When analyzing a pile or drilled shaft as a beam-column, how do you determine the unbraced length of the portion that is in the ground? Typically, the braced points on a beam or column are well defined, e.g. there are connections at discrete locations.

A pile in the ground will have some resistance to movement because of the soil. Is this resistance enough to consider the pile to be continuously braced (Lb = 0)?

Another way of considering the situation might be to look at the deflection diagram. Should the braced length be based off the first or second point where the deflection crosses zero? Or similarly where the moment diagram crosses zero?

  • $\begingroup$ I'm not a structural engineer so I'll let someone else answer proper. So this is just my gut feel. Deep down it's probably fully braced, as it emerges it's not braced at all. This bracing will reduce according to some function. This can probably be modeled as an infinite number of braces of decreasing rigidity along infinitely small sections of the beam. Therefore some integration function probably exists to give you what you need. $\endgroup$
    – jhabbott
    Feb 13, 2015 at 12:18
  • $\begingroup$ I'm with @jhabbott on this -- I'm not a structural engineer either, but it seems to me that it's going to be a function of the force that the soil exerts on the sides of the pile. The yield pressure of the soil is going to be a function of the soil type and the depth, and the net force on the pile is going to depend on the soil pressure and the geometry (shape) of the pile. It sounds like it could turn into a rather complex boundary value problem. $\endgroup$
    – Dave Tweed
    Feb 13, 2015 at 13:01

1 Answer 1


$\require{cancel}$ A common method was devised by Davisson and Robinson (1965). By this method, what you need to determine is the "length of fixity", which is given by

$$L_f = 1.8\left(\dfrac{EI}{n_h}\right)^{0.20}$$

Where $EI$ is the pile's stiffness and $n_h$ is the soil's coefficient of horizontal subgrade modulus. This is the relevant length for a buckling analysis, even if your pile is longer than $L_f$. Obviously, if your pile has an uncovered length ($L_u$) , that needs to be added to $L_f$, in which case your buckling length is $L_f + L_u$.

It is worth noting that this equation is not set to a specific set of units, as can be seen by dimensional analysis:

$$L_f = [L] = 1.8\left(\dfrac{EI}{n_h}\right)^{0.20} = \left(\dfrac{\left[\dfrac{\cancel F}{\cancel{L^2}}\right][L^{\cancel{4}2}]}{\left[\dfrac{\cancel F}{L^3}\right]}\right)^{\frac{1}{5}} = [L]$$

This article explains it in greater detail.

  • $\begingroup$ Please include the units for this empirical equation in the event that the linked reference becomes unavailable at some point. $\endgroup$
    – Rick
    Jul 12, 2017 at 15:31
  • $\begingroup$ @RickTeachey: The equation is actually unit-agnostic, but I've added a note to demonstrate this. $\endgroup$
    – Wasabi
    Jul 12, 2017 at 16:51
  • $\begingroup$ Oh of course! Very nice. $\endgroup$
    – Rick
    Jul 12, 2017 at 17:00

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