EUROCODE 2 WORKED EXAMPLES - Federbeton

EC2 – worked examples 7-10
EXAMPLE 7.4. Design formulas derivation for the cracking limit state
[EC2 clause 7.4]
8B7.4.1 Exact method
It is interesting to develop the (7.4) to use it as a design formula. In particular, stated b, h, d,
d’, b, and fixed M, we want to deduce the metal reinforcement amount As and its design
tension σs in order to have a crack amplitude wk lower than the fixed value w k . The
adimensional calculus leads to
1
2
Table of Content
( ) ( )
2
− ξ −αe⋅ρ s 1+β ξ+αe⋅ρs δ+β⋅δ ' = 0
(7.5)
αν e ( δ−ξ)
fctmkt ( ) ( )
σ s =
2 2 3
2⎡3n⋅ρ⎡ s δ−ξ + β δ−ξ ' ⎤+
ξ ⎤
⎣ ⎣ ⎦ ⎦
setting
M M
ν= = 0
Mcr
bh
kf t ctm
6
(7.6)
2
⋅ (7.7)
Deducing ρ s from (7.5) and with its substitution in the (7.6) we get
ξ
2 1 '
ρ s =
αe− 2
+β ξ+δ+βδ
⎡⎣ ( ) ⎤⎦
(7.8)
3
αν e ( δ−ξ) −2ξp 2
3ξp 2 2
( δ−ξ ) +β( δ'−ξ) ' ( 1 )
= δ +βδ − +β ξ
with p=σ s/(k tf ctm) (7.10)
From (7.4), where w = wk
, after some calculations we deduce
0
wk
λ
p= + + n
φ⋅λ
3.4 ⋅ c + 0.17
ρs
ρ
setting
w
Ew
s
(7.9)
(7.11)
s k
0k =
kf
(7.12)
t ctm
Combining (7.8) and (7.11), the (7.9) is
0 2
( )
wk
⋅ξ
2 2
( ) ( ' ) 3.4 c 0.34 e ' ( 1 )
αν e δ−ξ
⎡
⎣
δ−ξ +β δ−ξ
⎡
= ⎢
⎤
⎦
⎢⎣ 2
⋅ ⋅ξ + α ⋅φ⋅λ⎡⎣δ+βδ −
2α⋅λ
⎤
e + ⎡δ+βδ− ' 2 ⎣ ( 1+β)
ξ ⎤⎦+α
e⎥
×
+β ξ⎤⎦ ξ
⎥⎦
⎡ 2 3 ⎤
3ξ 2ξ
× ⎢ +
⎥
2 2
⎢⎡⎣δ+βδ'− ( 1+β) ξ⎤⎦ ⎡( δ−ξ ) +β( δ−ξ ' ) ⎤⎥
⎣ ⎣ ⎦⎦