Column Buckling Calculation Report

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IntroductionThis 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
:310UC158​
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﻿﻿Compressive Load
:100kN​
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﻿﻿Column Length
:5.00m​
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﻿﻿Elastic Modulus 
:200,000MPa​
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Outputs
﻿﻿Effective Length Factor, K
:1​
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﻿﻿Slenderness Ratio, Rs (1)
:126.74271229404309​
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﻿﻿Transition Ratio, R_trans
:118.74104117237255​
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﻿﻿Transition Length, L_trans
:9.36866814850023m​
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﻿﻿Critical Stress, σcr
:122.88072002741073MPa​
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﻿﻿Critical Force, Pcr  
:2469.90247255096kN​
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﻿﻿Critical Length
:15.7159233662899m​
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﻿﻿Factor of Safety
:2.27863248146284​
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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 
:Fixed-Fixed​
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﻿﻿Compressive Load 
:1000.0kN​
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﻿﻿Cross Section 
:Rectangle​
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﻿﻿Height/Radius 
:0.1m​
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﻿﻿Breadth/Radius
:0.1m​
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﻿﻿Column Length, L 
:6.0m​
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﻿﻿Elastic Modulus
:200000.0MPa​
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﻿﻿Yield Strength 
:500.0MPa​
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Outputs 
﻿﻿Radius of Gyration, r
:0.029m​
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﻿﻿Effective Length Factor, K 
:0.5​
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﻿﻿Slenderness Ratio, Rs
:208​
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﻿﻿Transition Ratio, R_trans 
:178​
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﻿﻿Transition Length, L_trans 
:5m​
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﻿﻿Critical Stress, σcr 
:183MPa​
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﻿﻿Critical Force, Pcr
:1828kN​
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﻿﻿Critical Length, Lcr
:8.111557351947226​
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﻿﻿Factor of Safety 
:3​
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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
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
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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 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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