Mechanics of Materials Flashcards

In the torsion formula τ = Tr/J, what does J represent?

Answer: Polar moment of inertia of the cross-section

J is the polar moment of inertia (second moment of area about the longitudinal axis), which governs resistance to torsion.

The polar moment of inertia J for a solid circular shaft of diameter D is:

Answer: πD⁴/32

J = πD⁴/32 for a solid circular shaft; note that the planar I = πD⁴/64 and J = 2I.

Euler's formula for the critical buckling load of a column is:

Answer: Pcr = π²EI/(KL)²

Euler's critical load Pcr = π²EI/(KL)², where KL is the effective length accounting for end conditions and I is the minimum second moment of area.

For a column with both ends pinned (pin-pin), the effective length factor K is:

Answer: 1.0

A pin-pin column buckles in a single half-sine wave so its effective length equals its actual length, giving K = 1.0.

The angle of twist φ (in radians) of a shaft of length L under torque T is given by:

Answer: φ = TL/GJ

φ = TL/(GJ), where G is the shear modulus (modulus of rigidity), L is the shaft length, and J is the polar moment of inertia.

When a structural member carries both an axial compressive load P and a bending moment M simultaneously, the critical total compressive stress is found by:

Answer: Adding P/A and My/I (principle of superposition)

By superposition, total normal stress = P/A ± My/I; the signs depend on whether the bending stress adds to or subtracts from the direct stress at the point of interest.

The slenderness ratio of a column is defined as:

Answer: Effective length (KL) divided by radius of gyration (r)

Slenderness ratio λ = KL/r, where r = √(I/A) is the radius of gyration; a high slenderness ratio indicates buckling will govern over material yielding.