Civil Engineering Reference
In-Depth Information
y
F
a
x
F
F
IGURE
2.19
Couple
produced by out-of-line forces
Plate
(there are no forces in the
y
direction so that
F
y
=
0 is automatically satisfied), but
form a couple
Fa
which will cause the plate to rotate in an anticlockwise sense so long
as its magnitude is sufficient to overcome the frictional resistance between the plate
and the surface. We have also seen that a couple exerts the same moment about any
point in its plane so that we may deduce a further condition for the statical equilib-
rium of a body acted upon by a system of coplanar forces, namely, that the sum of the
moments of all the forces acting on the body about
any
point in their plane must be
zero. Therefore, designating a moment in the
xy
plane about the
z
axis as
M
z
, we have
M
z
=
0
(2.9)
Combining Eqs (2.8) and (2.9) we obtain the necessary conditions for a system of
coplanar forces to be in equilibrium, i.e.
F
x
=
0
F
y
=
0
M
z
=
0
(2.10)
The above arguments may be extended to a three-dimensional force system which is,
again, referred to an
xyz
axis system. Thus for equilibrium
F
x
=
0
F
y
=
0
F
z
=
0
(2.11)
and
M
x
=
0
M
y
=
0
M
z
=
0
(2.12)
2.5 C
ALCULATION OF
S
UPPORT
R
EACTIONS
The conditions of statical equilibrium, Eq. (2.10), are used to calculate reactions at
supports in structures so long as the support system is statically determinate (see
Section 1.5). Generally the calculation of support reactions is a necessary preliminary
to the determination of internal force and stress distributions and displacements.
