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 = (π/4)U². 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.