases, the time history of the applied loading is not of
interest. The response of the strucktaffected structure depends on the time integration of the
loads (the momentum of pulse), in other words, initial velocities of the affected structure.
For large displacement analyses, an updated Lagrangian approach is adopted. At each load
step, the element stiffness matrices are reformed in the local coordinate systems and then
transformed to the global coordinate system. Here, the global stiffness matrix is assembled and
the increments of nodal displacements, measured in the global coordinate system, are
evaluated. Using the element transformation matrix, the increments of element displacements
can be calculated and the element displacements and forces are updated. The new
transformation matrices can then be evaluated and the updated element displacements and
forces transformed to the new local coordinate system and used in the following calculation of
new element forces and displacements for the next load step.
During the elastic-plastic analysis, the loading and unloading of nodes are checked carefully.
Once loading takes place in a node, a Newton-Raphson iteration is carried out in order to find
the exact load increments at which the element nodal forces may come to and thereafter move
along the yield surface. At each time step, the structural stiffness matrix is evaluated based on
the elastic-plastic status of the element nodes, at the end of the previous load increment.
However, as soon as the equations of motion are solved, a check is performed to analyze
whether unloading of the plastic nodes takes place. If this is the case, the structural stiffness
matrix is updated until no further unloading of plastic nodes is detected. Finally, the nodal
displacement increments are the solution to the equations of motion after the last iteration. The
nodal forces and the elastic-plastic status of elements are updated and unloading is re-
evaluated. In addition, when the elastic-plastic status of a node has changed, the unbalanced
forces are evaluated and transformed to the global coordinate system. The transformed
unbalanced forces are added to the load increments for the next time step.
For further cross-sections, there are two comers on the yield surface. The comers are at the
points where there are only axial forces acting on the beam element. When the forces at an
element node are at such a comer or close to a comer, then the element is treated as a truss
element, which is only subjected to an axial force. For such truss elements, unloading is
checked based on the axial forces and the axial displacement increments. The elements are
treated as normal three-dimensional beam-column elements once unloading is detected for
them.