Problem 2.45

2.45. Please, delete the last part of the question stating "Compare the result etc.  ", and add the following to the question:

Determine the displacement u at the load point, and verify that

u = ∂U/∂P

For a uniform bar in uniform axial behavior, U = N²L/2EA.  Bars BC, CD, and DE have the same axial force N = -P.

Bar CD: A_CD = p(nd)²/4 = n²A,  A = pd²/4

U_CD = P²L/8EA_CD = P²L/8EAn²

U_(BC+DE) = P²3L/8EA

U = P²L/8EAn² + P²3L/8EA = (P²L/2EA)(1/4n² + 3/4) coincides with stated formula

Displacement at load point = sum of decreases in length.

For a uniform bar in uniform axial behavior, decrease in length is DL = -NL/EA.

(DL)_CD = PL/4EA_CD = PL/4EAn²

(DL)_(BC+DE) = P3L/4EA

DL = PL/4EAn² + P3L/4EA = (PL/EA)(1/4n² + 3/4)

∂U/∂P = same value as above.