Structural Engineering & Statics
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The figure library — page 7

Every picture here is generated from code. This page lists the generator families, how many essays call each one, and how many separate drawings each has produced. Page 7 of 7, warping-torsion to yield-surface.
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.

warping-torsion

4 essays A fillet weld is stronger across than along. Capacity of a 4 mm throat over 100 mm, against the angle between the weld's axis and the load. Loaded along its length it carries 116.83 kN; loaded across it, 143.09 kN. The ratio is 1.22, which is √3/√2 exactly, and it comes out of the failure criterion rather than out of a test.

weld-direction

9 essays Throat stress round a fillet weld group. A c shape weld group carrying 100 kN at 150 mm from its centroid. The peak throat stress is 0.88 kN per mm of throat, at (79.5, -100); the worst point at maximum radius from the centroid carries 0.88. Checking by radius is right here, and points at identical radius differ by a factor of 1.

weld-group

8 essays Which limit arrives first. Utilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry.

which-governs

6 essays The wind pushes on one face and pulls on three. A 30 × 20 m building in plan, with the measured pressure coefficient on each face and the arrows drawn in the direction the pressure acts. Only the windward face is pushed; the other three are sucked, and the side faces are sucked hardest of all at c_p = -0.7. The horizontal resultant is 842 kN at a velocity pressure of 0.9 kPa, and the arithmetic of it is the whole point: the leeward suction pulls the building downwind, so it ADDS, supplying 38% of the answer, while the two side faces cancel each other exactly and supply none of it. The coefficients are wind tunnel data; what is computed is the free body they are applied to.

wind-pressure

2 essays Where the wind's energy is, and where the structure can reach it. The gust spectrum at a mean speed of 30 m/s, plotted as n·S(n) against frequency on a logarithmic axis so that equal areas are equal energy. Its total variance is 27 m²/s², which the closed form 6·K·U² gives as 27. Most of the energy is below a hundredth of a hertz — gusts lasting a minute or more. A structure at 0.2 Hz sits far out on the tail, and still takes 90% of its response from there, because the resonant part is amplified by π·f₀/(4·ζ) and the damping is 1.2%.

wind-spectrum

2 essays 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.

yield-surface

3 essays