Two opposite cantilevers shown below both are made of steel. They both have the
ID: 1822652 • Letter: T
Question
Two opposite cantilevers shown below both are made of steel. They both have the same length L of 1,044mm. However, their cross sections are different; the thinner one has the cross sectional area A of 13mm2 and the thicker one has the cross sectional area of 4A. The gap between the ends of the cantilevers is L/1,692. Determine the axial force induced in the thinner cantilever if the temperature of the thicker cantilever rises by 63ºC. Young's modulus for steel can be taken as 200GPa and the coefficient of thermal expansion as 12x10-6 mm/(mm ºC). Give your answer in N.
Two opposite cantilevers shown below both are made of steel. They both have the same length L of 1,044mm. However, their cross sections are different; the thinner one has the cross sectional area A of 13mm^2 and the thicker one has the cross sectional area of 4A. The gap between the ends of the cantilevers is L/1,692. Determine the axial force induced in the thinner cantilever if the temperature of the thicker cantilever rises by 63 degree C. Young's modulus for steel can be taken as 200GPa and the coefficient of thermal expansion as 12x10-6 mm/(mm degree C). Give your answer in N.Explanation / Answer
length of Thicker beam = 1044 mm
coefficient of thermal expansion, = 12*10-6 /0C
temperature change, T = 63 0C
A = 13 mm2 = 13 *10-6 m2
As per thermal expansion theory,
L/L = T
change in length of thicker beam, L = 1044*12*10-6*63 = 0.7893 mm
gape between beams = L/1692 = 1044/1692 = 0.6170 mm
means upto 0.6170 mm thicker beam can easily expand. after that it will exerted stress because of thinner beam.
then, modified length of thicker beam = 1044 + 0.6170 = 1044.6170 mm
after expansion, length of thicker beam = 1044 + 0.7893 = 1044.7893 mm
means (1044.7893 - 1044.6170) mm should causes the stress in thcker beam.
strain in thicker beam , = change in length/modified length
= (1044.7893 - 1044.6170)/1044.6170 = 1.6494 *10-4
Young's Modulus of steel, E = 200 GPa = 200*109 N/m2
then stress developed in thicker beam due to strain, = E
= (1.6494 *10-4) *(200*109 N/m2) = 3.2988 * 107 N/m2
force developed in thicker beam, F = *(4A)
F = 3.2988 * 107 *4*13 *10-6 = 1715.38 N
according to Newton's third law,
action = - reaction
that means, Axial force, F is also equal to the force acting on cross-section of thinner beam.
hence, Axial force induced in thinner beam is 1715.38 N
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