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:
M1i = εjik [mz1 pk y1j - my1 pk x1j + mx (x1j y2k - y1j x2k)/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
