Design aids for determining deflection
Eurocode 2 has two alternative methods for checking deflection. Either:
 a limiting spantodepth ratio may be used or
 the theoretical deflection can be assessed using the expressions given in the Code
Here the spantoeffectivedepth (L/d) method is explained. For the ‘rigorous’ method see 'How to Design Concrete Structures using Eurocode 2: 8 Deflection calculations'.
The spantoeffectivedepth (L/d) method
In simple terms, the current EN1992 L/d method means verifying that:
Allowable L/d = N x K x F1 x F2 x F3 ≥ actual L/d


where
N =

basic spantoeffectivedepth ratio derived for K = 1.0 and p'. from the formulae :


if ρ_{0} ≤ ρ
N = L/d = K[11 + 1.5f_{ck}^{0.5} ρ_{ 0}/ ρ + 3.2 f_{ck}^{0.5}(ρ_{ 0}/ ρ – 1)^{1.5}]
or if ρ_{0} > ρ
N = L/d = K[11 + 1.5 f_{ck} 0.5 ρ_{0}/( ρ – ρ') + f_{ck}^{0.5}(ρ'/ ρ_{0})^{0.5}/12]

For ρ’ =0, N may be determined from Figure 1

where



L =

span

d =

effective depth

ρ_{0} =

f_{ck}^{0.5}/1000

ρ =

A_{s,req}/bd

ρ_{’} =

A_{s2}/bd

K =

factor to account for structural system. See Table 1

F1 =

factor to account for flanged sections.
When b_{eff}/b_{w} = 1.0, factor F1 = 1.0. When b_{eff}/b_{w} is greater than 3.0, factor F1 = 0.80. For values of b_{eff}/b_{w} between 1.0 and 3.0, interpolation may be used. .

F2 =

factor to account for brittle partitions in association with long spans. Generally
F2 = 1.0 but if brittle partitions are liable to be damaged by excessive deflection, F2 should be determined as follows:
a) in flat slabs in which the longer span is greater than 8.5 m, F2 = 8.5/l_{eff}
b) in beams and other slabs with spans in excess of 7.0 m, F2 = 7.0/l_{eff}

F3 =

factor to account for service stress in tensile reinforcement = 310/s_{s}^{1,3a}.
It is considered conservative to assume that 310/s_{s }= 500 A_{s,prov}/(f_{yk}A_{s,req})
where
s_{s }= tensile stress in reinforcement at midspan (at support for cantilevers) under design load at SLS.

Notes

 In MC2010 this expression is given in Expression (7.623) as 250/s_{s }= 500 A_{s,prov}/(f_{yk}A_{s,req})
 Factors F1, F2 and F3 have been used here for convenience, they are not symbols used in EN199211.
 According to the Notes to Table NA.5 of the UK NA,
 the ratio, F3, is restricted to ≤ 1.5 where s_{s} is calculated using the characteristic value of serviceability load.
 Warnings are given that the values of K may not be appropriate when formwork is struck at an early age.
 L/d may not exceed 40K.



Figure 7 Basic spantoeffectivedepth ratios, N, for K = 1, ρ' = 0





Note: Extract from 'How to Design Concrete Structures using Eurocode 2' (page 29, figure 7)
Table 1
K factors to be applied to basic ratios of span to effective depth for structural system

Element

K

Simply supported beams or slabs

1.0

End span of continuous beams or slabs

1.3

Interior spans of continuous beams or slabs

1.5

Flat slabs (based on longer span)

1.2

Cantilevers

0.4

The use of L/d method ‘will be adequate for avoiding deflection problems in normal circumstances’
For more information on rigorous methods see
Deflection.
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