Question

A square duct made of a ductile material...

walls. The length of the duct is 4ft and is heated from room square duct made of ductile material is fixed between two A temperature Determine the following: a) Assuming all dimensions and material properties as variables, determine a formula for the change in temperature (AT) required to cause buckling of the duct, b) If the fixed wall supports are replaced by pin joints in both ends of the duct, what would be the temperature? Take, E=29x103ksi and a=158 10-6-1 C) If the yield strength of the material is 30 ksi, will the duct yield before Youckling ? Explain your answer mathematical procedure. What would be the AT change at yielding ATTO 12 in

Answer 1

AT >o Per tin pcr K 4ft 1.2 in 14 uft 48 inches 2 hence length of duet 48 inches 2. load * E I min Para Crippling 2 (le) I min 2 minimum cross section moment of inestia of effective of duct [ changes le = length som finture to finture > linead le length of change in length due to change in temperature Dla l x ot duet de coefficient of enpansion DT. change in temperature stres Young's modules Dele stress. Force Fone Area strain Area

At buckling Force applied a per Areaz Area of cross section of the duet Force 2 z) De e Area) XE E 2 Fone Area Dd e ex Fone Area) XE Por Force 2 Prve 2 ДА 2 n' E I min ce) 2 2 Dd also from 2 clue to ex Te I min (le) ²x (Ares) XE change in length change in temperature de la AT this supposed changed in length is suppressed by support force which in this case is given by both be are same Now shi (por hence &LOT. Xx n E I min (de) ² x (Area) XE 2 2 T I min ДТ а a f I min (le) 2 x (Aree) x fx & 2 (de) x Areaxa

Now duet (oobs section tin 1-2m And he or is' effective length for fined support le all { length of duet 2 2 DTZ I min 2 4 À Imin l 2 x Area xa (1/² x Area xa Now assa assuming dimension of Cross section of duet 3 dxd sxs² 2 I min - 2 12 12 4 4 S d Iabd 12 12 Area : 2 s-d 2) 2 2 DT 2 45 x ls ($9-09) ий 2 Lx 12 xe x (3-4) 2x12x &xis AD 4 7 (5+2) xX 12 x 22

ATC 2 x (5²+d) xx3 l ? 2 b Fined replaced by pin point o) le = (l) [length of dust we know 2 ЛТ Ăx I min Jound earlier 2 (le) ²x Areaxa nx Imin ax 1² x Area 2 I X I min axe ² x Area I min 4 sird 2 same Arean s²-d? 12 / ДГ 2 2 2 7 x (sh.d") nx (5² + 2) (d) o'x12 xé xes?-d) x/2x8*X($? A) a(std XX 12 e 2 DT - ²( 3²+d² ñ (1-25+(13') 15x100 x 12X (463² 2 x 12 x 1² 58.06781

30ksi a We know CE yield strength = AT at buckling Catyeild) 58.067 To find DT Weknow Dla ex Force z ea DT (pooored earlier Areave 2 &x Force Area XE XLDT > AT 2 Force gildstrength stren z Area & Exa Exa Force z [ given stressa gield stren area _ z Boksi yield stress 30 Exa 294103 x 15x106 [E. 29 x 8 (ksi), gield strength - 30 ksi) La Istiot Vof

Д1 30 68.9655°F 2 29x18x15x106 reach crippling So when we heat it will early for crippling & IT ind , hence it buckled before yielding Nil be like if you are please leave a Grywer with happy