MECH 237-002 Exam 1, Feb. 16, 2009
Name __ ______________________________
1. Link BC is 6 mm thik an! is ma!e "# stee$ %ith a &'0 M(a )$timate st*en+th in tensi"n. hat sh")$! be its %i!th w i# the st*)t)*e st*)t)*e sh"%n is bein+ !esi+ne! t" s)"*t a 20 kN $"a! %ith a #at"* "# sa#et "# 3.0/ Plan: Ultimate strength given. 600 mm Determine the Allowable Stress = P/A Where P = internal force along CB. 4 B Where A = (w(t of member CB. w
!iven "S = #.$ = %ltimate / allowable 90°
Allowable = %ltimate Allowable %ltimate / "S allowable = %ltimate / "S = &'$ Pa / #.$ σ = &'$ Pa / #.$ = )'$ * )$ + ,/m- = P / A
&0 mm C
'
s$"e
600
&0 &
5 3
Use Statics analsis to etermine the val%e for P in member CB. Call it 0 for tension.
( 20 kN
4
1 abo%t 2t. A = $ = 0(&3$ 4 P(+$$ 0 ( &3$ = 5$ 6, ( +$$ 0 = 5$ 6, (+$$ / &3$ 0 = 5' 6, = 5' * )$7 ,.
,ow anal8e member CB σ = )'$ * )$ + lb/in- = P / A Where P = internal tension9 0 = 5' 6,.
C 5 ( 20 kN
A = 0 / σ = 5' * )$7 , / )'$ * )$ + ,/m A = )++.++ * )$;+ m- = w t ∴ w = A / t = )++.+ )++.++ + * )$ ;+ m$.$$+ m ∴ w = 5.3 * )$ ;7 m ∴ w = 5.3 mm
h"% a$$ %"*k #"* #)$$ #)$$ *e!it. he FE Han!b""k ma be )se! )se! as ")* *es")*e.
-2-
2. %" %""!en membe*s " 3 x 6 inh )ni#"*m *etan+)$a* *"ss seti"n a*e 8"ine! b a sim$e +$)e! s$ie. n"%in+ that ( 2&00 $b, !ete*mine the n"*ma$ an! shea*in+ st*esses in the +$)e! s$ie.
( 3:
(
&0°
6:
0he cross;sectional area = (#<(+< = )3 s.in. B%t9 o% nee to o the analsis along the angle s2lice. (;; ( "s &0 ° &0°
4"
4x
( 2&00 $b.
&0°
(⊥ ( sin &0 °
,ormal stress = σ = P⊥ / Ao Shear stress = τ = P// / Ao P // = P (tangential com2 = 5&$$ cos &$ ° = ).3#3' * )$7 lb P ⊥ = P (normal com2 = 5&$$ sin &$° = ).'&5 * )$7 lb Using the geometr in the 2ict%re9 A* = Ao sin &$ °9 ∴ Ao = A* / sin &$° ,ormal stress = σ = P⊥ / Ao = 5&$$ sin &$° (sin &$° / )3 in- = ''.$>$ 2si Shear stress = τ = P// / Ao = 5&$$ cos &$° (sin &$° / )3 in- = +'.+'& 2si
h"% a$$ %"*k #"* #)$$ *e!it.
-3-
3. he stee$ #*ame E 200 <(a has a !ia+"na$ b*ae B5 %ith an a*ea "# 1920 mm=. 5ete*mine the $a*+est a$$"%ab$e $"a! ( i# the h an+e in $en+th "# membe* B5 is n"t t" exee! 1.6 mm. C
(
B
(
θ
6
6m
δ = P@ / A ≤ ).+ mm @ = √ (+- '- = .3)$ m A = )>5$ mm- given = 5$$ * )$ > ,/mP = internal force in BD. Use Statics to fin this force.
7.10 '
θ = tan;) +/' = '$.5°
4
5
E%ic6 ?t. C analsis: "BC = P "CD = $
'm
?oint B θ
( B5 "s θ
4B B5
1"* = $ = ;P BD cos θ ∴ BD = P / cos θ ∴ BD = P / cos '$.5° ∴ BD = ).'+5 P S%bstit%te this val%e into the e*2ression for eformation of Fchange in length< of BD.
B5 sin θ
δ = P@ / A ≤ ).+ mm =
( ).'+5 P (.3) m * )$ + mm- / m( )>5$ mm- ( 5$$ * )$> ,/m-
).+ * )$ ;7 m ( )>5$ m- (5$$ * )$> ,/m- ( ).'+5 ( .3) m ( )$ +
= P = '$.#+5 * )$7 , ∴ P = '$.#+5 6,
h"% a$$ %"*k #"* #)$$ *e!it.
-&-
&. his assemb$ "nsists "# an a$)min)m she$$ #)$$ b"n!e! t" a stee$ "*e an! is )nst*esse!. 5ete*mine a the $a*+est han+e in teme*at)*e i# the st*ess in the a$)min)m she$$ is n"t t" exee! 6 ksi, an! b the "**es"n!in+ han+e in $en+th "# the assemb$. 4$)min)m> tee$>
E 10.6 x 106 si, E 29 x 106 si,
α 12.9 x 10-6 ; °F, α 6.' x 10-6 ; °F,
1.2': ")tsi!e !iamete* 0.7': !iamete*
α al%m > α steel in
∴ al%m com2resses against the steel 1.2' in 0.7' in
0hin6 I Action = Jeaction P al%m = ; P steel
4$)min)m he$$
Bone I ∴ δ al%m = δ steel
tee$ C"*e
0otal eformation of al%min%m α @ ∆0 /; P@ / A Al%min%m: Steel: Al%min%m: Steel:
=
total eformation of steel
=
α @ ∆0 /; P@ / A
A = π/& ).5'- ; $.'- = $.3'& inA = π/& $.'- = $.&&)3 inA = ( $.3'& in- ()$.+ * )$ + lb/in- = 3.#5' * )$ + lb. A = ( $.&&)3 in- ( 5> * )$ + lb/in- = )5.3)5 * )$ + lb.
ach term has the length9 therefore o% can cancel the @Gs in all terms. α ∆0 ; α ∆0 = P / A steel ; P / A al%m ;+ ( )5.> 4 +.' )$ ∆0 = P ) / A ) / A Ho% have info abo%t the al%min%m shell: σ ≤ + * )$7 lb / in- = P / A P (al%min%m = σ A = ( + * )$7 lb/in- ( $.3'& in- = &.)5& * )$7 lb = P ∴ ∴ ∴
∆0 = &.)5& * )$7 lb / +.& * )$ ;+ ) / A ) / A ∆0 = #+.#)5' * )$ + ( ) / )5.3)5 * )$ + ( ) / 3.#5' * )$ + ∆0 = )&'.>° bac6 s%bstit%te this into either sie of the e%ation
Deformation of Al%min%m = eformation of Steel = )$.'# * )$ ;# in = $.$)$'# in = δ
h"% a$$ %"*k #"* #)$$ *e!it.
