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Mechanics: Statics




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Moment of force in 3D about a point

When an application is subjected to forces in space, the determination of the effective moment of forces about a reference point is much more complicated than application with forces in a plane. In most case, the effective moment in space can only be determined by moment vectors addition in space instead of algebraic addition of moment vectors in the 2D case.

image 

Moment Decomposition

Since the effective moment of concurrent forces about a reference point is equal to the sum of the moments of the concurrent forces as stated in Varignon's Theorem. Both the force vector and the position vector can be resolved into three rectangular components as in the 2D case.

A force vector, F in space can be resolved into three retangular components, Fx Fy and Fz. The position vector, r in space can also resolved into three retangular components, dx, dy and dz. Similarly, the moment vector, M of force, F can also be resolved into three retangular components, Mx, My and Mz.

image 

Therefore, the force vector, F and position vector, r can be expressed as:

image   and    image  

And express the resultant moment vector, M in the form of three retangular components

image

The moment M of a force F about a reference point can be defined as the vector product of r and F.

image

Imply.

image

Since the magnitude of vector product is the product of the magnitudes of the two vectors and the sine of the angle formed by the vectors, imply

image

Therefore, the vector product of a unit vector with itselt is equal to a zero vector, imply

image  ,  image  and image 

For the sense of the direction, the vector product of two different unit vectors is equal to the unit vector in the third dimension according to the right hand rule, imply

image ,   image  and  image 

image ,  image  and  image 

Substitute all variables into the moment vector equation, imply

image.

Each force component exerts zero moment along its own dimension, and exerts two moments with opposite sense along the other two dimensions. By grouping moments along the same dimension, moment, M can therefore be decomposed into three retangular components:

image.

The magnitude of the  three retangular components can be determined by:

image.

The sense or direction of the moment is indicated by the sign of the retangular resultant component.

The moment can be expressed in the determinant  form:

image.

Moment Addition

The effective moment of forces about a reference point in space can be represented by a moment vector, M, in three dimensional space.

image.

The most practical way of determining the resultant moment in space is to add up their rectangular components respectively.

image.

where:

image.


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References

  1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics, Saunders College Publishing, United States of America
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics, McGraw-Hill Companies, Inc., New York
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ID: 110600002 Last Updated: 6/13/2011 Revision: 0 Ref:

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