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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 L ef = 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 = 50,3 kNm 2,72 ∙ 106 mm3 = 18,5 N/mm2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 0,96 ∙ 44 Nmm 2 = 28,1 N/mm2
m,d ≤ m,0,edge,d → OK
m,crit = y,crit y = � 0,05 0,05 toref y (4.42)
m,crit = π�10600 N/mm2 ∙ 8,84 ∙ 106 mm4 ∙ 400N/mm2 ∙ 3,18 ∙ 10 7 ∙ mm4
1200 mm ∙ 2,72 ∙ 10 6 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, / ∙ ( )2/8 = 50,3 kNm
m,d = d = 50,3 kNm 2,72 ∙ 106 mm3 = 18,5 N/mm2
m,0,edge,d = mod M∙ h ∙ m,0,edge,k = 0,8 1,2 ∙ 0,96 ∙ 44 Nmm 2 = 28,1 N/mm2
m,d ≤ m,0,edge,d → OK
m,crit = y,crit y = � 0,05 0,05 toref y (4.42)
m,crit = π�10600 N/mm2 ∙ 8,84 ∙ 106 mm4 ∙ 400N/mm2 ∙ 3,18 ∙ 10 7 ∙ mm4
1200 mm ∙ 2,72 ∙ 10 6 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 ∙ 50,3 kN2 ∙ 40 800 mm2 = 1,9 N/mm2
v,0,edge,d = mod M∙ v,0,edge,k = 0,8 1,2 ∙ 4,2 Nmm2 = 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 ∙ 0,8 1,2 ∙ 6 N/mm 2 = 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 ∙ = 3 ∙ 50,3 kN2 ∙ 40 800 mm2 = 1,9 N/mm2
v,0,edge,d = mod M∙ v,0,edge,k = 0,8 1,2 ∙ 4,2 Nmm2 = 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 ∙ 0,8 1,2 ∙ 6 N/mm 2 = 4 N/mm2
c,90,d ≤ c,90 ∙ m,0,edge,d → OK (4.13)
l � , ,it N m 3 ,8 2 1, (4.41)
when 0,7 l, ,4 , i ,5 ,7 ∙ l,m , , ∙ , ,
i ∙ m,d 0,72 ∙ , N 2 0,1 N
, i ∙ m,d K
, ∙ ∙ = 2 ,1 ∙ ,0 5 ,3 k
,d = ∙ 2 ∙ ∙ 50,3 k
2 ∙ 4 8 m ,9 N
v,0,edge,d ∙ v,0,edge,k ,8 ,2 ∙ , ,
,d ,0,e ,d
,9 ,d = ,3
, 0, ,9 , f = , ,
∙� ort m �
, , 5 ,3k
∙ ∙ 0m ,7 N
, ∙ ,9 ,e , ,9 ∙ ∙ ,9 ,e ,k ,0 ∙ 0,8 1,2 ∙ N N
,9 ,d ,9 ∙ ,0,e , K
LVL Handbook Europe 189