The ana lysis of the load– settlement beha vi our of a single pile, accounting for non linearity, may be carried out by the trans fer curve approach (§5.2.2). It is also pos sible to predict a non linear response from elastic BEM solutions, by an incremental ana lysis imposing the con dition that the shear stress at the lateral pile–soil interface and the normal stress at the pile base cannot exceed an ultimate value, which can be evalu ated with the cri teria presented in Chapter 4. In structural ana lysis such a pro ced ure is known as Pushover Analysis.
This feature is implemented in the program SINGHYP (Russo 1996; Van Impe et al. 1998) which, in addition, introduces a hyper bolic load–settlement response at the pile base. Such a relation is completely defined by its initial tangent and its ultimate asymptotic value. The former of these quantities can be evalu ated by elasticity theory, while the ultimate bearing capa city by the cri teria presented in Chapter 4. Both evalu ations belong to design routine.
Predictions by SINGHYP are compared in Figure 5.30 to the same load– settlement curves used in Figure 5.9; it may be seen that the agreement between prediction and experimental data is even more satis fact ory than with the trans fer curves approach. As far as the group beha vi our is concerned, once the non linear beha vi our of the single pile is defined, in prin ciple there are no prob lems in describing the non linear beha vi our of the group by a stepwise incremental linear ana lysis in which the stiff ness of each pile is updated as a function of the load level; iteration within each loading step may improve the accuracy of the results. It is obvious, how ever, that the computing resources and time needed make such an ana lysis long and cumbersome. Caputo and Viggiani (1984) report some experimental data obtained by pile load tests in which, in addition to the settlement of the test pile, the settlement of unloaded adjacent piles had been meas ured. A sample of such data is reproduced in Figure 5.31. The load– settlement curves of the loaded (source) pile has the usual non linear trend, while the curve relating the settlement of the unloaded (receiver) pile to the load acting on the loaded (source) test pile is very nearly linear. It appears that, at some distance from an axially loaded pile, the deformation of the soil is essentially linear, and this applies to the displacement of any unloaded pile within the deformation field of the source pile. Such a beha vi our is consistent with the rapid decay of stress moving radially away from the pile–soil lateral interface, as argued by Cooke (1974) and Frank (1974) and expressed by Eq. 5.7.
In order to get a better insight into the phenomenon, let us con sider two ident ical adjacent piles (Figure 5.32) and ana lyse their beha vi our when pile 1 is subjected to
an axial load Q gradually increasing to the ultimate value Qlim while pile 2 is kept load free. The ana lysis is carried out incrementally, allowing local slip to occur at the pile–soil interface when the interface stress reaches its limiting value. The results obtained are reported in terms of the ratio between the inter action factor α12 evalu ated by the non linear ana lysis and the cor res ponding value from a linearly elastic ana lysis, plotted against the ratio Q / Qlim = 1 / FS. As observed in the experiments of Figure 5.31, the inter action factor keeps constant up to value of the safety factor of the order of 2.5 – i.e. the value usually adopted in design – and even for FS as low as 1.5 the decrease of the inter action factor is less than 10%. In a way, such a result could have been foreseen on the basis of the so called Saint Venant principle.
Figure 5.32 Nonlinearity effect in the interaction between two piles. Figure 5.31 Load–settlement relation of the
loaded pile and nearby unloaded piles during load tests.
On the basis of these and sim ilar results, Caputo and Viggiani (1984) suggested implementing the non linear incremental ana lysis of a pile group by updating, at each load increment, the coef fi cient αii expressing the effect of the load Qi on the
loaded pile while keeping constant the coef fi cient αij expressing the effect of the load
Qi acting on the pile i on the settlement of the pile j. In other words, in the
expression:
the terms on the prin cipal diagonal of the matrix of the inter action coef fi cients αii are
updated while the terms off diagonal αij are kept constant, strongly decreasing the
computational effort and improving the fidelity of the model. In this way, the non linearity is concentrated at the pile–soil interface, while the inter action between the piles are treated as linear. It may be shown that this pro ced ure is essentially equi val ent to the suggestion by Randolph (1994), to estim ate the group response on the basis of the initial small strain elastic stiffness and afterwards adding the plastic dis placement due to the slip at the pile–soil interface.
The load– settlement relation of the loaded piles may be assim il ated to a hyper bola (Chin 1970) whose equation is:
where b = 1 / Qlim is the inverse of the bearing capa city of the pile and is the inverse of the initial tangent stiffness of the pile (a = w1). It may be shown that, with this as sump tion, the updating of the inter action coef fi cients may be done according to the expression:
The pro ced ure is incorp or ated in the code of GRuPPALO and has been adopted for the back ana lysis of the same case histories of Figure 5.28; the results obtained are plotted in Figure 5.33. The NL ana lysis, which essentially consists in adding the non linear com pon ent of the settlement of the single pile to the settlement of the group, obtained as in the LE ana lysis, slightly improves the prediction of the average settle ment in all the cases where the LE ana lysis was already successful. In the cases where the non linearity plays a significant role, NL ana lysis significantly improves the prediction.