Understand Buckling Theory In Detail

Buckling analysis is the type of structural simulation wherein, the stability of a structure under compressive loading circumstances is assessed It will be necessary to use a weight lifting mechanism under compressive stresses to test the structure's stability. In buckling our main objective is to calculate the load factor or how much load it will require to collapse the beam for different mode shapes.

Buckling Load vs Bending Load

If the load is acting along the axis of the beam, it will be considered the buckling load.  If the load is acting perpendicular to the beam cross-section, it will be considered the bending load. Buckling is the reason for structural instability.

 

Buckling load formula:

 

E= Young’s modulus

I = Moment of inertia

L = length of the beam 

n = mode number

Buckling load factor:

The buckling load factor is the ratio of critical load(Pcr) and applied load. Mostly it should be more than 1, only then your structure is considered safe.

What is Mode or Mode Shape?

Mode shape describes the structure’s deformation. The first mode often defines the highest loads in a structure or how that structure will interact with the rest of the system.

Why Do We Need The Next Set Of Modes?

The calculated buckling load will be valid only if the beam is not attached to any other source but In reality, the beam will be attached with some other cross member so the beam can withstand the first mode’s buckling load. At that time, we can use the next set of modes

What Is The Need For Calculating The Buckling Load? 

While applying the buckling load to the beam, it loses its structural stiffness and due to this, it will not obey the actual stress-strain relation of the material. This allows a chance for permanent deflection before its yield point so buckling should be handled in a different way

How to Calculate the Buckling Load?

In FEA, there are only two methods to calculate the buckling load:

1. Linear buckling analysis or linear bifurcation analysis(LBA) or Eigenvalue buckling Analysis
2. Non-linear buckling analysis

Linear Buckling Analysis:

Here, the nonlinearity, p delta effect and imperfection of the geometry are neglected so the result that is obtained by this method has a 15 to 20% deviation from the actual buckling load. Due to its robustness and simplicity of this, it is used in various industries. 

• It will not predict the buckling direction, stress, displacement and final shape of the beam due to the load distribution in the simulation.
• As we already mentioned, it is also known as the eigenvalue buckling as it uses the eigenvalue method to calculate the buckling load factor. From that calculation, the eigenvalues are critical buckling load and eigenvectors are buckling load factor
• In this analysis, the permanent deflection is zero until it reaches the critical load. After that, we can not estimate the displacement of the structure, therefore it is drawn as a straight line in the below image,

 

 

Non-Linear Buckling analysis:

Here, all the instability factors such as non-linearity, pre-stress conditions, imperfections and p delta effects are considered. This will help predict the buckling load and show what happens after the buckling and also it can display the stress and displacement of this simulation at any time.

• In this type, the load will be distributed based on the load control method and will reduce the load step after the buckling happens to predict the effects on the beam completely.
• Sometimes, the load control method will not be sufficient to capture the effects of the buckling effects. At that time, the arc length control method will be used to predict the effects
• It will calculate the displacement even if it is not the critical load for capturing the buckling effect perfectly.

What is P delta?

While applying the load (P) on the beam, it will deflect the beam and that deflection will create extra momentum for the structure. The extra moment increases the instability of the structure.

Total moment = Actual moment induced in the beam + (P*deflection)