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Properties of Materials and Applications, CPE641

TUTORIAL -1

OCTOBER !14

1" A #!! I$ %o&o'eneo(s c)linder is s(pported $) t*o rollers as s%o*n in +i%(re

1" eter&ine t%e forces eerted $) t%e rollers on t%e c)linder" All s(rfaces are

s&oot%"

 

+i'(re 1. T%e %o&o'eneo(s c)linder and it/s s(pport

+ree Bod) ia'ra&.

+ind R A and RB.

∑ F  x+→=0

 R Acos60°− R Bcos60°=0

  R A= RB   0e" 12

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∑ F  y+↑=0

 R A sin 60 °+ RB sin 60 °−W =0(eq .2)

¿ (1 )→ (2 )

 RB sin 60°+ RB sin 60°−800=0

 2 [ RB (sin 60 ° ) ]=800

  R A= RB=461.8 lbf 

" A c(r3eed slender $ar is loaded and s(pported as s%o*n in +i'(re "

eter&ine t%e reactions at at s(pport A"

 

+i'(re T%e c(r3ed $ar s(pportin' ! L$ *ei'%t

+ree Bod) ia'ra&.

+ind M A.

∑ F  X +→=0

 − A X =0

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  ∑ F Y +↑=0

 A Y −250=0

 A Y =250 lbf 

∑ M  A+↓=0

 − M  A+250 (3 )=0

  M  A=750 lbf . ft 

5" T%e la*n &o3er s%o*n in +i'(re 5 *ei'%s 5 I$" eter&ine t%e force P

re(ired to &o3e t%e &o3er at a constant 3elocit) and t%e forces eerted on t%e

front and rear *%eels $) t%e inclined s(rface"

 

+i'(re 5 T%e la*n &o3er

ote. 1 l$f  74"4   1 in 7 !"!4 &

+ree $od) ia'ra&.

+ind P,R A and RB .

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8 7 5l$f 

∑ F  X +→=0

 Pcos30

°−

W  sin 15

°=0

 P=35sin 15°

cos30°

7 1!"46l$f 

∑ F Y +↑=0

 R A+ R B− Psin 30

°−W  cos15

°=0

 0e"12

∑ M  A+↓=0

W  cos15 ° (13 )−W  sin 15° (4 )− R B (27)− P sin 30° (34 )+ Pcos30 ( P sin 30° )=0

35cos15 ° (13 )−35sin15° (4 )− RB (27 )−10.46sin 30° (34 )+10.46cos30° (10.46sin 30° )=0

 RB=10.10 lbf 

9($ RB 7 1!"1! l$f to e"1

R A 7 -#":4 l$f 

4" eter&ine t%e forces in &e&$ers C, C+ and +; of t%e $rid'e tr(ss as s%o*n

in +i'(re 4"

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  +i'(re 4 T%e $rid'e tr(ss

  ote< 1 =ip 7 4"4 >

  1 ft 7 !"5!4# &

∑ F Y +↑=0

 R A+ R E=30  0e"12

∑ M  A+↓=0

10 (15 )+20 (30 )− R E (45 )=0

 R E=16.67 kip

9($ RE 7 16"6?=ip to e" 1

R A 7 15"55=ip

 At point A.

∑ F Y +↑=0

 R A+T  ABsin30 °=0

T  AB=−13.33

sin30°

¿−26.66kip

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∑ F  X +→=0

T  AG=−T  AB cos30 °

¿23.09 kip

 At point B.

T BA−T BC =0

T BA=T BC 

T BC =26.66kip

 At point E.

∑ F  X +→=0

−T  EF −T  EDcos30°=0  0e"12

∑ F Y +↑=0

 R E+T  ED sin 30°=0

T  ED=−

(

  R E

sin30

°

)

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¿−33.34kip  0e"2

9($ 3al(e in e" to e"1

T  EF =28.87 kip

 At point .

T  DE−T  DC =0

 T  DE=T  DC 

 T  DC =33.34 kip

 At point C.

∑ F  X +→=0

−T CBcos30°−T CGsin 30°+T CF sin 30°+T CD cos30°=0

−¿¿¿

26.66−T CG tan 30°+T CF tan 30°−33.34=0  0e"12

∑ F Y +↑=0

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−¿¿¿

26.66tan 30 °−T CG−T CF +33.34 tan 30 °=0(eq .2)

9ol3e e" 1 and e" si&(ltaneo(sl).

T CF tan 30°=T CG tan30°+6.68

T CF =−T CG+34.64  0e"52

9($ e"5 into e"1

(34.6−T CG ) tan 30°=T CG tan 30°+6.68

34.64−T CG=T CG+11.57

T CG=11.53 kip

T CF tan 30°=11.53 tan30°+6.68

T CF =23.10kip

 At point +.

∑ F 

 X 

+→=0

−T  FG−T  FC cos60 °+T  FDcos 60°+28.87=0  0e" 12

∑ F Y +↑=0

−20+T  FC sin 60°+T  FDsin 60°=0

T  FD=46.19

kip  0e"2

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9($ e" to e"1

−T  FG+23.10cos 60°+46.19cos60°−28.87=0

T  FG=5.775kip

" A dr(& of &ass !! >' is s(pported $) a pair of fra&es" T%e second fra&e is

$e%ind t%e one s%o*n" T%e s)ste& is as s%o*n in +i'(re " eter&ine t%e

forces actin' on &e&$er ACE"

 

+i'(re T%e dr(& and it/s s(pport

+ree Bod) ia'ra&.

 RC =W =m

¿( 200!9.811000   )k" 

¿1.962 k" 

∑ F  y+↑=0

 R D sin 45°+ R Esin 45°−W =0

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 R D sin 45°− R E sin 45°−1.962 k" =0  0e"12

∑ F  X +→=0

 R D cos45 °− R E cos 45°=0  

 R D= R E  0e"2

9($ e" to e"1

2⌊ R E (sin 45° ) ⌋=1.962

 R E=1.3873

k" 

 R D=1.3873k" 

∑ F Y +↑=0

 R A+ RB−W =0

 R A+ RB=1.962 k"    0e"12

∑ M  A+↓=0

− RB (2 )+W  (1 )=0

 RB=−1.962

−2

 

¿0.981k" 

  0e"2

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9($ e" to e"1

 R A=1.962−0.981

 R A=0.981 k" 

6" eter&ine t%e internal forces actin' on section a-a in t%e $ar rac= s%o*n in

t%e dia'ra& $elo*" Eac% $ar %as a &ass of ! >'

 

+ree Bod) ia'ra&.

W 1=(

50!9.811000   )k" 

¿0.4905 k" 

W 2=( 25!9.811000   )k" 

¿0.2453k" 

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∑ F  X +→=0

 A  X −B sin27°=0  0e"12

∑ F Y +↑=0

 A Y −2W 1+B cos27°=0  0e"2

∑ M  A+↓=0

−W 1(290cos27° )+W 

1 (120cos63° )+B (580 )=0

−0.49 (290 cos27 ° )+0.49 (120 cos 63 ° )+B (580 )=0

580B=−99.48

B=−0.172 k"   0e"52

9($ e"5 to e"1 and e"

 A Y −2 (0.49 )+0.172cos27°=0

 A Y =0.827

k" 

 A  X −0.172sin 27°=0

 A X =0.078 k" 

 A=√ ( A X )2

+( AY )2

¿√ (0.078 )2

+ (0.0827 )2

  ¿0.8308k" 

?" T%e reaction $et*een t%r cr(tc% and t%e 'ro(nd is 5 I$, as s%o*n in t%e

dia'ra& $elo*" eter&ine t%e internal forces actin' on section a-a

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+ree Bod) ia'ra&.

∑ F  X +→=0

 A  X −35sin25°=0

 A  X =14.79 ft 

∑ F Y +↑=0

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 AY −35cos25 °=0

 A Y =31.72 ft 

∑ M  A+↓=0

 M # .#+(35sin25° ) (2 )=0

 M # .#=29.58k" 

#" eter&ine t%e force P re(ired to p(s% t%e 11- >' c)linder o3er t%e s&all $loc=

s%o*n in t%e dia'ra& $elo*

 

+ree Bod) ia'ra&. 

eter&ine force P.

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W =115!9.81

¿1128.15k" 

 

∑ F Y +↑=0

− P sin 20°−1128.15cos20°+ R A+ RB sin41.23°=0   0e"12

∑ F  X +→=0

  P cos20 °− RB cos 41.23 °−1128.15 sin 20 °=0  0e"2

∑ M  A+↓=0

 P cos20° (220 )+1128.15sin20° (220 )− RB sin41.23° (165.45 )− RBcos41.23 ° (75)=0

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:" An aial load P is applied to a ti&$er $loc= as s%o*n" eter&ine t%e nor&al and

s%ear stress on t%e plane of t%e 'rain if P 7 1!! I$

+ree Bod) ia'ra&.

 Ass(&e Effecti3e Area7 1ft

 F  " = P sin 14°

  ¿362.883lb

 F t = P cos14°

¿1455.444 lb

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$ =(   F  "  Eff . A%e#)

¿362.883

1

¿362.883 lb

ft 2

& =(   F t  Eff . A%e# )

¿1455.444

1

¿1455.444 lb

ft 2

1!"T%e steel $ar s%o*n *ill $e (sed to carr) an aial tensile load of 4!! >" If t%e

t%ic=ness of t%e $ar is 4 &&, deter&ine t%e nor&al and s%earin' stresses on

t%e plane a-a

+ree Bod) ia'ra&.

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sin 53°=75

'

'=(   75sin 53° )m

¿93.91m

 Effe(ti)e . A%e#* A=45 (93.91 )

¿4225.96mm2

 F  " 

400k" =cos 37°

 F  " =319454.204 " 

 F t 

400k" =sin 37°

 F t =240726 " 

$ = F  " 

 Eff . A%e#

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¿319454.204

4225.96

¿75.59 MP#

& = F t 

 Eff . A%e#

¿  240726

4225.96

¿56.96 MP#

11"

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+ree Bod) ia'ra&.

sin 37°=  9

 AD

 AD=14.95

k" 

cos37°=  AB

14.95

 AB=11.94 k" 

2.25 MP#=11.94 k" 

100#

#=53.07mm

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 A%e# * A=4∈

!6∈¿

¿24¿2

in &&

1∈¿2

(25.42 )mm2

¿¿

24 ¿2! ¿

psi @&&

225 p+i !

0.0069  " 

mm2

1 p+i  =1.5525

  " 

mm2

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 F =1.5525  " 

mm2

!15483.84 mm2=24038.6616 " 

Co&presi3e force,P.12

13=

24038.6616

 P

 P=26.04 k"