Time Factor from Degree of Consolidation (U ≤ 60%)

The exact solution of the consolidation equation is an infinite Fourier series, but for the first two thirds of the process a single parabola fits it to better than 1%: $T_v = (\pi/4)U^2$. At 50% consolidation that gives $T_v$ = (π/4)(0.5)² = 0.196, which is the textbook T₅₀ of 0.197 to three figures; at 30% it gives 0.0707. Above 60% the series tail matters and the accepted approximation switches to $T_v$ = 1.781 − 0.933 log₁₀(100 − U%), which yields T₉₀ = 0.848.

The trap is confusing the average degree of consolidation with the local one. U here is the layer average, the fraction of the final settlement achieved, while the pore pressure at the centre of the layer is still far behind the edges. The second trap is treating U = 100% as an event with a date: the series converges asymptotically, so consolidation is never formally complete, which is why designers talk about t₉₀ and why Casagrande's and Taylor's graphical constructions on the oedometer curve exist to pull $c_v$ out of test data at t₅₀ and t₉₀ rather than at the end.