Course · lesson 3 of 7

Beams: bending, shear and lateral-torsional buckling

Most steel members are beams. This lesson covers their bending resistance, why an unrestrained beam twists sideways before it reaches it, shear in the web and deflection, to SANS 10162-1, by hand and in the STATIX steel member calculator.

1. Learning outcomes

After this lesson you can:

explain elastic and plastic moment and classify a section for bending;
find M_r of a laterally restrained beam;
explain lateral-torsional buckling, find M_cr and the reduced M_r, and choose ω₂;
check shear in the web and deflection under service loads;
check a beam in the STATIX steel member calculator.

2. Before you start

Lesson 2 (buckling of struts) and the free STATIX steel member calculator.
Your own copy of SANS 10162-1; this lesson gives clause numbers only.
A pocket calculator and about 55 minutes.

3. What a beam must do

What a beam must do. It must be strong enough in bending (largest at mid-span) and in shear (largest at the supports), and stiff enough that its sag under service loads does not crack finishes. This lesson checks all three.

4. Bending resistance of a restrained beam (SANS 10162-1, 13.5)

Elastic and plastic moment. At first yield the stress is a triangle and the moment is S F_y. If the section is stocky enough, it can go on until the whole depth has yielded: a rectangle of stress and the plastic moment Z F_y, about 10 to 15% more for an I-section.

The two plates that decide the class in bending: the flange outstand b/2t and the web h/w. The limits shrink as F_y rises, so a section that is class 1 in S275 may be class 2 or 3 in S355. STATIX classifies every section for you.

The section class for bending (SANS 10162-1, clause 11, Table 2). Class 1 and 2 sections reach the plastic moment (class 1 can also rotate enough for plastic analysis). Class 3 reaches only first yield, so M_r uses S. Class 4 plates buckle locally before that. Shape only.

class 1 and 2: M_r = φ Z F_y (plastic) class 3: M_r = φ S F_y (elastic) φ = 0.9

5. Lateral-torsional buckling (SANS 10162-1, 13.6)

Lateral-torsional buckling (SANS 10162-1, 13.6). The compression flange of a beam is a strut. If nothing holds it, it buckles sideways, and because the tension flange resists, the whole section twists. A beam can fail this way well below its plastic moment.

M_cr = (ω₂ π / L) √(E I_y G J + (π E / L)² I_y C_w) E = 200 000, G = 77 000 MPa M_cr > 0.67 M_p: M_r = 1.15 φ M_p (1 − 0.28 M_p/M_cr) ≤ φ M_p M_cr ≤ 0.67 M_p: M_r = φ M_cr (class 3: M_y = S F_y in place of M_p)

L is the unbraced length of the compression flange. For an I-section C_w ≈ I_y (d − t_f)²/4. Sections with one axis of symmetry use a different M_cr (13.6), which STATIX works out; this lesson stays with doubly symmetric I-beams.

How the bending resistance falls as the unbraced length grows. Short or fully restrained: φM_p (13.5). Longer: M_r = 1.15 φ M_p (1 − 0.28 M_p/M_cr) ≤ φ M_p while M_cr > 0.67 M_p; beyond that, φ M_cr (13.6). M_cr grows with the section's weak-axis stiffness, torsion constant J and warping constant C_w, and falls fast with length. Shape of one rolled beam; dashed: φ M_cr.

ω₂: the shape of the moment matters. A uniform moment along the whole unbraced length is the worst case (ω₂ = 1.0). If the moment drops towards one end, fewer parts of the flange are highly loaded and M_cr goes up. The STATIX page takes ω₂ typed (1.0 to 2.5, default 1.0) or works it out from κ, the ratio of the end moments (positive for double curvature).

The κ formula is for a straight-line moment between restraints. Where the moment between restraints is curved (a uniform load), 1.0 is always safe; check your copy of the code before using more.

6. Restraint

Restraint must act on the compression flange. A concrete slab or decking fixed to the top flange of a simply supported beam restrains it fully (in STATIX: Flange fully restrained). Restraint on the tension flange, or at a cantilever's top where the bottom is in compression, does not stop lateral-torsional buckling.

7. Shear and deflection

Shear (SANS 10162-1, 13.4.1). The web carries the shear. For a rolled section the shear area is the full depth times the web thickness, and for a stocky web (h/w up to 1 014/√F_y) the shear strength is 0.66 F_y. Shear governs only short, heavily loaded beams.

Deflection. Under a uniform load the sag grows with the fourth power of the span, but the limit only grows in proportion to it (span/n). So long spans are governed by stiffness, not strength. Check with service (unfactored) loads; the STATIX page's default limit is span/250, and you type a stricter n where brittle finishes need it.

8. In STATIX

The beam route in the STATIX steel member calculator: enter the factored M_x and V; give the LTB length (empty = KL) or tick Flange fully restrained; type ω₂ or tick auto with κ; and fill in the service load and span/n for the deflection check under More inputs.