FEA (hand calc) - beam buckling

• super sky
In summary, it seems like you have a good understanding of the problem and have made a good attempt at solving it. However, there may be some errors in your calculations or inputs in MATLAB, and it may be helpful to use a numerical method to solve the problem.
super sky

Homework Equations

KQ = PKGQ
where K = stiffness matrix,
Q = displacement vector,
KG = geometric stiffness matrix

det(K-PKG)=0

The Attempt at a Solution

Elements 1 (A-B), 2 (B-C) and 3 (vertical) each have element stiffness matrices where K1=K2 and K3 = (L)'(K1)(L) where L is a 6x6 transformation matrix. When assembled, the global stiffness matrix is a 12x12

The geometric stiffness matrix can also be assembled to a 12x12

Boundary conditions (where each node (in order of A,B,C,D) has 3 DOF: displacement in x, displacement in y, rotation about z) are:
Q2 = 0
Q4 = Q5 = 0
Q8 = 0
Q10 = 0

So eliminating columns and rows 2,4,5,8,10 reduces the K-KG matrix to a 7x7, which is still too large to solve by hand in half an hour (which is the time per question in the exam this question's from). Also, I tried solving det(__)=0 for the 7x7 matrix using MATLAB and unless I've made mistake in inputting it, it was unable to find a solution whereas it found a few solutions for the 12x12, some of which were zero.

Am I missing something?

Last edited:

Thank you for your post and for sharing your attempt at solving this problem. It seems like you have a good understanding of the problem and have made a good attempt at solving it. However, I would recommend double checking your calculations and inputs in MATLAB to make sure there are no errors.

Additionally, it is possible that the 7x7 matrix is too large to solve by hand in the given time frame. In this case, you may want to try using a numerical method, such as the finite element method, to solve the problem. This would involve breaking the structure into smaller elements and using numerical techniques to solve for the displacements and buckling load.

I hope this helps and good luck with your exam!

1. What is FEA and how does it relate to beam buckling?

FEA stands for Finite Element Analysis. It is a computer simulation method used to analyze the structural behavior of a complex system, such as a beam undergoing buckling. It breaks down the system into smaller, simpler elements and solves equations to predict how the system will behave under different loads and conditions.

2. What is hand calc and why is it important in beam buckling analysis?

Hand calc, short for hand calculation, involves using mathematical equations and principles to analyze the behavior of a beam under buckling. It is important in beam buckling analysis because it provides a quick and simplified way to estimate the critical load at which the beam will buckle, which can then be compared to results from FEA to validate the simulation.

3. How does beam geometry and material properties affect buckling?

Beam geometry, such as length, cross-sectional shape, and support conditions, can affect the buckling behavior of a beam. Generally, longer and thinner beams are more prone to buckling than shorter and thicker beams. Material properties, such as stiffness and yield strength, also play a role in buckling. Materials with lower stiffness and yield strength are more susceptible to buckling.

4. What are the different types of beam buckling?

There are three main types of beam buckling: Euler buckling, which occurs when the beam is subjected to an axial compressive load; lateral-torsional buckling, which occurs when the beam is subjected to a combination of bending and twisting; and local buckling, which occurs when a section of the beam is subjected to a high compressive stress and buckles locally.

5. How can beam buckling be prevented or controlled?

Beam buckling can be prevented or controlled by using appropriate beam geometry and material selection, ensuring proper support conditions, and incorporating reinforcement techniques such as bracing or stiffeners. Additionally, conducting FEA and hand calc analyses can help identify potential buckling issues and allow for modifications to be made before the beam is constructed.

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