Checking for deflection is an important serviceability limit state (SLS) requirement in the design of structures. This check ensures that the structure does not deflect excessively in a manner that will impair the appearance, cause cracking to partitions and finishes, or affect the functionality or stability of the structure. In Eurocode 2, the deflection of a structure may be assessed using the span-to-effective depth ratio approach, which is the widely used method. It is also allowed to carry out rigorous calculations in order to determine the deflection of a reinforced concrete structure, which is then compared with a limiting value.

According to clause 7.4.1(4) of EN 1992-1-1:2004, the appearance of a structure (beam, slab, or cantilever) may be impaired when the calculated sag exceeds span/250 under quasi-permanent loads. However, span/500 is considered an appropriate limit for good performance.

Using the span-to-effective depth approach, the deflection of a structure must satisfy the requirement below;

Allowable *l/d* = *N × K × F1 × F2 × F3 *≥ Actual *l/d*

Where;*N* is the basic span-to-effective depth ratio which depends on the reinforcement ratio, characteristic strength of the concrete, and the type of structural system. The expressions for calculating the limiting value of *l/d* are found in exp(7.16) of EN 1992-1-1:2004. The expressions are given as follows;

*l/d* = *K*[11 + 1.5√f_{ck}(ρ_{0}/ρ) + 3.2√f_{ck}(ρ_{0}/ρ – 1)^{1.5}] if ρ ≤ ρ_{0}*l/d* = *K*[11 + 1.5√f_{ck}(ρ_{0}/(ρ – ρ’)) + 0.0833√f_{ck}(ρ_{0}/ρ)^{0.5}] if ρ > ρ_{0}

Where:*l/d* is the limit span/depth ratio*K* is the factor to take into account the different structural systems*ρ _{0}* is the reference reinforcement ratio = √f

_{ck}/1000

*ρ*is the required tension reinforcement ratio at midspan to resist the moment due to the design loads (at supports for cantilevers)

*ρ’*is the required compression reinforcement ratio at midspan to resist the moment due to the design loads (at supports for cantilevers)

*f*is the characteristic compressive strength of the concrete in N/mm

_{ck}^{2}

The values of *K* for different structural systems are given in Table 1;

**Table 1**: Values of K for different structural systems

Structural System | K |

Simply supported beam, one or two way spanning simply supported slab | 1.0 |

End span of continuous beam or one-way continuous slab or two-way spanning slab continuous over one long side | 1.3 |

Interior span of beam or one way or two-way spanning slab | 1.5 |

Slab supported on columns without beams (flat slab) | 1.2 |

Cantilever | 0.4 |

Some design aids are available for the evaluation of the limiting span/effective ratio. This is shown in Table 2 below (culled from *Goodchild, 2009*) and has been derived for K = 1.0 and *ρ’* = 0.

**Table 2**: Basic ratios of span-to-effective-depth for members without axial compression (Goodchild, 2009)

For the table above, *ρ* *= A _{s}/bd* (note that

*A*is the area of steel required and not the area of steel provided). For T beams,

_{s}*ρ*is the area of reinforcement divided by the area of concrete above the centroid of the tension reinforcement.

F1 = factor to account for flanged sections.

When b_{eff}/b_{w} = 1.0, F1 = 1.0

When b_{eff}/b_{w} is greater than 3.0, F1 = 0.8.

Intermediate values of b_{eff}/b_{w }can be interpolated between 1.0 and 3.0

F2 = factor to account for brittle partition in long spans.

In flat slab where the longer span is greater than 8.5m, F2 = 8.5/l_{eff}

In beams and slabs with span in excess of 7.0m, F2 = 7.0/l_{eff}

F3 = factor to account for service stress in tensile reinforcement = 310/σ_{s} ≤ 1.5

Conservatively, if a service stress of 310 MPa is assumed for the designed reinforcement A_{s,req}, then F3 = A_{s,prov}/A_{s,req} ≤ 1.5

More accurately, the serviceability stress in the reinforcement may be stimated as follows;

σ_{s} = σ_{su}[A_{s,req}/A_{s,prov}](1/δ)

Where;

σ_{su} is the unmodified SLS steel stress taking account γ_{M} for reinforcement and of going from ultimate actions to serviceability actions.

σ_{su} = f_{yk}/γ_{s}(G_{k} + ψ_{2}Q_{k})/(1.25G_{k} + 1.5Q_{k})

A_{s,req}/A_{s,prov} = Area of steel required divided by the area of steel provided

1/δ = factor to un-redistribute ULS moments

**References**

(1) EN 1992-1-1:2004 – Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings. European Committee For Standardization

(2) Goodchild C.H. (2009): Worked Examples to Eurocode 2: Volume 1 (For the design of in-situ concrete elements in framed buildings to BS EN 1992-1-1:2004 and its UK National Annex 2005). MPA – The concrete Centre**Cover image credit**: Sharooz et al (2014): Flexural Members with High-Strength Reinforcement: Behavior and Code Implications. *ASCE Journal of Bridge Engineering* Volume 19 Issue 5