2. Force Systems. 2.1 Introduction. 2.2 Force

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1 2. Force Systems 2.1 Introduction 2.2 Force - A force is an action of one body on another. - A force is an action which tends to cause acceleration of a body (in dynamics). - A force is a vector quantity. - The complete specification of the action of a force must include its magnitude, direction, and point of application. External and internal effects - External forces can be either applied forces or reaction forces. - Internal forces are resultants of stresses caused by external forces. A A A N N F F F A, F : external N : internal 1

2 Principle of Transmissibility - Assume: rigid body - The principle of transmissibility states that a force may be applied at any point on its given line of action without altering the resultant effects of the force external to the rigid body The external effects at O and C (support reactions) remain the same regardless on which point of the line of action of the force, the force acts. - In this case, the force may be treated as sliding vector. - The principle of transmissibility is not valid for elastic bodes if we study the internal effects of a force. In this case, the force is a fixed vector Force Classification - Forces are classified as either contact or body forces - Contact forces are caused by the direct contact of one body with the surface of another - A body force occurs when one body exerts a force on another body without direct physical contact between the bodies. Examples include the effects caused by the earth s gravitation or its electromagnetic field. - Force may be further classified as either concentrated or distributed 2

3 Action and reaction - The action of a force is always accompanied by an equal and opposite reaction. - It is important to distinguish between the action and the reaction in a pairs of forces. F F A A B R F : Action force R : Reaction force, exerted on body A by body B. 3

4 Concurrent Forces - Two or more forces are said to be concurrent at a point if their lines of action intersect at that point. - Use parallelogram law to determine the resultant 4

5 - We can also use the triangle law to obtain R, but we need to move the line of action of one of the forces. If we add the same two forces, we correctly preserve the magnitude and direction of R, but we lose the correct line of action, because R obtained in this way does not pass through A. Therefore this type of combination should be avoided. - The relationship between a force and its vector components along given axes must not be confused with the relationship between a force and its perpendicular projections onto the same axes. 5

Vector Addition - Used when the resultant of two parallel forces is to be determined.

6 Vector Components - We often need to replace a force by its vector components in direction which are convenient for a given application Note: F a Fcosθ But F x = Fcosθ (rectangular components) A special Case of Vector Addition - Used when the resultant of two parallel forces is to be determined. - Applicable on rigid bodies only - R is the resultant of the forces F 1 and F 2 and is correct in magnitude, direction, and line of action. 6

7 2.3 Rectangular Components of a Force cos sin tan F is the magnitude of the force vector F and is always positive, where Fx and F y can be positive or negative. F x, F y are the vector components of F in the x- and y-directions. F x, F y are the scalar components of F. When both a force and its vector components appear in a diagram, it is desirable to show the vector components of the force with dashed lines and show the force with a solid line, or vice versa. (Only one force F, or two components F x and F y, but not 3 separate forces). 7

8 The choice of a coordinate system is arbitrary. A logical choice simplifies the solution of the problem. y y y β γ F α F x F x x cosβ sin cos sinβ cos sin 0 Note: In all three cases the magnitude F is the same and is defined as 8

9 Determining the Components of a Force 9

10 Determining the Resultant of two Forces 10

- The moment M of a force F about a point A is defined using cross product as where r is a position vector which runs from the

11 2.4 Moment - A force can tend to rotate a body about an axis which neither intersects nor is parallel to the line of action of the force. This rotational tendency is known as the moment M of a force. - The moment M of a force F about a point A is defined using cross product as where r is a position vector which runs from the moment reference point A to any point on the line of action of F. - Note. - Moment about a point A means here: Moment with respect to an axis normal to the plane and passing through the point A. 11

- The magnitude M of the moment is defined as: sin where d is a moment arm and is defined as the perpendicular distance between the line of action of the force and the moment center.

12 - The magnitude M of the moment is defined as: sin where d is a moment arm and is defined as the perpendicular distance between the line of action of the force and the moment center. - The moment M is a vector quantity. Its direction is perpendicular to the r-f-plane. - The sense of M depends on the direction in which F tends to rotate the body right-hand rule + : counterclockwise rotation. - : clockwise rotation. Sign consistency within a given problem is very important. - The moment M may be considered as a sliding vector with a line of action coinciding with the moment axis. 12

Varignon s Theorem - Valid for any number of concurrent forces - States that the moment of a force (R) about any point is equal to the sum of the moments of the components (P, Q) of the

13 Varignon s Theorem - Valid for any number of concurrent forces - States that the moment of a force (R) about any point is equal to the sum of the moments of the components (P, Q) of the force about the same point. - The theorem can be stated in another way as: The resultant moment M of the moments M i of the concurrent forces F i is equal to the moment of their resultant R. 13

2.5 Couple - The moment produced by two equal, opposite, parallel, and noncollinear forces is called a

- The moment M of a couple is defined as where r A and r B are position vectors which run from point O to

- The moment expression contains no reference to the moment center O and, therefore, is the same for all

14 2.5 Couple - The moment produced by two equal, opposite, parallel, and noncollinear forces is called a couple. - The force resultant of a couple is zero. Its only effect is to produce a tendency of rotation. - The moment M of a couple is defined as where r A and r B are position vectors which run from point O to arbitrary points A and B on the lines of action of F and F. - The moment expression contains no reference to the moment center O and, therefore, is the same for all moment centers the moment of a couple is a free vector. - The sense of the moment M is established by the right-hand rule. Counterclockwise couple (+) Clockwise couple (-) 14

Equivalent Couples - Changing the values of F and d does not change a given couple as long as the

15 - The magnitude M of the moment M of the couple is given as where d is the perpendicular distance between the couple forces (F, -F). - The magnitude of the couple is independent of the distance a. Equivalent Couples - Changing the values of F and d does not change a given couple as long as the product Fd remains the same. - A couple is not affected if the forces act in a different but parallel plane. 15

16 Force-Couple Systems - The effect of a force acting on a body is: a) the tendency to push or pull the body in the direction of the force, and b) to rotate the body about any fixed axis which does not intersect the line of action of the force ( force does not go through the mass center of the body). We can represent this dual effect more easily by replacing the given force by an equal parallel force and a couple to compensate for the change in the moment of the force. - Also we can combine a given couple and a force which lies in the plane of the couple to produce a single, equivalent force. 16

17 2/6 Resultants Reduce the system of forces and moments to a resultant force R only or to a resultant force R and a resultant moment M O. The resultant of a system of forces is the simplest force combination which can replace the original forces without altering the external effect on the rigid body to which the forces are applied. Assume: all forces act in one plane (x-y plane) and the moments act about an axis perpendicular to this plane. The resultant of a system of forces is the simplest force combination which can replace the original forces without altering the external effect on the rigid body to which the forces are applied. 17

18 a. Force Polygon Method Yields magnitude and direction of the resultant only Line of action of the resultant remains undetermined. b. Parallelogram Law Yields: Magnitude, direction, and the correct line of action 18

19 c. Algebraic Method Example: Three force-system 1. Choose a convenient reference point O and move all forces to that point with considering the couples M 1, M 2, and M 3 resulting from the transfer of forces F 1, F 2, and F 3 from their respective original lines of action to parallel lines of action through point O. 2. Add all forces at O to form the resultant force R, and add all couples to form the resultant couple M O. 19

20 3. Find the line of action of R by requiring R to have a moment of M O about point O. - For concurrent system of forces where the lines of action of all forces pass through a common point O, the moment sum M o about that point is zero. - For a parallel force system, select a coordinate axis in the direction of the forces. - If the resultant force R for a given force system is zero, the resultant of the system need not be zero because the resultant may be a couple. Principle of moments States that the moment of the resultant force about any point O equals the sum of the moments of the original forces of the system about the same point. The may be concurrent or nonconcurrent. 20

Engineering Mechanics: Statics

Engineering Mechanics: Statics Chapter 2: Force Systems Part A: Two Dimensional Force Systems Force Force = an action of one body on another Vector quantity External and Internal forces Mechanics of Rigid