SOLUCIÓN DE LA PRÁCTICA CALIFICADA-PASO A PASO-RESISTENCIA DE MATERIALES I, Gabriel D. Quispe Sanes.

FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

UNIVERSIDAD NACIONAL DE HUANCAVELICA FACULTAD DE CIENCIAS DE INGENIERÍA ESCUELA ACADÉMICA PROFESIONAL DE INGENIERÍA CIVIL-HVCA

I PRÁCTICA CALIFICADA EJERCICIOS RESUELTOS PASO A PASO ❖ ❖ ❖ ❖ ❖ ❖

CURSO DOCENTE ESTUDIANTE CICLO SECCIÓN CÓDIGO

RESISTENCIA DE MATERIALES I : Ing. Milton Huayllani Matamoros : QUISPE SANES, Gabriel David. :V :B : 2015151036 :

SOLUCIONARIO DE LA PRIMERA PRÁCTICA CALIFICADA RESISTENCIA DE MATERIALES I

100

Un bloque rígido tiene un peso de   y pende de tres varillas simétricamente colocados como se muestra en la figura. Si antes de colgar el bloque, los extremos inferiores de las varillas estaban al mismo nivel. EJERCICIO Nº 01.

a) Determinar el esfuerzo que se desarrolla en cada varilla después de suspender el bloque  y de una elevación de temperatura de . b) Determinar el alargamiento total en cada varilla después de suspender el bloque y de una elevación de temperatura.

80º

1

RESISTENCIA DE MATERIALES I

QUISPE SANES, Gabriel David

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ITEM

 /   1/º Área E

250 450 − − 200∙ 1 0 33∙ 1 0 11.7∙10 18.9∙10 ACERO

BRONCE

SOLUCIÓN: 1. PASO Nº O1: Convertimos las unidades para trabajar en un solo sistema. ITEM

 /  1/º Área E

250∙200∙1100− 11.7∙10−

450∙33∙1100− 18.9∙10−

ACERO

BRONCE

2. PASO Nº O2: Realizamos descomposición de cuerpo libre(DCL) del sistema: Donde: • • •

 ==    .  ..  =   

∑  = 0 → 2 ∙    =   = 10000∙∙ 10   2∙     11 3. PASO Nº O3: Simulamos la deformación deformación del bloque debido debido a la variación variación de la temperatura y el peso del bloque.

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 =     =    ( ∙ ∆ ∙     ∙∙∙∙ ) = (( ∙ ∆ ∙     ∙∙∙∙ ) 11.11.7∙10− ∙ 8080 ∙ 0.0.5  (250∙  100.−5 ∙∙20200 ∙ 10) = = 18.9∙10− ∙ 8080 ∙ 1  (450∙101−∙  ∙83 ∙ 10) 0.0.468∙10−  (0.5∙51∙0) = = 1.512∙10−  (3735∙ 10) (0.0.5∙51∙0)  (3735∙ 10) = = 1.512∙10−  0.0.468∙10− 3735∙1010 ∙ 0.5 ∙   5 ∙ 10 ∙  = 5 ∙ 10 ∙ 3735 ∙ 10 ∙ 0.00100104444 3735∙ 1867.5∙  5∙5 ∙ 10 ∙  = 1949 19496767∙∙ 10 1.8675∙  5∙ = 194967

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Reemplazamos

En (2):

1 2  en  en

:

1.8 =675∙10000∙∙  10 5 ∙ 2∙100100∙ 1010   1 1    2∙ = 194967  1.8675∙11.8675∙  5∙ 1=000019∙41906 710∙10∙5 ∙ 10 0=∙194967  1 0  11.8675∙11.8 675∙= 1949 =67694967  5 ∙ 100 ∙ 10   = 58.561 1. 8 675∙    5∙  = 194967 1.5 ∙8675∙ 675∙ = 1.58561 5885675∙ 61675∙ ∙ 58.510108.5 61∙ 5∙10 =194967 194967  5 ∙ = =17120. 85604.8367325   = 17.121

4. PASO Nº O4: Respondemos las preguntas: a) Calculo de esfuerzos en barras:

 = ( )  = 250∙ 58.51610−  = 234244 ⁄  = ( )  = 450∙  17.101−21

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b) Alargamiento total en cada varilla:

 =     = 4.68∙10−  (0.5 ∙510∙)  = 4.68∙10−  0.0.5 ∙ 58.585 ∙.5106161∙ ∙ 10  = 0.00105 = 1.053  =     = 1.512∙10−  (3735∙ 10)  17. 1 21∙ 1 0 −  = 1.512∙10    37373535 ∙ 1010   = 0.00197 = 1.97

Una barra de sección circular variable, varía diametralmente desde una   hasta hasta  como  como se muestra en la figura. Para el sistema mostrado, determine:

EJERCICIO Nº 02.

longitud

“” “”

““””

a) El alargamiento total debido a la carga exterior b) El alargamiento total debido a la carga exterior peso específico de la barra es .



aplicada axialmente.   y al peso propio de la barra, si el  y

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SOLUCIÓN: 1. PASO Nº O1: Calculamos

"ℎ"

 en función de variables, mediante relaciones geométricas.  en

ℎ   = ℎ ℎℎ∙  ∙∙ ==∙∙ℎℎ ∙∙== = ∙ ℎℎ∙∙ℎ  ∙∙  = ℎ “”

2. PASO Nº O2: Realizamos un corte a una distancia

.

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 = ℎ∙    = 4 ∙     = 4 ∙ (ℎ∙∙ )  =     = (13 ∙  ∙ )  (13 ∙  ∙ )  = (13 ∙  ∙ )  (13 ∙ 4 ∙  ∙ ℎ)  = 13 ∙  ∙   4 ∙  ∙ ℎ  =  ∙    = 3 ∙  ∙   4 ∙  ∙ ℎ “  ”  =  

3. PASO Nº O3: Calculamos área de la sección de corte.

4. PASO Nº O4: Calculamos el volumen del sólido.

5. PASO Nº O5: Calculamos el peso del sólido.

6.

PASO Nº O6: Calculamos el esfuerzo debido a la carga

.

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 =     =        ∙    ∙     ∙   3 4  =      ∙ ℎ        ∙      =   3 ∙     4 ∙  ∙ ℎ   =   3 ∙   4 ∙ 4 ∙ ∙ℎ ∙ℎ   ℎ  =   3 ∙    8.

PASO Nº O8: Respondemos las preguntas: a) Deformación debido a la carga axial “P”:

  ∙ ∆= ∫  ∙∙   ∙∆∙ ∆= ∫+ ∙ 

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 ∙∆∙ ∆= 4∙ ℎ∙∙  ∙ ∙ (ℎℎℎ∙ ℎ)    4 ∙ ℎ  ∙∆∙ ∆=  ∙∙  ∙ ∙ ℎ  ∙ ℎ  ∙∆∙ ∆= 4∙ ∙∙ℎ∙ ∙ ℎ      ∙    4 ∙      ∙∆∙ ∆=  ∙  ∙  ∙  ∙     ∙∆∙ ∆=  ∙4 ∙ ∙ ∙ ∙ ∙  ∙∙  ∙   ∙∙   ∙∆∙ ∆=  ∙ 4 ∙∙  ∙   ∙  ∙∙  ∙    ∙∆∙ ∆= 4 ∙∙  ∙  ∆= 4 ∙∙  ∙∙   ∙ 1   ∙ ∆= ∫  ∙∙ 

Reemplazamos el valor de “h”:

b) Deformación debido a su peso propio del sólido y la carga axial “P”:

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FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

 ∙∆∙ ∆= ∆  3 ∙  ∫+ ∙   ∫+ ℎ ∙  + +    1   ∙∆∙ ∆= ∆  3 ∙  2   ℎ  ∙  |  +     ∙∆∙ ∆= ∆  3 ∙  2   ℎ ∙  1|+           ℎ ℎ ℎ  ∙∆∙ ∆= ∆  3 ∙  2  2   ℎ ∙ (  ℎ1 ℎ  ℎ1)         2∙ 2∙  ∙ ℎ  ℎ ℎ    ℎ  ∙∆∙ ∆= ∆  3 ∙  2  ℎ ℎ ∙ (ℎ ℎℎℎ ∙ ℎℎ)     ∙∆∙ ∆= ∆  3 ∙    2∙22∙ ∙ℎ  ℎ ∙ ( ℎ ℎ)     ∙     ∙       2 2∙ ∙  ∙      ∙ ∙    ∙∆∙ ∆= ∆  3 ∙  2        ∙           ∙     2∙   ∙     ∙   ∙∆∙ ∆= ∆  3 ∙  2 ∙      ∙∙     ∙∙  ∙            ∙     ∙   2 ∙    ∙     ∙   ∙ ∆= ∆  3 ∙  2 ∙      ∙    ∙∙    ∙∙  ∙   

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FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

∆= 1 ∙ [4∙4∙∙  ∙∙∙     ∙∙6∙ ∙   3∙3∙∙ ∙ ∙ ] NOTA: La expresión anterior que está en función de variables que pueden tomar cualquier valor, se puede expresar de distintas formas (Simplificación). La cuestión es que cuando se toma valores nos debe dar la misma respuesta. La forma más simplificada de dicha expresión de desplazamiento es:

    1 4∙ 4∙  ∙   ∙ ∙    ∙   ∙ ∙  ∆=  ∙ [ ∙  ∙   3 ∙   6 ]

Una placa de acero delgada tiene la forma trapezoidal como se muestra en la figura. El espesor es  y   y varía uniformemente desde una anchura hasta otra   en una longitud . Si se aplica en cada extremo una fuerza axial , determine el alargamiento total de la placa. El módulo de elasticidad e lasticidad es . EJERCICIO Nº 03.

“”

“”



“  ” “”

“”

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FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

ℎ   = ℎ ℎ   ∙  =  ∙ ℎ ℎ∙∙ =∙ℎ  ∙  =  ∙ ℎ ℎ ∙∙ 

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FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

 = ℎ∙∙    = 4 ∙       ∙ ∙    = 4 ∙ ( ℎ ) “  ”  =    = 4 ∙ ℎ∙∙   4 ∙ ∙ℎ ℎ  =  ∙∙   ∙∙  4 ∙ ℎ ∙  1

3. PASO Nº O3: Calculamos área de la sección de corte.

4. PASO Nº O4: Calculamos el esfuerzo debido a la carga

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FACULTAD DE CIENCIAS DE INGENIERIA ESCUELA ACADEMICA PROFESIONAL DE INGENIERIA CIVIL -HVCA

   4 ∙ ℎ  ∙∆∙ ∆=  ∙∙  ∙ ∙ (ℎℎ ℎ ∙ ℎ)    4 ∙ ℎ  ∙∆∙ ∆=  ∙∙  ∙ ∙ ℎ  ∙ ℎ  ∙∆∙ ∆= 4∙ ∙∙ℎ∙ ∙ ℎ   Reemplazamos el valor de “h”:

   ∙    4 ∙   ∙∆∙ ∆=  ∙∙  ∙  ∙  ∙     ∙∆∙ ∆=  ∙∙4∙ ∙ ∙ ∙ ∙  ∙∙  ∙   ∙∙  