Cálculo integral 3 em engenharia cívil

7/25/2019 Cálculo integral 3 em engenharia cívil

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ESTADO DE MATO GROSSOSECRETARIA DE ESTADO DE CIÊNCIA E TECNOLOGIA

UNIVERSIDADE DO ESTADO DE MATO GROSSOCAMPUS UNIVERSITÁRIO DE TANGARÁ DA SERRA

DEPARTAMENTO DE ENGENHARIA CIVIL

APLICAÇÕES DE CÁLCULO DIFERENCIAL E INTEGRAL NA ENGENHARIA

CIVIL.

BORBA, Bruna1

PIETRAC! DE OLIVEIRA, Er"#$%n&

COSTA, I$a'()a*

VINICIUS, Mar+u$

BALDISSARELLI, T-"a% Lu"$/

Resumo

E$0( 0ra'a)-% 0( 2%r "n0u"0% a2r($(n0ar a)ua$ a2)"+a34($ 5( C6)+u)% D"7(r(n+"a) (In0(ra) n% 8'"0% 5a En(n-ar"a C"9"). E$06 +%n0"5% ua 'r(9( (:2)"+a3;% $%'r( %$+%n+("0%$ 5( M%(n0%$, ($7%r3%$ +au$a5%$ 2%r ua 7%r3a, ()a$0"+"5a5( 5( a0(r"a"$ '(+%% (:(2)%$. A"n5a n($0(, u 0<2"+% r()a+"%na5% =$ In0(ra"$ 5( L"n-a a2)"+a5a5(n0r% 5% +%n+("0% 5( Tra'a)-% 5( u 2%n0% 5( 9"$0a 7>$"+% ( a0(60"+%.

1 Acadêmico de Engenharia civil - UNEMAT/Tangará da Serra.


3 Acadêmico de Engenharia civil - UNEMAT/Tangará da Serra.4 Acadêmico de Engenharia civil - UNEMAT/Tangará da Serra.




Pa)a9ra$?+-a9(@ C6)+u)% A2)"+a34($ En(n-ar"a +"9") F>$"+a V"a$ Tra'a)-%

In0(ra"$ 5( )"n-a M%(n0% Ma$$a

SUMÁRIO1. INTRODUÇO3

&. APLICAÇÕES EM DEFLEO DE VIGAS.....................................................4

&.1. DEFINIÇO........................................................................................ 4

&.&. MDULO DE OUNG..........................................................................5

&.*. MOMENTO DE INRCIA DE ÁREA........................................................5

&.. MOMENTO DE INRCIA DE MASSA......................................................

&.. EEMPLO.......................................................................................... !

*. TRABALHO REALIADO POR UMA FORÇA...............................................1"

*.1. DEFINIÇO......................................................................................1"

*.&. EEMPLO........................................................................................11

. CALCULO DE MASSA DE UM CABO.........................................................13

.1. DEFINIÇO......................................................................................13

.&. EEMPLO........................................................................................13

/. CONSIDERAÇÕES FINAIS........................................................................14

. BIBLIOGRAFIA...................................................................................... 15

2



1. INTRODUÇÃOO +6)+u)%, a) 5( $(r ua 7(rra(n0a 5( (:0r(a "2%r08n+"a 2ara a

(n(n-ar"a, 0a', 5a ($a 7%ra, "n5"$2(n$69() 2ara 0a' 2ara 5"$+"2)"nar a$

(n0($ 5( 7u0ur%$ 2r%7"$$"%na"$ 5a (n(n-ar"a. E$0( +%n0(5% 2%$$u" a +a2a+"5a5( 5(5($(n9%)9(r u ra+"%+>n"% )<"+% n% "n5"9>5u%, % Jua) 0raK +%% '(n(7>+"% a ()-%r"a+a2a+"5a5( 5( (n+%n0rar $%)u34($ 2ara 2r%')(a$ 5% 5"a a 5"a 5( 7%ra r62"5a, 2r60"+a ((7"+aK.

E$0u5ar 5"7(r(n+"a3;%, "n0(ra3;%, +%n-(+(r r67"+%$, r(2r($(n0a34($, $a'(r %n5(a2)"+ar 2%n0%$ 6:"%$ ( >n"%$, r(5uK"r +u$0%$a$0%$ ( (n0r( %u0r%$, $;% a$$un0%$

2r"%r5"a"$ 2ara u '% (n(n-("r% Ju( ($06 a5(n0ran5% % (r+a5% 5( 0ra'a)-%.P%5(?$( 5"9"5"r ($0a a0r"a 'a$"+a(n0( ( 5ua$ 2ar0($ % +6)+u)% 5"7(r(n+"a) (

% +6)+u)% "n0(ra). E (ra), %0""Ka34($, (n+%n0rar 6:"%$ ( >n"%$ 5( 7un34($ $;% 2r%')(a$ r(7(r(n0($ a ($0a 2r"("ra. 6 % +6)+u)% "n0(ra) 2%5( $(r a2)"+a5% 2ara(n+%n0rar 9%)u($ 5( $<)"5%$, %(n0%$ 5( 7%r3a ( "nr+"a, +(n0r%$ 5( ra9"5a5( (

5"9(r$%$ "0(n$ 5(n0r% 5% un"9(r$% 5a 7>$"+a.

3



2. APLICAÇÕES EM DEFLEXÃO DE VIGAS

2.1. DEFINIÇÃOPar0"n5% 5% +%n+("0% 5a 2a)a9ra, 5(7)(:;% $"n"7"+a 'a$"+a(n0( ua

a)0(ra3;% na 5"r(3;% 5( a)%. Para % +6)+u)% ($0ru0ura), "$0% 2%5( $(r a2)"+a5%5uran0( % ($0u5% 5( 5(7%ra34($ 5( a0(r"a"$, ( ($2(+"a), 2ara a (n(n-ar"a+"9"), 5uran0( % ($0u5% 5( 9"a$ 5( 5(7%ra34($ 5( ua 9"a.

A 9"a ( u ()((n0% ($0ru0ura) u0")"Ka5% ( +%n$0ru34($ 2ara 0ran$"0"r %$ ($7%r3%$ 9(r0"+a"$ r(+('"5%$ 2()a )a( 2ara %$ 2")ar($. E$0( 2%5( $(r +%2%$0%5( a5("ra, 5( 7(rr% %u +%n+r(0% ara5%.

N($0( 0<2"+%, n;% $(r6 a'%r5a5% =$ 5(5u34($ 5( 7%ru)a$ ( (Jua34($, (%u0ra$ 2a)a9ra$, a2(na$ $(r;% a2r($(n0a5a$ ( +%nun0% +% 'r(9($ (:2)"+a34($.

Para ua 9"a, a Jua) 2%$$u" u 2($% a "ua) w ( x) w

+%rr($2%n5( = 2a)a9ra Qweight”, 2r%9(n"(n0( 5% "n)$, 2($%, a% "n0(r6?)%, 2%5(?$( (n+%n0rar % ($7%r3% +%r0an0( 0(n$;% 5( +"$a)-a(n0% Ju( a2)"+a5a na9"a, ( %u0ra$ 2a)a9ra$, 2%5( $(r +a)+u)a5% 5a $(u"n0( 7%ra@

V ( x )=−∫w ( x ) dx

O %(n0% +au$a5% 2()%$ 0a"$ ($7%r3%$ "n0(rn%$ ( M ( x )) n($0a 9"a

2%5( $(r +a)+u)a5% 2()a "n0(ra) 5% ($7%r3% +%r0an0(@ M ( x )=∫V ( x ) dx

Ou ( %u0ra$ 2a)a9ra$, 5( ua 7%ra a"$ r62"5a 5( $(r +a)+u)a5a,5"r(0a(n0( 2%5(?$( +a)+u)ar % %(n0% a 2ar0"r 5a $(u"n0( "n0(ra) 5u2)a@

M ( x)=−∬w( x )dxdx

O 8nu)% 5( r%0a3;% ( r()a3;% a% (":% -%r"K%n0a) (θ) 2%5( $(r

5(7"n"5% +%% a "n0(ra) 5% %(n0%, a Jua) 7%ra (n+%n0ra5a 2r(9"a(n0(,

5"9"5"5a 2()% 2r%5u0% 5% <5u)% 5( ()a$0"+"5a5( ( E) (ver item 2.2.) ( %

%(n0% 5( "nr+"a ( I ) (ver item 2.3.). E$0( 8nu)% 2%$$u" a un"5a5( 5(

(5"5a ar'"0r6r"a 5( (0r%(0r%, 5(9"5% = r%0a3;% $(r (5"5a 2()a 0a:a 5(9ar"a3;% 5( ua r(0a.

θ ( x)=

1

EI ∫ M ( x ) dx ∙

(m

m

)E -a9(n5% a "n0(ra3;% 5a 7un3;% Ju( r(2r($(n0a % 8nu)% 5( r%0a3;%5($0a 9"a, 2%$$>9() (n+%n0rar % 5($)%+a(n0% 9(r0"+a) 5($0a ($a, %u $(a, a5(7%ra3;% (:"$0(n0( n()a 5(9"5% a%$ ($7%r3%$ a2)"+a5%$ na 9"a@

v ( x )=∫θ( x )dx

4



2.. MOMENTO DE IN,RCIA DE MASSAE$0( 0<2"+% ($06 a2r($(n0a5% 2ara %$0rar %$ %(n0%$ 5( "nr+"a 5(

a$$a. P%5( )('rar 9"$ua)(n0( % M%(n0% 5( Inr+"a 5( Ár(a "0( &.*., 2%r % $"n"7"+a5% (:0r(a(n0( 5"7(r(n0(. A +%n7u$;% '( +%u, ua9(K Ju( (:"$0( "n(r%$ +%n+("0%$ +%n-(+"5%$ +%% Q%(n0%W.

Para %'(0%$ '"5"(n$"%na"$, % %(n0% 5( "nr+"a 5( a$$a 5(7"n"5% 2()a "n0(ra) 5u2)a@

I x=∬ R

y2 ρ( x , y )dA

I y=∬ R

x2 ρ( x , y )dA

Para %'(0%$ 0r"5"(n$"%na"$, % %(n0% 5( "nr+"a 5( a$$a 5(7"n"5%( r()a3;% a (":%$ ( yz , xz , xy) , ( ($+r"0% na 7%ra 5( "n0(ra) 0r"2)a@

x2+ y

2

(¿) ρ( x , y , z )dV

I z=∭ E

¿

y2+ z

2

(¿) ρ( x , y , z )dV

I x=∭ E

¿

x2+ z

2

(¿) ρ( x , y , z )dV

I y=∭ E

¿



2.. EXEMPLO Exemplo I: T(n5% +%% 'a$( a 7"ura a'a":%, $a'(?$( Ju( ua

5(0(r"na5a 9"a ($06 a2%"a5a 2%r u 2%n0% 7":% n% )a5% ($Ju(r5% ( 2%r u 2%n0% <9() = 5"r("0a.

S(n5% a$$", +a)+u)( % 5($)%+a(n0% 6:"% 5($0a 9"a n% $(u (":% 5(

$"(0r"a : Z ,/. M<5u)% 5( %un a $(r u$a5%@ 5 ∙103

kPa N;% -6

%(n0%$ (:0(rn%$ $(n5% a2)"+a5%.

Solução: D(9"5% a $"(0r"a 5a 9"a, $a'(?$( Ju( +a5a a2%"% $u2%r0a ua+ara 5( / #N n% $(n0"5% 9(r0"+a). U$an5% : +%% a 5"$08n+"a 5% 2r"("r%a2%"%, 2ara +a)+u)ar % ($7%r3% 0(%$ % $(u"n0(@

V ( x )=−∫w ( x ) dx

V ( x )=−∫10dx

V ( x )=−10 x+C 1

kN

C%% C1 r(2r($(n0a a +ara a2)"+a5a, % 2%n0% K(r% 5( a2%"% 2%$$u> ua+ara a2)"+a5a 5( /#N, )%% C1 Z /#N.

V ( x )=−10 x+75kN

A%ra, a2<$ (n+%n0rar%$ a 7%r3a Ju( r(2r($(n0a % ($7%r3% +%r0an0(,+a)+u)a%$ %$ ($7%r3%$ "n0(rn%$ 5a 9"a@

M ( x )=∫V ( x ) dx

M ( x )=∫(−10 x+75)dx

M ( x )=−5 x ²+75 x+C 2

kN ∙ m

P%r $a'(?$( Ju( n;% -6 %(n0%$ (:0(rn%$ a2)"+a5%$ na 9"a, )%%

C& Z X, (n0;%@ M ( x )=−5 x ²+75 x kN ∙ m

A%ra +a)+u)ar(%$ % 8nu)% 5( r%0a3;% ( r()a3;% a% (":% -%r"K%n0a) 5a

9"a. O <5u)% 5( %un E I ¿ (Ju"9a)( a 5000 kPa ( % %(n0% 5(

"nr+"a+ara un"7%r( a2)"+a5a ( I ) (Ju"9a)( a 1k N ∙ m .

θ ( x)= 1

EI ∫ M ( x ) dx

θ ( x)= 1

5000∫(−5 x ²+75 x )dx

!



θ ( x)= 1

5000∗(−5

3 x

3+75

2 x

2+C 3)

E 2ara 5($+%'r"r%$ % 5($)%+a(n0% 5a 9"a, 'a$0a u0")"Kar a 7%ru)a a$(u"r@

v ( x )=∫θ( x )dx

v ( x )=∫( 1

5000∗(−5

3 x

3+75

2 x

2+C 3))dx m

v ( x )= 1

5000∗(−5

12 x

4+75

6 x

3+ x C 3+C

4)m

D(9"5% a%$ a2%"%$ )a0(ra"$, % 5($)%+a(n0% 5a 9"a v ( x ) n%$ 2%n0%$

x=0m ( x=15m &e/e%0 $(r K(r%. Ou $(a@

ParaC 4

v (0 )=0= 1

5000∗(−5

1204+

75

603+0∗C

3+C

4)C

4

5000=0

C 4=0

ParaC 3

v (15 )=0= 1

5000∗(−5

1215

4+75

615

3+15∗C 3+0)

1

5000∗(−253125

12+253125

6+15∗C

3)=0

1

5000∗( 843754

+15∗C 3)=0

15∗C 3

5000=−135

32

C 3=−1406,25

#



S(n5% a$$", 0(%$ +%% (Jua3;% 2ara a anu)a3;% ( r()a3;% =5"$08n+"a@

θ ( x)= 1

5000∗(−5

3 x

3+75

2 x

2−1406,25)

E 2ara % 5($)%+a(n0% 9(r0"+a) 5a 9"a@

v ( x )= 1

5000∗(−5

12 x

4+75

6 x

3−1406,25 x)m

C%% % a"%r >n5"+( 5( 7)(:;% %+%rr( n% (":% 5( $"(0r"a 5a 9"a,

a2)"+ar(%$ na (Jua3;% 5( 5($)%+a(n0% 9(r0"+a), 0a) 2%n0%.v (7,5)= 1

5000∗(−5

127,5

4+75

67,5

3−1406,25∗7,5)m

v (7,5)≅ 1

5000∗(−5

12∗3164,06+

75

6∗421,87−1406,25∗7,5)m

v (7,5)≅ 1

5000∗(−1318,36+5273,44−10546,87 ) m

v (7,5)≅ 1

5000∗(−6591,80 )m

v (7,5)≅−1,32m

$



+. TRAALO REALI3ADO POR UMA FORÇA

+.1. DEFINIÇÃO Na 7>$"+a, % 0(r% 0ra'a)-% $( r(7(r( = (n(r"a 0ran$7(r"5a a u +%r2%

Juan5% ua 7%r3a a2)"+a5a ( u +%r2% 5uran0( u 5(0(r"na5%

5($)%+a(n0% S . Sua un"5a5( 5( (5"5a % OULE . U$ua)(n0(, n% (n$"n%

$u2(r"%r, u$a?$( a )(0ra W 5( Y%r#, %u $(a, 0ra'a)-% n% "5"%a "n)$ 2ara

5(7"n"?)%.

FaK(n5% % u$% 5( +a)+u)%, 2%$$>9() +a)+u)ar % 0ra'a)-% 2()a $(u"n0(

"n0(ra) 5( )"n-a@

W c=

∫C

⃗F ∙ d⃗ s

On5(@ W c % 0ra'a)-% r(a)"Ka5%, C representa a curva, F % 9(0%r 7%r3a,

( s % 9(0%r 5($)%+a(n0%.

A$ a2)"+a34($ $;% 5"$0% $;% "n(ra$, 2%r (:(2)%, +a)+u)ar a (n(r"a

n(+($$6r"a 2ara Ju( u u"n5a$0( (ra 9"a$ 5uran0( a +%n$0ru3;% 5( u

5(0(r"na5% 2r5"%. Ca)+u)ar a "n0(n$"5a5( Ju( u ')%+% u0")"Ka5% 5uran0( %

Standard Penetration Test SPT a0"n( % $%)%, a'( +%% 2%$$>9() +a)+u)ar

(7"+"n+"a 5( 2a"n"$ 7%0%9%)0a"+%$ $(n5% u$a5%$ ( +arr%$ ( a9"4($ %9"5%$ a()(0r"+"5a5( (!", 2#$%).

1"



+.2. EXEMPLO Exemplo II: U 0ra'a)-a5%r 5a +%n$0ru3;% +"9") n(+($$"0a +arr(ar u

')%+% 5( +%n+r(0% 5( u 2%n0% P, na %r"(, a0 u 2%n0% [ +% +%%r5(na5a

1,1 . Para "$0%, 7%ra u0")"Ka5% u +a"n-;% 5% 0"2% Qu+#W. O (n(n-("r% 2%r $ua 9(K, r(a)"K%u a)un$ +6)+u)%$ 2ara 5($+%'r"r Jua) 0(r"a $"5% % 0ra'a)-%

r(a)"Ka5% a% (ru(r 0a) ')%+%. Da$ ($2(+"7"+a34($ (n+%n0ra5a$ n% +a"n-;%, $a'(?

$( Ju( a 7%r3a a Jua) ()( (:(r+( 2%5( $(r r(2r($(n0a5a 2()a $(u"n0( 7un3;%

9(0%r"a)@ F = y i+2 x j na un"5a5( #N #")%n(Y0%n$. S(a % (n(n-("r% Ju( %

(:(r+>+"% 2r%24( ( r(a)"K( %$ +6)+u)%$ n(+($$6r"%$.

Solução: Sa'(?$( Ju( % 5($)%+a(n0% 5a 2ar0>+u)a ')%+% 5( +%n+r(0%

%+%rr( 5% 2%n0% X,X a0 % 2%n0% 1,1, )%% 2%$$>9() 0ra3ar % $(u"n0( r67"+%@

Gr67"+% 5a 7un3;% \ Z : G(ra5% (@ -002@Y%)7raa)2-a.+%

C% "$0% 2%$$>9() +%n+)u"r Ju( % 5($)%+a(n0% ua r(0a %n5( \ Z :.Ca$% n;% $(a (9"5(n0(, 5( %u0ra 7%ra, 2%5(%$ u$ar a $(u"n0( 7%ru)a 2ara

(n+%n0rar a (Jua3;% 5( ua r(0a y=mx+ .

m=! y

! x=

1−0

1−0=1

11



S(n5% a$$", a2)"+an5% ($0( 9a)%r na (Jua3;% 5a r(0a y=mx+ ,

0(?$( y= x+ , 2ara (n+%n0rar % ', a2)"+a%$ " (0 )=0=0+ , a$$"

=0 . S(n5% a$$", y= x .

O 0ra'a)-% na 7%ra 5( "n0(ra) 5( )"n-a, +%% 7%ra 5(7"n"5%

an0(r"%r(n0(, 2%5( $(r ($+r"0% na $(u"n0( 7%ra@

W c=∫C

⃗F ∙ d⃗ s

A% r(0"rar %$ 5a5%$ 5% (:(r+>+"% 2r%2%$0%, 0(%$ a$ $(u"n0($ (Jua34($

2ara u0")"Kar 5uran0( a r($%)u3;%@

F = y i⃗+2 x⃗ j

d⃗ s=dx i+dy j

En0;%, $(n5% a$$", r($%)9(?$( % (:(r+>+"% a2)"+an5% na 7%ru)a 5($+r"0a

( r($%)9(n5% % (:(r+>+"%@

W c=∫C

( y ⃗i+2 x⃗ j ) ∙(dx i⃗+dy⃗ j)

W c=∫C ( y dx+2 x dy ) ,#$d% y= x , dy=dx

W c=∫0

1

3 x dx

W c=∫0

1

3 x dx

W c=3∫0

1

x dx

W c=3 x ²

2 | x=1

x=0

W c=1.5k&

S(n5% a$$", $a'(?$( Ju( % 0ra'a)-% r(a)"Ka5% 2()% +a"n-;% (Ju"9a)( aW c=1.5 ∙10

3&

12



. CALCULO DE MASSA DE UM CAO

.1. DEFINIÇÃOA$ "n0(ra"$ 5( )"n-a 2%5( $(r u0")"Ka5a$ 2ara +a)+u)ar a 5(n$"5a5(,

5($5( Ju( a 7un3;% 5(n$"5a5( )"n(ar un"5a5( 5( a$$a 2%r +%2r"(n0% $(a

+%n-(+"5a. E$0( 0<2"+% n;% (n0rar6 na 2r%9a5(7"n"3;% 5( +%% $( Q(n+%n0ra a

7<ru)aW.

A a$$a 0%0a) 5( u +%r2% 2%5( $(r +a)+u)a5a 5a $(u"n0( 7%ra@

M =∫C

ρ( x , y )d s

Gr67"+% r(2r($(n0an5% a 5(n$"5a5( ( ua +ur9a, 7%n0(@ UNESP

.2. EXEMPLO Exemplo III: Su2%n-a%$ Ju( 5(0(r"na5a 2%n0( 2%$$ua +a'%$ n%

7%ra0% 5a 2ar6'%)a \ Z :] a% )%n% 5( X ^ : ^ , ( 2%$$u" a 7un3;% 5(n$"5a5(

ρ ( x , y )= x +a)+u)( a a$$a 5($0( a3%, $a'(n5% Ju( a ($0a 5(n$"5a5( ($06 na

un"5a5( 5( # g&m' .

Solução: A a$$a 5% +a'% 2%5( $(r 5(7"n"5a 2%r M =∫

C

ρ( x , y)d s,

2r"("ra(n0(, (n+%n0ra%$ a 7un3;% 5(n$"5a5(. N($0( (:(r+>+"% ()a ($06

(:2)"+"0a +%% ρ ( x , y )= x .

A%ra, (n+%n0rar(%$ d s:

13



'

1+[¿¿ ( ( x )] ²d x=√ 1+(2 x ) ²d x=√ 1+4 x ²d x

d s=√ ¿

Su'$0"0u"n5% %$ 5a5%$ 5% (:(r+>+"% 2r%2%$0% na "n0(ra)

M =∫C

ρ( x , y)d s, 0(r(%$@

M =∫C

ρ( x , y )d s=∫0

4

x √ 1+4 x ²d x=∫0

65

√ ) d )

M = 1

12)3 /2|)=65

)=0= 1

12[65

3

2−1

3

2 ]=[65√ 65 * 1]≅ 43.59 k'

4. CONSIDERAÇÕES FINAIST(n5% ( 9"$0a % 0ra'a)-% a2r($(n0a5%, 2%$$>9() (9"5(n+"ar +%% % un5% 5%

($0u5% 5a$ 0a:a$ 5( 9ar"a34($ a2)%. Ta"$ a2)"+a34($ 7%ra r(0"ra5a$ 5( $"0ua34($ 2r60"+a$ Ju( 2%5( $(r (n+%n0ra5a$ n% 5"a?a?5"a 5% 2r%7"$$"%na) 5( (n(n-ar"a, a$$"%$0ran5% a "2%r08n+"a ( (7"+"n+"a 5% +6)+u)% 5"7(r(n+"a) ( "n0(ra).

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Cálculo integral 3 em engenharia cívil

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