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 ∙ ℎ0 ∙ 0 ∙ 0 ser ∙ 0 2 = 1 1+π2 ∙ 7000 N/mm2 ∙ 22 mm∙ 100 mm∙ 200 mm 454N/mm∙ (5000 mm)2 =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing ∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 ∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104 2 ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103 2 ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103 2/m + 6,44∙ 104 2/m + 5,98∙ 103 2 ( )b = 7,35∙ 104 2 ,2 ( )b ( )l 4 6,44∙ 104 1,79∙ 106 4 =0,45 2 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106 2 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106 2 �0,6 mm 2,7 mm = 0,6 mm =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 ( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 10 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2 2 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( )decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42 0,45 1,79∙ 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK =0,27 ( )bracing =2∙ 2∙ 7 m 00 m 0N2 ∙ �100 mm∙ (22 mm) 3 12 +0,27∙ 100 mm∙ 22 mm∙ �2 2 40 + 2 2 2�2� 4,5 m ( ) bracing =6,44∙ 104Nm2/m ( )cross battens =7000 m N m2 ∙ 1400000 mmmm∙ 45 mm ∙ (45 mm)2 12 =5,98∙ 103Nm2/m ( )b = ( decking + ( )bracings + ( )cross battens ( )b = 3,11∙ 103Nm2/m + 6,44∙ 104 Nm2/m + 5,98∙ 103Nm2/m ( )b = 7,35∙ 104Nm2/m ,2 =�( )b ( )l = 4 �6,44∙ 104 1,79∙ 106 4 =0,45 2 =min⎪⎩⎨ ⎪⎪⎧ 1kN∙ (4,5m)2 42∙ 0,45∙ 1,79 106Nm2 m 1kN∙ (4,5m)3 48∙ 0,4∙ 1,79∙ 106Nm2 m =min�0,6 mm 2,7 mm = 0,6 mm→OK LVL Handbook Europe 183
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