LVL Handbook EUROPE

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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 = 22,3 kN 5400 mm2 = 4,1 N/mm2
c,0,d = mod M∙ c,0,SC1,k = 0,8 1,2 ∙ 26 Nmm2 = 17,3 N/mm2
y = √12 � c ℎ � = 3,46 ∙ 2700 mm 120 mm = 78 (4.37)
rel,y = y � c,0,k 0,05= 78 3,14∙ � 26 N/mm 2
8000 N/mm2 = 1,41 (4.35)
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 � = 0,5 ∙ (1 + 0,1 ∙ (1,412 − 0,3) + (1,41)2 = 1,56
c,y = 1
y+� y2− rel,y 2 = 1
1,56+�1,56 2−1,412 = 0,45 (4.31)
c,0,d c,z∙ c,0,d = 4,1 Nmm2
0,45 ∙ 17,3 Nmm2 = 0,52 ≤ 1,0 → OK
d = d,ULS ∙ z = 22,3 kN ∙ 0,03 m = 0,67 kNm
m,d = d = 0,67 kNm 1,08 ∙ 105mm3 = 6,2 N/mm2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 1,15 ∙ 27 Nmm2 = 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 Nmm2
0,45 ∙ 17,3 Nmm2 + 0,7 ∙ 6,2 Nmm2
20,7 Nmm2 = 0,52 + 0,7 ∙ 0,30 = 0,73 → OK
c,90,d = d = 22,3 kN
c,d = d,ULS = 22,3 kN 5400 2 = 4,1 N/ 2
c,0,d = mod M∙ c,0,SC1,k = 0,8 1,2 ∙ 26 N 2 = 17,3 N/mm2
y = √12 � c
ℎ � = 3,46 ∙ 2700 mm 120 mm = 78 (4.37)
rel,y = y � c,0,k 0,05= 78 3,14∙ � 26 N/mm 2
8000 N/mm2 = 1,41 (4.35)
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 � = 0,5 ∙ (1 + 0,1 ∙ (1,412 − 0,3) + (1,41)2 = ,56
c,y = 1
y+� y2− rel,y 2 = 1
1,56+�1,56 2−1,412 = 0,45 (4.31)
c,0,d c,z∙ c,0,d = 4,1 Nm 2
0,45 ∙ 17,3 N 2 = 0,52 ≤ 1,0 OK
d = d,ULS ∙ z 22,3 kN ∙ 0,03 = 0,67 kNm
m,d = d = 0,67 kN 1,08 ∙ 105 3 = 6,2 N/ 2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 1,15 ∙ 27 N 2 = 20,7 N/ 2
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 Nm 2
0,45 ∙ 17,3 Nm 2 + 0,7 ∙ 6,2 N2
20,7 N 2 = 0,52 + 0,7 ∙ 0,30 = 0,73 OK
c,90,d = d = 22,3 kN
c,d d,ULS = 22,3 kN 5400 2 = 4,1 N/ 2
c,0,d = mod M∙ c,0,SC1,k = 0,8 1,2 ∙ 26 Nmm2 = 17,3 N/ 2
y √12 � c ℎ � 3,46 ∙ 2700 mm 120 mm 78 (4.37)
rel,y = y � c,0,k 0,05= 78 3,14∙ � 26 N/mm 2
8000 N/mm2 = 1,41 (4.3 )
y 0,5�1 c� rel,y − 0,3� + rel,y 2 � = 0,5 ∙ (1 + 0,1 ∙ (1,412 − 0,3) + (1,41)2 = 1,56
c,y = 1
y+� y2− rel,y 2 1
1,56+�1,56 2−1,412 = 0,45 (4.31)
c,0,d c,z∙ c,0,d = 4,1 N2
0,45 ∙ 17,3 Nmm2 = 0,52 ≤ 1,0 OK
d = d,ULS ∙ z = 22,3 kN ∙ 0,03 = 0,67 kN
m,d d = 0,67 kN 1,08 ∙ 105 3 = 6,2 N/ m2
m,0,edge,d mod M ∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 1,15 ∙ 27 Nmm2 = 20,7 N/ 2
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 Nm 2
0,45 ∙ 17,3 Nm 2 + 0,7 ∙ 6,2 N2
20,7 N 2 0,52 0,7 ∙ 0,30 = 0,73 → OK
c,90,d = d 22,3 kN
c,d = d,ULS = 22,3 kN 5400 mm2 = 4,1 N/mm2
c,0,d = mod M∙ c,0,SC1,k = 0,8 1,2 ∙ 26 Nmm2 = 17,3 N/mm2
y = √12 � c
ℎ � = 3,46 ∙ 2700 mm 120 mm = 78 (4.37)
rel,y = � c,0,k 0,05= 78 3,14∙ � 26 N/mm 2
8000 N/mm2 = 1,41 (4.35)
y = 0,5�1 + c� rel,y − 0,3� + rel,y 2 � = 0,5 ∙ (1 + 0,1 ∙ (1,412 − 0,3) + (1,41)2 = 1,56
c,y = 1
y+� y2− rel,y 2 = 1
1,56+�1,56 2−1,412 = 0,45 (4.31)
c,0,d c,z∙ c,0,d = 4,1 Nmm2
0,45 ∙ 17,3 Nmm2 = 0,52 ≤ 1,0 → OK
d = d,ULS ∙ z = 22,3 kN ∙ 0,03 m = 0,67 kNm
m,d = d = 0,67 kNm 1,08 ∙ 105mm3 = 6,2 N/mm2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 1,15 ∙ 27 Nmm2 = 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 Nmm2
0,45 ∙ 17,3 Nmm2 + 0,7 ∙ 6,2 Nmm2
20,7 Nmm2 = 0,52 + 0,7 ∙ 0,30 = 0,73 → OK
c,90,d = d = 22,3 kN
c,d d,ULS = 22,3 kN 5400 mm2 = 4,1 N/mm2
c,0,d = mod M∙ c,0,SC1,k = 0,8 1,2 ∙ 26 Nmm2 = 17,3 N/mm2
y = √12 � c
ℎ� = 3,46 ∙ 2700 mm 120 mm = 78 (4.37)
rel,y = y � c,0,k 0,05= 78 3,14∙ � 6 N/mm2
8000 N/mm2 = 1,41 (4.35)
y = 0,5�1 + c� rel,y − 0,3 + rel,y 2 � = 0,5 ∙ (1 + 0,1 ∙ (1,412 − 0,3) + (1,41) 2 = 1,56
c,y = 1
y+� y2− rel,y 2 = 1,56+�1,562−1,412 = 0,45 (4.31)
c,0,d c,z∙ c,0,d = 4,1 Nmm2
0,45 ∙ 17,3 Nmm2 = 0,52 ≤ 1,0 → OK
d = d,ULS ∙ z = 22,3 kN ∙ 0,03 m = 0,67 kNm
m,d = d = 0,67 kNm 1,08 ∙ 105mm3 = 6,2 N/mm2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 1,15 ∙ 27 Nmm2 = 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 Nmm2
0,45 ∙ 17,3 Nmm2 + 0,7 ∙ 6,2 Nmm2
20,7 Nmm2 = 0,52 + 0,7 ∙ 0,30 = 0,73 → OK
c,90,d = d = 22,3 kN
196 LVL Handbook Europe