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 La vérification de la concentration des contraintes de cisaillement au bord du trou doit satisfaire à la condition suivante : τ_d=k_τ∙(1,5∙V_d)/(b∙(h-h_d ) )≤f_(v,d) (4.62) où 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 La contrainte de flexion à l’emplacement d’une ouverture rectangulaire est vérifiée à l’aide des équations suivantes : (M_d/W_n +M_(o,d)/W_o )/f_(m,d) ≤1 and (M_d/W_n +M_(u, (4.64 et 4.65) où 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 Étant donné que le trou se trouve sur la ligne médiane de la section, seule l’une des équations (4.64 et 4.65) doit être vérifiée. Le trou de 140 x 60 mm situé à 300 mm du bord du support répond aux exigences de la solive LVL 36 C. 9. EXEMPLES DE CALCUL DE STRUCTURES EN LAMIBOIS d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ �1−0,3 m +4,05,0m45 m/2� = 3,13 kN d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ (0 2 ,3 m + 0,045 m/2)∙ �4,5 m−�0,3 m + 0,04 2 5 m�� d,(x=323mm) = 1,1 kNm t,90,d = 3,13 kN∙ 4 0,06 m ∙ 0,24 m ∙ �3−�00,,0264mm�2� +0,008∙ 10,1,0k9Nmm = 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 =0,20m N m2 ≤ t,90,d →OK d = τ ∙ 1,5∙ d (ℎ−ℎd) ≤ v,d (4.62) τ =1,85∙ �1+ ℎ� ∙ �ℎdℎ �0,2 =1,85∙ �1+0,14 0 m ,24 m� ∙ �0,06 0 m ,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 m N m2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,5 m N m2 = 3,0 N/mm2 d ≤ v,0,edge,d →OK d n+ o,d o m,d ≤ 1 et 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 = ro ro+ ru ∙ d ∙ 2 (4.67) o,d = 45 mm∙ 90 mm 45 mm∙ 90 mm+ 45 mm∙ 90 mm∙ 3,13 kN∙ 1402mm = 0,11 kNm o = r2 o 6 =45 mm∙(90 mm)2 6 = 60750 mm3 (4.69) m,0,d = mod M ∙ h ∙ m,0,k = 0 1 , , 8 2∙ 1,03∙ 32 m N m2 = 22,1 N/mm2 d n + o,d o m,d = 4,05 1,1 kNm ∙ 105 mm3 + 600,71510kmNmm3 22,1N/mm2 = 4,5N/mm2 22,1N/mm2 =0,20≤1→OK d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ �1−0,3 m +4,05,0m45 m/2� = 3,13 kN d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ (0 2 ,3 m + 0,045 m/2)∙ �4,5 m−�0,3 m + 0,04 2 5 m�� d,(x=323mm) = 1,1 kNm t,90,d = 3,13 kN∙ 4 0,06 m ∙ 0,24 m ∙ �3−�00,,0264mm�2� +0,008∙ 10,1,0k9Nmm = 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 =0,20m N m2 ≤ t,90,d →OK d = τ ∙ 1,5∙ d (ℎ−ℎd) ≤ v,d (4.62) τ =1,85∙ �1+ ℎ� ∙ �ℎdℎ �0,2 =1,85∙ �1+0,14 0 m ,24 m� ∙ �0,06 0 m ,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 m N m2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,5 m N m2 = 3,0 N/mm2 d ≤ v,0,edge,d →OK d n+ o,d o m,d ≤ 1 et 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 = ro ro+ ru ∙ d ∙ 2 (4.67) o,d = 45 mm∙ 90 mm 45 mm∙ 90 mm+ 45 mm∙ 90 mm∙ 3,13 kN∙ 1402mm = 0,11 kNm o = r2 o 6 =45 mm∙(90 mm)2 6 = 60750 mm3 (4.69) m,0,d = mod M ∙ h ∙ m,0,k = 0 1 , , 8 2∙ 1,03∙ 32 m N m2 = 22,1 N/mm2 d n + o,d o m,d = 4,05 1,1 kNm ∙ 105 mm3 + 600,71510kmNmm3 22,1N/mm2 = 4,5N/mm2 22,1N/mm2 =0,20≤1→OK d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ �1−0,3 m +4,05,0m45 m/2� = 3,13 kN d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ (0 2 ,3 m + 0,045 m/2)∙ �4,5 m−�0,3 m + 0,04 2 5 m�� d,(x=323mm) = 1,1 kNm t,90,d = 3,13 kN∙ 4 0,06 m ∙ 0,24 m ∙ �3−�00,,0264mm�2� +0,008∙ 10,1,0k9Nmm = 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 =0,20m N m2 ≤ t,90,d →OK d = τ ∙ 1,5∙ d (ℎ−ℎd) ≤ v,d (4.62) τ =1,85∙ �1+ ℎ� ∙ � dℎ �0,2 =1,85∙ �1+014 0 m ,24 m� ∙ �006 0 m ,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 m N m2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,5 m N m2 = 3,0 N/mm2 d ≤ v,0,edge,d →OK d n+ o,d o m,d ≤ 1 et 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 = ro ro+ ru ∙ d ∙ 2 (4.67) o,d = 45 mm∙ 90 mm 45 mm∙ 90 mm+ 45 mm∙ 90 mm∙ 3,13 kN∙ 1402mm = 0,11 kNm o = r2 o 6 =45 mm∙(90 mm)2 6 = 60750 mm3 (4.69) m,0,d = mod M ∙ h ∙ m,0,k = 0 1 , , 8 2∙ 1,03∙ 32 m N m2 = 22,1 N/mm2 d n + o,d o m,d = 4,05 1,1 kNm ∙ 105 mm3 + 600,71510kmNmm3 22,1N/mm2 = 4,5N/mm2 22,1N/mm2 =0,20≤1→OK d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ �1−0,3 m +4,05,0m45 m/2� = 3,13 kN ,( ) = 0,4 m∙ 4,03 m kN2 ∙ (0 2 ,3 m + 0,045 m/2)∙ �4,5 m−�0,3 m + 0,04 2 5 m�� d,(x=323mm) = 1,1 kNm t,90,d = 3,13 kN∙ 4 0,06 m ∙ 0,24 m ∙ �3−�00,,0264mm�2� +0,008∙ 10,1,0k9Nmm = 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 =0,20m N m2 ≤ t,90,d →OK d = τ ∙ 1,5∙ d (ℎ−ℎd) ≤ v,d (4.62) τ =1,85∙ �1+ ℎ� ∙ �ℎdℎ �0,2 =1,85∙ �1+0,14 0 m ,24 m� ∙ �0,06 0 m ,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 m N m2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,5 m N m2 = 3,0 N/mm2 d ≤ v,0,edge,d →OK d n+ o,d o m,d ≤ 1 et 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 = ro ro+ ru ∙ d ∙ 2 (4.67) ,d = 45 mm∙ 90 mm 45 mm∙ 90 mm+ 45 mm∙ 90 mm∙ 3,13 kN∙ 1402mm = 0,11 kNm o = r2 o 6 =45 mm∙(90 mm)2 6 = 60750 mm3 (4.69) m,0,d = mod M ∙ h ∙ m,0,k = 0 1 , , 8 2∙ 1,03∙ 32 m N m2 = 22,1 N/mm2 d n + o,d o m,d = 4,05 1,1 kNm ∙ 105 mm3 + 600,71510kmNmm3 22,1N/mm2 = 4,5N/mm2 22,1N/mm2 =0,20≤1→OK d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ �1−0,3 m +4,05,0m45 m/2� = 3,13 kN d,(x=323mm) = 0,4 m∙ 4,03 m kN2 ∙ (0 2 ,3 m + 0,045 m/2)∙ �4,5 m−�0,3 m + 0,04 2 5 m�� d,(x=323mm) = 1,1 kNm t,90,d = 3,13 kN∙ 4 0,06 m ∙ 0,24 m ∙ �3−�00,,0264mm�2� +0,008∙ 10,1,0k9Nmm = 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 =0,20m N m2 ≤ t,90,d →OK d = τ ∙ 1,5∙ d (ℎ−ℎd) ≤ v,d (4.62) τ =1,85∙ �1+ ℎ� ∙ �ℎdℎ �0,2 =1,85∙ �1+0,14 0 m ,24 m� ∙ �0,06 0 m ,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 m N m2 v,0,edge,d = mod M ∙ v,0,edge,k = 0 1 , , 8 2∙ 4,5 m N m2 = 3,0 N/mm2 d ≤ v,0,edge,d →OK d n+ o,d o m,d ≤ 1 et 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 = ro ro+ ru ∙ d ∙ 2 (4.67) o,d = 45 mm∙ 90 mm 45 mm∙ 90 mm+ 45 mm∙ 90 mm∙ 3,13 kN∙ 1402mm = 0,11 kNm o = r2 o 6 =45 mm∙(90 mm)2 6 = 60750 mm3 (4.69) m,0,d = mod M ∙ h ∙ m,0,k = 0 1 , , 8 2∙ 1,03∙ 32 m N m2 = 22,1 N/mm2 d n + o,d o m,d = 4,05 1,1 kNm ∙ 105 mm3 + 600,71510kmNmm3 22,1N/mm2 = 4,5N/mm2 22,1N/mm2 =0,20≤1→OK 208 Manuel sur le Lamibois (LVL) – Europe
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