Application of strut and tie concepts to prestressed concrete bridge joints in seismic regions

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Application of Strut-and-Tie

Concepts to Presressed Concrete Bridge Joints in Seismic Regions Sri Sritharan, Ph.D Assistant Professor Department of Civil and Construction Engineering lowa State University Ames, lowa Jason M Ingham, Ph.D Senior Lecturer Department of Civil and Environmental Engineering University of Auckland Auckland, New Zealand

Over the past decade, comprehensive experimental and analytical studies on cap beam-to-column concrete bridge joints have been

conducted, with an emphasis on joint force transfer mechanisms based on strut-and-tie concepts Using the findings from these

studies, which focused on improving both detailing and seismic performance, a treatment for designing and assessing bridge joints

subjected to in-plane seismic actions using force transfer models is given in this paper Following an introduction to joint force conditions and potential failure modes, the force transfer method (FTM) suitable for design and assessment of bridge joints, including guidelines suitable for designers, is introduced Strut-and-tie concepts applicable to the modeling of bridge joints subjected to prestressing and seismic actions are then discussed, followed by a presentation of key joint mechanisms developed from these concepts Joint force

transfer models based on the proposed mechanisms and design

examples are also included to assist structural engineers with the

application of FTM

eginning with the pioneering

Br of Ritter! and Mérsch? about a century ago, numerous

researchers have examined the application of strut-and-tie model concepts to structural design problems.* Typical

applications have been directed at the detailing of deep beams, beam sup-

ports, frame corners or knee joints,

corbels, and membranes with open-

ings, when subjected to static loading More recently, strut-and-tie model concepts have been applied in order to

understand structural behavior and appropriately detail cap beam-to-column bridge joints, bridge footings and other bridge structural systems subjected to seismic loading In this regard, strut-and-tie models have direct application to prestressed concrete bridges

This paper presents a mcthodology

suitable for design and assessment of

bridge joints subjected to in-plane seis-

mic actions, which hereafter is referred to as the force transfer method (FTM)

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Table 1 Summary of large-scale in-plane seismic tests on bridge cap beam-to-column joints considered in the investigation of FTM

Description of joint tests Test scale Number of joints tested Reference

As-built, retrofitted, repaired and redesigned bridge knee joint systems J" " netomat ls

with columns having interlocking spirals

As-built, retrofitted, SBS ae redesigned bridge knee joint systems "1" ïm ale

having circular columns

As-built tee joint system having a circular column 75 percent 1 tee joint MacRae et al.” Redesigned tee joint systems having circular columns " Sritharan et al.!°

with varying amounts of cap beam prestressing

Two multiple column bridge bents consisted of circular columns 50 percent 2 knee joints and 2 tee joints Sritharan et al.!!

Knee joint system yee interlocking column spirals designed 95 percent | knee joint Ingham eral

with headed reinforcement

A three-column bent with cast-in-place steel shell circular columns 100 percent 2 knee joints and | tee joint Silva et al.! (Prestress)† lý (a) Tee joint (d) Moments up (e) Horizontal the column

Fig 1 Bridge tee

and knee joint

forces, and moment

and shear force diagrams at column overstrength condition.!” shear force

The FTM evolved from experimental and analytical studies of knee (exte- rior) and tee (interior) joints in concrete multiple column bridge bents

The bridge joint studies were moti- vated by the (a) use of inadequate joint details in practice and subsequent damage in earthquakes, and (b) unnec-

essarily congested details of bridge

joints resulting from the building joint design method

One major objective of the work has been to find sufficient and less conser-

July-August 2003

vative joint reinforcement details.*°

Encompassing details from as-built, retrofitted, and repaired joints, as well as joints designed to specific joint force transfer models, the investigation included seismic testing of 20 bridge joints at 33 to 100 percent scale

(see Table 1).”

Circular columns are generally pre-

ferred for bridge structures in seismic regions because they are efficient, easy to construct, and cost effective for confinement requirements Ac-

cordingly, circular columns were used in most of the test joints; five of them were designed with rectangular shaped columns with interlocking spirals

All of the bridge test joints were

subjected to cyclic loading with full

reversals to satisfactorily simulate seismic effects An extensive instru-

mentation scheme was adopted in cach test

The experimental studies were com- plemented with parallel analytical studies which focused on understand-

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ptrttrttt tt trtr teeta ty ¬ Ế | ¡ " + Kodo i] # = ae > ( Vie t 2 S ` : ` „5 ì ¿ * 1 y + wa sare ere a 7 “ý L | ' Hi tt Ay T A C N (a) Tee joint (b) Moment along cap beam (c) Variation of vertical shear force Fig 2 Comparison of maximum and average vertical joint shear forces.'7

ing the observed joint behavior using

experimental data and linear and nonlinear finite element analyses, as well as establishing or refining joint force

transfer models Results from various

detailed experimental and analytical studies were used to develop the FTM presented here

Seismic design procedures for bridge joints based on force transfer models have been recommended for

use in design practice.'*!° These docu- ments provide a prescriptive set of de-

sign steps which are based on one of several force transfer models presented by Priestley et al.'’ However, the selected force transfer model for use in design practice has been shown to be inadequate through experimental

and analytical studies.!':'872

Failure to provide a complete treatment of the joint force transfer is the cause for error in development of the design steps reported in References 14 to 16 Understanding of the FTM, as

detailed below, will enable both ap-

propriate improvements to be made to the existing, inadequate design models, and the introduction of new models suitable for different joint condi-

tions After selecting a force transfer

model for design in accordance with FTM, a satisfactory set of design steps may be established as illustrated as demonstrated in Reference 19

In the remainder of this paper, some familiarity with the basic strut-and-tie model framework, such as the method

outlined by Scleich et al.,”° is as-

sumed As a preamble to discussing FTM, the current bridge seismic design philosophy, resulting joint force condition and joint failure modes are first discussed An outline of the force transfer method, with guidelines for joint design and assessment, is then presented Application of strut-and-tie concepts in representing joint force transfer and key joint mechanisms, and joint force transfer models are fi- nally presented, along with examples in Appendix A

SEISMIC DESIGN PHILOSOPHY

Seismic design of concrete bridge structures is currently based on the ca-

pacity design philosophy,'’ in which

the locations of plastic hinges are preselected, most conveniently at the column ends, and inelastic actions developing outside these hinges are prevented by using strength hierarchy in the design Joints and other structural members are, therefore, designed for actions corresponding to develop-

ment of the overstrength moment ca-

pacities of the column plastic hinges Joint Forces

Typical forces acting in the bridge joint regions, consisting of the joint panel and end zones of the cap beam and column, are shown in Figs la to lc With plastic hinges developing at the column top adjacent to the joint interface, an average shear force acting upon the joint panel in the horizontal direction can be approximated assuming that the column overstrength moment uniformly diminishes over the full depth of the cap beam as illustrated in Figs 1d and le:!”

M’-AM_ M?

V =

" d-05a h, (1)

where

M? = overstrength moment capacity of the column at the joint interface and is obtained from a column section analysis with due consideration to the column axial force resulting from gravity and seismic actions

AM = resultant moment resistance due to beam shear at the joint interfaces [= 0.5h,(V 5,

+ V?,)|

d = effective beam depth

a = depth of the equivalent rectangular compression block in the beam

h, = beam depth

h = column section depth (or di-

ameter for circular columns)

in the plane of loading The corresponding average joint shear force in the vertical direction can be approximated by:

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h

The average joint shear forces in Eqs (1) and (2) are regarded as suitable for joint design, rather than using the maximum shear forces derived from forces acting upon the joints (see

Figs la to lc).'’?! The maximum

shear force, which is more useful for describing localized damage such as the initiation of joint cracking, is compared in Fig 2 with the corresponding average joint force in the vertical direction for a bridge tee joint

Joint Stresses

Using the average joint shear force, joint shear stress developed in the horizontal and vertical directions during in-plane loading can be obtained from: Vin bh Vi = Vay = Vin = (3) Cc

where 5; is the joint effective width and is taken as the lesser of /2D or b,, (see Fig 3) with D and b,, being, respec-

tively, the column diameter and the

beam width.'’ For joints with rectangu-

lar columns, /2D is replaced with (h, + b,), where b, is the column width

Using the column and beam axial forces, the joint normal stresses in the vertical and horizontal directions may

be estimated A 45-degree dispersion

of forces is assumed for calculating the vertical stress f, (see Fig 3a), while the beam gross area is used in estimating the joint horizontal stress f, With these estimates, the joint principal compression and tensile stresses are:

Since the principal stresses have better correlation to joint damage than

do other parameters such as the joint

shear force, p,; and p, are used as initial design parameters in FTM

JOINT FAILURE MODES When subjected to in-plane seismic loading, the failure of bridge joints July-August 2003 pa Shaded = b,*w, Plan view hy, Elevation view (a) Vertical direction axial stress Effective area = b,*h, Plan view (b) Horizontal shear stress

Fig 3 Effective areas for calculating stresses in joints with circular columns may occur in four different modes.°®

Each of these failure modes was ob-

served in large-scale testing of joints

and is shown in Fig 4 In each case, despite joint failure, the test unit was able to sustain the simulated gravity

load effects Descriptions of the joint

failure modes are given below

Compression Failure

In general, compression failure occurs in bridge joints in a brittle manner as a result of crushing of concrete struts in the joint This failure mode is typical in prestressed joints (see Fig 4a), and in reinforced concrete joints detailed with sufficient shear reinforcement such that they remain elastic during seismic response Compres- sion failure of joints will substantially reduce the lateral force resistance of the structure, most likely leading to total structural collapse with sufficient duration of earthquake shaking Tension Failure

Tension failure is typically developed in reinforced concrete joints when shear reinforcement responsible for mobilizing the joint compression

field is subjected to large inelastic

strains Since these inelastic strains are irreversible a growth of the joint panel occurs under seismic loading Consequently, the effective concrete strength of the joint core is significantly reduced, which often results in crushing of the joint strut at large displacement ductilities (Fig 4b) Al-

though significant lateral strength loss

is associated with such a joint failure, which may lead to structural collapse, strength degradation will occur in a gradual manner

In joints with wide cap beams, as currently adopted in practice,!® tension

failure can be triggered by crushing

and spalling of the thick lightly confined cover concrete, which participates in joint force transfer at initial

stages.°*? Tension failure is also ex-

pected in older bridge joints detailed with little or no shear reinforcement, as column longitudinal reinforcement provides some tensile resistance to the

joint at small shear strains.'”

Anchorage Failure

For satisfactory seismic performance of a bridge structure, it is essential that the column and cap beam

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(a) Compresssion failure (c) Anchorage failure

Fig 4 Four different joint failure modes longitudinal reinforcement be sufficiently anchored into the joint Inade- quate anchorage will result in bond slip of the reinforcement, introducing an additional member end rotation at the joint interface and thus reducing the lateral strength of the structure The bond slip rotation resulting from anchorage failure can contribute in excess of 40 percent to the total lateral

đisplacement.”

Given that the bond slip mechanism

does not provide adequate force resis-

tance, nor a profound energy dissipa-

tion system, the structure will exhibit

poor force-displacement hysteresis re-

sponse, characterized by gradual

strength deterioration and escalation of the loop pinching effect as displace- “as — `) (b) Tension failure q Saar

(d) Lap splice failure

ment ductility and/or number of load

reversals is increased However, there may be no apparent damage on the joint faces as shown in Fig 4c

The column longitudinal reinforcement is typically anchored into the joint with straight bar ends in order to

improve constructability.!*'’ These reinforcing bars are susceptible to bond

slip as they may be subjected to stresses up to 1.5 times the yield stress Hence, sufficient anchorage

length must be provided for the column longitudinal reinforcement based

on the maximum expected bar stress

Bond slip of the cap beam longitudinal reinforcement bars is most likely

to occur in bridge knee joints when

they are terminated within the joint

with straight bar ends,°° although it is

recognized that termination using a 90-degree hook at the bar end, as shown in Figs la and 1b, is typically

used in current practice

In seismic design, beam bars are not spliced within tee joints as this detail

causes additional reinforcement congestion Consequently, bond slip of

these bars is not expected in bridge tee joints unless significant inelastic stresses are developed in the beam longitudinal reinforcement at the column faces

Lap Splice Failure

Lap splice failure is most likely to occur in bridge knee joints subjected

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to closing moments As shown in Fig 5a, the column tension force may be transferred to the top beam reinforcement by bond if adequate confinement is provided for the lap splice If the confining pressure is not sufficient to prevent splitting of concrete between the reinforcement and straightening the hook of the beam bars, a failure may ensue as illustrated in Fig 5b (also see an example in Fig 4d)

Note that a lap splice failure can also occur in well-confined joints if the lap length between the reinforcement is not sufficient to transfer the column tension force to the beam reinforcement

FORCE TRANSFER METHOD Joint design has traditionally been performed based solely on the maximum shear force estimated within the joint panel, despite potential for the joint to experience different failure modes The joint shear is but one force of the complete force transfer action that develops in the joint region, which includes both the joint panel and the member regions directly adja-

cent to the joint

Therefore, it is conceivable that when the joint force transfer region is assumed to be limited to the joint panel and that shear, which is not directly correlated to damage, is treated as an independent force for design purposes to establish the joint reinforcement, unnecessarily conservative joint details are likely to result This notion is consistent with observations that bridge joint design based on the building code approach, using the joint shear force as the design parameter, led to congested, impracticable re-

inforcing details.>*!°

In FTM, the necessary joint reinforcement is viewed as that required to support sufficient anchorage of the column longitudinal reinforcing bars into the joint, eliminating the joint anchorage failure mode and permitting the plastic hinge capacity of the column to be fully developed Conse- quently, the necessary reinforcement

in the joint region is quantified by em-

ploying key mechanisms that satisfactorily anchor the column reinforcement into the joint and by estimating July-August 2003 Ị ea ae 111 Confining} 4° j! “| pressure (a) Force transfer by bond (b) Failure mode

Fig 5 Lap splice force transfer from column bars to top beam reinforcing bar and a failure mode due to inadequate confinement.”

various tension demands consistent with the selected mechanisms

Because the joint mechanisms account for all actions in the cap beam- to-column joint disturbed region (D- region), which includes the joints panel and the beam and column member ends, this design concept permits less conservative joint reinforcement details that significantly improve constructability

As shown subsequently, in addition

to transverse reinforcement within the

joint panel, the FTM may rely upon

transverse reinforcement placed in the cap beam region adjacent to the joint panel, and top and/or bottom beam longitudinal reinforcement across the joint to support force transfer In contrast, the conventional building joint design concept assumes that only the shear reinforcement provided within the joint panel is responsible for transfer of forces across the joint

In accordance with capacity design principles, the force transfer method of joint design or joint assessment is performed at the ultimate limit state for forces corresponding to the overstrength capacity of column plastic hinges The average joint principal stresses estimated at the ultimate limit state will be used as the initial design parameters in FTM

At the serviceable limit state, the

joint principal tensile stress is kept

below 0.25.) f’ (MPa) [or 3.0.) £’ (psi) ]

with no special detailing requirement, where f” is the specified unconfined

compressive strength of the joint con-

crete For a typical bridge column having longitudinal reinforcement content

in the range of 1.0 to 4.0 percent and a regular proportion for the column diameter and beam depth dimensions preva- lent in practice, the serviceability de-

sign criterion will be readily accomplished

At higher load levels, the force transfer across the joint initiates cracking in the joint region, which activates distinctive joint mechanisms and mo- bilizes reinforcement in the joint region Therefore, using the estimated average joint principal tensile stress to

gauge the extent of joint cracking, a

force transfer model consisting of appropriate joint mechanisms is selected and the required reinforcement in the joint region is then quantified consis-

tent with this design model

Reinforcement quantities in the joint

region will depend on the efficiency of

the adopted force transfer model However, when compared with the more traditional approach based directly on joint shear forces, the FTM

is expected to provide joint reinforce-

ment with reduced congestion regardless of the choice of the design model This expectation for FTM is a direct consequence of considering all actions in the joint region for quantifying the reinforcement

It is the authors’ opinion that the most efficient force transfer models for seismic joint design are those pro- ducing satisfactory joint performance while requiring the least amount of reinforcement within the joint panel Bearing this in mind, the remainder of this article addresses a formulation of the most efficient force transfer models for different joint conditions

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Guidelines for Joint Design

The following guidelines are sug-

gested for designing joints in new

bridges using FTM:

1 At the overstrength capacity of the plastic hinge, the column tension

force may be represented by: #1 T„=0.5A./ (5)

where A, and iG are, respectively, the total area and overstrength stress in the

column longitudinal reintorcement

The column overstrength stress may be

taken as 1.3 times the measured value

of f, or 600 MPa (87 ksi) for Grade 60 reinforcing bar Alternatively, an accu- rate estimate of 7, may be obtained from a section analysis of the column

2 Since the joint design procedure, which is aimed at protecting joints from any significant inelastic actions, is based on the overstrength moment capacity of the column plastic hinge and on conservative material proper- ties, a strength reduction factor of @ =

1.0 may be satisfactory

3 Using the principal tensile stress obtained at the ultimate limit state [from Eq (4)], the joint design is approached in the following manner:

(a) If p, < 0.25 f/(MPa) [or 3.0.) f’ (psi) ], only limited insignifi- cant joint cracking is expected Appli- cation of FTM is not required and the following nominal reinforcement is

provided within the joint paucl for sat-

isfactory force transfer:!”19

Total area of vertical joint reinforce-

ment:

4» ~ 0.084, (6)

Volumetric ratio of horizontal joint hoop or spiral reinforcement: 0.29, 7’ (SI units) 3.5J/ độn (7a) Ps = psi units) (7b)

The requirement in Fq_ (A) is in-

tended to assist bond transfer of top beam reinforcement and formation of joint diagonal struts while Eq (7) is based on providing hoop reinforcement sufficient to support a tension force equivalent to 50 percent of the

principal tension strength of 0.29,) £’ (MPa) [or 3.54) £7 (psi) ]."”

The nominal joint reinforcement in Eqs (6) and (7) may be viewed as equivalent to supporting a column tension force of (0.12 + B)T., where the first part of the expression is obtained by combining Eqs (5) and (6)

The second part of the expression is based on column tension force that can be supported by p, as in Eq (7) with:

B= 0.22.) f2 (MPa) 1?/Ascfy

[or 8= 0.22x103.//7 (psi) U7/Aschy |]

where /, is the anchorage length as defined in Eq (12)

(b) If p, > 0.42Vf0 (MPa) [or

5.0.) f’ (psi) ] joint design should be based on a force transfer model that supports the total column tension force, T The joint region is detailed identifying tension demands imposed

by the joint force transfer model (c) For joint principal tensile stresses between the above limits, satisfactory joint force transfer may be achieved by providing supplementary reinforcement to the nominal requirements in Eqs (6) and (7) The supple-

mentary reinforcement should be de-

termined using a force transfer model to anchor the unsupported component of the column tension force equal to

(0.88 — B)T., i.e., [1 — (0.12 + Ø)7,]

The advantage of this approach is that a suitable force transfer model may be

found using a single joint mechanism A higher limit of p, = 0.29) f” (MPa)

[or 3.5 ,/ f” (psi) ], was recommended in the past as a threshold value for detailing joints with nominal reinforce-

ment.*:!°!7 The more conservative ap-

proach suggested herein is due to the approximation made in Eq (1) for calculating the joint shear force, which in- fluences the value of p,

4 For joints with p, > 0.25) f7 (MPa)

[or 3.0,/ f’ (psi) ], nominal reinforcement will be adequate if it is shown that the column bars can be satisfactorily anchored into the joint main strut without the need for any special rein-

forcement.® This will often be satisfied

in joints designed with a fully prestressed cap beam The potential for satisfying this condition may be established using simple beam theory as illustrated for a tee joint in Fig 6 It will be necessary to show that for the overstrength condition, the beam neutral axis depth at the tension face of the of the column is equal to or greater than (g + la eg); where g is the distance between the end of the column bars and

the beam top surface, and /, vis the ef-

fective anchorage length as defined in Eq (13) The joint mechanism supporting force transfer in these joints is depicted in Fig 13b and its description is given under the clamping mechanism

5 The average joint principal compression stress should always be main-

tained below 0.3f; in order to prevent

compression failure as shown in Fig 4a For larger p, values, a study should

be conducted to verity that the average

stress demand does not exceed the capacity for all critical joint struts

6 Column bars should be anchored

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into the cap beam with straight bar ends The force transfer method ac- commodates the use of headed longitudinal reinforcement in columns, pro- ducing acceptable joint details (see distributed strut mechanism) How- ever, employing column bars with

hooks or tails should be avoided as

this detail causes reinforcement congestion in the joint

7 A minimum anchorage length for the beam and column longitudinal reinforcement into the joint should be pro-

vided assuming a uniform bond stress of

1.17, f/ (MPa) [or 14.) £7 (psi) ] along

the embedded portion of the bar.'” 8 Column bars should be extended as close as practicable to the height of the top beam reinforcement to maxi- mize embedment conditions for the extreme column tension bars into the joint diagonal strut

9 The last two guidelines described

above should be used to dimension the

minimum cap beam depth Guidelines for Joint Assessment

When compared to the design of joints in modern bridges, less conser-

vative guidelines can be adopted in

seismic assessment of joints for retrofit purposes This is consistent with recommendations by Priestley et

al.'” for joint assessment, who advo-

cate allowing limited joint damage to

occur as long as the damage does not

lead to total collapse of the structure or punching of columns through the deck In light of this philosophy, the following guidelines are recommended:

1 Considering the column and cap beam retrofit measures, a plastic collapse mechanism for the bridge bent should first be established Using Eq (1), estimate the joint shear demand based on the expected overstrength column moment at the joint interface

2 An estimate of the column tension force, 7., required to be anchored

into the joint should be based on the expected column overstrength moment Eq (5) may be used for this pur-

pose when the column plastic moment

capacity is expected to be fully developed adjacent to the joint Assessment of the joint should then follow assuming a strength reduction factor of @ =

July-August 2003

1.0

3 As part of the joint retrofit, joint dimensions may be increased This should be considered when estimating joint shear demand and principal

stresses

4 As with the design of new joints,

the principal tensile stress is used as

an initial assessment parameter as follows:

(a) If p, < 0.29J/(MP4) [or

3.5 7 (psi) |, the presence of nominal

reinforcement as given by Eqs (6) and 7) is adequate

th If p, > 0.42Vf/ (MPa) [or

0.42 / ¢’ (psi) ], the adequacy of the joint reinforcement must be established based on an efficient joint force transfer model supporting the column tension force T

(c) For joint principal tensile stresses between the above limits, adequacy of the existing joint reinforcement may be demonstrated by using a force transfer model Accordingly, the reinforcement in excess of the nominal requirements should be sufficient to

anchor the column tension force of (0.88 — B)T, into the joint

5 As discussed in the previous sub-

section, if 1t 1s Shown that the column

bars can be anchored into the joint

main strut without the need for any special reinforcement, then nominal joint reinforcement may be considered

adequate even if p; > 0.29.) f’ (MPa) [or 3.5.) f’ (psi) ]

6 The joint principal compression stress should always be maintained

below f/ unless it can be shown that

the demand on joint struts is not ex-

cessive This requirement is critical

when cap beam prestressing is used to improve joint and/or cap beam perfor-

mance

7 Premature termination of column bars is commonplace, particularly in

older bridge joints in California.°*? In-

creasing the column reinforcement embedment length will often be required as part of the retrofit procedure, for example, by haunching the joint, which should be reflected in the force transfer model

8 If necessary, permit limited in-

elastic action to take place in the cap

beam adjacent to the joint at larger

displacement ductilities (u, = 3 — 4) Also, permitting tensile strains of up

to 0.01 in the joint shear reinforcement may be acceptable when determining the capacity of joint ties

9 As discussed below, a realistic representation of concrete tension carrying capacity can be included in the force transfer model

Influence of Repeated Loading

In FTM, design is performed for the maximum possible forces that the joint can be subjected to during a repeated or seismic loading This is implied in

Eys (1), (4) and (5), in which joint

shear force, principal stresses and T, are obtained using estimated strain hardening and yield overstrength of the column longitudinal reinforce-

ment

The influence of seismic or cyclic type loading is not directly taken into account in FTM Strength deterioration of concrete struts resulting from such repeated loading is conveniently incorporated by defining appropriate permissible stress limitations These limitations were established empirically and are presented in the follow-

ing section

Since no significant hardening is ex-

pected for the joint reinforcement and cyclic inelastic excursions will be in the tension range, the stress-strain response envelope of steel under repeated loading is assumed to be the same as that obtained for monotonic

loading Therefore, for an estimated

strain in the joint reinforcement, the corresponding stress can be readily obtained

Columns with High Longitudinal

Steel Ratio

The force transfer method of design and assessment is applicable to all bridge joints, regardless of the longitudinal reinforcement ratio of the adja-

cent column As will be discussed

later, the required reinforcement for joint force transfer is determined as a function of the total area of column longitudinal reinforcement Therefore, high longitudinal column steel ratios

will result in larger reinforcement

quantities in the joint region

The higher column steel ratios also mean larger demand on the struts supporting the joint mechanisms Since

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(a) Arch: activ aud curved cracks

(b) Parallel sut mechanism and straight cracks

Fig 7 Different compression force paths in knee joints subjected to opening moments the effective strength of struts is not

increased proportionally, a high column longitudinal steel ratio will result in high demand to capacity ratios for the struts in the joint region

If the demand is kept below capacity in all critical struts, forces across the joint will be transferred satisfactorily For column steel ratios in the 1 to 4 percent range typically adopted in

practice,'’ satisfactory force paths for

the joint forces can be established using FTM

STRUT-AND-TIE CONCEPTS The fundamentals and application of strut-and-tie concepts to structural members subjected to static loading can be found in the literature [e.g., see References 20, 24 and 25] Due to dif- ferences in the design philosophy and the repetitive nature of seismic loads, some changes to the application procedure are necessary for successful modeling of bridge joint regions using struts and ties

These changes, as applicable to bridge joints subjected to seismic actions, are presented below Since the application of strut-and-tie concepts is

here focused on bridge joints only, the

procedure is simplified wherever possible

Compression Force Flow

Determining a suitable path for compression force flow across the joint is the most critical step in FTM

as this procedure essentially deter-

10

mines the node locations and orientation of struts Elastic analysis of the system using a finite element method- ology, observed crack patterns and past experience are generally considered as appropriate means for identifying the force paths in structural members subjected to static loading

Further, for simplicity, identical models for the ultimate limit state and for the cracked state of serviceability condition have been recommended in the literature (see, for example Refer- ence 20) However, a similar approach is not applicable to seismic design of bridge joints

Joints in a bridge bent are typically

subjected to axial, shear and flexural actions whose relative magnitudes and thus dominant action can be different at the service and ultimate limit states As demonstrated by Bhide and

Collins?® on shear panels with and

without an axial force, the force path and orientation of cracks in the joint region can be considerably different at the two limit states Also, elastic analysis ignores the force redistribution that occurs progressively with the de-

velopment of tensile cracks.”°

Therefore, the joint reinforcement

derived using a force path established

from an elastic analysis will be often unnecessarily conservative; failure of such joints is also possible since the joint behavior at the ultimate limit

state was not modeled Although it is

not required in FTM, it is acknowl- edged that force paths of the critical joint struts can be satisfactorily established using results from an elastic

analysis conducted at the onset of yielding of the column main reinforcement and good engineering judgment In this case, concrete cracking and strain penetration along the column bars into the joint must be accurately

modeled

The force paths identified for bridge joints in this paper as part of FTM are based on observed crack patterns, experimental data, linear and nonlinear finite element analyses, and the au-

thors’ experience Some issues rele-

vant to establishing force paths in

bridge joints are discussed below

Reinforcement layout and geometric constraints may significantly influence the compression force path in cracked joints This is illustrated in Fig 7 where two knee joints subjected to opening moments are compared In the first joint, with no stub, arch action

is expected to develop within the joint

and consequently curved cracks should result on the joint faces

In the second joint, with a stub and continuous cap beam longitudinal reinforcement detail as shown in Fig 7b, broadening of the joint diagonal

strut is possible by anchoring a joint

strut against the left bottom corner of the beam reinforcement Since this action reduces stresses in the critical struts of the joint, this mechanism, involving parallel struts, is likely to develop in the joint shown in Fig 7b in- stead of an arch mechanism A consequence of the parallel strut mechanism would be the formation of straight cracks on the joint faces

This argument, which is consistent with the cracked pattern observed on the joint faces during testing (see Fig 8), is in accordance with a suggestion

made by Collins and Mitchell? that

when cracking occurs and concrete tension carrying capacity is lost across the crack, the orientation of struts should be towards stiffer reinforcement so that the magnitudes of forces and deformations developed in the D- region are minimized

When joints are subjected to in- plane loading, struts are developed in

three dimensions The components of

the struts perpendicular to the loading plane can influence the crack pattern on the joint faces.° Therefore, it is noted that the observed or expected

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(a) Curved cracks Fig 8 Observed joint cracks in bridge knee joints indicating different force paths under opening moments (b) Straight cracks

crack pattern alone is not always sufficient to establish the compression force path in bridge joints

Furthermore, when establishing suitable force paths for bridge joints, a basic rule of strut-and-tie concepts

should not be forgotten That is, the

force transfer model resulting from the

compression force path should not re-

quire excessive deformation in any reinforcement ties supporting the joint

mechanism(s) in order to fully develop

the plastic state of the structure If this condition were not met, premature tension failure of joints and poor ductile performance for the bridge bent would be inevitable under seismic actions

Struts, Ties and Nodes

Compression forces in concrete structural members are transferred through three types of stress fields known as the “prism,” “fan” and “bot-

tle” as shown in Fig 9.2° The prism is

expected in B-regions (beam regions), while fan and bottle-shaped stress fields typically develop in D-regions (disturbed regions), with the struts in

beam-to-column connections gener-

ally being bottle-shaped When the joint compression force is transferred between two nodes through a bottle- shaped stress field, in-plane and out- of-plane tensile stresses are developed perpendicular to the force transfer di- July-August 2003 iH NTIS (b) Fan II (a) Prism VÀ CC 17, (11111111 TT (c) Bottle

Fig 9 Different stress fields identified in concrete struts (after Schlaich et al.?°)

rection, which reduce the strut capac-

ity

For simplicity, the struts in the joint region can be represented with single straight lines or with zones bounded by straight lines in 2D, ignoring the

in-plane and out-of-plane tensile

stresses (see Figs 10a and 10b) Fur- thermore, a uniform stress across the in-plane depth and in the out-of-plane direction at any section along the strut

is assumed

These assumptions, which simplify

the estimation of the demand on the

struts, are deemed satisfactory as long

as the allowable compression stresses in the struts are defined appropriately, taking the transverse tension field into

account This is dealt with in a subsequent section

The tensile resistance of the reinforcement or concrete is represented by ties in single or multiple one-di- mensional layers The tensile resistance of concrete can be adversely af- fected by microcracks induced by previous loads, thermal stresses and

shrinkage.’ Consequently, concrete

tension capacity is generally ignored in structural design

Nonetheless, it has been found that the tensile resistance of cracked concrete has a significant influence on joint force transfer, and that modeling its role is essential for accurately char- acterizing the seismic behavior of

Trang 11

CCC {c) nodal forces aS Remaining column |-“ bars are not shown for clarity Critical section Thee | (d) Dimensioning two joint struts

Fig 10 Dimensioning struts and nodes, and identifying strut critical sections ina bridge tee joint

bridge joints.°!*8

Several other researchers have also

promoted the influence of concrete

ties in structural response.”°??3° When the contribution of the concrete ties is

appropriately accounted for in the force transfer model, a reduced amount of joint reinforcement will be

required

Clearly, a designer can still choose

to conservatively neglect the contribution of concrete ties Incorporating concrete ties in the assessment of joints is especially encouraged as this can avoid unnecessary and expensive retrofit of bridge joints A procedure for estimating the joint concrete tension contribution is presented under “Contribution of Ties.”

Nodes represent the intersection points of three or more struts and/or ties, where change in direction of forces takes place It should be appre- ciated that such changes in a rein-

forced concrete structure typically occur over a zone, except where a

strut or tie delineates a concentrated stress field.“’ A node with gradual changes over a zone is identified as a smeared node, with its dimensions being determined by the effective

12

widths of struts and ties forming the

node A node having a concentrated stress field is generally referred to as a

singular node.”

Depending on the type of forces in- tersecting at nodes, they are identified

as CCC, CCT, CTT and TTT nodes,

where C and 7 stand for compression and tension, respectively In bridge

joint regions, CCC, CCT and CTT

nodes are commonplace, but TTT nodes are not expected

Dimensioning Struts and Nodes and Identifying Critical Sections

Consistent with the discussion presented above, the concepts of simple and detailed strut-and-tie joint models, different node types, the dimensioning

of struts and nodes, and identifying

the critical sections in joint struts are illustrated in Fig 10

Suppose that the anchorage of column tension force 7; in a tee joint is modeled with a simple mechanism as shown in Fig 10a The stress field

within the joint can be identified as

shown in Fig 10b, with strut dimensions dictated by the effective anchorage length of column reinforcement (discussed later) and by the depth of

equivalent beam flexural compression stress blocks

Adjacent to the tension face of the column, the equivalent stress block is required at the interface between the B- and D-region, located at a distance of h, from the column face Assuming that each stress field is bounded by straight lines, the node and strut dimensions can then be readily established

The Zones ABC and DEFG in Fig 10b, respectively, represent CCC and CCT nodes (identified in Figs 10a and 10c) while the joint strut is formed by stress field BDGC The nodal zones can be isolated as shown in Fig 10c and their stress state can be examined if necessary Also given consideration in Fig 10b is a multi-layer representation for column tension force 7; and the need for sufficient anchorage of each tie into the CCT nodal zone

As a result of the tension force increasing from Section EF to Section DG in the CCT node (Fig 10b), the resultant compression force in the direction of the joint strut gradually increases within the nodal zone and at- tains the maximum value at the

strut-to-node interface

Once the strut boundaries are established, the critical section(s) of the joint strut should be identified so that stability of the strut may be examined For the example in Fig 10b, the strut depth increases from DG to BC with no change in the magnitude of the compression force, and thus Plane DH perpendicular to the direction of the strut is a critical section

Further, due to the absence of significant confining stress along the sides (i.e., BD and CG in Fig 10b), the main strut in the joint typically has a bottle-shaped stress field, with the most adverse effects of the in-plane and out-of-plane tension field being present at the center of the joint Therefore, examining the stress state across the plane at the joint center is always essential This is consistent with experimental observations that crushing of struts typically develops at

the joint center

If two struts are identified within the

joint, the area bounded by the struts is

assumed to be participating in force transfer in proportion to the magni-

Trang 12

tudes of the struts as illustrated in Fig 10d Furthermore, the effective width of each strut at the joint center is taken as 2w, and 2w3, respectively

The procedure described above for dimensioning struts and nodes and identifying critical sections in tee joints can also be applied to bridge knee joints subjected to opening moments For knee joints under closing moments, critical sections can be identified as shown in Fig 11 using a similar concept

A critical section in a reinforced concrete knee joint, incorporating a stub and continuous top and bottom beam reinforcement (Fig 11a), is cho- sen such that the highest strut stress is at the section with the minimum depth, as for the tee joint in Fig 10b In addition, the stress state at the joint center should also be checked For a knee joint with a prestressed cap beam such as in Fig 11b, only one critical section at the center of the joint is se-

lected

From the above, it can be observed that although the strut depth is small close to the CCC node, with the joint strut force continuously increasing to-

wards this node (Fig 11b), the strut

capacity is significantly higher in this region due to the confinement provided by the CCC node

For reinforced and prestressed concrete bridge joints, where the column tension force is modeled with a single tie such as in Fig 10a, there is a ten- dency to select the critical section at the center of the joint This is satisfactory based on the discussion presented above However, in critical cases (e.g., assessment of joints with little or no

reinforcement), the designer is encour-

aged to perform checks at three sections along the strut; at the center, midway between the joint center and CCC node, and midway between the joint center and CCT node

In all joints, the width of the joint strut in the out-of-plane direction is taken as b; as defined in Eq (3) Allowable Stresses in

Cuncretle Struts

In order to preclude compression

failure of joints resulting from crushing of struts, it should be ensured that July-August 2003 Critical section (a) Reinforced concrete joint Critical section M4 £ | A iia iB ae i VN | bề ý i i EAI : RP a (prestress) (b) Prestressed joint Fig 11 Critical sections of joint main diagonal struts in bridge knee joints subjected to closing moments Table 2 Permissible stresses suggested for critical bridge joint struts under seismic conditions

Permissible stress Strut description

0.68/ For joint struts with only minor cracking,

such as that expected in prestressed joints

0.51f2 Struts in reinforced concrete joints with reinforcement

not subjected to significant strain hardening (¢, < 0.01)

Struts in unreinforced joints or in joints with potential for initiation of

0.34/⁄ tension failure following development of high inelastic strains in the

joint reinforcement (¢, 2 0.02)

strut capacities are sufficiently larger than the demands in the joint region Observed failure of joint struts and comparison of joint strut stress magnitudes with those in the beam and col-

umn ends adjacent to the joint re-

vealed that the struts bounded or anchored in the joint panel are most critical Therefore, limiting examina- tion of the stress state to these struts is sufficient

The strength of a concrete strut depends on its multi-axial stress state, confinement, damage caused by cracking, uniformity of cracking, dis- turbances from reinforcement and the influence of aggregate interlocking As noted previously, in-plane loading induces joint dilation in the out-of- plane direction, which, in turn, can reduce the strut capacity significantly

below the unconfined concrete

strength.”923931

Several different recommendations,

based either on beain/shear panel tests

or on engineering judgment, are found in the literature for estimating strut capacities They range from simple for-

mulas, in which the strut capacity is represented by the effective unconfined compressive strength, to detailed equations which account for the state of strain in the strut Among these recommendations, which are intended for monotonic loading, appreciable dis- crepancies exist between the permissible stresses suggested by different researchers for struts subjected to

similar conditions.°

From the seismic tests of bridge joints listed in Table 1 and subsequent analytical investigations, the stress limits shown in Table 2 are recommended for seismic design and assessment of bridge joints These limits were made to resemble those recom-

mended by Schlaich et al.”° for struts

in structural members subjected to static loads

In a recent study aimed at perform- ing push-over analyses of bridge bents based on strut and tie models, defining

suul Capacities using the permissible

stress values in Table 2 was found to

be satisfactory.*

Recall that in “Design and Assess-

Trang 13

0 z Se a ` Số) Cc (a) Idealized joint panel stresses A,f, Joint | _ reinforcement £ B £ ch Si Tơ Tử (c) Mohr’s circle for average strains

Fig 12 Estimating the tensile resistance of cracked concrete.°

ment Guidelines,” the average joint principal compression stress was lim-

ited to fZ This limitation, which was originally derived empirically based

on the performance of building joints,'’ is to keep the demand upon the joint struts within admissible limits

When compared to the procedure

described above which requires an es-

timation of stresses in the joint struts, limiting the joint principal stress to an allowable value is relatively simple [see Eq (4)] and is regarded as a more conservative approach However, it

should be kept in mind that the 0.3/7

stress limit, which is useful when designing or assessing prestressed joints, only addresses joint compression fail-

ure

Crushing of a concrete strut can take place at a joint principal compression stress considerably less than 0.3f2 when tension failure develops in a cap beam-to-column joint As indicated in Table 1, the capacity of a strut in a joint experiencing tension failure may be as low as 0.34f%, corresponding to a joint principal compression

stress in the range of 0.17 to 0.15f2

Contribution of Ties

Ties in joint force transfer models represent the tensile resistance of reinforcement and/or concrete It is straightforward to take the reinforcement contribution, 7,, into account as

14

described in Eq (8):

1; " As ef] (8)

where A, y is the effective steel area in the direction of the tie, and ƒ 1s the stress in the reinforcement

For design purposes, f, may be approximated to the yield strength f, for

reinforcement that participates in the

joint force transfer; this implies that developing a steel stress exceeding /,

is possible in localized regions For assessment purposes, a less conserva-

tive approach can be considered by approximating f, to 1.05f, for Grade 60 (414 MPa) reinforcement, which is obtained by allowing average steel strains of up to about 0.01

As noted previously, the cracked concrete in the joint region can also have a notable contribution to tensile resistance in the joint region This tension capacity may be estimated using

a blanketed approach.®°!* Drawing an

analogy to Vecchio and Collins’ method for estimating the tensile resistance of concrete that contributes to shear resistance,*° the tensile resistance provided by joint concrete can be found assuming a uniform joint stress and strain field as illustrated in Fig 12

Considering the forces in a joint

segment as shown in Fig 12b, the total tensile resistance of the joint panel in the vertical direction is: T,=DAS+T, =DAf+f(cos?)bl (9) where »A,f, = total force resisted by the reinforcement as defined in Eq (8)

T., = vertical component of the tension force carried by the cracked concrete ti = average joint principal tensile stress b; = effective joint width as defined in Eq (3)

l = length of the joint panel It is important to note that the development of 7 requires the presence of at least a minimal reinforcement

within the joint panel to distribute cracking.'®

For estimating /,, the following empirical relationship as suggested by

Collins and Mitchell*> may be used: Œ;¡Ø; ƒ.„ = 10 A 1+./500¢, 9) fy = cracking strength of concrete and is approximated to

0.33 J (MPa) [or 4.0 i (psi) ]

a, = a factor accounting for bond characteristics and is taken as 1.0 for deformed bars

Q> = a factor which depends on the

load history

€ = average tensile strain

For short-term monotonic loading Q = 1.0 and for sustained and/or repeated loads a) = 0.7 have been suggested A less conservative estimate for f, as given by Eq (11) is appropri-

ate for assessment purposes:

] Ti

es ot 11

⁄ 1+ 2/200, d1

Eq (11) shows the original relationship established between ƒ/¡ and £¡

using experimental data*® and Eq (10)

was later recommended as appropriate for design calculations Furthermore, the value of / in Eq (9) can be in-

ereased by 25 and 50 percent for as

sessment of knee and tee joints, respectively, in recognition that the beam regions adjacent to the joint also partic- ipate in force transfer across the joint

Trang 14

It should be noted that f,, defined above is slightly higher than the joint cracking strength defined previously under “Design Guidelines.” However,

0.33 / (MP4) [or 4.02/ /7 (psi) ] is re-

tained in Eqs (10) and (11) in order to remain consistent with the empirical equations suggested in References 25 and 30

Nodal Failure

Failure of a node in the joint region can lead to undesirable brittle behavior

of the structure This may develop due

to either concrete crushing, or anchorage failure of a reinforcement tie within the nodal zone From various

tests on bridge joint systems,”'?*?~* it

is found that failure of a CCC node seldom occurs

This is because struts acting upon the node simultaneously provide reli- able confinement, enabling it to sustain significantly high stresses Reduc- tion in the confinement effect is

possible when a compression force

ceases to exist due to premature fail-

ure of the column or cap beam, which should be dealt with prior to investi-

gating the joint

The most common nodal failure expected in bridge joints may occur when longitudinal column bars are anchored with straight bar ends (e.g., CCT node in Fig 10a) Such a nodal failure can be avoided in design and

predicted in assessment by appropriate

treatment of the column bar embedment length into the joint

As specified under “Design Guide- lines,” the required embedment length

for column bars can be obtained as-

suming a uniform bond stress of 0.33.) f’ (MPa) [or 4.0.) f’ (psi) ], which would result in a minimum anchorage lengths of: 1, = 0.30d,,f,/.f/ (mm, MPa) (12a) 1, =0.025d,,f,/.[f" (in.,psi) (12b)

where d;; and f, are, respectively, the diameter and yield strength of the column bar

In reality, column bar anchorage takes place over a much shorter length July-August 2003 (a) Reinforced concrete joint C Sle —— ` Am „ — AI ~3, 7 Uy ‘ “THỊ (b) Fully prestressed joint

Fig 13 Clamping mechanism

near the bar end due to strain penetration along the reinforcement into the joint

It was found from experimental data that an average bond stress of 2.5.) f/ (MPa) [or 30.) f’ (psi) ] is typically developed in well-designed

joints.**'’ Using this bond stress, the

effective anchorage length for the col-

umn reinforcement is thus defined as:

Log = 0.14d,,f, //f/ (mm, MPa)

(13a)

Lop = 0.012d,,f, //f" (in., psi)

(13h) To avoid anchorage failure, the column reinforcement should be effectively clamped at a minimum distance

of 0.5/, -¢ from the bar end This con-

dition will assist with locating critical

nodes within the joint (e.g., the CCT

node in Fig 10a)

In addition to providing a minimum required anchorage length, it must also be ensured that the column bars are extended into the joint as close to the

top beam bars as possible.®!”> If this

condition is not satisfied, adequate clamping of the column bars into the joint strut will not occur and nodal failure can develop despite satisfying the minimum anchorage length requirement

For assessing bridge joints with column longitudinal bars inadequately

embedded into the joint, as was typical until recently,*!’ the maximum

force that can be developed in the col-

um bars nay be estimated assuming

a uniform” bond - stress of

0.76, f/(MPa) [or 9.2.) f/ (psi) ]

along the embedded portion of the re-

inforcement This lower bond stress

value was inferred from ACI 318*

and has been found to give a good estimate of the maximum tension force that can be developed in the column

reinforcement.”

If column bar anchorage is ad- dressed as detailed above, it is suggested that no further check on nodal

failure is required

KEY MECHANISMS

Since the strut-and-tie representation

of structural members is based on equilibrium conditions alone, numerous al-

ternative strut-and-tie models are pos-

sible for a given reinforced concrete member In order to assist designers with developing efficient force transfer models for bridge joints, several key joint mechanisms are presented in this section in accordance with the general strut-and-tie concepts and related discussion presented above

Additionally, geometric considera-

tions, which are typically required for quantifying reinforcement, are provided for these mechanisms Develop- ing force transfer models using key joint mechanisms is discussed in the

subsequent section

Clamping Mechanism

The clamping mechanism anchors the column tension force directly at a CCT node using a joint diagonal strut and an external strut supported in the beam adjacent to the column tension side, as illustrated for tee joints in Fig 13 These two struts for a reinforced

and fully prestressed concrete joint are

identified in both Fig 13a and Fig 13b as C; and C)

Trang 15

Concrete’ tension tie ⁄ Fig 14 Splice transfer mechanism vi (a) Conventional mechanism anes eee ow Re 7 | 1 Spiral or z ` Top beam bar / 4 J] / Top beam cross-tie | Sole = / | / |) bar i} y ane ¥ | Concrete

_ Bottom 4/ | Bottom '? Tie or

#| 7 beam bar 7 LAI ‡ z beam bar ng stirrup Stirrup | (b) Single strut mechanism (c) Out-of-plane mechanism Headed ; a column bars : | | | A | | | i I | 1 h i i t : i ! F ere eS | i \ ! lý i iL] 1 II es oe (a) Example | | {| | Column bars with ||: nh threaded ends a | Cap beam (b) Example Z

Fig 15 Direct transfer mechanism The location of the CCT node

should be determined using the effec-

tive development length for the column bars and location of the resultant column tie If the column bars of an existing joint have insufficient anchor-

age length, this deficiency should be

reflected in the model by appropriately defining /, and determining the node location and magnitude of the column tension force anchored at that node as previously discussed

With sufficient development length for the column longitudinal reinforcement into the joint, the clamping mechanism can be used to transmit up to 50 percent of 7 in reinforced concrete joints and 100 percent of 7, in fully prestressed joints, as indicated in Fig 13.619

In reinforced concrete joints, a 45- degree incline is suggested for the external strut, C,, which may support a maximum column tension force of

16

0.157 '§ The orientation of the beam direct strut in prestressed joints will

depend on the amount of beam prestressing, and should be determined from equilibrium conditions

Splice Transfer Mechanism

Forming a CTT node between the beam and column longitudinal reinforcement is generally not possible as the column bars are usually terminated below the beam bars Utilizing joint vertical stirrups and/or concrete ties, the splice mechanism effectively transfers the anchorage location of the column tension force to the top beam bars This enables anchorage of the column tension force at a CTT node

using the beam top reinforcement and

a diagonal joint strut Three possible splice transfer mechanisms are illustrated in Fig 14 using struts and ties in two dimensions (2-D)

Comparable mechanisms in three dimensions (3-D) are also possible, and can be adequately represented using 2-D models More details of this mechanism and 3-D representation of

the models may be found in Reference

18 The splice transfer mechanism may be relied upon for supporting up to 50 percent of 7, with its participa- tion in joint force transfer diminishing as the magnitude of cap beam prestressing increases

To quantify the appropriate reinforcement for this mechanism, a sim- pler model representing all possible

mechanisms may be considered Ex-

ample 1 of Appendix A presents the simplified model and quantifies the corresponding tension demands Direct Transfer Mechanism

In contrast to the splice transfer

mechanism, the column longitudinal reinforcement is anchored directly in a CTT node involving the beam longitudinal reinforcement To develop a sta- ble CTT node, the column bars should be extended above the beam top longitudinal reinforcement and provided with a mechanical anchorage such that the tensile strength of the bars can be developed over a very short distance Two possible details are shown in Fig

15

In the first example, headed longitudinal column bars are mechanically in- terlocked with the beam longitudinal reinforcement In the next example, threaded column bars are mechanically anchored to the top of the cap beam using steel plates

The second detail will also be appropriate for seismic design of integral

Trang 16

bridge systems with ductile concrete columns and steel box-shaped cap beams, as used in a recent research project.*° Since this mechanism is an alternative to the splice transfer mechanism, the direct transfer mechanism

is recommended for anchoring up to

50 percent of 7 This implies that the mechanical anchorage detail is required only for the column reinforcing bars anchored into the CTT node

Due to direct anchorage of the column tension force, the joint reinforcement required for this mechanism will be less than that required for an equivalent splice transfer mechanism, and will be determined based upon the

confinement requirements to prevent

failure of joint struts

Haunched-Joint Mechanism

Haunching of joints, which is an effective means of retrofitting existing joints with poor column reinforcement anchorage and/or insufficient joint shear reinforcement, increases the joint size.’ The joint mechanisms responsible for force transfer in the expanded joint can mobilize relatively

more reinforcement Also, when

retrofitting joints using an external reinforced concrete jacket, additional shear reinforcement can be added without causing steel congestion in the joint region

The haunched-joint mechanism,

which anchors the column tension

force at a CTT node under closing moments or CCT node under opening moments, is illustrated for a bridge knee joint in Fig 16 A special feature of the haunched-joint mechanism is that it benefits from broadening of the joint main strut This reduces the demand in the strut, increases the strut capacity, and alleviates possible compression failure, particularly under closing moments

When subjected to opening moments, the expanded joint dimensions improve anchorage of the column bars into the joint by effectively lowering the position of the CCT node at which the column tension force is supported (see Fig 16b) A mechanism similar

to that shown in Fig 16b can also be

applied to a haunched bridge tee joint The contribution of this mechanism July-August 2003 (~~ \ SONS % ¬ ——- \ KT zx Vi ⁄⁄ ⁄ #‡ ⁄ | Ệ 1 (a) Under closing moments » ` # ft Ị ` ` ` ee = (b) Under opening moments Fig 16 Haunched-joint mechanism “14 (a) Under closing moments (b) Under opening moments

Fig 17 Distributed strut mechanism

for anchoring the column tension

force will depend on several factors,

including the embedment length of the column bars into the joint, anchorage detail of the cap beam top reinforcement, dimensions of the expanded joint, and the amount of joint reinforcement For typical bridge joints designed in the post-1971 era with premature termination of column bars into the joint, the haunched joint mechanism may be relied upon for supporting 0.57 provided the joint dimensions are enlarged by 30 percent or greater A larger contribution from this mechanism, providing anchorage for up to 1.07., is possible if supported by an appropriate model based on the joint condition

Distributed Strut Mechanism Using multiple layers of headed reinforcement in the cap beam and

strategically positioning the headed

ends of the reinforcement in the joint

region, the distributed strut mechanism can be developed as illustrated for a bridge knee joint with a short stub in Fig 17 A main advantage of this mechanism is broadening or fan- ning of the joint strut, thereby increas-

ing the joint strut capacity.”

This mechanism is feasible because the bar strength can be developed adjacent to the reinforcement head, which typically has a cross-sectional area ten times that of the bar, enabling development of struts directly against the reinforcement heads An example of strut formation against distributed beam headed reinforcement is shown in Fig 18, which was observed in a

test on a bridge knee joint.!?

Use of headed bars will result in al- most zero effective anchorage length for the column main reinforcement in this mechanism, as well as in the Di-

Trang 17

(a) Beam bars terminated in the stub are visible (b) Close-up view Fig 18 Formation of strut against the ends of headed reinforcing bars *I wi (a) Mechanism | * Nabe (b) Mechanism 2

Fig 19 Short-stub mechanisms for knee joints subjected to closing moments rect Transfer Mechanism (see Fig

15), enabling possible reduction of the cap beam depth To achieve this anchorage condition, the column bars

should be extended above the beam

top bars as shown in Fig 17

Further, when employing the distributed strut mechanism, it should be remembered that the beam longitudinal reinforcement quantity, the joint strut depth and demand on the struts depend on the number of beam reinforcement layers To employ this mechanism effectively in joint design, the top and bottom beam longitudinal beam bars should be placed in three or more layers as illustrated in Fig 17 Short-Stub Mechanism

The short-stub mechanism can as-

sist with anchoring of the column ten-

18

sion force into a knee joint subjected to closing moments As shown in Fig 19a, an external strut similar to that

described under the clamping mecha-

nism is relied upon for joint force transfer, which is supported by U- shaped reinforcing bars in the short stub

A detailing requirement of this mechanism is that the U-bars be part of the beam main reinforcement, en-

suring continuity between the top and

bottom longitudinal bars To improve constructability of this detail, it is recommended that short U-bars be employed in the joint region and that continuity of the beam reinforcement

be established using cither mechanical

connectors or competent splices outside of the joint.!!

With column bars extended as close

to the beam top reinforcement as pos-

sible, the column tension force can also be supported by the joint diagonal strut and another external strut an-

chored in the top corner of the stub as

shown in Fig 19b Since the horizontal component of this external strut is typically large, which will be determined by the yield strength of the beam top longitudinal reinforcement,

this mechanism can anchor an addi-

tional column tension force at this

CCT node without significantly increasing demand on the joint shear re-

inforcement

Based on test data, it is estimated that the total column tension force can be supported using the short-stub

mechanism.° The joint force transfer

mechanism described in Fig 19a will support a smaller proportion of 7, than the mechanism depicted in Fig 19b

The exact contribution of each mechanism can be determined from equilibrium conditions assuming the maximum out-of-balance force, T7,, that can be supported at Node C The value of 7, should be determined based on the reinforcement provided to support the joint force transfer when it is subjected to opening moments as illustrated in the second example in Appendix A

Therefore, it is suggested that when the short-stub mechanism is employed for closing knee joints, the required quantity of joint reinforcement will generally be governed by the mechanism responsible for force transfer under opening moments

Trang 18

Long-Stub Mechanism

The long-stub mechanism, which is conceptually similar to the short-stub mechanism, can be applied to the design of knee joints subjected to closing moments As shown in Fig 20, anchorage of the external strut in the stub is achieved using transverse ties and a strut anchored against the gravity load transferred through the long- stub.° With comparison to the clamping mechanism, a column tension force of up to 0.157 may be anchored

into the joint using the long stub

mechanism when adequate gravity loads and transfer ties are present in the stub

FORCE TRANSFER MODELS Joint force transfer models suitable for design or assessment can be formulated using a single mechanism or a combination of two or more of the key

mechanisms described above Limit-

ing the number of mechanisms to two is recommended to maintain simplic-

ity Some examples of design models

are presented below to demonstrate the application of the force transfer method

Modified External Strut Force Transfer Model

The external strut force transfer model, which was originally proposed

by Priestley*!9°!” and later modified by Sritharan,®!Š relies upon supporting 7,

equally by the clamping and splice transfer mechanisms Since this model makes the maximum use of the beam

stirrups in the joint force transfer, the

model is regarded as very efficient for seismic detailing of bridge joints Fig 21 illustrates the application of the modified external strut force transfer model to a bridge tee joint

The appropriate reinforcement required to develop the two mechanisms can be determined independently With an estimate for T., the reinforcement required for satisfactory joint force transfer can be determined (see the example in Appendix A and a pre-

scriptive set of design steps given in

Reference 18)

If a fully prestressed cap beam is used, the splice transfer mechanism is July-August 2003 TL Gravity load _ ‡11111131‡t1 : 2 ⁄ TỐ lá J ` ⁄⁄ \ A / = Ne Vy i * Ỹ Fig 20 Long-stub mechanism for knee joints subjected closing moments

not required and the tee joint can be detailed using the clamping mechanism shown in Fig 13b The possibil- ity of designing the joint utilizing only the clamping mechanism may be examined by estimating the main joint strut depth at the column tension face and comparing with the effective anchorage length of the column longitudinal bars as illustrated in Fig 6 In

joints with zero prestressing, the

splice transfer mechanism may be replaced with other mechanisms such as the direct transfer mechanism

If the cap beam is partially pre-

stressed, the combination of clamping

and splice mechanisms can still be employed, with the former supporting more than 0.57, based on the level of prestressing This will result in less joint reinforcement than would be required in an equivalent joint with zero prestressing The amount of column tension force that can be supported by the clamping mechanism may be obtained based on the percentage of beam negative moment at the column

face that is being resisted by the beam prestressing.°

Consequently, when the negative moment resisted by the beam prestressing is zero or 100 percent, the column tension force supported by the clamping mechanism is taken as 0.57,

or 1.07,, respectively, with the amount

of column tension force supported by any other level of beam prestressing being linearly interpolated between

0.57, or 1.07,

The modified external strut force transfer model can also be applied to bridge knee joints subjected to opening moments However, an alternative mechanism (e.g., the short-stub mechanism) is required for force transfer when the joint is subjected to closing moments The required joint reinforcement should be established following consideration of the reinforce-

ment details necessary to support the

force transfer across the joint for the two types of moment

As described above for tee joints, partially prestressed knee joints can

also be detailed using a combination

of clamping and splice mechanisms The column tension force supported by the two mechanisms, in this situa- tion, should be proportioned based on the beam positive moment resisted by prestressing at the column tension face

Haunched-Joint Force Transfer Model

This force transfer model combines

the haunched-joint mechanism with the splice transfer mechanism, unless the cap beam is fully prestressed The percentage of the column tension force supported by the haunched-joint mechanism should be established

based on the embedment length of the

column bars into the joint and the amount of horizontal and/or vertical shear reinforcement present in the joint As noted previously, a larger

Trang 19

Requirement for reinforcement additional top beam ~~ Splice transfer “” mechanism Fig 21 The modified external strut force transfer model

percentage of T, may be supported through the haunched-joint mechanism if proven using an appropriate strut-and-tie model

Distributed Strut Force Transfer Model

Relying solely on the distributed strut mechanism, the distributed strut force transfer model can be used for

supporting the column tension force

T The necessary joint reinforcement can be quantified using strut-and-tie models delineating the respective mechanisms shown in Fig 17 Al- though headed short bars may be used for resisting the tension force within

the joint in the vertical direction, use

of spiral or hoop reinforcement is recommended as joint ties in the horizontal direction, which serves to effectively enhance the strength of the joint

struts

The distributed strut force transfer model can also be developed by combining the distributed strut and direct transfer mechanisms The amount of reinforcement required within the joint of this model will typically be less than that required for the model based solely on the distributed strut mechanism

Short-Stub Joint Force

Transfer Model

The design of a force transfer model suitable for a knee joint with short- stub subjected to closing moments can be formulated by combining the short

20

stub mechanisms depicted in Fig 19 If necessary, the design model can be supplemented with the splice mechanism, as in the modified external strut

model

The reinforcement required for transferring the joint forces under opening moments may be derived from the modified external strut model illustrated in Fig 21 for a bridge tee joint

DESIGN PROCEDURE Following selection of a force transfer model, the design procedure suitable for a given joint type may be developed using the guidelines and

strut-and-tie concepts previously dis-

cussed Full development of these design procedures is constrained within the scope of this paper, but is presented elsewhere for the modified ex-

ternal strut force transfer model.!°

However, quantification of tension demands in the joint region following selection of a force transfer model is illustrated in Appendix A using exam-

ple problems

OUT-OF-PLANE AND BI-DIRECTIONAL LOADING

This paper has focused on application of the force transfer method to bridge joints subjected to in-plane

loading The force transfer method can

equally be applied to bridge joints

subjected to out-of-plane (i.e., the di-

rection parallel to the bridge longitudi-

nal axis) and bi-directional loading Experimental validation of the external strut force transfer model to bridge joints subjected to out-of-plane load-

ing may be found in Reference 28 For bi-directional loading, joint details may be established by employing

FTM in the transverse and longitudi-

nal directions independently When similar joint mechanisms are used in the transverse and longitudinal direc-

tions, there will be regions that require

reinforcement for each of the two loading cases

It has been suggested that the reinforcement placed in overlapping areas may be counted as effective for the

two directions.'’? This suggestion ap-

pears to be reasonable since, for example, the plastic moment capacity of a circular column remains the same for any loading direction Such simplifica- tion of the joint details must be exer- cised with sound engineering judg-

ment

The behavior of bridge joints under bi-directional loading has been experi- mentally studied.**3’7* However, the number of tests is limited for the pur- pose of systematically extending the

torce transter method to bridge joint

design under bi-directional seismic action More effort is required to complete this task, including an experimental investigation of bridge joints

designed based on carefully selected

joint mechanisms

CONCLUDING REMARKS A rational force transfer method for

seismic design and assessment of con-

crete bridge joints subjected to in- plane loading is presented in this paper This method determines the proper amount of joint reinforcement using simple analytical models based on strut-and-tie concepts, with consid-

eration to the repetitive nature of seis-

mic loading In order to assist with the practical application of this approach, several guidelines, efficient joint mechanisms and design/assessment models are also presented

Unlike the conventional joint design

approach, in which the joint shear is assumed to be an independent force, the force transfer method treats joint

shear as part of the complete force

Trang 20

transfer across the joint As a result, the force transfer method will provide reduced and less conservative reinforcement, thus improving the constructability of bridge joints

Expected Joint Performance

Based on the observed performance

of several joints designed using differ-

ent force transfer models, the following seismic performance is postulated for bridge joints designed with FTM:

¢ In small to moderate earthquakes,

which are expected frequently within

the projected lifetime of the structure, joints will respond elastically with joint reinforcement stresses significantly below the yield strength In- spection or repair of joints will not be necessary following such an event Minor joint cracking is, however, expected

* In moderate to large earthquakes, the joint reinforcement may be subjected to inelastic strains moderately exceeding the yield strain No structural repair of the joint will be necessary, although durability concerns may warrant measures such as injection of

grouting Joint deformation should not

significantly increase the lateral displacement of the bridge bent It is expected that the average joint shear

strain will not exceed 0.0025

¢ In the maximum credible earthquake, inelastic strains of up to 0.02 can be developed in the joint reinforcement, with average joint shear strains of up to 0.005 These limitations will satisfactorily control the contribution of joint deformations to the overall lateral displacement of the bent, ensuring inelastic actions within the preselected plastic hinge zones in the columns The joint damage resulting from such an event will be re- pairable

¢ When a joint is designed using a conservative force transfer model, the joint reinforcement will not be opti- mized In such cases, the average joint shear strains of up to 0.01 may be developed, with satisfactory overall joint behavior

The joints in Table 1 were subjected to quasi-static loading, and thus confir- mation of the expected joint performance needs to be validated using dy- namic load tests on laboratory bridge joints or by instrumenting bridge joints in field structures Nonetheless, the authors believe that the above informa- tion is valuable to the earthquake engi-

neering community 1n their current

efforts to establish performance-based seismic design procedures

ACKNOWLEDGMENTS The authors are indebted to Emeri- tus Professor M J Nigel Priestley, University of California at San Diego (UCSD), for his initiative towards developing the force transfer method concept for detailing of bridge joints, serving as the doctoral adviser for both authors in this area of research, and inspiring them to write this techni- cal article

The large-scale experiments on

bridge joints listed in Table 1 of this

paper were conducted at the Charles

Lee Powell Structural Laboratory at UCSD with financial support from the California Department of Transporta- tion, Alaska Department of Trans- portation and Headed Reinforcement Corporation of California

The authors are grateful to all the

sponsors of this research program for

their support

The opinions or recommendations expressed in this paper are those of the authors alone and do not necessarily reflect the views of the financial spon-

sors

The authors want to express their

gratitude to all the PCI JOLIRNAL re-

viewers for providing constructive comments on the original manuscript

Ritter, W., “Die Bauweise Hennebique,” Schweizerische Bauzeitung Bd, X XXIII, No 7, January 1899

Morsch, E., Der Eisenbeton, seine Theorie und Anwenung, Verlag Konrad Witter, Stuttgart, Germany, 1912

ACI Committee 445 - Shear and Torsion, “Strut-and-Tie Bibli- ography,” ACI Bibliography No 16, American Concrete Insti- tute, Farmington Hills, MI, September 1997, 50 pp

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Ingham, J M., “Seismic Performance of Bridge Knee Joints,” Doctoral Dissertation, Division of Structural Engineering, Uni- versity of California at San Diego, CA, 1995, 511 pp

Sritharan S., “Analysis of Concrete Bridge Joints Subjected to Seismic Actions,” Doctoral Dissertation, Division of Structural Engineering, University of California at San Diego, CA, 1998, 407 pp

Ingham, J M., Priestley, M J N., and Seible, F., “Seismic Re- sponse of Bridge Knee Joints Having Columns with Interlock- ing Spirals,” Bulletin of the New Zealand National Society for Earthquake Engineering, V 30, No 2, 1997, pp 114-132 July-August 2003

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14 lồ 16 Lis 18 Le 20 21 22 2s 24 25 26 22

“Seismic Design and Proof Test of a Bridge Bent Having

Three Steel Jacketed Columns,” Structural Systems Research Report No SSRP 98/13, University of California at San Diego,

CA, 1999,

ATC, “Improved Seismic Design Criteria for California Bridges: Provisional Recommendations,” ATC-32, Applied Technology Council, Redwood City, CA, 1996

ATC/MCEER Joint Venture, “Recommended LRFD Guide- lines for the Seismic Design of Highway Bridges, Part I: Spec- ifications and Part II: Commentary (Preliminary Reports),” ATC-49/MCEER-02-SP01, Applied Technology Council, Redwood City, CA, 2001

Caltrans, Seismic Design Criteria, Version 1.2, California De- partment of Transportation, Sacramento, CA, July 2001 Priestley, M J N., Seible, F., and Calvi, M., Seismic Design and Ketrofit of Bridges, John Wiley & Sons, NY, 1996, 686

Pp

Sritharan, S., “Strut-and-Tie Analysis of Bridge Joints Sub- jected to In-plane Seismic Actions,” Submitted for publication in the ASCE Journal of Structural Engineering

Sritharan, S., “An Improved Seismic Design Procedure for Concrete Bridge Joints,” Submitted for publication in the ASCE Journal of Structural Engineering

Schlaich, J., Schaefer, K., and Jennewein, M., “Toward a Con- sistent Design of Structural Concrete,” PCI JOURNAL, V 32, No 3, May-June 1987, pp 75-149

Sritharan, S., Priestley, M J N., and Seible, F., “Seismic Re- sponse of Column/Cap Beam Tee Connections with Cap Beam Prestressing,” Structural Systems Research, Report No SSRP 96/09, University of California at San Diego, CA, December

1996, 296 pp

Sritharan, S., Priestley, M J N., and Seible, F., “Seismic De- sign and Performance of Concrete Multi-Column Bents for Bridges,” Structural Systems Research, Report No SSRP 97/03, University of California at San Diego, CA, June 1997, 331 pp

Sritharan, S., Ingham, J M., Priestley, M J N., and Seible, F., “Bond Slip of Bridge Column Reinforcement Anchored in Cap Beams,” ACI Special Volume on Bond and Development of Reinforcement — A Tribute to Dr Peter Gergely, ACI SP-180,

American Concrete Institute, Farmington Hills, MI, pp 319- 345

MacGregor, J G., Reinforced Concrete Mechanics and Design, Prentice Hall, Upper Saddle River, NJ, 1988, 799 pp Collins, M P., and Mitchell, D., Prestressed Concrete Struc- tures, Response Publications, Toronto, Ontario, Canada, 1997, 766 pp

Bhide, S., and Collins, M P., “Tension Influence on Shear Ca- pacity of Members,” ACI Structural Journal, V 86, No 5, September-October 1989, pp 570-581 27 28 29, 30 31 32 ID 34 35 36 37 38

Bergmeister, K., Breen, J E., Jirsa, J O., and Kreger, M E., “Detailing in Structural Concrete,” Center for Transportation Research, Report No 1127-3F, The University of Texas at Austin, TX, May 1993

Holombo, J., Priestley, M J N., and Seible, F., “Continuity of Precast Spliced-Girder Bridges Under Longitudinal Seismic Loads,” PCI JOURNAL, V 45, No 4, March-April 2000, pp 40-63

Reineck, K., “Ultimate Shear Force of Structural Concrete Members Without Transverse Reinforcement Derived From a Mechanical Model,” ACI Structural Journal, V 88, No 5, September-October 1991, pp 592-602

Vecchio, F J., and Collins, M P., “The Modified Compression

Field Theory for Reinforced Concrete Elements Subjected to

Shear,” ACI Structural Journal, 83, No 2, March-April 1986,

pp 219-231

Hsu, T T C., Unified Theory of Reinforced Concrete, CRC Press, Boca Raton, FL, 1993, 313 pp

To, N H T., Ingham, J M., and Sritharan, S., “Montonic Non- linear Analysis of Reinforced Concrete Knee Joints Using Strut-Tie Computer Models,” Bulletin of the New Zealand So- ciety for Earthquake Engineering, Wellington, New Zealand, V 34, No 3, September 2001, pp 169-190

Thewalt, C R., and Stojadinovic, B., “Behavior of Bridge Out- rigger Knee Joint Systems,” Earthquake Spectra, V 11, No 3, August 1995, pp 477-509

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Sritharan, S., Abendroth, R E., Greimann, L F., Wassef, W G., and Vander Werff, J., “Seismic Performance of a Concrete Column/Steel Cap/Steel Girder Integral Bridge System,” Pro- ceedings of the Third National Seismic Conference & Work- shop on Bridges and Highways, Portland, OR, April 2002, pp 411-422

Priestley, M J N., Seible, F., and Anderson D L., “Proof Test of a Retrofit Concept for the San Francisco Double-Deck Viaducts,” Structural Systems Research, Report No SSRP 92/03, University of California at San Diego, CA, July 1992, 198 pp

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APPENDIX A — DESIGN EXAMPLES

Two design examples illustrating the application of the force transfer method are given in this Appendix In these examples, the tension demands in the joint region are determined following selection of a suitable force transfer model From the tension demands, the necessary reinforcement can be readily obtained

Unless otherwise noted, the reinforcement distribution should be such that the orientation and magnitude of the resultant force of the distributed reinforcement should coin- cide, respectively, with the direction and demand of the ties established for supporting the selected joint mechanisms

For simplicity, the following assumptions are made in

each of the examples The column that frames into the joint is 1.2 m (47.24 in.) in diameter and consists of 20 No 14 (d,; = 43 mm or 1.69 in.) longitudinal bars, corresponding to about 2.5 percent longitudinal reinforcement With adequate

confinement, as suggested in current design practice,'*'” the

moment capacity of the column is expected to fully develop in the plastic hinge region adjacent to the joint

An unconfined concrete strength of 30 MPa (4.35 ksi) and a reinforcement yield strength of 455 MPa (66 ksi) are assumed The column is subjected to an axial force of 1700 kN (382 kips) due to gravity loads, which represents a column axial load ratio of 5 percent (i.e., P/Agfz = 0.05, where P, A, and f’ are, respectively, the column axial load, gross section area, and concrete strength)

While this axial load may be assumed to be unaltered in

columns of tee joints, the axial loads in columns framing into knee joints will be modified due to seismic actions For the design examples, the combined axial load due to gravity and seismic actions is taken as 10 and 0 percent, when the joint is subjected to closing and opening moments, respec-

tively

The joint principal tensile stress is assumed to exceed

0.42,/ f’?(MPa) [or 5.0.,/ f’(psi)] in both examples, requir-

ing joint design based on a force transfer model that supports the total column tension force, T, This assumption is appropriate for the selected joint dimensions and column longitudinal steel ratio

For example, with an axial load ratio of 5 percent, the column framing into the tee joint is expected to develop an overstrength moment of 9090 kN-m (80460 kip-in) Ignoring the axial load in the cap beam, Eq (4) estimates p, = 16/2 and ?, = 0.737 (MPa) [or 8.8 £’ (psi) ] for the tee joint

From Eqs (12) and (13), it is determined that the minimum required /, = 1072 mm (42.2 in.) and J, 47 = 500 mm

(19.7 in.)

The value of 7, and locations of ties and struts in the column are established from analyses of the column section with the appropriate axial loads Inclinations of various struts are determined using the dimensions specified in fig- ures representing the various key mechanisms

Assuming single layers of top and bottom beam reinforce-

ment with a bar diameter of 43 mm (1.69 in.) and a cover concrete of 50 mm (1.97 in.) to all longitudinal bars, the cap

beam is dimensioned with a depth of 1250 mm (49.2 in.)

July-August 2003

and a width of 1500 mm (59.1 in) Consequently, the column bars are anchored into the joint with /, = 1100 mm (43.3 in.) and there is a gap of about 57 mm (2.24 in.) between top of the column bars and the underside of the beam longitudinal bars

Example 1 Bridge Tee Joint

The modified external strut force transfer model (Fig 21), utilizing the clamping mechanism and the splice transfer mechanism is selected for the joint design Each mechanism is assumed to support a column tension force of 0.57, as previously discussed Estimation of tension demands in the joint region are determined in terms of 7, which is estimated to be 11035 kN (2481 kips) from an analysis of the column section

The tension demands resulting from the clamping mechanism are shown in Fig Al Accordingly, the formation of an external joint strut with an incline of 45 degrees imposes tension demands in the beam region adjacent to the column tension face Using the maximum value of 0.157 for T.,, Ty, is estimated to be 0.157 In addition, a tension demand of 7, develops within the joint in the horizontal direction, where:

T, =(0.5 —0.15)7,tan48 — 0.157,

~ 0.25 (AI)

Quantification of joint spiral (or circular hoop) reinforce-

ment needed to support the tension demand estimated in Eq

Trang 23

—\ 0.25T,

Fig A2 Tension demands due to the splice mechanism (Dimensions are in mm.) {a) Transfer of column tension force (b) Anchorage of column tension force _ 2.4(0.257,) _ 0.67,

fala Soule (A2)

where f,,, is the yield strength of joint spiral and /, is the embedment length of column bars into the joint It is suggested that the spiral reinforcement ratio be maintained over the embedment length of the column bars starting above the bottom beam bars Also, the minimum spiral reinforcement

requirement of Eq (7) must be satisfied

The splice mechanism shown in Fig 14 is represented using simplified models in Fig A2 Fig A2a illustrates in a simplified manner the estimation of tension demands resulting from transfer of the column tension force to Node D, po- sitioned above the column bars at the location of the beam top longitudinal bars In this simplified figure, it is assumed that two vertical tie forces of 0.257, in magnitude, posi- tioned on each side of the joint and representing vertical

stirrups and concrete tensile resistance within the joint and

in the beam directly adjacent to the joint, will collectively introduce a tension force of 0.57, with a centroid acting at Node D This force will be anchored primarily with diagonal struts and beam top reinforcement as shown in Fig A2b

The demands induced by the splice mechanism may be estimated as follows:

¢ Assuming 50 percent contribution from the two ties located at the joint-to-beam interfaces in Fig A2a, a tension demand of 0.257, may be estimated within the joint panel in the vertical direction The tension carrying capacity of the cracked joint concrete may be assumed to support 25 per-

cent of 0.257 as discussed elsewhere.!”

¢ In addition to the demand of 0.157, previously calculated for the clamping mechanism, the bottom beam bars will be subjected to a demand of:

Tp, = 0.75 X 0.257,tan38 = 0.157 (A3)

Hence the total value of 7), is 0.30T

* Top beam bars will be subjected to a tension demand of

0.367 Derivation of this tension force assumes that 50 per-

24

cent of 7, estimated from the clamping mechanism participates in supporting the column force here Consequently:

Tp, = (0.57, — 0.57,tan40)tan31 + 0.57,

T, = 0.25T., thus, T,, = 0.36T, (A4)

The reason for using only 0.57, in the above calculation is that the horizontal joint shear reinforcement is partly relied upon for transfer of the column tension force to Node D as detailed in Fig 14

The splice transfer mechanism imposes an additional tension demand in the vertical direction in the beam region adjacent to the column compression face Although this demand could be quantified using the strut and tie forces in Fig A2b, an approximation for this demand may be taken as

0.257,.'? Hence, the value of 73, is approximated to 0.257,

With this estimate for 7%,, the beam regions adjacent to the joint should be designed with adequate reinforcement to

support a force of 7%, plus the calculated beam shear

Some of the assumptions made above are based on the experimental observations and the authors’ experience with the subject matter, which was primarily at minimizing shear reinforcement within the joint panel and improving constructability Strictly following the mechanisms for determining tension demands will also lead to satisfactory joint performance, but will result in somewhat larger reinforcement quantities In this case, the determination of 7 based on Eq (5) may be more appropriate as this gives about 25 percent less value than that reported above based on the section analysis

The estimates presented above are for one direction of loading, and the final estimates of the tension demands in the joint region should consider the loading in the two direc-

tions As a result, the following tension demands should be

used to quantify the reinforcement in the joint region: ¢ 0.197 to quantify the vertical joint shear reinforcement;

- 0.257, to quantify the spirals within the joint pancl;

¢ 0.257 to quantify the stirrups in the beam region adjacent to the joint;

* 0.307, to quantify additional bottom beam reinforce-

Trang 24

ment across the joint;

¢ 0.367 to quantify additional top beam reinforcement across the joint

Example 2 Bridge Knee Joint

The design of the knee joint should consider forces expected when the joint is subjected to both opening and closing moments The modified external strut force transfer model may be employed for transfer of joint forces under opening moments and the corresponding tension demands may be obtained as illustrated above for the tee joint in the first example For joint opening moments, the value of T, is found to be 12014 kN (2701 kips)

For joint closing moments, the short-stub mechanism is

relied upon for supporting the entire column tension force as shown in Fig A3, which combines Mechanisms 1 and 2 depicted previously in Fig 19 With a column axial load ratio of 10 percent estimated due to gravity and seismic actions combined, the column tension force J is estimated to be

10555 KN (2373 kips)

Assuming that the joint spiral reinforcement, which was provided for the force transfer when the joint is subjected to opening moments, can support an out-of-balance force 7, = 0.25T at Node C, the tension demands in the joint region can be quantified The vertical components of Struts Cy and C3 are equal (see Fig A3) Hence:

C›sin50 = Œ3sin26.7 (A5)

From an equilibrium of forces in the vertical direction

at Node C:

C,cos41.5 = 7, (A6)

From an equilibrium of forces in the horizontal direction at Node C:

T, = 0.257, = C,sin41.5 — C,cos50 — C3c0s26.7 (A7) From Eggs (A5 to A7), magnitudes of Struts C), C, and C3 are found to be 1.347,, 0.297, and 0.507., respectively Using these estimates, tension demands 7;,, 7;,, and 7), can be determined as follows: Tp, = Cscos50 ~ 0.207, (A8) duly-August 2003 Fig A3 Tension demands due to the short-stub mechanism (Dimensions are in mm.) Ty,= Cscos26.7 ~ 0.457, (A9) T,, = C38in26.7 = 0.257, (A10)

The incline of the struts in the short stub, especially C;, will be significantly influenced by the number of layers of

the beam longitudinal reinforcement In this example, one

layer of reinforcement was assumed When reinforcement is placed in multiple layers, the top beam tension chord should be located where the resultant tension force is expected which will reduce the inclination of C3; and increase the value of T;,

Extending the beam top and bottom longitudinal reinforcing bars through the joint and making them continuous in

the stub will be sufficient to support the tension demands es-

timated in Eqs (A8), (A9) and (A10) This can be ensured following the design of the beam longitudinal reinforcement Although the vertical portion of the beam reinforcement in the stub will adequately support the tension force 7;,,, nominal beam transverse reinforcement is recommended in the stub to adequately confine concrete in this region

More examples illustrating the application of FTM specific to different joint conditions may be found in Refer- ences 7, 8, 12 and 19

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26 APPENDIX B — NOTATION = depth of equivalent stress block in beam on left side of joint = depth of equivalent stress block in beam on right side of joint

= depth of equivalent stress block in column = gross section area

= area of joint vertical stirrup

= total area of column longitudinal reinforcement = area of vertical joint reinforcement

= column width = joint effective width = compression force

= resultant beam compression force due to flexure = resultant beam compression force due to flexure

on left side of joint

= resultant beam compression force due to flexure on right side of joint

= resultant column compression force due to flexure = effective beam depth

= diameter of reinforcing bar

= diameter of longitudinal reinforcing bar = effective beam depth on left side of joint = effective beam depth on right side of joint = column diameter

= unconfined concrete strength = cracking strength of concrete

= average joint normal stress in horizontal direction = average principal tensile stress for cracked con-

crete

= stress in steel reinforcement

= average joint normal stress in vertical direction = yield strength of steel reinforcement

= yield strength of column longitudinal bars yield strength of hoop reinforcement

= overstrength stress in column longitudinal reinforcement = ultimate strength of reinforcing bar = prestressing force = cap beam depth = column depth = length of joint panel in loading direction = anchorage length

= effective anchorage length = column overstrength moment = cap beam moment

= cap beam moment on left side of joint = cap beam moment on right side of joint = average joint principal compression stress = average joint principal tensile stress

|

column axial load

cap beam axial force on left side of joint cap beam axial force on right side of joint magnitude of strut

tension force

= tension force in bottom beam reinforcement = additional tension demand in bottom beam longitudinal reinforcement tension in bottom beam reinforcement on left side of joint tension in bottom beam reinforcement on right side of joint additional tension demand in top beam longitudinal reinforcement

total column tension force estimated at overstrength moment capacity

additional demand in external stirrups adjacent to column tension face

additional demand in external stirrups adjacent to column compression face

tensile resistance by concrete between tension cracks

transverse direction tension in a bottle-shaped

stress field

= total tensile resistance

out-of-balance tension force

average joint shear stress

average joint shear stress in horizontal direction

average joint shear stress 1n vertical direction

cap beam shear adjacent to left side of joint = cap beam shear adjacent to right side of joint

average joint shear force in horizontal direction = average joint shear force in vertical direction = column shear at overstrength condition

effective width of strut

effective joint width for computing /,

moment resistance due to beam shear

= factor representing bond characteristics factor representing type of load history average principal tensile strain

average principal compression strain average strain in x direction

average strain in y direction average joint shear strain displacement ductility = angle of principal strain

= volumetric ratio of horizontal hoop reinforcement

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