Transformation matrix for 3D frame? 1

[earthlink]

Hi all,

I have been searching transformation matrix for 3D frame element here and there, but could not find it.
Does anyone know what is the transformation matrix for 3D frame element in the direct stiffness matrix approach.

For the 2D beam it looks like this:

http://file.scirp.org/Html/16-8101383/15a51874-eba3-4044-95a7-0df24540dfe0.jpg

For the 3D truss (using directional cosine):

http://images.slideplayer.com/19/5716159/slides/slide_31.jpg

I imagine for anything correspond to the translational dof should all be the same as truss.
Should the stiffness contribution correspond to the moment should all be set as 1 just like in the 2D beam?

Thank you in advance for the help.

 

[Yt.]

Do you need pinned-pinned, fixed-fixed, fixed-pinned, pinned-fixed? or you are trying to get global to local axes and viceversa?

 

[Yt.]

This is not just about multiplying a rotation matrix to other stiffness matrix. What do you need to do?

If you just need to know what is it. It's a matrix which allows you to transform local stiffness of elements (which has x-axis along its longitudinal axe, for convenience) and express them in global coordinates. But transformation matrix alone isn't enough it's a full procedure in order to be usefull

 

[IDS]

Have a look here:

https://newtonexcelbach.wordpress.com/2016/12/09/3d-frames-axes-and-stiffness-matrices/

The blog post shows the transformation matrix and describes the process, and the download file includes an example and open-source VBA code for 3D frame analysis.

Please ask if anything is not clear.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[earthlink]

@Ytyus: yes, the transformation matrix the one I looked for is the one that is used in K=T'.k.T. For planar transformation is a 4x4 matrix from using 2D rotation matrix for x and y component of force and displacement

@IDS: thanks much for this reference. Do you know any reference how to derive this rotation matrix?
It seems clear to me the rotation matrix in planar truss, 3D truss and planar frame are pretty similar (only different a bit), the form has same appearance to a simple rotation matrix, or when using directional cosine. But why would this 3D frame rotation seems much different from those?
Seems XL, YL, and ZL are the dot products of beam local axis and global, and Den is the norm, what are Cg and Sg means when it says angle of 2 axis to XY plane? What 2 axis are they?

 

[IDS]

The axes I have used are the same as used in the Strand7 FEA package. They are defined in the text:

Beam principal axes are defined as i1 to i3, where:
i3 – is the unit vector directed from Node 1 to Node 2.
i2 – is the unit vector arising from i2 = Z × i3 where Z is the unit vector in the global Z direction
i1 – completes the right-handed system such that i1 × i2 = i3

The rotation matrix used in the spreadsheet originally came from Programming The Finite Element Method by Smith and Griffiths. I have modified it (as described in the blog post) to follow the Strand7 local axis convention. Note that the beam longitudinal axis is 3, not 1 (or x) as used in the book.

The 3D matrix is more complicated than the 2D or truss equivalent because you have to allow for rotation of the local axis about the longitudinal axis, as well as the angle of the longitudinal axis to the global axes.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Yt.]

I think i attached director cosine matrix:

If not i'll try again

javascript:attachFile()

it's something mit released

http://files.engineering.com/getfile.aspx?folder=ccbf1939-4eed-4561-9fed-be1c8f700a7a&file=HO2.PDF

 

[Yt.]

The r11,r12,... matrix is what you are looking for I believe. It's also works with 6 DOF and leaving 0 stiffness for other stresses

 

[earthlink]

@IDS: Thank you for the explanation. I was trying to understand the value of XL YL and ZL more.
Based on the data shown, I tried to find the value of XL YL and ZL.
You included the values of 2 end points, and the rotation in radian in your excel spreadsheet is gamma (1.047197551)?
So I do:

p1=[0 5 3]
p2=[5 0 3]
v=p1-p2
L=norm(p1-p2)
XL=L*0.6804138
YL=L*0.6804138
ZL=L*0.272165527
a=1.047197551
den=L*(XL^2 + YL^2)^0.5
T11=(XL*ZL*cos(a)-L*YL*sin(a))/den

The result of T11 is -0.5161, which is supposed to be 0.5632257 in your transformation matrix T.
I don't understand why its different than the one in your matrix.
Enlightenment regarding this matter would be much appreciated.

@Ytyus: Yes, I think that the formula provided in the transformation matrix by @IDS should be the equivalent version to that rotation matrix in 3D when using angle, the one you provide. Is it true @IDS? But if coordinates are known, usually directional cosine is preferable for transformation, as far as I know

 

[IDS]

earthlink - I'm not sure where you got your numbers but:

XL = x2-x1 = 5
YL = y2-y1 = -5
ZL = z2-z1 = 0
L = (5^2+-5^2+0^2)^0.5 = 7.071
Den = L*(XL^2 + YL^2)^0.5 = 50
Cos(gamma) = 0.500
Sin(gamma) = 0.866

T11 = (XL * ZL * Cg - L * YL * Sg) / Den = (5*0*0.5 - 7.071*-5*0.866)/50 = 0.612
T21 = -(XL * ZL * Sg + L * YL * Cg) / Den = -(5*0*0.866 + 7.071*-5*0.5)/50 = 0.356
T31 = XL/L = 5/7.071 = 0.707

etc. These are the values returned by the spreadsheet for the given beam coordinates. The numbers in the screen shots in the blog post are for a different beam, which may have caused some confusion.

I haven't checked, but I assume the matrices provided by Ytyus would give the same results, except they have different local axis conventions.





Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/