Beam serviceability — stresses, cracking, deflection

What this calculator checks

It covers all three serviceability checks of a reinforced concrete beam, on the same section and with the same transformed properties, to EN 1992-1-1 §7:

• §7.2 — stress limitation: \(\sigma_c \le 0.6 f_{ck}\) and \(\sigma_s \le 0.8 f_{yk}\) under the characteristic combination, \(\sigma_c \le 0.45 f_{ck}\) under the quasi-permanent one
• §7.3 — crack width \(w_k\), with the minimum reinforcement of §7.3.2
• §7.4.3 — deflection from curvature, interpolated with \(\zeta\) between the uncracked and cracked states

Creep and shrinkage are computed from Annex B, not entered as assumed values.

Why the quasi-permanent combination

Cracking and deflection are not judged on the ultimate limit state moment but on the quasi-permanent combination \(g + \psi_2 q\): that is the load which sits on the structure for years and drives creep. The characteristic moment \(M_k\) enters only the stress limitation, where the short-term peak is what matters.

Cracked and uncracked section

The section cracks when the stress at the tensile fibre exceeds \(f_{ctm}\) under the characteristic combination. From then on the tensile concrete no longer contributes, the neutral axis rises and the second moment of area drops — typically to a third or a quarter of the gross value.

The code does not ask you to pick one state, but to interpolate between them:

\(\zeta = 1 - \beta \left( \frac{M_{cr}}{M_{qp}} \right)^2\)

The physical reason is tension stiffening: between cracks the concrete still bonds to the bars and carries tension, so the real beam is stiffer than the fully cracked model.

Two different \(\alpha_e\) under one symbol

This is the easiest trap to miss in the whole of §7. The code uses the same notation for two different ratios:

With \(\varphi \approx 2\) the first is more than three times the second. Using the effective modulus in (7.9) inflates the term \((1 + \alpha_e \rho_{p,eff})\), reduces \(\varepsilon_{sm} - \varepsilon_{cm}\) and makes the crack width come out falsely small. The calculator keeps the two ratios apart and reports both.

Concrete stress peaks at first loading

Creep lowers the neutral axis and stiffens the transformed section, so \(\sigma_c\) decreases with time. On a typical example the concrete stress goes from 6.25 MPa right after loading to 3.61 MPa after creep — a factor of 1.7.

Checking §7.2 on the long-term state alone therefore reports about half the real concrete stress. The calculator evaluates both states and takes, for each quantity, the more severe value: the short-term state governs the concrete, the long-term one governs the steel.

Shrinkage bends the beam with no load at all

The tension reinforcement, sitting below the centroid, stops the concrete from shortening uniformly. The result is a curvature that adds to the one from the moment:

\(\left( \frac{1}{r} \right)_{cs} = \varepsilon_{cs} \, \alpha_e \, \frac{S}{I}\)

where \(S\) is the first moment of the reinforcement about the section centroid. The shrinkage curvature is computed separately for each state and only then interpolated with \(\zeta\) — it is not a correction applied at the end.

Input data

• Section — rectangular or tee, with the effective width \(b_{eff}\) already determined from §5.3.2.1
• Reinforcement — tension and compression bars, bar spacing and the cover to the bar surface (not to its centre: the \(c\) in \(s_{r,max}\) is to the surface)
• Moments — \(M_k\) and \(M_{qp}\), as effects rather than loads
• Span and static scheme — these give the factor \(k\) and the deflection limit (\(L/250\) for spans, \(L/125\) for cantilevers)
• Environment — relative humidity, age at loading, age at the start of drying, cement class

Assumptions and limits

• Pure bending: axial force and prestressing are not covered
• The flange is taken as compressed (sagging moment). Over a support, where the flange is in tension, enter the section as a rectangle of width \(b_w\)
• The deflection is the total long-term value under the quasi-permanent combination; any precamber is not deducted
• Imposed deformations (settlement, temperature) are not included

Related calculations

• RC section — ultimate limit state, \(M_{Rd}\) and \(V_{Rd}\)
• Creep and shrinkage — the coefficients \(\varphi\) and \(\varepsilon_{cs}\) in detail
• Slab crack width — the same §7.3 relation, applied to surface elements
• Load combinations — where \(M_k\) and \(M_{qp}\) come from