**3. Incremental back-calculation**

The function which converts axial strain to stress in a deep foundation is the multiplier consisting of Young's modulus of the foundation material *E* times the crosssectional area *A*. This function is often called the 'stiffness' of the foundation, although technically this is a misnomer since by definition the units of stiffness are force per length (*AE/L*), whereas the conversion of strain (unitless) to force must also be defined in units of force (*AE*), and is properly called the 'rigidity.' Composite axial rigidity calculations based on empirical relationships such as the ACI 318 formula [3] result in a constant value of *AE*. These types of empirical formulas are based on several assumptions, including average concrete strength *f'c* and knowledge of the cross-sectional area, which may be only nominally correct. In addition, confinement effects and the fact that the stress-strain relationship (modulus) of cementitious materials is not linear are also not considered.

The basic assumption of incremental rigidity back-calculation methods is that the non-linear stress-strain relationship of cementitious materials (drilled shaft concrete and augercast pile grout) can be adequately approximated with a quadratic function [4]. This assumption seems reasonable if a family of stress-strain curves is examined (**Figure 6**), with a parabola (red-dashed line in the figure) overlaid over the *f'c* = 4000 psi curve as an illustrative example. The approximation is quite good from the origin up to the peak stress (yield point), which is all that is required for the analysis of load test results. The value of Young's modulus is the slope of the stressstrain function curve at any given strain. As noted in the case histories in the previous section, it is not uncommon during axial load testing to measure strains on the order of 500 to 1000 *με* or more, especially in slender elements such as ACIP piles. Therefore, the non-linearity of Young's modulus of cementitious materials must be accounted for.

The axial rigidity contribution of steel reinforcement in the composite crosssection is typically relatively small, and the stress-strain curve for steel is assumed linear up to the yield point which means its modulus is relatively constant. Therefore, *Perspective Chapter: Interpretation of Deep Foundation Load Test Data DOI: http://dx.doi.org/10.5772/intechopen.105142*

**Figure 6.**

*Concrete stress-strain curves and quadratic approximation (red dashed line).*

the non-linear properties of the cementitious material govern the composite cross-section rigidity of the foundation element.

The equation of the parabolic quadratic function illustrated in **Figure 6** is given by:

$$
\sigma = a\varepsilon^2 + b\varepsilon + c \tag{4}
$$

Note the constant term *c* is always zero, which assures that the parabola intersects the origin. The cross-sectional area is assumed to remain constant throughout the analysis; the small effect of Poisson's ratio is neglected. Therefore, to convert from stress to force and modulus to rigidity, Eq. (4) is simply scaled by the cross-sectional area *A*:

$$F = A\sigma = A\left(ae^2 + be\right) \tag{5}$$

#### **3.1 Incremental back-calculation**

The Secant Modulus (SM) method is the simplest back-calculation technique. A strain gage (or set of strain gages) is installed in the deep foundation element at or above ground level, such that all of the applied force *P* at the head of the element must be registered by the gage(s). That is to say, no force is shed into the soil via skin friction between the point of load application and the strain measurement. At each incremental step *n* of the load test, the rigidity is computed as:

$$(AE\_n)\_{\text{secant}} = \frac{P\_n}{\varepsilon\_n} \tag{6}$$

Note that the method is typically discussed in terms of stress and strain [5, 6]. Starting with test load data, the foundation cross-sectional area is divided out in order to derive a function for the modulus. In order to recover forces, the cross-sectional area must be multiplied back into the analysis later. Herein, this intermediate step is eliminated since the ultimate objective is to convert strain data to force.

This method is called 'secant' because the resulting rigidity function is the slope from any point on the force-strain curve back to the origin. Once testing is complete, each of the rigidity values *AEn* are plotted versus *εn*, and linear regression is used to determine the slope and offset (*a* and *b* respectively) of the best-fit line through the data. Substituting Eq. (5) into Eq. (6), the result in terms of the quadratic function presented above is:

$$A(AE)\_{scam} \approx \frac{F}{\varepsilon} = \frac{A(ae^2 + b\varepsilon)}{\varepsilon} = A(ae + b) \tag{7}$$

To compute the force at any strain, Eq. (7) is rearranged:

$$F\_n = A\left(ae\_n^2 + b\varepsilon\_n\right) \tag{8}$$

One drawback of this method is that it cannot be utilized during bi-directional testing. The plane-strain assumption (the strain measured by the gages is an accurate representation of the average strain throughout the cross-section) means that the gages must be positioned at least one element diameter away from the point of loading, in order for local stress variations at the point of loading to even out. In a bidirectional test, significant force may be shed into the soil via skin friction within this span, invalidating the relationship in Eq. (6) because the force at the strain gage location is now an unknown.

Additionally, in cast-in-place foundation elements (with variable cross-section area and curing conditions), or those which have variable reinforcement with depth, the SM method may not yield accurate rigidity estimates for embedded strain gage levels because the ground-level gages may not be representative [7].

#### **3.2 Tangent modulus and incremental rigidity methods**

The Tangent Modulus (TM) method was initially derived by Fellenius explicitly for the modulus, with the cross-sectional area considered separately. This is best applicable for foundation elements with assured constant cross-section properties (such as driven piles). The Incremental Rigidity (IR) method discussed in [7, 8] recognized that the rigidity (modulus times area, *AE*) is a single function which can be identified without explicitly identifying the relative magnitude of either of the two components *A* and *E*. In this discussion the analysis focuses on the strain-force relationship (effectively, the Incremental Rigidity method), although from a strictly mathematical derivation standpoint, the TM and IR methods are equivalent.

In the IR method, rigidity is computed as the slope of the force-strain curve at a given strain. This slope is approximated as the change in applied load Δ*P* divided by change in strain Δ*ε* for successive load increments:

$$(AE\_n)\_{inrmmental} \approx \frac{\Delta P}{\Delta \varepsilon} = \frac{P\_n - P\_{n-1}}{\varepsilon\_n - \varepsilon\_{n-1}} \tag{9}$$

As with the SM method above, once testing is complete the rigidity value at each increment is plotted against its corresponding strain and a best-fit line plotted through the data. However, the incremental method requires that the side shear section between the point of load application and the strain gage elevation has reached or at least approached its ultimate capacity. Because of shear resistance, the force increase

*Perspective Chapter: Interpretation of Deep Foundation Load Test Data DOI: http://dx.doi.org/10.5772/intechopen.105142*

**Figure 7.** *Sample rigidity analysis from strain gage data.*

at the strain gage will be less than the applied load increase during the initial part of the test. Therefore, the resulting incremental rigidity values will be excessive. It is only after the side shear section between the point of load application and the strain gage reaches its ultimate shear capacity that subsequent applied load increments result in strain increments which give a true indication of the rigidity. This behavior becomes apparent on a plot of the analysis where the rigidity decreases from very high values at a small *ε* to a linear curve at high *ε* (see **Figure 7**, above). A linear regression through this ultimate portion of the incremental rigidity curve will yield slope and offset values *g* and *h*:

$$(AE)\_{incremental} = A(\lg \varepsilon + h) \tag{10}$$

The incremental rigidity, by definition is also the slope (first derivative) of the force-strain function (Eq. (5)):

$$(AE)\_{incremental} = dF/\_{dc} = A(2a\varepsilon + b) \tag{11}$$

Comparing Eqs. (7), (8) and (11) it becomes apparent that the incremental rigidity and secant rigidity analyses for the same load test will result in a different force-strain relationship, by a factor of 2 in the first term (slope) and that the second term constants (*b* and *h* respectively) are equivalent. This is illustrated in **Figure 7** with a sample data set from a series of top-down tests (the 'Texas' case history). Strain gage 1 is located just below the point of load application, and is analyzed using the secant rigidity method. Strain gage 2 is located some distance down within the shear embedment zone, and is analyzed using the incremental rigidity method. As expected from theory, the two linear regressions converge at the vertical axis (the zero-strain condition) but have significantly different slopes. For comparison purposes, the ACI

#### **Figure 8.**

*Non-linear force-strain curve with incremental (tangent) and secant moduli.*

rigidity (computed using the empirical relationship to the square root of concrete strength *f'c*) is also plotted as a horizontal line, showing that it does not produce a good result for this particular foundation element.

Because it is the slope from any point on the force-strain curve back to the origin, the secant rigidity can be multiplied by any measured strain to directly compute force. However, the incremental rigidity cannot be simply multiplied because it is by definition tangent to the force-strain curve at all points and does not intercept the origin. **Figure 8** illustrates this point graphically.

By simply multiplying the curve-fit incremental rigidity slope by 0.5, the equivalent secant rigidity function is recovered and Eq. (12) used to compute the force directly for each measured strain.

$$F\_n = A\left(0.5\left(ge\_n^2\right) + h\varepsilon\_n\right) = A\left(ae\_n^2 + b\varepsilon\_n\right) \tag{12}$$

Alternatively, or when dealing with highly non-linear rigidity relationships, the value of *F* at any loading point *n* may be approximated by a recursive summation formula [9]:

$$F\_n = F\_{n-1} + (AE\_n)\_{
in
mathcal{I}}(e\_n - e\_{n-1})\tag{13}$$

where *Fn-1* and ε*n-1* are the force and strain of the previous loading data point, respectively. This step-wise approximation will roughly follow the curved load-strain path.

This approach will give approximately correct results even if the rigidity function is highly non-linear, such as in the case of a tensile load test once the cementitious material begins to crack due to tensile strain, or in a compressive load test with pre-existing tension cracks in the cementitious material which are closed up by the compressive axial stress [9]. As noted above, foundation element axial rigidity *AE* is composed of two contributors, steel and cementitious material (*AsEs* and *AcEc*, respectively). For a cementitious material which is fractured (due to shrinkage during curing or applied tensile stress), the nominal area *Ac* is replaced with an effective area *A'c*.

**Figure 9** illustrates two idealized functions of nonlinear axial rigidity due to cementitious material fracturing in response to tensile stress (bold line segments). The full composite rigidity consists of *AsEs* + *AcEc*. The angular pathways to/from the reinforcing steel rigidity (*AsEs* only) indicate idealized changes in rigidity due to fracturing with increasing strain. With increased compressive strain pre-existing

*Perspective Chapter: Interpretation of Deep Foundation Load Test Data DOI: http://dx.doi.org/10.5772/intechopen.105142*

**Figure 9.**

*Possible non-linear rigidity function paths.*

**Figure 10.** *Incremental load calculation.*

fractures progressively close up, *A'<sup>c</sup>* increases from zero to *Ac* and the rigidity increases until the full composite rigidity is reached. Conversely, with increased tensile strain, the rigidity decreases from the full composite value down to the reinforcing steel rigidity only, as the cementitious material progressively fractures until only the reinforcing steel remains to transmit stress.

If Eq. (13) is employed, using each load test increment as a discrete step the nonlinear load-strain curve can be approximated by a series of small incremental increases in load, each of which is linear with its corresponding increase in strain, as illustrated in **Figure 10**.

Note that all the rigidity back-calculation methods depend on obtaining highquality strain gage data from relatively small, equal load increments to clearly define trends. Results obtained at one strain gage level may not apply at other levels, due to several factors including possible changes cross-sectional area, reinforcement details, confinement (within rock socket as opposed to overburden) and differing concrete curing conditions (hydrostatic pressure, water table elevation, environmental temperatures, etc.) among others.

Once the load at each strain gage level has been computed using the methods discussed above at every load increment, a family of load distribution curves can be generated (see **Figure 11**).

The difference between adjacent levels (a 'zone' of the foundation element), divided by the perimeter shear area of the zone, gives the unit shear, the 't' component of the desired t-z curve. A level of strain gages placed near the base of the foundation also allows for estimation of the bearing resistance *q*.

**Figure 11.** *Sample load distribution (Texas case history).*

Note that the analysis results presented herein are based on re-zeroing all strain gages prior to the start of loading, and account for the resistance of the as-built isolated test element to a relatively short-duration externally-applied load only. They do not account for any residual load present in the element at the start of testing, down-drag, long-term setup, creep or group effects.
