Beams: bending, shear and lateral-torsional buckling

How a steel beam is designed to EN 1993-1-1: the section class and the bending resistance, shear, lateral-torsional buckling and the restraint that prevents it, and deflection, by hand and on the Eurocode 3 route of the STATIX steel member calculator.

1. Learning outcomes

After this lesson you can:

classify an I-section in bending and find M_c,Rd with W_pl or W_el;
find the shear area and V_pl,Rd, and say when high shear reduces the bending resistance;
work out M_cr, λ̅_LT, χ_LT and M_b,Rd by 6.3.2.3 and 6.3.2.2;
place restraints where they work and judge the moment shape through C₁;
check a beam, including its deflection, in the STATIX steel member calculator.

2. Before you start

Lesson 2 (the buckling curves) and the moment and shear of a simply supported beam.
The free STATIX steel member calculator, set to Eurocode 3 (EN 1993-1-1).
A pocket calculator and about 55 minutes.
The same lesson on the SA code: Steel Design 1, lesson 3 (SANS 10162-1).

3. What a beam must do

What a beam must do. A beam must be strong enough in bending, where the moment is largest, and in shear, near the supports, and stiff enough not to sag too much under the service load. A floor beam is often decided by its deflection.

4. Bending resistance and the class

Elastic and plastic moment. When the outer fibres just reach f_y the section carries W_el f_y. If it can keep turning, the yield spreads through the depth until the whole section is at f_y: W_pl f_y, about 10 to 15% more for an I-section.

The section class (5.5, Table 5.2) says how far a section can bend before its flange or web buckles locally. Class 1 reaches M_pl and keeps rotating, class 2 reaches M_pl, class 3 only M_el, and class 4 buckles locally first. So M_c,Rd uses W_pl for class 1 and 2 and W_el for class 3 (6.2.5).

M_c,Rd = W_pl f_y / γ_M0 class 1 and 2 M_c,Rd = W_el,min f_y / γ_M0 class 3 (6.2.5) ε = √(235/f_y); outstand flange c/t and internal web c/t against the limits of Table 5.2

5. Shear

Shear (6.2.6). The web carries the shear; the shear area A_v of a rolled section adds a little of the flanges and the root radii. Where the shear is high and the moment is too (an interior support of a continuous beam, or a heavy point load), 6.2.8 reduces the bending resistance. The calculator applies 6.2.8 when V_Ed > 0.5 V_pl,Rd, taking the largest shear with the largest moment, on the safe side.

6. Lateral-torsional buckling

Lateral-torsional buckling. An I-beam bent about its strong axis is stiff in the plane of loading but slender sideways. If its compression flange is not held, it moves sideways and the section twists, at a moment that can be far below M_c,Rd.

What controls the elastic critical moment M_cr: the length between restraints, the weak-axis bending stiffness, the torsion and warping constants, and the shape of the moment diagram through C₁. EN 1993-1-1 does not give the formula; this is the common one for a doubly symmetric section loaded at its shear centre, which the calculator uses with E 210 000 and G 81 000 MPa.

The buckling resistance moment M_b,Rd = χ_LT W_y f_y/γ_M1 (6.3.2.1). Close restraints keep it at M_c,Rd; as the length between restraints grows it falls, fast at first. Restraints are often the cheapest way to strengthen a beam.

The moment shape. A uniform moment along the unrestrained length is the worst case: the whole compression flange is equally loaded. Any other shape raises M_cr, through C₁. The calculator takes C₁ as you type it (1.0, the safe value, by default) for end moments, with ψ for the factor f; or you choose the load in the segment (uniform or a central point load) and its height, and it works out M_cr and k_c itself.

From M_cr to M_b,Rd. The slenderness λ̅_LT compares the section’s moment resistance with M_cr; a curve gives χ_LT. For rolled I-sections 6.3.2.3 is less conservative than the general case of 6.3.2.2, and can be raised further by the factor f for the moment shape. The calculator uses 6.3.2.3 for rolled I and H sections, with the recommended values, unless you choose the general case 6.3.2.2.

λ̅_LT = √(W_y f_y / M_cr) 6.3.2.3: Φ_LT = 0.5 [1 + α_LT(λ̅_LT − 0.4) + 0.75 λ̅_LT²] χ_LT = 1 / [Φ_LT + √(Φ_LT² − 0.75 λ̅_LT²)] ≤ 1 and ≤ 1/λ̅_LT² χ_LT,mod = χ_LT / f, f from k_c (Table 6.6) M_b,Rd = χ_LT W_y f_y / γ_M1 (6.55)

Curves for 6.3.2.3 (Table 6.5): rolled I-sections with h/b ≤ 2 use curve b, deeper ones curve c. The general case 6.3.2.2 uses the curves of Table 6.4 with a plateau of 0.2 and no factor f.

7. Restraint

Restraint must hold the compression flange: in a simply supported beam under gravity load, the top flange. A slab properly fixed to it, or purlins and secondary beams with a positive connection, restrain it. A floor bearing on the bottom flange does not.

8. Deflection

δ = 5 w L⁴ / (384 E I_y) service (unfactored) uniform load, simply supported

EN 1993-1-1 7.2 leaves the deflection limits to the national annex and the client. The calculator checks the mid-span deflection of a simply supported span under the service load you type, against span/250 unless you choose another limit.

What the steel member calculator’s Eurocode 3 route uses for beams: γ_M0 = γ_M1 = 1.0 (recommended); Table 5.2 for the class; W_pl for class 1 and 2, W_el for class 3; A_v by 6.2.6(3) with η = 1.2 (the value EN 1993-1-5 recommends up to S460); 6.2.8 for high shear; M_cr with your C₁ (1.0 when blank) for end moments, or from the load in the segment and its height (shear centre, top or bottom flange), with E 210 000 and G 81 000 MPa; 6.3.2.3 for rolled I and H sections with λ̅_LT,0 0.4, β 0.75 and k_c from ψ or from the load (Table 6.6), or 6.3.2.2 by choice; the deflection with E 210 000 MPa.

9. In STATIX

The beam route in the STATIX steel member calculator: Eurocode 3, grade and section; the design moment and shear; the restraint (fully restrained, or the LTB length with C₁ and the method); and the unfactored service load for the deflection.