Unsymmetrical Bending

Why it matters

A beam may bend about two cross-sectional axes simultaneously. The normal stress then depends on both bending moments and the section's second moments of area. Using a polar moment in a general flexure formula gives incorrect results.

Assumptions and notation

Use a homogeneous linear-elastic straight beam, small deformation, and plane sections remaining plane. Let z run along the beam and let x and y be centroidal cross-sectional coordinates. There is no axial force in this example. Positive normal stress σ_z is tension. On the positive-z cut face, define M_x = ∫yσ_z dA and M_y = −∫xσ_z dA.

Define I_x = ∫y² dA, I_y = ∫x² dA, I_xy = ∫xy dA, and D = I_x I_y − I_xy². All area moments have units m⁴.

Stress equation

The linear stress distribution σ_z = ax + by must reproduce both bending resultants. Solving their two equations gives:

a = (−M_y I_x − M_x I_xy)/D

b = (M_x I_y + M_y I_xy)/D.

For centroidal principal axes, I_xy = 0 and the expression reduces to σ_z = −M_y x/I_y + M_x y/I_x. Signs change if a different cut-face moment convention is chosen; keep the convention consistent.

The neutral axis satisfies ax + by = 0. Under pure bending of a homogeneous section, it passes through the centroid. It need not coincide with a principal axis. An additional axial force shifts the zero-stress line. Principal-axis orientation is a geometric property; neutral-axis orientation also depends on the applied moments.

Worked example

Take M_x = 500 N·m, M_y = 300 N·m, I_x = 2000 cm⁴, I_y = 1500 cm⁴, and I_xy = 500 cm⁴. Find stress at x = 2 cm, y = 3 cm using the stated signs.

Since 1 cm⁴ = 10⁻⁸ m⁴:

• I_x = 2 × 10⁻⁵ m⁴
• I_y = 1.5 × 10⁻⁵ m⁴
• I_xy = 5 × 10⁻⁶ m⁴
• D = 2.75 × 10⁻¹⁰ m⁸.

a = −3.09091 × 10⁷ Pa/m and b = 3.27273 × 10⁷ Pa/m.

Thus σ_z = a(0.02) + b(0.03) = 363636 Pa = 0.364 MPa tension. The neutral axis is y = −(a/b)x ≈ 0.94444x.

Common mistakes

• Dividing a sum of bending terms by the polar area moment I_x + I_y.
• Treating cm⁴ as though it converts like cm².
• Assuming the neutral axis misses the centroid under pure bending.
• Mixing moment signs from opposite cut faces.

Quick check

1. What makes axes principal? I_xy = 0.
2. Does the neutral axis always align with the moment vector? No; its direction depends on the moments and section properties.
3. Is an eccentric axial force equivalent to pure bending? No. Include its axial force as well as its moment.