Static equilibrium - Biomechanics: Concepts and Computation - page 41

Requiring force and moment equilibrium provides for a limited number of equa-
tions only, and therefore only a limited number of unknowns can be determined.
For two-dimensional problems force equilibrium results in two equations, while
the requirement of moment equilibrium supplies only one equation, hence three
independent equations can be formulated. Only if the number of unknown reaction
loads equals three is the solution of the unknowns possible. Likewise, in the three-
dimensional case, imposing force and moment equilibrium generates six indepen-
dent equations, such that six unknown reactions can be computed. If a free body
diagram is drawn and all the reactions can be directly identified from enforcing
the equilibrium conditions, this is referred to as the statically determinate case.
If the reactions defined on a free body diagram cannot be calculated by
imposing the equilibrium conditions, then this is referred to as the statically
in determinate case. This is dealt with, if more than three forces or moments
for two-dimensional problems or more than six forces or moments for three-
dimensional problems need to be identified. It should be noted that the equilibrium
equations do not suffice if in the two-dimensional case more than one moment is
unknown and in the three-dimensional case more than three moments are
unknown.

Example 3.3 As a two-dimensional example, consider the body of a single cell as sketched in
Fig. 3.4 , that is loaded by a known force F P , while the body is supported at two
points, say A and B. The support is such that at point A only a force in the horizon-
tal direction can be transmitted. This is represented by the rollers, that allow point
A to freely move in the vertical direction. At point B, however, forces in both
the vertical and horizontal direction can be transmitted from the surrounding to
the body, in the figure indicated by the hinge. A free body diagram is sketched in
Fig. 3.5 . The supports are separated from the body. It is assumed that the supports
cannot exert a moment on the body, therefore only reaction forces in the horizon-
tal and vertical direction have been introduced. As a naming convention all forces
in the horizontal direction have been labelled H α

referring to the
point of application), while all vertical forces have been labelled V α . At each of
the attachment points, A and B, reaction forces have been introduced on both the
body and the support. According to the third law of Newton (see Section 2.3 ):
action =

(the subscript α

reaction, forces are defined in the opposite direction with respect to
each other, but have equal magnitude. The (three) reaction forces at point A and
point B are, for the time being, unknown. They can be calculated by enforcing
force and moment equilibrium of the body. Hence, both the sum of all forces in
the horizontal direction as well as the sum of all the forces in the vertical direction
acting on the body have to be equal to zero. For this purpose the load F P has been
decomposed into a horizontal force H P and a vertical force V P :

−