Dynamic driving formulas and static loadings in the light of wave equation solutions

Dynamic formulas have the appeal of their simplicity, especially those that depend on the set (s), the elastic rebound (K) and the efficiency (η) of the driving system. The trend today is to take advantage of the dynamic monitoring of a certain number of piles and obtain parameters such as η to be used in other piles of the work along with direct measurements of s and K, say, with pencil and paper. However, some of the most used dynamic formulas in Brazil, namely the Chellis-Velloso Formula, the Energy Approach Equation, and the Uto’s Formula have required adjustments considering the geometry and kind of piles, the types of soils, among other factors. The question that arises refers to the general validity of these formulas. Furthermore, in this context the simulation of static loadings through dynamic tests with increasing energy is discussed.


The Chellis-Velloso formula
The well-known formula of Chellis (1951) modified by Velloso (1987) is based on Hooke's law and uses measurements of elastic rebound to estimate static resistance, as shown in Equations 1 and 2.
In these equations R=RMX is the static mobilized resistance; K r , the pile stiffness; C 2 , the pile elastic shortening or pile compression; E, the dynamic Young's modulus; S, the area of the pile cross section; L, its length; K, the elastic rebound; C 3 , the toe "quake", usually taken equal to 2.5mm; and α, a factor dependent on the distribution of lateral friction and tip load, given by: where λ is the coefficient of Leonards & Lovell (1979) and β is the relationship between tip load and total load. Velloso (1987) suggested using α =0.7, an average value.

The energy approach equation
The Energy Approach Equation, as presented by Paikowsky & Chernauskas (1992) These authors assumed an elasto-plastic relation between resistance and displacement, as shown in Figure 1. The maximum energy delivered to the pile (EMX) was equated to the work done by the resistance (R u or RMX) offered by the soil to the penetration of the pile [RMX.(s+DMX)/2], where DMX=s+K. Based on a case study, Paikowsky & Chernauskas (1992) proposed a reduction parameter κ=0.8, arguing that part of the applied energy EMX is dissipated in the mobilization of viscous or dynamic resistances. Aoki (1997) interpreted the driving process in the light of the Hamilton's Principle of energy conservation and came up with a similar expression, using the ζ symbol instead of 2κ. The parameter ζ would depend on the magnitude and nature of the reaction forces (conservative or non-conservative) and could vary between 1 (permanent displacement predominates) and 2 (elastic displacement predominates). Uto et al. (1985) presented a simple formula based on the solution of one-dimensional wave equation, admitting as the boundary condition the displacement-time curves for the top and the tip of the pile. Several simplifying hypotheses were assumed, among which the following stand out: a) lateral friction and viscous resistance (damping) at the tip of the pile were neglected during driving; and b) the set (s) was taken equal to the toe quake (C 3 ). They came to the following equations:

The Utos's formula
where tip d R is the dynamic resistance mobilized at the pile tip; e o , a wavelength correction factor, is a function of both, (a) the relationship between the weights of the hammer (W H ) and the pile (W P ), and (b) the pile type, through the parameter ξ, that assumes a figure of 1.5 for steel piles and 2.0 for concrete piles.

Theoretical background
For the pile element in Figure 2a, the equation of the balance of the acting forces during driving can be written as follows (see attached list of symbols): According to Smith (1960), the shaft friction (f) is given by: valid for u smaller than the quake. If not, k.u is equal to the maximum shaft friction ( ) est max f ).
Equation 7 may be rewritten as: Note that: where R e , R d and R m are respectively the static, dynamic, and total resistances in the element. By Hooke's Law it follows: This equation shows that it is the dynamic force F that generates the elastic shortenings (du) in the element and not the static force R e , confirming the above-mentioned statement of Casagrande (1942).
By deriving both members from Equation 11 the term dF/dx is obtained, which, replaced in Equation 9, results in the Wave Equation: At time t, F varies as follows along depth (x): where F o stands for F(x) at pile top (x=0).

Numerical wave equation solutions for simple cases
Consider the solution of the wave equation presented in Figure 3a obtained through the methodology of Smith (1960) applied to a steel pipe pile (see Table 1), excited at the top by a speed v o =4.33 m/s at time t=0, due to the blow of a hammer. It was assumed that static maximum unit lateral frictions (Figure 2b) are known a priori just like the toe static resistance (R p ), the shaft (q s ) and tip (q t ) quakes, and "Smith dampings" of friction (J s ) and tip (J t ), indicated in Table 2. Under these conditions, the maximum static lateral (A lr ) and tip (Q pr = R p . S p ) loads are 8081 kN and 1225 kN, respectively, adding up 9306 kN.
From Figure 3a one may conclude that for t=t o =8 ms the speed at the top (v o ) is zero and therefore the displacement at the top reaches its maximum value, DMX in Figure 3b. This figure also displays the calculated C 2D by the difference of the top and tip pile displacements at each time. For the same time t=8 ms, Figure 4 shows the distribution along the shaft of the maximum static resistance and of the dynamic (F) axial force. From its analysis, it can be concluded that (see the list of symbols attached): a) the forces F for t=8 ms were lower than the maximum static resistances (Figure 4), with C 2D =10.1 mm ( Figure 3b) which is smaller than the corresponding static value, given by: and K D =C 2D +q t ≈11.8 mm against K E = C 2E +C 3 ≈20.9 mm. As DMX=17.1 mm (Figure 3b), it follows that s=DMX-K D =5.3mm; the values of λ and K r are given in Tables 1 and 2; and b) these differences between static and dynamic values ( Figure 4) result from Equation 13: the total resistances (R m ) interact with the inertial forces, due to acceleration, which acts either up or down, as illustrated in Figure 5. The same Pile E-1 was also submitted to a simulation of the dynamic loading test with increasing energy, as proposed by Aoki (1989). The speed at the top due to the impact of the hammer was varied between 1.08 and 6.49 m/s, which implied in EMX increasing from 6 to 228 kN.m, as shown in Table 3       with other data of this simulation; note that the set varies with time (s to < s tf ). The results in Table 4 confirms that the dynamic values of elastic compression (C 2D ) and rebound (K D ) are lower than the corresponding static values (C 2E and K E ).

Evaluation of Chellis-Velloso formula
The differences between static and dynamic values of C2 lead to the first conclusion about the Chellis-Velloso Formula, Equations 1 to 3. With the values of K r =316 kN/mm (Table 1), λ=0.60 ( Table 2) and β=1225/9306=13.2% it follows for blow 5 of Tables 3 and 4: . refers to static calculation. And these authors added that this formula may be valid for short piles, with lengths of the order of the wavelength and so the whole pile is compressed, which does not occur on long piles.

Force or resistance vs displacement.
Evaluation of the energy approach equation Figure 6 shows the progress of the mobilized static resistance (R e ) along the depth (x) and the time (t). For t=8 ms the static resistances in the elements (R e ) already reach the maximum available values. Figure 7 reveals that the total static (R eT ) and the total dynamic+static resistances (R mT =R eT +R dT ) reach maximum values at a time t≈7 ms, therefore close to 8 ms, at which time the maximum displacement (DMX) occurs, as seen above (Figure 3b). It is also interesting to note that as time proceeds, the portions of the dynamic resistances vanish, as the pile is no more in movement.
The dynamic displacement (D m ) progresses along the depth (x) and time t (from 2 to 8 ms) as shown in Figure 8.      concluded, therefore, that this equation does not represent reality: it is a fiction. Figure 11 is an extension of Figure 10, as it includes all blows of Tables 3 and 4, in addition to blow 5 (v o =4.33 m/s at t=0). It also includes the envelop representing the curve RMX as a function of DMX. This same curve is reproduced in Figure 12, along with two others: a) the RMX curve as a function of DMX plus the sets of the previous blows, as proposed by Aoki (1989) and Niyama & Aoki (1991); and b) the load-displacement curve for blow 6 of Table 3 simulating the static loading test (SLT) obtained through the Method of Coyle & Reese (1966), considering the maximum resistances, quakes, and pile stiffness. It is concluded that for the analyzed pile E-1 the Aoki-Niyama curve falls short of the simulated curve.

Blow # vo (t=0) (m/s) EMX (kN.m) to (ms) RMX (kN) DMX (mm) sto (mm) sf (mm)
At time t≈8 ms v o =0 (Figure 3a), D(t) and E(t) reach the maximum values, DMX and EMX, respectively. The F o .v o product assumes negative values between 8 and 13 ms (Figure 13a), hence the inflection in the E(t) curve as displayed in Figure 13-b. The value of EMX can be obtained as shown in Equation 18, where o F is an average value between t=0 and t=8 ms. In fact, the third term in this equation is a result of the application of the Mean Value Theorem (Pastor et al., 1958), because in the interval 0 to 8 ms the following inequation holds: Another conclusion arises from the ABCD curve of Figure 10, which represents the variation of F o with D o between 0 and t=8 ms. The area bounded by this curve corresponds

Massad
In the case of a pier of Santos (SP), below 6 m of water there was a layer of 20 m of a soft Holocene clay (SPT=1 to 5), followed by 10 m of fine clayey sand (SPT=7 to 33) and 8 m of a Pleistocene clay (SPT~7) over thick sand layer (SPT~40).
Finally, the subsoil in Cubatão (SP) consisted of sandy fill 6 m thick (SPT=1 to 10), followed by a layer of a Holocene marine clay (SPT=1 to 5) up to 24 m deep. Below there were two layers of sand (SPT=15 to 30 and 30 to 15, respectively) up to roughly 40 meters deep, followed by residual soil of gneiss. The water level was 1 m deep.
to the EMX value, which has nothing to do with the work of total static resistance R eT , putting again the Energy Approach Equation in question.

Evaluation of Uto's formula
Another conclusion refers to the application of the Uto's Formula, Equations 5 and 6. For the type of pile (E-1) the correction factor e o (Equation 6) assumes a value of the order of 1.10, so that for blow 5 of Table 4: It is interesting to mention the results indicated in Figure 14, related to pile E-1, blow 5 (v o =4.33 m/s at t=0), but assuming that the maximum static lateral and tip loads are 5000 kN and 4306 kN, respectively, adding up the same total static load 9306 kN. The distribution of the shaft friction (f) along depth was supposed to be the same (λ=0.6). Figure 14 shows that for t=8.8 ms the dynamic force F is resisted only by the tip, with tip d R =4078 kN. As K D =15.3, the Uto's Formula gives: . * .
about 8% more. This case fulfills one of the conditions of Uto´s Formula, i.e, practically no dynamic shaft friction.

Evidence from three case histories
Next, results of dynamic loading tests with increasing energy on three case histories comprising pipe piles will be presented. The piles were quite different as shown in Table 5.
The subsoil in the case of Osasco (SP) consisted of 3.5 m of a landfill (SPT=4 to 5), followed by layers of soft fluvial clays up to 9.1 m (SPT=1 to 3) and residual soil (SPT=25 to 55). The water level was 3 m deep.  Details of the hammers and of the instruments that were used and the sequence of blows in each pile are presented in the references shown in Table 5. The collected data were analyzed through the CAPWAP software by specialized    As can be seen, there are different behaviors in terms of: a) the rebounds K E and K D , on the one hand; and b) of the load-displacement curves, on the other hand. In fact: a) the elastic rebounds (K D ), measured during pile driving, are lower than the corresponding static values (K E ), again because they are related to the dynamic forces and not to the resistance of the soil, as Casagrande intuited; and b) the curves RMX-DMX plus the sets of the previous blows fall short of the simulated static curves with only 1 stroke. The pile in Santos ( Figure 16) was an exception, due to higher values of the set (s), accumulating about 5mm up to the last stroke. These differences depend on several factors, such as the distribution of the load in depth (friction and tip), the set values, among others. The phenomenon of pile driving is quite complex.

Conclusions
Elastic compression and rebound measured during pile driving may be lower than the corresponding static values because they are related to the dynamic forces and not to the static resistance of the soil, as Casagrande intuited.
This fact explains why the curve RMX-DMX plus the sets of the previous blows can fall short of the simulated static curve with one single stroke. Moreover, it conceptually invalidates the use of the Chellis-Velloso Formula to estimate the bearing capacity of a pile. This last conclusion extends to the Energy Approach Equation based on an elasto-plastic relationship without physical meaning; furthermore, it wrongly relates the transferred energy (EMX) to the work of the soil resistances instead of the involved dynamic forces.
The Uto's Formula has restricted use in view of the adopted hypotheses, allowing its application to determine the dynamic force at the tip in cases where lateral friction is exceedingly small.
This makes conceptually unsuccessful attempts to universally validate these formulas. But nothing prevents their use in engineering practice as empirical correlations with correction factors, supported by the dynamic monitoring of some piles of a given work and place.