**2. Optimal strain gage arrangement**

The discussion in this section was originally published in [2]. The assumption of axial plane-strain is significant to converting measured strain to axial force. Eccentric stress in the foundation element whether due to inclined or eccentric loading, uneven soil resistance, irregular foundation cross-section shape or other reasons will cause an uneven distribution of strain across the element cross-section. Based on Euler beam theory, the total strain is a superposition of axial strain (which is required to compute axial load) and bending strain which is disregarded. In axial compressive or tensile load testing of foundations it is presumed that axial strains due to applied loading will be significantly greater than incidental bending strains due to load eccentricity or other second-order causes. Total strain is assumed to be linearly distributed across the plane of the element, and the net average axial strain will intersect the centroid of the element. Therefore, obtaining the strain at the centroid is key to computing the net axial force.

The theoretical performance of gages arranged in various configurations and then averaged for the purpose of force calculation is validated using statistical results collected during the course of two large-scale test programs (the 'Florida' case history and the 'California/Nevada' case history, respectively), each involving multiple test foundations.

Normally, two or more strain gages are installed in a test foundation per level, attached to the steel reinforcement. Spacing the gages symmetrically around the perimeter allows for an estimate of the strain at the centroid to be computed as an average of the individual strain measurements. One opposed pair of gages is the typical arrangement. In the Florida case history test program, the owner specified three gages per level. It was not explicitly stated, but the implied arrangement was an equal spacing of 120° around the perimeter of the pile reinforcement cage (**Figure 1**).

Strain gages installed in cast-in-place foundation elements the field have a percentage mortality rate (probability of failure), designated *λ*. This is most often due to installation procedures for deep foundations. When constructing drilled shafts,

#### **Figure 1.**

*Typical arrangement of opposed pair and triplet strain gages in pile cross section with computed average (dashed lines).*

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

reinforcement cages must be lifted by crane, rotated from horizontal to vertical and then inserted into the shaft excavation. Concreting then takes place, either via the tremie method or by gravity pour, both of which are dynamic procedures with many opportunities to damage a gage. For auger-cast-in-place (ACIP) piles, the relatively slender and flexible reinforcement cage is lifted into the air at the head only for rapid insertion into fluid grout, which induces a 90° bend into the cage as it is lifted. In the Florida case history testing program, from a total of 657 sisterbar vibrating wire strain gages installed in eleven bi-directional ACIP test piles, seventeen gages failed to function during testing, yielding a mortality rate *λ* of 2.6%.

To estimate the strain at the centroid of the foundation element, the symmetrically-arranged gages at a given level are averaged. All the gages at a given level must function in order to compute the average at the centroid. Given *k* gages at a level, the probability of success *S* in this situation is computed as the simultaneous probability of survival of all the gages:

$$\mathbf{S}\_k = (\mathbf{1} - \lambda)^k \tag{1}$$

Even though in practice if a single gage fails the remaining gage(s) are often still utilized to estimate the average strain, this is not optimal since the resulting average is now off the centroid and therefore may not be representative of the average axial load if an uneven strain distribution is present in the cross-section due to bending stress.

To evaluate the potential significance of the difference between using an opposedpair average and a single gage (assuming its opposite malfunctioned), data from a total of 207 pairs of functioning opposed gage pairs in the eleven axial test piles in the Florida case history is analyzed. A relative difference is computed for each logged reading of each opposed gage pair:

$$d = \frac{\left| \varepsilon\_1 - \varepsilon\_{\text{avg}} \right|}{\varepsilon\_{\text{avg}}} \text{ where } \varepsilon\_{\text{avg}} = \frac{\varepsilon\_1 + \varepsilon\_2}{2} \tag{2}$$

For each gage pair, the differences are averaged for all increments of loading. The resulting 207 data points are plotted on a histogram, and a log-normal probability distribution function is fitted to the resulting data (**Figure 2**).

The results of this analysis indicate that for this data set, the mean difference between data from a single gage and the average of the opposed pair is 15.3%, a significant dissimilarity. The inset figure plots the difference between individual and averaged strains as a percentage versus maximum average strain, which ranged from single digits of microstrain in gage levels near the ground surface to over 1000 microstrain in the vicinity of the bi-directional jacks. Although several of the highest individual difference values correspond to the smallest maximum strains, there is a fairly even distribution and no strong correlation to absolute values of strain, indicating the high mean difference is not confined to gage levels recording relatively little total strain (in other words, due essentially to a low signal-to-noise ratio). Obtaining a good measure of the average strain, rather than relying on an off-center result is thus crucial to computing the correct axial force.

Using Eq. (1), the surprising conclusion is reached that installing three equallyspaced gages per level (presumably for additional redundancy) actually results in a lower probability of successfully in obtaining the average strain at the pile centroid (92.4%) than by using two gages in an opposed pair (94.9%, using the numeric

#### **Figure 2.**

*Histogram and estimated probability distribution for percent difference between individual and averaged strains (inset figure – Percent difference vs. maximum average strain).*

**Figure 3.**

*Strain gage triplet averaging results with defective gage (left), and with 0°, 90°, 180° arrangement (right).*

values for this case history). This is because in either arrangement, the average strain at the centroid is successfully computed only if all the gages function, and assuming each individual gage has an equal probability of malfunction, there is a higher cumulative probability of losing one gage out of three installed than one out of two installed.

In this test program the three specified gages were installed at 0°, 90° and 180° around the rebar cage at each level (see **Figure 3**). The gage at the 90° position was logged but the data was not used in the analysis of results unless one of the other gages malfunctioned. This resulted in a slight improvement in the overall test program; five of the seventeen malfunctioning gages were at the 90° position, resulting in no negative effect on the data analysis.

Substantial redundancy is achieved by installing four strain gages per level, if they are viewed as two independent sets of opposed pairs. If all four gages function properly, then the average strain is computed from all four. However, if any one gage malfunctions, it and its opposed twin is discarded and the average is computed from the remaining opposed pair only, which should still yield a good measure of strain at

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

**Figure 4.**

*Averaging results for two opposed pairs of strain gages.*

#### **Figure 5.**

*Scatterplot of ratio of maximum A-C average to B-D average versus overall maximum strain.*

the pile centroid. Note that the gages do not have to be spaced at 90° angularly; each pair needs only to be 180° opposed (**Figure 4**).

The probability of success *S2x2* for this arrangement is computed as one minus the probability of simultaneous failure of both opposed pairs:

$$\mathbf{S}\_{2 \times 2} = \mathbf{1} - \left(\mathbf{1} - \mathbf{S}\_2\right)^2 = \mathbf{1} - \left(\mathbf{1} - \left(\mathbf{1} - \lambda\right)^2\right)^2 \tag{3}$$

For the Florida case history, using the same value *λ* of 2.6% results in a probability of success of 99.7% (up from 94.9% using two gages in a single opposed pair).

The California/Nevada case history data set consisted of a total of 488 gages in 122 functioning quartets from sixteen drilled shaft tests. By convention, the gages are designated A, B C and D, clockwise in plan around the rebar cage perimeter. The two opposed pairs are then labeled A-C and B-D, respectively. **Figure 5** plots the ratio of the maximum average strain of the A-C pair to the B-D pair versus the average of all four gages.

The average of the ratios is 1.01, indicating that in general the A-C and B-D pairs converge on the same average strain value. However, the standard deviation is 0.10, meaning on average there is a potential for approximately 10% deviation in the

measured strain (and thus computed load) using one versus two pairs of gages. Depending on the test objectives, this value may be significant enough to justify specifying four gages per level.

The purpose of embedding strain gages in a test foundation is to determine the load distribution (see below) and from it, the t-z and q-z curves. As such, there are two possible strategies to consider when deciding on the location (depth in the foundation) for each level of strain gages. The first approach will seek to identify the shear capacity of distinct soil layers in the stratigraphy. Based on a nearby (or ideally, centerline) soil boring, gages should be positioned at the interfaces between various soil strata to separately identify the capacity of each. Alternatively, if the test data is to be used as input to a finite-difference computer model such as FB-MultiPier, the gages should be positioned at an even spacing corresponding to the node spacing in the computer model. Consultation with the design engineer during the planning phase of a load test program will help identify test objectives and inform the optimal layout of strain gage levels. As a general guideline, gages should not be located closer than one element diameter to boundaries of the foundation (top, base and/or embedded loading device for bi-directional tests), in order to assure a plane-strain condition.
