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 = 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 m,y,d ≤ crit ∙ m,y,d →OK (4.38) c,d = 1 0 840, 50kmN m2 = 0,42 N/mm2 c,0,d = mod M ∙ c,0,SC2,k = 0 1 , , 8 2∙ 29 m N m2 = 19,3 N/mm2 z =√12� c � =3,46∙ 2000 45 mm mm =154 (4.37) y =√12� cℎ � =3,46∙ 4204000 mmmm = 5 8 rel,z = z � c,0,k 0,05 =3 154 ,14∙� 29 N/mm2 11600 N/mm2 =2,45 (4.36) rel,y = y � c,0,k 0,05 = 3 5 ,1 8 4∙� 29 N/mm2 11600 N/mm2 =0,92 z =0,5�1+ c� rel,z −0,3�+ rel, 2 z� (4.34) 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 m,y,d ≤ crit ∙ m,y,d →OK (4.38) c,d = 1 0 840, 50kmN m2 = 0,42 N/mm2 c,0,d = mod M ∙ c,0,SC2,k = 0 1 , , 8 2∙ 29 m N m2 = 19,3 N/mm2 z =√12� c � =3,46∙ 2000 45 mm mm =154 (4.37) y =√12� cℎ � =3,46∙ 4204000 mmmm = 5 8 rel,z = z � c,0,k 0,05 =3 154 ,14∙� 29 N/mm2 11600 N/mm2 =2,45 (4.36) rel,y = y � c,0,k 0,05 = 3 5 ,1 8 4∙� 29 N/mm2 11600 N/mm2 =0,92 z =0,5�1+ c� rel,z −0,3�+ rel, 2 z� (4.34) , , ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ , , ∙ , , ,crit , , , , 1 rel, ∙ , ∙ , , ∙ , , ≤ ∙ , , , 1 0 840, 50kmN m , , ∙ , , , = 0 1 , , 8 2∙ ∙ � ∙ , , , , , , � , , , = 3 5 ,1 8 4 � , , 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, 2 y� y =0,5∙ (1+0,1∙ (0,92−0,3) + (0,92)2 =0,95 c,z = 1 z+� z2− rel, 2 z = 1 3,61+�3,612−2,452 =0,16 (4.32) c,y = 1 y +� y2 − rel, 2 y = 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 m Nm2 0,16∙ 19,3 m Nm2 +0,7∙ 13,1 m Nm2 30,3 m Nm2 + 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,83∙ 19,3 m Nm2 + 13,1 m Nm2 30,3 m Nm2 +0,7∙ 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,48∙ 30,3 m Nm2� 2 + 0,42 m Nm2 0,16∙ 19,3 m Nm2 = 0,82 + 0,14 = 0,96 →OK = ∙ ∙ /2 = 2,92 kN/m∙ 4 m/2 = 6,2 kN 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, 2 y� y =0,5∙ (1+0,1∙ (0,92−0,3) + (0,92)2 =0,95 ,z = 1 z+� z2− rel, 2 z = 1 3,61+�3,612−2,452 =0,16 (4.32) c,y = 1 y +� y2 − rel, 2 y = 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 m Nm2 0,16∙ 19,3 m Nm2 +0,7∙ 13,1 m Nm2 30,3 m Nm2 + 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,83∙ 19,3 m Nm2 + 13,1 m Nm2 30,3 m Nm2 +0,7∙ 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,48∙ 30,3 m Nm2� 2 + 0,42 m Nm2 0,16∙ 19,3 m Nm2 = 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, 2 y� y =0,5∙ (1+0,1∙ (0,92−0,3) + (0,92)2 =0,95 c,z = 1 z+� z2− rel, 2 z = 1 3,61+�3,612−2,452 =0,16 (4.32) c,y = 1 y +� y2 − rel, 2 y = 1 0,95+�0,952 −0,922 =0,83 c,0,d c,z∙ c,0,d + m∙ ,y,d m,y,d + ,z,d m,z,d ≤1 (4.29) 0,42 m Nm2 0,16∙ 19,3 m Nm2 +0,7∙ 13,1 m Nm2 30,3 m Nm2 + 4,7 m Nm2 32,0 m Nm2 = 0,14 + 0,30 + 0,15 = 0,59→OK c,0,d c,y∙ c,0,d + ,y,d m,y,d + m∙ ,z,d m,z,d ≤1 (4.30) 0,42 m Nm2 0,83∙ 19,3 m Nm2 + 13,1 m Nm2 30,3 m Nm2 +0,7∙ 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,48∙ 30,3 m Nm2� 2 + 0,42 m Nm2 0,16∙ 19,3 m Nm2 = 0,82 + 0,14 = 0,96 →OK z ∙ (1+0,1∙ (2,45−0,3) + (2,45)2 =3,61 �1+ c� rel,y −0,3�+ rel, 2 y� y =0,5∙ (1+0,1∙ (0,92−0,3) + (0,92)2 =0,95 c,z = 1 z+� z2− rel, 2 z = 1 3,61+�3,612−2,452 =0,16 (4.32) c,y y +� y2 − rel, 2 y = 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 m Nm2 0,16∙ 19,3 m Nm2 +0,7∙ 13,1 m Nm 30,3 m Nm2 + 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,83∙ 19,3 m Nm2 + 13,1 m Nm 30,3 m Nm2 +0,7∙ 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,48∙ 30,3 m Nm2� 2 + 0,42 m Nm2 0,16∙ 19,3 m Nm2 = 0,82 + 0,14 = 0,96 →OK z (2,45 0,3) + (2,45) =3,61 y =0,5�1+ c� rel,y −0,3�+ rel, 2 y� y =0,5∙ (1+0,1∙ (0,92−0,3) + (0,92)2 =0,95 ,z = 1 z+� z2− rel, 2 z = 1 361+�3,612−2,452 =0,16 (4.32) c,y = 1 y +� y2 − rel, 2 y = 1 0,95+�0,952 −0,922 =0,83 c,0,d c,z∙ c,0,d + m∙ ,y,d m,y,d + ,z,d m,z,d ≤1 (4.29) 0,42 m Nm2 0,16∙ 19,3 m Nm2 +0,7∙ 13,1 m Nm2 30,3 m Nm2 + 4,7 m Nm2 32,0 m Nm2 = 0,14 + 0,30 + 0,15 = 0,59→OK c,0,d c,y∙ c,0,d + ,y,d m,y,d + m∙ ,z,d m,z,d ≤1 (4.30) 0,42 m Nm2 0,83∙ 19,3 m Nm2 + 13,1 m Nm2 30,3 m Nm2 +0,7∙ 4,7 m Nm2 32,0 m Nm2 = 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 m Nm2 0,48∙ 30,3 N � 2 + 0,42 m Nm2 0,16∙ 19,3 N = 0,82 + 0,14 = 0,96 →OK ,y,crit 2 ∙ 1,82∙ 6 4 ∙ 2 ∙ 6,56∙ 6 ∙ 4 ∙ 4,32∙ 5 3 ,y,crit 2 rel h∙ ,k ,y,crit 1,03∙44 / 2 21,6 / 2 (4.41) when 1,4 < rel, , crit 1 rel,m 2 ∙ ,452 (4.40) crit ∙ ,y,d ∙ 2 2 ,y,d ≤ crit ∙ ,y,d (4.38) c,d 1 0 840, 50kmN m2 2 c,0,d mod ∙ c,0,SC2,k = 0 1 , , 8 2∙ 29 m N m2 2 z � c � ∙ 2000 (4.37) y � cℎ � ∙ 4204000 mmmm = 5 8 rel,z z c,0,k 0,05 3 154 ,14 29 / 2 11600 / 2 (4.36) rel,y y � c,0,k 0,05 = 3 5 ,1 8 4 2 2 z � c� rel,z 0,3� rel, � (4.34) LVL Handbook Europe 193
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