Matlock and Reese (1960) have given equations for the determination of y, S, M, V, and p at any point x along the pile based on dimensional analysis. The equations are

where T is the relative stiffness factor expressed as

In Eqs (16.9) through (16.13), A and B are the sets of non-dimensional coefficients whose values are given in Table 16.2. The principle of superposition for the deflection of a laterally loaded pile is shown in Fig. 16.4. The A and B coefficients are given as a function of the depth coefficient, Z, expressed as

The A and B coefficients tend to zero when the depth coefficient Z is equal to or greater than 5 or otherwise the length of the pile is more than 5T. Such piles are called long or flexible piles. The length of a pile loses its significance beyond 5T.
Normally we need deflection and slope at ground level. The corresponding equations for these may be expressed as

Figure 16.4 Principle of superposition for the deflection of laterally loaded piles

yg for fixed head is

Table 16.2 The A and B coefficients as obtained by Reese and Matlock (1956) for long vertical piles on the assumption Es = nhx

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