Non-Linear Analysis of Precast Concrete Segment Joints

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 9 (2018) pp. 6979-6988

© Research India Publications. http://www.ripublication.com

6979

Non-Linear Analysis of Precast Concrete Segment Joints under High

Compressive Loads

Brian Adjetey Boye1, *Samuel Jonah Abbey2, Adegoke Omotayo Olubanwo3 and Joao Fonte4

1Principal Tunnel Engineer, Jacobs (CH2M Hill)

2Lecturer in Structures/Geotechnics, University of South Wales, UK. (* corresponding author)

3Lecturer in Civil Engineering and Finite Element, Coventry University, Coventry, UK.

4Senior Tunnel Engineer, Jacobs (CH2M Hill)

Abstract

When a precast concrete tunnel lining is loaded, the joints rotate

as the lining deforms leading to varying contact lengths. This

paper presents a numerical study on a precast concrete segment

joint subjected to a high compressive loading, using nonlinear

finite element analysis. The precast concrete segment joint was

idealised as a two -dimensional (2D) prismatic block and

subjected to a 6000kN compressive load. The varying contact

lengths due to deformation of the linings and joint rotation

under loading was represented as varying load widths in the

idealised model. The Mohr Coulomb model was used to

approximate the non-linear behaviour of concrete in tension and

compression with an introduction of tension cut-off to the Mohr

Coulomb model to represent the tensile capacity of concrete based

on mean tensile strength. The transverse tensile bursting stresses

in the idealised block was analysed for different load width

ratios (a/d) and compressive strengths of concrete and

presented as proportions of the uniformly distributed basic

stress on the idealised block. The tensile bursting stress

distributions for varying load widths and characteristics

concrete strengths was developed into design charts for use by

engineers in industries to eliminate the need for extensive finite

element analysis and use of empirical relations. The area under

the bursting stress distribution curve is the total bursting force

and was found to decreases with increase in load width ratio.

Keywords: tensile bursting, precast concrete, tunnel segments,

radial joint, TBM ram, circumferential joint, reinforced

concrete, finite element analysis.

INTRODUCTION

Precast concrete segments are subjected to various load

actions from the time they are cast in the factory, through

various stages of handling, storage and transportation to the

final long-term loads in the ground after installation. One of the

most critical modes of failure is known as bursting. When a

concentrated load acts on one of the joint faces of the segment,

tensile bursting forces develop beneath the applied

concentrated load. These tensile bursting forces are

perpendicular to the applied load. The radial joints are stressed

due to the axial loads in the lining resulting from applied external

ground or hydrostatic loads and should be designed to prevent

failure due to bursting. The circumferential joints are stressed

under the action of the Tunnel Boring Machine (TBM) thrust

rams during construction. Bursting stresses can be estimated

using published design standards and empirical equations and

the accuracy of the various equations are assessed by

comparison with a validated high resolution two-dimensional

(2D) finite element analysis (FEA) model. The key parameters

for bursting design are the total force, peak stress and centroid

of bursting forces. The earliest work in the literature to propose

an equation for the bursting force was by Morsch (1922,1924).

Further theoretical investigations are given in the literature by

Bortsch (1935); Magnel (1949); Sievers (1952); Guyon

(1953,1972); Iyengar (1962); Yettram and Robbins

(1969,1970); Sarles and Itani (1984). Of these the works by

Guyon (1953,1972) and Iyengar (1962) are widely used in the

present-day industry in the design of precast segments. Of these

the works by Guyon [14, 15] and Iyengar [19] are the ones

widely used in the present day industry in the design of precast

segments. The theoretical solutions by Morsch [24, 25] can be

derived using the strut and tie method (STM). The STM involves

representing the flow of forces through the block with a truss

and calculating the forces in the truss with one of the forces

being the bursting force. This conceptual tool has been used for

over a 100 years. For a concentrically loaded block, the STM

solution is as presented in Equation 1.

𝑻𝒃

𝑷= 𝟎.25(𝟏 −

𝒂

𝒅) Eqn. 1

The solution by Iyengar (1962) has been presented in the form

of a chart for use in design. These charts have been reproduced

in Figure 1a and the parameters describing the bursting stresses

have been fitted to equations in Figure 1b. This is also to enable

comparison of the study in this paper to the theoretical

solution.

mailto:[email protected]



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(a) Stress distribution (b) Summary parameters and fitting equations

Figure 1.0(a-b): Theoretical solution according to Iyengar (Traced from Leonhardt (1964))

More recently, He and Liu (2010); Zhou et al. (2015) used

polynomial equations defining the load transfer paths of force

(isostatic lines of compression (ILC)) to derive equations for the

bursting force and centroid of the force. Their equations for

bursting force included the effects of eccentricity, however, the

peak stress was not included in their study (Table 1). Carlo et

al. (2016) presents a procedure for the design of precast

tunnel segments for mechanical excavated tunnel lining in

fibre reinforced concrete, without any traditional steel

reinforcement. Based on photo-elastic tests carried out by

Sargious (1960), who was working in the same department as

Leonhardt (1960), a formula for the bursting force is given by

equation 2.

𝑻𝒃

𝑷= 𝟎.𝟑(𝟏 −

𝒂

𝒅) Eqn. 2

Leonhardt (1964) proposed this linear relationship (Equation 2)

because the theoretical solution by Iyengar (1962) under-

estimated the bursting force as compared to the experimental

data from Sargious (1960). Leonhardt (1960) also summarised

the previous publication by Iyengar (1962), which was based

on theoretical solutions, in a convenient form (charts) for

reinforcement design. Williams (1982) carried out a series of

tests on the joints of reinforced tunnel segments. The failure

modes were identified. The study concluded that the provision

of transverse steel in the area of maximum tensile stress delays

the onset of cracking and that treating the tunnel-lining segment

as an end-block and designing it to the CP110 design code led

to very conservative results. In the early 1990s, the University

of Texas at Austin carried out dozens of model tests of

anchorage zones (Burdet (1990); Zhou et al (2015), based

upon which the design equations in the AASHTO (2014) and

ACI318 (2014) were developed (Table 1). Moccichino et al.

(2010) carried out flexural and point load tests on fibre

reinforced concrete tunnel segments. These tests assessed the

performance of fibre reinforced concrete as compared to

steel reinforcement concrete and did not include assessment of

the joint bursting capacity. Liao et al. (2015) conducted an

experimental program using small scale specimens with and

without fibre reinforcement. The experimental results were

compared with analytical expressions based on a parabolic

distribution of tensile stress. Burdet (1990) used elastic FE

analysis in his study at the University of Texas in Austin to

propose design equations which were later adopted in the

American Standards. Hengprathanee (2004) carried out linear

and non-linear FE analysis to derive equations for precast

beams (rectangular, "T" and "I" shaped sections)

considering the effects of a support reaction. A convergence

test was carried out to confirm that the mesh size was

appropriate. Load width ratios (a/d) of up to 0.5 and eccentricity

ratios (e/d) of up to 0.4 were used in the study by

Hengprathanee (2004). From the FE results, the bursting force

(Tb) was found to agree with Equation 1 for load width ratios

(a/d) greater than 0.2. However, for a knife edge load (a/d =

0), the bursting force was found to be about (0.42P), which

is much higher than 0.3P according to Leonhardt (1964).

Hengprathanee (2004) proposed a modified equation (xc = 0.5

+ 0.25a/d) for the position of the peak stress. However, the

magnitude of the peak stress was not compared with the FE



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results in the study. Gupta and Khapre (2008) used a FE

computer code on the platform of a supercomputer known as

"PARAM 10000" in their study. Gupta and Khapre (2008)

showed that the peak bursting stress increases with eccentricity

but did not include the effects of eccentricity in their proposed

equation. Francis and Mangione (2012) verify the design charts

by Leonhardt (1964) using a numerical analysis for a

concentrically applied load. In their study, the effect of the

contact stress distribution at the joint of a precast tunnel segment

on bursting is highlighted and a procedure for the design of

steel fibre reinforced concrete (SFRC) tunnel segments

involving the use of non-linear numerical analysis is proposed.

The British Standard (BS), BS8110 (1997), gives a table of

values (Tb/P ratios) for estimating the total bursting force (Tb).

The values from this table have been presented as an equation

in this paper for comparison (See Table 1.0). The Eurocode

(2004) provides equations (See Table 1.0) for estimating the

bursting force (Tb). The equations in the Eurocode (2004) take

the effects of the height-to-width (h/d) in account. Therefore,

this paper will proposed a graphical solution for estimation of

bursting force and bursting stress for different load-width ratios

and compressive concrete strength for use in design and

industries.

Table 1.0 Comparison of published equations from literature

Bursting force, Tb Peak Stress, σp Centroid. xc

Guyon (1953, 1972) 1.1 × 0.25(1 −𝑎

𝑑)𝑃 1.1 × 0.47(1 −

𝑎

𝑑)𝑃

ab

-

Leonhardt [20] 0.3(1 −𝑎

𝑑)𝑃

- -

BS8110 (1997) Min of 0.23P and (0.32 − 0.3𝑎

𝑑)𝑃 - -

Eurocode (2004) h/d >= 2, 0.25(1 −𝑎

𝑑)𝑃

h/d < 2, 0.25(1 − 0.7𝑎

ℎ)𝑃

- -

Gupta and Khapre (2008) (0.239 − 0.267𝑎

𝑑+ 0.075𝑣)𝑃

- -

He and Liu (2011) 0.22(1 +2𝑒

𝑑)2(1 −

2𝑒

𝑑−

𝑎

𝑑)∑𝑃 - -

ACI318 (2014) 0.25(1 −𝑎

𝑑)𝑃

- 0.5 (d-2e)

AASHTO (2014) 0.25(1 −𝑎

𝑑)∑𝑃

- 0.5 (d-2e)

Zhou et al (2015) 0.25(1 +

2𝑒

𝑑)2(1 −

2𝑒

𝑑−𝑎

𝑑)∑𝑃

- -

METHODOLOGY

When concentrated forces act on a prismatic precast concrete

member, transverse tensile and compressive stresses are

generated (Leonhardt, 1964; Guyon, 1953,1972; Sarles and

Itani, 1984). The same analogy can be applied to the radial and

circumferential joints of a precast concrete segmental lining.

The contact problem at the radial or circumferential joints of a

precast tunnel lining was idealised as a two -dimensional (2D)

prismatic block as shown in Figure 2a. Where a is the width of

the applied load, d is the depth of the prismatic block, e is the

eccentricity of the applied load, h is the height of the prismatic

block and P is the magnitude of the applied load. When the

tunnel lining is loaded, the joints rotate as the lining deforms

leading to varying contact lengths at the joint, which can be

represented by varying load widths, a, in the idealised model.

The distribution of the tensile stress is non-linear as shown in

Figure 2b. This non-linear curve can be defined by the first

position (x0) of zero tensile stress from the loaded area and the

position (xp) of the peak stress (σp). The area under the bursting

stress distribution is the total bursting force (Tb). The centroid

of the bursting force occurs at a distance, xc from the loaded

area. Values of bursting stress in this study are presented as

proportions of the uniformly distributed basic stress:

𝜎0 =𝑃

𝑏𝑑 Eqn. 3



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Figure 2(a-b): Idealisation of precast segment joint

A finite element solver with a nonlinear material model (Mohr-

Coulomb) was used in this study. The use of nonlinear material

model in FEA provides near actual behaviour of materials such as

concrete and cement-soil columns (Abbey et al., 2015, 2016, 2017

and Olubanwo et al., 2017, 2018). The Mohr Coulomb material

model was used to approximate the non-linear behaviour of

concrete in tension and compression. The relationship between

the uniaxial compressive strength (fck) and the equivalent

cohesion (c) in terms of the friction angle was established using

Equation 4 given by according to Prinja and Puri (2005) as:

𝑐 = 𝑓𝑐𝑘1−𝑠𝑖𝑛(𝜑)

2𝑐𝑜𝑠(𝜑) Eqn. 4

The Mohr Coulomb model failure function has been defined in

Equation 5. A tension cut-off was added to the Mohr Coulomb

model to represent the tensile capacity of the concrete based on

the mean tensile strength (fctm) as defined in Eurocode 2, see

Equation 6 and 7. The elastic modulus of concrete was defined as

a function of characteristic compressive strength of concrete (See

Equation 8) based on Table 3.1 of Eurocode 2.

𝐹 =𝜎1+𝜎3

2sin(𝜑) −

𝜎1−𝜎3

2− 𝑐 cos(𝜑) = 0 Eqn. 5

𝑓𝑐𝑡𝑚 = 0.3𝑓𝑐𝑘2

3 if 𝑓𝑐𝑘< 50MPa Eqn. 6

𝑓𝑐𝑡𝑚 = 2.12𝑙𝑛 (1 +𝑓𝑐𝑘+8

10) if 𝑓𝑐𝑘> 50MPa Eqn. 7

𝐸𝑐 = 22000 (𝑓𝑐𝑘+8

10)0.3

Eqn. 8

Table 1: Summary of material properties for different concrete strengths

Cylinder strength,

fck (MPa)

Cohesion, c

(MPa)

Friction

angle,φ (deg)

Tension cut-off,

fctm (MPa)

Elastic Modulus,

Ec (MPa)

Poisson

ratio

30 12.6 10 2.9 32837 0.2

40 16.8 10 3.5 35220 0.2

50 21.0 10 4.1 37278 0.2

60 25.2 10 4.4 39100 0.2

70 29.4 10 4.6 40743 0.2



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The dimensions of the model were set up using typical values for

precast concrete tunnel linings (PCTL). A block depth of

300mm was chosen for this study as many segmental linings

are close to 300 mm thick (See Table 1). The block height was

assumed to be equal to the block depth, as the zone of high

tensile stress was not expected to extend more than one block

depth from the loaded face. The block depth was divided into

100 elements in each dimension to give a mesh size of 6 mm x

6 mm (grid dimension of 50 x 50), leading to a model made up

of 7701 nodes and 2500 elements. To ensure second order

accuracy, eight (8) node quadratic plain strain elements were

used. In terms of boundary conditions, the model was fixed

vertically at the bottom, with the leftmost node on the bottom also

fixed horizontally. The FEA model was set up with 9 No

different load widths (a) ranging from 3 0 mm (0.1d) to 270

mm (0.9d) in increments of 30 mm (0.1d). The uniaxial

concrete strength was varied from 30MPa to 70MPa in

increments of 10MPa, which represents the range of strengths

typically used in industry. This lead to a total of 45No different

load combinations analysed in this study. The eccentricity (e)

of the applied load was assumed to be zero. A non-linear

analysis was carried out using the visco-plastic strain method

with the load applied in increments of 2% of the total load (50

load steps). A visco-plastic strain method was adopted for less

computational time and memory requirements. The total load

applied, P was 6000kN greater than the maximum compressive

load of the prismatic block at 70MPa concrete strength as

presented in Table 2 with other model parameters.

RESULTS AND DISCUSSION

Total force is important for determination of the amount of

reinforcement needed to prevent bursting and the position of

the peak stress determines an idealized location of the

reinforcement. Incorrect placement of reinforcement may lead

to bursting failure. The peak stress is essential for the design of

fibre reinforced precast segments, as the value of this parameter

needs to be lower than the tensile strength of the concrete. A

non-linear analysis of the idealised precast concrete segment

joints under high compressive load has considered the effect of

load-width ratio on force and peak stress ratios for different

concrete strengths and compared with previous studies by

(Iyengsr, 1962, Zhou, 2015, Leohardt, 1964 and Eurocode 2)

as presented for different load width ratio (a/d) in Figure 3.

From Figure 3(a-i), the variation of bursting stress (mean

tensile strength) with characteristic compressive strength of

concrete has been established. Figures 3(a-i) show that peak

tensile bursting stress increases as compressive strength of

concrete increases. The effect of increase in concrete strength

on the peak tensile bursting stress diminishes at higher

characteristics compressive strength values of concrete. Figures

3(a-i) also show that increase in load width ratio (a/d) reduces

the amount of concrete at failure. This implies that at higher

load width ratios (a/d) the area under the curve (bursting force)

reduces as shown in Figure 4(a-b).

Table 2: Summary of model geometry and applied loads

Load width, a, (mm) Block depth, d (mm) Load-width ratio (a/d) Force, P(kN) Distributed load (kN/m)

30 300 0.1 6000 200

60 300 0.2 6000 100

90 300 0.3 6000 67

120 300 0.4 6000 50

150 300 0.5 6000 40

180 300 0.6 6000 33

210 300 0.7 6000 29

240 300 0.8 6000 25

270 300 0.9 6000 22

Figure 3(a-b): Variation of bursting stress with distance for different characteristic strengths of concrete.

(a) (b)

a)



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Figure 3(c-f): Variation of bursting stress with distance for different concrete strength

Figure 3(g and h): Variation of bursting stress with distance for different concrete strength.

(c) (d)

(e) (f)

(g) (h)



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Figure 3(i): Variation of bursting stress with distance for different concrete strength

The effect of load-width ratio (a/d) on bursting force and peak

stress ratios was investigated on the assumption of zero

eccentricity for the different characteristic strengths of concrete

considered and the FEA results show good agreement with

previous studies. The FEA results presented in Figures 4(a-b)

show that increase in load width ratio causes reduction in force

and peak stress ratios and in good agreement at higher

characteristic concrete strength with theoretical solutions by

Zhou, (2015) compared to the theoretical solutions by

Leonhardt (1964) and Eurocode, 2. From Figure 4a, the results

for the nonlinear material analysis are much lower than that of

Leonhardt at high load concentrations (low load width ratios).

This demonstrates that the Leonhardt solution are conservative

as compared to the nonlinear analysis. Figure 4b shows that the

theoretical solution of peak stress according to Iyengar, 1962

agrees more with the FEA results of the present study at higher

load width ratios however, higher peak stresses were obtained

at lower load width ratios for both solutions due to reduction in

contact area. The nonlinear material analysis can therefore be

used to optimise the design of segmental lining joints and

improve equations in the design codes for practising engineers.

Figure 4(a-b): Variation of peak stress and force ratios with load width ratio for different concrete strength

(i)

(a) (b)



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CONCLUSION

From the non-linear analysis of the idealised precast concrete

segment joints under the applied compressive load, the

following conclusions has been drawn:

The variation of bursting stress (mean tensile strength)

with characteristic compressive strength of concrete

depends on the magnitude of characteristics

compressive strength of concrete.

Peak tensile bursting stress increases as compressive

strength of concrete increases.

The effect of increase in concrete strength on the peak

tensile bursting stress diminishes at higher

characteristics compressive strength values of

concrete.

An increase in load width ratio (a/d) reduces the

amount of concrete at failure leading to reduction in

bursting force.

The plotted charts can be adopted as design charts by engineers

in industry to eliminate the need of conducting extensive finite

element analysis and use of empirical relations in estimation of

bursting stress. The charts are recommended for plain concrete

design and can be developed further for Fibre Reinforced

concrete.

List of abbreviations;

AASHTO American Association of State Highway and

Transportation Officials

ACI - American Concrete Institute

a - width of load area

b - width of prismatic block

BS - British Standard

d - thickness of prismatic block

e - Eccentricity

Ec - Concrete elastic modulus

FE - Finite Element

FEA - Finite Element Analysis

EC - European Code

h - Height of prismatic block

ILC - Isostatic Lines of Compression

𝑷 - Applied force

𝝈𝒑 - Peak stress

𝝈𝟎 - reference stress

STM - Strut and Tie Model

TBM - Tunnel Boring Machine

𝑻𝒃 - Bursting force

xc - Centroid of bursting stress

x0 - distance from load to first position of zero

stress

2D - two-dimensional

REFERENCES

[1] AASHTO, AASHTO LRFD Bridge Design

Specifications. Standard, American, 2014 Association

of State Highway and Transportation Officials

(AASHTO). 7th Edition.

[2] Abbey, S.J., Ngambi, S., Coakley, E. (2016) ‘Effect of

Cement and by-product material inclusion on

plasticity of deep mixing improved soils’.

International Journal of Civil Engineering and

Technology, 7 (5), pp. 265-274.

[3] Abbey, S.J., Ngambi, S., Olubanwo, A.O. (2017)

‘Effect of overlap distance and chord angle on

performance of overlapping soil-cement columns’.

International Journal of Civil Engineering and

Technology, 8 (5), pp. 627-637.

[4] Abbey, S.J., Ngambi, S., Ngekpe, B.E. (2015)

‘Understanding the performance of deep mixing

improved soils - A Review’ International Journal of

Civil Engineering and Technology, 6 (3), pp. 97-117.

[5] ACI318, Building Code Requirements for Structural

Concrete (ACI 318M-14) and Commentary (ACI

318RM-14). Standard, American Concrete Institute

(ACI), 2014.

[6] Bortsch, R., Die Spannungen in Wa1zge1enkquadern.

Beton und Eisen, 1935, 35(4), 61– 66.

[7] BS8110, Structural use of concrete - Part 1: Code of

practice for design and construction. Standard, British

Standards Institute (BSI), 1997.

[8] Burdet, O. L., Analysis and Design of Anchorage

Zones in Post-Tensioned Concrete Bridges. Phd

thesis, The University of Texas at Austin, 1990.

[9] Carlo, F. D., Meda, A. and Rinaldi, Z., Design

Procedure of precast fiber reinforced concrete

segments for tunnel lining construction. Structural

Concrete, 2016, 17(5).

[10] Christodoulides, S. P., A Two-Dimensional

Investigation of the End Anchorage of Post-Tensioned

Concrete Members. Structural Engineer, 1955, 33(4),

120 – 133.

[11] Christodoulides, S. P., Three-Dimensional

Investigation of the Stresses in the End Anchorage

Blocks of a Prestressed Concrete Gantry Beam.

Structural Engineer, 1957, 35(9), 349 – 356.

[12] DAUB, Recommendations for the design, production

and installation of segmental rings, German

Tunnelling Committee, 2013.



6987

[13] Eimer, C., Doroskiewicz, R., Michalski, B. and Szule,

J., Anchorage Zone in Pre-stressed Concrete Elements

in the Light of Photo-elastic Investigations on

Reinforced Models. Archiwum Inzynierii Ladowej,

1966, 12, 149.

[14] Eurocode, Eurocode 2: Design of concrete structures

- Part 1-1: General rules and rulesfor buildings.

Standard, British Standards Institute (BSI). 2004,

Incorporating corrigenda January 2 0 0 8 , November

2010 and January 2014.

[15] Francis, O. and Mangione, M., Developments in joint

design for steel fibre reinforced concrete segmental

tunnel linings. In: ITA - AITES World Tunnel

Congress, 2012.

[16] Gupta, P. K. and Khapre, R. N., A study on

development of stresses in anchorage zone using

parallel processing. Asian Journal of civil engineering

(building and housing), 2008, 9(1), 47 – 59.

[17] Guyon, Y., Pre-stressed Concrete. John Wiley and

Sons, Inc., New York, 1953.

[18] Guyon, Y., Limit-state Design of Pre-stressed

Concrete Translated by P. Chambon and F. H. Turner.

New York: Halsted Press Division, Wiley, 1972.

[19] He, Z.-Q. and Liu, Z., Investigation of Bursting Forces

in Anchorage Zones: Compression-Dispersion

Models and Unified Design Equation. Journal of

Bridge Engineering, 2011, 16(6), 820–827.

[20] Hengprathanee, S., Linear and nonlinear finite

element analysis of anchorage zones in post-

tensioned concrete structures. Ph.D. Thesis,

Virginia Polytechnic Institute and State University,

Blacksburg, Virginia, 2004.

[21] Ibell, T. and Burgoyne, C., Behaviour of Pre-stressed

Concrete End Blocks. In: Proceedings ofthe 9th

Conference on Engineering Mechanics (Edited by

Lutes, L. D. and Niedzwecki, J. M.). ASCE: American

Society of Civil Engineers., Texas, 1992, 135–138.

[22] Iyengar, K. T. S. R., Two-Dimensional Theories of

Anchorage Zone Stresses in Post-Tensioned Pre-

stressed Beams. ACI Journal, 1962, 59(10), 1443 –

1466.

[23] Leonhardt, F., Pre-stressed concrete: design and

construction. W. Ernst, 1964.

[24] Liao, L., Fuente, A. D. L., Cavalaro, S. H. P. and

Carbonari, G., Experimental and analytical studyof

concrete blocks subjected to concentrated loads with

an application to TBM-constructed tunnels.

Tunnelling and Underground Space Technology,

2015, 49, 295 – 306.

[25] Magnel, G., Design of the Ends of Pre-stressed

Concrete Beams. Concrete & Construction

Engineering, 1949, 44(5), 141 – 148.

[26] Moccichino, M., Romualdi, P., Perruzza, P., Meda, A.

and Rinaldi, Z., Experimental Tests on Tunnel Precast

Segmental Lining with Fiber Reinforced Concrete. In:

World Tunnel Congress (WTC). Vancouver, Canada,

2010.

[27] Morsch, E., Reinforced Concrete Construction -

Theory and Application (Der Eisenbetonbau-Seine

Theorie und Anwen- dund). 1. Stuttgart, 1922, 112 pp.

5th Edition.

[28] Morsch, E., Uber die Berechnung der Ge1enkquader.

Beton und Eisen, 1924, 12, 156 – 161.

[29] Okada, K. and Fujii, M., Some Considerations on

Stress Concentration in the Notched Anchorage Zone

of Post-Tensioned Concrete Members. Japanese

Prestressed Concrete Engineering Association

(JPCEA) Journal, 1964, 6(6), 32 – 39.

[30] Olubanwo, A.O., Karadelis, J.N., & Abbey, S.J.

(2017). Evaluation of Equivalent Delamination

Driving Coefficient in Bonded Concrete Overlays.

International Journal of Civil Engineering &

Technology (IJCIET), 8(5), 1436-1444.

[31] Olubanwo, A. O., Karadelis, J.N., Saidani, M.,

Khorami, M. and Abbey, S.J. (2018) ‘Investigation of

intrinsic de-bonding in bonded concrete overlays:

Material characterisation and numerical Study’

Engineering Solid Mechanics 6(2) pp. 155-174.

[32] Sargious, M., Beitrag zur Ermitt1ung der

Hauptzugspannungen arn Endauf-1ager

vorgespannter Betonba1ken. Ph.D. thesis, University

of Stuttgart, 1960.

[33] Sarles, D. and Itani, R. Y., Effect of End Blocks on

Anchorage Zone Stresses in Prestressed Concrete

Girders. Precast / Pre-stressed Concrete Institute

Journal, 1984, 29(6), 100 – 114.

[34] Sievers, H., Die Berechnung von Auf1ager-banken

und Auf1agerquadern von Bruckenpfei1ern.Der

Bauingenieur, 1952, 27(6), 209 – 213.

[35] Stone, W. C. and Breen, J. E., Analysis of post-

tensioned girder anchorage zones. Research Report

FHWA/TX-81/13+208-1, Center for Transportation

Research, the University of Texas, Austin, Texas,

78712. AA (Federal Highway Administration, Austin,

TX.), 1981

[36] Tesar, M., Determination experimentale des tensions

dans les extremites des pieces prismatiques munies

d’une semi-articulation. International Association

of Bridge and Structural Engineering (IABSE), 1932,

1, 497–526.

[37] Williams, A., Radial joints for precast tunnel linings.

Technical Report 552, Cement and Concrete

Association, 1982.

[38] Yettram, A. L. and Robbins, K., Anchorage Zone

Stresses in Axially Post-Tensioned Members of

Uniform Rectangular Section. Magazine of Concrete

Research (MCR), 1969, 21(67), 103 –112.



6988

[39] Yettram, A. L. and Robbins, K., Anchorage zone

stresses in post-tensioned uniform members with

eccentric and multiple anchorages. Magazine of

Concrete Research, 1970, 22(73), 209–218.

[40] Zhou, L.-Y., Liu, Z. and He, Z-Q., Further

investigation of transverse stresses and bursting forces

in post-tensioned anchorage zones. Structural

Concrete, 2015, 16(1), 84–92.

Documents

Non-Linear Analysis of Precast Concrete Segment Joints