The Peck formula describes tunnel-induced ground settlement using a Gaussian curve. The settlement is maximum directly above the tunnel centerline and decreases exponentially with distance. The key parameters are S-max, the maximum settlement, and i, the trough width parameter representing the distance to the inflection point.
The Peck formula is expressed as S of x equals S-max times e to the power of negative x squared over 2 i squared. This exponential function creates the characteristic bell-shaped settlement trough. As we move the point along the curve, you can see how settlement decreases exponentially with distance from the tunnel centerline.
Volume loss is the key parameter in tunnel settlement prediction. It represents the percentage of excavated tunnel volume that is lost due to ground deformation. The relationship between volume loss, maximum settlement, and trough width follows a mathematical formula. As we increase volume loss from 1% to 3%, you can see how the maximum settlement increases proportionally.
The trough width parameter i controls how wide the settlement spreads from the tunnel centerline. It represents the distance to the inflection points where settlement is about 60% of maximum. The i parameter is empirically related to tunnel depth and soil type. Clay soils typically have larger i values, creating wider settlement troughs, while sandy soils have smaller i values with more concentrated settlement.
To apply the Peck formula in practice, follow these systematic steps. First, estimate the volume loss based on your construction method. Second, determine the trough width parameter using empirical ratios for your soil type. Third, calculate the maximum settlement from the volume relationship. Fourth, apply the Peck formula to calculate settlement at any distance. Finally, plot the complete settlement profile. Always validate your predictions with monitoring data when available.