Calostro y leche materna

Let us con sider (Figure 5.3) a pile subjected to an axial load Q, resisted by a distri­ bution of shear stress τ at the pile–soil lateral interface and by a normal stress p at the pile base. The beha vi our of the compressed pile is de scribed by the fol low ing dif­ ferential equation

(5.2)

where E is the Young modulus of the pile mater ial, A the area of its section, Ω its perimeter. To complete the ana lysis, a relation between the ver tical displacement w of a point at the pile–soil interface and the resulting shear stress τ is needed.

Table 5.2 Values of M, Eq. 5.1

Pile type Soil type M

Displacement Cohesionless 80

Cohesive 120

Small displacement (driven H or tube; large stem auger piles) Cohesionless 50

Cohesive 75

Replacement Cohesionless 25

Two pos sible strategies can be adopted:

• relating the displacement at a point only to the shear stress acting at the same point. Such an as sump tion, gen erally associated to the name of Winkler, assimi­ lates the pile–soil connection to a set of mutually inde pend ent springs; the rela­ tion connecting the displacement w to the shear stress τ is known as “trans fer curve” or t–z curve (with our notations, τ–w curve or p–w curve for the pile base);

• modelling the soil as a continuum, and imposing the compatibility of the dis­ placement at the interface between the soil and the pile.

In the present paragraph, we shall ex plore the first strategy. In the case of a homo­ gen eous soil, we can assume:

(5.3) where τ is the ver tical shearing stress acting over a circumferential section of the pile and k [FL–3_{] is the spring constant applying to the soil alongside the pile. In other} words, the trans fer curve is as sumed to be a straight line. The total force acting on the area Ωdz is:

From Eq. 5.2 one obtains: � � �d

� Q

A

94 Present practice: vertical loads Defining a charac ter istic length:

(5.4)

Eq. 5.2 becomes:

(5.5)

whose solution is:

The boundary con ditions are:

• at the top of the pile (z = 0; ):

• at the tip of the pile (z = L; P = pA = AkwL)

Closed form solutions for this and other simple cases are given by Scott (1981), Mylonakis and Gazetas (1998) and Salgado (2008).

A better approximation of the actual pile beha vi our may be obtained con sidering that the actual relation between lateral stress and settlement is non­ linear, and adopting non­ linear empirical trans fer curves instead of Eq. 5.3. As recalled above, load trans fer curves are usually referred in the liter at ure as t–z curves; with our nota­ tions they have to be called τ–w and p–w curves. They are obtained from load tests on piles with the shaft instrumented for the meas ure ment of axial strain with depth (see Figure 7.13). The curves are usually reported in dimensionless form, with the displacement w normalized to the pile dia meter d and shear stress τ or base normal stress p normalized to their ultimate value.

Typical trans fer curves for side and base resistance of bored piles in clay and in sand are reported in Figures 5.4 and 5.5 (Reese and O’Neill 1988). The secant modulus of a τ–w curve gives the vari able value of k making pos sible the solution of Eq. 5.5 by numerical methods (Figure 5.6). For instance, in terms of finite dif fer­ ences, it is pos sible to express the deriv at ive of the function w(z) at a point i as follows (Figure 5.7):

Settlement 95

where δ = L / n. Eq. 5.5 becomes:

(5.6)

where λi is obtained with the value of k of the soil surrounding point i; this allows an easy con sidera tion of stratified soils. Eq. 5.6 may be written in the n points and, together with the proper boundary con ditions, furnishes a system of linear equations

Side load

Ultimate side resistance

Side load

Ultimate side resistance

Ultimate side resistance

Side load

Ultimate side resistance

Figure 5.4 Transfer curves of side resistance for cohesionless (a) and cohesive (b) soils.

End bear

ing

Ultimate end bear

ing

End bear

ing

Ultimate end bear

ing

End bear

ing

Ultimate end bear

ing

End bear

ing

Ultimate end bear

ing

96 Present practice: vertical loads

whose unknowns are the values of the displacement at the pile top and along the pile shaft. The system may be solved for each loading step, thus allowing the construc­ tion of the (non­ linear) load–settlement relation for the pile top.

Randolph (1989) has de veloped a code named RATZ in which the shape of τ–w curves is that reported in Figure 5.8. The solution of the equation is obtained adopt­ ing the so­ called expli cit time approach (Cundall and Strack 1979), which involves the introduction of time as a vari able, artificially in the case of static loading.

k = p/w p w plim k w k _s k = t/w t k = p/w p w plim k w k _s k = t/w t _{k = p/w} p w plim k w k _s k = t/w t k = p/w p w plim k w k _s k = t/w t

Figure 5.6 Typical transfer curves and definition of the secant modulus; (a) lateral and (b) base resistance.

In addition to mono tonic and cyclic loading of the pile, RATZ may con sider exter­ nal soil movements causing downdrag, thermal strains in the pile, residual load de veloped during pile installation and matching of a meas ured load displacement response of the pile.

In Figure 5.9 two load– settlement curves obtained by pile load tests (Van Impe et al. 1998) are compared to the predictions performed by RATZ. The para meters needed have been evalu ated fitting the experimental data. The agreement between predictions and experiments is rather satis fact ory, and this occurs in the large major­ ity of cases.

�� � �

�

Figure 5.8 Transfer curves adopted in the code RATZ (after Randolph 1989).

Figure 5.9 Comparison between pile load tests and prediction by RATZ (after Randolph 1989).

98 Present practice: vertical loads