Loading
/custom-emojis/emojis/contour-map.png
Templates
📚
Articles & Resources
📖
Guides & Support
🌵
CalcTree
Bust Common Myths About Java Programming
Loading
/custom-emojis/emojis/calculator.png
Tensile Strength and Capacity Control of the W-Shape Sections According to AISC 360-16
Estados de Vigas de Concreto
Loading
/custom-emojis/emojis/calculator.png
Concrete Cylinder Strength Vs Cube Strength
Loading
/custom-emojis/emojis/calculator.png
Earthquake Design Action Calculation
Sıvılaşma Verileri Tablosu
EM Wave Propagation Calculator
Loading
/custom-emojis/emojis/rc-beam.png
Concrete Column Designer to AS3600
section properties with units
Forward Kinematics of Robotic Arm with 6 Degrees of Freedom
İKSA YAPILARI PROJELENDİRME HİZMET BEDELİ (2024)
GEOTEKNİK RAPOR (EK-B) ASGARİ HİZMET BEDELİ (2024)
ZEMİN İYİLEŞTİRME/DERİN TEMEL PROJELENDİRME ASGARİ HİZMET BEDELİ (2024) (İMO)
🚀
Projectile motion
Loading
/custom-emojis/emojis/bending-moment.png
Dezi et. al (2010)
🤾
Projectile motion
Sitemap
Elastic Critical Moment Calculator for Lateral Torsional Buckling to EC3's banner
/custom-emojis/emojis/steel-beam.png

Elastic Critical Moment Calculator for Lateral Torsional Buckling to EC3

This calculator determines the elastic critical moment for lateral torsional buckling of steel sections.
All calculations are performed in accordance with:
  1. EN 1993-1-1:2005 Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings. This code is typically referred to as "EC3".
  2. NCCI SN003b-EN-EU: Elastic critical moment for lateral torsional buckling
EC3 does not include a formula for the elastic critical moment and hence the NCCI (Non-contradictory Complementary Information) is used here. The critical elastic moment (

) is an input into the buckling resistance moment (

) which is crucial for assessing the capacity of a section in bending.

Calculation

📃Technical notes

  1. The calculator is only valid for
    
    . The effective length factor,
    
    depends on the restraint conditions at supports. A value of
    
    assumes both flanges at the supports are free to rotate on plan.
  1. When either uniform or linear moment distribution is selected for "Applied load type", the calculator sets a default value for "Load application" while the "Support type" does not impact the calculation. The "Load application" defaults to top flange.
  1. When either parabolic or triangular moment distribution is selected for "Applied load type", users must define the "Support type" (for calculation of
    
    and
    
    ) and "Load application" (for calculation of
    
    ).
  1. The moment factor,
    
    is only applicable for linear moment distributions, that is bending moment diagrams that are either uniform or have a linear slope. For uniform linear bending moment diagrams, the calculator automatically sets
    
    . For linear bending moment diagrams with a slope,
    
    must be defined within the range of -1 to 1.
  1. The user can select from the following European & UK steel sections: UB (universal beams), UC (universal columns), IPE (parallel faced flange beams) or HE (wide/very wide flanged beams). The geometric properties of the cross-section are provided as per the standard: BS EN 10365:2017 - Hot rolled steel channels, I and H sections - Dimensions and masses.


Inputs

Section Properties



Section size
:HE 700 A​



L
:5.00m​



Applied Load Type



Applied load type
:UDL (parabolic BM)​

💡Note, the below inputs are applicable based on your selected "Applied load type". Otherwise, they will not affect the calculation.
Selected "Applied load type":
  1. UDL (parabolic BM) or Point load at midspan (triangular BM)
Applicable user input:


Support type
:Pinned/Pinned​



Load application
:Section centroid​

Selected "Applied load type":
  1. End moments (linear BM)
Applicable user input:


ψ
:-0.268​


Outputs



C1 factor
:1.127​



C2 factor
:0.454​



Mcr
:4,437kN m​



Explanation

What is lateral torsional buckling?

Lateral torsional buckling occurs in members in bending. It is where the beam bends in it's minor axis and twists, as this behavior is the least stiff bending failure.
Lateral torsional buckling is particularly relevant in beams that are slender, have a significant length between supports or are not laterally supported along their length. These conditions make the beam more susceptible to twisting and lateral displacement when subjected to bending moments.
This is what lateral torsional buckling looks like:



What is the elastic critical moment?

The critical elastic moment,

is the critical bending moment that leads to the onset of elastic lateral torsional buckling (LTB) in a beam. It is analogous to how the Euler buckling concept identifies the critical compression causing a column to succumb to elastic flexural buckling,


is a parameter in EC3 and serves as an input into the buckling resistance moment,

which is crucial for assessing the capacity of a section in bending.



in Eurocode 3

The following code snippets of EC3 show

as an input to the factor

in the design buckling resistance moment,




The engineer's role is to design a beam that has an adequate bending strength to resist applied moments without experiencing LTB. This involves selecting a beam size and positioning lateral supports such that the bending stresses are within acceptable limits.

NCCI Parameters and Equations

Below are the parameters and equations as provided in NCCI SN003b, that are used in this calculator.

Elastic critical moment (

equation)

The Elastic critical moment,

is determined by:

Mcr=C1π2EIz(kL)2[(kkw)2IwIz+(kL)2GItπ2EIz+(C2zg)2−C2zg]\large M_{cr} = C_1\dfrac{\pi^2EI_z}{(kL)^2}\left[ \sqrt{(\dfrac{k}{k_w})^2\dfrac{I_w}{I_z}+\dfrac{(kL)^2GI_t}{\pi^2EI_z}+(C_2z_g)^2}-C_2z_g \right]
Where:
  1. 
    
    = Youngs modulus
  2. 
    
    = shear modulus
  3. 
    
    = second moment of area about the minor axis
  4. 
    
    = torsion constant
  5. 
    
    = warping constant
  6. 
    
    = length of beam between points of lateral support
  7. 
    
    = effective length factor related to end rotation on plan
  8. 
    
    = effective length factor related to end warping, unless special provision for warping fixity is made take
    
    
  9. 
    
    = distance between the point of load application and the shear centre
In the typical scenario of a normal support conditions,

and

are 1. When the bending moment diagram is linear between lateral constraints,

. When the load is applied at the shear center,

. Under these conditions the equation for

simplifies to:

Mcr=C1π2EIzL2IwIz+L2GItπ2EIz\large M_{cr} = C_1\dfrac{\pi^2EI_z}{L^2}\sqrt{\dfrac{I_w}{I_z}+\dfrac{L^2GI_t}{\pi^2EI_z}}

Load application point (

parameter)

The load is generally applied to the top of the section. The parameter

is positive for loads acting towards the shear centre and negative for loads acting away from the shear centre as shown in the figure below.
Point of application of the load as per NCCI Section 2

A beam with load applied to the top of the section will have a lower

value then a beam with load applied to the bottom of the section, and hence a lower resistance to lateral torsional buckling.


Applied load type (

and

parameters)

The factors

and

are influenced by the bending moment distribution in the beam and it's support conditions. The types of bending moment distributions are listed below and shown in the image below.
  1. Uniform or linear bending moment, when there is applied end moments only
  2. Parabolic bending moment, when there is an applied uniformly distributed load (UDL)
  3. Triangular bending moment, when there is an applied concentrated load
Types of bending moment distributions in a beam (Source: eurocodeapplied.com)



and

values for the types of moment distributions are:
  1. Uniform or linear bending moment,
    
    is taken from Table 3.1 of NCCI (provided below) and
    
    
  2. Parabolic bending moment,
    
    and
    
    is taken from Table 3.2 of NCCI (provided below) for both pinned and fixed support conditions
  3. Triangular bending moment,
    
    and
    
    is taken from Table 3.2 of NCCI (provided below) for both pinned and fixed support conditions

Table 3.1 of NCCI

C1 values for beams with a linear moment distribution as per NCCI Section 3.2



Table 3.2 of NCCI

C1 and C2 values for beams with a parabolic or triangular moment distribution as per NCCI Section 3



Ratio of end moments (

parameter)

The parameter

is the ratio of applied end moments for beams with a linear bending moment diagram.

is the maximum end moment and therefore the range of

is:

−1≤ψ≤1-1 \leq \psi \leq 1
If

is equal to 1, the moment distribution is uniform.
Beam with a linear moment distribution has as per NCCI Section 3.2



Acknowledgements

This calculation was built in collaboration with Hakan Keskin. Learn more.

Related Resources

  1. Column Buckling Calculator
  2. Steel Section Designer to EC3
  3. Steel Beam and Column Designer to AS4100
  4. Steel Beam and Column Designer to AISC
Check out our full library of CalcTree templates here!

Check out our library of engineering tools here!