Animate in real time the "axial stress N/A + bending stress M·c/I" produced by an eccentric axial load on a column section. Live view of the normal stress distribution, neutral axis, kern (no-tension core) and safety factor.
What exactly is "combined" stress? Is it just adding two numbers together?
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Basically, yes! It's the superposition of stresses from different load types acting on the same point. A structural member is rarely loaded by pure tension or pure bending alone. A column supporting a roof is under axial compression, but if the load is off-center, that eccentricity e produces a bending moment M = N·e automatically. Drag the "Eccentricity e_z" slider above and the load point (the white "P" marker) moves; the axial stress N/A and the bending stress ±M·c/I superpose, and the section color shifts from compression (blue) toward tension (red) in real time.
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What is the small green diamond (the kern)? What changes when the load point is inside it versus outside?
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Great catch! That green diamond is the "kern" (core). As long as the compressive load point stays inside the kern, the whole section stays in compression — no tension anywhere. For a rectangle the kern limit along the height is exactly h/6. The moment the load point leaves the kern, tension (red) appears on one side and a "⚠ tension" warning is shown. For materials weak in tension, such as concrete, keeping the load inside the kern is a critical design rule. Use the "Within kern" and "Outside kern → tension" presets to see the difference instantly.
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The "neutral axis" moves on the diagram. Is there a rule to how it shifts?
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Good question! It is set by the balance between the axial stress N/A and the bending stress. When the eccentricity e is small (inside the kern), the neutral axis (yellow dashed line) lies outside the section and the whole section has one sign. Once e exceeds the kern, the neutral axis enters the section, splitting it into a compression side and a tension side. As you increase eccentricity, watch the yellow line travel from outside the section toward the inside.
Physical Model & Key Equations
The core principle is linear superposition. The total normal stress at any point (y,z) in the cross-section is the sum of the axial stress and the bending stresses about the y and z axes. With an eccentric load, the moments come from the eccentricity: $M_y = N\,e_z$ and $M_z = N\,e_y$.
Where:
• $N$ = Axial Force (Tension +)
• $A$ = Cross-sectional Area
• $M_y, M_z$ = Bending Moments about the y and z axes ($=N e_z,\ N e_y$)
• $I_y, I_z$ = Second Moments of Area (Moment of Inertia)
• $y, z$ = Coordinates of the point where stress is calculated
The geometry of the cross-section defines A, I_y, and I_z. For the two shapes in this simulator:
The neutral axis is found by solving $\sigma(y,z) = 0$. Its location changes directly with the ratio of axial force to bending moment, i.e. with the eccentricity.
Frequently Asked Questions
Since the second moment of area and cross-sectional area change, the stress distribution changes even under the same load. A circular cross-section tends to have a smaller second moment of area compared to a rectangular one, making bending stress more likely to increase. The kern of a circle is a disc of radius D/8. The safety factor also changes accordingly, so choose the cross-section based on design conditions.
The neutral axis is the line where stress is zero, and due to the combination of axial force and bending, it does not necessarily pass through the centroid. In the simulator, it is shown automatically as a yellow dashed line. As the axial force increases relative to the moment (smaller eccentricity), the neutral axis tends to move outward from the cross-section.
A safety factor below 1 indicates a risk of yielding, so design changes such as increasing the cross-sectional area, changing to a higher-strength material, or reducing the eccentricity / bending moment are necessary. It is recommended to use the simulator to adjust parameters and find a combination where the safety factor is 1.5 or higher.
When the compressive load point leaves the kern, tension develops on part of the section. For materials weak in tension — concrete, masonry, soil — that region cracks, separates or lifts off. This is why footing design requires keeping the resultant inside the kern (the "middle-third rule"); for a rectangle the eccentricity must stay within 1/6 of the dimension.
Real-World Applications
Eccentrically Loaded Columns: This is the most direct application. A column supporting a crane rail or an off-center floor beam has an axial load that doesn't pass through the centroid, creating a bending moment (M = N·e). Engineers use this exact calculation to check stress distribution, neutral axis and safety, often before running a full FEM analysis.
Footings & Retaining Walls (the kern): Under a footing or wall base, what matters is whether the resultant stays inside the kern (the middle-third for a rectangle). If it leaves, part of the base lifts off the soil (tension = zero bearing pressure) and support becomes one-sided. The kern visualization here maps directly onto the "middle-third rule".
Machine Shaft Analysis: A rotating shaft transmitting power might also carry gear-induced bending moments and axial thrust. Eccentric loads add bending M = N·e to the axial stress; this superposition is the first step in more complex combined loading (later adding shear and torsion).
Pre-FEM Verification: Before a full 3-D finite element simulation, engineers pick a representative section and hand-check the combined stress. It catches modeling or boundary-condition mistakes early and provides a quick "sanity check" of the overall stress level.
Common Misunderstandings and Points to Note
There are several points beginners often stumble on. First and foremost is the "sign combination of axial force and eccentricity". The tool defines "tension positive, compression negative". To simulate a realistic column being compressed and bent, set N to a negative value and apply an eccentricity e_z or e_y. Getting the signs wrong produces a completely reversed stress pattern that would not occur in reality.
Secondly, a misunderstanding of the "relationship between the kern and tension". Even a compressive axial force produces tension if the eccentricity exceeds the kern. Conversely, if the eccentricity stays inside the kern, the entire section remains in compression even if the offset looks large. Watch the green diamond (kern) and the load point P together to grasp exactly where tension appears.
Third is the interpretation of "Safety Factor SF=1.0". SF=1.0 is theoretically the limit where yielding begins somewhere in the material. In practice, considering uncertainties in loads and material variations, you should aim for a value with margin (e.g., SF=1.5 to 3 or more). Use this as a rule of thumb: if you select "Steel S235" and the SF falls below about 1.2, it's almost certainly a failure.
Select section type: enter rectangular dimensions (width b in mm, height h in mm) or circular diameter D in mm
Set axial force N (kN; compression is negative) and eccentricities e_z, e_y (mm); the moments M_y = N·e_z and M_z = N·e_y are computed automatically
Choose the material yield stress, then read the live σ_max, σ_min, axial stress N/A, bending stress, kern limit and safety factor SF. Use presets and pause/play to watch the load-application animation
Worked Example
S235 steel rectangular column (b=100 mm, h=200 mm, Sy=235 MPa), compression N=-100 kN at eccentricity e_z=33 mm (≈h/6). A=20,000 mm², I_y=66.67×10⁶ mm⁴. Axial stress N/A=-5.0 MPa, bending stress M_y·(h/2)/I_y = N·e_z·100/I_y = ±4.95 MPa. One fiber is ≈0 (kern edge), the other ≈-10 MPa (limit of full-section compression). Increase the eccentricity beyond the kern (e_z=70 mm) and tension appears on one side. Safety factor SF = Sy / |σ_max|.
Practical Notes
Even a compressive axial load produces tension once the eccentricity exceeds the kern (1/6 of the dimension for a rectangle); keep the load inside the kern for tension-weak materials such as concrete and soil
For circular sections use I=π·d⁴/64 and the kern radius D/8; stress concentration factors apply at stress risers (Kt=2.0–3.5 for fillets in ductile steel)
Neutral axis location depends on load eccentricity; eccentric axial loads create additional bending and shift the stress-free line off the geometric centroid
Use safety factor ≥2.5 for static ductile materials, ≥3.0 for brittle cast iron or fatigue-prone aluminium alloys