Abstract
When designing pile groups subjected to lateral loading, modeling and analysis of the connections between piles and the overlying pile caps require careful consideration. For a given pier foundation, characterization of the pile-to-cap connections typically corresponds to one of: pinned (no moment transfer); fixed (moment transfer up to pile rotational capacity); or, partially fixed pile head conditions. The present study aims to provide bridge engineers with practical resources for modeling laterally loaded pile group behaviors, in association with partially fixed pile head conditions. Modeling strategies are presented for both relatively simple (i.e., highly idealized) configurations, as well as for an array of conceivable pier foundations. In particular, an analytical approach is presented, which is suitable for preliminary design of configurations analyzed in a linear elastic manner, and on a representative-pile basis. For nonlinear analysis of pile groups (including soil–structure interaction), a search algorithm is developed that converges on a given target level of partial pile head fixity and associated pile group response (e.g., group lateral displacements, pile head moments). The search algorithm is utilized to investigate 20 unique pier foundation configurations under various lateral load intensities and across target levels of partial fixity. Pairing of the computed foundation responses with the levels of partial fixity, in the ensemble, is intended to serve as a resource to bridge engineers when making initial estimates to model partially fixed pile head conditions.
Keywords
The ability of pier foundations to transmit moments between the topmost pile portions (pile heads) and immediately overlying pile caps is dependent on factors such as the length to which piles extend into the pile caps. As indicated in Wilson et al. ( 1 ), if a pile extends only a short distance (relative to width) into the pile cap, then the pile-to-cap interface can be idealized for design purposes as incapable of transmitting moment (i.e., pinned). In contrast, if a pile is embedded two to three diameters ( 2 )—or even four diameters ( 3 )—into the pile cap, then the interface can be idealized as capable of transmitting moments equal to or less than the pile moment capacity (i.e., fixed). In practice, scenarios may arise where the pile embedment into the pile cap lies somewhere between fewer than one and two (or more) pile diameters. For such scenarios, piles may behave in a manner that is somewhere between the idealizations of pinned and fixed conditions.
When designing pile groups to resist lateral loads, representation of pile head fixity conditions can significantly affect computed responses. For example, assuming fixed-head conditions during design can lead to underestimation of pile head deflections ( 4 ). Further, computed lateral deflections of pile groups obtained under fixed-head conditions afford, on average, deflection magnitudes that are approximately 25% as large as those obtained under pinned-head conditions ( 5 ). Although accounting for pinned or fixed pile head conditions in design applications is relatively straightforward (e.g., through use of design-oriented finite element analysis [FEA] software), additional considerations are necessary to account for modeling of those foundation configurations that are anticipated to exhibit partial fixity.
Background
The American Association of State Highway and Transportation Officials (AASHTO) LRFD specifications for bridge design ( 6 ) require that deep foundation members be embedded some (unspecified) distance into overlying pile caps, and furthermore, that the extent of embedment (the degree of fixity) be considered in design. Quantitative requirements concerning minimum pile embedment lengths into pile caps are provided in state-specific design manuals, such as Pennsylvania Department of Transportation ( 7 ). The need for additional modeling considerations specific to pile head conditions arises for configurations that fall near to minimum prescribed pile embedment lengths into overlying pile caps.
For lateral load analysis of configurations where partial fixity conditions are applicable, only limited standardized modeling strategies are available. For example, per guidance in Pennsylvania Department of Transportation ( 7 ), to model pile head fixity at 50% (relative to fixed-head conditions), the pile configuration is first analyzed under fixed-head conditions and the computed pile head moments are then cataloged. Subsequently, fixed-head moments at one-half the magnitude of the cataloged moments are applied (as opposing moments) and the configuration is re-analyzed. Although standardized reaction-based approaches undoubtedly possess utility, such approaches may become cumbersome when many load combinations are required for design.
Experimental investigations previously conducted on laterally loaded pile groups ( 8 ) found that currently used methods for calculating moment-transfer capacities of embedded piles can under-predict the effect of embedment. In addition, a previous study ( 9 ) of detailed elastic and inelastic finite element (FE) models of pile groups subjected to pushover loading analyses, and validated against physical load-test measurements, revealed that the relationship between the extent of pile (into cap) embedment lengths and the corresponding pile group displacements resulting from lateral loading is intrinsically nonlinear. The present study is motivated by: (1) shortcomings identified in previous studies concerning presently available strategies for modeling partial fixity conditions; and (2) given that the relationship between the extent of pile head fixity to lateral response is not trivial to discern (e.g., linear).
Objective and Scope
Investigated in the present study are partial fixity behaviors of laterally loaded pile groups, where the study objective is to establish practical modeling strategies. To increase robustness relative to limitations identified in previous studies, such as Richards et al. ( 8 ), the strategies presented here rely on scaling of the rotational stiffnesses allotted to the tops of piles within numerical models, relative to the rotational stiffnesses associated with fixed-head conditions. An analytical approach is developed for use in preliminary design of relatively simple configurations (viz., configurations amenable to modeling via a single, representative pile). A more broadly applicable iterative approach (i.e., search algorithm) is also developed for use in conjunction with use of design-oriented FEA software. The iterative approach facilitates quantification of target-level partial pile head fixities for laterally loaded pile groups when inclusion of nonlinear structural behaviors and soil–structure interaction (SSI) phenomena is necessary.
A parametric study is conducted by employing the search algorithm and performing nonlinear lateral load analyses of 20 different pier foundation configurations. Each of the 20 models is subjected (one at a time) to multiple magnitudes of lateral load (10%, 25%, and 50% of the respective pushover load magnitudes). In turn, responses are computed at several levels of partial pile head fixity (25%, 50%, and 75% of the fixity associated with fixed-head conditions). Computed foundation responses are then paired with the various levels of partial fixity, where the collected pairings facilitate selection of initial estimates by practicing engineers when designing pier configurations that possess partially fixed pile head conditions.
Analytical Approach for Modeling Partial Pile Head Fixity in Simple Configurations
The following pertains to pile groups that exhibit partial fixity and are amenable to being modeled using a representative pile (Figure 1a). Further, the analytical approach is limited to scenarios where fixed-head conditions are associated with a finite, maximum rotational stiffness at the pile head (i.e., the approach cannot be used for scenarios where rotation is fully restrained at the pile head). Consider a pile (with equivalent length, L) that contains gross cross-section properties (elastic modulus, E; moment of inertia, I) and is assumed to behave in a linear elastic manner. Also, the pile base is fixed and the pile head is assigned a rotational spring (with stiffness, kr). The schematic in Figure 1a is highly idealized and is applicable to scenarios where an equivalent point of fixity ( 10 – 12 ) is judged to be suitable (in contrast to hand methods that more directly include simplified representations of lateral soil resistance such as Davisson [ 13 ] and Norris [ 14 ]). Even so, the highly idealized analytical approach—with focus given to pile head rotational stiffness—serves to inform the more generalized (nonlinear, computational) approach presented later.
Analytical approach for idealized (linear elastic, representative, point-of-fixity) piles subjected to lateral loads: (a) idealized, representative pile, (b) laterally loaded pile under fixed-head conditions, (c) laterally loaded pile under partial fixity conditions, (d) demonstration case, and (e) semi-log plot of Equation 5 using demonstration-case parameters.
Under application of a lateral load (F) acting at the pile head, the pile undergoes lateral translation and rotation until equilibrating reactions form, such as the moment in the pile head rotational spring, Mtop and the moment at the fixed base, Mbot. Routine application of a hand method on this highly idealized system, such as direct stiffness, produces an expression for the moment in the pile-head rotational spring:
By statics, the moment at the fixed base of the pile is:
The ratio of the top and bottom moments, η, can then be expressed as:
For scenarios where the rotational spring stiffness can be estimated under fixed-head conditions, defined here as kfh, Equation 3 can be evaluated with substitution of kfh for kr. In this way, a moment ratio (ηfh) is established, which in this context is the moment ratio associated with fixed-head conditions (Figure 1b). (Note that if the fixed-head rotational stiffness is not amenable to being estimated, then force-based methods such as that described in Pennsylvania Department of Transportation [ 7 ] may prove useful.) By rearranging the terms in Equation 3, the fixed-head rotational stiffness (kfh) is given as:
As the final step in this derivation, given a desired level of partial fixity (Figure 1c), the corresponding rotational spring stiffness (kpartial) can be evaluated by introducing Λ into Equation 4:
where Λ is a decimal-fraction measure (from 0 to 1, exclusive) of proximity to fixed-head conditions. Pinned-head conditions are converged on as Λ approaches zero; fixed-head conditions are converged on as Λ approaches unity. Within the scope of applicability for the configuration shown in Figure 1a, and because the above formulation operates on stiffness within the linear elastic system, said approach is robust to the magnitude of the lateral load applied to the pile head.
As demonstration of the analytical approach, consider an idealized pile configuration, with model parameters indicated in Figure 1d. Here, purely for demonstration, kfh is set equal to 2.44E+06 kN-m/rad such that ηfih is approximately equal to unity. Also for this demonstration, the pile head rotational spring stiffness (kpartial) needed to achieve pile head conditions halfway between pinned-head and fixed-head conditions (i.e., 50% fixity) is determined using Equation 5 (Figure 1e). The corresponding magnitude of kpartial—2.20E+04 kN-m/rad at 50% fixity—is less than 1% of kfh. As an extension of the demonstration case, listed in Table 1 are the pile head rotational stiffness (kpartial) values that, in turn, correspond to lateral load responses 25%, 50%, and 75% of the way toward fixed-head conditions (i.e., Equation 5 is evaluated, one at a time, using Λ equal to 0.25, 0.50, and 0.75).
Pile Model Parameters for Demonstration of the Analytical Approach
Observations in Relation to the Analytical Approach
Owing to the nonlinearity of Equation 5, and by examining values of kpartial needed to achieve a desired level of partial fixity in comparison to kfh, it is apparent that introduction of a relatively small proportion of rotational stiffness at the pile head (in relation to modeling) brings about considerable rotational resistance. Alternatively stated, for numerical modeling of simple configurations, incorporation of relatively small fractions of pile rotational stiffness (relative to fixed-head conditions) can lead to computed responses that rapidly converge on those of fixed-head conditions.
Although the analytical formulation presented above may prove useful for configurations that are amenable to significant idealization, additional considerations are necessary when modeling nonlinear structural behavior and SSI phenomena. Accordingly, focus is given in the remainder of the present study to establishing practical resources for bridge designers to employ when modeling partial pile-head fixity and incorporation of nonlinear phenomena is of interest.
Computational Approach for Modeling Partial Pile Head Fixity in Pier Foundations
In current design practice, nonlinear behaviors of laterally loaded pier foundations often need to be incorporated into modeling and analysis procedures. Capturing such behaviors (structural, SSI) extends beyond the scope of the analytical approach presented above. Discussed in the following is a more broadly applicable, iterative (search) algorithm. This latter approach can be used to quantify a key modeling parameter for pier foundations such that partial fixities associated with target response levels are achieved when performing nonlinear analysis of said foundation systems under lateral loads. More specifically, the computational approach consists of iteratively scaling the rotational stiffness terms at pile heads throughout a pier foundation FE model, relative to fixed-head conditions, until a target response under lateral loading is computed (along with the associated level of partial fixity). Note that a prerequisite for use of the computational approach is that the target level of partial fixity must be known (e.g., based on physical aspects of the configuration such as length of pile embedment into the pile cap). By operating on pile head rotational stiffness, the overall approach is amenable to use within design-oriented bridge FEA software, and furthermore, attains increased robustness over force-based approaches (which hold greater dependence to a single load case). For these purposes, the term pile head rotational stiffness (PHRS) factor is utilized, and here defined as: the ratio of pile-head rotational stiffness of piles within a given configuration (e.g., when partial fixity applies), relative to the pile-head rotational stiffness that would be attributed to piles of said configuration under fixed-head conditions. This definition is applicable when the domain of interest for modeling partial fixity lies between pinned-head and fixed-head conditions.
Search Algorithm
The search algorithm is delineated in Figure 2 and can be carried out given any pier FE model subjected to lateral loading, assuming that the pile head rotational stiffnesses can be manipulated. Additional prerequisites include: a specified demand type (e.g., group lateral displacements; maximum pile head moments); the desired proximity of response between pinned-head and fixed-head conditions (0% corresponds to pinned-head; 100% to fixed-head). Two search parameters are also required: allowable percent error with respect to converging on the PHRS factor, and the maximum number of search attempts.
Search algorithm to determine pile head rotational stiffness (PHRS) factors for modeling of partial pile head fixity conditions in laterally loaded pier foundations.
The first major step in the algorithm (Figure 2) is to establish a target magnitude of computed response. That is, lateral load analyses are performed on the pier, separately, under pinned-head (PHRS = 0) and fixed-head (PHRS = 1) conditions. Demands (of the specified type) are then cataloged. Next, given desired proximity of response and the cataloged demand magnitudes, the target response magnitude is calculated in proportion to the respective pinned-head and fixed-head response magnitudes.
Subsequently, the search portion of the algorithm is carried out. Here, the PHRS factor is iterated on using a binary search technique. In particular, iterations continue until (if successful) the computed response of the laterally loaded pier model is found to contain demands that are commensurate with the target response magnitude. If the number of search attempts performed exceeds a preset limit (10 was the limit set for all scenarios considered in the current study) without converging to a solution then the search algorithm exits, not having converged on the desired PHRS factor. Discussed later are the results from a parametric study, where the search algorithm is employed and this limit (a maximum of 10 attempts) is demonstrated to be sufficiently large across all combinations of loadings and configurations considered. To initiate the iterative process, a search interval is initialized with lower and upper bound PHRS factors, respectively, set to 0 (pinned-head) and 1 (fixed-head). For example, in the first iteration, given the search interval [0, 1], the midpoint PHRS factor is 0.5. Accordingly, in the pier FE model, the pile head rotational stiffnesses are scaled by a factor of 0.5 (as emphasis, relative to the stiffnesses computed under fixed-head conditions). The pier configuration (with PHRS = 0.5) is then analyzed under lateral loading.
For the specified demand type of interest, the difference between the magnitudes of computed (current iteration) demand and target demand is then calculated and compared with the allowable percent error for convergence of the PHRS factor. If the calculated difference is less than the allowable percent error, then the algorithm terminates (i.e., the PHRS factor has been converged on). Otherwise (Figure 2, lower right) the demand computed for the current iteration is compared with the target demand. If the current-iteration demand is larger than the target demand, then the upper bound of the search interval is set equal to the PHRS factor used in the current iteration. If the opposite is true, then the lower bound of the search interval is set equal to the current-iteration PHRS factor. Regardless of which interval bound is updated, the next iteration is undertaken (i.e., another round of bisection is carried out using the updated interval) unless the maximum number of search attempts have occurred (Figure 2, bottom).
Demonstration Case
As demonstration, consider the pier FE model presented in Figure 3. The column in this hammerhead configuration is rectangular and tapers from 2.4 m by 2.4 m at the base to 1.5 m by 3.7 m at the top. Also, the pier cap tapers from base dimensions of 1.5 m by 2.7 m to tip dimensions of 1.5 m by 1.2 m. Framing together the hammerhead and the underlying foundation members is a 7.6 m square pile cap, which is 1.8 m thick. Foundation members supporting the hammerhead pier consist of four 1.70 m-diameter drilled shafts, which are partially encased. Each shaft extends 19.4 m from the pile cap and down through a layered profile of cohesionless, cohesive, and limestone materials. All pier member reinforcement consists of mild steel (yield stress of 410 MPa) and all pier members contain concrete with a compressive strength of 34 MPa.
Pier configuration used for demonstration of the search algorithm.
Modeling and nonlinear analysis of the pier configuration is carried out using a research version of the design-oriented bridge FEA software, FB-MultiPier version 5.8.1, and soil resistance parameters are estimated using tables given in the software manual ( 15 ). The software tool makes use of the beam on nonlinear Winkler foundation approach for modeling SSI. Use of the software has previously been shown it to be capable of producing computed lateral load responses that agree with physical testing of deep foundation members in a laboratory setting (centrifuge testing of drilled shafts embedded in limestone, Taghavi et al. [ 16 ]). Furthermore, the software has been validated against measurements from full-scale pushover tests performed on a variety of pile types embedded in layered profiles of soil and rock media ( 17 ).
Under fixed-head conditions, the configuration for the demonstration case is found to fail at 1068 kN of lateral load—applied at the mid-depth of the pile cap, Figure 3. Purely for demonstration purposes, a lateral load of 534 kN (half of the pushover load) is utilized. As context, a pier lateral displacement (the average lateral displacement of the shaft heads) of 80.3 mm is computed under pinned-head conditions, whereas 25.4 mm is computed under fixed-head conditions.
The objective of this demonstration case is to employ the search algorithm (Figure 2) and quantify PHRS factors applicable to the pier shown in Figure 3 under three conceivable states of partial fixity: 25%, 50%, and 75%, relative to fixed-head conditions. The demand type of interest is selected as the group lateral displacement, and so, the three states of partial fixity, respectively, correspond to lateral displacements of 66.6 mm, 52.9 mm, and 39.1 mm. (Note that if pile head moments are instead selected as the demand type of interest, then target magnitudes of displacement response, as just listed, would only be approximations.) The algorithm is shown to produce PHRS factors that correspond to target-level shaft-head displacements, and correspondingly, desired proximities of response to fixed-head conditions (i.e., away from pinned-head conditions, toward fixed-head conditions). Also, for this demonstration case, the allowable percent error for search convergence is set equal to 1%, and the maximum number of search attempts (iterations) is set to 10. This process is repeated a total of three times, once for each of three target percentages (25%, 50%, and 75%) of group lateral displacements relative to that computed under fixed-head conditions. Results obtained from the three usages of the search algorithm (Figure 2) are plotted along with respective percent errors in Figure 4.
Demonstration case results: pile head rotational stiffness (PHRS) factors versus group lateral displacements.
For each of the three scenarios making up the demonstration case, the search algorithm successfully quantifies appropriate values of PHRS factors to within 1% error in fewer than 10 search attempts. Of note, the relationship between PHRS factors and group lateral displacements (Figure 4) is clearly nonlinear, where such nonlinearity is consistent with Duncan et al. ( 5 ). Further, the presence of nonlinearity in this context evidences that relatively small increases in pile head rotational stiffness (relative to fixed-head conditions) rapidly brings about computed responses that converge on those of fixed-head conditions. For example, consider when a partial fixity of 50% (halfway between pinned-head and fixed-head conditions) is desired for the pier shown in Figure 3. When the pier is subjected to a lateral load equal to half of the pushover load, per Figure 4, the pile head rotational stiffness supplied to the model only needs to be approximately 4% as large as that associated with fixed-head conditions (i.e., PHRS≈ 0.04).
Parametric Study: Setup
A parametric study is carried out to characterize typical ranges of PHRS factors, where such characterization is intended to aid in preliminary design of laterally loaded pile groups that are anticipated to exhibit partial fixity conditions. Nonlinear lateral load responses are computed (via FB-MultiPier v5.8.1) using FE models of 20 pier and pile group systems, the configurations of which were adapted from real-world structures. For each configuration, three unique load lateral levels are considered. In turn, for each loading, PHRS factors are quantified as corresponding, respectively, to three target reaction magnitudes (viz., group lateral displacements). In total, the parametric study comprises 180 scenarios, each carrying several lateral load simulations.
Bridge Pier Foundations Considered
Descriptions of the foundation configurations selected for study are listed in Table 2 along with associated model numbers (1–20). Correspondingly, structural member layouts and soil layerings are depicted in Figure 5. When available, both pier column and foundation member data are indicated.
Bridge Pier and Foundation Configuration Data
Note: na = not applicable.
Pier foundations analyzed (not to relative scale).
Foundations included in the parametric study entail a mixture of four mudline, 10 waterline, and six above-waterline footings (Table 2). Alternatively stated, pile free lengths vary from 0 m (for mudline footings) up to 9.5 m. Pier column data are available for 10 of the foundations, including hammerhead and two-column configurations of varying dimensions. Half (ten) of the model configurations make use of drilled shaft foundation members, with diameters ranging from 0.18 m to 2.74 m. Eight models contain prestressed concrete piles—including, collectively, both square and circular cross-sections—of widths (or diameters) from 0.61 m to 1.37 m. Two models possess steel H-piles for foundation members. The number of deep foundation members across the 20-model array vary from two (i.e., large drilled shafts) up to more than 40 members (e.g., H-piles).
Lateral Loads Considered
To inform the lateral loading scheme adopted for the parametric study, pushover analyses are first carried out for each of the 20 pier foundation model under fixed-head conditions. The ranges of lateral loads imparted to the pier FE models in the parametric study are then set proportional (per configuration) to the respective maximum lateral load that can be equilibrated by each nonlinear pier FE model. Namely, and to accommodate conditions being investigated that are less than those of fixed-head conditions, three intensities of lateral loading are considered for each foundation model: 10%, 25%, and 50% of the pushover load associated with fixed-head conditions. To further contextualize the array of foundations investigated, the lateral load magnitudes and corresponding group lateral displacements (averaged across the pile, or shaft, heads of a given footing) are listed in Table 3. For all analyses conducted, the lateral loads are applied head-on at the midplanes of the pile caps.
Applied Lateral Loads and Corresponding Group Displacements Computed Under Pinned-Head and Fixed-Head Conditions
A visual scan of Table 3 reveals the range of lateral loads that are included in the parametric set of analyses, where applied lateral loads (collectively) vary on the order 100 kN to more than 10,000 kN. Correspondingly, computed lateral displacements, approximately, range from 1 mm up to 500 mm.
Parametric Study: Results
Equipped with the loading scheme listed above in Table 3, the search algorithm (Figure 2) is utilized to identify PHRS factors for each of the 20 models. In particular, three levels of partial fixity are investigated for each model: 25%, 50%, and 75% proximity to fixed-head conditions (based on assessments of lateral group displacements). Further, three lateral load intensities are investigated for each state of partial fixity: 10%, 25%, and 50% of the foundation-specific pushover capacity. As aforementioned, 180 scenarios are investigated in total, including quantification of the PHRS factors. For brevity, selected results associated with five cases are listed in each of Table 4 (for driven piles) and Table 5 (for drilled shafts).
Computed Pile Head Rotational Stiffness (PHRS) Factors Corresponding to 25%, 50%, and 75% Proximity to Fixed-Head Conditions (With Respect to Lateral Displacements, Relative to Fixed-Head Conditions) for Five Finite Element Models That Contain Driven Piles
Computed Pile Head Rotational Stiffness (PHRS) Factors Corresponding to 25%, 50%, and 75% Proximity to Fixed-Head Conditions (With Respect to Lateral Displacements, Relative to Fixed-Head Conditions) for Five Finite Element Models That Contain Drilled Shafts
As a narrative illustration of the interrelations between the foundation configurations considered and the respective response quantities obtained under lateral loading, consider Model 1 (Figure 5, upper-left). The average pile head displacement for Model 1, for example, at 10% of the pushover load (2800 kN, Table 3, row 1) is computed as 70.7 mm (Table 3, row 1) when analyzed under pinned-head conditions and 19.2 mm (Table 3, row 1) when analyzed at the same load level under fixed-head conditions. The lateral pile head displacement corresponding to 50% of the way toward fixed-head conditions is approximated as the average of 70.7 mm and 19.2 mm, which is 45.0 mm. Making use of the search algorithm (Figure 2), it is determined that the PHRS factor required to obtain a lateral pile head displacement of 45.0 mm, and thus achieve 50% proximity to fixed-head conditions, is 0.027 (Table 4, row 1). As discussed later, repeating this tabulation process over the collective body of results obtained from the parametric study (180 scenarios) allows for identification of trends in values of PHRS factors that may, in turn, approximately correspond to achievement of a given, desired partial fixity state.
Search Algorithm Performance
In relation to performance of the search algorithm, use of the procedure (delineated in Figure 2) is demonstrated to successfully find values of PHRS factors to within 1% error for each of the 180 scenarios considered. No more than 10 iterations of the search algorithm are found to be necessary; more specifically, five or fewer iterations are typically required. In addition, the wall-clock time required for the search algorithm to complete (with use of FB-MultiPier v5.8.1 on an ordinary laptop) is typically less than 4 min (the maximum wall-clock time required is 7 min). Given the typical duration over which the search algorithm reached convergence across the 180 scenarios, no further optimization of the algorithm is undertaken. Note though that additional considerations are necessary if further reductions in wall-clock time are desired (e.g., use of a parallel solver; selecting a more deliberate initial value of PHRS when first entering the search loop, recall Figure 2). Furthermore, if highly nonlinear phenomena such as pile buckling are anticipated to be prominent for a pier foundation under a given lateral (combined with axial) loading, then increases in the algorithm control parameters may become necessary (e.g., increases in error tolerance; increases in the maximum search attempts). The full collections of computed PHRS factors, packaged into box plots, is presented in Figure 6.
Computed pile head rotational stiffness (PHRS) factors: (a) 10% of pushover load, (b) 25% of pushover load, and (c) 50% of pushover load.
Comparison of Partial Fixity Behaviors for Driven Pile and Drilled Shaft Foundations
Concerning selected results associated with configurations supported by driven piles (Table 4) versus those specific to drilled shafts (Table 5), mean values of the computed PHRS factors are markedly similar in magnitude. For example, for the selected results presented in both Table 4 (driven piles) and Table 5 (drilled shafts), mean values of computed PHRS factors range approximately 0.02 (at 10% of pushover loads) to 0.09 (at 50% of pushover loads). Even so, dispersions (standard deviations) in computed PHRS factors among the selected driven pile results (0.02 to 0.07) are approximately twice as large as those attributed to drilled shaft foundations (0.01 to 0.04). As an additional difference of note, maximum values of computed PHRS factors tend to be of greater magnitude for the driven pile foundations (0.06 to 0.23) versus those of drilled shaft foundations (0.04 to 0.15).
Trends and Dispersions among Computed PHRS Factors
As can be generally observed among the parametric study results (Figure 6), PHRS factors obtained from usage of the search algorithm tend to increase both with respect to loading intensity and as the target percentage of the fixed-head reaction increases. For the range of scenarios considered in the parametric study, no significant correlations were identified between the dimensions of the pile caps (e.g., thickness) and the computed values of PHRS factors. (Note: more detailed modeling approaches, and even physical testing, may be warranted to explore such correlations.) However, increases in computed PHRS factors associated with load magnitudes at 10% (Figure 6a) to 25% (Figure 6b) of the pushover loads are more pronounced than those associated with 25% (Figure 6b) versus 50% (Figure 6c) of the model-specific pushover loads. Differences in relative increases in pile head rotational stiffness are consistent with prior observations of the nonlinearity associated with characterization of partially fixed pile head conditions ( 18 ).
Despite the broad differences present within the set of pier foundations examined (recall Figure 5), relatively low dispersions are observed in the ensembles of computed PHRS factors. More specifically, and independent of both lateral load intensity and target state of partial fixity, the interquartile ranges (IQRs) of PHRS factors throughout Figure 6 span across approximately 0.1 or less. Such relatively low dispersions in computed PHRS factors—given the overall interval of [0, 1]—indicate that the box plots of Figure 6 may be of use when estimating partial fixity modeling parameters in preliminary design applications where nonlinear phenomena are pertinent. However, if a foundation is being analyzed under lateral loading, and said foundation configuration deviates from the 20 configurations considered here, then additional iterations may be necessary to arrive at an appropriate PHRS factor (relative to the observable trends in Figure 6).
Typical Magnitudes of Computed PHRS Factors
The collective set of computed PHRS factors, as presented in Figure 6, indicate yet again that incorporation of relatively small proportions of pile head rotational stiffness are necessary (relative to fixed-head conditions) to bring about computed responses, in turn, that increasingly approximate fixed-head conditions. For example, the maximum value from among all IQRs plotted in Figure 6 is approximately 0.15 (at 75% proximity to fixed-head conditions, Figure 6a). Stated more plainly, 15% (or less) of the pile head stiffness associated with fixed-head conditions is typically necessary to bring about a computed (nonlinear) lateral load response that is 75% of the way toward fixed-head conditions. Recall that a similar phenomenon is found (albeit more pronounced) as part of the previously discussed analytical approach (Figure 1). These outcomes are particularly of interest given findings from previously conducted field-test studies. For example, as found in Rollins and Stenlund ( 19 ): many pile–pile cap connections that are assumed to be pinned, because of the shallowness of the connection, in fact behave as fixed connections in the field—an experimental finding that is at odds with what analysis software often predicts.
Resource for Preliminary Design
Table 6 is presented as an aid to practicing engineers when making initial selections of PHRS factors for the purpose of modeling partial pile head fixity, and for modeling scenarios where manipulation of pile head rotational stiffness is feasible. Specifically, summary statistics of the computed PHRS factors are listed at 25%, 50%, and 75% proximities to fixed-head conditions. As emphasis, these percentages express behaviors between (away from) pinned-head conditions and (toward) fixed-head conditions (e.g., in this context, 25% signifies that the computed lateral displacement is 25% of the way toward the displacement computed under fixed-head conditions, relative to the displacement computed under pinned-head conditions). As listed, mean values of the PHRS factors range from approximately 0.02 up to 0.1, with respective standard deviations of (approximately) 0.01 to 0.07. To give an example, if a pier configuration is anticipated to exhibit partial fixity halfway between pinned-head and fixed-head conditions (i.e., 50% fixity), then an initial estimate of the PHRS factor is approximately 0.05 per Table 6. However, the value of the PHRS factor that may actually bring about the target level of partial fixity may fall between approximately 0.01 and 0.08. Nonetheless, the values listed in Table 6 may greatly reduce the iterations needed to achieve the desired level of partial fixity for a given pier FE model subjected to lateral loading (including nonlinear behaviors). Note that the listings in Table 6 are based on selection of lateral displacements as the demand type of interest. If other demand types (e.g., pile head moments) are of particular interest, then the listings in the table should be utilized more as starting-point approximations for modeling partial fixity conditions. Alternatively stated, use of the search algorithm is recommended for design scenarios beyond preliminary design, and where pile head moments constitute the demand type of interest.
Summary Statistics of Computed Pile Head Rotational Stiffness (PHRS) Factors Corresponding to 25%, 50%, and 75% Proximity to Fixed-Head Conditions (With Respect to Lateral Displacements, Relative to Fixed-Head Conditions)
Conclusions
Designing pier foundations to resist lateral loads necessitates considerations for the extent to which rotational resistance can be developed at pile–pile cap interfaces. When modeling pier foundations subjected to lateral loading, the ability of the topmost portions of piles to develop moment reactions is commonly idealized as pinned (incapable of transmitting moment) or fixed (capable of transmitting moment up to pile capacity). However, design scenarios can arise where the embedment lengths of piles (or shafts) into overlying pile caps are not sufficient to incorporate one of pinned or fixed conditions into associated numerical models.
The present study was undertaken to provide bridge engineers with insights into the computed responses of laterally loaded pier foundation models when partial fixity conditions apply. As emphasis, the findings summarized below are limited to analytical and computational aspects of laterally loaded pier foundation design, where additional research is necessary in relation to distinctly physical aspects of pile-to-pile-cap configurations (e.g., correlations between relative, physical embedment length of piles into caps and the resulting foundation behaviors under lateral loading). Modeling strategies in this vein were developed on two fronts. First, an analytical derivation was presented and demonstrated as a convenient method for practicing engineers when modeling partially fixed pile head conditions. The scope of the analytical approach pertained to preliminary design of those foundations that are amenable to considerable idealization. In particular, the scope included linear elastic behaviors of representative piles, modeled using equivalent point-of-fixity methods, and fitted with pile-head rotational springs. Demonstrated use of the analytical approach revealed that only a relatively small fraction of the rotational stiffness associated with fixed-head conditions was necessary to deviate substantially away from pinned-head conditions, and toward fixed-head conditions.
A second approach was also investigated to establish a more broadly applicable algorithm for modeling partial fixity behaviors of laterally loaded pier foundations. Further, this latter approach was developed to be robust to capturing (computed) nonlinear behaviors such as those associated with structural members and SSI. Given the presence of nonlinear phenomena in the underlying lateral load analyses, occurring at various stages in the algorithm, an iterative (search-based) procedure was implemented, with use of a binary search technique. For a given demand type of interest (displacements, moments), the algorithm was also constructed to be amenable to use in conjunction with design-oriented FEA software. Key outputs of the search algorithm included relative, quantitative measures of pile head rotational stiffnesses associated with desired levels of partial fixity.
The search algorithm was demonstrated to be a robust tool for carrying out a parametric study, which in turn involved 20 FE models of unique pier foundations subjected to lateral loading. The collection of configurations modeled allowed for inclusion of considerable variations in pile free lengths (e.g., mudline, waterline footings), numbers and types of foundation members (piles, shafts). A total of 180 scenarios comprised the parametric study, which consisted of subjecting the 20 pier models to three unique lateral load intensities for each of three conceivable levels of partial fixity (relative to fixed-head conditions).
Outcomes from the parametric study included quantitative characterization of relative pile head rotational stiffnesses. Consistent with findings from demonstration of the analytical approach, it was found that relatively small proportions of pile head rotational stiffness (relative to fixed-head conditions) can bring about increasingly close-proximity response to fixed-head conditions. The as-quantified relative stiffnesses were then distilled down to summary listings of typical values that may be of use in preliminary designs involving partial fixity conditions. In particular, pairings of the computed foundation responses with desired levels of partial fixity were produced. These pairings were formulated to constitute a resource to bridge engineers when making initial estimates for modeling pile head rotational stiffnesses, as part of routine modeling activities when partial fixity is anticipated to be present at a pile–pile cap interface. For example, if modeling a laterally loaded pier foundation that is consistent with the 20 configurations studied here, then use of the quantitative ranges of pile head rotational stiffness factors recommended above may aid in bringing about the desired (partial) pile head fixity conditions. More broadly, findings from this study better equip practicing engineers with modeling strategies for addressing partially fixed pile head conditions in laterally loaded pier foundations.
Footnotes
Author Contributions
The authors confirm paper contributions as follows: study conception and design: H. Bollmann; M. Davidson; data collection: A. Shishlov; analysis and interpretation of results: A. Shishlov; M. Davidson; H. Bollmann; draft manuscript preparation: A. Shishlov; M. Davidson; H. Bollmann. All authors reviewed the results and approved the final manuscript.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.