LVL Handbook EUROPE

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when 1,4<λ_(rel,m) ,k_crit=1/〖λ_(rel,m)〗^2 ∙=1/〖1,45〗^2 =0,4
8 (4.40)
k_crit∙ f_(m,y,d)=0,48 ∙30,3 N/mm^2=14,4 N/mm^2
σ_(m,y,d)≤k_crit∙ f_(m,y,d)→OK (4.38)
Axial compression
N_(c,d)/A= 4,5kN/(10800 mm^2 )=0,42 N/mm^2
f_(c,0,d)=k_mod/γ_M ∙f_(c,0,SC2,k)=0,8/1,2∙29 N/mm^2 =19,3 N/mm^2
Buckling, buckling length lc = 2000 mm in z-direction and 4000 mm in y-direction
λ_z=√12 (l_c/b)=3,46∙(2000 mm)/(45 mm)=154 (4.37)
λ_y=√12 (l_c/h)=3,46∙(4000 mm)/(240 mm)=58
λ_(rel,z)=λ_z/π √(f_(c,0,k)/E_0,05 )=154/3,14∙√((29 N/mm^2)/(1
(4.36)
λ_(rel,y)=λ_y/π √(f_(c,0,k)/E_0,05 )=58/3,14∙√((29 N/mm^2)/(11
k_z=0,5(1+β_c (λ_(rel,z)-0,3)+λ_(rel,z)^2 )
(4.34)
k_z=0,5∙(1+0,1∙(2,45-0,3)+(2,45)^2=3,61
k_y=0,5(1+β_c (λ_(rel,y)-0,3)+λ_(rel,y)^2 )
k_y=0,5∙(1+0,1∙(0,92-0,3)+(0,92)^2=0,95
k_(c,z)=1/(k_z+√(k_z^2-λ_(rel,z)^2 ))=1/(3,61+√(〖3,61〗^2-〖2,4
(4.32)
k_(c,y)=1/(k_y+√(k_y^2-λ_(rel,y)^2 ))=1/(0,95+√(〖0,95〗^2-〖0,
The following expressions shall be satisfied with km = 0,7 for rectangular cross sections:
σ_(c,0,d)/〖k_(c,z)∙f〗_(c,0,d) +k_m∙σ_(m,y,d)/f_(m,y,d) +σ_(m,z,d)/f_(m,z,d) ≤1 (4.29)
(0,42 N/mm^2 )/(0,16∙19,3 N/mm^2 )+0,7∙(13,1 N/mm^2 )/(30,3 N/mm^2 )+(4,7 N/
)=0,14+0,30+0,15=0,59→OK
σ_(c,0,d)/〖k_(c,y)∙f〗_(c,0,d) +σ_(m,y,d)/f_(m,y,d) +k_m∙σ_(m,z,d)/f_(m,z,d) ≤1
(4.30)
(0,42 N/mm^2 )/(0,83∙19,3 N/mm^2 )+(13,1 N/mm^2 )/(30,3 N/mm^2 )+0,7∙(4,7 N
)=0,03+0,43+0,10=0,56→OK
(σ_(m,y,d)/(k_crit 〖∙f〗_(m,o,edge) ))^2+σ_(c,0,d)/(k_(c,z)∙f_(c,0,d) )≤1
9. CALCULATION EXAMPLES OF LVL STRUCTURES m,y,crit 2
rel = � h∙ m,k m,y,crit= �1,03∙44 N/mm2
21,6 N/mm2 = 1,45 (4.41)
when 1,4 < rel,m , crit = 1
rel,m 2 ∙= 1 1,452 = 0,48 (4.40)
crit ∙ m,y,d = 0,48 ∙ 30,3 N/mm 2 = 14,4 N/mm2
m,y,d ≤ crit ∙ m,y,d → OK (4.38)
c,d = 4,5kN 10800 mm2 = 0,42 N/mm2
c,0,d = mod M∙ c,0,SC2,k = 0,8 1,2 ∙ 29 Nmm2 = 19,3 N/mm2
z = 12 � c� = 3,46 ∙ 2000 mm 45 mm = 154 (4.37)
y = √12 � c ℎ � = 3,46 ∙ 4000 mm 240 mm = 58
rel,z = z � c,0,k 0,05= 154 3,14∙ � 29 N/mm2
11600 N/mm2 = 2,45 (4.36)
rel,y = y � c,0,k 0,05= 58 3,14 ∙ � 29 N/mm2
11600 N/mm 2 = 0,92
z = 0,5�1 + c� rel,z − 0,3� + rel,z 2 � (4.34)
m,y,crit = 21,6N/mm2
rel = � h∙ m,k m,y,crit= �1,03∙44 N/ m2
21,6 N/mm2 = 1,45 (4.41)
when 1,4 < rel,m , crit = 1
rel,m 2 ∙= 1 1,452 = 0,48 (4.40)
crit ∙ m,y,d = 0,48 ∙ 30,3 N/mm 2 = 14,4 N/mm2
m,y,d ≤ crit ∙ m,y,d → OK (4.38)
c,d = 4,5kN 10800 mm2 = 0,42 N/mm2
c,0,d = mod M∙ c,0,SC2,k = 0,8 1,2 ∙ 29 Nmm2 = 19,3 N/mm2
z = √12 � c� = 3,46 ∙ 2000 mm 45 mm = 154 (4.37)
y = √12 � c ℎ � = 3,46 ∙ 4000 mm 240 mm = 58
rel,z = z � c, ,k 0,05= 15 3,14∙ � 29 N/mm2
11600 N/mm2 = 2,45 (4.36)
rel,y = y � c,0,k 0,05= 58 3,14 ∙ � 29 N/mm2
11600 N/mm 2 = 0,92
z = 0,5�1 + c� rel,z − 0,3� + rel,z 2 � (4.34)
, , it = 21,6N/mm
l ∙ ,
, , it , ∙
,6 , .
, l, , it l, ∙ , , .
it ∙ , , = 0,48 ∙ , ,
, , it ∙ , , .
, = 4,5kN 0 = 0,42 N/mm
, , d ∙ , , , , , ∙ ,
12 � � = 3,46 ∙ 200 = 154 .
� , ∙ = 58
l, = � , , ,= , ∙ � 29 N/mm = 2,45 .
l, = � , , , 58 ,∙ � 29 N/mm 2 ,
, l, , l, 2 . z = , ∙ (1 + 0,1 ∙ (2,45 − 0,3) + (2,45)2 = 3,61
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 �
y = 0,5 ∙ (1 + 0,1 ∙ (0,92 − 0,3) + (0,92)2 = 0,95
c,z = 1
z+� z2− rel,z 2 = 1
3,61+�3,612−2,452 = 0,16 (4.32)
c,y = 1y + � y2− rel,y 2 = 1
0,95 + �0,952 − 0,922 = 0,83
c,0,d c,z
∙ c,0,d + m ∙ m,y,d m,y,d+ m,z,d m,z,d≤ 1 (4.29)
0,42 Nmm2
0,16 ∙ 19,3 Nmm2 + 0,7 ∙ 13,1 Nmm2
30,3 Nmm2 + 4,7 Nmm2
32,0 Nmm2 = 0,14 + 0,30 + 0,15 = 0,59 → OK
c,0,d c,y∙ c,0,d + m,y,d m,y,d+ m ∙ m,z,d m,z,d≤ 1 (4.30)
0,42 Nmm2
0,83 ∙ 19,3 Nmm 2 + 13,1 Nmm 2
30,3 Nmm2 + 0,7 ∙ 4,7 Nmm2
32,0 Nmm2 = 0,03 + 0,43 + 0,10 = 0,56 → OK
� m,y,d
crit∙ m,o,edge�2 + c,0,d c,z∙ c,0,d ≤ 1 (4.39)
� 13,1 Nmm2 �2 + 0,42 Nmm2 = 0,82 + 0,14 = 0,96 → OK
z = 0,5 ∙ (1 + 0,1 ∙ (2,45 − 0,3) + (2,45)2 = 3,61
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 �
y = 0,5 ∙ (1 + 0,1 ∙ (0,92 − 0,3) + (0,92)2 = 0,95
c,z = 1
z+� z2− rel,z 2 = 1
3,61+�3,612−2,452 = 0,16 (4.32)
c,y = 1y + � y2− rel,y 2 = 1
0,95 + �0,952 − 0,922 = 0,83
c,0,d c,z
∙ c,0,d + m ∙ m,y,d m,y,d+ m,z,d m,z,d≤ 1 (4.29)
0,42 Nmm2
0,16 ∙ 19,3 Nmm2 + 0,7 ∙ 13,1 Nmm2
30,3 Nmm2 + 4,7 Nmm2
32,0 Nmm2 = 0,14 + 0,30 + 0,15 = 0,59 → OK
c,0,d c,y∙ c,0,d + m,y,d m,y,d+ m ∙ m,z,d m,z,d≤ 1 (4.30)
0,42 Nmm2
0,83 ∙ 19,3 Nmm 2 + 13,1 Nmm 2
30,3 Nmm2 + 0,7 ∙ 4,7 Nmm2
32,0 Nmm2 = 0,03 + 0,43 + 0,10 = 0,56 → OK
� m,y,d
crit∙ m,o,edge�2 + c,0,d c,z∙ c,0,d ≤ 1 (4.39)
z = 0,5 ∙ (1 + 0,1 ∙ (2,45 − 0,3) + (2,45)2 = 3,61
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 �
y = 0,5 ∙ (1 + 0,1 ∙ (0,92 − 0,3) + (0,92)2 = 0,95
c,z = 1
z+� z2− rel,z 2 = 1
3,61+�3,612−2,452 = 0,16 (4.32)
c,y = 1y + � y2− rel,y 2 = 1
0,95 + �0,952 − 0,922 = 0,83
c,0,d c,z∙ c,0,d + m ∙ m,y,d m,y,d+ m,z,d m,z,d≤ 1 (4.29)
0,42 Nmm2
0,16 ∙ 19,3 Nmm2 + 0,7 ∙ 13,1 Nmm2
30,3 Nmm2 + 4,7 Nmm2
32,0 Nmm2 = 0,14 + 0,30 + 0,15 = 0,59 → OK
c,0,d c,y∙ c,0,d + m,y,d m,y,d+ m ∙ ,z,d m,z,d≤ 1 (4.30)
0,42 Nmm2
0,83 ∙ 19,3 Nmm 2 + 13,1 Nmm 2
30,3 Nmm2 + 0,7 ∙ 4,7 Nmm2
32,0 Nmm2 = 0,03 + 0,43 + 0,10 = 0,56 → OK
� m,y,d
crit∙ m,o,edge�2 + c,0,d c,z∙ c,0,d ≤ 1 (4.39)
13,1 Nmm2 2 0,42 Nmm2
z ∙ (1 + 0,1 ∙ (2,45 − 0,3) + (2,45)2 = 3,61
,5�1 + c� rel,y − 0,3� + rel,y 2 �
y = 0,5 ∙ (1 + 0,1 ∙ (0,92 − 0,3) + (0,92)2 = 0,95
c,z = 1
z+� z2− rel,z 2 = 1
3,61+�3,612−2,452 = 0,16 (4.32)
c,y = 1y + � y2− rel,y 2 = 1
0,95 + �0,952 − 0,922 = 0,83
c,0,d c,z
∙ c,0,d + m ∙ m,y,d m,y,d+ m,z,d m,z,d≤ 1 (4.29)
0,42 Nmm2
0,16 ∙ 19,3 Nmm2 + 0,7 ∙ 13,1 Nmm 30,3 Nmm2 + 4,7 Nmm2
32,0 Nmm2 = 0,14 + 0,30 + 0,15 = 0,59 → OK
c,0,d c,y∙ c,0,d + m,y,d m,y,d+ m ∙ m,z,d m,z,d≤ 1 (4.30)
0,42 Nmm2
0,83 ∙ 19,3 Nmm 2 + 13,1 Nmm 2
30,3 Nmm2 + 0,7 ∙ 4,7 Nmm2
32,0 Nmm2 = 0,03 + 0,43 + 0,10 = 0,56 → OK
� m,y,d
crit∙ m,o,edge�2 + c,0,d c,z∙ c,0,d ≤ 1 (4.39)
2
z 2 45 , 2 45 3 61
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 �
y = 0,5 ∙ (1 + 0,1 ∙ (0,92 − 0,3) + (0,92)2 = 0,95
,z = 1
z+� z2− rel,z 2 = 1
3,61+�3,612−2,452 = 0,16 (4.32)
c,y = 1y + � y2− rel,y 2 = 1
0,95 + �0,952 − 0,922 = 0,83
c,0,d c,z
∙ c,0,d + m ∙ m,y,d m,y,d+ m,z,d m,z,d≤ 1 (4.29)
0,42 Nmm2
0,16 ∙ 19,3 Nmm2 0,7 ∙ 13,1 Nmm2
30,3 Nmm2 + 4,7 Nmm2
32,0 Nmm2 = 0,14 + 0,30 + 0,15 = 0,59 → OK
c,0,d c,y∙ c,0,d + m,y,d m,y,d+ m ∙ m,z,d m,z,d≤ 1 (4.30)
0,42 Nmm2
0,83 ∙ 19,3 Nmm 2 + 13,1 Nmm 2
30,3 Nmm2 + 0,7 ∙ 4,7 Nmm2
32,0 Nmm2 = 0,03 + 0,43 + 0,10 = 0,56 → OK
m,y,d
crit∙ m,o,edge�2 + c,0,d c,z∙ c,0,d ≤ 1 (4.39)
,y,crit 2
rel h∙ m,k ,y,crit 1,03∙4 N/ 2
21,6 N/ 2 ,4 (4.41)
en 1,4 rel,m , crit 1
rel,m 2∙= 1 1,452 ,4 ( . )
crit ∙ ,y,d ,4 ∙ ,3 N/m 2 ,4 N/ 2
,y,d crit ∙ ,y,d ( . )
c,d = 4,5kN 800 m 2 ,4 N/ 2
c,0,d od ∙ c,0,SC2,k ,8 ,2 ∙ N m2 ,3 N/ 2
z = 1 � c� = ,4 ∙ 2000 m 45 m 154 ( . )
y � c ℎ � ,4 ∙ m m = 58
rel,z = z c,0,k 0,05 154 3,14∙ � 29 N/ 2
1 600 N/ 2 = ,4 ( . )
rel,y y � c,0,k 0,05 58 ,1 ∙ � 29 N/mm2
0 N/m 2 ,9
z ,5� c� rel,z ,3� + rel,z 2 � ( . )
LVL Handbook Europe 193