Tube element loads are calculated with respect to the local tube axis system, which is defined in Morison Equation and displayed in Figure 6.1: Local Tube Axis System.
The tube diameter is 
, the tube wall thickness is 
, and the tube structural material
density is 
. In the local tube axis system, the structural mass
and moment of inertia with respect to the geometric center of the
tube are
 | (13–73) |
where 
, as shown in Figure 6.1: Local Tube Axis System. The matrix of structural mass and moment of inertia is
 | (13–74) |
where the 3×3 sub-matrices are 
and 
Denoting 
as the directional cosine matrix of the local tube
axis system with respect to the fixed reference axes and 
as the position of the
geometric center of the tube in the fixed reference axes, the velocity
and acceleration at the tube geometric center in the fixed reference
axes can be found from Equation 13–71 and Equation 13–72:
 | (13–75) |
The total tube element structural force and moment components with respect to the geometric center of the tube, in the local tube axes, are
 | (13–76) |
where 
represents the acceleration due to gravity in the
fixed reference axes. The above forces and moments are the sum of
the structural inertia force and moment, the gravitational force,
and the structural gyroscopic moment.
Denoting the distance between the geometric center of the tube
and the origin of the local tube axes (i.e. the first node of tube)
as 
, the force and moment with respect to the origin
of the local tube axes may be expressed as
 | (13–77) |
where 
The tube element fluid force and moment matrix 
consists of the following
components:
Morison drag, Froude-Krylov, wave inertia and radiation (added mass-related) force and moment, as expressed in Equation 6–2, where the fluid mass and moment of inertia of an internal flooded tube are included in the added mass matrix when the radiation force and moment are calculated.
Fluid gyroscopic and momentum force, as given in Equation 13–26 and Equation 13–27, respectively.
Slamming force and moment on tube, as defined in Equation 6–21.
Hydrostatic force and moment, as calculated from Equation 3–4.
The total tube element force and moment with respect to the origin of the local tube axes are
 | (13–78) |
The equivalent nodal forces and moments at the two tube ends are defined in the local tube axes as
 | (13–79) |
Considering the terms of Equation 13–79 individually:
 | (13–80) |
which are the forces in the x-axis direction and the moment about the x-axis.
 | (13–81) |
where 
and 
are the equivalent forces
per unit length in the local xy-plane.
 | (13–82) |
where 
and 
are the equivalent forces per unit length in the
local xz-plane.
The tube end cap forces, which are due to the sum of the hydrostatic and incident wave pressures over the tube end cross-sections, can also be split from the nodal forces in Equation 13–80. The end cap forces in the local tube x-axis direction are
 | (13–83) |
where the subscripts 1 and 2 indicate the node number, 
and 
(j =
1, 2) are the incident wave and hydrostatic pressures at the j-th node of a tube element, respectively,
and the end cap area is
At the position of a user-defined node number k, where more than two elements of a space frame may be joined, the total tube element load is the summation of the nodal loads for all of the joined elements at that node, and is represented in the local structure axes (LSA) as:
 | (13–84) |
where 
is the Euler transformation matrix
from the local structure axes to the fixed reference axes.