A rectangular steel bar shown is fixed at the right end has a thickness of 10 mm

Question

A rectangular steel bar shown is fixed at the right end has a thickness of 10 mm. The temperature is increased by 20 C and a force F0 is applied as shown mid way along the bar. Use the properties E = 200 GPa , alpha = 12 (10-6)/ C. Determine the value of the force F0 which results in zero displacement at the left end (x= 0). For the value of force F0 determined in part (a) and the temperature rise of 20 C, determine the following: the stress at x = 50 mm, the displacement x = 100 mm, and the stress at x = 150 mm

Explanation / Answer

First find the expansion of last point due to heating

L = LxxT = .2x12x10^-6x20 = 48x10^-6

Now the displacement of whole left half will be same as of mid point compression

therefore for ist part F/A = ExL/ L => F = 200x10^9x48x10^-6x10x10x10^-6/.1 = 9600 N

Now second part

Expansion of point x = 50 due to tempe => L = LxxT = .15*12*10^-6*20 = 3.6x10^-5

but right side displacement due to F at x =50 is 4.8x10^-5

net L = 1.2x10^-5 towards right

stress =  200x10^9x1.2x10^-5/.15 = 16x10^6

Now for Displacement at x = 100

due to Force it is 48x10^-6

due to expansion it is .1x12x10^-6x20 = 24x10^-6

net displacement is 24x10^-6 towards right

Now for displacement at x = 150

due to expansion it is .05x12x10^-6x20 = 12x10^-6

due to Force it is 9600/(10x10x10^-6)x(.05/200*10^9) => 24x10^-6

net displacement = 12x10^-6 towards right

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