Proceedings of the Institution ofCivil EngineersStructures & Buildings 160February 2007 Issue SB1Pages 31–35

Paper 14594Received 19/01/06Accepted 21/09/06

Keywords:beams & girders/cables & tendons/concrete structures

Bilal El-ArissAssistant Professor, UnitedArab Emirates University,United Arab Emirates

Strut–tie analysis of beams with external tendons

B. El-Ariss Phd, PEng, MASCE

Strengthening of concrete beams by means of external

tendons has been increasingly used. Analysis of such

beams is more difficult than that of beams with internal

bonded tendons because the stress in external tendons

depends on the deformations of the whole beam. In the

current paper, a simple model based on the strut-and-tie

method is presented to predict the capacity of simply

supported externally prestressed concrete beams

subjected to equal concentrated loads at third-points.

The model defines the struts and ties that form a load

transfer mechanism in the beams. It predicts the

ultimate loads the beam can carry and the force in the

tendons at failure using the section method and

equilibrium equations. When comparing the results

obtained using the proposed model with others in the

literature, the model was found to be conservative. The

proposed model is intended to help designers in the

inspection and structural assessment of beams.

NOTATION

Ac area of the concrete compression zone at failure

As area of tensile normal steel bars

a one-third of the beam span

C resultant compressive force in concrete at failure

D compressive force in the diagonal strut

ds effective depth of normal steel bars

dp effective depth of prestressing tendons

F ultimate load (at failure)

f 9c compressive strength of concrete

fy yielding strength of normal steel bars

jc depth of the centroid of the concrete compression zone

(area) at failure

M moment

m point where moment is taken

P prestressing force at failure

T resultant tensile force in normal steel bars at failure

� the angle measured clockwise from a horizontal line to

the diagonal strut

�cmax concrete maximum compressive strain at the extreme

fibre

�cu crushing strain of concrete

1. INTRODUCTION

The use of external prestressing technique has been growing

rapidly in recent years. Analysis and design of structures with

external tendons is conceptually different from that of

structures with internal tendons and is still not fully

understood. The main difference in behaviour between

members with internal tendons and those with external

tendons lies in the deflected shape of the beam and the tendons

and in the strain incompatibility between the concrete and

external tendons. This makes the deformation and hence the

stress in the external tendon member dependent rather than

section dependent. Many investigations have been carried out

on externally prestressed members. Muller and Gauthier,1

developed a finite-element computer program for the ultimate

response of simply supported and continuous beams with

external tendons. Their model requires information regarding

the moment against curvature or moment against rotation

relationship. Alkhairi and Naaman,2 have proposed a simplified

methodology to compute the stress in unbonded internal/

external steel tendons in the elastic range as well as the

ultimate resistance. The methodology introduces strain

reduction coefficients to convert the analysis of a beam with

unbonded tendons to analysis of a beam with bonded tendons,

hence allowing a conventional sectional (fictitious section)

analysis to be performed. This methodology requires the

calculations of different strain reduction coefficients for

different spans, loads and tendon profiles. Xiao-Han Wu and

Xilin Lu,3 have proposed a model for non-linear analysis of

externally prestressed beams, which is capable of simulating

the slip of the steel tendons at the deviators but under-predicts

the nominal strength of the beams. Their study did not account

for the effects of span-to-depth ratios on the flexural

behaviour of the beam.

The current study is one of the continuing efforts to describe a

simple analytical model to understand better the flexural

behaviour of beams with external prestressing tendons, and to

better inspect and assess the above beams in a better way.

2. OBJECTIVES

The basic objective of the present paper is to develop a simple

model to predict the load-carrying capacity of simply

supported, externally prestressed concrete beams subjected to

equal concentrated loads at third-points; it is also to predict the

force in the external tendons at the ultimate limit state (failure)

using the section method and equations of equilibrium. The

model is intended to help designers in the inspection and

structural assessment of statically determinate beams.

Structures & Buildings 160 Issue SB1 Strut–tie analysis of beams with external tendons El-Ariss 31

3. PROPOSED ANALYTICAL MODEL

The strut-and-tie method (STM) is based on the lower-bound

theory of limit analysis. In the STM, the flow of internal forces

is idealised as a truss carrying the external loading through the

region to its supports. Like a real truss, a strut-and-tie model

consists of struts and ties interconnected at nodes (also referred

to as nodal zones or nodal regions). As shown in the figures,

struts are symbolised using broken lines, and ties are denoted

using solid lines. Struts are the compression members of a

strut-and-tie model and represent concrete stress fields whose

principal compressive stresses are predominantly along the

centre-line of the strut. Ties are the tension members of a strut-

and-tie model. In the present paper, the ties represent the

reinforcing and prestressing steel. The nodes are analogous to

joints in a truss and are where forces are transferred between

struts and ties. As a result, these regions are subject to a

multidirectional state of stress. The strut-and-tie model has to

be in equilibrium externally with the applied loading and

reactions and internally at each node. Equations of equilibrium

and the section method are used to solve for the ultimate loads

applied on the beams and the corresponding prestressing force

in the external tendons. In the analysis, the external tendons

are replaced by their action on the beam at the anchorage and

deviator locations, Figs 1(b) to 3(b), and the tendons are not

part of the section at any other locations. This is because the

external tendons are unbonded to the concrete, except at

anchorages and deviator locations, and the stress in such

tendons depends on the deformations of the whole member and

is assumed uniform at all sections.

There is no unique strut-and-tie model for a given problem,

and more than one strut-and-tie model may be developed for

each loading case as long as the selected truss is in equilibrium

with the boundary forces and the stresses in the struts, ties and

nodes are within the acceptable limits. The STM selection and

the acceptable limits consist of rules for defining the

dimensions and ultimate stress limits of struts and nodes as

well as the requirements for the distribution and anchorage of

reinforcement. Guidelines for these limits and for selecting

(N.A.)

ds

(b)

TP

F

F

C

m

a

dp

jc

e

0·85 fc�

F F

F F

PP

a aa

Ap AsExternaltendons

Section A–A

(a)

A

A

FNeutralaxis (N.A.)

S

aa a

Externaltendons

F

Anchorage

Eccentricity, e

Fig. 1. (a) Beam with external straight tendons; (b) strut-and-tie model of the beam

(a)

Ap AsExternaltendons

Section A–A

A

A

FNeutralaxis (N.A.)

Saa a

Externaltendons

F

AnchorageDeviator

Eccentricity, e( )xx

(b)

FP

F

P

F Fa aa

P α)sin(

P α)sin(

P α)sin(

P α)sin(

P α)sin(

P α)sin(

Pcos( )α

P αcos( )

P αcos( )Deviators

F

F

P

T

Cdp

jce( )x

m

0·85 f �cn

a

α

α

α

(N.A.)

ds

Fig. 2. (a) Beam with external two-point deviated tendons; (b)strut-and-tie model of the beam

α

�

α

Ap AsExternaltendons

Section A–A

(a)

A

A

FNeutralaxis (N.A.)

Saa a

Externaltendons

F

AnchorageDeviator

Eccentricity, e( )xx

(b)

F

P

F

P

F Fa aa/2 a/2

Deviator

F

F

Pe( )x

T

C

D

dp

jc

m

0·85f �c

a a/2

Cut is just to theleft of deviator

Psin( )α

Psin( )α

2 sin( )P α

Psin( )α

Pcos( )α

Pcos( )α

Pcos( )α

(N.A.)ds

α

Fig. 3. (a) Beam with external one-point deviated tendons;(b) strut-and-tie model of the beam

32 Structures & Buildings 160 Issue SB1 Strut–tie analysis of beams with external tendons El-Ariss

STM have been incorporated in the Canadian Concrete Design

Code,4 and in the 2002 American Concrete Institute (ACI)

code.5

3.1. Beams with straight external tendon profiles

The strut-and-tie model for a beam with external tendons

whose profile is a straight line is shown in Fig. 1. The beam is

subjected to equal concentrated loads at third-points. For the

analysis of this beam, the section method and three equations

of equilibrium are applied as follows (Fig. 1(b))

�Fx ¼ 0; P þ T � C ¼ 01

�Mm ¼ 0; F(a)� C dp � jcð Þ � T (ds � dp) ¼ 02

T ¼ As fy3

C ¼ 0:85 f 9cAc4

where components of equations (1) to (4) are defined in the

Notation.

By solving equations (1) and (2) ultimate load F, the

prestressing force P and the distance jc at failure can be

obtained.

Equations (1) and (2) are solved by trial and error as follows.

(a) Assume a value for P.

(b) Knowing T at ultimate using equation (3), C can be

obtained from equation (1).

(c) Using C in equation (4), Ac can be computed and therefore

jc is calculated since the width of the section is given.

(d ) With jc known, and therefore the location of the neutral

axis is known, compute the concrete maximum

compressive strain at the extreme fibre, �cmax, using the

section strain distribution; at ultimate the yielding strain in

the steel bars is �y ¼ 0.002.

(e) If �cmax is close or equal to the crushing strain of concrete,

�cu ¼ 0.003, then compute F using equation (2) and the

solution has been obtained.

( f ) If �cmax 6¼ �cu, repeat steps (a)–(e) (by assuming a new

value for P) until �cmax is close or equal to �cu. F and P are

then obtained.

3.2. Beams with deviated external tendon profiles

3.2.1. Two-point deviated tendon profiles (deviations at third-

points). The strut-and-tie model for a beam with external

tendons whose profile is deviated at two points along the span

of the beam is shown in Fig. 2. The beam is subjected to equal

concentrated loads at third-points. For the analysis of this

beam, the section method and three equations of equilibrium

are applied as follows (Fig. 2(b))

�Fx ¼ 0; P cos(Æ)þ T � C ¼ 05

�Mm ¼ 0;

F að Þ � P sin Æð Þ að Þ � C dp � jcð Þ � T (ds � dp) ¼ 06

By solving equations (5) and (6) ultimate load F, the

prestressing force P and the distance jc at failure can be

obtained. Equations (5) and (6) are solved by trial and error

using steps (a)–(f ) described previously and equations (5) and

(6) in place of equations (1) and (2), respectively.

3.2.2. Single-point deviated tendon profile (deviation at

midpoint). The strut-and-tie model for a beam with external

tendons whose profile is deviated at midpoint of the beam is

shown in Fig. 3. The beam is subjected to equal concentrated

loads at third-points. For the analysis of this beam, the section

method and three equations of equilibrium are applied as

follows (Fig. 3(b))

�Fy ¼ 0; P sin(Æ)� D sin(�) ¼ 0;

D ¼ P sin(Æ)

sin(�)

7

where D is the compressive force in the diagonal strut and � is

the angle measured clockwise from a horizontal line to the

diagonal strut, Fig. 3(b).

tan �ð Þ ¼ ds � jcð Þa=2ð Þ8

�Fx ¼ 0; P cos(Æ)þ T � C � D cos(�) ¼ 09

�Mm ¼ 0; F að Þ þ D cos(�)(ds � dp)

� D sin(�)3a

2

� �� C dp � jcð Þ � T (ds � dp) ¼ 0

10

By solving equations (9) and (10) the ultimate load F, the

prestressing force P and the distance jc at failure can be

obtained.

Equations (9) and (10) are solved by trial and error as follows.

(a) Assume a value for P and assume jc equals zero.

(b) Compute D using equations (7) and (8).

(c) Knowing T at ultimate using equation (3), C can be

obtained from equation (9).

(d ) Using C in equation (4), Ac can be computed and therefore

jc is calculated since the width of the section is given.

(e) With jc known, and therefore the location of the neutral

axis is known, compute the concrete maximum

compressive strain at the extreme fibre, �cmax, using the

Structures & Buildings 160 Issue SB1 Strut–tie analysis of beams with external tendons El-Ariss 33

section strain

distribution; at ultimate

the yielding strain in the

steel bars is �y ¼ 0.002.

( f ) If �cmax is close or equal

to the crushing strain of

concrete, �cu ¼ 0.003,

compute F using equation

(10) and the solution has

been obtained.

(g) If �cmax 6¼ �cu, repeat steps(b)–(f ) (by assuming a

new value for P and

using jc from step (d)

until �cmax is close or

equal to �cu. F and P are then obtained.

4. ANALYSIS AND RESULTS

The relatively limited number of tests available does not cover

all the aspects related to external prestressing of concrete

beams, such as different tendon profiles. Besides, most of the

studies reported in the literature do not give complete test data

and therefore could not be used to validate the model. Hence,

the current author carried out the verification and the reliability

of the analytical model proposed in this paper with the test

results found in the literature where the full data and all

parameters needed in this model were available. These reported

test results were those of beams tested by Aparicio et al.6

The test programme of Aparicio et al. included simply

supported and continuous beams, monolithic and segmental,

tested up to failure by flexure and by flexure and shear. In the

current study the test results of beams tested in flexural failure

by Aparicio et al. were considered. The variables studied in

their flexural tests were directly involved in the evaluation of

the nominal flexural resistance. The characteristics of the tested

beams are shown in Table 1.

The span of the monolithic simply supported beams (beams

M2, M3 and M4) was 7.20 m, and the depth was 0.60 m as

shown in Fig. 4. The cross-section was a box girder, Fig. 5,

with webs and flanges 10 cm wide. The tendons were deviated

at third-points by concrete deviators and placed outside the

box girder to make monitoring possible. The strands were with

no duct, and the deviators were steel tubes embedded in the

concrete. The monolithic beams were reinforced with 8 mm

diameter bars, in order to resist self weight during transport

and placing at the laboratory. The flexure tests were performed

by loading symmetrically the beams with two loads applied

over the diaphragms used for deviating the tendons at third-

points. The shear reinforcement was ignored in this research.

Since the model described in this paper deals with monolithic

simply supported beams with flexural behavior, beams, M2, M3

and M4 tested by Aparicio et al. were selected for the

verification of the model. The normal reinforcement bars have

yielding strength of 400 MPa. The compressive strengths of the

concrete in the beams M2, M3 and M4 were 30.0, 33.1 and

36.0 MPa, respectively. It can be seen from Table 2 that the

analytical results obtained from the model and the

experimental results of Aparicio et al. are comparable.

5. CONCLUSIONS

A simple model based on the STM is presented to predict

the load-carrying capacity of simply supported, externally

prestressed concrete beams subjected to equal concentrated

loads at third-points when flexural failure occurs. The

approach considers only the flexural behaviour of beams

and neglects the shear reinforcement. The model defines

Beam Type Erection Statics scheme Prestressing

M2 Bending Monolithic Simply supported 4 � 15 mmM3 Bending Monolithic Simply supported 6 � 15 mmM4 Bending Monolithic Simply supported 8 � 15 mm

Table 1. Characteristics of the tested beams, Aparicio et al.6

External prestressing

720

200 205 200

240 240 240

45

22·5 22·5

4535 35

Fig. 4. Geometry of the simply supported beams (dimensions in cm)

120

25

25 70 25

10

40 60

10

10 50 10 25

Fig. 5. Geometry of the cross-section of the beams(dimensions in cm)

34 Structures & Buildings 160 Issue SB1 Strut–tie analysis of beams with external tendons El-Ariss

the struts and ties that form a load transfer mechanism in

the beams at the ultimate limit state. It predicts the

ultimate loads the beam can carry and the force in the

external tendons at the ultimate limit state using the

section method and equations of equilibrium. Table 2

shows that the analytical and experimental results are

comparable.

The model predicts the solution of the equations of equilibrium

that show that the ultimate loads as well as the prestressing

force in the tendons depend on the compressive strain of the

concrete and the depth of the neutral axis, the compression

zone of the concrete. This indicates that the ultimate loads are

larger than those calculated using the compressive strain limit

for the concrete under bending moment only, and no axial

force. Therefore, this model shows the capabilities of external

prestressing for repairing structures (already well known) and,

hence, can be used by designers as a tool for preliminary

inspection and structural assessment of statically determinate

beams.

The relatively limited number of tests available and the lack of

studies in the literature that report a complete set of needed

data show the need for more experimental results and data to

validate the analytical results from the model presented in the

present paper and to provide more complete and satisfactory

results.

A parametric study should be carried out to investigate the

influence of some parameters such as span-to-depth ratio,

shear reinforcement and tendon eccentricity on the capacity of

externally prestressed concrete beams.

REFERENCES

1. MULLER J. and GAUTHIER Y. Ultimate behavior of precast

segmental box-girders with external tendons. Proceedings of

the International Symposium: External Prestressing in

Bridges, ACI SP 120–17 (NAAMAN A. E. and BREEN J. E.

(eds)), American Concrete Institute (ACI), Detroit, Michigan,

1989, pp. 355–373.

2. ALKHAIRI M. and NAAMAN A. E. Analysis of beams

prestressed with unbonded internal or external tendons.

Journal of Structural Engineering, 1993, 119, No. 9,

2680–2700.

3. XIAO-HAN W. and XILIN L. Tendon model for nonlinear

analysis of externally prestressed concrete structures.

Journal of Structural Engineering, 2003, 129, No. 1,

96–104.

4. CANADIAN STANDARDS ASSOCIATION. Technical Committee

A23.3, Design of Concrete Structures CSA A23.3–94. CSA

Rexdale, Ontario, December 1994.

5. AMERICAN CONCRETE INSTITUTE. Building Code Requirements

for Structural Concrete (ACI 318–02) and Commentary (ACI

318R-02). ACI, Farmington Hills, Michigan, 2002.

6. APARICIO A. C, RAMOS G. and CASAS J. R. Testing of

externally prestressed concrete beams. Engineering

Structures, 2002, 24, No. 1, 73–84.

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Flexural test results (Aparicio et al.6) Flexural analytical results

Beam Ultimate load F: kN Prestressing force at failure,P: kN

Ultimate load, F: kN Prestressing force at failure,P: kN

M2* 255 842 224 812M3* 373 1284 330 1195M4* 441 1712 418 1601

*Beams are simply supported with external tendons deviated at third-points

Table 2. Experimental and analytical results

Structures & Buildings 160 Issue SB1 Strut–tie analysis of beams with external tendons El-Ariss 35