FE Mechanical Mechanics of Materials Practice Problems

The current FE Mechanical specification assigns 9-14 of the 110 exam questions to Mechanics of Materials. Form A contains 10 original problems in this area. The sample below is published in full, with the same worked-solution format used throughout the book.

A stepped tie rod of high-strength alloy steel with a 690 MPa yield strength hangs from a rigid ceiling and carries a 40 kN axial tensile load at its lower end. The upper segment is 900 mm long with a 20 mm diameter; the lower segment is 600 mm long with a 14 mm diameter. The modulus of elasticity is 200 GPa. The total elongation of the rod is most nearly:

(A)0.21 mm
(B)0.95 mm
(C)1.35 mm
(D)1.95 mm

Show worked solution

Answer: (C)

Axial flexibility varies inversely with area, so the 14 mm segment supplies more than half of the 1.35 mm total elongation.

Both segments transmit the full 40 kN, so their elongations add and each cross-sectional area must be evaluated separately.

A_{1} = \frac{π ( 20 ) ^{2}}{4} = 314.2 mm^{2}

A_{2} = \frac{π ( 14 ) ^{2}}{4} = 153.9 mm^{2}

The higher of the two axial stresses occurs in the slender segment and stays well below the 690 MPa yield strength, so the linear elastic relation governs throughout.

σ_{2} = \frac{40 , 000}{153.9} = 259.8 MPa < 690 MPa

Deformation scales inversely with area, so the slender lower segment stretches more even though it is shorter, and the two contributions sum to the downward movement of the loaded end relative to the ceiling.

δ_{1} = \frac{( 40 , 000 ) ( 900 )}{( 314.2 ) ( 200 , 000 )} = 0.573 mm

δ_{2} = \frac{( 40 , 000 ) ( 600 )}{( 153.9 ) ( 200 , 000 )} = 0.780 mm

δ = 0.573 + 0.780 = 1.35 mm

FE Reference Handbook — Mechanics of Materials: Uniaxial Loading and Deformation, δ = PL/(AE)

Why the other choices appear

(A)Subtracting the segment elongations instead of adding them, 0.780 - 0.573 = 0.21 mm.
(B)Carrying the 20 mm diameter over the full 1500 mm length, giving 0.95 mm.
(D)Carrying the 14 mm diameter over the full 1500 mm length, giving 1.95 mm.