LVL Handbook EUROPE

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V_(d,(x=323mm))=0,4 m∙4,03 kN/m^2 ∙(1-(0,3 m+0,045 m/2)/(4,5 m))=3,13 kN
M_(d,(x=323mm))=(0,4 m∙4,03 kN/m^2 ∙(0,3 m+0,045 m/2))/2∙(4,5 m-(0,3 m+(0,045 m)/2))
M_(d,(x=323mm))=1,1 kNm
F_(t,90,d)=(3,13 kN∙0,06 m)/(4∙0,24 m)∙[3-((0,06 m)/0,24m)^2 ]+0,008∙(1,1 kNm)/(0,09 m)=0,57 kN+0,10
kN=0,67 kN
σ_(t,90,d)=F_(t,90,d)/(0,5∙l_(t,90)∙b∙k_(t,90) )=(0,67 kN)/(0,5∙0,15 m∙0,045 m∙1,0)=0,20 N/mm^2 ≤f_
(t,90,d)→OK
The verification of shear stress concentration at the hole edge shall fulfil the condition:
τ_d=k_τ∙(1,5∙V_d)/(b∙(h-h_d ) )≤f_(v,d) (4.62)
Where
k_τ=1,85∙(1+a/h)∙(h_d/h)^0,2=1,85∙(1+(0,14 m)/(0,24 m))∙((0,06 m)/(0,24 m) (4.63)
τ_d=k_τ∙(1,5∙V_d)/(b∙(h-h_d ) )=2,22∙(1,5∙3,13 kN)/(45 mm∙(240 mm-60 mm) )=
f_(v,0,edge,d)=k_mod/γ_M ∙f_(v,0,edge,k)=0,8/1,2∙4,5 N/mm^2 =3,0 N/mm^2
τ_d≤f_(v,0,edge,d)→OK
Bending stress at the location of a rectangular hole is verified by the equations:
(M_d/W_n +M_(o,d)/W_o )/f_(m,d) ≤1 and (M_d/W_n +M_(u, (4.64 and 4.65)
Where
W_n=(b∙(h^2-h_d^2 ))/6=(45 mm∙((240 mm)^2 〖-(60 mm)〗^2 ))/6=4,0 (4.66)
M_(o,d)=A_o/(A_u+A_o )∙V_d∙a/2=(b∙h_ro)/(b∙h_ro+b∙h_ru )∙V_d∙a/2
(4.67)
M_(o,d)=(45 mm∙90 mm)/(45 mm∙90 mm+45 mm∙90 mm)∙3,13 kN∙(140 mm
W_o=(b∙h_ro^2)/6=(45 mm∙(90 mm)^2)/6=60750 mm^3
(4.69)
f_(m,0,d)=k_mod/γ_M ∙k_h∙f_(m,0,k)=0,8/1,2∙1,03∙32 N/mm^2 =22,1 N/mm^2
(M_d/W_n +M_(o,d)/W_o )/f_(m,d) =((1,1 kNm)/(4,05∙〖10〗^5 mm^3 )+(0,
(22,1N/mm^2 )=(4,5N/mm^2)/(22,1N/mm^2 )=0,20≤1→OK
Since the hole is at the centre line of the cross section, only one of the equations (4.64 and 4.65) needs to be
verified. The 140x60 mm hole 300 mm from the support edge fulfils the requirements in LVL 36 C joist.
9. CALCULATION EXAMPLES OF LVL STRUCTURES
d,(x=323mm) = 0,4 m ∙ 4,03 kNm2 ∙ �1 − 0,3 m + 0,045 m/24,5 m � = 3,13 kN
d,(x=323mm) = 0,4 m ∙ 4,03 kNm2 ∙ (0,3 m + 0,045 m/2)2 ∙ �4,5 m − �0,3 m + 0,045 m 2 ��
d,(x=323mm) = 1,1 kNm
t,90,d = 3,13 kN ∙ 0,06 m 4 ∙ 0,24 m ∙ �3 − � 0,06 m 0,24m �2 � + 0,008 ∙ 1,1 kNm 0,09 m = 0,57 kN + 0,10 kN = 0,67 kN
t,90,d = t,90,d
0,5∙ t,90 ∙ ∙ t,90 = 0,67 kN
0,5∙0,15 m∙0,045 m∙1,0 = 0,20 Nmm2 ≤ t,90,d → OK
d = τ ∙ 1,5∙ d
∙(ℎ−ℎd) ≤ v,d (4.62)
τ = 1,85 ∙ �1 + ℎ� ∙ �ℎ d ℎ �0,2 = 1,85 ∙ �1 + 0,14 m 0,24 m� ∙ �0,06 m 0,24 m�0,2 = 2,22 (4.63)
d = τ ∙ 1,5 ∙ d ∙ (ℎ − ℎd) = 2,22 ∙ 1,5 ∙ 3,13 kN
45 mm ∙ (240 mm − 60 mm) = 1,3 Nmm2
v,0,edge,d = mod M∙ v,0,edge,k = 0,8 1,2 ∙ 4,5 Nmm2 = 3,0 N/mm2
d ≤ v,0,edge,d → OK
d n+ o,d o m,d≤ 1 and d n+ u,d u m,d≤ 1 (4.64 and 4.65)
n = ∙�ℎ2−ℎd2� 6= 45 mm∙�(240 mm)2−(60 mm)2�
6 = 4,05 ∙ 105 mm3 (4.66)
o,d = o u
+ o ∙ d ∙ 2 = ∙ℎro
∙ℎ ro+ ∙ℎru ∙ d ∙ 2 (4.67)
o,d 45 mm ∙ 90 mm
45 mm ∙ 90 mm + 45 mm ∙ 90 mm ∙ 3,13 kN ∙ 140 mm 2 = 0,11 kNm
o = ∙ℎ ro2
6 = 45 mm∙(90 mm)2
6 = 60750 mm3 (4.69)
m,0,d = mod M∙ h ∙ m,0,k = 0,8 1,2 ∙ 1,03 ∙ 32 Nmm2 = 22,1 N/mm2
d n+ o,d o m,d= 1,1 kNm 4,05 ∙ 105 mm3 + 0,11 kNm 60750 mm3
22,1N/mm2 = 4,5N/mm2
22,1N/mm2 = 0,20 ≤ 1 → OK
d,(x=323mm) = 0,4 m ∙ 4,03 kNm2 ∙ �1 − 0,3 m + 0,045 m/24,5 m � = 3,13 kN
d,(x=323mm) = 0,4 m ∙ 4,03 kNm2 ∙ (0,3 m + 0,045 m/2)2 ∙ �4,5 m − �0,3 m + 0,045 m 2 ��
d,(x=323mm) = 1,1 kNm
t,90,d = 3,13 kN ∙ 0,06 m 4 ∙ 0,24 m ∙ �3 − � 0,06 m 0,24m �2 � + 0,008 ∙ 1,1 kNm 0,09 m = 0,57 kN + 0,10 kN = 0,67 kN
t,90,d = t,90,d
0,5∙ t,90 ∙ ∙ t,90 = 0,67 kN
0,5∙0,15 m∙0,045 m∙1,0 = 0,20 Nmm2 ≤ t,90,d → OK
d = τ ∙ 1,5∙ d ∙(ℎ−ℎd) ≤ v,d (4.62)
τ = 1,85 ∙ �1 + ℎ� ∙ �ℎ d
ℎ �0,2 = 1,85 ∙ �1 + 0,14
0,24 m� ∙ �0,06
0,24 m�0,2 = 2,22 (4.63)
d = τ ∙ 1,5 ∙ d ∙ (ℎ − ℎd) = 2,22 ∙ 1,5 ∙ 3,13 kN
45 mm ∙ (240 mm − 60 mm) = 1,3 Nmm2
v,0,edge,d = mod M∙ v,0,edge,k = 0 8 1,2 ∙ 4,5 Nmm2 = 3,0 N/mm2
d ≤ v,0,edge,d → OK
d n+ o,d o m,d≤ 1 and d n+ u,d u m,d≤ 1 (4.64 and 4.65)
n = ∙�ℎ2−ℎd2� 6= 45 mm∙�(240 mm)2−(60 mm)2�
6 = 4,05 ∙ 105 mm3 (4.66)
o,d = o u
+ o ∙ d ∙ 2 = ∙ℎro
∙ℎ ro+ ∙ℎru ∙ d ∙ 2 (4.67)
o,d = 45 m ∙ 90 mm
45 mm ∙ 90 mm + 45 mm ∙ 90 mm ∙ 3,13 kN ∙ 140 mm 2 = 0,11 kNm
o = ∙ℎ ro2
6 = 45 mm∙(90 mm)2
6 = 60750 mm3 (4.69)
m,0,d = mod M∙ h ∙ m,0,k = 0,8 1,2 ∙ 1,03 ∙ 32 Nmm2 = 22,1 N/mm2
d n+ o,d o m,d= 1,1 kNm 4,05 ∙ 105 mm3 + ,11 kN 60750 mm3
22,1N/mm2 = 4,5N/mm2
22,1N/mm2 = 0,20 ≤ 1 → OK
,(x 323m ) , ∙ , 2 ∙ 0,3 m + 0, /
, ,
,(x 323 ) = , ∙ , kN2 ∙ ( , , ) ∙ , , , 2
d,(x=323mm) = 1,
t,90, , ∙ ,
∙ , ∙ � , ,�2 � , ∙ , , , , ,
t,90,d = t,90,d
0,5∙ t,90 ∙ ∙ t,90 0,67 k
0,5∙0,15 ∙0,045 m∙1,0 ,2 mm2 t,90,
τ ∙ 1,5∙ d ∙(ℎ ℎd) v, (4.62)
τ , ∙ ℎ ∙ ℎ d
ℎ 0,2 , ∙ 0,14
0,24 m ∙ 0,06
0,24 m 0,2 , ( . )
d τ ∙ 1,5 ∙ d ∙ ( ) , ∙ , ∙ , 3 N
45 mm ∙ (240 mm − 60 mm) = 1,3 Nm 2
v,0,e ge, = o ∙ v,0,e ge,k 0,8 1,2 ∙ 4,5 N 2 , 2
v,0,e ge,
d n o,d o ,d ≤ 1 d n u,d u ,d 1 (4.64 and . )
n = ∙�ℎ2 ℎd2� 6= 45 ∙�(240 )2 (60 )2�
6 , ∙ 5 mm3 ( . )
o,d o u o∙ d ∙ 2 ∙ℎro
∙ℎ ro ∙ℎ ru ∙ d ∙ 2 ( . )
o, ∙
∙ 9 ∙ ∙ , ∙ m ,
o ∙ℎro2 6 = 45 ∙(90 )2
6 3 ( . )
,0, mo ∙ h ∙ ,0,k , , ∙ ,03 ∙ 32 2 = 22,1 N/mm2
o, o , ,
,0 ∙ 5 3 ,
6 3
, 2 , N 2
, 2 , K
d,(x=323mm) = 0,4 m ∙ 4,03 kNm2 ∙ �1 − 0,3 m + 0,045 m/24,5 m � = 3,13 kN
d,(x=323mm) 0,4 m ∙ 4,03 kNm 2 ∙ (0,3 m + 0,045 m/2)2 ∙ �4,5 m − �0,3 m + 0,045 m 2 ��
d,(x=323mm) = 1,1 kN
t,90,d = 3,13 kN ∙ 0,06 m 4 ∙ 0,24 m ∙ �3 − � ,060,24m �2 � + 0,008 ∙ 1,1 kNm 0,09 m = 0,57 kN + 0,10 kN = 0,67 kN
t,90,d = t,90,d
0,5∙ t,90 ∙ ∙ t,90 = 0,67 kN
0,5∙0,15 m∙0,045 m∙1,0 = 0,20 Nmm2 ≤ t,90,d → OK
d = τ ∙ 1,5∙ d
∙(ℎ−ℎd) ≤ v,d (4.62)
τ = 1,85 ∙ �1 + ℎ� ∙ �ℎ d ℎ �0,2 = 1,85 ∙ �1 + 0,14
0,24 m� ∙ �0,06
0,24 m�0,2 = 2,22 (4.63)
d = τ ∙ 1,5 ∙ d ∙ (ℎ − ℎd) = 2,22 ∙ 1,5 ∙ 3,13 kN
45 mm ∙ (240 mm − 60 mm) = 1,3 Nmm2
v,0,edge,d = mod M∙ v,0,edge,k = 0,8 1,2 ∙ 4,5 Nmm2 = 3,0 N/mm2
d ≤ v,0,edge,d → OK
d n+ o,d o m,d≤ 1 and d n+ u,d u m,d≤ 1 (4.64 and 4.65)
n = ∙�ℎ2−ℎd2� 6= 45 mm∙�(240 mm)2−(60 mm)2�
6 = 4,05 ∙ 105 mm3 6
o,d o u
+ o ∙ d ∙ 2 = ∙ℎro
∙ℎ ro+ ∙ℎru ∙ d ∙ 2 (4.67)
o,d = 45 mm ∙ 90 mm
45 mm ∙ 90 mm + 45 mm ∙ 90 mm ∙ 3,13 kN ∙ 140 mm 2 = 0,11 kNm
o = ∙ℎ ro2
6 = 45 mm∙(90 mm)2
6 = 60750 mm3 (4.69)
m,0,d = od M∙ h ∙ m,0,k = 0 8 1,2 ∙ 1 03 ∙ 32 Nmm2 = 22,1 N/mm2
d n+ o,d o m,d= 1,1 kN 4,05 ∙ 105 mm3 + ,11 kN 60750 mm3
22,1N/mm2 = 4,5N/mm2
22,1N/mm2 = 0,20 ≤ 1 → OK
d, Nm 5 m
, = ,3 m + 0, 045 m
, 23mm)
, ,
, , , ,
, , , ,
, , , , ,2 , ,
, ∙ v,
, , 4 m , 4 m , , ,
∙ (ℎ − ℎ ) − 60 m
, , , = ∙ , , , = ∙ 4,5 = 3,0 N/mm
≤ , , , → OK
+ , o , n ,,
6 6 = 4,05 ∙ 105 mm3
, + 2 ∙ℎ + ℎ
, 9
, , , , ,8 ,2 m
,
, 4,0 ∙ 10 7
,1N/mm 22,1N/m
208 LVL Handbook Europe