4: Notes

Force System Resultants​

1) Moment of Force ​

A force produces a turning effect or moment about a point O that does not lie on its line of action
The direction of the moment is defined using the right hand rule. M_O always acts along an axis perpendicular to the plane containing F & d, and passes through the point O. 

The easiest way I remember the right hand rule is by the bottle cap method that my professor mentioned to me. If you're holding a soda bottle in your left hand and unscrewing it with your right, the cap spins from left to right and moves in an upward direction. 

a) Scalar Definition​

The magnitude of the moment of force is the product of the force and the moment arm, or perpendicular distance from point O to the line of action of the force. 
Rather than finding d, it is normally easier to resolve the force into its x and y components, determine the moment of each component about the point, and then sum the results. This is called the principle of moments.

M_O = Fd = F_xy - F_yx

b) Vector Definition​

Most common with 3D moment analysis. 
The moment is determined by taking the cross product of the vectors. 
if r is a position vector extending from point O to any point A, B, or C on the line of action of a force 
F, 

M_O = r_A × F = r_B × F = r_C × F ​

Cross Product: 
if U = <a, b, c> & V = <d, e, f>, then U × V = <bf - ce, cd - af, ae - bd>

2) Moment about an Axis​

If the moment of a force F is to be determined about an arbitrary axis a, then for a scalar solution the moment arm, or shortest distance d_a from the line of action of the force to the axis must be used. This distance is perpendicular to both the axis and the force line of action. 

In 3D, the scalar triple product should be used. It's best to use a unit vector, u_a, that specifies the direction of the axis and a position vector, r, that is directed from any point on the axis to any point on the line of action of the force. 

If M_a is calculated as a negative scalar, then the sense of direction of M_a is opposite to u_a

M_a = Fd_a = u_a⋅(r × F)

3) Couple Moment​

Consists of two equal but opposite forces that act at a perpendicular distance d apart. Couples tend to produce a rotation without translation. 

The magnitude of the couple moment is M = Fd, and its direction is established using the right hand rule. 

If the vector cross product is used to determine the moment of a couple, then r extends from any point on the line of action of one of the forces to any point on the line of action of the other force F that is used in the cross product. 
M = r × F

4) Simplification of a Force and Couple System​

Any system of forces and couples can be reduced to a single resultant force and resultant couple moment acting at a point. The resultant force is the sum of all the forces in the system, F_R = ΣF, and the resultant couple moment is equal to the sum of all the moments of the forces about the point and couple moments.

M_RO = ΣM_O + ΣM​

Equate the moment of the resultant force about the point to the moment of the forces and couples in the system about the same point

If the resultant force and couple moment at a point are not perpendicular to one another, then this system can be reduced to a wrench, which consists of the resultant force and collinear couple moment. 

5) Coplanar Distributed Loading​

A simple distributed loading can be represented by its resultant force, which is equivalent to the area under the loading curve. This resultant has a line of action that passes through the centroid or geometric center of the area or volume under the loading diagram.