Loading combinations Own weight in z-direction gk,z: cos15° ∙ 0,9m ∙ 0,3 kN/m2 = 0,26 kN/m Own weight in y-direction gk,y: sin15° ∙ 0,9m ∙ 0,4 kN/m2 = 0,07 kN/m. Snow load at roof level qk = μ1 ∙ Ce ∙ sk Form factor μ1 = 0,8, when roof angle is less than 30° and in normal conditions Ce = 1,0. qk = 0,8 ∙ 1,0 ∙ 2,5 N/m2 = 2 kN/m2 (horizontal projection). qk,z = cos15° ∙ cos15° ∙ 2kN/m = 1,68 kN/m qk,y = cos15 ∙ sin15° ∙ 2kN/m = 0,45 kN/m The most critical ultimate limit state (ULS) load combination: Ed,z,ULS = γG ∙ gk,z + γQ∙qk,z Ed,z,ULS = 1,15 ∙ 0,26 kN/m2 + 1,5 ∙ 1,68 kN/m2 = 2,82 kN/m Ed,y,ULS = γG ∙ gk,y + γQ ∙ qk,y Ed,y,ULS = 1,15 ∙ 0,07 kN/m2 + 1,5 ∙ 0,45 kN/m2 = 0,76 kN/m Axial compression Nc,d = γQ ∙ Nc,k = 1,5 ∙ 3 kN/m2 = 4,5 kN Note: Safety factors γG and γQ are according to Finnish national annex of Eurocode 0. Most critical serviceability limit state (SLS) load combination: Ed,z,SLS = γG ∙ gk,z + γQ∙qk,z Ed,z,ULS = 1,0 ∙ 0,26 kN/m2 + 1,0 ∙ 1,68 kN/m2 = 1,94 kN/m ULS design Bending moment resistance in y-direction M_(d,z) = E_(d,z,ULS)∙L2/8 = 2,82kN/m∙〖(4m)〗^2/8 = 5,64 kNm σ_(m,y,d)=M_(d,z)/W_y =(5,64 kNm)/(4,32〖∙10〗^5 mm^3 )=13,1 N/mm^2 f_(m,0,edge,d)=k_mod/γ_M ∙k_h∙f_(m,0,edge,k)=0,8/1,2∙1,034∙44 N/mm^2 =30,3 N/mm^2 Bending moment resistance in z-direction at centre support of a 2-span beam M_(d,y)=E_(d,y,ULS)∙〖(L/2)〗^2/8 = 0,76 kN/m∙〖(4 m/2)〗^2/8 = 0,38 kNm σ_(m,z,d)=M_(y,d)/W_z =(0,38 kNm)/(8,10〖∙10〗^4 mm^3 )=4,7 N/mm^2 f_(m,0,flat,z,d)=k_mod/γ_M ∙f_(m,0,flat,z,k)=0,8/1,2∙48 N/mm^2 =32,0 N/mm^2 Lateral torsional buckling (LTB) is prevented at the middle of the span. The purlin is loaded from the compression side and supported against torsion at the main supports and in the middle of the span. According to Table 6.1 of EN1995-1-1, for uniformly distributed load, the effective length is Lef = 2000 mm+2∙240 mm = 2480 mm. σ_(m,y,crit)=M_(z,crit)/W_y =(π√(E_0,05 I_z G_0,05 I_tor ))/(l_ef W_y ) (4.42) σ_(m,y,crit)= (π√(10600 N/mm^2∙1,82∙〖10〗^6 〖 mm〗^4∙400 N/mm^2∙6,5 mm∙〖4,32∙10〗^5 mm^3 ) σ_(m,y,crit)=21,6N/mm^2 λ_rel=√((k_h∙f_(m,k))/σ_(m,y,crit) )=√((1,03∙44 N/mm^2)/(21,6 N/mm^2 ))=1,45 (4.41) 9. CALCULATION EXAMPLES OF LVL STRUCTURES d,y = d,y,ULS ∙ (L/2)2/8 = 0,76 kN/m∙ (4 m/2)2/8 = 0,38 kNm m,z,d = y,d z = 8,1 0 0 ,38 kNm ∙ 104mm3 = 4,7 N/mm2 m,0,flat,z,d = mod M ∙ m,0,flat,z,k = 0 1 , , 8 2∙ 48 m N m2 = 32,0 N/mm2 m,y,crit = z,crit y = � 0,05 0,05 tor ef y (4.42) m,y,crit = π�10600 N/mm2 ∙ 1,82∙ 106 mm4 ∙ 400 N/mm2 ∙ 6,56∙ 106 ∙ mm4 2480 mm∙ 4,32∙ 105mm3 m,y,crit =21,6N/mm2 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/mm2 = 14,4 N/mm2 d,y d,y,ULS ∙ (L/2)2/8 = 0,76 kN/m∙ (4 m/2)2/8 = 0,38 kNm ,z,d y,d z = 8,1 0 0 ,38 kNm ∙ 104 3 = 4,7 N/mm2 ,0,flat,z,d mod ∙ ,0,flat,z,k = 0 1 , , 8 2∙ 48 m N m2 = 32,0 N/mm2 ,y,crit z,crit y � 0,05 0,05 tor ef y (4.42) ,y,crit 10600 N/mm2 ∙ 1,82∙ 106 4 ∙ 400 N/mm2 ∙ 6,56∙ 106 ∙ 4 2480 mm∙ 4,32∙ 105 3 ,y,crit 2 rel h∙ m,k m,y,crit 1,03∙44 N/ 2 21,6 N/ 2 =1,45 (4.41) when 1,4 < rel, , crit 1 rel,m 2 ∙ 1 1,452 =0,48 (4.40) crit ∙ ,y,d =0,48 ∙ 30,3 N/mm2 = 14,4 N/mm2 d,z = d,z,ULS ∙ 2/8 = 2,82kN/m∙ (4m)2/8 = 5,64 kNm m,y,d = d,z y = 4,3 5 2 ,64 kNm ∙ 105mm3 = 13,1 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,034∙ 44 m N m2 = 30,3 N/mm2 d,y = d,y,ULS ∙ (L/2)2/8 = 0,76 kN/m∙ (4 m/2)2/8 = 0,38 kNm m,z,d = y,d z = 8,1 0 0 ,38 kNm ∙ 104mm3 = 4,7 N/mm2 m,0,flat,z,d = mod M ∙ m,0,flat,z,k = 0 1 , , 8 2∙ 48 m N m2 = 32,0 N/mm2 m,y,crit = z,crit y = � 0,05 0,05 tor ef y (4.42) m,y,crit = π�10600 N/mm2 ∙ 1,82∙ 106 mm4 ∙ 400 N/mm2 ∙ 6,56∙ 106 ∙ mm4 2480 mm∙ 4,32∙ 105mm3 m,y,crit =21,6N/mm2 rel =� h∙ m,k m,y,crit =�1,03∙44 N/mm2 21,6 N/ 2 =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/mm2 = 14,4 N/mm2 ≤ ∙ →OK 192 LVL Handbook Europe
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