Structural Engineering & Statics
where the load goes
Generator

Two mechanisms, and they add up to the torque at every section

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
Two mechanisms, and they add up to the torque at every section. Saint-Venant torque and warping torque along a 305 by 165 mm I-section of 6 m, twisted by 0.5 kN·m with the ends fixed-free. J is 1.31×10⁵ mm⁴ and I_w 1.63×10¹¹ mm⁶, so k = √(GJ/EI_w) gives kL = 3.34 and a decay length of 1.80 m — 30% of the member. At the built-in end the shearing mechanism is exactly zero and all 0.5 kN·m is carried by the flanges bending in opposite directions; a decay length along, that share has fallen to 37%, and at the far end it is 7.1%. The two curves sum to the flat line at 0.5 kN·m at every one of the 161 stations, to the last bit of the arithmetic, which is the equilibrium of a slice of the member and is the only reason the split may be believed.

Two mechanisms, and they add up to the torque at every section. Saint-Venant torque and warping torque along a 305 by 165 mm I-section of 6 m, twisted by 0.5 kN·m with the ends fixed-free. J is 1.31×10⁵ mm⁴ and I_w 1.63×10¹¹ mm⁶, so k = √(GJ/EI_w) gives kL = 3.34 and a decay length of 1.80 m — 30% of the member. At the built-in end the shearing mechanism is exactly zero and all 0.5 kN·m is carried by the flanges bending in opposite directions; a decay length along, that share has fallen to 37%, and at the far end it is 7.1%. The two curves sum to the flat line at 0.5 kN·m at every one of the 161 stations, to the last bit of the arithmetic, which is the equilibrium of a slice of the member and is the only reason the split may be believed.

4 essays call warping-torsion. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

The flanges go opposite ways, and the pair of them is the bimoment. A 305 by 165 mm I-section held against warping and twisted by 0.5 kN·m, with the section on the left and the two flanges seen in plan on the right. At the built-in end each flange bends in its own plane, one way at the top and the other at the bottom, through 29.3 mm at the free end — drawn 20 times its true size against the 6 m length. The pair of flange shears is 1.69 kN each, and 1.69 × 295 mm is 0.500 kN·m — the whole torque at that section, carried by two forces neither of which is a torque. The pair of flange moments is 3.04 kN·m each, and 3.04 × 295 mm is 0.897 kN·m², which is the bimoment. It puts 67.0 N/mm² into two diagonally opposite flange tips and takes the same out of the other two, so its net force and its net moment about every axis are zero — which is exactly why no member diagram has a place for it. Sections and stress

The section that cannot stay flat

Twist an I-section and its cross-section dishes out of its own plane. Stop that happening at one end and the member finds a second way to resist — the flanges bend in opposite directions — and the stress resultant that describes it has units nothing else in statics has.

 A brace on the wrong flange never gets there, however stiff it is. The critical moment of an 8 m beam against the stiffness of a single midspan brace, drawn three times for the three heights the brace could sit at. On the compression flange it climbs from 143 kNm to the two-half-wave plateau of 447 — the beam braced into two 4.0 m beams — and reaches 99% of it at 447 kN/m. At the shear centre it needs 2252 kN/m, 5.0 times as much. On the tension flange it never arrives at all: at the stiffness that would have done the job on the other flange it has bought a factor of 1.068, and a stiffer brace in the same place buys the same nothing. Past the plateau the beam stops using the brace, which is where the idea of an ideal stiffness comes from. Stability

The brace on the wrong flange

A brace on a column has one property that matters, and it is stiffness. A brace on a beam has two, and the second decides whether the first is worth anything: put the identical restraint on the tension flange and it does not reach the answer at any stiffness whatever.

 One slit, and the torsional stiffness falls by a factor of hundreds. A 200 by 200 box of 8 mm wall, drawn closed and then slit along its length. Closed, the torque runs round the wall as a shear flow and the torsion constant is 5.66×10⁷ mm⁴; slit, the loop is broken and only each wall's own thickness resists, giving 1.31×10⁵ mm⁴. The ratio is 432 to one, so the same torque twists the slit section 432 times as far and raises a peak shear stress 36 times as high. Nothing about the material changed. Sections and stress

The slit that costs a factor of six hundred

Bending stiffness cares where the material is, and changes by a factor of two or three between sensible sections of the same area. Torsional stiffness cares whether the material forms a closed loop, and the penalty for not doing so is an order of magnitude squared.

 Warping stiffens a short member and nothing at all a long one. The stiffening 1/[1 − tanh(κ)/κ] against kL, both axes logarithmic, over kL from 0.05 to 200. At the low end the curve is a straight line of slope −2, because for small kL the bracket is κ²/3 and the stiffening is 3/kL²: it reaches 1201 at kL = 0.05, falls to 1.005 at the top, and every open section ever rolled sits somewhere on it. The same three plates arranged three ways are marked: the 533 by 190 mm I-section at kL 2.76 and ×1.562, the tee at kL 37 and ×1.027, the angle at kL 34 and ×1.030. A tee's warping constant is 288 times smaller than the I-section's and an angle's 236 times, because their plates meet at a point and there is no pair of flanges to bend against each other — so they have no warping resistance to offer at all, and that is the reason an angle is a poor thing to twist. Stability

The restraint that beats the gradient

A moment-gradient factor is worth up to 2.7 on a beam's critical moment and is tabulated everywhere. Holding the ends against warping is worth more, is achieved by a detail rather than by a load case, and appears in no table at all.

The library, page 7 of 7 — where warping-torsion sits