TRELIÇAS

TRELIASSo estruturas formadas por barras, ligadas entre si atravs de ns.

Consideramos para efeito de clculo os esforos aplicados nos ns.

Existem alguns tipos de calculo para determinao dos esforos nas barras, como o Mtodo dos Ns, Mtodo Ritter ou Mtodos das sees.

Nesta apostila, sero resolvidos apenas exerccios de trelias pelo Mtodo dos Ns.

Para determinar os esforos internos nas barras das trelias plana, devemos verificar a condio de Isosttica da Trelia, sendo o primeiro passo.

Depois calculamos as reaes de apoio e os esforos normais axiais nos ns. Tais esforos sero denominados de N.

1 Condio de Trelia Isosttica:

2 . n = b + Sendo 2 Calcular as Reaes de Apoio (Vertical e Horizontal):Fx = 0

Fy = 0

M = 0 (Momento fletor)

Por conveno usaremos: no sentido horrio no sentido anti-horrio +

-

3 Mtodos dos Ns

Quando calculamos os esforos, admitimos que as foras saem dos ns e nos prximos ns usamos os resultados das foras do n anterior fazendo a troca de sinais.

Importante lembrar que somente o jogo de sinais devero ser feitos na equao dos ns, pois as foras das reaes horizontais e verticais devem ser inseridos na equao considerando-se exclusivamente os sinais que possuem, ou seja, no fazer jogo de sinais para tais reaes.

Calma, nos exercicios ver que fcil.

Por Conveno os sinais das foras das barras so: + TRAO

- COMPRESSOTrelia Esquemtica

Exerccios

1) Calcule as reaes de apoio e as foras normais nas barras atravs do Mtodo dos Ns.

1 Passo Condio de Isosttica

2.n = b+

2.6 = 9+3

12 = 12 OK2 Passo Reaes de Apoio

Fx = 0

Fy = 0

M = 0 (Momento fletor)

HE = 0

VA+VE = 50+100+50

VA.4-50.4-100.2 = 0 VA+VE = 200 KN VA = 4004 100+VE = 200 KN VA = 100 KN VE = 200-100

VE = 100 KN

3 Passo Mtodo dos Ns

Decomposio das foras

N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VA+NAB = 0

NAF = 0

100+NAB = 0

NAB = -100 KN

N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-50-NBA-NBF.cos45 = 0 NBC+NBF.sen45 = 0

-50-(-100)-NBF.cos45 = 0 NBC+70,7.sen45 = 0

-NBF = -50cos45

NBC = - 50 KN

NBF = 70,7 KNN C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-100-NCF = 0

-NCB+NCD = 0

NCF = -100 KN

-(-50)+NCD = 0

NCD = - 50 KN

N F

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NFC+NFB.sen45+NFD.sen45 = 0 -NFB.cos45+NFD.cos45-NFA+NFE = 0

-100+70,7.sen45+NFD.sen45 = 0 -70,7.cos45+70,7.cos45-0+NFE = 0

NFD = 50sen45

NFE = 0 KN

NFD = 70,7 KNN E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NED+100 = 0

0-HE = 0

NED = -100 KN HE = 0 KNN D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-50-NDF.sen45-NDE = 0 -NDC-NDF.cos45 = 0

-50-70,7.sen45+100 = 0-(-50)-70,7.cos45 = 0

-50-50+100 = 0

50-50 = 0

0 = 0

0 = 0

BARRAFORAS NORMAIS AXIAIS (KN)ESFORO

NAB-100COMPRESSO

NED-100COMPRESSO

NAF0-

NEF0-

NBC -50COMPRESSO

NDC-50COMPRESSO

NBF70,7TRAO

NDF70,7TRAO

NCF-100COMPRESSO



2.n = b+

2.5 = 7+3

10 = 10 OK2 Passo Reaes de Apoio

Fx = 0

Fy = 0

M = 0 (Momento fletor)HA+HB = 40

VB = 20 KN -HA.2+20.4+40.1 = 0

60+HB = 40

-HA.2+120 = 0HB = 40-60

HA = 1202HB = -20 KN

HA = 60 KN3 Passo Mtodo dos Ns


N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VB-NBA-NBC.sen26,57 = 0

-HB+NBC.cos26,57 = 0

20-NBA-22,36.sen26,57 = 0 -20+NBC.cos26,57 = 0

NBA = 10 KN

NBC = 20cos26,57

NBC = 22,36 KN

N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NAB+NAC.sen26,57 = 0

HA+NAC.cos26,57+NAE = 0

10+NAC.sen26,57 = 0 60+(-22,36).cos26,57+NAE = 0

NAC = -10sen26,57 NAE+60-20 = 0

NAC = -22,36 KN

NAE = -40 KN

N E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NEC = 0 -NEA+NED = 0

-(-40)+NED = 0

NED = -40 KN

N C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0NCB.sen26,57-NCA.sen26,57-NCE-NCD.sen26,57=0 -40-NCB.cos26,57-NCA.cos26,57+NCD.cos26,57 = 0 22,36.sen26,57-(-22,36).sen26,57-0-NCD.sen26,57=0 -40-22,36.cos26,57-(-22,36).cos26,57+44,7.cos26,57=0 10+10-NCD.sen26,57=0 -40-20+20+40 = 0

NCD = 20sen26,57

0 = 0

NCD = 44,7 KN

N D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-20+NDC.sen26,57 = 0

-NDC.cos26,57-NDE = 0

-20+44,7.sen26,57 = 0 -44,7.cos26,57-(-40) = 0

-20+20 = 0 -40+40 = 0

0 = 0

0 = 0


NAB10TRAO

NBC22,36TRAO

NAC-22,36COMPRESSO

NAE-40COMPRESSO

NEC0-

NED-40COMPRESSO

NCD44,7TRAO



2.n = b+

2.8 = 13+3

16 = 16 OK2 Passo Reaes de ApoioFx = 0

Fy = 0

M = 0 (Momento fletor)HA = 0

VA+VB = 2+2+2

-VB.16+2.12+2.8+2.4=0

VA = 6-3

VB = 4816

VA = 3 t

VB = 3 t3 Passo Mtodo dos Ns


N 1

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N13.sen36,87+VA = 0

HA+N13.cos36,87+N12 = 0

N13.sen36,87+3 = 0 0+(-5).cos36,87+N12 = 0

N13 = -3sen36,87

N12 = 4 t

N13 = -5 t

N 2

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N23 = 0

-N21+N24 = 0

-4+N24 = 0

N24 = 4 t

N 3

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-2-N34.sen36,87-N32-N31.sen36,87+N35.sen36,87 = 0 +N34.cos36,87-N31.cos36,87+N35.cos36,87 = 0-2- N34.sen36,87-0-(-5).sen36,87+N35.sen36,87 = 0 +N34.cos36,87-(-5).cos36,87+N35.cos36,87 = 0 -N34.sen36,87+N35.sen36,87-2+3 = 0 +N34.cos36,87+N35.cos36,87+4 = 0 (-N34+N35).sen36,87 = -1 (+N34+N35).cos36,87 = -4

N34-N35 = 1sen36,87 N34+N35 = -4cos36,87 N34-N35 = 1,67 1 N34+N35 = -5 2

Sistema de Equaes

Substituindo na equao 1 ou 2 1 N34-N35 = 1,67

N34+N35 = -52 N34+N35 = -5

-1,67+N35 = -5

2N34 = -3,33

N35 = -5+1,67 N34 = -3,332

N35 = -3,33 t N34 = -1,67 tN 5

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 -2-N53.sen36,87-N57.sen36,87-N54 = 0 -N53.cos36,87+N57.cos36,87 = 0 -2- (-3,33).sen36,87-(-3,33).sen36,87-N54 = 0 -(-3,33).cos36,87+N57.cos36,87 = 0 -2+2+2-N54 = 0 2,66+N57.cos36,87 = 0 N54 = 2 t N57 = -2,66cos36,87

N57 = -3,33 tN 4

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 N43.sen36,87+N45+N47.sen36,87 = 0 -N42+N46-N43.cos36,87+N47.cos36,87 = 0

+(-1,67).sen36,87+2+N47.sen36,87 = 0 -(+4)+N46-(-1,67).cos36,87+(-1,67).cos36,87 = 0

-1+2+N47.sen36,87 = 0 N46-4+1,34-1,34 = 0

N47 = -1sen36,87 N46 = 4 t

N47 = 1,67 t

Por simetria dos carregamentos e das caractersticas das barras (dimenses, ngulos), as barras dos ns 7, 6 e 8 so iguais as dos ns 1, 2 e 3, conforme tabela.

BARRAFORAS NORMAIS AXIAIS (t)ESFORO

N13 = N87-5COMPRESSO

N12 = N864TRAO

N24 = N644TRAO

N23 = N670-

N34 = N74-1,67COMPRESSO

N35 = N75-3,33COMPRESSO

N542TRAO



2.n = b+

2.7 = 11+3


Fy = 0

M = 0 (Momento fletor)HA+HB =0

VB = 0

-HA.3+2.6+2.4+2.2=0

HA = -HB

HA = 243

HB = -8 t

HA = 8 t3 Passo Mtodo dos Ns


N 5

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VB-N51-N56.sen26,57 = 0

-HB+N56.cos26,57 = 0

6-N51-8,94.sen26,57 = 0 -8+N56.cos26,57 = 0

-N51+6-4 = 0 N56 = 8cos26,57

N51 = 2 t

N56 = 8,94 t

N 1

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N15+N16.sen45 = 0

HA+N12+N16.cos45 = 0

2+N16.sen45 = 0 8+N12+(-2,83).cos45 = 0

N16 = -2sen45 N12+8-2 = 0

N16 = -2,83 t

N12 = - 6 tN 6

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-2+N65.sen26,57-N61.sen45-N62-N67.sen26,57=0-N65.cos26,57-N61.cos45+N67.cos26,57=0

-2+8,94.sen26,57-(-2,83).sen45-N62-6,7.sen26,57=0 -8,94.cos26,57-(-2,83).cos45+N67.cos26,57=0

-2+4+2-3-N62 = 0 -8+2+N67.cos26,57 = 0

N62 = 1 t

N67 = 6cos26,57

N67 = 6,7 tN 2

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N26+N27.sen26,57 = 0 -N21+N23+N27.cos26,57 = 0

1+N27.sen26,57 = 0 -(-6)+N23+(-2,23).cos26,57 = 0

N27 = -1sen26,57 N23+6-2 = 0

N27 = -2,23 t

N23 = -4 t

N 3

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N37 = 0 -N32+N34 = 0

-(-4)+N34 = 0

N34 = -4 tN 7

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-2+N76.sen26,57-N72.sen26,57-N73-N74.sen26,57=0 -N76.cos26,57-N72.cos26,57+N74.cos26,57=0

-2+6,7.sen26,57-(-2,23).sen26,57-0-N74.sen26,57=0 -6,7.cos26,57-(-2,23).cos26,57+4,47.cos26,57=0

-2+3+1-N74.sen26,57 = 0 -6+2+4 = 0

N74 = 2sen26,57

0 = 0

N74 = 4,47 t

N 4

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-2+N47.sen26,57 = 0 -N43-N47.cos26,57 = 0

-2+4,47.sen26,57 = 0 -(-4)-4,47.cos26,57 = 0

-2+2 = 0 +4-4 = 0

0 = 0

0 = 0

BARRAFORAS NORMAIS AXIAIS (t)ESFORO

N512TRAO

N568,94TRAO

N16-2,83COMPRESSO

N12-6COMPRESSO

N621TRAO

N676,7TRAO

N27-2,23COMPRESSO

N23-4COMPRESSO

N370-

N34-4COMPRESSO

N744,47TRAO



2.n = b+

2.5 = 7+3


Fy = 0 M = 0 (Momento fletor)HA = 0

VA+VB = 10+20

-VB.2,4+20.1,8+10.0,6=0

VA+17,5 = 30

VB = 422,4

VA = 30-17,5

VB = 17,5 KN

VA = 12,5 KN


N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VA+NAC.sen53,13 = 0 HA+NAE+NAC.cos53,13 = 0

12,5+NAC.sen53,13 = 0 0+NAE+(-15,63).cos53,13 = 0

NAC = -12,5sen53,13 NAE = 9,38 KN

NAC = -15,63 KN

N C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-10-NCA.sen53,13-NCE.sen53,13=0 -NCA.cos53,13+NCD+NCE.cos53,13=0-10-(-15,63).sen53,13-NCE.sen53,13=0 -(-15,63).cos53,13+NCD+3,13.cos53,13=0

-10+12,50-NCE.sen53,13 = 0 9,38+NCD+1,88 = 0

NCE = 2,5sen53,13

NCD = -11,26 KN

NCE = 3,13 KN

N E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 NEC.sen53,13+NED.sen53,13 = 0 -NEA+NEB+NED.cos53,13-NEC.cos53,13 = 0

3,13.sen53,13+NED.sen53,13 = 0 -9,38+NEB+(-3,13).cos53,13-(+3,13).cos53,13 = 0

NED = -2,5sen53,13 NEB-9,38-1,88-1,88 = 0

NED = -3,13 KN NEB = 13,14 KN

N D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-20-NDE.sen53,13-NDB.sen53,13=0 -NDE.cos53,13-NDC+NDB.cos53,13=0-20-(-3,13).sen53,13-NDB.sen53,13=0 -(-3,13).cos53,13-(-11,26)+(-21,88).cos53,13=0

-20+2,50-NDB.sen53,13 = 0 1,88+12,26-13,13 = 0

NDB = -17,50sen53,13

0 = 0

NDB = -21,88 KN

N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VB+NBD.sen53,13 = 0

-NBD.cos53,13-NBE = 0

17,5+(-21,88).sen53,13 = 0 -(-21,88).cos53,13-13,14 = 0

0 = 0

0 = 0


NAC-15,63COMPRESSO

NAE9,38TRAO

NCE3,13TRAO

NCD-11,26COMPRESSO

NED-3,13COMPRESSO

NEB13,14TRAO

NDB-21,88COMPRESSO



2.n = b+

2.5 = 7+3


Fy = 0 M = 0 (Momento fletor)HA+HB = 0 VA = 225

-HB.0,9+75.2,4+150.1,2 = 0

HB = -HA

HB = 3600,9HA = - 400 KN

HB = 400 KN


N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NBA = 0 HB+NBD = 0

400+NBD = 0

NBD = -400 KN

N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VA-NAB-NAD.sen36,87 = 0 -HA+NAC+NAD.cos36,87 = 0

225-0-NAD.sen36,87 = 0 -400+NAC+375.cos36,87 = 0

NAD = 225sen36,87 NAC = 100 KN

NAC = 375 KN

N D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NDA.sen36,87+NDC = 0 -NDA.cos36,87-NDB+NDE = 0

375.sen36,87+NDC = 0

-375.cos36,87-(-400)+NDE = 0

NDC = -225 KN -300+400+NDE = 0

NDE = -100 KN

N C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-150-NCD-NCE.sen36,87=0 -NCA+NCE.cos36,87=0-150-(-225)-NCE.sen36,87=0

-100+125.cos36,87 = 0 -150+225-NCE.sen36,87 = 0 -100+100 = 0

NCE = 75sen36,87

0 = 0

NCE = 125 KN

N E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-75+NEC.sen36,87 = 0 -NEC.cos36,87-NED = 0

-75+125.sen36,87 = 0 -125.cos36,87-(-100) = 0

-75+75 = 0 -100+100 = 0

0 = 0

0 = 0BARRAFORAS NORMAIS AXIAIS (KN)ESFORO

NBA0-

NBD-400COMPRESSO

NAD375TRAO

NAC100TRAO

NDC-225COMPRESSO

NDE-100COMPRESSO

NCE125TRAO



2.n = b+

2.8 = 13+3


Fy = 0 M = 0 (Momento fletor)HE = 0 VA+VE = 8

VA.4a-4.3a-4.1a = 0

4+VE = 8

VA.4a-12a-4a = 0

VE = 8-4

VA = 16a4a

VE = 4 KN

VA = 4 KN3 Passo Mtodo dos Ns

N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VA+NAB.sen30 = 0 NAB.cos30+NAF = 0

4+NAB.sen30 = 0 +(-8)cos30+NAF = 0

NAB = -4sen30 NAF = 6,9 KN

NAB = -8 KN

N F

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NFB-4 = 0

-NFA+NFG = 0

NFB = 4 KN

-6,9+NFG = 0

NFG = 6,9 KNN B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0NBC.sen30-NBA.sen30-NBF-NBG.sen30 = 0 -NBA.cos30+NBC.cos30+NBG.cos30 = 0 NBC.sen30-(-8).sen30-4-NBG.sen30 = 0 -(-8).cos30+NBC.cos30+NBG.cos30 = 0 NBC.sen30-NBG.sen30+4-4 = 0 6,9+NBC.cos30+NBG.cos30 = 0

NBC.sen30-NBG.sen30 = 0 NBC.cos30+NBG.cos30 = -6,9

(NBC-NBG).sen30 = 0 (NBC+NBG).cos30 = -6,9

NBC-NBG = 0sen30

NBC+NBG = -6,9cos30

NBC-NBG = 0 1 NBC+NBG = -8 2

Sistema de Equaes

Substituindo na equao 1 ou 2 1 NBC-NBG = 0

NBC-NBG = 02 NBC+NBG= -8

-4 - NBG = 0

2NBC = -8

NBG = -4 KN NBC = -82

NBC = -4 KNN C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 -NCB.sen30-NCD.sen30-NCG = 0 -NCB.cos30+NCD.cos30 = 0 -(-4).sen30-(-4).sen30-NCG = 0 -(-4).cos30+NCD.cos30 = 0 2 + 2- NCG = 0 3,5+NCD.cos30 = 0

NCG = 4 KN NCD = -3,5cos30

NCD = -4 KN

N G

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NGB.sen30+NGC+NGD.sen30 = 0 -NGB.cos30+NGD.cos30-NGF+NGH = 0

-4.sen30+4+NGD.sen30 = 0 -(-4).cos30+(-4).cos30-6,9+NGH = 0

NGD = -2sen30 +3,5-3,5-6,9+NGH = 0 NGD = -4 KN NGH = 6,9 KN

Por simetria dos carregamentos e das caractersticas das barras (dimenses, ngulos), as barras dos ns H, D e E no precisam ser calculadas.


NAB = NED-8COMPRESSO

NAF = NEH6,9TRAO

NFG = NHG6,9TRAO

NFB = NHD4TRAO

NBC = NDC-4COMPRESSO

NBG = NDG-4COMPRESSO

NCG4TRAO



2.n = b+

2.10 = 17+3


Fy = 0 M = 0 (Momento fletor)HJ = 0 VF+VJ = 2000 VF.4a-400.4a-400.3a-400.2a-400.1a = 0

1000+VJ = 2000 VF = 4000a4a

VJ = 2000-1000

VF = 1000 N VJ = 1000 N


N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-400-NAF = 0

NAB = 0

NAF = -400 N

N F

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NFB.sen45+NFA+VF = 0 NFB.cos45 + NFG = 0

NFB.sen45+(-400)+1000 = 0 -848,5.cos45+NFG = 0

NFB = -600sen45 NFG = 600 N

NFB = -848,5 N

N G

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NGB = 0 -NGF+NGH = 0

-600+NGH = 0

NGH = 600 NN B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-400-NBF.sen45-NBG-NBH.sen45=0 -NBA+NBC-NBF.cos45+NBH.cos45 = 0

-400-(-848,5).sen45-0-NBH.sen45=0 -0+NBC-(-848,5).cos45+282,8.cos45=0

-400+600-NBH.sen45=0 NBC+600+200 = 0

NBH = 200sen45

NBC = -800 N

NBH = 282,8 N

N C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-400-NCH=0

-NCB+NCD = 0

NCH = -400 N

-(-800)+NCD=0

NCD = -800 NPor simetria dos carregamentos e das caractersticas das barras (dimenses, ngulos), as barras dos ns H, D, I, E e J no precisam ser calculadas.

BARRAFORAS NORMAIS AXIAIS (N)ESFORO

NAB = NED0-

NAF = NEJ-400COMPRESSO

NFB = NJD-848,5COMPRESSO

NFG = NJI600TRAO

NGB = NID0-

NGH = NIH600TRAO

NBH = NDH282,8TRAO

NBC = NDC-800COMPRESSO

NCH-400COMPRESSO



2.n = b+

2.6 = 8+4


Fy = 0 M = 0 (Momento fletor)HE = 18 KN VE+VF = 0 VE.3,6-9.5,4-9.2,7 = 0

HF = 0

VE = -VF VE.3,6-48,6-24,3 = 0

VF = -20,25 KN

VE = 72,93,6

VE = 20,25 KN


N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NAC = 0

-9+NAB = 0

NAB = 9 KN

N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 -NBD-NBC.sen36,87=0

-NBA-NBC = 0

-NBD-(-11,25).sen36,87=0 -NBA-NBC.cos36,87=0

-NBD+6,75=0 -9-NBC.cos36,87 = 0

NBD = 6,75 KN

NBC = -9cos36,87

NBC = -11,25 KN

N C

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NCA-NCE+NCB.sen36,87 = 0 -9+NCB.cos36,87+NCD = 0

0-NCE+(-11,25).sen36,87 = 0 -9+(-11,25).cos36,87+NCD = 0

NCE = -6,75 KN -9-9+NCD = 0 NCD = 18 KN

N D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0 NDB-NDE.sen36,87-NDF=0 -NDE.cos36,87-NDC = 0

6,75-(-22,5).sen36,87-NDF=0 -NDE.cos36,87-18 =0

6,75+13,5-NDF=0 NDE = -18cos36,87

NDF = 20,25 KN

NDE = -22,5 KN

N E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VE+NEC+NED.sen36,87 = 0 HE+NED.cos36,87 = 0

20,25+(-6,75)+(-22,5).sen36,87 = 0 HE+(-22,5).cos36,87 = 0

20,25-6,75-13,5 = 0 HE = 18 KN Ok0 = 0

N F

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NFD-VF = 0 -HF = 0

20,25-20,25 = 0 HF = 0 Ok

0 = 0 BARRAFORAS NORMAIS AXIAIS (KN)ESFORO

NAB9TRAO

NAC0-

NCE-6,75COMPRESSO

NCD18TRAO

NBD6,75TRAO

NBC-11,25COMPRESSO

NDF20,25TRAO

NDE-22,5COMPRESSO



2.n = b+

2.6 = 9+3


Fy = 0 M = 0 (Momento fletor)HC+HF = 0 VC = 100 KN HF.4,5-100.7,2 = 0

HF = -HC

HF = 7204,5HC = -160 KN

HF = 160 KN


N D

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NDA-100 = 0 NDE = 0

NDA = 100 KN

N A

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0NAB.sen22,62-NAD-NAE.sen22,62 = 0 NAB.cos22,62+NAE.cos22,62 = 0NAB.sen22,62-100-NAE.sen22,62 = 0 (NAB+NAE).cos22,62 = 0 (NAB-NAE).sen22,62 = 100 NAB+NAE = 0cos22,62 NAB-NAE = 100sen22,62

NAB-NAE = 260 1 NAB+NAE = 0 2

Sistema de Equaes

Substituindo na equao 1 ou 2 1 NAB-NAE = 260

NAB+NAE = 02 NAB+NAE= 0

130+NAE = 0

2NAB = 260

NAE = -130 KN NAB = 2602

NAB = 130 KN

N E

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NEB+NEA.sen22,62 = 0 NEF-NED-NEA.cos22,62 = 0

NEB+(-130).sen22,62 = 0 NEF-0-(-130).cos22,62 = 0

NEB = 50 KN NEF = -120 KN

N F

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

NFC+NFB.sen39,81 = 0 -NFE-HF-NFB.cos39,81 = 0

NFC+(-52).sen39,81 = 0 -(-120)-160-NFB.cos39,81 = 0

NFC = 33,3 KN NFB = -40cos39,81

NFB = -52 KN

N B

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0+NBC.sen22,62-NBA.sen22,62-NBE-NBF.sen39,81=0 -NBA.cos22,62+NBC.cos22,62+NBF.cos39,81 = 0

173.sen22,62-130.sen22,62-50-(-52).sen39,81=0 -130.cos22,62+NBC.cos22,62+(-52).cos39,81=0

66,5-50-50+33,5=0 NBC.cos22,62-120-40 = 0

0=0

NBC = 160cos22,62

NBC = 173 KNN C

Foras Verticais (V)

Foras Verticais (H)

FV= 0

FH = 0VC-NCF-NCB.sen22,62=0 -NCB.cos22,62+HC = 0

100-(+33,3)-(+173).sen22,62=0 -(+173).c os22,62+160 = 0

100-33,3-66,7=0

-160+160 = 0

0=0

0 = 0BARRAFORAS NORMAIS AXIAIS (KN)ESFORO

NDA100TRAO

NDE0-

NAE-130COMPRESSO

NAB130TRAO

NEB50TRAO

NEF-120COMPRESSO

NFC33,3TRAO

NFB-52COMPRESSO

NBC173TRAO



2.n = b+

2.10 = 17+3


Fy = 0 M = 0 (Momento fletor)HA = 0 VA+VB = 30 KN -VB.10+15.6+15.3 = 0

VA+13,5 = 30

VB = 13510

VA = 30-13,5

VB = 13,5 KN

VA = 16,5 KN3 Passo Mtodo dos Ns

N 1

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

VA+N12.sen45 = 0 HA+N13+N12.cos45 = 0

16,5+N12.sen45 = 0 0+N13+(-23).cos45 = 0

N12 = -16,5sen45 N13 = 16,3 kN

N12 = -23 KN

N 3

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N32 = 0 -N31+N32.cos71,57+N34 = 0

-16,3+0.cos.71,57+N34 = 0

N34 = 16,3 kN N 4

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N42 = 0 -N43+N42.cos71,57+N45 = 0

-16,3+0.cos.71,57+N45 = 0

N45 = 16,3 kNN 2

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-15-N21.sen45-N23.sen71,57-N24.sen71,57-N25.sen45=0 -N21.cos45-N23.cos71,57+N26+N25.cos45+N24.cos71,57 = 0

-15-(-23).sen45-0-0-N25.sen45=0 -(-23).cos45-0+N26+2,29.cos45+0=0

N25 = 1,62sen45 N26+16,26+1,62 = 0

N25 = 2,29 KN

N26 = -17,88 KN

N 6

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

-15-N65 = 0 -N62+N67 = 0

N65 = -15 KN -(-17,88)+N67 = 0

N67 = -17,88 KNN 5

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N52.sen45+N56+N57.sen45 = 0 -N52.cos45-N54+N57.cos45+N58 = 0

2,29.sen45+(-15)+N57.sen45 = 0 -2,29.cos45-16,3+18,9.cos45+N58 = 0

N57.sen45+1,62-15 = 0 N58-1,62-16,3+13,36 = 0

N57 = 13,38sen45 N58 = 4,5 KN

N57 = 18,9 KNN 8

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N87 = 0 -N85+N87.cos71,57+N89 = 0

-4,5+0.cos.71,57+N89 = 0

N89 = 4,5 KN

N 9

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N97.sen71,57+VB = 0 -N97.cos71,57-N98+N910 = 0

N97.sen71,57+13,5 = 0 -(-14,23).cos71,57-4,5+N910 = 0

N97 = -13,5sen71,57

N910 = 0

N97 = -14,23 KN N 10

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0

N107 = 0 -N107-N109 = 0

-0-N109 = 0

N109 = 0N 7

Foras Verticais (V)

Foras Verticais (H)

FV = 0

FH = 0-N75.sen45-N78.sen71,57-N79.sen71,57-N710.sen45 = 0 -N76-N75.cos45-N78.cos71,57+N79.cos71,57+N710.cos45 = 0

-18,9.sen45-0-(-14,23).sen71,57-0=0 -(-17,88)-(+18,9).cos45-0+(-14,23).cos71,57+0=0

-13,5+13,5 = 0 17,88-13,36-4,5 = 0

0 = 0

0 = 0


N12-23COMPRESSO

N1316,3TRAO

N3416,3TRAO

N420-

N320-

N4516,3TRAO

N522,29TRAO

N26-17,88COMPRESSO

N67-17,88COMPRESSO

N65-15COMPRESSO

N584,5TRAO

N5718,9TRAO

N870-

N894,5TRAO

N97-14,23COMPRESSO

N9100-

N1070-

Bibliografia

INTRODUO ISOSTTICA, autor Eloy Ferraz Machado Junior, 2007

Apostila MECNICA DOS MATERIAIS, autor Ricardo Gaspar, 2005

n = n de ns

b = quantidade de barras

= n de reaes (Verticais e Horizontais)

NAB

NAB

NAF

NAF

VA

VA

50

50

NBC

NBF

NBC

NBF

NBA

NBF

NBA

100

100

NCB

NCD

NCB

NCD

NCF

NCF

NFC

NFD

NFB

NFB

NFC

NFD

NFB

NFD

NFA

NFE

NFA

NFE

NED

NED

NEF

HE

NEF

HE

VE

VE

50

50

NDC

NDC

NDF

NDF

NDE

NDE

NDF

VB

VB

HB

HB

NBC

NBC

NBC

NBA

NBA

NAC

NAB

NAB

NAC

NAC

HA

HA

NAE

NAE

NEC

NEC

NED

NEA

NEA

NED

NCB

NCB

NCB

40

NCD

NCA

NCD

NCA

40

NCD

NCA

NCE

NCE

NDC

20

NDC

20

NDC

NDE

NDE

N13

N13

N13

HA

HA

N12

N12

VA

VA

N23

N23

N21

N24

N21

N24

2

N35

2

N35

N35

N31

N31

N34

N34

N32

N31

N34

N32

+

2

2

N57

N53

N57

N53

N54

N53

N57

N54

N43

N47

N45

N43

N45

N47

N46

N43

N47

N42

N42

N46

VB

VB

N56

HB

HB

N56

N56

N51

N51

N16

N15

N15

N16

N16

HA

HA

N12

N12

N65

2

2

N65

N65

N67

N61

N67

N61

N62

N61

N67

N62

N27

N26

N26

N21

N27

N27

N21

N23

N23

N37

N37

N34

N32

N32

N34

N76

2

2

N76

N76

N74

N72

N74

N72

N73

N72

N74

N73

N47

N47

2

2

N47

N43

N43

NAC

HA

NAC

NAC

HA

NAE

NAE

VA

VA

10

10

NCD

NCA

NCD

NCE

NCE

NCA

NCA

NCE

NEC

NED

NEC

NED

NEB

NEC

NED

NEA

NEA

NEB

20

20

NDC

NDB

NDC

NDE

NDB

NDE

NDE

NDB

NBD

NBD

NBD

NBE

NBE

VB

VB

NBA

NBA

NBD

HB

HB

NBD

VA

VA

HA

NAC

HA

NAD

NAD

NAC

NAB

NAD

NAB

NDC

NDA

NDC

NDA

NDE

NDA

NDB

NDE

NDB

150

150

NCA

NCA

NCEW

NCE

NCD

NCD

NCE

75

NEC

NEC

75

NEC

NED

NED

NAB

NAB

NAB

NAF

NAF

VA

VA

NFB

NFB

NFG

NFA

NFA

NFG

4

4

NBC

NBC

NBC

NBA

NBA

NBG

NBG

NBF

NBA

NBG

NBF

+

NCD

NCB

NCB

NCD

NCG

NCB

NCD

NCG

NGB

NGD

NGC

NGB

NGC

NGD

NGB

NGD

NGF

NGH

NGF

NGH

400

400

NAB

NAB

NAF

NAF

NFB

NFA

NFA

NFB

NFB

NFG

NFG

VF

VF

NGB

NGB

NGF

NGH

NGF

NGH

400

400

NBC

NBA

NBA

NBH

NBF

NBH

NBF

NBC

NBH

NBF

NBG

NBG

400

400

NCB

NCB

NCD

NCD

NBG

NCH

Por se tratarem de foras de reao horizontal e estarem na mesma linha de ao, bem como as foras externas de 9 KN serem aplicadas no segmento AE, a reao horizontal HE sofre sozinha a ao dos 18 KN enquanto a HF no solicitada.

Os clculos mostraro essa teoria.

9

NAB

9

NAB

NAC

NAC

NBA

NBA

NBC

NBC

NBD

NBC

NBD

NCB

NCA

NCA

9

9

NCB

NCB

NCD

NCD

NCE

NCE

NDB

NDB

NDC

NDC

NDE

NDE

NDE

NDF

NDF

NEC

NED

NEC

HE

NED

NED

HE

VE

VE

HE confirmada

NFD

NFD

HF

HF

VF

VF

HF confirmada

NDA

NDA

NDE

NDE

100

100

NAB

NAB

NAB

NAE

NAE

NAE

NAD

NAD

+

NEA

NEB

NEA

NEB

NEF

NEA

NED

NED

NEF

NFB

NFC

NFB

NFC

HF

NFE

HF

NFB

NFE

NBC

NBA

NBC

NBC

NBF

NBF

NBA

NBF

NBA

NBE

NBE

VC

VC

NCB

HC

HC

NCB

NCB

NCF

NCF

N12

HA

HA

N12

N12

N13

N13

VA

VA

N32

N31

N31

N32

N32

N34

N34

N42

N42

N31

N32

N43

N34

N45

15

15

N21

N25

N26

N26

N23

N24

N21

N25

N25

N24

N23

N21

N23

N24

15

15

N62

N67

N62

N67

N65

N65

N52

N57

N56

N52

N56

N57

N52

N57

N54

N58

N54

N58

N87

N85

N85

N87

N87

N89

N89

N97

N97

N910

N97

N98

N910

N98

VB

VB

N107

N107

N107

N109

N109

N76

N710

N76

N75

N710

N75

N78

N79

N78

N79

N710

N75

N78

N79

TRELIÇAS

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