9. CALCULATION EXAMPLES OF LVL STRUCTURES Loading combinations The most critical ultimate limit state (ULS) load combination: Ed,ULS = γG∙gk+ γQ∙qk = 1,15∙5,0 kN+1,5∙11 kN = 22,3 kN Note: Safety factors γG and γQ are according to Finnish National annex of eurocode 0. ULS design Axial compression N_(c,d)/A=E_(d,ULS)/A= (22,3 kN)/(5400 mm^2 )=4,1 N/mm^2 f_(c,0,d)=k_mod/γ_M ∙f_(c,0,SC1,k)=0,8/1,2∙26 N/mm^2 =17,3 N/mm^2 Buckling, buckling length lc = 2700 mm in z-direction λ_y=√12 (l_c/h)=3,46∙(2700 mm)/(120 mm)=78 (4.37) λ_(rel,y)=λ_y/π √(f_(c,0,k)/E_0,05 )=78/3,14∙√((26 N/mm^2)/(8000 N/mm^2 ))=1,41 (4.35) k_y=0,5(1+β_c (λ_(rel,y)-0,3)+λ_(rel,y)^2 )=0,5∙(1+0,1∙(1,412-0,3)+(1,41)^2=1,56 k_(c,y)=1/(k_y+√(k_y^2-λ_(rel,y)^2 ))=1/(1,56+√(〖1,56〗^2-〖1,41〗^2 ))=0,45 (4.33) (4.31) σ_(c,0,d)/〖k_(c,z)∙f〗_(c,0,d) =(4,1 N/mm^2 )/(0,45∙17,3 N/mm^2 )=0,52≤1,0→OK Bending moment resistance M_d=E_(d,ULS)∙e_z= 22,3 kN∙0,03 m = 0,67 kNm σ_(m,d)=M_d/W=(0,67 kNm)/(1,08〖∙10〗^5 mm^3 )=6,2 N/mm^2 f_(m,0,edge,d)=k_mod/γ_M ∙k_h∙f_(m,0,edge,k)=0,8/1,2∙1,15∙27 N/mm^2 =20,7 N/mm^2 σ_(m,d)≤f_(m,0,edge,d) →OK 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) ≤1 (4.30) (4,1 N/mm^2 )/(0,45∙17,3 N/mm^2 )+0,7∙(6,2 N/mm^2 )/(20,7 N c,d = d,ULS = 542020,3mkNm2 = 4,1 N/mm2 c,0,d = mod M ∙ c,0,SC1,k = 0 1 , , 8 2∙ 26 m N m2 = 17,3 N/mm2 y =√12� cℎ� =3,46∙ 2700 120 mm mm =78 (4.37) rel,y = y � c,0,k 0,05 =3 78,14∙� 26 N/mm2 8000 N/mm2 =1,41 (4.35) y =0,5�1+ c� rel,y −0,3�+ rel, 2 y� =0,5∙ (1+0,1∙ (1,412−0,3) + (1,41)2 =1,56 c,y = 1 y+� y2− rel, 2 y = 1 1,56+�1,562−1,412 =0,45 (4.31) c,0,d c,z ∙ c,0,d = 4,1 m Nm2 0,45∙ 17,3 m Nm2 =0,52≤1,0→OK d = d,ULS ∙ z = 22,3 kN∙ 0,03 m = 0,67 kNm m,d = d = 1,0 0 8 ,67 kNm ∙ 105mm3 = 6,2 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,15∙ 27 m N m2 = 20,7 N/mm2 m,d ≤ m,0,edge,d →OK c,0,d c,z∙ c,0,d + m∙ m,y,d m,y,d ≤1 (4.30) 4,1 m Nm2 0,45∙ 17,3 m Nm2 +0,7∙ 6,2 m Nm2 20,7 m Nm2 =0,52+0,7∙ 0,30=0,73→OK c,90,d = d = 22,3 kN c,d = d,ULS = 542020,3mkNm2 = 4,1 N/mm2 c,0,d = mod M ∙ c,0,SC1,k = 0 1 , , 8 2∙ 26 m N m2 = 17,3 N/mm2 y =√12� cℎ� =3,46∙ 2700 120 mm mm =78 (4.37) rel,y = y � c,0,k 0,05 =3 78,14∙� 26 N/mm2 8000 N/mm2 =1,41 (4.35) y =0,5� c� rel,y −0,3�+ rel, 2 y� =0,5∙ (1+0,1∙ (1,412−0,3) + (1,41)2 =1,56 c,y = 1 y+� y2− rel, 2 y = 1 1,56+�1,562−1,412 =0,45 (4.31) c,0,d c,z ∙ c,0,d = 4,1 m Nm2 0,45∙ 17,3 m Nm2 =0,52≤1,0 OK d = d,ULS ∙ z = 22,3 kN∙ 0,03 m = 0,67 kNm m,d = d = 1,0 0 8 ,67 kNm ∙ 105 3 = 6,2 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,15∙ 27 m N m2 = 20,7 N/mm2 m,d ≤ m,0,edge,d OK c,0,d c,z∙ c,0,d + m∙ m,y,d m,y,d ≤1 (4.30) 4,1 m Nm2 0,45∙ 17,3 m Nm2 +0,7∙ 6,2 m Nm2 20,7 m Nm2 =0,52+0,7∙ 0,30=0,73 OK c,90,d = d = 22,3 kN c,d d,ULS = 542020,3mkNm2 = 4,1 N/mm2 c,0,d mod M ∙ c,0,SC1,k = 0 1 , , 8 2∙ 26 m N m2 = 17,3 N/mm2 y √12� cℎ� =3,46∙ 2700 120 mm mm (4.37) rel,y y c,0,k 0,05 3 78,14∙� 26 N/mm2 8000 N/mm2 =1,41 (4.35) y =0,5� c� rel,y 0,3� rel, 2 y� =0,5∙ (1+0,1∙ (1,412 0,3) + (1,41)2 =1,56 c,y 1 y+� y2− rel, 2 y 1 1,56+�1,562−1,412 =0,45 (4.31) c,0,d c,z ∙ c,0,d = 4,1 m Nm2 0,45∙ 17,3 m Nm2 =0,52 1,0 OK d d,ULS ∙ z = 22,3 kN∙ 0,03 m = 0,67 kNm m,d d = 1,0 0 8 ,67 kNm ∙ 105 3 = 6,2 N/mm2 m,0,edge,d mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,15∙ 27 m N m2 = 20,7 N/mm2 m,d ≤ m,0,edge,d OK c,0,d c,z∙ c,0,d m∙ m,y,d m,y,d 1 (4.30) 4,1 m Nm2 0,45∙ 17,3 m Nm2 +0,7∙ 6,2 m Nm2 20,7 m Nm2 =0,52+0,7∙ 0,30=0,73 OK c,90,d d = 22,3 kN c,d = d,ULS = 542020,3mkNm2 = 4,1 N/mm2 c,0,d = mod M ∙ c,0,SC1,k = 0 1 , , 8 2∙ 26 m N m2 = 17,3 N/mm2 y =√12� cℎ� =3,46∙ 2700 120 mm mm =78 (4.37) rel,y = y � c,0,k 0,05 =3 78,14∙� 26 N/mm2 8000 N/mm2 =1,41 (4.35) y =0,5�1+ c� rel,y −0,3�+ rel, 2 y� =0,5∙ (1+0,1∙ (1,412−0,3) + (1,41)2 =1,56 c,y = 1 y+� y2− rel, 2 y = 1 1,56+�1,562−1,412 =0,45 (4.31) c,0,d c,z ∙ c,0,d = 4,1 m Nm2 0,45∙ 17,3 m Nm2 =0,52≤1,0→OK d = d,ULS ∙ z = 22,3 kN∙ 0,03 m = 0,67 kNm m,d = d = 1,0 0 8 ,67 kNm ∙ 105mm3 = 6,2 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,15∙ 27 m N m2 = 20,7 N/mm2 m,d ≤ m,0,edge,d →OK c,0,d c,z∙ c,0,d + m∙ m,y,d ≤1 (4.30) 4,1 m Nm2 0,45∙ 17,3 m Nm2 +0,7∙ 6,2 m Nm2 20,7 m Nm2 =0,52+0,7∙ 0,30=0,73→OK c,90,d = d = 22,3 kN c,d = d,ULS = 542020,3mkNm2 = 4,1 N/mm2 c,0,d = mod M ∙ c,0,SC1,k = 0 1 , , 8 2∙ 26 m N m2 = 17,3 N/mm2 y =√12� cℎ� =3,46∙ 2700 120 mm mm =78 (4.37) rel,y = y � c,0,k 0,05 =3 78,14∙� 26 N/mm2 8000 N/mm2 =1,41 (4.35) y =0,5�1+ c� rel,y −0,3 + rel, 2 y� =0,5∙ (1+0,1∙ (1,412−0,3) + (1,41)2 =1,56 c,y = 1 y+� y2− rel, 2 y = 1 1,56+�1,562−1,412 =0,45 (4.31) c,0,d c,z ∙ c,0,d = 4,1 m Nm2 0,45∙ 17,3 m Nm2 =0,52≤1,0→OK d = d,ULS ∙ z = 22,3 kN∙ 0,03 m = 0,67 kNm m,d = d = 1,0 0 8 ,67 kNm ∙ 105mm3 = 6,2 N/mm2 m,0,edge,d = mod M ∙ h ∙ m,0,edge,k = 0 1 , , 8 2∙ 1,15∙ 27 m N m2 = 20,7 N/mm2 md ≤ m,0,edge,d →OK c,0,d c,z∙ c,0,d + m∙ m,y,d m,y,d ≤1 (4.30) 4,1 m Nm2 0,45∙ 17,3 m Nm2 +0,7∙ 6,2 m Nm2 20,7 m Nm2 =0,52+0,7∙ 0,30=0,73→OK c,90,d = d = 22,3 kN 196 LVL Handbook Europe
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