Chapter 3 Rigid Bodies : Equivalent Systems of Forces Part -2

Couple Two forces F and -F having the same magnitude, parallel lines of action, and opposite sense are said to form a couple.

Moment of A Couple Moment of the force F, r MO= rA x F B rB Moment of the force F, - F r MO= rA x F F A rA Moment of the force -F, MO= rB x (-F) O Moment of the couple, MO= rA x F + rB x (-F) MO= (rA - rB) x F MO= r x F

Moment of A Couple Moment of the couple, Where d is the perpendicular distance between the lines of action of F and -F The moment vector of the couple is independent of the choice of the origin of the coordinate axes, i.e., it is a free vector that can be applied at any point with the same effect.

Problem 3.69 on page 116 A piece of plywood in which several holes are being drilled has been secured to a workbench by means of two nails. Knowing that the drill applies a 12 Nm couple of moment on the plywood, determine the magnitude of resulting forces applied to the nails if they are located to At A and B At A and C

Problem 3.69 on page 116 Nails are at A and B 12 Nm = (0.45m) * F ---> F = 26.67 N F=26.67 N -F= 26.67 N

Equivalent Couples Two couples will have equal moments if the two couples lie in parallel planes, and the two couples have the same sense or the tendency to cause rotation in the same direction.

Equivalent Couples 1 m 120 N M M 20 N 20 N 6 m 1 m 120 N M

Non- Equivalent Couples M 1 m 120 N 20 N 120 N M 20 N 6 m

Addition of Couples Consider two intersecting planes P1 and P2 with each containing a couple Resultants of the vectors also form a couple Sum of two couples is also a couple that is equal to the vector sum of the two couples By Varigon’s theorem

Couples Can Be Represented by Vectors A couple can be represented by a vector with magnitude and direction equal to the moment of the couple. Couple vectors obey the law of addition of vectors. Couple vectors are free vectors, i.e., the point of application is not significant. Couple vectors may be resolved into component vectors.

Sample Problem 3.6 on page 113 Determine the components of the single couple equivalent to the couples shown.

Sample Problem 3.6 on page 113 SOLUTION: compute the sum of the moments of the four forces about D.

Sample Problem 3.6 on page 113 ALTERNATIVE SOLUTION: Attach equal and opposite 20 lb forces in the +x direction at A, thereby producing 3 couples for which the moment components are easily computed.

Sample Problem 3.6 on page 113 Attach equal and opposite 20 lb forces in the +x direction at A The three couples may be represented by three couple vectors,

Resolution of a Force Into a Force at O and a Couple Force vector F can not be simply moved to O without modifying its action on the body. Attaching equal and opposite force vectors at O produces no net effect on the body. The three forces may be replaced by an equivalent force vector and couple vector, i.e, a force-couple system.

Resolution of a Force Into a Force at O and a Couple MO r A A O O Any force F acting at a point A of a rigid body can be replaced by a force-couple system at an arbitrary point O. It consists of the force F applied at O and a couple of moment MO equal to the moment about point O of the force F in its original position. The force vector F and the couple vector MO are always perpendicular to each other.

Sample Problem 3.7 on page 114 Replace the couple and force shown by an equivalent single force applied to the lever

Sample Problem 3.7 on page 114 First calculate the moment created by the couple Secondly resolve the given force into a force at O and a couple We move the force F to O and add a couple of moment about O

Sample Problem 3.7 on page 114

Problem 3.80 on page 118 A 135 N vertical force P is applied at A to the bracket shown. The bracket is held by two screws at B and C. Replace P with an equivalent force-couple system at B Find the two horizontal forces at B and C that are equivalent to the couple of moment obtained in part a

Problem 3.79 on page 118

Problem 3.79 on page 118 Find the two horizontal forces at B and C that are equivalent to the couple of moment obtained in part a F= 225 N -F= 225 N

Reduction of a System of Forces to One Force and One Couple Any system of forces can be reduced to a force-couple system at a given point O. First, each of the forces of the system is replaced by an equivalent force-couple system at O. Then all of the forces are added to obtain a resultant force R, and all of couples are added to obtain a resultant couple vector MO. In general, the resultant force R and the couple vector MO will not be perpendicular to each other.

Reduction of a System of Forces to One Force and One Couple First, each of the forces of the system is replaced by an equivalent force-couple system at O. Then all of the forces are added to obtain a resultant force R, and all of couples are added to obtain a resultant couple vector MO.

Sample Problem 3.8 on page SOLUTION: For the beam, reduce the system of forces shown to (a) an equivalent force-couple system at A, (b) an equivalent force couple system at B, and (c) a single force or resultant. Note: Since the support reactions are not included, the given system will not maintain the beam in equilibrium. SOLUTION: Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about A. Find an equivalent force-couple system at B based on the force-couple system at A. Determine the point of application for the resultant force such that its moment about A is equal to the resultant couple at A.

Sample Problem 3.8 on page SOLUTION: Compute the resultant force and the resultant couple at A.

Sample Problem 3.8 on page Find an equivalent force-couple system at B based on the force-couple system at A. The force is unchanged by the movement of the force-couple system from A to B. The couple at B is equal to the moment about B of the force-couple system found at A.