Abstract
Static load tests are seen by many practitioners as the best techniques to approximate in-service conditions for deep foundation elements and to validate analytical model predictions of capacity and settlement. Full-scale static load tests are fairly expensive to implement, especially as part of a pre-construction investigation when equipment and personnel must be mobilized to site separately to specifically install the test element(s). Test elements are often instrumented with strain gages to determine the load distribution during the test. Correct installation of gages and interpretation of the resulting data is critical to properly evaluate the test results and recoup the significant investment made in conducting the test. This paper discusses several key points in the interpretation of strain gage data in deep foundation load tests.
Keywords
- strain gages
- deep foundation load testing
- tangent stiffness
- incremental rigidity
- t-z curves
1. Introduction
Static load tests are seen by many practitioners as the premier techniques to approximate in-service conditions for deep foundation elements and to validate analytical model predictions for load-bearing capacity and settlement. Full-scale static load tests are fairly expensive to implement, especially as part of a pre-construction investigation when equipment and personnel must be mobilized to site separately to specifically install the test element(s). Test elements are often instrumented with strain gages to determine the load distribution during the test. Correct installation of gages and interpretation of strain data is critical to properly evaluate the test results and recoup the significant investment made in conducting the test.
The ultimate product of a foundation test strain data analysis is often a set of curves which model the non-linear unit soil response to shear and bearing load (typically called ‘t-z’ and ‘q-z’ curves, respectively). These curves are useful to model the foundation response to load [1]. Strain gage data is utilized to compute both the shear and bearing (‘t’) and displacement (‘z’) portions of the curves.
In the first section, statistical results collected by the author in two large-scale test programs involving multiple test foundations each are analyzed to investigate the optimal positioning of strain gages in a test element. In the second section, the conversion of strain to force via the rigidity function is discussed. In the third section, use of strain data to properly calculate zone displacement is derived.
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
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
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 opposed-pair 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:
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 equally-spaced 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 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 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
For the Florida case history, using the same value
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.
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
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
The axial rigidity contribution of steel reinforcement in the composite cross-section 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, 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:
Note the constant term
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
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
To compute the force at any strain, Eq. (7) is rearranged:
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 bi-directional 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,
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
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 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
The incremental rigidity, by definition is also the slope (first derivative) of the force-strain function (Eq. (5)):
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 (
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.
Alternatively, or when dealing with highly non-linear rigidity relationships, the value of
where
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
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
If Eq. (13) is employed, using each load test increment as a discrete step the non-linear 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 high-quality 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
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.
4. Compression and displacement
To a first-order approximation, the displacement of each zone of an axially-loaded deep foundation is the measurement
For each zone, the zone strain is computed as the average of the measured strains at the top and bottom of the zone. Change in length (compression or elongation)
Zones which do not have strain gage levels both at the top and bottom, but rather are located next to boundary changes (the zone(s) adjacent to the load-application device and/or the top and bottom of the foundation element) must be evaluated differently. Depending on the situation, the one available strain gage level may be assumed to be representative of strain throughout the zone, strain data from two or more levels may be extrapolated, or strain may be estimated by correlation to extensometer telltale rod data.
The total displacement of each zone
This calculation will then result in the ‘z’ component of the t-z curve for every shear zone. Figure 12 is a schematic of the zone displacement calculation.
In certain circumstances, the elastic compression of the test foundation may be a minor contributor to the total computed displacement. However, in situations with very stiff soils or rock, and/or with slender elements with a relatively small rigidity and large strains, the elastic deformation can be a significant if not major portion of the total displacement of each shear zone. While the interpretation of data described in this section is simpler than in the previous sections, it is no less critical to constructing the t-z and q-z curves from strain data correctly.
5. Conclusions
Strain gages are critically important instruments for monitoring the performance of a deep foundation element undergoing a load test. Strain gage data must be properly analyzed in order to gain insight into the true soil-structure interaction. In order to verify design assumptions and particularly to optimize computer models, the unit capacity curves (so-called t-z and q-z curves) must be obtained from the load test. Properly analyzed strain gage data contributes to both components of these curves. However, strain gages are not load cells – the conversion of strain to load is not as straightforward as linear elastic theory may lead one to believe. Careful attention must be given to the selection, placement, monitoring and interpretation of strain gages in deep foundation load testing. The techniques described herein have been developed and successfully deployed by the author and others for over two decades to utilize strain gages in deep foundation load testing.
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