Three-dimensional analysis of two-pile caps Análise numérica tridimensional de blocos sobre duas

© 2012 IBRACON This paper compares the results between a non-linear three-dimensional numerical analysis of pile caps with two piles and the experimental study conducted by Delalibera. It is verified the load-carrying capacity, the crack pattern distribution, the principal stress in concrete and steel, the deflection and the fracture of the pile cap. The numerical analysis is executed with the finite-element software ATENA 3D, considering a perfect bond between concrete and steel. The numerical and experimental results are presented and have demonstrated a good approximation, reasserting the results of the experimental model and corroborating the theory.


Introduction
The primary function of pile caps is to transfer superstructure forces to the infrastructure elements.In the last decades there was a significant progress in the study of the structural behavior of pile caps.With the introduction of computing systems and the Finite Element Method (FEM) new refined techniques of analysis were developed.The current theory of pile caps is due to the original work devel-oped by Blévot & Frémy [3], whose authors published a study which came to be a reference for all posterior works and served as the framework for the formulation of most of the national codes in this field up to date.Three-dimensional analysis of two-pile caps not submitted to flexure.The principal design guidelines are provided by an array of codes such as the Canadian National Code (CSA Standard A23.3-94), the Australian National Code (AS3600-1994), the New Zealand National Code (NZS3101: Part2: 1995) and the FIB Model Code 1990.Despite that, each code has its own materials and loading safety factors and different design methodologies.In particular the NBR-6118: 2007 [11] only mentions the preference for tridimensional strut and tie models in spite of linear and non-tridimensional models.

Justification
Researches have been evolving to a consensus that the struts and tie model is assumed to be the most accurate method to represent the structural behavior of pile caps.Notwithstanding, there still remains a dissent in the literature, for example, in the subject of the configuration of compressive struts and distribution of stresses inside the specimen.Delalibera [1] affirms that "there is a lack of knowledge on the geometrical shape of compressive stresses that forms the struts in pile caps submitted to axial and eccentric loads" and "numerical analysis of rigid pile caps have demonstrated that the distribution of stress in the struts on the surface of piles is not uniform, demanding an adaptation to the chosen hypothesis".Therefore, this paper has the main objective to verify the behavior of reinforced concrete pile caps with two piles by comparing, on one side, Delalibera's [1] experimental results and, on the other side, non-linear numerical analysis results.Experimental and numerical results comparison has the merit to expose discrepancies and convergences between them, and, hence, validates the finite elements method software analysis.Moreover, numerical analysis contributes to the corroboration of the pile caps theoretical fundaments.The crack pattern is analyzed in this paper, including initial crack formation in Stadium II and its propagation within the structure, besides stress and strain distribution in pile caps and steel bars.In addition, the load carrying capacity and pile caps collapse by concrete splitting and compressive struts crushing in the nodal zones is verified.

Analysis Method
A numerical analysis of five pile caps with two piles is developed, as described on Table 1.For that, the finite element software AT-ENA 3D [2], from Cervenka Consulting, was used.The pile caps perpendicular tension stresses within the structure) with the formation of several cracks before the collapse.In relation to anchorage, Blévot & Frémy [3] proved that the reinforcing bar slipping of ridged steel bars without hooks only occurs after the strut crushes.Mautoni [4] noted that most pile caps were subjected to a fragile collapse by struts splitting in nodal zones.Before the structural collapse the cracks ran parallel to the struts.This fact was also noted in experimental tests conducted by Clarke [5] and Sabnis & Gogate [6].Studying the strut and tie model, Adebar et.al. [7] proved that the collapse of pile caps is due to the concrete splitting with the expansion of compressive stresses (concrete crushing and increase of cracks) and the subsequent yielding of ties rebars.Delalibera & Giongo [8] demonstrated the formation of cracks parallel to the struts in pile cap models.The authors verify that pile caps collapse by concrete splitting and crushing in the superior nodal zone (C-C-C), on the interface between the column and the pile cap, and in inferior nodal zones (C-C-T), on the interface between the piles and the pile cap.Moreover, the authors support that in the design of pile caps the column's forces are equally divided in two halves at the cross-section between the column and the pile cap.Reinforcing bars adherence analysis conducted by Delalibera [1] demonstrated that there is no tie steel bar slipping due to struts compressive stresses favorable contribution.This reduces tensile stress values and significantly diminishes reinforcing bar strains in inferior nodal zones.According to Souza et. al. [9] it is important to mention that there is not yet a general procedure accepted to pile caps design.Despite the existence of several design procedures, there is still a large difference among them.Most of national codes recommend deep beams, bending beams or truss models for the design pile caps.Notwithstanding, Souza et.al. [9] demonstrated that many pile caps designed to flexural collapse presented a fragile rupture by shear.The authors also attested that pile caps are submitted to a complex tridimensional non-linear strain distribution nominated as D-region.In general, D-regions are developed due to static order (caused by load actions) and geometric disturbances (caused by abrupt geometric changes).Particularly in the case of pile caps, all the structure behaves as a D-region due to the concentration of stresses both in the superior and inferior sections caused by the intersection between the column and the pile cap, and between the pile and the pile cap.It is worth mentioning that in the last decades, according to Su & Chandler [10], struts and tie models have been one of the most popular and rational methods of structural analysis of structures  3. (1) For steel bars, an elastic-perfectly plastic behavior is adopted.Steel bar materials properties are listed in Table 4.The yielding criterion is based in von Mises definitions.Finally for steel plates pile supports an isotropic elastic material was defined as shown in Table 5.In the model 1, all vertical displacements at the base of the piles were restrained (Table 1, model 1).In the models 2 and 3 vertical displacements are restrained respectively on 50% and 25% of the base of the piles, as shown in Table 1.The reduction of piles supports (vertical displacements restrain) has the main purpose to study its influence on pile caps stiffness.Model 4 characteristics are equal to model 1 except for splitting steel bar addition as shown in Table 1.The objective of this reinforcement is to observe its contribution to pile caps load bearing capacity.Another aspect analyzed is the concentration of the struts compressive stress in the cross-section on the surfaces of the piles.High compressive stress concentration was observed on cross-section region of the pile closer to the column.Compressive stresses were more intensive in the beginning of inferior nodal zone, being very low in the end of the inferior nodal zone.To detect this phenomenon and observe the behavior of specimens (compressive stress flow) a pile cap with the pile's cross-section width reduced was modeled (Table 1, model 5).

Computational software
Numerical analysis was carried out with computational software ATENA 3D [2].The software's basic concept operation is based on finite element theory and non-linear analysis of reinforced concrete structures.The software simulates the behavior of real structures using either linear or non-linear analysis.Maximum load is obtained by increments of time force integration, applying Arc-Length and Newton-Rhapson methods.To determine concrete strain structural behavior it is used the Lagrangian or Euler's formulations.

Materials properties
A plastic fracture model is adopted for concrete as described by Cervenka [2] and shown in Table 3. Concrete principal characteristics are presented in Table 2. Three-dimensional analysis of two-pile caps (3) Where: R ct, min : minimum tensile force; h f : vertical dimension -strut and tie model (Delalibera [1]); h y : column cross-section; f ctk, inf : inferior value of concrete characteristic tensile resistance; L est : piles span; A x : pile dimension in the considered direction.

Analysis method
The Newton-Rhapson analysis method was adopted with a concentrated load at the center of the column's superior cross-section and force increments of 25 kN.Added to this, an hexahedral finite elements mesh was adopted to pile caps, piles and columns as shown in Figure 5.And a tetrahedral finite elements mesh was adopted for the steel plates.
Absolute (100%) and partial (50% and 25%) vertical movement restraints were imposed to piles base supports.At piles-pile cap and column-pile cap contact surfaces a finite elements 3D interface was used based on Mohr-Coulomb criterion.Their properties are presented in Table 6.

Divergences between numerical and experimental models
One of the most discrepant structural behaviors observed between numerical and experimental models was in the stiffness pile caps, which was much higher in numerical models.This fact demonstrates the inherent complexity of laboratory experiments.Delalibera [1] points out three main reasons for this stiffness difference, which are prototype accommodation at the beginning of the experiment, perfect bond assumption between steel bars and concrete in numerical models and perfect connection between piles and pile cap.
In reference to the first reason, [1] cites pile cap accommodation at the beginning of the experiment, which was verified in load versus displacement curve.About the second reason, the author did not confirm this hypothesis after the preliminary tests.And in respect to the third reason, [1] affirms that this is probably what has mainly collaborated to the augment in pile cap's stiffness, since a detachment between piles-pile cap interface occurred.Therefore, [1] suggests the use of interface finite elements on the contact surfaces of the structural elements.
Following the recommendation of [1], all numerical pile caps were modeled using a finite elements interface between column-pile cap contact and piles-pile cap contacts.Moreover, the supporting area of the piles was reduced in order to observe pile caps stiffness behavior.
The reduction of the supporting area of the piles enabled an increase in pile caps strains and displacements.This fact is in accor-   [12] observations who demonstrated, through computational analysis, that "pile caps structural behavior is highly influenced by piles type of support and by pile cap stiffness".

A resume of Delalibera [1] experimental results
Delalibera's [1] results demonstrated that pile caps resist until the beginning of concrete crushing, when a fracture plane along the struts is started due to shear forces action.Ruin occurred by concrete crushing in nodal zones and by pile caps splitting along the compressive struts.In most cases, concrete collapse occurred before reinforcement yielding.The strains on the ties were not constant over the reinforcing bars.A significant stress reduction has taken place in the inferior nodal zones.And the strains at the ends of the steel bars of the ties were close to zero, despite the existence of anchorage hooks.In the model with splitting reinforcing bars (steel bars disposed perpendicularly in relation of struts and with the purpose to absorb tensile stresses and to resist to concrete splitting) proposed by [1], an increase in pile caps resistance was observed.This model presented intensive strains in struts cross-section.All experimental models presented a similar structural behavior, with initial cracking beginning in the inferior nodal zone, at the pilepile cap interface, and propagating up to the superior nodal zone, at the column-pile cap interface.Cracks were propagated over the struts forming a clear rupture plane.From principal compressive stress, [1] detected a greater concentration of stresses at the column-pile cap contact surface and in the piles region located at the beginning of inferior nodal zone.

Results and discussion
of the numerical models

Crack pattern
In all numerical models, first cracks appeared in the inferior nodal zone at the piles-pile cap contact surface, and propagated along the compressive struts up to the superior nodal zone.During load increments a wide range of cracks parallel to the struts developed, as shown in Figures 6 and 7.
In models 2 and 3, with piles supports reduction, an augment of cracking intensity occurred during the loading.The formation of cracks was observed cracks in the base of the piles and in its adjacencies due to geometric eccentricity.This eccentricity generated a rotation in the axis of the piles and consequently a region with stress concentration as it can be noted in Figures 7 (a

Three-dimensional analysis of two-pile caps
In model 5, with reduction of pile's cross-section area, the results were very similar to those observed in model 1.
In general, a good approximation between experimental and numerical results was observed, as it can be verified in the results presented in Table 7.

Stress-flow
In all numerical models prismatic compressive struts were formed.At the bottom surface of the column stress flow was divided equally in two halves, proving [1] statement that it is correct to consider that half pile cap-column interface receives half of the column's forces.Besides, all compressive stresses have

Three-dimensional analysis of two-pile caps
propagated up to piles surfaces, forming compressive struts and concentrating at the piles cross-section closest to the column, as shown in Figures 8 and 9.
Maximum compressive stresses, presented in Table 8, occurred in the intersection between column-pile cap and piles-pile cap.This indicates the collapse of pile caps in the nodal zones region, similarly to [1] specimen rupture.
In models 2 and 3, the eccentricity provoked by piles supports area reduction, which resulted in an augment in structure's displacement, caused struts stress expansion towards the inferior nodal zone.This resulted in a redistribution of pile cap stresses at the piles surface.
In model 3, struts stress flow became to concentrate in the pile's surface that is closer to the pile caps border, as shown in Figure 9 (b) In model 5, presented in Figure 9 (d), struts compressive stresses were distributed in all piles cross-section area.Notwithstanding, maximum stresses were similar in value to model 1, as shown in Table 8, proving that in model 1 piles cross-section surfaces were only partially solicited by struts compressive stresses.
Tensile stresses perpendicular to compressive struts were also formed, characterizing concrete splitting, which is indicated by tensile vectors perpendicular to compressive vectors shown in details in

Ultimate load and ruin
In all models a fragile collapse occurred with compressive struts formation and concrete splitting and crushing.Also intensive cracks were observed.Ultimate load in model 1 was 1900 kN.Comparing to the experimental model the results were very similar, as demonstrated in Table 9.
In model 3, with supports in only 25% of the pile's basis, the principal reinforcing bars of the ties yielded.There was also a reduction in pile cap load carrying capacity.In the other models, yielding occurred only after the model's ruin.
In model 4, with splitting reinforcing bars, an increase in pile cap ultimate load capacity was observed, as shown in Table 9.In addition, no significant variation was observed in pile cap's ruin, stiffness and bearing capacity patterns.The numerical results are similar to those obtained by [1] in experimental tests with pile caps with splitting reinforcement where an augment in pile cap's ultimate load was obtained.
In model 5, despite piles cross-section area reduction, no significant difference was observed in pile cap bearing capacity.Ultimate load capacity of the numerical model was 1825 kN, which is close to the experimental model.The results restate that piles of numerical model 1 are only partially solicited, occurring compressive stress concentration in the pile's cross-section area that are closer to the column (in the beginning of the inferior nodal zone).

Reinforcement strains and stresses
Steel bars, except in model 3, did not yield until the pile caps ruin.Moreover, the ties steel bar stresses were not uniform, T.E.T. BUTTIGNOL | L.C ALMEIDA occurring significant reduction in the inferior nodal zones due to struts compressive effect, as shown in Figure 13.The reduction of the supporting area of the piles caused an increase on pile cap's inferior surface strains due to the rotation of the piles axis.Consequently, the ties steel bar tensile stresses augmented.
Horizontal stirrups have absorbed part of the tensile stresses in the struts region.In model 3 stresses in the superior longitudinal steel bars achieved 360 MPa, as shown in Figure 13 (c).Besides, in model 3, due to stress concentration near the piles supports, an increase in piles stirrups stresses located near the supports was observed, as shown in Figure 13 (c).Splitting reinforcement of the model 4 absorbed part of pile cap's stresses, proving to be effective against pile caps tensile stress, as shown in Figure 14.In Table 10  In model 4, with splitting reinforcing bars, intensive stresses in the ties steel bars at the beginning of the inferior nodal zone were observed.However, a considerable reduction in steel bar stresses occurred along the nodal zone.Very low stress values rose at the steel bar ends, as shown in Figure 18.
In model 5, despite the reduction of the piles cross-section area, zero strain value occurred at the end of the ties steel bars, as shown in Figure 19.

Stiffness and bearing capacity
In relation to the experimental specimen, model 1 was more rigid, presenting lower displacements.This was a result of the vertical restraints imposed to the piles in the numerical models, which has produced a clamping effect, limitating the rotation of the piles and the displacements of the pile caps.
The reduction of the pile's supporting area caused a decrease in the pile caps structural stiffness, leading to a growing convergence of the force versus displacement curve among the models 1, 2 and 3.In model 3, as shown in Figure 20, the force versus displacement curve overlapped with the experimental model's curve which is already adjusted to expunge displacements due to specimen accommodation in the first load stages.This proves that in numerical models the supports of the piles directly affect the stiffness of structural elements.At the same time, load bearing capacity was not significantly changed.Both numerical and experimental models achieved ultimate load capacity with very similar load intensity and crack pattern.
In addition, as the supporting area of the piles was reduced, there was a shift in the maximum plastic strain location.In Figure 21 it is possible to notice plastic strain shift from the inferior nodal zones to the superior nodal zones.If this reduction in the pile caps supporting area, on one side, allowed increased structural displacements, on the other side, it generated a high compressive zone in the pile cap's superior nodal zone.In general, there was a reduction up to 30% in stresses in the piles-pile cap contact surfaces with an increase up to 28% in the stresses in the column-pile cap's contact surface, as shown in Table 8.In model 4, splitting reinforcement contributed to load bearing capacity and to the increase in the pile cap resistance.

Three-dimensional analysis of two-pile caps
In model 5, with the reduction in the cross-section area of the piles, stiffness and load bearing capacity were similar to model 1.

Conclusions
The results of the numerical models showed that the reduction in the piles supporting area has a direct influence in the pile caps load bearing capacity and a non-negligible influence in pile caps stiffness.
In model 1, vertical displacement restraints in all pile's basis generated a clamping effect that lead to structural stiffening.Despite piles confining effect due to soil reaction along the piles, this fact is not observed in pile caps.Pile caps are not tensile structures.Therefore small displacements of the piles could cause great influence in pile caps stiffness.Therefore, this could become a critical question and deserves considerable attention in laboratory tests and numerical analysis.

Acknowledgements
To FEC-UNICAMP for the provision of the necessary tools to develop this work.And to CAPES for its support to the first author's participation in the 52º Brazilian Concrete Congress which resulted in an article publication and presentation which originated this paper.

T
Based on experimental results, Blévot & Frémy [3] formulated a theory on struts and ties to explain the structural behavior of pile caps.The authors observed that the collapse of the majority of pile caps specimens have occurred by concrete splitting (opening cracks parallel to principal compressive stresses as a result of 253 IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 2

Figure 1
Figure 1 shows the geometric characteristics of the pile caps.The distribution of pile caps is presented in Figures 1, 2, 3 and 4. For the pile cap with splitting reinforcing bars (model 4) two 16mm steel bars were used passing perpendicularly through the struts, following Delalibera's [1] recommendations.The reinforcing bars were designed according to requirements proposed by [1], due to equations 2, 3 and 4.
from Delalibera's [1] experimental model nº B35P25E25e0.The numerical models maintained the shape of the structural elements, the distribution of the reinforcing bars and the materials properties.
) and 7 (b).In model 4, splitting reinforcing bars effectively contributed to pile cap's cracking control.A reduction in crack opening intensity also occurred.

Figure 1 -Figure 2 -
Figure 1 -Details of pile caps geometry and reinforcement bars (stirrups and ties)

Figure 6 -Figure 7 -
Figure 6 -Crack evolution in model 1 -(a) at the beginning of load steps; (b) at the middle of load steps; (c) at the collapse (ATENA)

Figure 8 .
Splitting steel bars of model 4 absorbed partially these tensile stresses, contributing to pile caps resistance increase, as predicted by Delalibera [1].A significant reduction in tensile stress was also observed on the pile cap's inferior surface (between piles) as shown in Figure12(a).These results confirm splitting steel bars favorable action to pile caps tensile and shear resistance increase.In the other models, tensile stress in the pile cap inferior surface was similar, as shown inFigures 10,11 and 12 (a and b).

Figure 10 -Figure 11 -
Figure 10 -Details of model 1 tensile stresses [MPA] at the bottom of pile cap (ATENA)

Figure 15 -Figure 18 -
Figure 15 -Anchorage length and ties stresses [MPa] in the inferior nodal zone of model 1

Table 2 -
Concrete properties

Table 5 -
Steel plates properties s 210 GPa T.E.T. BUTTIGNOL | L.C ALMEIDA dance with Ramos maximum strain and stress values of both numerical analysis and experimental models are presented.The results demonstrated a good approximation between them.In Figures 15 to 19 the pile caps inferior nodal zones are shown in details, demonstrating the ties anchorage region (l anc ) and the ties stress graphic values.In all models (Figures15 to 19) strains at the ties steel bar ends were very low.Notwithstanding, as the piles supporting area was being reduced, there was a progressive increase in the area of the nodal zones.An increase of the stresses in the ties steel bar at the end of the inferior nodal zone also occurred with repercussion in the steel bar ends as can be seen in Figures 15, 16 and 17.

Table 8 -
Maximum stress values (MPa) on ultimate load of numerical models and Delalibera [1] experimental specimen

Table 9 -
Ultimate load of numerical models and Delalibera [1] experimental specimen

Table 10 -
Stress and strain values on ultimate load of numerical models and Delalibera [1] experimental specimen