Three round, copper alloy bars having the same length L but different shapes are
ID: 1939148 • Letter: T
Question
Three round, copper alloy bars having the same length L but different shapes are shown in the figure. The first bar has a diameter d over its entire length, the second has a diameter d over one-fifth of its length, and the third has a diameter d over one-fifteenth of its length. Elsewhere, the second and third bars have diameter 2d. All three bars are subjected to the same axial load P. Use the following numerical date: p = 1400 kN, L = 5 m, d = 80 mm, E = 110 GPa, and upsilon = 0.33. Find the change in length of each bar. Find the change in volume of each bar.Explanation / Answer
Crosssectional area for d = 80mm, A1 = d2/4 = 1600 mm2 =16x10-4 m2
Crosssectional area for 2d = 100mm, A1 = (2d)2/4 = 6400mm2 =64x10-4 m2
(a) Change of length
From L=PL/(EA), we have
L1 = PL/(EA1) = 1400,000*5/(110x109*16x10-4) = 0.01266 m = 12.66 mm
L2 = P(L-0.2L)/(EA2)+P(0.2L)/(EA1) =(PL/E)(0.8/A2+0.2/A1)
=1400,000*5/(110x109)*[0.8/(64x10-4)+0.2/(16x10-4)] = 0.00506 m = 5.06 mm
L3 = P(L-L/15)/(EA2)+P(L/15)/(EA1) =(PL/E)(0.933333/A2+0.066667/A1)
=1400,000*5/(110x109)*[0.933333/(64x10-4)+0.066667/(16x10-4)] = 0.00380 m = 3.80 mm
(gt_grad's work, please do not copy)
(a) Change of volume
V=eV
where e = (1-2)/E*(x+y +z) = (1-2)/E=(1-2)P/(EA) (since y =z =0 here), and V = AL
V=(1-2)P/(EA)*AL = (1-2)PL/EA
V1 = (1-2)PL/E =(1-2*0.33)* 1400,000*5/(110x109) = 21636.36x10-9 m4 = 21636.36 mm4
V2 =(1-2) P(L-0.2L)/E+(1-2)P(0.2L)/E =(1-2)PL/E = 21636.36 mm4
V3 =(1-2) P(0.933333L)/E+(1-2)P(0.066667L)/E =(1-2)PL/E = 21636.36 mm4
The answer in the book may have too much error.
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