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.

0

0

0

0

kMjMiM

M

kRjRiR

FR

zyx

zyx

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.

0

0 0 0

kMM

FRFRjRiRFR

z

yyxxyx

or in scalar form,

0 0 0 Oyx MFF

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.

0F

0,0,0 zyx FFF

A frame of reference where Newton’s first law is valid

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.

0/,0 dtLdL

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.

THE END