Thermal stress questiontopic strengths of materials

[gunta]

1.The center rod CD of the assembly shown in the figure is heated from T1 = 30oC to T2 = 180oC using electrical resistance heating. The two end rods AB and EF are heated from T1 = 30oC to T2 = 50oC. At the lower temperature T1, the gap between C and the rigid bar is 0.7 mm. Rods AB and EF are made of steel and each has a cross-sectional area of 125 mm2. CD is made of aluminum and has a cross-sectional area of 375 mm2. ESteel = 200 GPa, EAluminum = 70 GPa, alpha of steel = 12x10-6/oC, alpha of Aluminum = 23 x10-6 /oC.

a. Calculate the force in rods AB, EF and CD caused by the increase in temperature.

b. Calculate the stress in rods AB, EF and CD caused by the increase in temperature2. i tried using the equations for thermal stress i.e. stress = E x alpha x change in temp this is a copy of the question's diagram in case the attachment doesn't open
http://www.chegg.com/homework-help/questions-and-answers/1-center-rod-cd-assembly-shown-figure-heated-t1-30oc-t2-180oc-using-electrical-resistance--q2928326

 

[SteamKing]

Show your work. Where did you get stuck?

 

[gunta]

i tried using the equation where (alpha of aluminium - alpha of steel) x (change in temp) = F [(1/(EaxAa) + (1/(EsxAs)]

and i did that for both temp changes and then used the equation stress = F/A to find both stresses

please tell me if this is correct please... i cannot take a pic of my work right now because my phone data cable isn't working

 

[SteamKing]

I think this problem has some subtleties which you have overlooked. The cylinder CD has a gap between its top and the cross member supported by the two rods. Until CD expands and makes contact, there is no stress present.

 

[gunta]

i think i figured it out... i used the 30 deg temp for the equation to find the temp change using the elongation... that is 0.7 = 23x10^-6 x 70 x (T - 30) and then solved for T and then from that temp i used the long equation from above and solved F ... you think that is correct?

 

[Chestermiller]

i think i figured it out... i used the 30 deg temp for the equation to find the temp change using the elongation... that is 0.7 = 23x10^-6 x 70 x (T - 30) and then solved for T and then from that temp i used the long equation from above and solved F ... you think that is correct?

In my judgement, this is not the correct way to proceed. You can solve this problem correctly using a two step approach. In step 1, you assume that, rather than the three bars being constrained mechanically, they are completely free to expand thermally (i.e., with no stress). You determine how much the center bar would expand with a temperature rise of 150C, and how much the other two bars expand with a temperature rise of 20 C. I'll discuss what to do in step 2 after you have reported back the results for step 1.

Chet

 

[gunta]

but if they said that there is a space of 0.7mm wouldn't that mean that that is the expansion when the temp rise before it goes under stress at the bar ?

 

[Chestermiller]

but if they said that there is a space of 0.7mm wouldn't that mean that that is the expansion when the temp rise before it goes under stress at the bar ?

Yes. But that temperature is not of interest in the solution of this problem as stated. Please indulge me, and calculate the expansions of the bars as if they were not constrained. I promise that you will soon see where his is leading.

Chet

 

[SASKUE995]

So what is the step 2..i have completed the expansions...

 

[Chestermiller]

So what is the step 2..i have completed the expansions...

Hi Saskue995. Welcome to physics forums!
So, you completed the calculation of the free expansion of the rods. What changes in lengths did you calculate? Was this enough to more than close the 0.7 mm gap? If so, by how much?

What you have found so far are the new unstressed lengths of the three rods. If the changes in length would have been more than enough to close the gap, the middle rod is going to have to be compressed, and the two outer rods are going to have to be stretched so that the middle rod just fits between the plates. The total compression of the middle rod plus the total stretch of the outer rods is going to have to just match the excess length from step 1.

Let T represent the tension required in each of the outer rods. If you isolate the upper plate as a free body, what upward force must the middle bar exert on the plate in order for the upper plate to be in equilibrium? By Newton's third law, what is the downward force that the upper plate exerts on the middle rod? In terms of T, what is the tensile stress in the outer rods? In terms of T, what is the compressive stress in the center rod? What are the tensile strains in terms of T? What are the changes in length in terms of T? What does T have to be in order to close down the excess length from step 1?