Eccentric Loading, Core of Section & Theories of Failure
Combined direct + bending stress under eccentric axial load (one axis and both axes), the middle-third no-tension rule, core/kernel of a section, principal stresses via Mohr's circle, and the five classical yield/failure criteria.
8.1 Eccentric load acting on one axis
When an axial load P is applied to a column not through the centroid but offset by a distance e along one principal axis, it is statically equivalent to a concentric load P plus a moment M = P·e. Every fibre of the cross-section therefore carries direct stress and bending stress together.
For a rectangular section b (width) × d (depth), with the eccentricity e measured along the depth d, A = bd, I = bd³/12, ymax = d/2:
The trapezoidal stress block above is only valid while σmin stays compressive. As e increases, σmin shrinks; once e exceeds a critical value, σmin would need to be negative — i.e. tensile — to satisfy equilibrium. Sections 8.2–8.4 pin down exactly where that limit lies.
8.2 Condition for no tension in the section
Masonry, brickwork, plain concrete, and soil under a footing cannot reliably carry tension — the joint simply opens up. So the eccentricity of the resultant load must be limited to whatever keeps σmin ≥ 0 everywhere in the section.
So the load must act within the middle third of the depth — a band of width d/3 straddling the centroid. This is the classical middle-third rule used for masonry dams, footings, and unreinforced piers.
Beyond e = d/6, a no-tension material doesn't develop the negative stress shown by the dashed red triangle — instead the material simply loses contact over part of the base, and the effective bearing area shrinks until equilibrium is restored on a smaller triangular stress block (relevant for footings and gravity dams, covered separately under eccentric footing design).
8.3 Eccentricity about both axes (biaxial)
If the load P is applied at a point offset by ex from the y–y axis and ey from the x–x axis simultaneously, it produces two moments at once: Mx = P·ey (bending about x–x) and My = P·ex (bending about y–y). Stress at any point (x, y) superposes all three effects:
For a rectangular section b × d, the four corners are always where the extreme stresses occur (x = ±b/2, y = ±d/2):
Four combinations of the ± signs give the four corner stresses. The single largest value (both signs positive, load-side corner) governs compression design; the smallest value governs the no-tension check.
8.4 Core (kernel) of a section
Generalising §8.2, the no-tension limit for eccentricity along any single axis is:
The core (or kernel) of a section is the locus of all such points, in both axes at once — if the load acts anywhere inside this core, no tension develops anywhere on the section.
| Section shape | Core shape | Core dimension |
|---|---|---|
| Rectangle b × d | Rhombus (diamond) | Diagonals b/3 and d/3 — i.e. vertices at b/6, d/6 from centroid |
| Circle, diameter D | Circle | Radius D/8 (derived from e = I/(A·ymax) = (πD⁴/64)/(πD²/4 · D/2) = D/8) |
| Hollow circular (tube) | Circle (larger than solid) | Radius = (D² + di²)/(8D) |
Eccentric-load instrument
9.1 Why we need a theory of failure
A tension test gives one clean number: the yield stress σy or ultimate stress σu at which a bar fails under uniaxial stress. But a real machine shaft carries bending and torsion together, a pressure-vessel wall carries hoop and longitudinal stress together, and a column carries axial load with eccentricity. In every one of these the material is under compound (combined) stress — two or three principal stresses acting at once, not one.
A theory of failure is simply a rule that converts a complicated multi-axial stress state into a single "equivalent" number that can be compared directly against the simple, single-number yield or ultimate strength from a tension test.
| Theory | Also known as | Best suited to |
|---|---|---|
| Maximum Principal Stress | Rankine's theory | Brittle materials (cast iron, ceramics, rock, concrete) |
| Maximum Shear Stress | Guest's / Tresca's theory | Ductile materials — conservative, simple to apply |
| Maximum Principal Strain | St. Venant's theory | Rarely governs; occasionally used for brittle materials |
| Total (Maximum) Strain Energy | Haigh's theory | General elastic materials; historically used, less common now |
| Maximum Shear Strain Energy | Von Mises' / Distortion Energy theory | Ductile materials — most accurate, industry standard |
9.2 Stresses on an oblique plane
Consider a small rectangular element carrying normal stresses σx, σy and complementary shear stress τxy on its faces. Cutting the element on a plane inclined at angle θ to the x-axis and resolving forces gives the normal stress σθ and shear stress τθ on that plane.
The principal planes are the two orthogonal planes where τθ = 0. On these planes σθ is either maximum (σ₁) or minimum (σ₂) — these are the principal stresses, and it is these two numbers that every failure theory below acts on.
9.3 Mohr's Circle — the graphical shortcut
Mohr's circle plots every possible (σθ, τθ) pair for a stress state as a circle in the σ–τ plane, so principal stresses can be read off directly without repeated trigonometry.
- Plot point A = (σx, τxy) and point B = (σy, −τxy).
- The centre C lies on the σ-axis at (σx+σy)/2 — join A to B; this line is a diameter.
- Radius R = √[((σx−σy)/2)² + τxy²] = the maximum shear stress τmax.
- σ1 = C + R (right-most point), σ2 = C − R (left-most point). Both intersections with the σ-axis have τ = 0, confirming these are principal planes.
Top 3 theories of failure engineers actually use
Von Mises (Distortion Energy)
Best experimental fit to real ductile-metal yield data. Default criterion in FEA software, ASME & most mechanical design codes. σe = √(σ1² − σ1σ2 + σ2²).
Tresca (Max Shear Stress)
Hand-calculation friendly, always errs on the safe side vs Von Mises. Basis of Guest's Law and many older machine-design codes. σ1 − σ2 ≥ σy.
Rankine (Max Principal Stress)
The only theory of the five that correctly predicts brittle fracture on the plane of maximum tension. σ1 ≥ σy. Simple but unsafe for ductile metals under biaxial stress of opposite sign.
9.4 Maximum Principal Stress Theory
Failure occurs when the largest principal stress reaches the uniaxial yield (or ultimate) stress from a simple tension test — the other principal stress is ignored entirely.
In the σ1–σ2 plane this criterion is a square box of side 2σy centred at the origin — very simple, but it ignores σ2 completely, which makes it unsafe in states of biaxial stress with opposite signs (e.g. torsion, where σ1 = −σ2).
9.5 Maximum Shear Stress Theory
Failure (yielding) occurs when the maximum shear stress in the complex state reaches the maximum shear stress at yield in a simple tension test, which is σy/2.
Plots as a hexagon inscribed inside the Rankine square — always more conservative (smaller safe region) than Rankine and than the Von Mises ellipse in §9.8. Because it is simple and always errs safe, it is the most widely used theory in machine design codes (e.g. ASME shaft design).
9.6 Maximum Principal Strain Theory
Failure occurs when the largest principal strain reaches the strain at yield in simple tension. Using Hooke's law for a biaxial state:
The Poisson's ratio term rotates and skews the failure boundary into a parallelogram-like shape relative to Rankine's square. It rarely governs design today, but historically was applied to brittle materials as an alternative to Rankine.
9.7 Total (Maximum) Strain Energy Theory
Failure occurs when the total elastic strain energy stored per unit volume in the complex stress state equals the strain energy per unit volume at yield in simple tension.
This traces an ellipse in the σ1–σ2 plane — but because it includes the energy of pure volume change (hydrostatic strain energy) as well as shape change, it predicts yielding even under equal all-round hydrostatic stress, which real ductile metals do not exhibit. This flaw is exactly what the next theory fixes.
9.8 Maximum Shear Strain Energy Theory
Total strain energy can be split into (a) energy that changes volume and (b) energy that changes shape (distortion). Experiments show ductile metals yield due to shape-distortion energy only — hydrostatic stress alone does not cause yielding. Von Mises' theory keeps only that distortion component.
This also traces an ellipse (slightly larger than Haigh's, tilted at 45°), circumscribing the Tresca hexagon. It gives the best experimental fit to real ductile-metal yield data and is the default criterion in most modern FEA software ("Von Mises stress" contour plots).
| Theory | Shape in σ1–σ2 plane | Relative safety margin |
|---|---|---|
| Rankine | Square | Least conservative — largest safe zone |
| Von Mises | Ellipse (45° tilt) | Middle — best fit to test data |
| Tresca | Hexagon inside the ellipse | Most conservative for ductile metals |
All three envelopes coincide exactly at the uniaxial points (σy, 0) and (0, σy) — they agree with the tension test by definition. They diverge most in the third quadrant (σ1, σ2 of opposite sign, i.e. pure shear), which is exactly why the choice of theory matters most for torsion-dominated shafts.
Stress-state drafting instrument
References
- Timoshenko, S. & Young, D. H. — Elements of Strength of Materials, 5th ed., East-West Press.
- Beer, F. P., Johnston, E. R., DeWolf, J. T. & Mazurek, D. F. — Mechanics of Materials, McGraw-Hill.
- Hibbeler, R. C. — Mechanics of Materials, Pearson.
- Khurmi, R. S. — Strength of Materials, S. Chand Publishing.
- Punmia, B. C., Jain, A. K. & Jain, A. K. — Mechanics of Materials, Laxmi Publications.
- Budynas, R. G. & Nisbett, J. K. — Shigley's Mechanical Engineering Design, McGraw-Hill (theories of failure, distortion-energy criterion).
- Pytel, A. & Singer, F. L. — Strength of Materials, Harper & Row.
- Bureau of Indian Standards — IS 800: 2007, General Construction in Steel — Code of Practice (design stress limits).