Assistance with LTB Calculations in FEM

[engr_mkhalid]

Hi everyone,

I’m implementing Lateral Torsional Buckling (LTB) calculations for beams using the Finite Element Method (FEM). My approach involves formulating the problem in matrix form, starting with the Elastic Stiffness Matrix and then the Geometric Stiffness Matrix. However, I’m encountering a singularity error when using the eigenvals function.

I’ve attached the Mathcad Prime 9.0 file for your reference. I would greatly appreciate any help or insights you can provide. Thank you!

[Werner_E]

I have no expertise whatsoever in your field of working, but when I look at your definition of K.g, I see that the rank of this matrix is not maximal as the matrix contains dependent rows.

The third row is the negative of the first row. So regardless of the value of L the determinant of this matrix is zero and so the inverse is not defined.

That's the reason Prime chokes on K.g^-1 and matrix A remains undefined!

Maybe "geninv" can be of help!?

You may want to look at these help pages

Inverting Matrices

Rank and Linear Systems Properties of Matrices

[terryhendicott]

Hi,

The structural model in FEM consists of discrete element stiffness matrices usually more than one assembled into a global stiffness matrix with structural restraints applied.  Once assembled and adequately structurally restrained global stiffness matrix is invertible.

Of itself the stiffness of one element is singular (third row negative of first row).

Look at a structure of two elements 3000 long.  With restraints applied vertically at each end.  Then the global stiffness matrix can be inverted.

There are two ways to handle restraints.

One is to drop the row and column of the restraint from the global stiffness matrix and keep track of what was deleted

The other easier method is to apply a large stiffness relative to the values in the stiffness matrix values on the diagonal at the correct location for the deflection/rotation under consideration.

So two elements each 3000 long with a point load of -1000 applied at the center span is:

Answer when solved is correct.

Cheers

Terry

I have version 10 so cannot save back the files I have.

[terryhendicott]

Hi,

Had to do something else and am returning to finish off.

Use chatGPT with the following two questions to get an overall view of how to use both Ke and Kg

Formulate elastic stiffness matrix and geometric stiffness of 2D frame bending?

How to use the elastic stiffness matrix and geometric stiffness matrix in analysis of a 2D frame?

First how to examine the two element 6000 long simply supported beam under a compression of -10000

Note as has been pointed out this is singular and cannot be inverted.

The geometric stiffness is assembled just like the elastic stiffness into a global geometric stiffness

Now the combined action under the influence of the vertical point load downward and a compression inwards can be determined,

Due to the P-Delta effect of the compression applied over a sagging beam will decrease the stiffness of the structure and deflect a little more.

You can see the deflection has increased from 4.238 of previous post to a little more 4.438.

Now you can do a buckling analysis to determine how large the compression can be to buckle the beam (not lateral torsional buckling but member buckling)

Hope this helps

Cheers

Terry

[engr_mkhalid]

Hi,

Thank you for the detailed explanation. I am trying to solve a beam by splitting it into a couple of elements to solve MCRIT, which is the critical moment load before LTB starts. Lateral supports need to be taken into consideration, fx. at a fixed support, the beam can’t rotate about its axis. One beam element will have 8 degrees of freedom: Torsion, Rotation out of the plane, Translation out of the plane, and Warping.

Is it possible for you to help me solve the attached example?

[terryhendicott]

Finite-Element Formulation for the Lateral Torsional Buckling of Plane Frames
April 2013Journal of Engineering Mechanics 139(4):512-524
Liping Wu, Magdi Emile Mohareb
University of Ottawa

Abstract
A finite-element formulation is developed for the lateral torsional buckling analysis of plane frames with moment connections consisting of two pairs of welded plate stiffeners. The finite element provides a realistic representation of the partial warping restraint provided by the joint to the adjoining members. The element consists of two nodes with four generalized buckling degrees of freedom per node and is thus devised to interface with two classical beam buckling elements connected at right angles. The new finite element extends the functionality of the classical beam finite element to predict the lateral buckling load for noncollinear structures, such as portal frames.A comparison with results based on shell finite-element analysis demonstrates the ability of the new formulation to reliably predict the lateral buckling resistance of plane frames at a fraction of the computational and modeling cost of shell-based finite-element solutions.

[terryhendicott]

Hi,

Two references.

One has Ke and Kg of seven degree of freedom per node

Other has solution of Ke and Kg of four degree per node.

Cheers

Terry

[engr_mkhalid]

Thank you. Can you help me with the matrice formation? How do you develop a matrice depending on the boundary and loading condition?

[terryhendicott]

Hi,

This is not really the forum for individual lessons on the finite element method as I would have to prepare much material to aid you understanding.

I suggest looking on the internet for pdf of the following simple introduction.  It takes you progressively through how to assemble a global stiffness matrix from element stiffness matrices, how to apply boundary conditions, how to apply forces to the structural model.

A First Course in the Finite Element Method

Daryl L. Logan

University of Wisconsin–Platteville

[terryhendicott]

Hi Here is a full sample of how to assemble matrix, apply restraints, and apply loads.

Once you have spent some time with the book what is in the worksheet will make sense.

[terryhendicott]

Hi,

File last post is Prime 10.  Here is pdf of file.

[terryhendicott]

Hi,

Here is Prime 9 version.  I had to go to a different computer for this.