SOLICITACIONES_EJEMPLOS

Abril 2014 Material elaborado con fines didácticos por Arq. Alejandra Palmieri y Arq. Ana Ostera

ESTRUCTURAS I B 2014

EJERCITACIÓN N°3

ELEMENTOS ESTRUCTURALES SOMETIDOS A FLEXIÓN SIMPLE. SOLICITACIONES:

Esfuerzo de corte y Momento Flector

Objetivos:

1-Calcular las solicitaciones en distintas secciones de elementos

estructurales dados (cabios y vigas)

2-Graficar el diagrama de Esfuerzos de Corte y de Momentos Flectores en

una escala adecuada.

3-Evaluar las solicitaciones máximas.

Enunciado:

A partir de cada esquema de cargas, se pide:

Calcular las solicitaciones (Esfuerzo de Corte y Momento Flector)

Dibujar los diagramas correspondientes.

Los gráficos de cada una de las solicitaciones deben estar a escala y

corresponderse con esquema de cargas.

Se recomienda realizarlos uno debajo de otro en la misma hoja y seleccionar una

escala adecuada para las longitudes y para las solicitaciones. Designar los valores en todos los puntos notables que considere necesarios con las unidades correspondientes. Diagrama de esfuerzo de corte: DEC Diagrama de momento flector: DMF EJERCICIO 1

Esquema de Cargas:

32°

1.50 0.90

145kg/m

B

RB

A

RA


Cálculo de Esfuerzo de Corte

VAi = 0

VAd = +82.56kg x cos 32º

VAd = +70.01kg

VBi = +70.01kg – 172kg/m x 1.50m x cos 32º

VBi = -148.78kg

VBd = -148.78kg + 330.24kg x cos 32º

VBd = +131.28kg

VCi = +131.28kg – 172kg/m x 0.90m x cos 32º

VCi = 0

Cálculo de Momento Flector máximo

Distancia a momento flector máximo (cálculo adicional)

VD=0

+82.54kg – 172kg/m x “x” = 0

+82.54kg = 172kg/m x “x”

82.54kg = x

172kg/m

x = 0.48m

MA=0 MD = - 172kg/m x 0,48m x 0.24m + 82.56kg x 0.48m

MD = +19.81kgm

MB = -172kg/m x 0.90m x 0.45m

MB = -69.66kgm

MC=0


Gráficos:

EJERCICIO 2

Esquema de Cargas:

32°

1.50 0.90

172kg/m

B

RB

A

RA

x = 0.48m

+

+-

0

70.01kg

148.78kg

131.28kg

B

A

C

D

C

0

B

A

C

D

-

+

19.81kgm

69.66kgm

145kg/m

330.24kg

82.56kg

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=100kgVmáx. tramo: 148.78kg

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=100kgmM máx. tramo: 19.81kgmM máx. voladizo: 69.66kgm

Vmáx. voladizo: 131.28kg



VAi = 0

VAd = +518.04kg – 355kg

VAd = +163.04kg

VCi =+ 163.04kg – 47kg/m x 1.15m

VCi = +108.99kg

VCd = +108.99kg – 467kg

VCd = -358.01kg

VDi = -358.01kg – 47kg/m x 1.15m

VDi = -412.06kg

VDd = -412.06kg – 355kg

VDd = -767.06kg

VBi = -767.06kg – 47kg/m x 1.15m

VBi = -821.11kg

VBd = -821.11kg + 2097.21kg - 467kg

VBd = +809.10kg

VEi = +809.10kg – 47kg/m x 1.15m

VEi = +755.05kg

VEd = 755.05kg – 467kg

VEd = +288.05kg

VFi = +288.05kg – 47kg/m x 1.15m

VFi = +234kg

VFd = +234kg – 234kg

VFd = 0


MC = + 518.04kg x 1.15m – 355kg x 1.15m – 47kg/m x 1.15m x 0.575m

B

A

RA

q=47kg/m

355kg

3.45

RB

467kg 355kg 467kg 467kg 234kg

1.15

2.30

1.15 1.15 1.15 1.15


MC =+ 156.42kgm

MB = + 518.04kg x 3.45m – 355kg x 3.45m – 467kg x 2.30m – 355kg x 1.15m - 47kg/m x 3.45m x

1.725m

MB = -1199.57kgm

Gráficos:


EJERCICIO 3 Esquema de Cargas:

BA

RA

q=47kg/m

355kg

3.45

RB

467kg 355kg 467kg 467kg 234kg

1.15

2.30

1.15 1.15 1.15 1.15

D.E.C.

Esc.Long.: 2cm=1m


D.M.F.

Esc.Long.: 2cm=1m


Vmáx. voladizo:809.10kg

+

+

-

0

A

163.04kg

108.99kg

358.01kg412.06kg

767.06kg 821.11kg

809.10kg755.05kg

288.05kg 234kg

C

D

B E F

E FDC

0

A

C D

B E F

-

+

1199.57kgm

156.42kgm

BA

RA

q=47kg/m

375kg 375kg 300kg 375kg 300kg

1.30

190kg

1.30 1.30 1.30 1.30

2.60 3.90

RB



VCi = 0

VCd = -190kg

VDi = -190kg – 47kg/m x 1.30m

VDi = = -251.10kg

VDd = -251kg – 375kg

VDd= -626.10kg

VAi = -626kg – 47kg/m x 1.30m

VAi = -687.20kg

VAd = -687.20kg + 1771.25kg – 375kg

VAd = 709.05kg

VEi = 709.05kg – 47kg/m x 1.30m

VEi = 647.95kg

VEd = 64.95kg – 300kg

VEi = 347.95kg

VFi = 347.95kg – 47kg/m x 1.30m

VFi = 286.85kg

VFd = 286.85kg – 375kg

VFd = -88.15kg

VBi = -88.15kg – 47kg/m x 1.30m

VBi = -149.25kg

VBd = -149.25kg + 449.25kg - 300kg

VBd = 0


MA = - 190kg x 2.60m – 375kg x 1.30m - 47kg/m x 2.60m x 1.30m

MA = -1140.36kgm

MF = - 300kg x 1.30m +449.25kg x 1.30m - 47kg/m x 1.30m x 0.65m

MF = +154.31kgm


Gráficos:

BA

RA

q=47kg/m

375kg

2.60

RB

375kg

D

300kg

F

375kg 300kg

E

1.30

3.90

0A B

C

x = 2.60m

190kg

C

1.30 1.30 1.30 1.30

D F

E

-

-

+

709.05kg

190kg

251.10kg

626.10kg

687.20kg

647.95kg

347.95kg

286.85kg

88.15kg

149.25kg

0

A BC D

FE

-

+

1140.36kgm

154.31kgm

D.E.C.

Esc.Long.: 2cm=1m


Vmáx. voladizo:687.20kg

D.M.F.

Esc.Long.: 2cm=1m



EJERCICIO 4

Esquema de Cargas:


VAi = 0

VAd =+ 4.81t

VBi = +4.81t – 2.20t/m x 5.45m

VBi = -7.18t

VBd = -7.18t + 10.18t

VBd = +3.00t

VCi = +3.00t – 3.00t

VCi = 0

Cálculo de Momento Flector

Distancia al momento máximo (cálculo adicional)

+ 4.81t -2.20t/m x “x” = 0

x = 2.186m

MA=0

MD = + 4.81t x 2.18m – 2.20t/m x 2.186m x 1.093m

MD = + 5.25tm

MB = + 3.00t x 2.15m

MB = -6.45tm

MC=0

BA

5.45 2.15

q=2.20t/m 3.00t

RBRA


Gráficos:

B

RB=10.18t

A

RA=4.81t5.45 2.15

q=2.20t/m

C

4.81t

7.18t

3.00t

0

+ +

-

A

D

C

B

A

C

0

-

+

5.25tm

6.45tm

3.00t

x = 2.186m

B

3.00t

D

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=3T

Vmáx. tramo: 7.18tVmáx. voladizo: 3.00t

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=3TmM máx. tramo: 5.25tmM máx. voladizo: 6.45tm


EJERCICIO 5

Esquema de Cargas:


VAi = 0

VAd = + 1.24t

VCi = + 1.24t

VCd = + 1.24t – 3.00t

VCd = -1.76t

VBi = -1.76t

VBd = -1.76t + 6.49t

VBd = + 4.73t

VDi = + 4.73t – 2.20t/m x 2.15m

VDi = 0


MA= 0

MC =+ 1.24t x 1.50m

MC = + 1.86tm

MB = - 2.20t/m x 2.15m x 1.075m

MB = -5.08tm

MD=0

B

RB

A

RA

5.45 2.15

q=2.20t/m3.00t

1.50


Gráficos:

EJERCICIO 6

Esquema de Cargas:

B

RB=6.49t

A

RA=1.24t

5.45 2.15

q=2.20t/m

C

1.24t

4.73t

++

-A D

C

B

A0

-

+

5.08tm

3.00t

B

1.76t

1.50

D

1.24t

1.76t

C

D

1.86tm

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=4tVmáx. tramo: 1.76tVmáx. voladizo: 4.73t

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=4tmM máx. tramo: 1.86tmM máx. voladizo: 5.08tm



VCi = 0

VCd = 0

VDi = - 24kg/m x 1.20m x cos 12º

VDi = -28.17kg

VDd = -28.17kg

VAi = -28.17kg – 107kg/m x 0.40m x cos 12º

VAi = -70kg

VAd = -70kg + 313.55kg x cos 12º

VAd = +236.66kg

VBi = +236.66kg – 107kg/m x 4.40m x cos 12º

VBi = -223,85kg

VBd = -223.85kg + 271.65kg x cos 12º

VBd = 41.87kg

VEi = 41.87kg – 107kg/m x 0.40m x cos 12º

VEi = 0


Distancia al momento flector máximo del tramo (cálculo adicional)

+313.55kg -24kg/m x 1.20m – 107kg/m x 0.40m – 107kg/m x “x”=0

x = 2.261m

MC=0

MD = -24kg/m x 1.20m x 0.60m

MD = -17.28kgm

A

RA

B

RB

1.20 5.20

12°

1.60 4.40 0.40

104.66kg/m

23.48kg/m


MA = -24kg/m x 1.20m x 1.00m – 107kg/m x 0.40m x 0.20m

MA = -37.36kgm

MF = -24kg/m x 1.20m x 3.26m – 107kg/m x 0.40m x 2.46m + 313.55kg x 2.26m – 107kg/m x

2.26m x 1.13m

MF = 235.88kgm

MB = -107kg/m x 0.40m x 0.20m

MB = -8.56kgm

ME=0

Gráficos:


EJERCICIO 7

A

RA =313.55kg

B

RB = 271.65kg

107kg/m

24kg/m

1.20 5.20

12°

1.60 4.40 0.40

C

D

E

12°

+ +

-

-A

B

C

D

28.17t

70.03t

236.66t

223.85t

41.87t

E

0

A

B

C

D

E

0

+

-

235.88tm

8.56tm

37.36tm

17.28tm

F

F

x = 2.261m

104.66kg/m

23.48kg/m

F

-

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=100tVmáx. tramo:236.66t

Vmáx. voladizo der.: 41.87tVmáx. voladizo izq.: 70.03t

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=100tmM máx. tramo: 235.88tmM máx. voladizo izq.: 37.36tmM máx. voladizoder.: 8.56tm


Esquema de Cargas:


VAi = 0

VAd = +4.77t

VCi = +4.77t – 0.60t/m x 0.70m

VCi = +4.35t

VCd = +4.35t – 1.20t

VCd = +3.15t

VDi = +3.15t-0.60t/m x 0.70m

VDi = +2.73t

VDd = +2.73t – 0.80t

VDd = +1.93t

VEi = +1.93t – 0.60t/m x 0.70m

VEi = +1.51t

VEd = +1.51t – 0.80t

VEd = +0.71t

VFi = +0.71t – 0.60t/m x 0.70m

VFi = +0.29t

VFd = +0.29t – 1.00t

VFd = -0.71t

VGi = -0.71t – 0.60t/m x 1.00m

VGi = -1.31t

VGd = -1.31t

VHi = -1.31t

VHd = -1.31t

VBi = -1.31t – 0.60t/m x 2.00m

VBi = -2.51t

BA

RA

q=0.60t/m

RB

1.20t

0.70

7.30

0.80t 0.80t 1.00t

0.70 0.70 0.70

q=0.60t/m

1.00 1.50 2.00


VBd = -2.51t + 2.51t

VBd = 0


MA=0

MF = +4.77t x 2.80m – 1.20t x 2.10m – 0.80t x 1.40m – 0.80t x 0.70m – 0.60t/m x 2.80m x 1.40m

MF = 6.80tm

MC =+ 4.77t x 0.70m – 0.60t/m x 0.70m x 0.35m

MC = +3.19tm

MD = +4.77t x 1.40m – 0.60t/m x 1.40m x 0.70m – 1.20t x 0.70m

MD = 5.25tm

ME = +4.77t x 2.10m – 1.20t x 1.40m – 0.80t x 0.70m – 0.60t/m x 2.10m x 1.05m

ME = 6.45tm

MG = +0.60t/m x 2.00m x 2.50m – 2.51t x 3.50m

MG = +5.78tm

MH = -0.60t/m x 2.00m x 1.00m + 2.51t x 2.00m

MH =+3.82tm

MB= 0

Gráficos:


BA

RA = 4.77t

q=0.60t/m

RB = 2.51t

1.20t

0.70

7.30

0.80t 0.80t 1.00t

0.70 0.70 0.70

C D E F

q=0.60t/m

1.00 1.50 2.00

D.E.C.

D.M.F.

Esc.Long.: 2cm=1m

Esc.Fza.: 1cm=1t

Esc.Long.: 2cm=1m

Esc. Mom..: 1cm=1tm

Vmáx. : 4.77t

M máx.: 6.80tm

G H

B

A C D E

F G H

4.77t

4.35

2.73t

3.15t

1.93t

1.51t

1.31t 1.31t

2.51t

0.71t

0.71t

0.29t0

-

+

BA C D E F G H0

3.19tm

5.25tm

6.45tm6.80tm

5.79tm

3.82tm

+


EJERCICIO 8

Esquema de Cargas:

Cálculo de Esfuerzo de Corte VAi = 0

VAd = +1.35t

VCi = +1.35t – 0.40t/m x 1.00m

VCi = +0.95t

VCd = +0.95t – 0.60t

VCd = +0.35t

VDi = +0.35t - 0.40t/m x 2.60m

VDi = -0.69t

VDd = -0.69t

VBi = -0.69t – 0.20t/m x 1.40m

VBi = -0.97t

VBd = -0.97t + 1.49t

VBd = +0.52t

VEi = +0.52t - 0.20t/m x 1.10m

VEi = +0.30

VEd = +0.30t – 0.30t

VEd = 0

VFi = 0


Distancia al momento flector máximo (cálculo adicional)

+1.35t – 0.60t – 0.40t/m x “x” = 0

+0.75t = 0.40t/m x “x”

0.75t = x

0.40t/m

BA

R A

q = 0 , 4 0 t / m

R B

5 . 0 0

0 , 3 0 t

0 , 6 0 t

q = 0 , 2 0 t / m

2 . 1 0

2 . 6 0 1 . 1 0 1 . 0 01 . 4 01 . 0 0


x = 1.875m

MA=0

MB = -0.20t/m x 1.10m x 0.55m - 0.30t x 1.10m

MB = -0.45tm

MC =+ 1.35t x 1.00m – 0.40t/m x 1.00m x 0.50m

MC = 1.15tm

MQ = +1.35t x 1.875m – 0.60t x 0.875m – 0.40t/m x (1.875m)²/2

MQ = + 1.30tm

MD = +1.35t x 3.60m – 0.60t x 2.60m – 0.40t/m x (3.60m) ² /2

MD = + 0.71tm

ME = MF = 0


Gráficos:

EJERCICIO 9

Esquema de Cargas:

BA

RA = 1.35t

q=0,40t/m

RB = 1.49t

5.00

0,30t

0,60t

q=0,20t/m

2.10

2.60 1.10 1.00

C D E F

0

-

1.35t

0.95t

BA C

D

E F

+

+

0.69t

0.97t

0.52t

0.30t

1.40

D.E.C.

D.M.F.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=1t

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=1tm

x=1.875m

Q

Q

0.35t

1.00

1.875


M máx. tramo: 1.30tmM máx. voladizo: 0.45tm

0

1.15tm

B

A Q D

0.45tm

1.30tm

0.71tm

+

-F

C

E



VAi = 0

VAd = +0.54t

VCi =+0.54t – 0.30t/m x 1.70m

VCi = +0.03t

VCd = +0.03t

VDi = +0.03t - 0.12t/m x 1.50m

VDi = -0.15t

VDd = -0.15t – 0.90t

VDd = -1.05t

VEi = -1.05t – 0.12t/m x 0.60m

VEi = -1.12t

VEd = -1.12t

VFi = -1.12t

VFd = -1.12t – 0.50t

VFd = -1.62t

VGi = -1.62t

VGd = -1.62t – 0.40t

VGd = -2.02t

VBi = -2.02t

VBd = -2.02t + 3.85t

VBd = +1.82t

VHi = +1.82t

VHd = +1.82t

Vii = +1.82t – 0.16t/m x 0.80m

Vii = +1.69t

Vid = +1.69t – 0.70t

0,50t

0,80t0,70t 0,90t

q=0,16t/m

2.40

0.60 1.200.60

q=0,12t/m

1.50 0.60 0.80

0,40t

0.601.70 0.40

A

RA

q=0,30t/m

RB

5.60


Vid = +0.99t

VJi = +0.99t – 0.16t/m x 1.20m

VJi = +0.80t

VJd = +0.80t – 0.80t

VJd = 0


Distancia a momento flector máximo (cálculo adicional) +0.54t – 0.30t/m x 1.70m – 0.12t/m x “x” = 0

+0.03t = 0.12t/m x “x”

0.03t = x

0.12t/m

x = 0.25m

MA=0

MC = + 0.54t x 1.70m – 0.30t/m x 1.70m x 0.85m

MC = + 0.48tm

MS = + 0.54t x 1.95m – 0.30t/m x 1.70m x 1.10m – 0.12t/m x 0.25m x 0.125m

MS = +0.49tm

MD = + 0.54t x3.20m – 0.30t/m x 1.70m x 2.35m – 0.12t/m x 1.50m x 0.75m

MD = +0.39tm

ME = +0.54t x 3.80m – 0.30t/m x 1.70m x 2.95m – 0.12t/m x 2.10m x 1.05m – 0.90t x 0.60m

ME = -0.26tm

MF = +0.54t x 4.40m – 0.30t/m x 1.70m x 3.55m – 0.12t/m x 2.10m x 1.65m – 0.90t x 1.20m

MF = -0.93tm

MG = +0.54t x 5.00m – 0.30t/m x 1.70m x 4.15m – 0.12t/m x 2.10m x 2.25m – 0.90t x 1.80m –

0.50t x 0.60m

MG = -1.90tm

MB = - 0.16t x 2.00m - 0.70t x 1.20m - 0.80t x 2.40m

MB = -3.21tm

MH = -0.70t x 0.80m - 0.80t x 2.00m - 0.16t/m x 2.00m x 1.00m

MH = - 2.48tm


MI = - 0.80t x 1.20m - 0.16t/m x 1.20m x 0.60m

MI = -1.08tm

Gráficos:


EJERCICIO 10

0

0.48tm

3.21tm

0.49tm

0.26tm

1.08tm

+

-

0.39tm

0.93tm

2.48tm

BA D

F HC

E G I

S

1.90tm

B

A

RA = 0.54t

q=0,30t/m

RB = 3.85t

5.60

0,50t

0,80t0,70t 0,90t

q=0,16t/m

2.40

0.60 1.20

D F H J

0

D.E.C.

-

+

+

D.M.F.

0.60

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=1t

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=1tm

q=0,12t/m

1.50 0.60 0.80

C E G

0,40t

I

0.601.70

BA

D F

H JC

E G

I

0.54t

S

S

0.03t

0.15t

1.05t 1.12t 1.12t

1.62t

2.02t 2.02t

1.82t 1.82t

1.69t

0.80t

1.62t

0.99t

1.95

x=0.25m

Vmáx. tramo: 2.02t

Vmáx. voladizo: 1.82t

M máx. tramo: 0.49tm

M máx. voladizo: 3.21tm

0.40

J


Esquema de Cargas:


VCi = 0

VCd = 0

VAi = -0.44t/m x 1.20m x cos 25°

VAi = -0.48t

VAd = -0.48t + 1.72t x cos 25°

VAd = +1.08t

VDi = +1.08t

VDd = +1.08t – 1.10t x cos 25°

VDd = +0.08t

VEi = +0.08t

VEd = +0.08t – 1.00t x cos 25°

VEd = -0.83t

VFi = -0.83t

VFd = -0.83t

VBi = -0.83t – 0.33t/m x 1.30m x cos 25°

VBi = -1.22t

VBd = -1.22t + 1.34t x cos 25°

VBd = 0

4.50

25°

1.10t

1.00t

2.00 0.50 0.70

0.40t/m

A

RA

B

RB

0.30t/m

1.30

1.20



MC = 0

MA = -0.44t/m x 1.20m x 0.60m

MA = -0.32tm

MD = -0.44t/m x 1.20m x 2.60m + 1.72t x 2.00m

MD = +2.07tm

ME = -0.44t/m x 1.20m x 3.10m + 1.72t x 2.50m - 1.10t x 0.50m

ME = +2.11m

MF = - 0.33t/m x 1.30m x 0.65m + 1.34t x 1.30m

MF = -1.46tm

MB = 0


Gráficos:

D.E.C.

4.50

25°

1.10t

1.00t

2.00 0.50 0.70

0.40t/m

A

RA = 1.72t

B

RB = 1.34t

0.30t/m

1.30

0.48t

0

0.83t

0.83t

1.08t

0.08t

0.08t

+

-

-

0

Esc.Long.: 2cm=1mEsc.Fza.:2cm=1t


D.M.F.

C

D

E

F

C

0.33t/m

1.08

1.22t

C

A

D

E

E

F

F

B

B

A

D

2.07tm

0.32tm

2.11tm

1.46tm

-

+

1.20

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=1tm




EJERCICIO 11

Esquema de Cargas:


VCi = 0

VCd = 0

VAi = -0.40t/m x 1.20m

VAi = -0.48t

VAd = -0.48t + 1.66t

VAd = +1.18t

VDi = +1.18t

VDd = +1.18t – 1.10t

VDd = +0.08t

VEi = +0.08t

VEd = +0.08t – 1.00t

VEd = -0.92t

VFi = -0.92t

VFd = -0.92t

VBi = -0.92t – 0.30t/m x 1.30m

VBi = -1.31t

VBd = -1.31t + 1.31

VBd = 0


MC = 0

MA = -0.40t/m x 1.20m x 0.60m

MA = -0.29tm

MD = -0.40t/m x 1.20m x 2.60m + 1.66t x 2.00m

MD = +2.07tm

4.501.20

1.10t 1.00t

2.00 0.50 0.70

q=0.40t/m

A

RA

B

RB

q=0.30t/m

1.30


ME = -0.40t/m x 1.20m x 3.10m + 1.66t x 2.50m + 1.10t x 0.50m

ME = +3.21tm

MF = - 0.30t/m x 1.30m x 0.65m +1.31t x 1.30m

MF = +1.45tm

MB = 0

Gráficos:

D.E.C.

1.18t

0

0.48t

1.31t

0.92t0.92t

1.18t

0.08t 0.08t

+

-

-

0

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=1tVmáx. tramo: 1.31tVmáx. voladizo: 0.48t

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=1tm



C A

B

D E

F

4.501.20

1.10t 1.00t

2.00 0.50 0.70

q=0.40t/m

A

RA = 1.66t

B

RB = 1.31t

q=0.30t/m

1.30C

D E F

-

+

C A

0.29tm

2.07tm3.21tm

1.45tm

BD E F


EJERCICIO 12

Esquema de Cargas:

Cálculo de Esfuerzo de Corte VAi = 0

VAd = +1689.10kg

VDi = +1689.10 – 17.55kg/m x 0.78m

VDi = +1675.41kg

VDd = +1675.41 – 683.60kg

VDd = +991.81kg

VEi = +991.81kg – 17.55kg/m x 0.78m

VEi = +978.12kg

VEd = +978.12kg – 669.50kg

VEd = +308.62kg

VCi = +308.62kg – 17.55kg/m

VCi = +294.93kg

VCd = +294.93kg – 655.60kg

VCd = -360.67kg

VFi = -360.67kg – 17.55kg/m x 0.78m

VFi = -374.36kg

VFd = -374.36kg – 641.80kg

VFd = -1016.16kg

VGi = -1016.16kg – 17.55kg/m x 0.78m

VGi = -1029.85kg

VGd = -1029.85kg – 541.80kg

VGd = -1571.65kg

VBi = -1571.65kg – 17.55kg/m x 0.78m

VBi = -1585.33kg

VBd = -1585.33kg + 1585.33kg

B

RB

A

RA4.68

q=17.55kg/m

683.60kg 669.50kg 655.60kg 641.80kg 541.80kg

0.78 0.78 0.78 0.78 0.78 0.78


VBd = 0

Cálculo de Momento Flector MA = 0

MD = +1689.10kg x 0.78m – 17.55kg/m x 0.78m x 0.39m

MD = +1312.16kgm

MD = +1689.10 x 2.34m – 17.55kg/m x 2.34m x 1.17m – 683.60kg x 1.56m – 669.50kg x 0.78m

MD = +2315.82kgm

ME = 1689.10kg x 1.56m – 17.55kg/m x 1.56m x 0.78m – 683.60kg x 0.78m

ME = +2080.43kg

MF = +1585.33kg x 1.56m - 17.55kg/m x 1.56m x 0.78m + 541.80kg x 0.78m

MF = +2874.36kgm

MG = +1585.33kg x 0.78m + 17.55kg/m x 0.78m x 0.39m

MG = +2874.36kgm

MB = 0


Gráficos:

EJERCICIO 13

Esquema de Cargas:

B

RB =1585.33kg

A

RA = 1689.10kg 4.68

q=17.55kg/m

0

+

-

A

C

A

0

+

2315kgm

683.60kg

2.34m

C

669.50kg 655.60kg 641.80kg 541.80kg

0.78 0.78 0.78 0.78 0.78 0.78

1689.10kg

B

1675.41kg

991.81kg978.12kg

294.93kg

360.67kg374.36kg

1016.16kg1029.85kg

1571.65kg 1585.33kg

308.62kg

1312kgm

2080kgm 2029kgm

1231kgm

B D E F G

CD E F G

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=1000Kg

Vmáx. tramo: 1689.10kg

D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom..: 2cm=1000kgmM máx. tramo: 2315kgm

D E

F G



VAi = 0

VAd = +1056.19kg

VCi = +1056.19kg – 21kg/m x 0.375m

VCi = +1048.32kg

VCd = +1048.32kg

VDi = +1048.32kg – 551.13kg/m x 3.00m

VDi = - 605.07kg

VDd = - 605.07kg

VEi = - 605.07kg – 21kg/m x 0.375m

VEi = - 612.96kg

VEd = - 612.96kg – 487.30kg

VEd = -1100.25kg

VBi = -1100.25kg – 21kg/m x 0.75m

VBi = -1116.00kg

VBd = -1116.00kg+1116.00kg

VBd = 0


Cálculo adicional:

1056.19kg – 21kg/m x 0.375m – 551.13kg/m x “x” = 0

x = 1.90m

MA = 0

MC = +1056.19kg x 0.375m – 21kg/m x 0.375m x 0.375m/2

MC = +394.59kgm

MF = 1056.19kg x 2.275m -21kg/m x 0.375m x (0.375m/2 + 1.90m) – 551.13kg/m x 1.90m x 0.95m

MF = 1364.21kgm

MD = +1056.19kg x 3.375m – 551.13kg/m x 3.00m x 1.50m – 21kg/m x 0.375m x (0.375m/2 +

3.00m)

RBRA

BA

q=21kg/m

0.75

397.60kg 397.60kg 397.60kg 397.60kg 487.30kg

0.750.750.750.750.75

4.50


MD = +1059.46kgm

ME = +1056.19kg x 3.75m – 21kg/m x 0.375m x (0.375m/2 + 3.375m) –

- 551.13kg/m x 3.00m x 1.875m – 21kg/m x 0.375m x 0.375m/2

ME = +831.08kgm

MB = 0

Gráficos:

BA

1.125

487.30kg

0.375 3.00

RB =1116.00kgRA = 1056.19kg

C D E

B

A CD E

BA C D E

+

-

1056.19kg

1048.32kg

605.07kg

612.96kg

1116.00kg

397.60kg/0.75m=530.13kg/m

551.13kg/m

4.50

q=21kg/m

487.30kg

530.13kg/m

q=21kg/mq=21kg/m

0.75

x=1.90m

394.59kgm

1364.21kgm

1059.46kgm

831.08kgm

F

F

F

D.E.C.

Esc.Long.: 2cm=1mEsc.Fza.: 2cm=1000Kg


D.M.F.

Esc.Long.: 2cm=1m

Esc. Mom.: 2cm=1000kgmM máx. tramo: 1364.21kgm

+

BA

q=21kg/m

0.75

397.60kg 397.60kg 397.60kg 397.60kg 487.30kg

0.750.750.750.750.75

4.50 RBRA

0

1100.25kg


EJERCICIO 14

Esquema de Cargas:


VCi = 0

VCd = -210kg

VAi = -210kg – 47kg/m x 1.30m

VAi = -271.10kg

VAd = -271.10kg – 300kg + 1092.38kg

VAd = +521.28kg

VDi = +521.28kg – 47kg/m x 1.30m

VDi = +460.18kg

VDd = +460.18kg – 420kg

VDd = +40.18kg

VEi = +40.18kg

VEd = +40.18kg – 300kg

VEd = -259.82kg

VBi = -259.82kg – 47kg/m x 1.30m

VBi = -320.92kg

VBd = -320.92kg -420kg + 740.92kg

VBd = 0


MC=0

MA= -210kg x 1.30m – 47kg/m x 1.30m x 0.65m

MA = -312.72kgm

MD = -210kg x 2.60m – 47kg/m x 2.60m x 1.30m – 300kg x 1.30m + 1092.38kg x 1.30m

MD = +325.23kgm

ME = -47kg/m x 1.30m x 0.65m - 420kg x 1.30m + 740.92kg x 1.30m

BA

RA

q=47kg/m

210kg

RB

300kg 420kg 300kg 420kg

1.30 1.30 1.30 1.30


ME = +377.48kgm

MB = 0

Gráficos:

EJERCICIO 15

Esquema de Cargas:


VAi = 0

BA

RA=1092.38kg

q=47kg/m

210kg

RB=740.92kg

300kg

C

420kg 300kg 420kg

1.30 1.30 1.30 1.30

D E

BA

B

A

D.E.C.

Esc.Long.: 2cm=1m

Esc.Fza.: 2cm=1000Kg


D.M.F.

Esc.Long.: 2cm=1m


0

0

D E Vmáx. voladizo: 271.10kg

M máx. voladizo: 312.72kgm

521.28kg460.18kg

210.00kg 271.10kg

40.18kg

259.82kg320.92kg

40.18kg

C

C

-

+

-

-

+

312.72kgm

325.23kgm 377.48kgm

BA

q=23kg/m

0.60

675kg675kg675kg675kg675kg675kg675kg675kg675kg

0.60 0.60 0.60 0.60 0.60 0.60 0.60 RBRA

4.80


VAd = + 2755.20kg

VAd = + 2755.20kg

VBi = + 2755.20kg– 1148kg/m x 4.80m

VBi = -2755.20kg

VBd = -2755.20kg + 2755.20kg

VBd = 0

Cálculo de Momento Flector MA = 0

MC = 1148 x 4.80² /8

MC = +3306.24kgm

MB = 0

Gráficos:

EJERCICIO 16

Esquema de Cargas:

BA

q=23kg/m

RB=2755.20kgRA=2755.20kg 4.80

BA

1148kg/m

RB=2755.20kgRA=2755.20kg

4.80

675kg/0.60m=1125kg/m

A

A

D.E.C.



D.M.F.

Esc.Long.: 2cm=1mEsc. Mom.: 1cm=2000kgmM máx. tramo: 3306.24kgm

0

0

C

C

2.40

3306.24kgm

2755.20kg

2755.20kg

+

+

-B

BC



VAi = 0

VAd = -245kg + 576.47kg

VAd = +331.47kg

VCi = +331.47kg – 99kg/m x 1.15m

VCi = +217.62kg

VCd = +217.62kg – 326kg

VCd = -108.38kg

VDi = -108.38kg – 99kg/m x 1.15m

VDi = -222.23kg

VDd = -222.23kg – 155kg

VDd = -377.23kg

VBi = -377.23kg – 99kg/m x 1.15m

VBi = -491.08kg

VBd = -491.08kg + 1276.93kg - 538kg

VBd = +247.85kg

VEi = +247.85kg – 99kg/m x 1.15m

VEi = +134kg

VEd = +134kg

VFI = +134kg

VFd = +134 – 134kg

VFd = 0


MA= 0

MC = -245kg x 1.15m + 576.47kg x 1.15m – 99kg/m x 1.15m x 0.575m

MC = +315.73kgm

MD = -245kg x 2.30m + 576.47kg x 2.30m – 326kg x 1.15m – 99kg/m x 2.30m x 1.15m

MD = +125.63kgm

B

A

RA

q=99kg/m

245kg

3.45

RB

326kg 155kg 538kg

134kg

1.15

2.30

1.15 1.15 1.15 1.15


MB = -134kg x 2.30m - 99kg/m x 1.15m x 0.575m

MB = -373.66kgm

ME = -134kg x 1.15m

ME = -154.10kgm

MF = 0

Gráficos:

EJERCICIO 17

Esquema de Cargas:

BA

RA=576.47kg

q=99kg/m

245kg

3.45

RB=1276.93kg

326kg 155kg 538kg

134kg

1.15

2.30

1.15 1.15 1.15 1.15

DC E F

A

A

D.E.C.



D.M.F.

Esc.Long.: 2cm=1m


0

0

B

D

C E F

B

DC

E F

331.47kg217.62kg

108.38kg 222.23kg

377.23kg491.08kg

247.85kg 134kg134kg

-

+ + Vmáx. voladizo: 491.08kg


315.73kgm

125.63kgm

373.66kg

154.10kg-

+



VCi = 0

VCd = -190kg

VDi = -190kg – 47kg/m x 1.30m

VDi = -251.10kg

VDd = -215.10kg – 375kg

VDd = -626.10kg

VAi = -626.10kg – 47kg/m x 1.30m

VAi = -687.20kg

VAd = -687.20kg + 1771.25kg – 375kg

VAd = +709.05kg

VEi = +709.05kg – 47kg/m x 1.30m

VEi = +647.95kg

VEd = +647.95kg – 300kg

VEd = +347.95kg

VFi = +347.95kg – 47kg/m x 1.30m

VFi = +286.85kg

VFd = +286.85kg – 375kg

VFd = -88.15kg

VBi = -88.15kg – 47kg/m x 1.30m

VBi = -149.25kg

VBd = -149.25kg – 300kg + 449.25kg

VBd = 0

Cálculo de Momento Flector MC= 0

MD = -190kg x 1.30m – 47kg/m x 1.30m x 0.65m

MD = -286.72kgm

BA

RA

q=47kg/m

375kg 375kg 300kg 375kg 300kg

1.30

190kg

1.30 1.30 1.30 1.30

2.60 3.90

RB


MA = -190kg x 2.60m – 47kg/m x 2.60m x 1.30m – 375kg x 1.30m

MA = -1140.36kgm

ME = -190kg x 3.90m – 375kg x 2.60m – 375kg x 1.30m + 1771.25kg x 1.30m – 47kg/m x 3.90m x

1.95m

ME = -258.31kgm

MF = -300kg x 1.30m + 449.25kg x 1.30m - 47kg/m x 1.30m x 0.65m

MF = +154.31kgm

MB = 0

Gráficos:


EJERCICIO 18

Esquema de Cargas:


VCi = 0

VCd = 0

VAi = -197.80kg/m x 0.80m x cos 29°

VAi = -138.40kg

RA=1771.25kg RB=449.25kg

A

A

D.E.C.



D.M.F.

Esc.Long.: 2cm=1m


0

0B

251.10kg-

+

-

+



BA

q=47kg/m

375kg 375kg 300kg 375kg 300kg

1.30

190kg

1.30 1.30 1.30 1.30

2.60 3.90

B

F

F

E

E

D

D

C

C

DC FE

-190kg

626.10kg687.20kg

709.05kg 647.95kg

347.95kg 286.85kg

88.15kg149.25kg

1140.36kgm

258.31kgm

154.31kgm

286.72kgm

29°

2.000.80

173kg/m

A

RA

B

RB

0.80


VAd = -138.40kg + 356.04kg x cos 29°

VAd = +173kg

VBi = +173kg – 197.80kg/m x 2.00m x cos 29°

VBi = -173kg

VBd = -173kg + 356.04kg x cos 29°

VBd = +138.40kg

VDi = +138.40kg – 197.80kg/m x 0.80m x cos 29°

VDi = 0

Cálculo de Momento Flector MC = 0

MA = 197.80kg/m x 0.80m x 0.40m

MA = -63.30kgm

MB = -63.30kgm

ME = -197.80kg/m x 1.80m x 0.80m + 356.04kg x 1.00m

ME = +35.60kgm

MD = 0


29

°

2.000.80

173kg/m

A

RA = 356.04kg

B

RB = 356.04kg

0.80

197.80kg/m

A

B

A

B

C

D

C

D

C

D

E

173kg

138.40kg

173kg

138.40kg

63.30kgm

35.60kgm

63.30kgm

D.E.C.


Vmáx. tramo: 173kg

D.M.F.

Esc.Long.: 2cm=1m



M máx. voladizo: 63.30kgm0

0

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