Column Buckling Calculator

This calculator plots a Critical Buckling Force vs Length graph for a given steel section for the purpose of understanding the buckling phenomenon. Buckling occurs in compression members, such as columns.
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Illustration of column buckling 
Standards across the world address buckling in slightly different ways, but they are all based on the fundamental Euler buckling equation. For short columns, the Euler buckling equation does not represent real-life buckling accurately so modifiers are implemented, one example being the Johnson parabola. 
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Euler-Johnson graph for buckling behaviour
This calculator does not intend to replace Standards, but is to help understand the physics behind buckling.
CalculationThis calculation has two types:
For steel sections found in Australia's OneSteel 300 Plus® catalogue: One Steel - Hot Rolled and Structural Steel products  
For steel sections of any shape (rectangular, circular, pipe)
OneSteel 300 Plus® Steel Column Sections✏️ Note: If an output error occurs, update the designation selection 
Inputs
﻿﻿End Condition
:Pinned-Pinned​
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﻿﻿Steel Column Type
:UC​
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﻿﻿Designation
:150UC37.2​
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﻿﻿Es
:{"mathjs":"Unit","value":200000,"unit":"MPa","fixPrefix":false}​
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﻿﻿N*
:100 kN​
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﻿﻿L
:{"mathjs":"Unit","value":5,"unit":"m","fixPrefix":false}​
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Outputs
﻿﻿K
:1​
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﻿﻿Rs (1)
:129.87012987012986​
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﻿﻿R_trans
:114.71474419090949​
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﻿﻿ L_trans
:4.41651765135m​
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﻿﻿σcr
:117.0337689881175MPa​
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﻿﻿Pcr  
:553.569727313798kN​
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﻿﻿Lcr (1)
:3.7201133293012574m​
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﻿﻿Factor of Safety
:2.5633627165374624​
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Critical Buckling Force Graph
If your applied compressive load is below the line, then your column will PASS buckling checks. 
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Other Steel Column SectionsInputs 
﻿﻿End Condition 
:Pinned-Pinned​
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﻿﻿Compressive Load 
:{"mathjs":"Unit","value":1000,"unit":"kN","fixPrefix":false}​
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﻿﻿Cross Section 
:Rectangle​
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﻿﻿Height/Radius 
:{"mathjs":"Unit","value":0.1,"unit":"m","fixPrefix":false}​
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﻿﻿Breadth/Radius
:{"mathjs":"Unit","value":0.1,"unit":"m","fixPrefix":false}​
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﻿﻿ L 
:10.0 m​
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﻿﻿Elastic Modulus
:{"mathjs":"Unit","value":200000,"unit":"MPa","fixPrefix":false}​
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﻿﻿Yield Strength 
:{"mathjs":"Unit","value":500,"unit":"MPa","fixPrefix":false}​
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Outputs 
﻿﻿r
:0.029m​
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﻿﻿K 
:1​
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﻿﻿Rs
:346​
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﻿﻿ R_trans 
:89​
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﻿﻿L_trans 
:3m​
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﻿﻿ σcr 
:16MPa​
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﻿﻿Pcr
:164kN​
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﻿﻿Lcr
:4.055778675973584​
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﻿﻿Factor of Safety 
:30​
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Critical Buckling Force Graph
If your applied compressive load is below the line, then your column will PASS buckling checks. 
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ExplanationWhat is Buckling?Buckling occurs when slender elements suddenly collapse under compressive loads. Try holding a ruler at both ends and push your hands together. Notice how the ruler bends or snaps in the middle? That's what we call buckling. 
Buckling occurs due to a loss of stability which\ means a compression member will displace and continue to displace, that is, it is unstable. 
💡Learn more about stability
To understand why buckling occurs a good understanding of the three stages of equilibrium (or stability) is very helpful.
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Three stages of equilibrium
Stable Equilibrium 
Here, structures with minor perturbations will not produce large movements or displacements. Removing the disturbance will allow the structure to return to its original position. 
Picture this, you drop a marble from the side of a bowl and it rolls from side to side. Eventually, after a while the marble will settle at the bottom centre of the bowl. 
Neutral Equilibrium
The disturbances on the structure neither bring it back to its original position nor is it driven away from its position, resting where it is displaced.
Picture this, you are on a skateboard but lean over so slightly on one leg. You remain in that position but if you push against the ground you will move away from your previous position.
Unstable Equilibrium
Small perturbations on a structure will cause significant movement. Here, the structure will not return to its original position. Instability is generally caused by large deformations of the structure and elasticity of the structural materials.
Picture this, you fill a bowl upside down and drop a marble from the top. The marble will slide down the bowl but will never return to the top of the bowl.  
It is important to note that this failure mode is instantaneous, which is why it is very dangerous and can be seemingly deceptive.
What is slenderness?The classification of slender means the member will buckle before it yields, while on the other hand an intermediate (shorter) member will yield before it buckles. Buckling is an instantaneous failure mode, as oppose to yielding which is ductile, and so buckling is important to avoid.
To classify a member as slender, it's slenderness ratio is more than it's transition slenderness ratio.
The slenderness ratio is determined by: 

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Rs=Lrwhere:L = length of the columnr = radius of gyration of the cross-sectionR_s=\dfrac{L}{r}\\\text{where:}\\L\ \text{=\ length\ of\ the\ column}\\r\ \text{=\ radius\ of\ gyration\ of\ the\ cross-section}Rs​=rL​where:L = length of the columnr = radius of gyration of the cross-section
The transition slenderness ratio is determined by: 
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Rtrans=Ltransr=2π2EK2fywhere:Ltrans = transition lengthE= modulus of elasticityK=effective length factor (determined by the end condition)fy=yield strengthR_{trans}=\dfrac{L_{trans}}{r}=\sqrt{\dfrac{2\pi^2E}{K^2f_y}}\\\text{where:}\\L_{trans}\text{\ =\ transition\ length}\\E=\ \text{modulus\ of\ elasticity}\\K=\text{effective\ length\ factor\ (determined\ by\ the\ end\ condition)}\\f_y=\text{yield\ strength}Rtrans​=rLtrans​​=K2fy​2π2E​​where:Ltrans​ = transition lengthE= modulus of elasticityK=effective length factor (determined by the end condition)fy​=yield strength
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💡More on effective length factor, K
The effective length factor, K helps to account for how the the column is constrained or supported at either end. 
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Theoretical values for the effective length factor, K (image: MechaniCalc)
Standards around the world provide different recommendations for effective length factors. This calculation uses the theoretical values provided above.
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Euler-Johnson Buckling EquationThe Euler buckling equation is the solution to a second-order differential equation of a compression member. 
The Euler buckling equation is:
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σcr=PcrA=π2E(KLr)2where:σcr=critical stressPcr=critical forceA=cross-section areaE=elastic modulusK=effective length factorLr=slenderness ratio\sigma_{cr}=\dfrac{P_{cr}}{A}=\dfrac{\pi^2E}{\left(K\frac{L}{r}\right)^2}\\\text{where:}\\\sigma_{cr}=\text{critical\ stress}\\P_{cr}=\text{critical\ force}\\A=\text{cross-section\ area}\\E=\text{elastic\ modulus}\\K=\text{effective\ length\ factor}\\\dfrac{L}{r}=\text{slenderness\ ratio}\\σcr​=APcr​​=(KrL​)2π2E​where:σcr​=critical stressPcr​=critical forceA=cross-section areaE=elastic modulusK=effective length factorrL​=slenderness ratio
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The Euler critical stress approaches infinity as the column length approaches zero. Of course, this doesn't appropriately reflect the failure load that has been observed in practice. 
Therefore, the Euler buckling equation is modified by the Johnson parabola for shorter compression members. 
The Johnson parabola equation is:
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σcr=PcrA=fy−(fy2πKLr)2(1E)where:σcr=critical stressA=cross-section areafy=yield strength of materialK=effective length factorL=column lengthr=radius of gyrationE=elastic modulus\sigma_{cr}=\dfrac{P_{cr}}{A}=f_y-\left(\dfrac{f_y}{2\pi}\dfrac{KL}{r}\right)^2\left(\dfrac{1}{E}\right)\\\text{where:}\\\sigma_{cr}=\text{critical\ stress}\\A=\text{cross-section\ area}\\f_y=\text{yield\ strength\ of\ material}\\K=\text{effective\ length\ factor}\\L=\text{column\ length}\\r=\text{radius\ of\ gyration}\\E=\text{elastic\ modulus}σcr​=APcr​​=fy​−(2πfy​​rKL​)2(E1​)where:σcr​=critical stressA=cross-section areafy​=yield strength of materialK=effective length factorL=column lengthr=radius of gyrationE=elastic modulus
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Thus, the Critical Buckling Force for a slender member is calculated from the Euler buckling equation and for an intermediate member is calculated from the Johnson parabola. The combined Euler-Johnson equation has demonstrated a good correlation with column buckling failures in practice. 
This calculator plots this Euler-Johnson equation for a given steel section. It defines the following output parameters:
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Euler-Johnson equation
Euler-Johnson Graph
Critical Force, Pcr: The maximum compressive axial load your column of a given length can support until it buckles.
Critical Stress, σcr: The maximum compressive stress your column of a given length can support until it buckles.
Critical Length, Lcr: The maximum unsupported length of your column for a given compressive stress, before it buckles.
Transition Length, L_trans: The length that distinguishes a long column defined by buckling and an intermediate (short) column by squashing. This is the length where the Johnson and Euler lines intersect.
Factor of Safety, FoS: Ratio of material yield stress to critical stress. This is considered for the allowable stress design and must always be greater than or equal to 1. The required FoS will vary depending on the application of the column. 
Explore more on bucklingBuckling Examples in Practice
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(Image: Engineering Discoveries)
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(Image: Grzeskykowski & Szmigiera)
AutoFEM Buckling Analysis
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Forms of equilibrium states corresponding to the fifth and eighth critical loads of construction (Image: AutoFEM)
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