Confirm the narrow model conditions
Use only a straight prismatic doubly symmetric I beam with uniform bending moment. The moment-gradient factor is fixed at C1 = 1.
BEAMS & SECTIONS / TOOL 15
Ideal Beam Elastic Critical Moment Calculator
Use this to examine the classical elastic lateral–torsional critical moment of an ideal, doubly symmetric I beam under uniform pure major-axis bending.
Before you start: required inputs
- Confirm that the model has fork supports at both ends, free warping and no intermediate lateral or torsional restraint.
- Obtain E, G, weak-axis inertia, Saint-Venant torsion constant J and warping constant Iw for the actual section. These are not all provided by the basic section-properties tool.
Calculation diagram
A PRACTICAL WALKTHROUGH
How to use this calculator
Supply the correct section constants
Use the weak-axis second moment, Saint-Venant J and warping Iw in the displayed units. J is not the polar second moment of area.
Interpret Mcr as an elastic model result
Review how unbraced length changes the ideal bifurcation moment. Do not treat Mcr as an allowable working moment or a code design resistance.
Ideal I beam with a 6 m unbraced length
THE INPUTS
E = 200,000 MPa; G = 77,000 MPa; weak-axis Iz = 8,000,000 mm⁴; J = 150,000 mm⁴; Iw = 1,000,000,000,000 mm⁶; L = 6 m.
THE RESULT
The ideal elastic critical moment Mcr is approximately 170.640 kN·m for the stated model with C1 = 1. This is not a rated or permissible working moment.
What does the result mean?
01
Mcr is the elastic bifurcation value for the stated support and loading assumptions.
02
The unbraced length is the segment with no intermediate restraint; it need not be the complete fabricated member length.
03
Yielding, residual stress, initial imperfections, local buckling and load-height effects are excluded. A design-resistance calculation needs an appropriate design method beyond this result.
Watch for this
Substituting the polar second moment for Saint-Venant J, or applying this uniform-moment equation to an arbitrary crane girder with unspecified restraints and load height.
Formula & model conditions +
Mcr = (π / L) × √[E × Iz × G × J + π² × E² × Iz × Iw / L²]
Use N and mm consistently; divide N·mm by 1,000,000 for kN·m.
- An ideal straight, prismatic, doubly symmetric I beam is in uniform pure major-axis bending; the moment-gradient factor is exactly C1 = 1.
- Both ends provide fork supports: lateral displacement and twist restrained, warping free. No intermediate lateral or torsional restraint is present.
- The model assumes linear elasticity and supplied section properties. Residual stress, initial imperfections, yielding, local buckling and load-height effects are excluded.
- Mcr is an elastic bifurcation value, not a permissible working moment or a code design resistance. No safety factor or pass/fail criterion is applied.
Common questions
Can I compare Mcr directly with a working moment and declare the beam safe?
No. Mcr is not code resistance or an allowable moment. Imperfections, yielding, relevant factors and other failure modes need the appropriate separate design checks.
Can I change the support or moment-gradient factor?
Not in this tool. It fixes C1 = 1 and ideal end fork supports with free warping. Other restraint and loading conditions need a different model.