ED 6° semestre

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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 = √3.018,580
 L = 54,94 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
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