Analytical solutions for piles subjected to a passive load

The paper analyses a single free-head flexible passive pile embedded in a soil profile consisting of an unstable layer overlying a stable one. The pile segment in the unstable layer is subjected to a linear distribution of the load, whereas the pile response is evaluated by assuming a uniform modulus of subgrade reaction in the stable layer. Closed-form expressions are derived for the pile head deflection and tabulated values are provided to evaluate the maximum bending moment as a function of dimensionless length of the pile segment in the unstable and stable layers. Analytical solutions for infinitely flexible and rigid piles have been obtained as special cases of the general solution. A novel rational rigidity criterion for passive piles is also proposed. A numerical example is finally presented to illustrate the application of the derived solution.