Steel · Members & beams
Beam lateral–torsional buckling (restraint options) calculator
Segment capacity with restraint arrangement, load height and rotational restraint factors and αm options, for library or custom I-sections.
Standard: AS 4100 Cl 5.6.1, 5.6.3 (kt, kl, kr, αm)
Inputs in the worked example
The live calculator starts with these values; change any of them and every result, check and diagram updates instantly.
| Input | Value | Unit |
|---|---|---|
| Section (library or custom) | 360UB50.7 | |
| Custom d | 310 | mm |
| Custom bf | 165 | mm |
| Custom tf | 10 | mm |
| Custom tw | 6 | mm |
| Steel grade (AS/NZS 3679.1) | Grade 300 | |
| Segment length | 6 | m |
| End restraints | FF / FP / PP (kt from twist) | |
| Load height factor kl | 1.0 shear-centre loading | |
| Rotational restraint kr | 1.0 none | |
| Moment modification αm | 1.13 UDL simply supported | |
| Design moment M* | 90 | kNm |
Worked example — results
| Result | Value | Unit |
|---|---|---|
| Depth d | 356.0 | mm |
| Flange width bf | 171.0 | mm |
| Flange thickness tf | 11.50 | mm |
| Web thickness tw | 7.30 | mm |
| Yield stress (governed by tf) | 300.0 | MPa |
| Clear web depth d1 | 333.0 | mm |
| Area (plates, no root radii) | 6,364 | mm² |
| Ix | 139,199,040 | mm⁴ |
| Iy | 9,594,533 | mm⁴ |
| Elastic modulus Zx | 782,017 | mm³ |
| Plastic modulus Sx | 879,832 | mm³ |
| rx | 147.9 | mm |
| ry | 38.83 | mm |
| Torsion constant J | 216,561 | mm⁴ |
| Warping constant Iw | 284,670,392,930 | mm⁶ |
| Flange slenderness λe | 7.80 | |
| Web slenderness λe (bending) | 49.97 | |
| Governing ratio λs/λsy | 0.487 | |
| Compact modulus min(S, 1.5Z) | 879,832 | mm³ |
| Effective modulus Ze (AS 4100 5.2) | 879,832 | mm³ |
| Form factor kf (web effective width, simplified) | 0.886 | |
| Twist restraint kt = 1 + (d1/l)(tf/2tw)³ | 1.03 | |
| le = kt kl kr l | 6.16 | m |
| Ms = fy Ze | 263.9 | kNm |
| Mo = √((π²EIy/le²)(GJ + π²EIw/le²)) | 126.6 | kNm |
| αs = 0.6(√((Ms/Mo)² + 3) − Ms/Mo) | 0.375 | |
| φMb = 0.9 min(αm αs Ms, Ms) | 100.7 | kNm |
- ✓ M* ≤ φMb
Method, formulas and assumptions
- Depth d = IF(4=0,310,CHOOSE(4,203,252,304,356,406,457,533,158,203,254,308)) = 356 mm
- Flange width bf = IF(4=0,165,CHOOSE(4,133,146,165,171,178,190,209,153,203,254,305)) = 171 mm
- Flange thickness tf = IF(4=0,10,CHOOSE(4,7.8,8.6,10.2,11.5,12.8,14.5,15.6,9.4,11.0,14.2,15.4)) = 11.5 mm
- Web thickness tw = IF(4=0,6,CHOOSE(4,5.8,6.1,6.1,7.3,7.8,9.1,10.2,6.6,7.3,8.6,9.9)) = 7.3 mm
- Yield stress (governed by tf) = IF(1=1,IF(11.5<=11,320,IF(11.5<=17,300,280)),IF(11.5<=11,360,IF(11.5<=17,340,330))) = 300 MPa
- Clear web depth d1 = 356-2*11.5 = 333 mm
- Area (plates, no root radii) = 2*171*11.5+333*7.3 = 6363.9 mm²
- Ix = (171*356^3-(171-7.3)*333^3)/12 = 1.392e+8 mm⁴
- Iy = 2*11.5*171^3/12+333*7.3^3/12 = 9.595e+6 mm⁴
- Elastic modulus Zx = 2*1.392e+8/356 = 7.820e+5 mm³
- Plastic modulus Sx = 171*11.5*(356-11.5)+7.3*333^2/4 = 8.798e+5 mm³
- rx = SQRT(1.392e+8/6363.9) = 147.9 mm
- ry = SQRT(9.595e+6/6363.9) = 38.828 mm
- Torsion constant J = (2*171*11.5^3+333*7.3^3)/3 = 2.166e+5 mm⁴
- Warping constant Iw = 9.595e+6*(356-11.5)^2/4 = 2.847e+11 mm⁶
- Flange slenderness λe = (171-7.3)/2/11.5*SQRT(300/250) = 7.7967
- Web slenderness λe (bending) = 333/7.3*SQRT(300/250) = 49.97
- Governing ratio λs/λsy = MAX(7.7967/16,49.97/115) = 0.48729
- Compact modulus min(S, 1.5Z) = MIN(8.798e+5,1.5*7.820e+5) = 8.798e+5 mm³
- Effective modulus Ze (AS 4100 5.2) = IF(AND(7.7967<=9,49.97<=82),8.798e+5,IF(0.48729<=1,IF(7.7967/16>=49.97/115,7.820e+5+(16-7.7967)/(16-9)*(8.798e+5-7.820e+5),7.820e+5+(115-49.97)/(115-82)*(8.798e+5-7.820e+5)),7.820e+5/0.48729)) = 8.798e+5 mm³
- Form factor kf (web effective width, simplified) = MIN(1,(6363.9-333*7.3+MIN(333,35*7.3/SQRT(300/250))*7.3)/6363.9) = 0.88556
- Twist restraint kt = 1 + (d1/l)(tf/2tw)³ = IF(1=1,1+(333/(6*1000))*(11.5/(2*7.3))^3,1) = 1.0271
- le = kt kl kr l = 1.0271*1*1*6 = 6.1627 m
- Ms = fy Ze = 300*8.798e+5/1E6 = 263.95 kNm
- Mo = √((π²EIy/le²)(GJ + π²EIw/le²)) = SQRT((PI()^2*200000*9.595e+6/(6.1627*1000)^2)*(80000*2.166e+5+PI()^2*200000*2.847e+11/(6.1627*1000)^2))/1E6 = 126.56 kNm
- αs = 0.6(√((Ms/Mo)² + 3) − Ms/Mo) = 0.6*(SQRT((263.95/126.56)^2+3)-263.95/126.56) = 0.37526
- φMb = 0.9 min(αm αs Ms, Ms) = 0.9*MIN(1.13*0.37526*263.95,263.95) = 100.73 kNm
- Section properties computed from catalogue d, bf, tf, tw as three plates (root radii ignored, so A and J are a few percent low). Use the published AS/NZS 3679.1 values for final design.
- kt uses the AS 4100 Table 5.6.3(1) twist-restraint expression with nw = 1.
Related steel calculators
- Steel beam (UB) — Section and member moment capacity (lateral-torsional buckling), shear and deflection, with built-in UB sizes.
- Steel angle lintel — Equal-angle lintel over a masonry opening with triangular (arching) or full masonry load.
- Steel I-beam in torsion with warping — UB or UC simply supported with fork ends (twist prevented, warping free) carrying an eccentric point load at…
- Tension member: net area & stagger — Gross yield and net-section fracture with staggered holes (s²t/4g) and the correction factor kt for…
- Web bearing: yield & buckling — Unstiffened web bearing yield (1.25 bbf tw fy) and bearing buckling with αc at le/r = 2.5 d1/tw for library…
- Welded plate girder (flexure & shear) — Section properties from plates, effective section modulus with flange/web slenderness, web shear buckling…
- Castellated beam (net tee checks) — Parent I-section expanded by the cut depth: global bending carried as tee axial forces, Vierendeel bending…
- Composite steel–concrete beam — Effective width, plastic neutral axis in the slab, positive moment capacity and stud count for full shear…
- Stair stringer — Inclined stringer carrying half the stair: horizontal-span moment, shear, and deflection along the incline.
- SHS / RHS beam: bending & shear — Section modulus and compactness of a rectangular hollow section in major-axis bending, web shear capacity…
For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.