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Zhejiang Normal University Developed by A.Kryzhanovskyi

ENGINEERING STATICS

Chapter 3 Equilibrium of a rigid body

Lecturer: PhD Andrii Kryzhanovskyi

Zhejiang Normal University

Zhejiang Normal University Developed by A.Kryzhanovskyi  Any body can be subjected to an external force and couple moment system that is the result of the effects of gravitational, electrical, magnetic, or contact forces caused by adjacent bodies.  The force and couple moment system acting on a body can be reduced to an equivalent resultant force and resultant couple moment at any arbitrary point O on or off the body.

Conditions for rigid-body

equilibrium

Zhejiang Normal University Developed by A.Kryzhanovskyi  Two last equations are not only necessary for equilibrium, they are also sufficient.  To prove this, consider summing moments about some other point A  Since r ≠ 0 , this equation is satisfied if F R = 0 and ( M R

O

Conditions for rigid body

equilibrium

A R R O

M  r  F  ( M )  0

Zhejiang Normal University Developed by A.Kryzhanovskyi  When applying the equations of equilibrium, we can generally assume that the body will remain rigid and not deform under the applied load without introducing any significant error.  In this case the direction of the applied forces and their moment arms with respect to a fixed reference remain the same both before and after the body is loaded.

Conditions for rigid body

equilibrium

Zhejiang Normal University Developed by A.Kryzhanovskyi  Successful application of the equations of equilibrium requires a complete specification of all the known and unknown external forces that act on the body.  The best way to account for these forces is to draw a free-body diagram.  This diagram is a sketch of the outlined shape of the body, which represents it as being isolated or “free” from its surroundings.  On this sketch it is necessary to show all the forces and couple moments that the surroundings exert on the body.

Free body diagram

Zhejiang Normal University Developed by A.Kryzhanovskyi  If a support prevents the translation of a body in a given direction then a force is developed on the body in that direction.  If a support prevents the rotation of a body in a given direction then a couple moment is exerted on the body.

Support reactions

or

Zhejiang Normal University Developed by A.Kryzhanovskyi  (^) The internal forces that act between adjacent particles in a body always occur in collinear pairs such that they have the same magnitude and act in opposite directions.  Since these forces cancel each other, they will not create an external effect on the body.  Due to this reason the internal forces should not be included on the free-body diagram if the entire body is to be considered.

Internal forces

Zhejiang Normal University Developed by A.Kryzhanovskyi  When a body is within a gravitational field, then each of its particles has a specified weight.  Such a system of forces can be reduced to a single resultant force acting through a specified point.  We refer to this force resultant as the weight W of the body and to the location of its point of application as the center of gravity.  When the body is uniform or made from the same material, the center of gravity will be located at the body’s geometric center or centroid. Weight and the center of gravity

Zhejiang Normal University Developed by A.Kryzhanovskyi The steel beam is to be used to support the three roof joists of a building.  (^) Material is rigid.  A pin can be considered for support A because bolted connection at A allow for any slight rotation  A roller can be used for support B since it offers no resistance to horizontal movement.  (^) Building code requirements are used to specify the roof loading  The weight of the beam is generally neglected

Idealized models

Zhejiang Normal University Developed by A.Kryzhanovskyi To construct a free-body diagram for a rigid body, the following steps should be performed:  (^) Draw outlined shape. Imagine the body to be isolated from its constraints and connections and draw its outlined shape.  (^) Show all forces and couple moments. Identify all the known and unknown external forces and couple moments that act on the body. Those generally encountered are due to ̶ applied loadings, ̶ reactions occurring at the supports or at points of contact with other bodies, ̶ the weight of the body. Procedure for drawing a free-body diagram

Zhejiang Normal University Developed by A.Kryzhanovskyi (^) 16

Example 3.

Draw the free-body diagram of the uniform beam. The beam has a mass of 100 kg. Solution

Zhejiang Normal University Developed by A.Kryzhanovskyi (^) 17

Example 3.

Two smooth pipes, each having a mass of 300 kg, are supported by the forked tines of the tractor. Draw the free-body diagrams for each pipe and both pipes together. Solution

Zhejiang Normal University Developed by A.Kryzhanovskyi  (^) When the body is subjected to a system of forces, which all lie in the x–y plane, then the forces can be resolved into their x and y components.  (^) Consequently, the conditions for equilibrium in two dimensions are where ΣF x and ΣF y represent the algebraic sums of the x and y components of all the forces acting on the body, and ΣM O represents the algebraic sum of the couple moments and the moments of all the force components about the z axis , and passes through the arbitrary point O.

Equations of equilibrium

O

M  0

x 

F  0;

 y

F  0;

Zhejiang Normal University Developed by A.Kryzhanovskyi  Two alternative sets of three independent equilibrium equations may also be used.  One such set is  When using these equations it is required that a line passing through points A and B is not parallel to the y axis.

Alternative sets of

equilibrium equations

x

F  0;

A

M  0;

B

M  0;