Design of Deep Pile Caps by Strut-and-Tie Models

Comparisons with results from 48 pile cap tests demonstrate that the one-way shear design provisions of the present ACI Building Code are excessively conservative for deep pile caps, and that the traditional flexural design procedures for beams and two-way slabs are unconservative for pile caps. Flexural design can best be accomplished using a simple strut-and-tie model, and test results demonstrate that the longitudinal reinforcement should be concentrated over the piles as suggested by strut-and-tie models. A simple shear design procedure is proposed in which maximum bearing stress is considered the best indicator of shear strength for deep pile caps. The maximum bearing stress that can be applied without causing splitting of compression struts within pile caps depends on the amount of confinement, as well as the aspect ratio (height-to-width) of compression struts. The influence of confinement is more gradual than suggested by the ACI Code bearing strength provisions.

The ACI Building Code procedure for the shear design of footings supported on piles (pile caps) is the same sectional approach used for footings supported on soil and for twoway slabs.The procedure involves determining the section thickness that gives a concrete contribution V c greater than the shear force applied on the code-defined critical section.While this approach is reasonable for slender footings supported on numerous piles, it is not appropriate for deep pile caps.
A change recently introduced in the ACI Building Code 1 means that the critical section for one-way shear in deep pile caps is now at the column face rather than d from the column face.This relatively small change in location of the critical section has resulted in a very significant increase in the required depth of many deep pile caps.The fact that a small change in location of the critical section has such a large consequence is a demonstration that a sectional approach is not appropriate in this case.It is also important to note that the drastic increase in the ACI Code shear requirements for deep pile caps implies that either the present method is overly conservative or that previously designed deep pile caps may be unsafe.
As the ACI Code shear design procedures are not appropriate for deep pile caps (they were not developed for that purpose), the CRSI Handbook 2 suggests an alternate one-way shear design procedure when the center of the nearest pile is within d from the column face, and an alternate twoway shear design procedure when the center of the nearest pile is within d/2 from the column face.The CRSI Handbook alternate procedures involve a critical section along the column face for both one-way and two-way shear, as well as modified expressions for the concrete contribution.
Another approach for deep pile caps is to use strut-and-tie models 3,4,5 that consider the complete flow of forces rather than the forces at any one particular section.The internal load path in cracked reinforced concrete is approximated by an idealized truss, where zones of concrete with primarily unidirectional compressive stresses are modeled by compression struts, tension ties are used to model the principal reinforcement, and the areas of concrete where strut and ties meet (referred to as nodal zones) are analogous to joints of a truss.While the concept of using a truss analogy for the flexural design of deep pile caps (i.e., determining the required amount of longitudinal reinforcement) is well known, 6,7,8 a sectional approach has invariably been used for the shear design of pile caps.
Unlike traditional design procedures, strut-and-tie models do not separate flexural and shear design; however, it may be said that the "shear design" of deep members using strutand-tie models involves limiting the concrete stresses to insure that the tension tie reinforcement yields prior to a concrete shear failure.If sufficient distributed reinforcement is provided to insure crack control, thereby allowing internal redistribution of stresses after cracking, the compressive stresses in the concrete struts should be limited depending on the biaxial strains. 4On the other hand, if little or no reinforcement is provided for crack control, the concrete tensile stresses should be limited to avoid diagonal cracking of compression struts. 5In pile caps it is usually not practical to provide sufficient distributed (horizontal and vertical)

Design of Deep Pile Caps by Strut-and-Tie Models
by Perry Adebar and Luke (Zongyu) Zhou Title no.93-S41 reinforcement to insure crack control; therefore, diagonal cracking of the compression struts should be avoided.Adebar and Zhou 9 have recently developed bearing stress limits to avoid transverse splitting in concrete compression struts confined by plain concrete, similar to the situation that occurs in pile caps.Utilizing these concrete stress limits, strutand-tie models can be used for both "flexural design" and "shear design" of deep pile caps.
In this paper the methods commonly used in North America for the design of deep pile caps are briefly reviewed.This includes the ACI Building Code with and without the recent modifications, as well as the method suggested in the CRSI Handbook.A shear design procedure for deep pile caps based on the strut-and-tie model concept is presented, and results from 48 deep pile cap tests are reviewed and compared with predictions from the different design methods.

RESEARCH SIGNIFICANCE
Deep pile caps are important structural elements that are not adequately covered by the ACI Building Code.Many pile caps are designed by design aids with rule-of-thumb procedures and what are hoped to be conservative allowable stresses, but considerable disparity exists between the various procedures.
The information presented in this paper should prove useful to the organizations who publish design aids for deep pile caps and practicing engineers who must design appropriate pile cap designs.

DESIGN METHODS ACI Building Code
The ACI Building Code (ACI-318) does not contain any provisions specifically for deep pile caps.Thus, designs are based on the procedure for slender footings that can be divided into three separate steps: 1) shear design, which involves calculating the minimum pile cap depth so that the concrete contribution to shear resistance is greater than the shear applied on the code-defined critical sections for shear; 2) flexural design, in which the usual assumptions for reinforced concrete beams are used to determine the required amount of longitudinal reinforcement at the critical section for flexure; and 3) a check of the bearing stress at the base of the column and at the top of the piles.
The special provisions for the shear design of slabs and footings (Section 11.12) requires that designers consider both one-way and two-way shear.In the 1977 and earlier editions of the ACI Code, 10 the special provisions for slabs and footings specifically stated that the critical section for oneway shear was located at a distance d from the face of the concentrated load or reaction area.In addition, Section 11.1 of the ACI Code stated that sections located less than a distance d from the face of support may be designed for the same shear as that computed at a distance d.The commentary to Section 11.1 warned that if the shear at sections between the support and a distance d differed radically from the shear at distance d, as occurs when a concentrated load is located close to the support, the critical section should be taken at the face of the support.Designers of pile caps could ignore this warning, however, since the specific statement in the code for slab and footings superseded the more general statement made in the commentary.In addition, a number of technical reports (e.g., Reference 11) described how the shear strength of deep members is much greater than the shear strength of slender members.
In the 1983 and subsequent editions of the ACI Code, the statement about the location of critical section for one-way shear was removed from the special shear provisions for slabs and footings, and the general statement about the critical section being at the face of the support when a concentrated load occurs within d from the support was moved from the commentary to the code.In addition, the commentary was modified to include a footing supported on piles as an example of when the critical section is commonly at the face of the support.The result is that designers of deep pile caps now have no choice but to take the critical section for oneway shear at the face of the column.
The ACI Building Code procedures for two-way shear have not been modified recently.The critical section remains at d/2 from the perimeter of the column regardless whether there is a concentrated load applied within the critical section.Section 15.5.3 states that any pile located inside the critical section is considered to produce no shear on the critical section and describes how to calculate the contribution from any pile that intercepts the critical section.The commentary on Section 15.5.3 contains a statement (since 1977) that when piles are located within the critical section, analysis for shear in deep flexural members, in accordance with Section 11.8, needs to be considered.Unfortunately, Section 11.8 of the ACI Code addresses only one-way shear in deep members, where the critical section is taken midway between the concentrated load and the support and the concrete contribution is increased due to deep beam action.
The ACI Building Code specifies that the critical section for moment in footings is at the face of concrete columns.The quantity of longitudinal reinforcement required at this critical section is determined by the usual procedures for reinforced concrete members, assuming plane sections remain plane and assuming that there is uniform flexural compression stresses across the entire width of the member.The designer is told to distribute the required longitudinal reinforcement uniformly across the footing (except that the short-direction reinforcement of rectangular footings must be somewhat more concentrated near the center).
According to the ACI Code, the maximum bearing strength of concrete is 0.85 f c ′, except when the supporting surface area A 2 is wider on all sides than the loaded area A 1 , the bearing strength is multiplied by but not more than 2.

CRSI Handbook
The CRSI Handbook 2 makes use of the general design procedures in the ACI Building Code for the design of pile caps, with the exception of the shear design procedures for deep pile caps.When the center of the nearest pile is within d from the column face, the CRSI Handbook suggests that the one-way shear capacity should be investigated at the face of the column (similar to recent ACI Codes), but suggests that the concrete contribution should be significantly increased to account for deep beam action.The suggested relationship for one-way shear is (1)   where w is the distance from the center of the nearest pile to the face of the column.The CRSI Handbook suggests that to include the effect of M/Vd for several piles at varying spans, the more complex ACI Code expression for V c [Eq. (11-6)] should be used.
When the center of the nearest pile is within d/2, the CRSI Handbook suggests that the two-way shear capacity should also be investigated at the perimeter of the column face (this is different than the ACI code), and again, the concrete contribution should be increased to account for deep (two-way shear) action.The suggested relationship for two-way shear is

Strut-and-tie model
The influence of a concentrated load within d from the face of the support of a member subjected to one-way shear is summarized in Fig. 1.The sectional shear force in such a member is very different depending on which side of the concentrated load the "critical section" is located on [see Fig.The "shear design" of a deep pile cap using a strut-and-tie model involves limiting the concrete stresses in compression struts and nodal zones to insure that the tension tie (longitudinal reinforcement) yields prior to any significant diagonal cracking in the plain concrete compression struts.Schlaich et al. 5 suggest that the concrete stresses within an entire disturbed region can be considered safe if the maximum bearing stress in all nodal zones is below a certain limit.Based on an analytical and experimental study of compression struts confined by plain concrete, 9 it is proposed that the maximum bearing stresses in nodal zones of deep pile caps be limited to (3a) (3b) where f c ′ and f b have units of psi.If MPa units are used, the 72 in Eq. (3a) should be replaced by 6.The parameter β accounts for confinement of the compression strut.The ratio A 2 /A 1 in Eq. (3b) is identical to that used in the ACI Code to calculate bearing strength.The parameter β accounts for the geometry of the compression strut, where the ratio h s /b s is the aspect ratio (height-to-width) of the compression strut.To calculate the maximum bearing stress for the nodal zone below a column, where two or more compression struts meet, the aspect ratio of the compression strut can be approximated as (4)   where d is the effective depth of the pile cap and c is the dimension of a square column.For a round column, the diameter may be used in place of c.To calculate the maximum bearing stress for a nodal zone above a pile, where only one compression strut is anchored, the aspect ratio of the compression strut can be approximated as (5)   where d p is the diameter of a round pile.Note that the ratio h s /b s should not be taken less than 1 (i.e., β ≥ 0).
The lower bearing stress limit of 0.6 f c ′ in Eq. ( 3) is appropriate if there is no confinement (A 2 /A 1 ≈ 1), regardless of the height of the compression strut, as well as when the compression strut is short (h s /b s ≈ 1), regardless of the amount of confinement.The upper limit of Eq. ( 3) results in similar maximum bearing strengths as the ACI Code.
The proposed strut-and-tie model approach is intended for the design of deep pile caps, not slender pile caps.As it is not always obvious whether a pile cap is slender or deep, and some pile caps may be somewhere in between, a general shear design procedure for pile caps can be accomplished by the following.First, choose the initial pile cap depth using the traditional ACI Code one-way and two-way shear design procedures.In the case of one-way shear, the critical section should be taken at d from the column face, and any pile force within the critical section should be ignored (i.e., the ACI procedure prior to 1983).Second, the nodal zone bearing stresses should be checked using Eq. ( 3).If necessary, the pile cap depth may be increased (β increased), or the pile cap dimensions may be increased to increase the confinement of the nodal zones (α increased), or else the bearing stresses may need to be reduced by increasing the column or pile dimensions.Thus, the shear strength of slender pile caps will be limited by the traditional sectional shear design procedures, while the shear strength of deep pile caps will be limited by the nodal zone bearing stress limits.

Comparison of design methods
To compare the one-way shear design procedures, Fig. 3 summarizes the relationship between the maximum column load and the width b and depth d of a two-pile cap.When the width of the pile cap is the same as the column width (b = c), the pile cap is essentially a deep beam [see Fig. 3(b)].When the width of the pile cap is increased, larger shear forces can be resisted by the increased concrete area at the critical section, and the maximum bearing stress (and hence, maximum column load) is larger as a result of increased confinement [see Fig. 3(c) and (d)].
Three different ACI Code predictions for one-way shear are given in Fig. 3.The least conservative prediction, entitled "ACI '77," is what designers of pile caps could have used prior to the 1983 edition of the ACI Building Code (any pile within d of the column face is assumed to produce no shear on the critical section); the "ACI '83" procedure is what designers must use since the 1983 edition of the ACI Code (critical section at the column face).This method gives very conservative predictions.The procedure from Section 11.8 for deep flexural members, "ACI [11.8]," gives an intermediate result.The CRSI Handbook method, in which the critical section is also at the face of the column, is much less conservative than "ACI '83" due to an enhanced concrete contribution, but it's more conservative than when the critical section is taken at d from the column face ("ACI '77").
All methods predict that when the pile cap is very deep, the maximum column load is limited by bearing strength (indicated by the horizontal lines in Fig. 3).When the pile cap is twice as wide as the column (b = 2c), the ACI Code predicts that confinement is sufficient so that the bearing strength has reached the upper limit of 2 × 0.85 f c ′ = 1.7 f c ′. Results from numerous bearing strength tests and the procedure proposed Fig. 4 compares the influence of pile cap depth on twoway shear strength predictions for a typical four-pile cap.Although the CRSI Handbook expression gives a considerably larger concrete contribution for deep pile caps than the ACI Code, the maximum column load is always smaller than the ACI Code method.This is because in the ACI Code method, the critical section is at d/2 from the column face and any pile that intercepts the critical section is assumed to transmit part of the load directly to the column.For example, if a pile is centered on the critical section, only half of the pile reaction must be resisted by the critical section according to the ACI Code method.It is interesting to note that as the CRSI Handbook method suggests that the ACI Code procedures be used until the center of the nearest pile is at d/2 from the column face, there is an abrupt reduction in maximum column load at that point (d = 22 in. in Fig. 4).This problem can be corrected by applying the CRSI Handbook procedure when the face of the pile is within d/2 of the column face so that none of the pile shear bypasses the critical section; the result is shown by the dashed line in Fig. 4.
The proposed method, which combines the "ACI '77" procedure for pile caps with smaller depths (slender pile caps) with the more conservative bearing stress limit in Eq. ( 3) gives a very reasonable result.Note that for the particular example shown in Fig. 4, the pile bearing stress is slightly more critical than column bearing stress.That is, according to the proposed method, the confinement around the pile is not sufficient to reach the maximum bearing stress limit.

EXPERIMENTAL RESULTS
The first results from tests on pile caps were reported by Hobbs and Stein 13 who tested numerous small-scale models of two-pile caps.In all cases, the simulated column and piles were the same width as the "pile cap," so the models were really wide deep beams.The models had various amounts of either straight or curved nondeformed reinforcing bars that were anchored by a number of different methods.Shear failure occurred when a diagonal crack formed between the column and a pile.
Deutsch and Walker 14 tested four full-scale two-pile cap specimens.The objective of the tests was to investigate the influence of pile cap depth and the amount of reinforcing steel.Specimens were stronger than anticipated, and two of the specimens did not fail.All pile caps behaved similarly with one main vertical (flexural) crack forming at midspan.
Blévot and Frémy 7 tested two series of pile caps.The first series consisted of 94 models at about half-scale, while the second series consisted of 22 approximately full-scale specimens (eight four-pile caps, eight three-pile caps, and six two-pile caps).The main objective of the tests was to determine the influence of pile cap depth and longitudinal reinforcement layout.The longitudinal reinforcement was either concentrated over the piles, as suggested by a truss model, or distributed in a uniform orthogonal grid, as required by the ACI Code.
Bunching the longitudinal reinforcement resulted in higher capacities (for a given quantity of steel), even though some parts of the specimens had poor crack control.Distributing an equal amount of reinforcement in a uniform grid resulted in the four-pile caps being 20 percent weaker and the threepile caps being 50 percent weaker.The capacities were not significantly influenced by whether the bunched reinforcement was provided around the perimeter of the pile cap or diagonally across the pile cap; however, the best crack control under service loads occurred when a combination of the two was used.
Clarke 8 tested 15 four-pile caps, all approximately halfscale.The longitudinal reinforcement layout and anchorage were the parameters studied.Similar to Blévot and Frémy, the reinforcement was either bunched over the piles or distributed in a uniform grid.In the study, "nominal anchorage" involved extending the longitudinal reinforcement just beyond the piles, while "full anchorage" meant providing a 90deg hook and extending the longitudinal reinforcement to the top of the pile cap.
The behavior of all pile caps was similar.Vertical cracks formed near the center of the pile cap sides, extending to near the top of the pile caps.Prior to failure, the pile caps had usually split into four separate pieces hinged below the column base.According to the author, most specimens failed in "shear" after the longitudinal reinforcement yielded.The author also classified the failure modes as either one-way (beam) shear or two-way (punching) shear, depending on the appearance of the failed specimen.Bunching the reinforcement over the piles resulted in a 14 percent increase in capacity compared to spreading the reinforcement uniformly.The so-called "full anchorage" resulted in approximately a 30 percent increase in capacity.
Sabnis and Gogate 15 tested nine very small ( 1 / 5 ) scale models of four-pile caps to study how the quantity of uniformly distributed longitudinal reinforcement influences the shear capacity of deep pile caps.Similar to Clarke, 8 the longitudinal reinforcement was hooked and extended to the top surface.The tests showed that varying the reinforcement ratio between 0.0014 and 0.012 had little influence on the shear capacities of the models; however, no details were given about how artificial restraint was eliminated at the base of the simulated piles.
Adebar, Kuchma, and Collins 16 tested six full-scale pile caps (five four-pile caps and one six-pile cap).The largest specimen weighed more than 7 ton (6.4 tonne).All pile caps were statically indeterminate (piles in four-pile caps were arranged in a diamond shape), and the actual pile loads were measured throughout the test.Sliding bearings were used under the pseudo-piles to simulate the lateral flexibility of piles.External and internal strain measurements taken during the tests demonstrated that the behavior of pile caps is very different from two-way slabs.Plane sections do not remain plane, and strut action is the predominant mechanism of shear resistance.Deep pile caps deform very little before failure and thus, have virtually no ability to redistribute pile loads.
Strain gages in two of the specimens indicated that the longitudinal reinforcement had definitely yielded prior to failure; however, the failure mode still looked very much like a "shear failure" because the plain concrete in the pile caps had very little ductility.The authors believed that true shear failures (prior to steel yielding) were a result of compression struts splitting longitudinally.Depending on the geometry of the pile cap, the final failure mechanism resembled either a one-way or two-way shear failure.The maximum bearing stress in specimens that failed in shear varied from 1.13 to 1.27 f c ′.

COMPARATIVE STUDY
Table 1 summarizes the properties of 48 pile cap specimens that are used in the comparative study.Specimens not considered include the small wide-beam models tested by Hobbs and Stein, the small-scale specimens (first series) tested by Blévot and Frémy, and the two specimens tested by Deutsch and Walker that did not fail.
Table 2 summarizes the details of the ACI Code and CRSI Handbook predictions.In the case of one-way shear, three different predictions are given from the ACI Building Code: 1) the 1977 edition of the ACI Building Code (critical section at d from the column face); 2) the 1983 ACI Building Code (critical section at the column face); and 3) the special provisions for deep flexural members (Section 11.8 of the ACI Code).Table 3 presents the ratio of measured pile cap capacity to predicted capacity for the three ACI Code predictions, as well as the CRSI Handbook prediction.The predicted failure mode and reported failure mode are also given.It is interesting to note that many pile caps predicted to fail in flexure were reported to have failed in shear.As previously mentioned, the likely reason for this is that pile caps are large blocks of plain concrete that do not have the ductility to un- dergo significant flexural deformations without triggering a shear failure.Table 4 summarizes the predictions 17 from the proposed strut-and-tie model and compares the predictions with the ex-Table 3-Comparison of ACI Code and CRSI Handbook predictions: ratio of measured capacity to predicted capacity and failure mode* perimental results.The "shear" capacity is the maximum column load limited by the nodal zone bearing stresses given by Eq. (3), while the "flexural" capacity is the maximum column load limited by yielding of the longitudinal reinforcement.The flexural capacity depends strongly on the inclination of the compression strut that is defined by the location of the nodal zones.The lower nodal zones were located at the center of the piles at the level of the longitudinal reinforcement, while the upper nodal zones were assumed to be at the top surface of the pile cap at the column quarter points.compression is concentrated near the compression face is inappropriate.Assuming the flexural compression is uniform across the entire pile cap, which strain measurements have shown to be incorrect, 16 leads to a further overprediction of the flexural capacity.
While the proposed strut-and-tie method gives the least amount of scatter between experimental results and predictions, the amount of scatter is nonetheless still relatively high (COV = 28 percent).This can be explained by the fact that the shear failure of pile caps involves a tension failure of the concrete.It is the author's opinion that a further refinement of the design procedure to reduce this scatter is not warranted.The most important issue is that the proposed design method is simple, rational, and conservative, and unlike the other design methods, it does not overpredict any of the pile cap test results.

SUMMARY AND CONCLUSIONS
Recent editions of the ACI Building Code require that the critical section for one-way shear be taken at the support face if a concentrated load exists within d from the support.While this is appropriate for heavily reinforced deep beams (Fig. 1), where a shear failure may occur due to diagonal crushing of concrete, it is excessively conservative for pile caps [Fig.5(b)], which do not fail as a result of diagonal compression.The more appropriate one-way shear design procedure for pile caps in the 1977 and earlier editions of the ACI Building Code results in two-way shear and flexure being more critical for most pile caps (except for two-pile caps) [Fig.5(a)].
The ACI Building Code procedure for two-way shear involves a critical section at d/2 from the face of the column, and any pile reaction within d/2 from the column face does not produce shear on the critical section.This results in an "infinite" two-way shear capacity for some deep pile caps (Table 2).The CRSI Handbook suggests an alternate twoway shear design procedure for deep pile caps, where the critical section is at the column face.Since the critical section must resist much larger shear forces, the concrete contribution is greatly enhanced to account for deep two-way action.While the sectional shear resistance is larger according to the CRSI Handbook method, the maximum column load is usually smaller than the ACI Code method, where a significant portion of the column load does not produce shear on the critical section.
The CRSI Handbook suggests an upper limit of 32 for the shear stress on two-way critical sections in very deep members and others 18 have suggested reducing this limit to 24.Neither suggestion is based on any experimental results; however, an upper limit is actually not needed since the maximum load that can be applied to very deep pile caps is always limited by bearing stress at either the base of the column or the top of the piles (see Fig. 3).
In this paper a simple rational design method for deep pile caps is proposed in which the maximum bearing stress is considered a better indicator of shear strength than the "shear stress" on any prescribed critical section.In deep pile caps the shear stress is concentrated in zones (compression struts) between the column and piles, and is not uniform over the height, which makes it difficult to calculate a meaningful shear stress.The procedure suggested herein is based on the premise proposed by Schlaich et al. 5 that an entire D-region of a concrete structure can be considered safe if the maximum bearing stress is maintained below a certain limit.
Based on a study of idealized compression struts confined by plain concrete, 9 Eq.( 3) is proposed for the maximum bearing stress in pile caps.The maximum bearing stress is a function of confinement (similar to the ACI Code), as well as the aspect ratio (height-to-width) of the compression struts that transmit shear between the column and piles.The influence of confinement is much more gradual in the proposed relationship than in the ACI Code procedure (i.e., more confinement is needed before reaching the maximum bearing stress).
A general shear design procedure for all pile caps (deep or slender) can be accomplished by combining the ACI Code shear design procedure with the maximum bearing stress limit of Eq. ( 3); the more critical one controls.As the bearing stress limit will always control the "shear strength" of very deep pile caps, the shear force from any pile within the critical section (d or d/2) can be ignored with confidence.
Comparisons with experimental results indicate that the traditional flexural design procedures for beams and twoway slabs are unconservative for deep pile caps [Fig.5(a)].The flexural compressive stresses within pile caps are concentrated near the column (not spread uniformly across the section), and pile caps are large blocks of plain concrete that cannot undergo significant flexural deformations without triggering brittle shear failure.A more appropriate flexural design procedure for deep pile caps can be achieved by using strut-and-tie models.Reasonably conservative designs are obtained [Fig.5(e)] when the upper nodal zones are located on the top surface of the pile cap at c/4 from the column center.Previous experimental results have demonstrated that concentrating the longitudinal reinforcement over the piles, as suggested by strut-and-tie models, results in considerably higher flexural capacities compared to when the longitudinal reinforcement is distributed in a uniform grid; however, some of the longitudinal reinforcement should be uniformly distributed to help control cracking.
The method proposed in this paper for the design of deep pile caps has been implemented in the 1995 CPCA Concrete Design Handbook. 19The pile cap design tables were developed using the method proposed herein, and a number of examples are provided to show how to apply the method in manual calculations.

( 2 )
where b o equals 4 × c for a square column of dimension c.As the critical section is at the perimeter of the column, the CRSI two-way shear strength equation is much more sensitive to the dimensions of the column compared to the ACI approach, where the critical section is at d/2 from the column perimeter [b o equals 4 × (c + d)].The term (1 + d/c) in the CRSI equation is a factor that compensates for this difference.
1(b)].The truss model shown in Fig. 1(d) indicates that the concentrated load is transmitted directly to the support by a compression strut.No stirrups are required to resist the "shear" created by the concentrated load [see Fig. 1(f)].The concentrated load does, however, increase the diagonal compression stresses in the concrete immediately above the support [see Fig. 1(e)], as well as the required tension force in the longitudinal reinforcement at the face of the support [see Fig. 1(g)].Fig. 2 depicts a simple three-dimensional strut-and-tie model for a four-pile cap.The concentrated column load is transmitted directly to the support by inclined compression struts.Horizontal tension ties (longitudinal reinforcement) are required to prevent the piles from being spread apart.

Fig. 1 -
Fig. 1-Truss model for simply supported beam with concentrated load close to support: (a) geometry and loading; (b) sectional shear forces; (c) sectional bending moments; (d) truss model; (e) discontinuous stress field; (f) required stirrup resistance per unit length of beam; (g) required longitudinal reinforcement (adapted from Marti 3 )

Fig. 3 -Fig. 4 -
Fig. 3-Comparison of one-way shear design methods for two-pile caps with f c ′ = 25 MPa: (a) plan view of pile cap; (b) to (d) influence of pile cap depth on column load for various pile cap widths (1 in.= 25.4 mm; 1 kip = 4.45 kN)

Fig. 5
Fig. 5-Ratio of experimentally measured-to-predicted pile cap capacities from: (a) 1977 ACI Building Code (critical section for one-way shear at d from column face); (b) 1983 ACI Building Code (critical section for one-way shear at column face); (c) ACI Building Code special provisions for deep flexural members; (d) CRSI Handbook; (e) proposed strut-and-tie model