Environmental Engineering Reference
In-Depth Information
“BL” in eqn (12) corresponds to the loads at the boundaries and is related to the
connection conditions that must be satisfi ed when assembling the component
equations into the fi nal system. This point is explained in Section 4.3.
4.2 Connection conditions
Consider fi rst two bodies connected at one point as in Fig. 5. If the connection is
completely rigid, then the displacement and rotation at the connection point must
be the same. Also the loading should be the same. The fi rst constraint corresponds
to a kinematic condition while the second to a static condition. Note that while it is
allowed to either specify a kinematic or a static condition at an end point, we can-
not do both. In fact each kinematic condition has a corresponding static condition
associated to it. So if a displacement (or rotation) is specifi ed then the associated
reaction force (or moment) becomes part of the solution and therefore must remain
free. Clearly the reaction will depend on the value specifi ed for the displacement.
If instead the force is specifi ed, then the displacement will depend on the input
load. So conditions appear in pairs and for each pair we can only specify one. At
connection points the situation is somehow different because there are at least two
bodies connected, each requiring its own boundary condition at the connection
point. Also the entries to both conditions are unknown. Let
q
denote collectively
the displacements and rotations of the connection point and
Q
the corresponding
loads. By considering each body separately, it is possible to formulate separate
solutions by setting
q
as kinematic condition to body 1 and
Q
as static condition to
body 2. The solution for body 1 will provide
Q
as a function of
q
while body 2 will
provide
q
as a function of
Q
. These two relations defi ne the connection conditions
needed. Note that any kind of combination is possible provided that the pairs of
conditions are properly split.
If the connection is not rigid but allows free motion in certain directions, then
the connection conditions do not apply in these specifi c directions. For example a
(2)
Body 2
F,M
(1)
q
(2)
Body 1
(1)
Figure 5: Realization of kinematic and dynamic coupling conditions.
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