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01) Qc = 2,5 (1*1) Qalv.= 2,0 (0,8*8) Qt= 2,5 Tf/m + 12,8 Tf/m Qc = 2,5 Tf/m Qalv.= 12,8 Tf/m Qt= 15,3 Tf/m M = QL²/8 I = bh³/12 Ymáx = h / 2 M = 2,5*12²/8 I = 1 * 1³/12 Ymáx = 1 / 2 M = 45 Tf/m I = 0,8333 m4 Ymáx = 0,5 σ máx = M/I * Ymáx σ máx = 45 / 0,8333 * 0,5 σ máx = 270 Tf/m² 02) Qc = 2,5 (1*1) Qalv.= 2,0 (0,8*8) Qt= 2,5 Tf/m + 12,8 Tf/m Qc = 2,5 Tf/m Qalv.= 12,8 Tf/m Qt= 15,3 Tf/m σ máx = M/W M = QL²/8 W = bh²/6 σ máx = 275,3 / 0,1667 M = 15,3*12²/8 W = 1/6 σ máx = 1652,46 Tf/m² M = 275 Tf/m W = 0,1666 m4 03) Carga da Viga M máx (Viga) Tensão de Compressão Qc = ɤc (b * h) M máx = QL²/8 σc = FP / A = > FP = σc * A Qc = 2,5 (0,6*0,9) M máx = 1,35 *10²/8 FP = 120 (π * D² / 4) Qc = 1,35 Tf/m M máx = 16,875 Tf/m FP = 84,823 Tf/m M máx (Coluna) ΣM máx = M máx (Viga) + M máx (Coluna) M máx = 84,823 * 2 ΣM máx = 16,875 + 169,646 M máx = 169,646 Tf/m ΣM máx = 186,521 Tf/m σ máx = ΣM máx / I * Ymáx I = bh³/12 Ymáx = h / 2 σ máx = 186,521 / 0,03645 *0,45 I = 0,6 * 0,9³/12 Ymáx = 0,9 / 2 σ máx = 2302,72 Tf/m I = 0,03645 m4 Ymáx = 0,45 σ máx = 230,3 Kgf/cm² 04) Qc = 25 (2*1) Qalv.= 20 (0,8*H) Qt= 50 kN/m + 16 H Qc = 50 kN/m Qalv.= 16 H Qt= 66 H kN/m σ adm. = σ rup./2 M = QL²/8 W = bh²/6 σ adm. = 30 MPa./2 M= (50 + 16 H)*18² W = 1*2²/6 σ adm. = 15 MPa M= 2,025 + 648 H W = 0,666 m³ σ = M / W 15 * 10³ = 2,025 + 648 H / 0,666 15 * 10³ *0,666 = 2,025 + 648 H 648 H = 9.990 * 10³ - 2,025 H = 7.965 * 10³ / 648 H = 12,29 m 05 ) Qalv.= 20 (0,5*H) M máx = QL²/93 Tabela da Viga Qalv.= 10 H kN/m M máx = 10 H *6293 W=S= 667 X 10³ mm³ => 6,67 x 10^-4 m³ M máx = 23,094 H kN/m σ adm. = 300 MPa => 3000 Kgf/cm² => 30000 Tf/m² => 300000 kN/m² => 30 x 10^-4 kN/ m² σ máx = M máx/W 30 x 10^4 = 23,094 H /6,67 x 10^-4 H = (30 x 10^4 * 6,67 x 10^-4) /23,094 H = 8,66 m 06) M máx = PL/4 M máx = P*8/4 M máx = 2P W = 3630 x 10³ mm³ = 3,63x10^6.10^-9 m³ = 3,63x10^-3 σ adm = 2P/W σ máx = P/S 3,3 x 10^4 = 2 P/3,63x10^-3 σ máx = P /π * D² / 4 3,3 x 10^4 * 3,63x10^-3 = 2P σ máx = 59895 /π * 23² / 4 P = 59,895Tf/m σ máx =144162 kN /m² P = 59895 Kgf/m σ máx = 144,162 Kgf/cm² 07) Qc = 25 (1,5*0,8) => Qc = 30 kN/m Qalv.= 20 (0,6*6) => Qalv.= 72 kN/m Qt = Qc + Qalv => Qt = 30 + 72 => Qt = 102 kN/m M máx = QL²/8 => Mmáx = 102 * 16 ²/8 => Mmáx = 3264 kN/m Qalv. (∆) = 20 (0,6*H) => Qalv.(∆)= 12H kN/m M máx = QL²/12 M máx = 12 H * 16 ²/12 => M máx = 256 H kN/m ΣMmáx = 3264 + 256 H σadm =16 MPa => 160 Kgf/cm² => 1600 Tf/m² => 16000 kN/m² σadm = ΣMmáx. / I *Ymáx I = bh³/12 Ymáx = h/2 16000 = (3264 + 256 H) * 0,75/0,225 I = 0,8 * 1,5³/12 Ymáx = 1,5/2 16000 * 0,225/0,75 = 3264 + 256H I = 0,225 m^4 Ymáx = 0,75 m 48000 - 3264 = 256H H = 1536/256 => H = 6m 08) Viga Qc = 2,5 (2*0,8) => Qc = 4 Tf/m M máx = QL²/8 => M máx = 4*20²/8 => M máx = 200 Tf/m Pilar Qp = 100 (0,30*0,30) => Qp=90 Tf/cm² M máx = 90 * 3 => M máx = 270 Tf/m V = Q*b/L(a + b/2) => 10 Q/20(5 + 10/2) => 5Q M(10) = 5 Q * 10 – Q (10 - 5)²/2 => M máx = 50 Q – 12,5Q M máx = 37,5 Q Qalv. = 0,8 * 2* H => Qalv. =1,6 H Qalv. = 37,5 * 1,6 H => 60H ΣM máx = 200 + 270 + 60 H Y máx = h/2 I = bh³/12 ΣM máx = 470 + 60 H Y máx = 1 m I = 0,8 * 2³/12 I = 0,5333 σ adm. = ΣM máx * Y máx /I 30000 = 470 + 60 H * 1/0,5333 15.999 - 470 = 60 H H = 18,83 m 09) σadm = 120 Kgf => 1200 Tf/cm²; C.S.P. = 3,0; E = 300Tf/cm²=> 3000Tf/m²; L = 9 m; I = π D^4 / 64; A = πD ²/4. ( Engastada + Articulada) => Le = 0,7 L = 0,7 *9 => Le = 6,3 m σadm = P/S => P = σadm * A A = π* D ²/4 P = 1200* π* D ²/4 Pcr = π ² * E * I/Le² Pcr = π ² * 3x10^6 * π D^4 / 64 * 6,3² Pcr = 36619,2799 D^4 C.S.P = Pcr/P => 3 = Pcr/P => Pcr = 3* P 36619,2799 D^4 = 3 * 1200 * π * D²/4 D = √3 * 1200 * π /4 *36619,2799 D =√ 11309,733/146477,120 D = 0,2779 m => D = 27,79 cm 10 ) D = 0,2779 m Pcr = 3* P => Pcr = 3 * 1200 * π * D²/4 Pcr = 3 *(1200 * π * 0,2779² /4) => Pcr =218,36 Tf 11) σadm = 18 Mpa => 180 Kgf/cm² => 1800 Tf/m²; C.S.F = 2,5; A = π* D ²/4 E = 300 Tf/cm²=> 3000 Tf/m² => E = 3x10^6 Tf/m²; σadm = P/S => P = σadm * A C.S.F = Pcr/P => Pcr = C.S.F * P P= 1800 * π * D ²/4 Pcr = 2,5 *1710,6 P= 1800 * π * 1,1²/4 Pcr = 4276,5 Tf P = 1710,6 Tf (Articulado + Engaste) => Le = 0,7 L => L = Le/0,7 I = π D^4 / 64 Pcr = π ² * E * I/Le² I = π * 1,1 ^4 / 64 4276,5 = π ²* 3x10^6* 0,0719/Le² I = 0,0719 m^4 Le = √ π ² * 3x10^6 * 0,0719/4276,5 Le =22,31 m L = Le/0,7 => L= 22,31/ 0,7 => L = 31,87 m => L = 31,90 m 12) b = 1,1m; h = 3,2 m ; σadm = 18 Mpa; C.S.F =2,8; (Bi-Articulado) => K = 1 . Obs. : 1 Tf => 1000 Kg =. 10000 N ; E = 260 Tf/cm² => 260 x 10^8 N/m² Ix = bh³/12 Iy = b³h/12 Ix = 1,1 * 3,2³/12 Iy = 1,1³3,2/12 Ix = 3 m4 Iy = 0,355 m4 Ix < Iy - A flambagem ocorrerá em cima desse eixo. σ adm = Padm/A A = b*h 18 x 10^6 = Padm / 3,52 A = 1,1 *3,2 Padm = 63,36 x 10^6 N A = 3,52 m² Padm = Pcr / C.S.F 63,36 x 10^6= Pcr /2,8 => Pcr = 177,408 x 10^6 N (Bi-Articulado) => K = 1 Pcr = π ² * E * Iy /K* L² 177,408 x 10^6 = π ²* 260 x 10^8 * 0,355/L² L = √9,1096 x 10^10/177,408 x 10^6 L = √513,486 => L= 22,66 m => L= 22,7 m 13) σadm = 380 Mpa; D=17 cm; d=15 cm; C.S.F = 2,5; E = 21000 kN/cm² => 21000 x 10^7 Ix = Iy = π (D^4 - d^4)/64 I = π (0,17^4 – 0,15^4)/64 I= 16,15 x 10^-6 m^4 σadm = Padm/A A = π (D^2 - d^2)/4 380 x 10^6 = Padm/5,026 x 10^-3 A = π (0,17^2 - 0,15^2)/4 Padm = 191 x 10^6 N A = 5,026 x 10^-3 m² Padm = Pcr / C.S.F 1,91 x 10^6= Pcr /2,5 Pcr = 4.775 x 10^6 N (Bi-Articulado) => K= 1 Pcr = π ² * E * I /(K*L)² 4.775 x 10^6 = π ²* 21000 x 10^7 * 16,15 x 10^-6 /L² L= √33,47 x 10^6/4,775 x 10^6 L = √7,010 L= 2,64 m => L= 2,7 m 14) a = 7m; b = 3m; L = 85 m; E = 2600 kN/cm² => 2600 x 10^7 N/m²; σadm = 16 Mpa; A elipse = π. a. b; (Engaste + Articulado) => K = 0,7 Ix = π* bh³/4 Iy = π* b³h/4 Ix = π * 7* 3³/4 Iy = π* 7³* 3/4 Ix = 148,44 m4 Iy = 808,17 m4 Ix < Iy - A flambagem ocorrerá em cima desse eixo. Pcr = π ² * E * I / (K*L) ² Pcr = π ²* 26000 x 10^7 * 148,44 /(0,7* 85)² Pcr =1.076 x 10 ^10 N σ adm = Padm/A A =π* b*h 16 x 10^6 = Padm / 65,97 A = π* 7* 3 Padm = 1.056 x 10^9 N A = 65,97 m² Padm = Pcr / C.S.F 1.056 x 10^9 = 1,076 x 10 ^10/ C.S.F C.S.F = 1,076 x 10 ^10/ 1.056 x 10^9 C.S.F = 10,2 O pilar apresentou C.S.F = 10,2, bem superior ao coeficiente mínimo de C.S.F = 3,0, portanto é resistente à flambagem . 15) a = 7m; b = 3m; L = 85 m; E = 2600 kN/cm² => 2600 x 10^7 N/m²; σadm = 16 Mpa; A elipse = π. a. b; (Engaste + Articulado) => K = 0,7 Ix = π* bh³/4 Iy = π* b³h/4 Ix = π * 7* 3³/4 Iy = π* 7³* 3/4 Ix = 148,44 m4 Iy = 808,17 m4 Ix < Iy - A flambagem ocorrerá em cima desse eixo. Pcr = π ² * E * I / (K*L) ² Pcr = π ²* 26000 x 10^7 * 148,44 /(0,7* 85)² Pcr = 3809/3540 Pcr =1.076 x 10 ^10 N Pcr =10.760 kN 16) Perfil : W310X129 => (A= 16500 mm²; I = 100 x 10^-6 mm^4); C.S.F = 2,8 E = 2100 kN/cm² => 2100 x 10^7 N/m²; σadm = 380 Mpa; (Bi-Articulado) => K = 1 σadm = Padm/A σadm = Padm/C.S.F 380 x 10^6 = Padm/16500 6,27 x 10^6 = Padm/2,8 Padm = 6,27 x 10^6 N Padm = 17,556 x 10^6 (Bi-Articulado) => K = 1 Pcr = π ² * E * I / (K*L) ² 17,556 x 10^6 = π ²* 21000 x 10^7 * 100 x 10^-6 / L² L =√ π ²* 21000 x 10^7 * 100 x 10^-6/17,556 x 10^6 L =√11,81 L = 3,44 m 17) D = 120 mm; G=26 Gpa; ζmáx = 140 Mpa; F.S = 2; ρ= R= 60 mm +> 0,06 m. I = π D^4/32 I = π * 0,12^4/32 I = 20,36 x 10 ^-6 m4 ζadm = ζmáx / F.S ζadm = T * ρ / I ζadm = 140 x 10^6 /2 70 x 10^6 = T * 0,06/20,36 x 10^-6 ζadm = 70 x 10^6 T = 1.425,2/0,06 => T = 23,75 kN/m 18) ζmáx = 152 Mpa; ρ= R= 38 mm => 0,038 m. J = π R^4/2 J = π * 0,038^4/2 J= 3,28 x 10 ^-6 m4 ζmáx = Tmáx* ρ / J 152 x 10^6 = Tmáx* 0,038/3,28 x 10^-6 Tmáx = 498,56 /0,038 T = 13,12 kN/m 19) ζmáx = 152 Mpa; ρ= R= 22 mm => 0,022 m; Tmáx = = 13,12 kN/m; J = 3,28 x 10 ^-6 m4 ζ = Tmáx* ρ / J ζ = 13,12 x 10^3* 0,022 / 3,28 x 10 ^-6 ζ = 88 x 10^3 ζ = 88.000 kN/m² 20) ρ= R= 22 mm => 0,022 m ζmáx = 152 Mpa; G = 73 Gpa; ζmáx = G * ɤmáx 152 x 10^6 = 73 x 10^9 * ɤmáx ɤmáx = 152 x 10^6 / 73 x 10^6 ɤmáx = 2,08 x 10^-3 rad 21 – Obs: 1 Gpa=> 10^9 N/m² => 10^6 kN/m²; G = 73 Gpa =>73 x 10^9 N/m²; ρ = R= 38 mm => 0,038 m; ζmáx = 152 Mpa; J = (π * R^4) / 2 T = J * ζmáx/ R J= π * 0,038^4 / 2 T = 3,275 x 10^-6 * 152 x 10^6 / 0,038 J= 3,275 x 10^-6 m4 T = 13,10 KN m Ø = T* L / G*J Ø = 13,10*10³*1,60 / 3,275 x10^6 * 73 x 10^6 Ø = 0,08769 rad 22) I = bh³/12 PCR= (π²*E*I)/Le² => Le²= (π²*E*I)/PCR I = 20*20^3/12 Le²= π²* 3 x10³ *1,333 x 10^-4/1000*10³ I = 20^4/12 Le²= 39,468 => I = 13,3 x 10-³ Le=√39,468 I = 1,33 x 10^-4 m^4 Le=6,28 m 23) – E = 3000 Kn/cm² => 3 x 10^6/10^-4=>3 x 10^10 ; L = 14 m; CSF = 3,0; P = 1600 KN. CSF = PCR/ P => PCR = CSF * P PCR= (π²*E*I)/Le² => I = PCR * Le² /π²*E PCR = 1600 * 3,0 I = 4800 X 10³(0,7*14²)/π²*3 X 10^10 PCR = 4800 kN I = 0,00155 I = π*d^4/64 => d^4 = I * 64 / π d^4 = 0,00155 * 64 / π d = 4√0,03157641 d = 0,42 cm 24) E = 700 kN/cm² ; L = 6,4 m; C.S.F = 2; P = 120 KN. I = π*d^4/64 P=Pcr/C.S.F I = 0,05 D^4 m4 Pcr = 120000 * 2 Pcr = 240000 (Bi-Articulado – Le = L) PCR= (π²*E*I)/Le² 240000= π² * 700 x 10^3 * 0,05*D^4/6,4² 9830,400 = π² * 700 x 10^3 * 0,05*D^4 0,05D^4 = 9830,400 /6908,723 D^4 = 1422896822/0,05 D =4√28457936 D = 23,10 m 25 ) e = 0,2m; L= 1 m; σadm = 15 Mpa; CSF = 3,0; h = e = 0,2m σadm = P /A A = b * e P = σadm * A A = 1 * 0,2 P = 15 x 10^6 * 0,2 A = 0,2 m² P = 3000 kN 26) C.S.F = 3,0; P=3000 kN; P = Pcr / C.S.F Pcr = 3 * 3000 Pcr = 9000 kN 27) Resposta D 24 cm e 5,68 m D – L = 10 m; CSF = 3,0; σcomp = 15x10^6; e = 0,8 m E = 3000 Kn/cm² => 3 x 10^6/10^-4 => 3 x 10^10 ; (Pilar) CSF = PCR/ P => PCR = CSF * P => PCR = 3,0 * 15x10^ 6 => PCR = 45x10^6 kN/m Qc =25(1,0 * 1,0)*10 => Qc = 250 kN/m Qalv= 20(9 * 0,8)* 10 => Qalv = 1440 kN/m Fcomp = Qc + Qalv/2 => Fcomp = 1440 + 250 /2 => Fcomp = 845 kN/m σ = FCOMP/A => A = FCOMP/ σ b =√A I = bh³/12 => A = 845x10³/ 15x10³ b =√0,05633 I = 24^4 /12 A = 56,333 => A = 0 ,0563 m² b = 24 cm I = 27,648 I = 2,76x10^-4 (Viga +Alvenaria) CSF = PCR/P => PCR = CSF*Fcomp PCR= (π²*E*I) /Le² => Le²= π²*E* I /PCR PCR = 3,0 * 845 Le² = π²* 3X10^10* 2,76x10^-4/2535x10³ PCR = 2535 kN/m Le = √π²* 3x10^10 * 2,76x10^-4 /2535x10³ Le = 5,68 m 28) Resposta C - (Bi-articulado) => LE = L (Bi-engastado) => LE = 0,5 L 2*(LE = 0,5L) = L => 2 LE = L Portanto conclui que a carga crítica de flambagem de um pilar bi-engastado é o quadruplo da carga critica do pilar biarticulado 29 – D = 0,8 m; H = 20 m; (Bi-Articulada => Le² = L) I = π*d^4/64 I = π * 0,8 D^4 m4 I = 0,0201 m4 Pcr = π²*E*I /Le² Pcr = π² * 3000 x 10^3 * 0,0201/20 Pcr = 29.608,813 kN σadm = Pcr / C.S.F C.S.F = 29.609 x 10^6 / 10 x 10^6 C.S.F = 2,96 30) – C.S.F = 3,0; H = 18 m σ = 3.200 kN => 3.200 x 10^3 E = 2.800 kN => 2.800 x 10^3 (Bi- Articulado => Le = 0,5 L); Ix = b * h³ / 12 Iy = b³ * h/12 Ix = b * 18³/12 Iy = b³ * H^4/12 Ix = 486 b m4 Iy = 1,5 b³ m4 Ix< Iy P = Pcr / C.S.F Pcr = 3200 * 3,0 Pcr = 9600 kN Pcr = π ² * E * I / Le ² 9600 x 10^3 = π ²* 2.800 x 10^3* 486 b / 0,5 *18 86.400 x 10^3 / π ²* 2.800 x 10^3 = 486 b b = ³√24,512/1,5 b=0,0504 b = 50,4 cm 31 – Resposta D D- P =PCRIT / CFS I=Pi D^4/64 I=0,049087385 PCRIT= (PI² * E * I) / Le² Questão letra B P =PCRIT / CFS 200* 3 = PCRIT 600= (PI²* 300*104*I) / (22,4) ² I= 0,101677m4 IY= (B³ * H) / 12 0,1017= b³ * 3² / 12 Le=0,70l Le= 0,70*32 Le= 22,4 31 – D = 1,00 m I =π* D^4/64 I = π*1,00^4/64 I = 0,05 m4 Resposta D D- P =PCRIT / CFS I=0,049087385 PCRIT= (PI² * E * I) / Le² Questão letra B P =PCRIT / CFS 200* 3 = PCRIT 600= (PI²* 300*104*I) / (22,4) ² I= 0,101677m4 IY= (B³ * H) / 12 0,1017= b³ * 3² / 12 Le=0,70l Le= 0,70*32 Le= 22,4 32) E = 300 Tf/cm² => 300 x 10^4 Tf/m² b = 10 m; L = 32 m; Padm = 2000 Tf; C.S.F = 3,0; h = e = ? (Engaste + Articulado => K = 0,7); Ix = be³/12 Iy = b³e/12 Padm = Pcr / C.S.F Ix = 10 * e³ /12 Iy = 10³ * e /12 Pcr = 2000 * 3 Ix = 0,833 e³ m4 Iy = 808,17 m4 Pcr = 6000 Tf e < b, logo Ix < Iy - A flambagem ocorrerá em cima desse eixo. Pcr = π ² * E * I / (K*L) ² 6000 = π ²* 300 x 10^4 * 0,833 e³ /0,7² * 32 ² e³ = = 6000 * 0,7² * 32 ²/ π ²* 300 x 10^4 * 0,833 e = ³√3,01 x 10 ^6 / 24,66 x 10 ^6 e = ³√0,122 => e = 0,496 m => e = 49,6 cm 33) b = 1,2; h = 4,2; L =40 m; Ɣ = 25 kN/m³ Qvg = Ɣ * (b* h) Qvg = 25 x 10^6 (1,2*4,2) Qvg = 126 kN/m M = QL²/8 I = bh³/12 Ymáx = h/2 M = 126* 40²/8 I = 1,2 * 4,2³/12 Ymáx = 4,2/2 M = 25,20 kN I = 7,41 m4 Ymáx = 2,1 σ máx = M/I * Ymáx σ máx = 25,20 x 10^6 * 2,1 / 7,41 σ máx = 7,14 Mpa 34) Seção circular => D = 80 cm => 0,8 m I = π* D^4/64 I = π*0 ,8^4/64 I = 0,02 m4 35) Padm = 800 kN; C.S.F = 3,0; σadm = 10 Mpa; E = 300 kN/cm² => 3000 x 10^7 N/m²; σadm = Padm/A A = π* d²/4 A = 800 x 10^3/10 x 10^6 D = √0,08 * 4/ π A = 0,08 m² D = 0,319 m I = π *D^4/64 Padm = Pcr / C.S.F I = π * 0,319^4/64 Pcr = 3,0 * 800 x 10^3 * 3,0 I = 0,000508 m4 Pcr = 2400 kN (Bi- Articulado – K = L) Pcr = π²*E*I /K*L² 2400 x 10^3 = π² * 3000 x 10^7 * 0,000508/ L² L = √ π² * 3000 x 10^4 * 0,000508 / 2400 x 10^3 L = √62,671 L = 7,92 m 36) Pcr = 13000 kN; D = 1,3 m; E = 2840 kN/cm² (Bi- Articulado – K = L) I = π *D^4/64 I = π * 1,3^4/64 I = 0,140 m4 Pcr = π ² * E * I / Le ² 13000 = π² * 2840 x 10^7 * 0,140 / L² L²= π² * 2840 x 10^7 * 0,140 / 13000 L = √301.858 L = 17,37 m 37) b = 1; h = 10; σ adm = 12 Mpa; C.S.F = 3,0; (Engastado + Articulado – K = 0,7 L) E = 300 kN/cm² => 3000 x 10^7 N/m²; Ix = bh³/12 Iy = bh³/12 Ix = 1 * 10^3/12 Iy = 1³ * 10/12 Ix =83,33 m4 Iy = 0,833 m4 Ix > Iy => A flambagem ocorrerá no momento menor. σadm = Padm /A A = b * h Padm = 12 x 10^6 * 10 A = 1 * 10 P = 120 x 10^6 N A = 10 m² Padm = Pcr / C.S.F Pcr = 120 x 10^6 * 3,0 Pcr = 360 x 10^6 N Pcr = π²*E*I /K*L² 360 x 10^6 = π² * 3000 x 10^7 * 0,833/0,7² * L² L = √ π² * 3000 x 10^7 * 0,833/360 x 10^6 * 0,7² L = √ 2.466 x 10^4/176,4 x 10^6 L =√1.398 x 10^3 L = 37,4 m 38 ) D = 40 cm=> 0,4 m; d = 38 cm => 0,38 m; e = 1 cm; h = 6 m; ζmáx = 300 Mpa; ρ= R= 20 cm => 0,2 m. I = π (D^4 - d^4)/32 I = π (0,4^4 - 0,38^4)/32 I = 466,2 x 10^-6 ζmáx = T * ρ /I 300 x 10^6 = T * 0,2/466,2 x 10^-6 T = 300 x 10^6 * 466,2 x 10^-6/0,2 T = 139,860/0,2 T = 699,3 kN/m 39) E = 3000 kN/cm²; V = 0,2 (Módulo de Poison). G = E/2(1 + v) G = 3000/2(1 + 0,2) G = 1250 Kn/cm² 40) D = 1 m; T = 2000 Kn/m; ρ= R= 0,5 σmáx = T * ρ /J J = π * R^4/2 σmáx = 2000 X 10^3 * 0,5 / 0,098 J = π *0,5 ^4 σmáx = 10,20 Mpa J = 0,098 m4 41) L = 10 m; ɤ= 25 kN => 25x10^3; E = 3000 kN/m => 3 x 10^10; ΣFy = 0 Q = ɤ * A RA = 25 x 10^3 * 10 Q = 25 x 10^3 * 1² RA = 25 x 10^4 Q = 25 x 10^3 N/m² ΣMa = 0 ΣFy = 0 M – Q * 5 =0 RA – Q x – V =0 M = 25 x 10^4 * 5 V = - 25 x 10^4 + 25 x 10^3 x M = 125 x 10^4 ΣMa = 0 - 25 x 10^4 + 25 x 10^3 x * x/2 + 125 x 10^4 + M = 0 M = 25 x 10^4 x - 25000 x²/2 - 125 x 10^4 M = - 125000 x² + 25 x 10^4 x - 125 x 10^4 M = EI d²v/dx² EI d²v/dx² = - 125000 x² + 25 x 10^4 x - 125 x 10^4 EI dv/dx =- 125000 x^3/3 + 25 x 10^4 x²/2 - 125 x 10^4 x + C1 EI v ==- 125000 x^4/12 + 25 x 10^4 x^3/6 - 125 x10^4 x²/2 + C1X + C2 C1 = 0 => X = 0 (no engaste) C2 = 0 => V = 0 (no engaste) X = 5 (no balanço) I = a^4/12 => I =1^4/12 => I = 0,08333 EI V = - 125000 (5^4) /12 + 25 x 10^4 (5^3) /6 - 125 x10^4 (5²)/2 3 x 10^10 * 0,08333 V = 651041,66 + 5208333,33 - 15625000 0,25 x 10 ^10 V = - 11067708,33 x 10 ^6/3 x 10^10 V = 11067708,33 /0,25 x 10 ^10 V = 4,427083 x 10 ^ -3 V = 4,43 mm 42) Qc = 25000 (0,8 * 1,2) Qc = 24000 kN/m M = QL²/8 I = bh2/6 Ymáx = h / 2 M = 24000*8²/8 I = 0,8 * 1,2 ²/6 Ymáx = 1,2 / 2 M = 192 kN/m I = 0,192 m4 Ymáx = 0,6 σ máx = M/I * Ymáx σ máx = 192000 / 0,192 * 0,6 σ máx = 6 Mpa 43) I = bh³/12 I = 1 * 2³/12 I = 0,666 2* I = bh³/12 2 * 0,666 = 1 * h³/12 1.332 * 12 = h³ h= ³√15984 h= 2,52 m44) 45) 46) 47) 48) 49)
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