Experimental and numerical study on the load-bearing behavior of steel fiber reinforced concrete for precast tunnel lining segments under concentrated loads

FRC 2014 Joint ACI-fib International Workshop

Fibre Reinforced Concrete: from Design to Structural Applications

417

Experimental and numerical study on the load-bearing

behavior of steel fiber reinforced concrete for precast tunnel

lining segments under concentrated loads

Rolf Breitenbücher, Günther Meschke, Fanbing Song, Michael Hofmann, Yijian Zhan

Rolf Breitenbücher 1, Günther Meschke

2, Fanbing Song

1, Michael Hofmann

2, Yijian

Zhan 2

1 : Institute for Building Materials, Ruhr-University Bochum, Bochum, Germany.

2 : Institute for Structural Mechanics, Ruhr-University Bochum, Bochum, Germany.

Abstract

To investigate the behavior of precast tunnel lining segments subjected to concentrated

loads on a small scale, laboratory tests on concrete prisms under partial-area loading in

conjunction with numerical analyses were performed. Various parameters influencing the

load-bearing and fracture behavior of plain concrete (PC) and steel fiber reinforced concrete

(SFRC) under concentrated loads, such as the area ratio, loading eccentricity, fiber properties

(dimension, aspect ratio, tensile strength), dosage and fiber orientation are considered. Effects

of those parameters on the ultimate bearing capacity, stress-displacement behavior, failure

mode and crack characteristics are analyzed and discussed. Parallel to the experimental

investigations, numerical simulations using a continuum coupled damage-plasticity model for

triaxially loaded cementitious material were performed. It is shown that the numerical

analysis is capable to realistically capture the structural behavior and the crack pattern of

partially loaded PC and SFRC specimens.

Keywords

Concentrated loads, steel fiber reinforced concrete, load-bearing behavior, failure mode,

finite element analysis.



418

1 Introduction

In mechanized tunneling, precast segmental tunnel linings are subjected to various loading

conditions during the production, construction and service stages. Critical loading situations

appear mainly in the construction stage rather than in service stage, since concentrated loads

are primarily induced in the joint areas (edges and corners) of segments by the hydraulic jacks

of tunnel boring machines (TBMs), dimensional imperfections of segments, assembly

inaccuracies or by the offset of segments. Under such concentrated loading, both compressive

and splitting tensile stresses (along directions perpendicular to the load) develop directly

beneath the partially loaded area. If the splitting tensile stresses exceed the concrete tensile

strength, cracking and spalling occurs in the concrete. This phenomenon is particularly critical

in the case of high-strength concrete as used for precast segmental linings, where the

compressive strength typically ranges from 70 to 90 MPa. Therefore, concrete members under

concentrated loads are reinforced with transverse steel reinforcement (hoops or stirrups) to

resist splitting tensile stresses. An alternative approach is to incorporate steel fibers into the

concrete. Due to the crack-bridging effect, steel fibers introduce a ductile post-cracking

behavior and distributed cracks with limited crack width. Furthermore, the structure can be

effectively strengthened even in the concrete cover.

In the last two decades steel fiber reinforcement has been frequently used for precast

segmental tunnel linings (Moyson, 1995; Kasper, 2007; Winterberg, 2009). Segmental tunnel

linings are predominantly subjected to high compressive normal forces combined with

relatively small bending moments. This allows for the application of steel fibers even without

any conventional reinforcement (Kasper, 2007). Compared to traditionally reinforced

segments, segments produced with steel fibers exhibit distinct advantages in terms of

resistance against impact and concentrated loads, control of crack width and distribution,

simplification of production and construction (Hansel, 2011).

In a collaborative research project on Interaction Modeling in Mechanized Tunneling at

Ruhr-University Bochum, one direction of research is targeted towards the improvement of

the robustness of tunnel lining segments adopting a hybrid experimental-numerical approach.

A series of laboratory tests on the structural behavior of plain and steel fiber reinforced

concretes are carried out, where the influence of various parameters is systematically

investigated. A finite element model for SFRC subjected to triaxial loading is developed.

Following a multiscale approach, the enhanced fracture energy due to crack bridging of the

fibers is computed directly from an analytical model for single fiber pull out. This modeling

strategy, supported by specifically designed tests, will allow following the influence of certain

design variables, such as type, content and orientation of fibers, on the structural performance.

2 Laboratory tests

2.1 Materials, specimens and testing scope

For the production of SFRCs, a base concrete mixture corresponding to a typical concrete

composition for precast tunnel lining segments was used throughout the experiments (Table

https://www.researchgate.net/publication/228926500_Lining_design_for_the_district_heating_tunnel_in_Copenhagen_with_steel_fibre_reinforced_concrete_segments?el=1_x_8&enrichId=rgreq-7ee37ff1-2987-4574-b4d3-04032f233c83&enrichSource=Y292ZXJQYWdlOzI2ODk3OTA3NTtBUzoxNjk1Njc4NjMxODk1MDRAMTQxNzQzOTUzNzA5NA==

https://www.researchgate.net/publication/228926500_Lining_design_for_the_district_heating_tunnel_in_Copenhagen_with_steel_fibre_reinforced_concrete_segments?el=1_x_8&enrichId=rgreq-7ee37ff1-2987-4574-b4d3-04032f233c83&enrichSource=Y292ZXJQYWdlOzI2ODk3OTA3NTtBUzoxNjk1Njc4NjMxODk1MDRAMTQxNzQzOTUzNzA5NA==



419

1) The cement used was a Portland cement CEM I 52.5 R (DIN EN 197-1). The aggregates

consisting of Rhine river sand and gravel with a maximum size of 16 mm exhibited a grading

curve of A/B 16 (DIN EN 12620, DIN 1045-2). Fly ash used conformed to the requirements

of DIN EN 450-1 and DIN 1045-2. The types and properties of the steel fibers investigated

are summarized in Table 2. To maintain an adequate workability, the SFRCs were slightly

modified by adding higher dosage of the superplasticizer conformed to DIN EN 934-2.

Table 1: Proportions of the base concrete mixture (plain concrete: PC)

Cement

[kg/m³]

Fly ash

[kg/m³]

Aggregate

[kg/m³]

Water

[kg/m³] w/c-ratio

Superplasticizer

[kg/m³]

330 90 1849 148.5 0.45 1.3

Table 2: Types and properties of the steel fibers

Fiber

index Fiber Type Shape

Length

[mm]

Diameter

[mm]

Aspect ratio

[l/d]

Tensile strength

[MPa]

F1 RC-80/60-BN

hook-ended

60 0.75 80 1250

F2 RC-65/60-BN 60 0.90 67 1160

F3 RC-80/60-BP 60 0.71 85 2600

F4 ZP305 30 0.55 55 1345

The properties of the fresh as well as the hardened PC and SFRCs were determined in

accordance with DIN EN 12350, DIN EN 12390 and DIN 1048, as presented in Table 3. With

exception of fiber F1 (fiber dosage: 40, 60 and 80 kg/m³), for the other fiber types, the fiber

content in SFRC was uniformly 60 kg/m³ (approximately 0.75% by volume).

Table 3: Basic properties of the PC and SFRCs

Properties Series index

PC F1_40 F1_60 F1_80 F2_60 F3_60 F4_60

Flow consistency [cm]

(DIN EN 12350-5) 42 41 43 40 42 40 45

Air void content [%]

(DIN EN 12350-7) 3.0 2.4 1.9 1.7 2.2 2.1 2.8

Bulk density [kg/m³]

(DIN EN 12350-6) 2356 2360 2437 2455 2383 2430 2382

Compressive strength [MPa]

(DIN EN 12390-3) 84.5 85.5 87.4 94.5 77.4 89.0 81.4

Splitting tensile strength [MPa]

(DIN EN 12390-6) 4.0 5.3 6.7 7.4 6.5 6.3 6.0

Young's modulus [GPa]

(DIN 1048-5) 36 36 37 37 38 35 36



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For the partial-area loading tests, each test group consisted of 3 prisms. These prismatic

specimens manufactured in either standing or lying wooden moulds had a cross-section of

150 mm × 150 mm and a height of 300 mm. To prevent any stress concentrations caused

through surface roughness, the testing surface (Ac1 = 150 mm × 150 mm) of prism was plane

parallel ground shortly prior to the testing. All tests were carried out at an age of 28 days.

The scope of the experiments is described in Table 4 and 5 for centric and eccentric

loading cases. Three area ratios were considered: 9, 4 and 2.25, corresponding to Ac0 = 50

mm × 50 mm, 75 mm × 75 mm and 100 mm × 100 mm, respectively. To investigate the effect

of casting direction on the fiber orientation and consequently on the load-bearing behavior,

prisms were additionally produced in lying moulds with 60 kg/m³ of F1 fiber. The fiber

orientation was determined by a BSM 100 device based on the magnetic-inductive measuring

method. For eccentric loading, SFRCs with 60 kg/m³ of F1 fiber were tested. For comparison,

PC specimens were tested centrically and eccentrically under the same conditions.

In the case of centric loading, 60 specimens (20 series × 3 per series) were tested in total,

and 36 specimens (12 series × 3 per series) were tested eccentrically.

Table 4: Experimental program for centric loading case

Area ratio

[Ac1/Ac0]

Series code

PC F1_40 F1_60 F1_80 F2_60 F3_60 F4_60 F1_60_l*

9 + + + + + + + +

4 + + + + + + + +

2.25 + + + + * Prisms which were produced in lying moulds (l: lying).

Table 5: Experimental program for eccentric loading case (SFRCs with 60 kg/m³ of F1

fiber)

Series index Area ratio

[Ac1/Ac0]

Loading eccentricity

e = 15 mm e = 30 mm corner edge

PC 9 + + + +

4 + +

F1_60 9 + + + +

4 + +

2.2 Test set-up and testing procedure

All tests were performed using a servo-hydraulic universal testing machine with a

maximum load of 5 MN. The load was transmitted onto the upper surface of the specimen by

a steel plate (Figure 1, left). By using LVDTs (HBM 1-WA/20MM-T) the deformations of the

specimen were measured. As shown in Figure 1, two vertical LVDTs were placed diagonally

around the specimen to measure the relative longitudinal displacement between the upper

steel plate and the lower platen of the testing machine. The other two vertical LVDTs were



421

attached onto the prism to measure the displacement at the midpoint of the top edges. Four

horizontal LVDTs were positioned around the prism to measure the transverse deformation.

The load was continuously applied at a loading rate of 0.5 mm/min. The testing process

was automatically terminated by the software under the condition that a load drop by 60% of

the maximum load was detected. In the case of eccentric loading, to avoid an overturning of

the prism, the specimen was placed eccentrically on the lower machine platen (Figure 1,

right).

Figure 1: Partial-area loading test in progress (left) and positioning of the specimen for

centric and eccentric loading (right)

3 Results of laboratory tests

3.1 Ultimate local compressive stress

As listed in Table 6 and 7, by the addition of steel fibers, the mean values of ultimate local

compressive stress (σmax, defined as the ultimate force divided by Ac0) of concrete increase

remarkably. For both PCs and SFRCs, the σmax decreases with decreasing area ratio (see also

Hawkins, 1968; Niyogi, 1973; Klotz, 2008) or with increasing eccentricity (also reported in

Hawkins, 1968 and Niyogi, 1973), as a result of reduced confinement effect of the

surrounding concrete. Furthermore, the variation of σmax due to different fiber properties,

dosages and eccentricities is less pronounced for low area ratios or large eccentricities (Figure

2); in other words, the fiber reinforcement has more influence for high area ratios or small

eccentricities as is concluded from comparing the values of σmax/fc for a given area ratio in

Table 6 and 7 (ΔSFRC-PC represents the differences in σmax/fc induced by adding fibers).

With increasing fiber content, σmax increases steadily up to 31% (F1_80_9 vs. F1_40_9). In

the case of identical fiber dosage, specimens with high-strength fibers show only an increase

in σmax of up to 7% (F3_60_9 vs. F1_60_9). Increasing the aspect ratio from 67 (F2) to 80

(F1) leads to a slight increase of σmax up to 4.8%. For short fibers, a reduction of approx. 9 %

in σmax was observed (F4_60_9 vs. F1_60_9). At small eccentricities, for both PCs and

SFRCs, σmax drops only slightly (e.g. stress drop: F1_60_9_e15, 4.7%) compared with the

values of the samples tested centrically However, under edge or corner loading, a remarkable

decrease in σmax in the range of 43.5% to 46.7% is observed. Regarding the influence of fiber

orientation, SFRC specimens produced in lying forms show a considerable stress drop of up

to 23% compared with the samples cast in standing moulds (F1_60_4_l vs. F1_60_4).

centric

edgecorner

Machine platen

Loaded area

(Steel plate)



422

Table 6: Results of centric partial-area loading tests

Series index

Area ratio

[Ac1/Ac0]

Concrete compressive

strength fc [MPa]

Ultimate local compressive stress σmax [MPa]

σmax/fc ΔSFRC-PC

(σmax/fc)

PC

9

84.5

152 1.80 -

4 101 1.23 -

2.25 84 0.96 -

F1_40

9

85.5

194 2.27 0.47

4 132 1.55 0.32

2.25 97 1.13 0.17

F1_60

9

87.4

214 2.45 0.65

4 153 1.75 0.52

2.25 113 1.30 0.34

F1_60_l 9 173 1.98 0.18

4 118 1.35 0.12

F1_80

9

94.5

254 2.69 0.89

4 171 1.81 0.58

2.25 121 1.28 0.32

F2_60 9

84.1 206 2.45 0.65

4 146 1.74 0.51

F3_60 9

89.0 229 2.57 0.77

4 164 1.84 0.61

F4_60 9

81.5 196 2.40 0.60

4 139 1.71 0.48

Table 7: Results of eccentric partial-area loading tests

Series index

Area ratio [Ac1/Ac0]

Loading eccentricity [e] Ultimate local compressive stress σmax [MPa]

σmax/fc ΔSFRC-PC

(σmax/fc)

PC

9 15 mm

144 1.70 -

4 100 1.18 -

9 30 mm 135 1.60 -

9 edge

113 1.34 -

4 81 0.96 -

9 corner 83 0.98 -

F1_60

9 15 mm

204 2.33 0.63

4 146 1.67 0.49

9 30 mm 185 2.12 0.52

9 edge

154 1.76 0.42

4 118 1.35 0.39

9 corner 121 1.38 0.40



423

Figure 2: Effect of area ratio (left, centric loading) and loading eccentricity (right, area ratio

= 9 and 4) on the ultimate load-bearing capacity of PCs and SFRCs

3.2 Stress-displacement responses

As shown in Figure 3 (left), the structural stiffness for both PCs and SFRCs is nearly

identical in the pre-peak branch for a given area ratio and increases with increasing area ratio.

After the peak load, an abrupt drop of the stress is observed for PCs indicating a sudden

failure, whereas SFRCs exhibited a continuous decline of stress, implying a ductile material

behavior. For different SFRCs, similar shapes of the stress-displacement curves are observed

(Figure 3, right). However, with increasing fiber dimension (F4_60_9 vs. F1_60_9) or

strength (F1_60_9 vs. F3_60_9), the ultimate stress reaches higher values and the

corresponding SFRCs show a more ductile post-cracking behavior associated with larger

deformations. Increasing the aspect ratio from 67 (F2_60_9) to 80 (F1_60_9) had merely

effect on the peak stress as well as on the post-cracking behavior. With increasing fiber

dosage, the ultimate stress as well as the corresponding displacement increases and the SFRCs

show more enhanced post-cracking behavior (F1_80_9 vs. F1_40_9).

Figure 3: Effect of area ratio (left) and fiber properties and dosages (right) on the stress-

displacement behavior under centric loading

In the case of small eccentricities of up to 30 mm (Figure 4, left), the stiffness of the PCs

and SFRCs is nearly identical in the pre-peak phase and the post-peak branches of the curves

of the SFRCs are more or less parallel. Similar tendencies are also observed for Acl/Ac0=4.

0

50

100

150

200

250

300

9 4 2.25

Ult

imate

lo

cal c

om

pre

ssiv

e st

ress

[MP

a]

Area ratio [Ac1/Ac0]

PC F1_40F2_60 F1_60F3_60 F1_80F4_60 F1_60_l

0

50

100

150

200

250

300

0 mm 15 mm 30 mm edge corner

Ult

imate

loca

l co

mp

ress

ive

stre

ss

[MP

a]

Loading eccentricity [e]

PC_9 F1_60_9

PC_4 F1_60_4

0

50

100

150

200

250

300

0 1 2 3 4 5

Lo

cal c

om

pre

ssiv

e st

ress

[M

Pa

]

Longitudinal displacement [mm]

I: PC_9 1: F1_60_9

II: PC_4 2: F1_60_4

III: PC_2.25 3: F1_60_2.25

I

II

III3

2

1

0

50

100

150

200

250

300

0 1 2 3 4 5

Lo

cal c

om

pre

ssiv

e st

ress

[M

Pa

]

Longitudinal displacement[mm]

1: F1_80_9 I: F2_60_9

2: F1_60_9 II: F3_60_9

3: F1_40_9 III: F4_60_9

II

1

I

2

3 III



424

For large eccentricities (Figure 4, right), the SFRCs show remarkably higher stiffness

compared with the PCs. However, compared to centric or small eccentric loading, the SFRCs

loaded at the edge or at one corner exhibit a relatively quick stress drop after the peak stage.

This is related to an excessive reduction in confinement effect of the surrounding concrete.

Such a rapid drop of stress did not occur for SFRCs loaded at one corner with low area ratio

of 4.

Figure 4: Effect of eccentricities on the stress-displacement behavior (left: small

eccentricities and right: large eccentricities for Ac1/Ac0=9)

In Figure 5 (left), the fiber orientation with respect to the loading direction is depicted by

percentage (provided by the BSM 100 device) in three spatial directions both for prisms cast

in standing and lying forms. Each value describes the possibility of the orientation of fibers in

the corresponding spatial direction in the sample. For prisms produced in lying moulds, about

46% of the steel fibers are oriented towards the loading direction; consequently, compared

with prisms cast in standing moulds, fewer fibers are oriented in the direction of the major

tensile stresses (perpendicular to the loading direction).

As is well known, fibers aligned to the acting direction of tensile stresses have the best

crack-bridging capacity. As a result of a preferred fiber orientation, the specimens produced in

lying moulds exhibit considerably lower values of maximum stress (stress drop up to 23%,

Figure 5, right) and a relatively abrupt stress drop after the peak stage indicating a less ductile

post-cracking behavior. This adverse effect is independent of the area ratios.

Figure 5: Effect of casting direction on the fiber orientation (left) and the load-bearing

(right) behavior of SFRCs under centric partial-area loading

0

50

100

150

200

250

300

0 1 2 3 4 5

Lo

cal c

om

pre

ssiv

e st

ress

[M

Pa

]


I: PC_9 1: F1_60_9

II: PC_9_e15 2: F1_60_9_e15

III: PC_9_e30 3: F1_60_9_e30

1

23

III

III

0

50

100

150

200

250

300

0 1 2 3 4 5

Lo

cal c

om

pre

ssiv

e st

ress

[M

Pa

]


I: PC_9 1: F1_60_9

II: PC_9_edge 2: F1_60_9_edge

II: PC_9_corner 3: F1_60_9_corner

1

23

I

III

II

loading

direction

transverse

diretion-1

transverse

diretion-2

lying 45.8 17.2 37

standing 23.2 37.9 38.9

0

10

20

30

40

50

Fib

er o

rien

tati

on

w.r

.t.

loa

din

g d

irec

tio

n [%

]

y z z y x x

23.2

45.8

37.9

17.2

38.937

0

50

100

150

200

250

300

0 1 2 3 4 5

Lo

cal c

om

pre

ssiv

e st

ress

[M

Pa

]


I: F1_60_9_l 1: F1_60_9_s

II: F1_60_4_l 2: F1_60_4_s

1

2

I

II

Casting direction Loading direction

lying standing

zy

x

xz

y



425

3.3 Failure mode and crack pattern

For both PCs and SFRCs, no visible cracking or spalling is observed until shortly before

reaching the peak load. Shortly afterwards, without any exception, all PC specimens fail

explosively by splitting, particularly in the case of low area ratio and/or large eccentricities,

whereas all SFRC prisms show a ductile fracture behavior with a warning of multi-cracking

before complete failure.

Under centric loading, with reduced area ratio, the PC samples tend to lose their integrity

more easily. For Ac1/Ac0 = 9, the prisms fail with 3-5 main radial cracks on the testing surface

with some of them spreading through the lateral surfaces; however, at a ratio of 2.25, all

specimens split into several parts with an inverted pyramid wedge generated beneath the

loaded area (Figure 6a). For the SFRC specimens, the average number of cracks on the testing

surface varies in the range of 6-12, depending on the testing parameters. At a high ratio of 9,

only SFRCs produced with short fiber (F4_60_9) or low fiber dosage (F1_40_9) exhibit a

crack propagation reaching from top to bottom of the specimens, while, for the other SFRCs,

cracks only spread to approximately the half-height of the specimens (Figure 6b). With

decreasing area ratio, fatal cracking and spalling tends to develop in the lower half of the

specimens, especially at a ratio of 2.25 (Figure 6c). In the case of SFRCs with different aspect

ratios, no distinct differences are found in the failure pattern. For specimens cast in lying

moulds, the crack pattern is characterized by two main cracks propagating through the lateral

surfaces (parallel to yz cross-section), independently of the area ratio.

a) PC_2.25 b) F1_60_9 c) F1_60_2.25 d) PC_9_edge e) PC_4_edge f) F1_60_9_edge

Figure 6: Typical examples of failed PC and SFRC specimens (a)-c): centric; d)-f):

eccentric)

In the case of eccentric loading, with increasing eccentricities and/or decreasing area ratio,

the PC prisms show severe cracking before complete failure. For edge loading at a ratio of 9,

the concrete was one-sidedly punched out beneath the loaded area (Figure 6d). For Ac1/Ac0=4,

the specimens failed through splitting into several parts, even at small eccentricity (Figure

6e). For the SFRC specimens, concrete damage is restricted to local regions adjacent to the

loaded area, as shown in Figure 6f.

4 Numerical modeling and finite element analysis

In addition to the experimental investigation, analytical modeling and numerical simulation

are performed as well, which provides a fast and convenient approach to the analysis of the

structural behavior of PC/SFRC under partial-area loading.



426

4.1 Crack bridging effect

For an opening crack in PC and SFRC using different fiber types, the relation between

tractions, i.e. the stresses normal to the crack plane and the separation displacement is

illustrated in Figure 7. These curves are obtained from integration of the pullout responses of

all fibers intercepting the crack:

∑𝑃

(1)

Here, represents the contribution of the plain concrete; is the fiber bridging stress,

which is calculated as the sum of the pullout forces of fibers divided by the unit area of

a crack. The single fiber pullout force 𝑃 is obtained according to the analytical model

proposed in Zhan and Meschke (2014).

a) b)

Figure 7: a) An opening crack with bridging fibers, b) traction-separation laws for SFRC

with hooked and straight fibers and for plain concrete.

4.2 Finite Element simulation of the partial-area loading tests

The Finite Element Method is used for the simulation of test cases. Based upon the

multisurface elastoplastic damage model originally proposed for plain concrete in Meschke,

Lackner and Mang (1998), an improved constitutive model describing the stress-strain

relationship of PC and SFRC in triaxial loading conditions is implemented. The major

characteristics of this constitutive model are described as follows.

A multi-surface yield criterion, consisting of the Rankine and Menétrey-Willam yield

functions (Menétrey and Willam, 1995), defines the range of elastic stress state:

( ) 𝜃 (2)

Here, indicates the yield function corresponding to the Rankine criterion; is the stress

tensor and is the back stress tensor, which governs the post-cracking behavior as a function

of the tensor of internal variables . For plain concrete, the functional dependence is

calibrated according to the fracture energy from experimental stress-strain curves obtained



427

from uniaxial tension tests (Meschke, Lackner and Mang, 1998). For fiber reinforced

concrete, this relation is modified in order to take into account the crack bridging effect of

fibers. By considering the characteristic length of a cracked finite element, the evolution law

is obtained based on the relation (Equation 1). Correspondingly, the tensile fracture

energy of SFRC is enhanced.

In equation (2), stands for the Menétrey-Willam yield function (Figure 8a). , and 𝜃

represent the hydrostatic stress, the deviatoric stress and the Lode angle of the stress tensor in

Haigh-Westergaard stress space, respectively:

√ √ 𝜃

√

(3)

Here, , and stand for the first principal invariant of the stress tensor , the second

and third principal invariants of the deviatoric stress tensor , respectively (Menétrey and

Willam, 1995). A non-associated flow rule using the plastic potential function 𝜃 and

the hardening/softening law 𝑞 for concrete are used according to Červenka and

Papanikolaou (2008), accounting for the enhanced fracture energy (Figure 8b).

a) b)

Figure 8: a) Menétrey-Willam yield surface. b) Evolution law for Menétrey-Willam yield

function.

This material model is implemented as a user-subroutine in the Finite Element program

MSC-Marc. The parameters (e.g. the elasticity modulus, Poisson’s ratio, tension/compression

strength, softening/hardening laws, etc.) for the constitutive model are directly determined

from the experimental data and/or computed using the sub-models (e.g. the single fiber

pullout model, the crack bridging model, etc.), in the framework of a multiscale modeling

approach. Figure 9 shows the comparison between the results obtained from the numerical

simulation and the experiments of the specimens subjected to centric loading in a 50 mm × 50

mm area. In Figure 9a, the stress vs. the longitudinal displacement of the F1_60_9 specimen

(with 60 kg/m³ of fiber RC-80/60-BN) is illustrated in comparison with respective results for

plain concrete PC_9. Good agreement between the numerical results and the experimental

data is observed, despite the overestimation of pre-peak structural stiffness (approx. 30% for

both cases) and the maximum stress in SFRC (approx. 10%) by the numerical simulation. The

ductile response of the specimen due to the existence of steel fibers is well captured. In the

experiments, it is observed that, unlike the explosive failure of PC specimen due to splitting,

the failure mode of the SFRC specimen shows multiple cracking and crack branching

phenomena, which could be well captured by the developed numerical model (Figure 9b).



428

a) b)

Figure 9: Comparison between the results obtained from experiments and the numerical

simulation for centric partial-area loading test PC_9 and F1_60_9: a) Stress-

displacement relations for SFRC and plain concrete, b) crack pattern in the SFRC-

prism.

5 Conclusions

A series of partial-area loading tests on plain concrete and steel fiber reinforced concrete

prisms were carried out. The influence of area ratio, eccentricity, fiber properties (type,

dimension and tensile strength), fiber dosage and orientation were systematically investigated.

From the experimental results, the following conclusions can be drawn: The load-bearing

capacity of concrete under concentrated loads can be substantially improved by the addition

of steel fibers, changing the failure mode from a brittle to a ductile one. With growing area

ratio, the load-bearing capacity of both PCs and SFRCs increases considerably and the failure

pattern changes and the efficiency of fiber reinforcement increases as well. The load-bearing

capacity can be positively influenced by increasing fiber dimension, tensile strength and

dosage to some extent. Varying the casting direction exerts a preferred fiber orientation in the

concrete which considerably affects the load-bearing behavior and crack pattern. With

increasing eccentricity, the load-bearing capacity and the efficiency of fiber reinforcing

decline. Fatal concrete damage occurs already at small eccentricities in the case of PC; while,

for SFRC, cracking and spalling is restricted locally to the regions adjacent to the loaded area.

A selection of the partial-area loading tests has been analyzed numerically by means of the

Finite Element Method. To this end, a multi-surface plasticity-damage model for triaxially

loaded concrete has been adapted to SFRC. In both cases, the failure characteristics and the

stress-displacement curves could be obtained in good agreement with the corresponding test

results. These analyses demonstrate the potential of performing “virtual experiments” for the

optimized design of structures made of SFRC.

The present work serves as the foundation of a collaborative research topic aiming at the

analysis, design and optimization of segmental tunnel linings made of SFRC. With this

experimental-numerical platform, various design parameters can be assessed by means of

tracing their effect from single fiber pullout to the structural behavior. This will allow

designing new types of robust hybrid segments, for instance, characterized by prefabricated

elements of high-performance SFRC in the vulnerable regions combined with ferroconcrete in

the interior of segment.



429

6 Acknowledgement

Financial support was provided by the German Science Foundation (DFG) in the

framework of subprojects B1 and B2 of the Collaborative Research Center SFB 837. This

support is gratefully acknowledged.

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Documents

Experimental and numerical study on the load-bearing behavior of steel fiber reinforced concrete for precast tunnel lining segments under concentrated loads