The collection

Every essay — page 9

Essays 193 to 196 of 196, in the same order.
00.20.40.60.8100.20.40.60.81along the memberdensity ÷ its peak, and running sharewhere it comes fromrunning totalhalf nearest the root: 87.5% of it · worth 7.0× stiffening the other half Deflection

Where a deflection comes from

The unit-load method gives a deflection as an integral, and this collection has treated that integral as a number to evaluate. It is not a number. It is a density, and it says which millimetres of the member produced the answer — which is not the same map as where the moment is largest.

10 figures · deflection distribution
-1-0.50.51-1-0.50.51σ₁ ÷ f_yσ₂ ÷ f_yvon MisesTrescapure shearthe widest gap is 2/√3 = 1.1547, at pure shear Materials

The shear strength nobody measured

Every web on this site 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.

9 figures · yield criterion
02040608005101520angle to the grain (degrees)strength (N/mm²)Hankinsona straight linetension4.47 at 45°, not 11.821 per cent of the strength along the grain, at half a right angle Materials

The material that has a direction

Every material in this collection so far has had one modulus and one strength. Timber has three of each, differing by more than an order of magnitude, and the consequence is not a correction to steel design — it is a different set of checks with a different one governing.

10 figures · anisotropy
-150-100-505010015000.20.40.60.811.21.4the redundant — horizontal reaction at the right base (kN)load factorbeam 1.00sway 1.25combined 0.71best lower bound 0.714every mechanism is at or above the answer; every admissible field is at or below it Equilibrium

Two ways of being wrong

Plastic analysis has two theorems and they point in opposite directions. Any equilibrium field that nowhere exceeds the plastic moment gives a load at or below the collapse load; any mechanism gives one at or above it. Only one of those errors is safe.

9 figures · bound theorems