Pile Cap Design Using Strut-and-Tie Strut-and- Tie Modelling 1
2
William Buswell and Vinh Dao 1 Graduate Structural Engineer, Arup 2 Lecturer, School of Civil Engineer, The University of Queensland Abstract: Pile caps are an important structural component which connect superstructure with substructure and exhibit non-flexural behaviour. As a result, strut-and-tie modelling (STM) has been increasingly accepted as the preferred design option. In this paper, existing STM techniques for pile caps are first reviewed. It is shown that current STM fails to accurately predict pile cap response under the combined actions of column axial force, shear force and bending moment. It is also shown that the strut efficiency factor in AS3600-2009 may not accurately describe the compressive stress field behaviour within within pile caps. The paper then presents a new concept using effective concrete stress blocks and concentrated reinforcement forces to describe pile cap response. The proposed concept is subsequently used to modify a set of existing pile cap design equations, in which column and pile bearing stresses are used as indicators for pile cap capacity. New coefficients are developed which enable these bearing stresses to be predicted when column moment and shear is present. Testing against existing pile cap data provides encouraging results, however further verification is still required. Keywords: Pile cap, strut-and-tie modelling, bursting, moment, bearing stress.
1
Introduction
Pile caps interface superstructure with substructure and exhibit non-flexural behaviour when loaded. STM is generally considered the most appropriate technique for their design (1). This lower bound plasticity method is a simplified application of stress field field theory. The designer is is required to approximate internal stress fields with an idealised truss, which is constructed from uniaxial concrete struts and reinforcement ties, and concrete nodal zones. The capacity of ideal, uniaxial concrete struts must be moderated for the effects of bursting forces. Bursting forces develop as compressive stress fields expand along their length into surrounding concrete. These tensile forces act across a strut, and can cause shear failure, failure, or bursting at a load which is below the axial capacity (1). This study examines pile caps which are loaded through columns only.
2
Review of Literature
Pile cap design using STM can be extremely design intensive and prone prone to variability. It is thus highly desirable to develop methods that simplify the design process. One such method is to provide a set of equations which remove the need for the designer to develop their own three dimensional STM (1 - 8). The common approach is to use conventional strut-and-tie theory to develop a simple set of equations which calculate strut and nodal zone capacities. The limitation to this approach is that these equations are only genuinely applicable to the cases for which they were derived (1). While these equations have been developed using STM theory, removing the need for the designer to create their own STM means that it is very difficult to manipulate the models to suit different problems. By simplifying the theory and trying to reduce variation between pile cap designs, the scope of application of the equations has been limited Pile cap design equations have been developed for simple or ideal cases only, and begin to progressively break down when the: Column is not square,
Piles are not circular, Pile cap is not square, STM does not form a square-based pyramid, Column is not concentrically positioned on the pile cap (1).
Models also either ignore loads other than uniform column compression, or represent them inaccurately (1). While not present in every design, column moments and shear can be transferred into a pile cap and must be considered (5).
2.1
G aps in S TM for P ile Cap A nalys is and Des ig n
Published literature generally represents column moment as a statically equivalent point load applied at an eccentricity to the column centreline (1), as illustrated in Figure 1. e M N
N
≡
C.L.
such that
C.L.
Figure 1. Replacing Moment with a Statically Equivalent Eccentric Point Load Problems with the approach include: 1. 2. 3. 4.
5.
Tensile stresses which may develop in the column reinforcement are ignored. If any column reinforcement is strained in tension, the compressive force in the column compressive zone is greater than the net column compression. Nodal zone geometry beneath the column is unclear. A statically equivalent eccentric load cannot be achieved for all combined actions that a column can support. This is due to the boundary condition which restricts available eccentricity to half of the column depth. Combined actions which cannot be represented are shaded on the column interaction diagram in Figure 2. Figure 2 was produced for the column section described in Appendix A. The net compressive load is applied in the incorrect location.
Figure 2. Typical Column Interaction Diagram Certain publications recognize that a more complex model is required if the effects of column moment are to be understood. Shirato et al. (5) propose a model which allows tension in the column to be considered. Concentrated compressive and tensile forces are applied in opposing column faces, and a centreline STM is developed to provide a load path to the piles. This concept is extended by Chantelot & Mathern (4), who propose that the concentrated forces in the column be replaced with the corresponding linear stress distribution. Regions of the column which are in compression and tension can now be identified, giving a better indication to the three-dimensional STM geometry. Each of these concepts presents improved descriptions of the load transfer between a column under moment, and a pile cap. They however fail to describe these load paths in a way which clearly defines the geometry of a nodal zone, which is loaded through a uniform block of compressive stress from the column. The geometry of this nodal zone is fundamental as it is used to define the entire STM. The reviewed publications did not investigate pile cap reaction to lateral load.
2.2
C ritique of AS 3600 S TM C ontent for Pile C ap Des ig n
AS3600 recognises that pile caps are non-flexural elements, and directs the designer to use STM. It also defines the STM methodology which shall be used. This methodology features a strut efficiency factor, which moderates strut capacity to account for bursting, as well as minimum criteria for bursting reinforcement (9). Each of these may be considered inappropriate when applying STM to pile cap design. Observations from the failure of 135 planar, non-flexural elements were used to empirically calibrate the AS3600 strut-efficiency factor (10). Unlike planar elements such as deep beams and corbels, pile caps feature large dimensions in each direction. This creates uncertainty as to whether compressive stress fields will behave similarly in each type of structure. For example, unlike planar elements, pile cap geometry provides large volumes of lowly stressed, unreinforced concrete to surround and confine any transverse strain in a strut (4). This surrounding concrete also allows the compressive stress fields to diverge in all radial directions, which also limits the maximum transverse strain in any one direction (1). Each observation suggests that the AS3600 strut efficiency factor may actually under-estimate pile cap strut capacity. However, the difference in strut behaviour between pile caps and planar elements warrants further investigation to confirm the relevance of the strut efficiency factor for pile caps. Clause 7.2.4 in AS3600 stipulates that bursting reinforcement must be detailed within a strut if the bursting force exceeds half of the strut capacity (9). As it is impractical to detail bursting reinforcement within pile caps, the acceptable capacity of a strut is halved. This will deliver excessive and inefficient pile cap design. Further consideration should be given to the relevance of this clause, in its current form, to pile caps.
3
Addressing Gaps in the Current Theory
The gaps identified in section 2 may be addressed by re-examining plasticity and stress field theories. These theories instruct the designer to follow the flow of stress through a structure. As load is transferred from superstructure to substructure via the pile cap, understanding the stress state at the base of the column, during loading conditions which relate to the current gaps, will allow a total STM approach to be developed. AS3600 assumes that linear str ain and stress distributions are present throu gh a column cross-section at ultimate capacity. Equivalent compressive stress blocks are then used to approximate the stress profile with a uniform block of compressive stress in the concrete, and concentrated compressive and tensile forces in any reinforcement (9). Linear strain and stress distributions can only be assumed in flexural regions. A non-flexural or disturbed region develops at the geometric discontinuity between column and pile cap, and propagates 1 – 1.5 times the section depth into the column (11). AS3600 permits the designer to ignore this non-flexural behaviour, and produce the entire column design using equivalent stress blocks. Equivalent stress blocks can therefore also be used to estimate the stress profile at the base of the column. Figure 3 illustrates these simplified stress profile at the base of a typical column section under different loading conditions (1). The nodal zones and concentrated tensile forces which develop within the pile cap are also shown.
Figure 3. Column Stress Profiles Figure 3 demonstrates that an increase in both column load and moment will cause larger stresses to develop within a pile cap. Moment increases stress by altering STM geometry. This is evident in Figure 3, which shows that the size of the nodal zone beneath the column reduces as moment increases. Load increases stress by channelling additional force through the existing STM. Column shear can therefore be assumed to not affect STM geometry. Existing STM elements must instead become more greatly stressed to equilibrate the new load. Figure 4 illustrates two typical centreline STMs which may be used to model pile cap responses for different magnitudes of column moment (1; 5).
Figure 4. Typical STM Geometries for Pile Caps under Column Moment
4
Modifications to the Adebar & Zhou (2) Equations
More complex STMs are required to apply the concepts developed in section 0 to model pile cap response to column moment and shear. This study therefore selected one of the simplified design approaches referred to in section 2, and developed the components necessary to expand the original scope in response to section 0. The equations developed by Adebar & Zhou (2) were selected.
4.1
Ori g inal E quations
Adebar & Zhou (2) based their pile cap design equations on the Schlaich, Schἂfer et al. (11) concept that, assuming adequate reinforcement has been detailed, the stress state within an entire disturbed region can be considered safe if the maximum bearing stress acting on that region is below a known value. Adebar & Zhou (3) found that the safe column or pile bearing stress is influenced by the confinement of the strut by surrounding concrete, and the strut aspect ratio. Experimental observations verified that these equations provide an accurate but conservative capacity prediction when applied within their original scope (2).
√ ( )
(1) (2) (3) (4) (5)
Permissible column and pile bearing pressures are limited by equation (1). Equations (2) and (3) measure the contributions of confinement and the strut aspect ratio. These strut aspect ratios are approximated using equations (4) and (5). AS3600 defines the ratio A2/A1 in clause 12.6. The definitions for each term have been extracted and discussed below. “ earing area” ( 9). For pile caps, this is equal to the area of the compressive region A1: beneath a column or above a pile. Te “larget area o te uorting urae tat i geometrially imilar to an A2: concentric with A1” (9).
4.2
S ug g ested Modifications
These suggested modifications to the original Adebar & Zhou (2) equations were developed using the concepts from section 0. They aim to provide a simple pass or fail check for a pile cap under any loading condition.
4.2.1 B earing Pres s ure f b Section 0 explained that the entire column cross-section will not be in compression if moment is present. Column bearing pressure must therefore be calculated using the area of the equivalent stress block. Figure 5 illustrates generic column bearing areas for different moments. Column bearing stress is calculated using equation (6).
Figure 5. Generic Column Bearing Areas
m
m
(6)
Figure 6 maps the change in bearing pressures along the section capacity line for the typical section described in Appendix A. The column is loaded under uniaxial moment. This demonstrates that for uniaxial moment, the greatest bearing pressure occurs during uniform compression (1).
1.0 0.8 o u . b
f /
0.6
Ultimate Capacity
u . b
f
0.4
Reduced Capacity
0.2 0.0 0.0
0.4
0.8
1.2
1.6
Mu /Muo Figure 6. Relationship between Bearing Stress and Moment for a Typical Column Section under Uniaxial Bending
4.2.2 R educed Capacity due to Column S hear Section 0 demonstrated that column shear does not affect STM geometry. The factor shown in equation (7) can therefore be applied to adjust the column bearing stress due to vertical load only, to account for any additional lateral load. This factor is the ratio between strut
compression due to vertical load and shear, and to the compression generated by vertical load only (1).
(7)
4.2.3 B earing P ress ure R atio A 2 /A 1 Section 4.2.1 explained that when a column moment is present, the dimensions of the bearing area term A1 become the dimensions of the equivalent stress blocks. Figure 7. Typical A1 and A2 ExamplesFigure 7 illustrates how A2 will vary for typical A 1 examples.
Figure 7. Typical A1 and A2 Examples
4.2.4 As pect R atio Equivalent stress block dimension should replace the column dimension when estimating the strut aspect ratio. This is shown in equations (8) and (9). As the original equations were developed for square columns, it is conservative to take the largest stress block dimension when calculating the aspect ratio (1). Further investigation may be able to justify using a less conservative value such as the theoretical thickness.
m m
(8) (9)
4.2.5 Assumptions Both column moment and shear alter the stress state within a pile cap. These modified equations were developed under the assumption that any changes do not affect the relationship between the safe bearing pressure and safe internal stress distribution, as well as the approximate relationship to the strut aspect ratio.
4.3
Testing ag ains t E xis ting E xperimental R esults
These modified pile cap design equations were tested against two, separate four-pile cap specimens, which were loaded with a column moment and shear until failure (5). Material properties, column
capacities and the column stress profiles were calculated using AS3600. A reinforcement rate of 1.65% was assumed in each face of the column, as the actual value was not reported. The modified equations correctly predicted that each pile cap had failed at the reported loading. Governing bearing stresses were however calculated as being 225% and 237% of the predicted safe bearing pressure. These predictions are significantly overconservative and suggest that the current modified equations do not yet accurately describe the pile cap behaviours. Photos showing crack patterns at failure indicate that the pile caps had achieved an internal stress distribution similar to the second STM shown in Figure 4 (5). These patterns highlighted that pile cap failure was initiated by bursting in a particular strut. A revised estimate of the aspect ratio was performed for this strut, and was inserted into the modified equations. This improved the critical bearing stress predictions to 158% and 165% of the safe bearing pressure. This improvement in accuracy suggests that the assumptions stated in section 4.2.5 may be incorrect. Further analytical or computational investigation is required to fully understand the effects that column moment and shear have on the safe column and pile bearing stress. These modified equations may then be refined in response to any findings. A comprehensive table of results is included in Appendix B.
5
Conclusion
Gaps in current STM applications for pile cap design have developed from attempts to simplify the STM process. These simplifications have taken from the designer the ability to create their own STM to suit expected loading conditions. As a result, the pile cap response to more complex column loads, such as moment and shear cannot be predicted. This paper re-examined plasticity and stress field theories to understand how pile caps behave under all loading conditions, prior to attempting to simplify the approach for future use. This permitted a simple pass or fail approach, which uses an accepted method of approximating the column stress profile, to check pile cap capacity under all loading conditions. Sufficient data is not currently available to verify the accuracy of the modified pile cap design equations that are presented in this paper. Further analytical or computational investigation is first required to fully understand the effects that a column moment has on the safe column and pile bearing stresses. The modified equations developed in this paper may then be refined to reflect any findings. While it is not suggested that these equations should remove the need for the designer to use STM, they do have the potential to provide a relatively simple pass or fail check for pile cap capacity. Pile cap geometry influences compressive stress field capacity in a way which may not be accurately predicted by AS3600. While it is expected that the current standard may actually be more conservative when applied to pile caps, further investigation is recommended.
6
References 1.
Buswell, W , “Pile a Deign Uing trut -and-Tie Moelling”, Undergraduate Thesis, The University of Queensland, 2012, Brisbane.
2.
Adebar, P. & Zhou, Z, “Deign o Dee Pile a y trut -and-Tie Moel”, ACI Structural Journal, 93(4), 1996, pp 1-12.
Adebar, P. & Zhou, Z, “Bearing trengt o omreie trut onine y Plain onrete”, ACI Structural Journal, 90(5), 1993, pp 534-541. 4. Chantelot, G. & Mathern, A., “trut-and-Tie Moelling o einore onrete Pile a”, Mater’ tei, a lmers University of Technology, 2010, G ὃteborg. 3.
5.
Shirato, M, ukui, J et al, “Ultimate ear trengt o Pile a,” Technical Memorandum of Public Works Research Institute, 39(20), 2003, pp 355-368.
6.
Souza, RA., et al., "Non-Linear Finite Element Analysis of Four-Pile Caps Supporting Columns Subjected to Generic Loading", Computers and Concrete, 4(5), 2007, pp 142-150.
7.
Souza, RA., et al., "Adaptable Strut-and-Tie Model for Design and Verification of Four-Pile Caps", ACI Structural Journal, 106(2), 2009, pp 142-150.
8.
Park, JW., et al., "Strength Predictions of Pile Caps by Strut-and-Tie Model Approach", Canadian Journal of Civil Engineering, 35(1), 2008, pp 1399-1413.
9.
tanar utralia, “onrete truture ( -9)”, tanar utralia, 9, Sydney.
10. Foster, SJ. & Malik, AR, “Ealuation o Eiieny ator Moel ue in trut -and-Tie Moeling o Nonlexural Memer”, Journal of Structural Engineering, 128(5), 2002, pp 569577. 11. Schlaich, J., Schἂfer, K et al, “Towar a onitent Deign o trutural onrete”, PCI Journal, 31(3), 1987, pp 75-149.
7 Ac,m A1 A2 bs c Cc Cs d dp D f b f b.u f b.uo f b,m f ’c Fh Fv hs ku Mu Muo M* Rh T V* 2 Φ γ
Nomenclature bearing area of a column under moment column bearing area, equal to Ac,m the largest area of the supporting surface which is geometrically similar to and concentric with A1 (9) strut width dimension of a square column column compressive force in concrete column compressive force in reinforcement column section effective depth circular pile diameter column section geometric depth bearing stress bearing stress in a column at ultimate capacity bearing stress in a column section under uniform compression bearing stress in a column under moment concrete compressive strength component of strut compression from horizontal column force component of strut compression from vertical column force height of strut a neutral axial parameter in a column at ultimate capacity ultimate moment capacity of a column moment capacity of a column section under pure bending design bending moment ratio measuring the compression in a strut due to both horizontal and vertical load, to the compression generated by vertical load only column tension force in reinforcement design shear force confinement coefficient, presented by Adebar & Zhou (2) coefficient for calculating stress in an equivalent stress block aspect ratio coefficient, presented by Adebar & Zhou (2) capacity reduction factor coefficient for calculating the depth of an equivalent stress block
Appendix A The column interaction diagram shown in Figure 2, and bearing stress relationship in Figure 6, were constructed using the typical column section described in Figure 8.
Figure 8. Typical column section
Appendix B Table 1 refers to the checks performed in section 4.3. Table 1. Results from Review of Existing Pile Cap Test Data
V* (kN) M* (kNm) Assumed column reinforcement f'c (MPa) ΦMu (kNm) 2
A1 (mm ) 2 A2 (mm ) hs/bs Safe bearing pressure (MPa) Rh Fb (MPa) Capacity Revised hs/bs estimate Revised capacity
Pile Cap C-1 681 1362
Pile Cap C-2 575 1150
1.65% each face 26.2 1389 Column 45978 413804 0.667 1.8 0.267
Pile 31416 196349 0.5 2.25 0.419
29.4 1391 Column 43761 393850 0.667 1.8 0.267
Pile 31416 196349 0.5 2.25 0.417
21.2
23.4
22.1
24.4
1 50.2 237%
18.4
1 52.8 225%
13.1
3.16
-
3.16
-
165%
158%
