Combined Bending and Axial Stress Calculator
Combined Bending and Axial Stress Calculator
σ = P/A ± M·c/I with both extreme fibres reported, so you can see whether the section is entirely in tension, entirely in compression or both — and the kern derived rather than quoted, which is where the middle-third rule comes from.
Combined bending and axial stress
A 60 × 25 mm solid rectangular member carrying 60 kN of tension applied 12 mm off the centroid
Superposition, both signs, and the kern
- P/A
- the uniform part. Same stress at every point of the section, and it carries the sign of the load
- M·c/I
- the bending part. Zero at the centroid, largest at the extreme fibres, and of OPPOSITE sign at the two of them. That opposite sign is the whole page
- e
- eccentricity: the distance from the load’s line of action to the centroid. An offset load is not a special kind of load, it is an axial load plus P·e of moment
- Z/A
- the kern half-width, and the answer to “how far off centre can the load be before one face goes into tension”. It is a property of the shape alone — it does not depend on the load
- h/6
- the kern of a rectangle, which is the middle-third rule. Derived, not asserted: Z/A = (bh²/6)/(bh) = h/6
- d/8
- the kern of a solid circle: Z/A = (πd³/32)/(πd²/4) = d/8. The middle QUARTER, not the middle third
Worked example
A 60 × 25 mm solid rectangular member carrying 60 kN of tension applied 12 mm off the centroid
Section properties first, because everything else divides by them: A = 60 × 25 = 1,500 mm², I = 25 × 60³/12 = 450,000 mm⁴, c = 30 mm either way, so Z = I/c = 15,000 mm³
The uniform part: P/A = 60,000/1,500 = 40 N/mm². Every point of the section sees this, tensile
The offset becomes a moment. M = P·e = 60,000 × 12 = 720,000 N·mm = 720 N·m. This is the step worth being explicit about: there is no separate theory for an eccentric load, it is an axial load plus a moment, and the moment is the load times the offset
The bending part: M·c/I = 720,000 × 30 / 450,000 = 48 N/mm². Equivalently M/Z = 720,000/15,000, which is the same number and is why Z exists
NOW THE PLUS AND MINUS, which is the point. The fibre on the same side as the eccentricity: 40 + 48 = 88 N/mm² tensile. The fibre on the other side: 40 − 48 = −8 N/mm², which is COMPRESSIVE. A member loaded purely in tension has one face in compression, because the load is off centre
Is that inevitable? No, and the kern says exactly when it happens. Setting P/A = M·c/I and solving for e gives e = I/(A·c) = Z/A = 15,000/1,500 = 10 mm, which for a rectangle is h/6 — the middle-third rule, derived rather than quoted. At 12 mm the load is outside the kern by 20 per cent, so the far face reverses. At 10 mm exactly the far face would sit at precisely zero
What it costs. At 150 N/mm² allowable this section could carry 225,000 N with no eccentricity at all. At 12 mm off centre it can carry 102,273 N — 55 per cent less, for a 12 mm offset on a 60 mm deep part. Eccentricity is expensive and it is usually free to remove
One check this page does not do for you. If that 60 kN were COMPRESSIVE and the member were slender, the deflection caused by the moment would increase the eccentricity, which would increase the moment. That is a stability problem rather than a stress one; the radius of gyration for it is on the section properties page and the buckling check itself belongs to a structural code
The kern, shape by shape — and only the rectangle gives a middle third
| Section | Closed form | Kern half-width (mm) | … ÷ depth | Full kern width (mm) | … as a percentage of the depth |
|---|---|---|---|---|---|
| Solid rectangle, depth h | h/6 | 10.000 | 0.1667 | 20.000 | 33.3 % |
| Solid round, diameter d | d/8 | 7.500 | 0.1250 | 15.000 | 25.0 % |
| Tube, D = 60, wall 5 | (D² + d²)/8D | 12.708 | 0.2118 | 25.417 | 42.4 % |
| Box, 60 × 25, wall 5 | Z/A | 13.056 | 0.2176 | 26.111 | 43.5 % |
| I-section, 60 × 25, 5/6 | Z/A | 15.278 | 0.2546 | 30.556 | 50.9 % |
Walking the eccentricity out, on this page’s own 60 × 25 section at 60 kN
| e as a multiple of the kern | e | M = P·e | Near fibre (N/mm²) | Far fibre (N/mm²) | Stress signs on the section | Peak ÷ P/A |
|---|---|---|---|---|---|---|
| 0.0× | 0.00 mm | 0.0 N·m | 40.0 | 40.0 | one sign only | 1.00× |
| 0.5× | 5.00 mm | 300.0 N·m | 60.0 | 20.0 | one sign only | 1.50× |
| 1.0× | 10.00 mm | 600.0 N·m | 80.0 | 0.0 | one sign only | 2.00× |
| 1.5× | 15.00 mm | 900.0 N·m | 100.0 | -20.0 | BOTH signs | 2.50× |
| 2.0× | 20.00 mm | 1,200.0 N·m | 120.0 | -40.0 | BOTH signs | 3.00× |
| 3.0× | 30.00 mm | 1,800.0 N·m | 160.0 | -80.0 | BOTH signs | 4.00× |
Where combined loading actually shows up
| The part | The axial load | The eccentricity | What usually governs |
|---|---|---|---|
| C-frame press or punch | the press force, compressive in the frame | the throat depth — often larger than the frame’s own depth | the tension face of the frame’s back, which is where a C-frame cracks |
| Bolted bracket under an offset load | the bolt tension | the distance from the bolt line to the load | whether the joint face goes into tension at one edge, which the kern answers directly |
| Crane hook shank | the lifted load, tensile | the offset from the shank axis to the load point | the inner fibre of the curved part, which also needs a curved-beam correction this page does not apply |
| Eccentrically loaded bolt in a joint | the bolt preload plus its share of the external load | the offset of the load path from the bolt axis | the bolt’s peak stress, and its fatigue life once the load cycles — the bolt fatigue page owns that case |
| Machine frame column carrying an offset head | the head weight and cutting force | the overhang | frame deflection at the tool, long before any stress limit |
| Screw jack or leadscrew under side load | the axial thrust | the side load times the unsupported length | buckling, which is out of scope here, and then the combined stress |
The plus-or-minus, the kern, and why a 12 mm offset costs half the section
The whole page is one plus-or-minus sign. σ = P/A ± M·c/I is just superposition: a uniform stress from the axial load, and a linearly varying one from the moment that is zero at the centroid and equal and opposite at the two extreme fibres. Add them and the two faces of the section see different stresses. Which of the two governs depends on the sign of P and on which face you are asking about, and there are three qualitatively different outcomes: the section can be entirely in tension, entirely in compression, or have one face of each. This page always reports both extremes and says which case you are in, because reporting only the larger one hides the answer to the question that usually matters.
An eccentric load is not a different kind of load. Shift an axial load a distance e off the centroid and you have exactly an axial load through the centroid plus a moment P·e. That is not an approximation, it is a statics identity — a force at one point is equivalent to the same force at another point plus the couple between them. So there is no separate eccentric-loading theory to learn, and every offset in a machine, from a bracket’s overhang to a press frame’s throat to a bolt that is not on the load line, reduces to the same two terms.
The kern is the most useful thing here and it is usually quoted without its derivation. Ask for the eccentricity at which the far fibre reaches exactly zero stress. That means P/A = M·c/I with M = P·e, so e = I/(A·c), and I/c is Z, so e = Z/A. The load cancels, which is the important part: the kern is a property of the SHAPE alone and has nothing to do with how hard you pull. For a rectangle Z/A = (bh²/6)/(bh) = h/6, so the kern is the middle third of the depth. For a solid circle it is (πd³/32)/(πd²/4) = d/8, so the kern is the middle quarter of the diameter and the middle-third rule would be one third too generous. And for a rectangle offset in both directions at once the kern is a rhombus with diagonals h/3 and b/3, not a middle-third rectangle — the two eccentricities add as a normalised sum. That last point catches people and it errs on the unsafe side.
Where this arrives from. A C-frame press or punch, where the throat depth is the eccentricity and the frame’s back face is the tension side that cracks. A bolted bracket under an offset load, where the question is whether the joint face lifts at one edge. A crane hook, where the shank carries the load off axis. A bolt in an eccentric joint. A machine column with an overhanging head. In every case the geometry that makes the part convenient to use is the geometry that loads it eccentrically, and the eccentricity costs more capacity than its size suggests: at the kern limit you have already given away half the section.
Two checks deliberately left out, and where they live. BUCKLING: a slender member in compression plus bending is a stability problem, and a second-order one — the deflection increases the eccentricity which increases the deflection. Column design of a structural member belongs to a structural code and is out of scope for this vertical; the radius of gyration such a check needs is on the section properties page. CURVED BEAMS: a crane hook or a C-frame with a sharply curved section has its neutral axis displaced from its centroid, and the inner fibre stress is higher than M·c/I gives — by tens of per cent when the radius of curvature is comparable with the depth. Both are named here and neither is computed here.
Frequently asked questions
What exactly is the kern of a section?
The region of the cross-section within which a compressive axial load produces no tension anywhere on the section — or equivalently, within which a tensile load produces no compression. Its boundary is where the far fibre’s stress is exactly zero, and the half-width along any axis is e = Z/A. Because the load cancels out of that equation, the kern depends only on the shape. For a rectangle it is the middle third of each dimension; for a solid circle the middle quarter of the diameter.
Is the middle-third rule always right?
It is exactly right for a rectangle with the load offset in one direction only, and it is wrong in two common situations. For any other shape the fraction is different — a solid circle gives the middle QUARTER, a tube more than that. And for a rectangle offset both ways at once the kern is a rhombus, so the two eccentricities trade against each other: at h/12 and b/12 together the corner is already at zero stress even though each is only half of its own limit. Use the normalised sum |e_y|/k_y + |e_x|/k_x and the boundary is 1.0.
Can a member in pure tension really have a compressive face?
Yes, and that is the whole reason this page exists. If the tensile load is applied further off the centroid than Z/A, the bending stress at the far fibre exceeds the uniform P/A and the net stress there changes sign. On this page’s default — 60 kN of tension 12 mm off the centroid of a 60 mm deep section — one face is at +88 N/mm² and the other at −8. Nothing is wrong with the arithmetic; the load is simply outside the kern.
Does it matter whether the axial load is tension or compression?
For the stress arithmetic, no — only the signs move. For what the answer means, very much. A tensile axial load stabilises the member: the bending deflection reduces the eccentricity, so superposition is conservative. A compressive one destabilises it: the deflection increases the eccentricity, which increases the moment, which increases the deflection, and a slender member can lose stability at a load well below the one this page’s stress check would allow. Change the sign and the numbers stay valid as a first-order estimate but you also need a buckling check, which is not on this page.
How do I get the section properties for a shape that is not in the dropdown?
Use the section properties page, which covers T, channel and angle as well, and take its A, I and c straight into σ = P/A ± M·c/I. For an asymmetric section be careful which c you use: there are two, the two section moduli differ, and the bending stress at the two faces is not equal and opposite. This page’s four-corner calculation assumes the section is symmetric about both axes, which the five shapes in its dropdown are.
Why is the eccentricity so much more expensive than it looks?
Because it enters linearly while the section’s resistance to it does not grow at all. The bending stress is P·e·c/I, so doubling the offset doubles the bending term, while the axial term stays put. At the kern limit the peak stress is exactly twice P/A, meaning half the section’s capacity has gone; at three times the kern it is four times P/A. On a 60 mm deep member a 12 mm offset costs 55 per cent of the capacity. Removing an offset is nearly always cheaper than thickening a part to survive it.
Does this page handle a crane hook properly?
It handles the shank, and not the hook. The straight part of a hook is exactly this calculation: an axial load with an eccentricity. The curved part is not, because in a sharply curved member the neutral axis moves away from the centroid towards the inner fibre and the inner-fibre stress is higher than M·c/I predicts — by tens of per cent once the radius of curvature is comparable with the section depth. That needs Winkler’s curved-beam theory, which this page does not implement and says so rather than quietly being wrong on the unsafe side.
Related calculators
References
- Eigenplus, Eccentrically loaded columns — kern of the section, for the kern limits and their derivation: e ≤ b/6 and d/6 for a rectangle, whose kern is a RHOMBUS with diagonals b/3 and d/3 rather than a middle-third rectangle, and e = d/8 for a solid circle. Both were re-derived here from e = Z/A and both were then found independently by bisecting the computed stress distribution until the far fibre reached exactly zero. The rhombus is the part usually left out: two eccentricities at once do not each get their own sixth.
- R. C. Hibbeler, Mechanics of Materials, on the superposition of axial and bending stress, on Mohr’s circle and the stress-transformation equations, and on the absolute maximum shear stress in plane stress — the point that the third principal stress is zero rather than absent, so the governing shear may be σ₁/2 and not (σ₁ − σ₂)/2.
- ASTM A36/A36M, Standard Specification for Carbon Structural Steel. Cited by number, not reproduced. The two figures this page uses from it — a minimum yield of 250 MPa (36 ksi) for plate, bar and shapes under 200 mm and a tensile range of 400–550 MPa — are the specification’s headline values and are quoted here from the public summary of the standard rather than from the document.
- EN 1993-1-1 (Eurocode 3) and AISC 360 are named here only to say what this page is NOT. Structural member design — beam reactions, shear and moment diagrams, column buckling of building members, weld groups, plate design — belongs to those codes and to a structural calculator, not to a machine-part stress page. A cantilever bracket on a machine frame is in scope here; a floor beam is not.