EQUILIBRIUM OF
RIGID BODIES
RIGID BODIES
• Rigid body—Maintains the relative position of
any two particles inside it when subjected to
external loads. All objects deform to some
degree in response to their environment, but for
rigid bodies, these deformations are so small
compared to the size of the body that they can
be ignored. For example, the small deformations
that occur in bone under standard conditions are
ignored, and bone is considered to be a rigid
body.
Cont…
• Mechanical Equilibrium—A system is in mechanical
equilibrium when the sums of the forces and moments
are zero. A body in mechanical equilibrium is undergoing
neither linear nor rotational acceleration; however, it
could be translating or rotating at a constant velocity.
Static equilibrium describes the special case of
mechanical equilibrium of an object that is at rest.
When a rigid body is in equilibrium, both the resultant
force and the resultant couple must be zero.


R  F  0



Rx i  R y j  Rz k  0

M  0



M x i  M y j  M z k  0
Forces and moments acting on a rigid body could be external
forces/moments or internal forces/moments.
Forces acting from one body to another by direct physical
contact or from the Earth are examples of external forces.
Fluid pressure acting to the wall of a water tank or a force
exerted by the tire of a truck to the road is all external forces.
The first step in the analysis of the equilibrium of
rigid bodies must be to draw the “free body
diagram” of the body in question.
Common Support / Connection Element Types in Two
Dimensional Analysis
In rigid bodies subjected to two dimensional force systems, the
forces exerted from supports and connection elements are shown
in the free body diagram as follows:
It should be kept in mind that reaction will occur along the
direction in which the motion of the body is restricted.
Equations of Equilibrium in Two Dimensional Case
If all the forces acting on the rigid body are planar and all the couples are
perpendicular to the plane of the body, equations of equilibrium become two
dimensional.




R  F  R x i  R y j  0
R x  Fx  0


M  M z k  0
R y  Fy  0
Fy  0
M O  0
or in scalar form,
Fx  0
At most three unknowns can be determined.
Chapter 12: Static Equilibrium
Conditions for equilibrium
 First condition for equilibrium
• When a particle is in equilibrium (no acceleration), in an inertial
frame of reference, the vector sum of all the forces acting on the
particle is zero.
A frame of reference where Newton’s first law is valid

F  0
F
x
 0,  Fy  0,  Fz  0
The sum includes external forces only.
Conditions for equilibrium (cont’d)
 Second condition for equilibrium
• For an extended body to be in equilibrium the body must have no
tendency to rotate.


L  0 , dL / dt  0
about any point in the body

  0 about any point in the body
• The sum of the torques due to all external forces acting on the
body with respect to any specified point must be zero.
• Note that the above conditions apply to a rigid body in uniform
translational motion or to constant rotation (if L is constant) as well.
But in the following we will mostly consider situations in which a
rigid body is at rest, i.e., in a static equilibrium.