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Zhejiang Normal University Developed by A.Kryzhanovskyi

ENGINEERING STATICS

Chapter 2 Force Systems

Lecturer: PhD Andrii Kryzhanovskyi

Zhejiang Normal University

Zhejiang Normal University Developed by A.Kryzhanovskyi  The state of equilibrium or motion of a rigid body depends upon the nature of its mechanical interactions with other bodies.  The quantity which allows to determine action of one body on another is called a force.  Force is a vector quantity , because its effect depends on the direction of action as well as on the magnitude.

Force

Zhejiang Normal University Developed by A.Kryzhanovskyi  System of forces is a group of some forces which act on the body.  If line of action of all forces are in one plane we deal with a planar system of forces, otherwise a system is spatial.  If one system of forces can be replaced by the other without changing the state of rest or motion of a body we deal with equivalent systems of forces

Force

Zhejiang Normal University Developed by A.Kryzhanovskyi  If under action of forces a rigid body is in a state of rest the system of forces is called as a balanced system.  Resultant force is a force which is equivalent to the whole system of forces.  Balanced (equilibrium) force is a force which is equal to the resultant force and is oppositely directed along the line of action.

Force

Zhejiang Normal University Developed by A.Kryzhanovskyi  (^) Contact and body forces ̶ Contact force is produced by direct physical contact between two objects (force exerted on a body by a supporting surface) ̶ Body force is a force which acts throughout the volume of a body (gravity, electromagnetic forces)

Classification of forces

Zhejiang Normal University Developed by A.Kryzhanovskyi  Concentrated and distributed forces ̶ Concentrated force is applied at the point of a body ( pressure force of a ball on a surface). ̶ Distributed force is a force which act on all points of a line, surface or volume of a body (gas pressure in the cylinder of the internal combustion engine).

Classification of forces

Zhejiang Normal University Developed by A.Kryzhanovskyi Axiom 1 - about equilibrium of two forces  A rigid body acted upon by two forces is in a state of static equilibrium if and only if the two forces are of the same intensity, lie along the same line of action, and are oriented in opposite directions along the line.

Axioms of statics

F' = -F''

 F' + F''  0

F''

F'

A B

Zhejiang Normal University Developed by A.Kryzhanovskyi 2

F

n

F

Axiom 2 - about adding and extracting of forces in equilibrium  If a system of two forces in equilibrium is added to or extracted from a given system of forces, the way that the system of forces acts on a rigid body undergoes no change.

Axioms of statics

F''

3

F

F'

F' = -F''

1 3

F = - F

1

F

Zhejiang Normal University Developed by A.Kryzhanovskyi Axiom 3 - about parallelogram of forces  The resultant of two forces acting at the same material point is equal to the vector sum of the two forces. The line of the resulting force's action contains the material point. This axiom obeys the principle of vector summation.

Axioms of statics

R  F 1 + F 2

1

F

2

F

R

A φ    2 2 1 2 1 2

R F F 2F F cos 

Zhejiang Normal University Developed by A.Kryzhanovskyi 1

F

F 2 

A 3

F

 (^) If more than two forces are to be added, successive applications of the parallelogram law can be carried out in order to obtain the resultant force.

Axioms of statics

R

1

F F

Zhejiang Normal University Developed by A.Kryzhanovskyi Axiom 5 - about solidification  If a deformable body is in a state of static equilibrium, it would also be in static equilibrium if the body were rigid. This axiom is also known as the principle of solidification.

Axioms of statics

Zhejiang Normal University Developed by A.Kryzhanovskyi  The resultant of several coplanar forces can easily be determined if an x, y coordinate system is established and the forces are resolved along the axes.  The direction of each force is specified by the angle its line of action makes with one of the axes.

System of coplanar forces

x y

F F F

θ A y x F F y F x

x

F F cos  

y

F F sin 

2 2 x y

F F F 

y x

F

arctan F

Zhejiang Normal University Developed by A.Kryzhanovskyi  (^) To determine the resultant of several coplanar forces, we need to do the following ̶ Each force should be resolved into its x and y components, ̶ The respective components have to be added using scalar algebra since they are collinear. ̶ The resultant force is then formed by adding the resultant components using the parallelogram law.

Coplanar force resultants

Zhejiang Normal University Developed by A.Kryzhanovskyi

Coplanar force resultants

1 1 x 1y F F i F j   2 2 x 2 y F F i F j   3 3 x 3 y F F i F j     R 1 2 3

F F F F

1 x 2 x 3 x 1y 2 y 3 y ( F F F )i ( F F F ) j   Rx Ry ( F )i ( F ) j 

Rx  x

F F ,

Ry  y

F F