9. CALCULATION EXAMPLES OF LVL STRUCTURES ULS design Bending moment resistance M_d = E_(d,ULS)∙s∙L2/8 = 25,1 kN/m∙〖(4m)〗^2/8 = 50,3 kNm σ_(m,d)=M_d/W=(50,3 kNm)/(2,72〖∙10〗^6 〖 mm〗^3 )=18,5 N/mm^2 f_(m,0,edge,d)=k_mod/γ_M ∙k_h∙f_(m,0,edge,k)=0,8/1,2∙0,96∙44 N/mm^2 =28,1 N/mm^2 σ_(m,d)≤f_(m,0,edge,d) →OK Lateral torsional buckling The ridge beam is loaded by the roof rafters connected at the sides of the beam at 1200 mm spacing and they act as supports against lateral torsional buckling, so the effective length is Lef = 1200 mm. 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∙8,84∙〖10〗^6 〖 mm〗^4∙400N/m 〖2,72∙10〗^6 〖 mm〗^3 ) σ_(m,crit)=34,8 N/mm^2 λ_rel=√(f_(m,k)/σ_(m,crit) )=√((44 N/mm^2)/(34,8 N/mm^2 ))= 1,12 (4.41) when 0,75<λ_(rel,m)≤1,4 ,k_crit=1,56-0,75∙λ_(rel,m)=1,56-0,75∙1,12=0,72 k_crit∙ f_(m,d)=0,72 ∙28,1 N/mm^2=20,1 N/mm^2 σ_(m,d)≤k_crit∙ f_(m,d)→OK (4.38) Shear resistance V_d = E_(d,ULS)∙s∙L/2 = 25,1kN/m∙4,0m/2 = 50,3 kN τ_(v,d)=〖3∙V〗_d/(2∙A)=(3∙50,3 kN)/(2 ∙40 800 mm^2 )=1,9 N/mm^2 f_(v,0,edge,d)=k_mod/γ_M ∙f_(v,0,edge,k)=0,8/1,2∙4,2 N/mm^2 =2,8 N/mm^2 τ_(m,d)≤f_(v,0,edge,d) →OK Compression perpendicular to grain σ_(c,90,d)=F_(c,90,d)/A_ef =F_(c,90,d)/(b∙(l_support+15 mm) ) (4.14) σ_(c,90,d)=50,3kN/(2∙51mm∙(120mm+15mm))=3,7 N/mm^2 k_(c,90)∙f_(c,90,edge,d)=k_(c,90)∙k_mod/γ_M ∙f_(c,90,edge,k)=1,0∙0,8/1,2∙6 N/mm^2=4 N/mm^2 σ_(c,90,d)≤k_(c,90)∙f_(m,0,edge,d) →OK (4.13) d,SLS = G ∙ ( 1,k + 2,k)+ Q∙ k (4.1) d,SLS =1,0∙ (6m ∙ 1,0 kN/m2 + 0,2 kN/m )+ 6,0∙ 1,0∙ 2,0 kN/m2 d,SLS = 18,2 kN/m d = d,ULS ∙ ∙ 2/8 = 25,1 kN/m∙ (4m)2/8 = 50,3 kNm m,d = d = 2,7 5 2 0,3 kNm ∙ 106 mm3 = 18,5 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 0,96∙ 44 m N m2 = 28,1 N/mm2 m,d ≤ m,0,edge,d →OK m,crit = y,crit y = � 0,05 0,05 tor ef y (4.42) m,crit = π�10600 N/mm2 ∙ 8,84∙ 106 mm4 ∙ 400N/mm2 ∙ 3,18∙ 107 ∙ mm4 1200 mm∙ 2,72∙ 106 mm3 m,crit = 34,8 N/mm2 d,SLS = G ∙ ( 1,k + 2,k)+ Q∙ k (4.1) d,SLS =1,0∙ (6m ∙ 1,0 kN/m2 + 0,2 kN/m )+ 6,0∙ 1,0∙ 2,0 kN/m2 d,SLS = 18,2 kN/m d = d,ULS ∙ ∙ 2/8 = 25,1 kN/m∙ (4m)2/8 = 50,3 kNm m,d = d = 2,7 5 2 0,3 kNm ∙ 106 mm3 = 18,5 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 0,96∙ 44 m N m2 = 28,1 N/mm2 m,d ≤ m,0,edge,d →OK m,crit = y,crit y = � 0,05 0,05 tor ef y (4.42) m,crit = π�10600 N/mm2 ∙ 8,84∙ 106 mm4 ∙ 400N/mm2 ∙ 3,18∙ 107 ∙ mm4 1200 mm∙ 2,72∙ 106 mm3 m,crit = 34,8 N/mm2 rel =� m,k m,crit =�44 N/mm2 34,8 N/mm2 = 1,12 (4.41) when 0,75 < rel,m≤1,4 , crit =1,56−0,75∙ rel,m=1,56−0,75∙ 1,12=0,72 crit ∙ m,d =0,72 ∙ 28,1 N/mm2 = 20,1 N/mm2 m,d ≤ crit ∙ m,d →OK (4.38) d = d,ULS ∙ ∙ /2 = 25,1kN/m∙ 4,0m/2 = 50,3 kN v,d = 3∙ d 2∙ = 3∙ 2 50,3 kN ∙ 40 800 mm2 = 1,9 N/mm2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,2 m N m2 = 2,8 N/mm2 m,d ≤ v,0,edge,d →OK c,90,d = d = 50,3 c,90,d = c,90,d ef = c,90,d ∙� support+15 mm� (4.14) c,90,d = 50,3kN 2∙ 51mm∙ (120mm + 15mm) = 3,7 N/mm2 c,90 ∙ c,90,edge,d = c,90 ∙ mod M ∙ c,90,edge,k =1,0∙ 0 1 , , 8 2∙ 6 N/mm2 = 4 N/mm2 c,90,d ≤ c,90 ∙ m,0,edge,d →OK (4.13) rel =� m,k m,crit =�44 N/mm2 34,8 N/mm2 = 1,12 (4.41) when 0,75 < rel,m≤1,4 , crit =1,56−0,75∙ rel,m=1,56−0,75∙ 1,12=0,72 crit ∙ m,d =0,72 ∙ 28,1 N/mm2 = 20,1 N/mm2 m,d ≤ crit ∙ m,d →OK (4.38) d = d,ULS ∙ ∙ /2 = 25,1kN/m∙ 4,0m/2 = 50,3 kN v,d = 3∙ d 2∙ ∙ 2 50,3 kN ∙ 40 800 mm2 = 1,9 N/mm2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,2 m N m2 = 2,8 N/mm2 m,d ≤ v,0,edge,d →OK c,90,d = d = 50,3 c,90,d = c,90,d ef = c,90,d ∙� support+15 mm� (4.14) c,90,d = 50,3kN 2∙ 51mm∙ (120mm + 15mm) = 3,7 N/mm2 c,90 ∙ c,90,edge,d = c,90 ∙ mod M ∙ c,90,edge,k =1,0∙ 0 1 , , 8 2∙ 6 N/mm2 = 4 N/mm2 c,90,d ≤ c,90 ∙ m,0,edge,d →OK (4.13) , , , , ∙ , ∙ ∙ , ∙ , ≤ ∙ , , ∙ ∙ ∙ , ∙ ∙ ∙ ∙ , , , ∙ , , , = 0 1 , , 8 2∙ , ≤ , , , , , , , , , , , ∙� , , ∙ ∙ , ∙ , , , , ∙ ∙ , , , ∙ 0 1 , , 8 2∙ , , ≤ , ∙ , , , LVL Handbook Europe 189
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