Bending moment transformation from local to global coordinate system using the 3D permutation tensor

[Cagemine]

Hi, I am seeking advice, direct or indirect (link to helpful resources), on transforming bending moments from local to global coordinate system using 3-D permutation tensor.

 

For example, I have a series of bending moments acting on a node but in a local coordinate system and I want to transform these to the global coordinate system. From my reading on this the 3-D permutation tensor is used in conjunction with a function for the local bending moments and transforms them into global bending moments at that node.

 

Global coordinate system [1 0 0 ; 0 1 0 ; 0 0 1]

Local coordinate system [ 0.707 0.707 0 ; -0.707 0.707 0 ; 0 0 -1]

 

A vector P <px, py, px> represents the beam and is parallel to the global X-axis, P = < 1 0 0 >

 

The following local bending moments are applied (counter clockwise positive)

Moment about local x-axis, mx = 0 units

Moment about local y-axis, my = 2 units

Moment about local z-axis, mz = 0 units

 

What are the resulting global moments.

 

The general formula used is:

 

M_1i = εjik [m_z1 p_k y_1j - m_y1 p_k x_1j + m_x (x_1j y_2k - y_1j x_2k)/2]

 

1 or 2 mean the near or far ends of a beam

ijk take the form of 1,2,3 (representing x,y,z)

 

I have determined the answer to be:

Moment about global X-axis, Mx = -1.41 units (anti cyclic permutation)

Moment about global Y-axis, My = 1.41 units (cyclic permutation)

Moment about global Z-axis, Mz = 0 units

 

I have an issue with my understanding of the 3-D permutation tensor.

 

Thus my true question being, How does one determine what the permutation should be, cyclic (123) or anti-cyclic (321) in this Cartesian coordinate axis environment?

 

Regards,

Cage