3D Frame Interpolation Functiones 2

[alex9wyne]

Hello,

I have a question. I'm currently implementing a 3D spatial frame matlab FEM code based on Bernoulli beam. I'm building upon AJM Ferreira's book ''Matlab Codes for finite element Analysis'' formulations, because I want to automatize stress calculations on different configurations of a simple 3D frame, using matlab.

The issue is I can obtain the node displacements and reactions on restrictions, but I can't obtain the strain field, and thus the tension on the nodes, because I don't know the interpolation functions (or even better the strain displacement matrix B).

My question is, does anyone know any reference where I can get the interpolation functions of the 3d frame case? (or the B matrix). Or does anybody know another way to obtain the stresses on the nodes?

I've spent a lot of time looking for these interpolation functions but havn't had any luck on the net or in books.

Thanks a lot to any one who can give me a hand.

Alex

 

[alex9wyne]

I almost finished the matlab implementation of a particular 3d frame for multiple configuration scenarios,in matlab, as I stated on my previous post. I shall only check displacement criteria because I can't interpolate the strain and stress fields.

But I'm still curious about how to do this 3d intrepolation. Does no one know enough about the math behind the FEA to give any insight into this particular problem?

 

[IDS]

I'm not sure what you mean by strain and stress fields, in the context of a frame analysis, but for each frame member you have the stiffness matrix, and you have the deflections at each end. If you multiply the stiffness matrix by the deflection vector that gives you the actions at each end (3 forces and 3 moments at each end), and from that it is straightforward to calculate the forces and moments along the beam, taking account of any intermediate applied forces or moments.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[alex9wyne]

I’m not an expert and FEA is not a primary aspect in the work I do (it’s a nice to have). But from what I know from university this is what I mean. In general terms, considering x as the normal direction from the beams section, what I mean by displacement field, is the ‘’elastic equation’’ (the deflection of the beam in the y direction at some position x) of each beam element, call it V(x). In 2d FEA beam theory at least, this function can be interpolated as I will show.

The strain field (call it epsilon(x)) would be, according to Euler-Bernoulli beam model, the second derivative of the displacement field with respect to the beam axis direction times the normal coordinate in the cross section: epsilon(x)=y*V(x)’’

The stress field would be simply: sigma(x)=E* y*V(x)’’

Now, for the 2d frame FEA, the deflection field (equation of the elastic) of a beam element is obtained by supposing it follows a cubic polynomial on x with four constants ( V(x)=a1 + a2*x + a3*x^2 + a4*x^3)) which are obtained by applying boundary conditions at the elements edges (x=0, x=L) of both the displacement and slope field ( theta(x)=V(x)’).

This gives you four equations in which the constants, the vertical displacements and slope on the edges are the variables. By expressing in matrix form, solving for the constants and rearranging the equations, you can obtain the displacement field V(x) as the product of a 1x4 first order Hermite polynomial vector ( [N] ) and the nodal displacement vectors (call it q = [v1, theta1, v2, theta2] ):

V(x) = [N]q
Where [N] = [ N1 , N2, N3, N4]

N1=1-x2/L2+2x3/L3, N2=1-2x2/L+x3/L2, N3=3x2/L2-2x3/L3, N4=-x2/L+x3/L2

So the rest is simple, the strain field is: epsilon(x)= y*V(x)’’= y (d2/dx2)[N]q=q
Where is called the strain-displacement vector, = y d2([N])/dx2

And the stress field is
sigma(x) = E q

If anyone wants a reference for interpolation of 2d beam elements in FEA a quick google search brings this lecture slides from Profesor Nam Ho Kim, of the Mechanical & Aerospace Engineering department of the University of Florida:

http://www2.mae.ufl.edu/nkim/eml5526/Lect05.pdf

The stiffness matrix of an element can be obtained by performing a volume integration (in this case a line integral) of the product of the transpose of the strain-displacement vector times the regular vector times E, that is: E*integral( ’ ) from 0 to L (element length), by doing this you obtain the conventional stiffness matrix [K]=EI/L[12/L2 6L -12/L2 6/L; 6/L 4 -6/L etc….]

So this is how strain and stress fields are obtained for the 2d Bernoulli beam. What I want is to do the same for the 3d case. That is, I want the vector (or matrix) that will give me the stress and strain field.

 

[alex9wyne]

That's what I meant with the strain and stress fields, but what you propose is a good solution and not that more difficult to code. Thank you for your input.

 

[IDS]

The link from the University of Florida is essentially following the same procedure, in 2D, (in more detail) as I summarised in my previous post. The only difference is that he is converting all applied loads to the equivalent nodal loads. This gives exactly the same results for global deflections (i.e. deflections at the nodes), but the stress and strain distribution between nodes is obviously different for beams with any load applied between the nodes.

To apply the method to 3D you just need the stiffness matrix for a 2D beam (i.e. a line beam in 3D space), then apply the procedure shown in the last slide of the lecture.

You might like to have a look at my 3D frame analysis program at:

http://newtonexcelbach.wordpress.com/2013/01/22/3dframe-3d-frame-analysis-for-excel/

The code is all open source; VBA, but you should be able to extract the required equations.

My main reference is Programming the Finite Element Method by Smith Margetts and Griffiths, which contains details of the 3D stiffness matrix, as well as code for many sample applications (plate and brick elements, as well as beams).

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[alex9wyne]

Thank you very much for your response. Yes, you're absolutly right, in slide 23 he uses the vector to calculate the bending moment and shear forces by multiplying with the nodal deflectiones, but in thinking about your reply I realize that the vector is essentially the same thing as the stiffness matrix (as I even said in my previous post [K]=E*integral( ’ ) from 0 to L). So I was just over-complicating myself with the math.

I actually stumbled upon and used your frame spreadsheet programs a couple of years ago when I studied FEM in university. They were very useful! I used the 2d programs but never checked out the 3d frame one but it looks very good and the large frame model you posted looks amazing! I'll definitley play around with it. Also thank you for the book reccomendation.