5003 Lectures - Faculty of Engineering and Applied Science

E5003 - Ship Structures I 164
© C.G. Daley
dy = d + f x , d , d , d ) (why is this more
2
( 3 5 6
complex?)
For this beam element, we made use of what is
called ‘beam theory’, to solve for the loads and
deflections under certain loading conditions.
However, in the case of most finite elements, such
as 2D planar elements, plate elements, and solid
elements, we will not start from some general
analytical solution of a loaded membrane, plate or
solid. These solutions are too complex and will not
give practical results. Instead, we assume some
very simple behaviors, highly idealized, but which
satisfy the basic requirements for equilibrium (i.e.
forces balance, energy is conserved). With this
approach, the single element does not really model
the behavior or a comparable real solid object of
the same shape. This is ok, because the aggregate
behavior of a set of these simple elements will
model the behavior quite well. This is something
like modeling a smooth curve as a series of
straight lines (even horizontal steps). This is
locally wrong, but overall quite accurate.
Constant Stress Triangle
To illustrate the way that finite elements are
formulated, we will derive the full description of
an element called the constant stress triangle (cst).
This is a standard 2D element that is available in
most finite element models.
Consider a 2D element which is only able to take
in-plane stress. The three corners of the triangle
can only move in the plane.
For this element the force balance is;
δ
e
F = K
6x 1 = 6x6
6x1
{ } [ ]{ }
We want to determine the element stiffness matrix
Ke , and we want it to be valid for any triangle;