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9. CALCULATION EXAMPLES OF LVL STRUCTURES
K_ser=〖ρ_k〗^1,5∙d^0,8/30=454 N/mm
γ=1/(1+(π^2∙E_0∙h_0 〖∙b〗_0∙s_0)/(K_ser∙〖L_0〗^2 ))=1/(1+(π^2∙7000 N/mm^2∙22 mm∙100 mm∙200 mm)/
(454N/mm∙(5000 mm)^2 ))
γ
= 0,27
(EI)_bracing=2∙2∙7000N/mm^2 ∙((100 mm∙(22 mm)^3/12+0,27∙100 mm∙2)
(EI)_bracing=6,44∙〖10〗^4 Nm^2/m
The stiffness of the 45x45 mm cross battens C14 solid wood underneath the floor:
(EI)_(cross battens)=7000 N/mm^2 ∙(1000 mm)/(400 mm)∙(45 mm ∙(45 mm)^2)/12=5,98∙〖10〗^3 Nm^2/m
〖(EI)〗_b= 〖(EI)〗_decking + 〖(EI)〗_bracings + 〖(EI)〗_(cross battens)
〖(EI)〗_b = 3,11∙〖10〗^3 Nm^2/m + 6,44∙〖10〗^4 Nm^2/m + 5,98∙〖10〗^3 Nm^2/m
〖(EI)〗_b = 7,35∙〖10〗^4 Nm^2/m
k_(δ,2)=∜((EI)_b/(EI)_l =) ∜((6,44∙〖10〗^4)/(1,79∙〖10〗^6 ))=0,45
w_2=min{█((1kN∙(4,5m)^2)/█(42∙0,45∙1,79∙(〖10〗^6 Nm^2)/m@ )@(1kN∙(4,5m)^3)/(48∙0,4∙1,79∙(〖10〗^6
Nm^2)/m))┤=min{█(0,6 mm@ @2,7 mm)┤=0,6 mm→OK
The transfer bracing lines and cross battens underneath the joists improve the floor stiffness under a 1kN
point load significantly and the vibration design requirements are fulfilled. The deflection could be decreased
~0,2 mm more, by gluing the decking onto the joists.
ser
=
k 1,5
∙
0,8
30 = 454 N/mm
=
1
1 +
2
∙
0
∙ ℎ
0
∙
0
∙
0
ser
∙
0 2
=
1
1 + π
2
∙ 7000 N/mm
2
∙ 22 mm ∙ 100 mm ∙ 200 mm
454N/mm ∙ (5000 mm)
2
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1000 mm 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪ ⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 70 0N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cros bat ens
= 7000 Nmm
2
∙ 1000 mm 400 mm ∙ 45 mm ∙ (45 m )
2
12
= 5,98 ∙ 10
3
N
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross bat ens
( )
b
= 3,11 ∙ 10
3
Nm
2
/ + 6,4 ∙ 10
4
N
2
/ + 5,98 ∙ 10
3
Nm
2
/
( )
b
7,35 ∙ 10
4
Nm
2
/m
,2
� ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪ ⎨ ⎪ ⎧ 1k ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 m → K
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 2 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1000 mm 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪⎪⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1000 mm 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3, 1 ∙ 10
3
N
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪⎪⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7 00N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1000 m 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
= 0,45
2
= min
⎩⎪ ⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ ,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ ,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7 00 Nmm
2
∙ 1000 mm 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
(
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪ ⎨ ⎪⎪⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
∙ �100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 100 mm ∙ 22 mm ∙ � 2402 + 22 2 �
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
N
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1 00 mm 400 mm ∙ 45 m ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
N
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
N
2
/m + 5,98 ∙ 10
3
N
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪⎪⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 m 2,7 mm = 0,6 mm → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 700 Nm
2
∙ �100 m ∙ (22 m)
3
12 + 0,27 ∙ 100 m ∙ 22 m ∙ � 2402 + 22 2
2
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 1000 m 400 m ∙ 45 mm ∙ (45 m)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪⎪⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
2 0,45 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 m 2,7 m = 0,6 m → OK
= 0,27
( )
bracing
= 2 ∙ 2 ∙ 7000N mm
2
�100 mm ∙ (22 mm)
3
12 + 0,27 ∙ 10 mm ∙ 22 mm ∙ � 2402 + 22 2
2
�
4,5 m
( )
bracing
= 6,44 ∙ 10
4
Nm
2
/m
( )
cross battens
= 7000 Nmm
2
∙ 100 mm 400 mm ∙ 45 mm ∙ (45 mm)
2
12
= 5,98 ∙ 10
3
Nm
2
/m
( )
b
= ( )
decking
+ ( )
bracings
+ ( )
cross battens
( )
b
= 3,11 ∙ 10
3
Nm
2
/m + 6,44 ∙ 10
4
Nm
2
/m + 5,98 ∙ 10
3
Nm
2
/m
( )
b
= 7,35 ∙ 10
4
Nm
2
/m
,2
= � ( )
b
( )
l
=
4
� 6,44 ∙ 10
4
1,79 ∙ 10
6
4
= 0,45
2
= min
⎩⎪⎪⎨ ⎪ ⎧ 1kN ∙ (4,5m)
2
42 ∙ 0,45 ∙ 1,79 ∙ 10
6
Nm
2
m
1kN ∙ (4,5m)
3
48 ∙ 0,4 ∙ 1,79 ∙ 10
6
Nm
2
m
= min � 0,6 mm 2,7 mm = 0,6 mm → OK
LVL Handbook Europe
183