Beams in bending

How a reinforced concrete beam is designed for bending to EN 1992-1-1, with EN 1990 load combinations, on the Eurocode 2 route of the STATIX RC beam calculator: design strengths, the rectangular stress block, K and K′, the lever arm, cover and the bars.

1. Learning outcomes

After this lesson you can:

combine the loads on a beam with EN 1990 and find the design moment;
work out the design strengths of concrete and steel, and say which national choices the calculator makes;
find the nominal cover and the effective depth;
check K against K′ and design the tension steel;
check the bar spacing and the minimum and maximum steel, and find the moment a given section can carry.

2. Before you start

Statics: the moment and shear of a simply supported beam.
The RC beam calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 50 minutes.
The same lesson on the SA code: RC Design 1, lesson 1 (SANS 10100-1).

3. Loads and materials

The load on the beam. The characteristic dead load G_k (with the self-weight) and imposed load Q_k are combined with EN 1990 equation 6.10. The calculator uses the recommended partial factors 1.35 and 1.5; it does not offer the alternative pair 6.10a and 6.10b, which some national annexes choose.

Materials. A concrete class names two strengths: C30/37 has a characteristic cylinder strength f_ck of 30 MPa and a cube strength of 37 MPa; the Eurocode designs with f_ck. B500 reinforcement has f_yk = 500 MPa. The design strengths divide by the partial factors for materials.

The national choices. EN 1992-1-1 leaves some values to each country’s national annex. The calculator’s Eurocode 2 route uses the recommended values throughout, except α_cc: 0.85 for bending and axial load (the recommended value is 1.0; many annexes choose 0.85) and 1.0 for shear. Check what your country’s annex says; the SA annex is a separate route on the page.

4. The section at the ultimate limit state

Three stages. At small loads the whole section works. Then the concrete below the neutral axis cracks and the bars take all the tension. At the ultimate limit state the concrete in compression is close to crushing, and the Eurocode replaces its curved stress with a rectangle: ηf_cd over a depth λx (3.1.7(3)). Lesson 1 designs for stage 3.

Strain and stress at the ultimate limit state. The concrete crushes at a strain of 0.0035; the strain is a straight line through the neutral axis at depth x. The compression is a rectangle ηf_cd over 0.8x (for classes up to C50/60), and the bars carry A_sf_yd once they have yielded. The two forces are equal, and their lever arm z gives the moment.

Why K′ matters. A beam with a modest amount of steel fails slowly: the bars yield and the beam sags and cracks first. EN 1992-1-1 5.5(4) keeps the neutral axis depth x_u within 0.45d when the moments are not redistributed. With α_cc 0.85, that limit gives K′ = 0.168, the value the calculator uses.

The lever arm. The harder the section works, the deeper the compression block and the shorter the lever arm. The 1.134 comes from η, λ and α_cc 0.85 with γ_c 1.5; for small K the calculator caps z at 0.95d. With the SA annex’s α_cc 1.0 the number changes, which is why one route at a time matters.

5. Cover, depth and bars

The effective depth. d runs from the top face to the centre of the tension bars: the depth less the cover to the links, the link and half a bar. Before the bars are known, assume a size (Y20 is common) and check again once they are chosen.

Cover. The nominal cover is the minimum cover plus an allowance for deviation. The minimum is the largest of the bond requirement (the bar size), the durability requirement for the exposure class (Table 4.4N, structural class S4 on the page) and 10 mm. The calculator’s cover input is the cover to the links, so it takes the link as the bar for c_min,b.

Space between the bars. The concrete must flow between the bars and grip them, so the clear gap must be at least the bar size, the largest aggregate plus 5 mm, and 20 mm. The calculator lays the bars out in up to three layers and checks the gap.

Minimum and maximum steel. Too little steel and the beam breaks the moment it cracks; too much and the bars crowd out the concrete. f_ctm is the mean tensile strength of the concrete class (Table 3.1).