3.2 Statics II: Friction, Centroids & Moment of Inertia

Dry (Coulomb) Friction

Friction resists relative sliding between dry surfaces in contact. The maximum static friction force just before motion begins (impending motion) is:

• F_max = μs·N

where μs is the coefficient of static friction (dimensionless) and N is the normal (perpendicular) reaction force. Once sliding, kinetic friction governs: F_k = μk·N, with μk < μs. Key facts tested on the FE Civil exam:

• Friction always opposes the tendency of motion; below impending motion, the actual friction force is whatever equilibrium requires — it is ≤ μs·N, not automatically equal to it.
• The angle of friction φ satisfies tan φ = μs. A block on an incline is on the verge of sliding when the incline angle equals the friction angle, θ = φ.
• The normal force N is not always equal to the weight — on an incline N = W·cos θ.

Friction trap

A frequent error is setting F = μN before confirming impending motion. First check whether the applied force exceeds the available friction. If equilibrium is possible with F < μs·N, the body does not move and F is found from ΣF = 0.

, belt friction T₁/T₂ = e^(μβ), where β is the wrap angle in radians). On an incline, resolve forces along and perpendicular to the surface, not horizontal and vertical — this isolates N = W·cos θ and the driving component W·sin θ cleanly. A block resting on an incline stays put as long as W·sin θ ≤ μs·W·cos θ, which simplifies to tan θ ≤ μs; the block slides once the incline angle exceeds the friction angle. Because the weight cancels, the verge-of-slipping angle depends only on the coefficient of friction, not on how heavy the block is — a result candidates frequently get wrong by assuming a heavier block is harder to start moving.

Centroids of Areas & Composite Shapes

The centroid is the geometric center of an area — where it would balance. For a composite area made of simple parts (handbook table of shapes), the centroid coordinates are area-weighted averages:

• x̄ = Σ(Aᵢ·x̄ᵢ) / ΣAᵢ, ȳ = Σ(Aᵢ·ȳᵢ) / ΣAᵢ

Treat holes or cutouts as negative areas. Measure each part's x̄ᵢ, ȳᵢ from a single common reference axis. The NCEES FE Reference Handbook gives centroid locations for rectangles, triangles, circles, semicircles, and quarter-circles.

A triangle's centroid sits one-third of the height up from the base (two-thirds from the apex) — a value the exam loves to test.

The procedure for any composite shape is mechanical: (1) divide the figure into standard sub-areas, treating cutouts as negative; (2) tabulate each Aᵢ and the centroid distances x̄ᵢ, ȳᵢ from one chosen reference; (3) compute the area-weighted sums. Keeping an organized table prevents sign and bookkeeping errors, which are the dominant cause of wrong centroid answers under time pressure.

Moment of Inertia & Parallel-Axis Theorem

The area moment of inertia (second moment of area) measures resistance to bending about an axis; units are length⁴ (in⁴ or mm⁴). Standard centroidal values:

• Rectangle about its centroid: Ī = bh³/12
• Circle: I = πd⁴/64 = πr⁴/4

Note the cube on h: doubling the depth of a rectangle increases I by 8×, which is why beams are oriented with the long dimension vertical. To find I about a non-centroidal axis, use the parallel-axis theorem:

• I = Ī + A·d²

where Ī is the moment of inertia about the centroidal axis, A is the area, and d is the perpendicular distance between the two parallel axes. For composite sections, compute each part's Ī, transfer with A·d² to the composite centroid, then sum.

Worked Example — Composite I

Find I about the centroidal x-axis of a rectangle 4 in wide × 6 in tall.

• Ī = bh³/12 = 4·(6³)/12 = 4·216/12 = 72 in⁴

Now find I about its base (6 in from the centroid is wrong — d is to the base, 3 in):

• A = 24 in², d = 3 in → I_base = 72 + 24·(3²) = 72 + 216 = 288 in⁴

Radius of gyration

The radius of gyration is r = √(I/A) (units: length). It compactly represents how far from the axis the area is effectively located and is used directly in the column slenderness ratio KL/r for buckling. A larger r means a stiffer, more buckling-resistant section for the same area.

A practical caution: the parallel-axis theorem only transfers I between an axis through the centroid and a parallel axis — you cannot jump directly between two non-centroidal axes in one step. Always transfer down to the centroidal axis first, then back out to the target axis. For composite sections, this means locating the overall centroid before summing the A·d² transfer terms, because each d is measured from that overall centroid.