Combined axial-bending stress calculator
This calculator superposes axial and bending stress to give the peak normal stress at the extreme fibre of a member: sigma = P/A + M c / I. It is aimed at engineering students working through mechanics of materials, and at engineers making a quick first-pass check on a beam, column or shaft that carries both a direct load and a moment. The result tells you the highest normal stress the section sees, which is what you compare against an allowable stress to decide whether a section is plausible.
When this is valid. Superposition holds only for linear elastic behaviour: stress below the proportional limit, small deflections, a prismatic member and a material that behaves the same in tension and compression. It assumes bending about a principal axis of the cross-section and that plane sections remain plane (Euler-Bernoulli). It is a stress calculation, not a design check.
When not to rely on it. It does not check buckling, and for a slender compression member buckling, not stress, usually governs: a section that passes this check can still fail. It ignores stress concentrations at holes, fillets and welds, shear and torsional stress, biaxial bending, plastic redistribution, fatigue and the load factors your design code requires. The most common input mistakes are mixing units (using mm and m in the same calculation, or entering the second moment of area in mm4 while area is in m2), and entering the distance to the centroid rather than to the extreme fibre. Sign matters too: a negative axial load is compressive, and the opposite face of the section sees P/A minus the bending term.
With Axial load = 10000 N, Area = 0.01 m2, Moment = 500 Nm, Distance to fibre = 0.05 m, Inertia = 0.00001 m4, the result is 3500000 Pa.
Applies to: prismatic members in linear elastic bending about a principal axis, below the proportional limit. Method: superposition of axial and bending stress, a standard result of engineering mechanics. Checked 2026-06-22.
The formula
sigma = P/A + M c / I
Worked example
A member carrying a 10 kN axial load and a 500 Nm bending moment: area = 0.01 m2, distance from the neutral axis to the extreme fibre = 0.05 m, second moment of area = 0.00001 m4.
- Axial stress: P/A = 10,000 / 0.01 = 1,000,000 Pa (1.0 MPa)
- Bending stress at the extreme fibre: M c / I = (500 x 0.05) / 0.00001 = 25 / 0.00001 = 2,500,000 Pa (2.5 MPa)
- Superpose: 1,000,000 + 2,500,000 = Combined stress = 3500000 Pa (3.5 MPa)
Note that bending contributes two and a half times more than the axial load here, which is typical: a modest moment often dominates the stress at the extreme fibre. This is the peak stress on the compression or tension face; the opposite face sees P/A minus the bending term, which is 1.0 - 2.5 = -1.5 MPa, so that face is in net tension even though the applied load is compressive.
This worked example is one of the automated golden-value tests this calculator must pass before it can publish.
What this assumes
- Inputs are real numbers in the units shown.
- The result is the exact value of sigma = P/A + M c / I; general information, not professional advice.
Frequently asked questions
What formula does this use?
sigma = P/A + M c / I: the axial stress plus the bending stress at distance c from the neutral axis. This is linear elastic superposition, a standard result of engineering mechanics, valid below the proportional limit.
Does this tell me if my section is safe?
No. It gives peak elastic normal stress at one section. A design check also needs buckling (which often governs slender compression members), shear and torsion, stress concentrations at holes, fillets and welds, fatigue if the load cycles, and the load and resistance factors of the code you are designing to.
Why is the stress on the other face different?
Bending puts one face in tension and the other in compression, so the opposite fibre sees P/A minus Mc/I. With a small axial load and a large moment that face can go into net tension even when the applied axial load is compressive, which matters for materials that are weak in tension, such as concrete and cast iron.
What are the most common input mistakes?
Mixing units (second moment of area in mm4 with area in m2 is the classic one), and using the distance to the centroid instead of the distance to the extreme fibre. Both produce a plausible looking number that is wrong by orders of magnitude.
Official sources and verification
- Method: linear elastic superposition of axial and bending stress (sigma = P/A + Mc/I), a standard result of engineering mechanics rather than a jurisdiction-specific rule. How we document and test our methods. Checked 2026-06-22.
Reviewed by the CalculatorHub team, edited by James Graham, 2026-06-22. See our methodology. General information, not professional advice.