Biaxial and Unsymmetrical Flexure

Biaxial and Unsymmetrical Flexure

Draw the axes: A biaxial stress calculation is a signed stress plane, not M_x/S_x + M_y/S_y with every term assumed positive. Label the centroid, positive axes, positive moments, and every candidate point before substituting numbers.

Biaxial Bending About Principal Axes

For centroidal principal axes x and y, the product of inertia is zero and the two bending contributions can be superimposed directly. Adopt positive x to the right, positive y upward, tensile normal stress positive, and a moment convention giving

σ(x,y) = -M_x y/I_x + M_y x/I_y.

Under this stated convention, positive M_x compresses the +y side and positive M_y tensions the +x side. Another handbook convention can be used, but moments, coordinates, and signs must all change consistently. Section-modulus shortcuts work at selected extreme fibers for one-axis bending; for simultaneous biaxial bending, evaluate both terms at the same physical point.

Signs of eccentric-load moments come from the cross product and the chosen axial-load direction. Do not attach e_x to M_x merely because the subscripts match. A transverse load that misses the shear center can also produce torsion; keep that action separate from the axial-eccentricity mapping above.

Worked Four-Corner Stress Check

A solid rectangle is 8 in wide along x and 12 in deep along y. It carries M_x = +80 kip-ft and M_y = +30 kip-ft under the convention above. Its centroidal properties are

I_x = bh³/12 = 8(12³)/12 = 1,152 in⁴

and

I_y = hb³/12 = 12(8³)/12 = 512 in⁴.

Convert moments: M_x = 960 kip-in and M_y = 360 kip-in. At a corner, x = ±4 in and y = ±6 in, so

σ = -(960/1,152)y + (360/512)x

σ = -0.8333y + 0.7031x in ksi.

Evaluate signs instead of adding magnitudes blindly:

The governing compression is -7.81 ksi at (-4,+6); governing tension is +7.81 ksi at (+4,-6). At the other two corners the contributions partly cancel. Adding both extreme magnitudes everywhere would falsely report 7.81 ksi at all corners.

The neutral axis satisfies σ = 0:

y = (M_y I_x)/(M_x I_y) x = 0.844x.

Its slope is not M_y/M_x = 0.375 because bending stress also depends on unequal inertias. The neutral axis still passes through the centroid for pure elastic bending without axial force.

Eccentric Axial Loading

An axial force P applied away from the centroid produces axial stress plus bending. An offset in y produces moment magnitude P e_y about x; an offset in x produces P e_x about y, with signs determined from the force vector. Taking P algebraically positive in tension, the elastic stress becomes

σ = P/A - M_x y/I_x + M_y x/I_y

under the same sign convention. This superposition is a service-level elastic stress calculation, not a material-code strength interaction equation. Keep that distinction when later checking AISC 15th design resistance.

Unsymmetrical Sections

An angle, channel about non-symmetry axes, or arbitrary built-up section may have product of inertia I_xy ≠ 0. Then bending about the chosen geometric x axis can create stress variation involving both coordinates; the independent principal-axis equation cannot be used on those axes unchanged.

A robust exam method is:

1. Find the centroid of the actual section.
2. Calculate I_x, I_y, and signed I_xy using one coordinate convention.
3. Use the current PE Civil handbook's transformation relationship to find principal axes x_p,y_p and properties I_xp,I_yp, where I_xpyp = 0.
4. Resolve the applied moment vector into signed M_xp,M_yp components.
5. Transform each candidate point into (x_p,y_p) coordinates consistently.
6. Evaluate σ = -M_xp y_p/I_xp + M_yp x_p/I_yp at all extreme vertices and compare tension and compression.

The coordinate labels in step 5 must match the adopted principal-axis equation; sketching them prevents a silent swap. Alternatively, use the handbook's full coupled unsymmetrical-bending formula with the same I_xy sign convention. Do not rotate properties but leave moments or point coordinates unrotated.

Axis and Corner Audit

For a polygonal elastic section, stress is a linear plane, so an extreme value occurs at a boundary extreme point; check all relevant vertices rather than only “top” and “bottom.” Mark each term + or − before calculating its magnitude. Verify that stress changes linearly, the zero-stress line is plausible, and opposite corners of a centrally symmetric section have opposite stresses under pure bending.

For a 2026 exam, use the PE Civil handbook active for the test date and AISC Steel Construction Manual, 15th edition, where steel provisions are required. The April 2027 edition set is future-only.