Structural Engineering & Statics
where the load goes
Concept

Residual stress — where it appears

7 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither

Strong enough and still falls over

A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.

stability · buckling 00.0050.010.0150.020.02500.20.40.60.81lateral deflection at mid-heightload ÷ P꜀ᵣP ÷ P꜀ᵣ = 1.00, approached and never reachedinitial bow: δ₀ = 0.002

The column that was never straight

Euler's load is the load at which a perfectly straight column becomes indifferent to being bent. No column is perfectly straight, so no column ever reaches it — and the load it never reaches can still be measured.

stability · buckling neutral axisI-sectionstrainalways a straight linestressthe material's own curve, sidewaysC = 322.5 kN · T = 322.5 kN · lever arm 180 mm · M = 58.1 kNm0% of the area has yielded — 0 mm from the top, 0 mm from the bottom · Mp = 72.6 kNm · shape factor 1.09

The stress that was there before the load

A rolled steel section leaves the mill carrying eighty N/mm² of stress with nothing applied to it, in a pattern that sums to no force and no moment. It is invisible to every calculation and it is the knee in every column curve.

materials · residual stress flange outstandk = 0.4318.6 at 23515.2 at 35513.3 at 460quoted: 14ε1.33× the quoted limit, at every gradeweb, in bendingk = 456.8 at 23546.2 at 35540.6 at 460quoted: 42ε1.35× the quoted limit, at every grade0102030405060width ÷ thickness

The section that cannot reach its own strength

A section classification looks like a table of arbitrary numbers. Set a plate's buckling stress equal to the yield stress and the numbers fall out of a formula written three phases ago — larger than the quoted ones by a constant factor, at every grade.

materials · section classification -0.6%-0.4%-0.2%0.2%0.4%0.6%-300-200-100100200300strainstress, N/mm²0.469% of the strain never came back

What is left when the load comes off

Unload a section that has yielded and it does not return to nothing. It returns to a self-equilibrating stress field it did not have before, a permanent set, and an elastic range wider than the one it started with.

materials · unloading 024681012141600.511.52cyclescurvature gained ÷ curvature at first yieldM/Mp = 0.30, ΔT = 20°C — elasticM/Mp = 0.60, ΔT = 120°C — shakedownM/Mp = 0.85, ΔT = 200°C — ratcheting

The structure that settles down, and the one that walks

A load that is safe applied once may not be safe applied ten thousand times. Nothing about that is fatigue — the structure never breaks, it simply arrives somewhere slightly further round every cycle, until it has arrived somewhere unusable.

materials · shakedown 10⁴10⁵10⁶10⁷10⁸2050100200500cycles to failurestress range, N/mm²category 160category 90category 366.8e+74.3e+62.7e+5working range 70 N/mm² — the lives are marked

The load that never came near failing anything

A detail survives sixty-eight million cycles at a stress range that another detail in the same steel survives two hundred and seventy thousand of. The two lie a factor of two hundred and fifty apart, and the material is not on the plot anywhere.

materials · fatigue

Named alongside it

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

ImperfectionSelf equilibratingCritical loadEuler bucklingKinematic hardeningMoment curvatureAmplification factorAutofrettageBauschinger effectBifurcationCollapseColumn curve

All concepts