Page 192 - Dynamics of Mechanical Systems

0593_C06_fm Page 173 Monday, May 6, 2002 2:28 PM

Forces and Force Systems 173

S S
F
2
T (= M )
F i F (= R) O
F 1

O O
F
N
FIGURE 6.5.7
Equivalent force systems.

the basis of Saint Venant’s principle, which states that if two equivalent force systems are
applied to a physical body, the stress distributions resulting from the force systems are
different, but the difference is only significant in the regions of application of the force
systems. At points distant from the regions of force applications, the stress distributions
are essentially the same. That is, equivalent but distinct force systems produce distinct
effects locally, but the same effects globally. This means that even for nonrigid, nonideal
bodies equivalent force systems may often be interchanged for the purpose of global
analysis.
Finally, observe that for every force system there is an equivalent force system consisting
of a single force passing through an arbitrary point together with a couple. That is, for
rigid (or nearly rigid) bodies every force system may be replaced by an equivalent force
system consisting of a single force and a couple.
To see this let S be a given force system. Let R be the resultant of S and let M be the
O
ˆ
moment of S about some point O. Let be a force system consisting of a single force F,
S
equal to R, with the line of action passing through O together with a couple with torque
ˆ
T equal to M (see Figure 6.5.7). Then, S and are equivalent because they have equal
S
O
resultants and equal moments about point O.

6.6 Wrenches

Consider again the example of the equivalent force system of the forces on the box of the
foregoing section (see Figures 6.5.1 and 6.5.2). Recall, in that example, that we replaced
the original force system by a force system consisting of a single force F passing through
O together with a couple with torque T (see Figure 6.5.2). Suppose that instead of having
the line of action of F pass through O we pass it through Q as in Figure 6.6.1. Then, for
the force systems to be equivalent, we must adjust the torque of the couple, so that now
the torque is M , which is the moment of the original system of forces about Q.
Q
Observe that aside from the placement of the force F the only difference in the two
ˆ
T
equivalent force systems is in the torques T and of the accompanying couples. For the
ˆ
system of Figure 6.5.2, the torque T is M , whereas here (Figure 6.6.1) the torque T is
O
M . From Eqs. (6.3.15) and (6.3.16), M and M are:
Q
Q
O
M = 160 n − 30 n − 168 n ftlb and M = −32 n − 42 n − 132 n ftlb (6.6.1)
O 1 2 3 Q 1 2 3