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9. CALCULATION EXAMPLES OF LVL STRUCTURES
Loading combinations
Snow load at roof level
q
k
=
μ
1
∙
C
e
∙
s
k
. Form factor
μ
1
= 0,8, when roof angle is less than 30° and in normal
conditions
C
e
= 1,0 →
q
k
= 0,8 ∙ 1,0∙2,5 N/m
2
= 2,0 kN/m
2
.
Accidental load combination of fire in the ultimate limit state (ULS):
E
d,ULS,fi
=
γ
G
∙ (
g
1,k
+
g
2,k
) +
ψ
1
∙
γ
Q
∙
q
k
E
d,ULS,fi
=1,0 ∙ (8m ∙ 1,0 kN/m
2
+ 0,2 kN/m ) + 0,4 ∙ 1,0 ∙ 8m ∙ 2,0 kN/m
2
E
d,ULS,fi
= 14,6 kN/m
Note: Safety factors γ
G
, ψ
1
and γ
Q
are according to Finnish National annex of Eurocode 0.
ULS design
Bending moment resistance
M_d = E_(d,ULS,fi)∙s∙L2/8 = 14,6 kN/m∙(4m)^2/8 = 29,2 kNm
σ_(m,d)=M_d/W=(29,2 kNm)/(1,52〖∙10〗^6 〖 mm〗^3 )=19,2 N/mm^2
f_(m,d,fi)=(k_(mod,fi) 〖∙k〗_fi∙k_h)/γ_(M,fi) ∙f_(m,edge,k)
f_(m,d,fi)=(1,0∙1,1∙(300mm/344mm)^0,15)/1,0∙44 N/mm^2 =47,4 N/mm^2
σ_(m,d)≤f_(m,d,fi) →OK
Lateral torsional buckling
The beam is loaded on the top side and the purlins won’t act as supports against lateral torsional buckling
for 30min fire exposure. Therefore according to Table 4.9 and EN1995-1-2, clause 4.3.2 (1) and the effec-
tive length Lef of the beam is
L_ef=0,9∙L + 2∙h = 0,9 ∙ 4000mm + 2 ∙ 344mm = 4288mm.
σ_(m,crit)=M_(y,crit)/W_y =(π√(E_0,05 I_z G_0,05 I_tor ))/(l_ef W_y )
(4.42)
σ_(m,crit)= (π√(10600 N/mm^2∙1,31∙〖10〗^7 〖 mm〗^4∙400N/mm^
〖1,52∙10〗^6 〖 mm〗^3 )
σ_(m,crit)= 25,8 N/mm^2
λ_rel=√(f_(m,k)/σ_(m,crit) )=√((44 N/mm^2)/(25,8 N/mm^2 ))= 1,36
(4.41)
when 0,75<λ_(rel,m)≤1,4 ,k_crit=1,56-0,75∙λ_(rel,m)=1,56-0,75∙1,36=0,58
k_crit∙ f_(m,d,fi)=0,58 ∙47,4 N/mm^2=27,5 N/mm^2
σ_(m,d)=19,2 N/mm^2≤k_crit∙ f_(m,d)→OK
(4.38)
d
=
d,ULS,fi
∙ ∙ 2/8 = 14,6 kN/m ∙ (4m)
2
/8 = 29,2 kNm
m,d
=
d
= 29,2 kNm 1,52 ∙ 10
6
mm
3
= 19,2 N/mm
2
m,d,fi
=
mod,fi
∙
fi
∙
h
M,fi
∙
m,edge,k
m,d,fi
= 1,0 ∙ 1,1 ∙ � 300mm 344mm �
0,15
1,0
∙ 44 Nmm
2
= 47,4 Nmm
2
m,d
≤
m,d,fi
→ OK
ef
= 0,9 ∙ + 2 ∙ ℎ = 0,9 ∙ 4000mm + 2 ∙ 344mm = 4288mm.
m,crit
=
y,crit y
=
�
0,05 0,05 tor
ef y
(4.42)
m,crit
= π�10600 N/mm
2
∙ 1,31 ∙ 10
7
mm
4
∙ 400N/mm
2
∙ 4,71 ∙ 10
7
∙ mm
4
4288 mm ∙ 1,52 ∙ 10
6
mm
3
m,crit
= 25,8 N/mm
2
d
=
d,ULS,fi
∙ ∙ 2/8 = 14,6 kN/m ∙ (4m)
2
/8 = 29,2 kNm
m,d
=
d
= 29,2 kNm 1,52 ∙ 10
6
mm
3
= 19,2 N/mm
2
m,d,fi
=
mod,fi
∙
fi
∙
h
M,fi
∙
m,edge,k
m,d,fi
= 1,0 ∙ 1,1 ∙ � 300 344mm �
0,15
1,0
∙ 44 Nmm
2
= 47,4 Nmm
2
m,d
≤
m,d,fi
→ OK
ef
= 0,9 ∙ + 2 ∙ ℎ = 0,9 ∙ 4000mm + 2 ∙ 344mm = 4288mm.
m,crit
=
y,crit y
=
�
0,05 0,05 tor
ef y
(4.42)
m,crit
= π�10600 N/mm
2
∙ 1,31 ∙ 10
7
mm
4
∙ 400N/m
2
∙ 4,71 ∙ 0
7
∙ mm
4
4288 mm ∙ 1,52 ∙ 10
6
mm
3
m,crit
= 25,8 N/mm
2
rel
= �
m,k m,crit
= �
44 N/mm
2
25,8 N/mm
2
= 1,36
(4.41)
when 0,75 <
rel,m
≤ 1,4 ,
crit
= 1,56 − 0,75 ∙
rel,m
= 1,56 − 0,75 ∙ 1,36 = 0,58
crit
∙
m,d,fi
= 0,58 ∙ 47,4 N/mm
2
= 27,5 N/mm
2
m,d
= 19,2 N/mm
2
≤
crit
∙
m,d
→ OK
(4.38)
d
=
d,ULS,fi
∙ /2 = 14,6 kN/m ∙ 4,0m/2 = 29,2 kN
v,d
= 3 ∙
d
2 ∙
= 3 ∙ 29,2 kN
2 ∙ 26488 mm
2
= 1,7 N/mm
2
v,d,fi
=
mod,fi
∙
fi
M,fi
∙
v,0,edge,k
= 1,0 ∙ 1,1 1,0 ∙ 4,2 Nmm
2
= 4,6 N/mm
2
m,d
≤
v,d,fi
→ OK
212
LVL Handbook Europe