Structural Engineering & Statics
where the load goes
Concept

Interaction diagram — where it appears

A locus of the combinations of two or more actions a section can carry at once, drawn in the plane of those actions. It replaces a pair of independent checks with one boundary, and its shape says whether the two actions simply add or whether the section carries more of the combination.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The neutral axis obeys neither the load nor the moment. A 305 × 102 mm I-section carrying a moment 5° out of the plane of its web. The moment vector is the short arrow; the neutral axis is the long line, at 69.7° to the strong axis. They do not line up, and the reason is that the neutral axis follows the moment ratio scaled by the stiffness ratio: tan α = (M_z/M_y)(I_y/I_z), and I_y ÷ I_z is 30.8 here. So a 5° tilt of the load puts the neutral axis 70° over, the corner that ends up furthest from it carries 489 N/mm² against the 258 the straight-down case would give, and the section has lost 47 per cent of its capacity to a misalignment nobody would draw on a detail.

Two moments and a neutral axis that obeys neither

Tilt the load on a rolled beam by five degrees and the neutral axis swings by seventy. The section is doubly symmetric, its product of inertia is exactly zero, and none of that helps — because what decides the axis is the moment ratio multiplied by a stiffness ratio of thirty.

sections · Biaxial bending One criterion inside the other, touching at six points. The two yield criteria in principal stress space with the third principal stress zero, both normalised by the yield stress. Von Mises is the ellipse — σ₁² − σ₁σ₂ + σ₂² = f_y², which is a circle seen at an angle — and Tresca is the hexagon inscribed in it, touching at the six points where one principal stress is zero or the two are equal. Everywhere else Tresca is the smaller, by up to 15.5 per cent, and the widest gap is at pure shear, where σ₁ = −σ₂ and the two answers are 205 and 178 N/mm². The ratio there is exactly 2/√3, computed rather than quoted, and it is the whole reason a web is checked against f_y over root three.

The shear strength nobody measured

Every steel web in a design check is checked against the yield stress divided by the square root of three, and no test produced that number. It is a consequence of a decision about what makes a metal yield, and the alternative decision gives a different answer by fifteen per cent.

materials · Yield criterion The four corners of an opening, on the tee's own interaction diagram. The plastic interaction of the tee left above and below a 400 × 300 mm opening — every point on it computed by sweeping the plastic neutral axis through the section rather than from an interaction formula. Its squash load is 1519 kN and its plastic moment 32.2 kNm. At the working load the global moment puts 301 kN into each tee — compression above the hole, tension below — and the shear puts the same Vierendeel moment of 6.3 kNm into all four corners. The line is the path the demand takes as the load rises, and it reaches the surface at a load factor of 3.89 against 1.49 for first yield at one corner: the elastic check is finding one corner and the mechanism needs all four.

Four corners and a mechanism

The check on a hole in a web adds an axial stress to a local bending stress and compares the sum with the yield stress at one corner. What ends the opening is four hinges arriving together — and the gap between the two is a factor that varies along the span from two and a half to exactly one.

sections · Web opening

Named alongside it

The objects these essays reach for when they reach for this one.

Shape factorBending stressBiaxial bendingCellular beamConfinementDeviatoric stressDuctilityEquivalent stressHydrostatic stressLoad factorMechanismMohr circle

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