Engineering Structures 100 (2015) 164–177

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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct

Analytical solutions for flexural design of hybrid steel fiber reinforced

concrete beams
Barzin Mobasher a,⇑, Yiming Yao a, Chote Soranakom b
a
School of Sustainable Engineering and Built Environment, Arizona State University, Tempe, AZ 85287-8706, United States
b
IMMS Co., Ltd., Bangkok 10110, Thailand

a r t i c l e i n f o a b s t r a c t

Article history: Hybrid reinforced concrete (HRC) is referred to as a structural member that combines continuous rein-
Received 11 February 2015 forcement with randomly distributed chopped fibers in the matrix. An analytical model for predicting
Revised 29 April 2015 flexural behavior of HRC which is applicable to conventional and fiber reinforced concrete (FRC) is pre-
Accepted 4 June 2015
sented. Equations to determine the moment–curvature relationship, ultimate moment capacity, and min-
Available online 20 June 2015
imum flexural reinforcement ratio are explicitly derived. Parametric studies of the effect of residual
tensile strength and reinforcement ratio are conducted and results confirm that the use of discrete fibers
Keywords:
increases residual tensile strength and enhances moment capacity marginally. However improvements in
Cracking
Hybrid reinforced concrete
post-crack stiffness and deformation under load is substantial in comparison to conventional steel rein-
Flexural reinforcement forcement. Quantitative measures of the effect of fiber reinforcement on the stiffness retention and
Strain softening reduction of curvature at a given applied moment are obtained. The approach can also be presented in
Deflection hardening a form of a design chart, representing normalized moment capacity as a function of residual tensile
Residual tensile strength strength and reinforcement ratio. Numerical simulations are conducted on the steel fiber reinforced con-
Moment–curvature crete (SFRC) and HRC beam tests from published literature and the analytical solutions predict the exper-
Analytical model imental flexural responses quite favorably.
Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction amount of reinforcement is needed along the column strips to pre-

vent progressive failure, while the amount of rebar in precast seg-
For more than forty years FRC has been used in many construc- mental sections is substantially reduced.
tion applications such as slabs on grade, industrial floors, tunnel The enhancement in the load capacity and ductility depend on
linings, precast and prestressed concrete products. Use of discrete the fiber parameters such as type, shape, aspect ratio, bond
fibers significantly improves fracture toughness, ductility, fatigue strength and volume fraction [6]. Tensile characteristics are
resistance, as well as tensile and shear strength. Recent advances defined in terms of strain softening and hardening, and within
in performance of FRC have been based on a sufficiently high fiber the strain softening category, sub-classes of deflection-softening
content (0.5% < Vf < 1%) to gain significant ductility and strength. A and -hardening may be defined based on the behavior in bending
fiber content of 0.75% without stirrups is considered sufficient to [7]. Several building codes provide guidelines on design with FRC
achieve the equivalent ultimate resistance of a conventional RC materials [8–11]. Combinations of FRC and rebars or welded wire
flexural member with stirrups [1]. The use of fiber also enhances mesh may be used to meet the strength criteria, hence HRC is
the behavior at service life conditions by increasing the stiffness referred to as a section that combines a continuous reinforcement
and residual strength in the serviceability loading stage by means with randomly distributed chopped fibers. Many available models
of restraining the crack opening and limiting excessive deforma- for FRC [12–15] require a strain compatibility analysis of the lay-
tions [2]. This has led to development of structures such as ele- ered beam section in order to obtain moment capacity, which
vated SFRC slabs and precast tunnel lining segments that use a may be impractical for general users. Development of a unified
hybrid reinforcement approach [3–5]. Portions of the conventional approach for both continuous and discrete reinforcements is there-
reinforcement are replaced by steel fibers in most parts to address fore needed.
the flexural capacity. In the case of elevated slabs only a small Post-cracking tensile behavior of FRC materials have been sim-
ulated by either a stress–strain (r–e) relationship in a smeared
⇑ Corresponding author. Tel.: +1 480 965 0141; fax: +1 480 965 0557. crack continuum model, or a stress–crack width (r–x) discrete
E-mail address: barzin@asu.edu (B. Mobasher). model using non-linear fracture mechanics. The original discrete

http://dx.doi.org/10.1016/j.engstruct.2015.06.006
0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 165

Notation

As area of steel rebar qg,bal steel reinforcement ratio per gross area at balance
b beam width failure
B1–5 coefficients for neutral axis depth ratio in Table 5 qg,min minimum flexural reinforcement per gross section
C1–11 coefficients for normalized moment in Table 5 qg,min,rc minimum flexural reinforcement per gross section for
d effective depth at location of steel rebar conventional reinforced concrete
E elastic tensile modulus of concrete qmin minimum flexural reinforcement ratio per effective
Ec elastic compressive modulus of concrete section
Es elastic modulus of steel qmin,rc minimum flexural reinforcement ratio per effective
f0 c cylindrical ultimate compressive strength of concrete section for conventional reinforced concrete
f stress components in stress diagram r concrete stress
F force components in stress diagram rc concrete compressive stress
G1, G2 coefficients for minimum flexural reinforcement in Eq. rp residual tensile strength
(21) rt concrete tensile stress
h full height of a beam section or height of each compres- x normalized concrete compressive yield strain (ecy/ecr)
sion and tension zone in stress diagram v normalized steel strain (es/ecr)
K effective flexural stiffness of a beam section
k neutral axis depth ratio Subscripts
M moment 1 at stage 1, elastic compression–elastic tension
Mn nominal moment capacity 21 at stage 2.1, elastic compression–residual tension, steel
Mu ultimate moment is elastic
n modulus ratio (Es/E) 22 at stage 2.2, elastic compression–residual tension, steel
R coefficient of resistance is yield
y moment arm from force component to neutral axis 31 at stage 3.1, plastic compression–residual tension, steel
a normalized depth of steel reinforcement (d/h) is elastic
b normalized tensile strain (et/ecr) 32 at stage 3.2, plastic compression–residual tension, steel
b1 coefficient for the depth of ACI rectangular stress block is yield
e strain c1 elastic compression zone 1 in stress diagram
ec concrete compressive strain c2 plastic compression zone 2 in stress diagram
ec0 concrete compressive strain at peak stress cr at first cracking
ectop concrete compressive strain at top fiber cu at ultimate concrete compressive strain
et concrete tensile strain cy at concrete compressive yielding
etbot concrete tensile strain at bottom fiber i at stage i of normalized concrete compressive strain and
/ curvature tensile steel condition
c normalized concrete compressive modulus (Ec/E) s refer to steel
j normalized steel yield strain (esy/ecr) sy at steel yielding
k normalized compressive strain (ec/ecr) t1 elastic tension zone 1 in stress diagram
kR1 normalized compressive strain at the end of elastic re- t2 residual tension zone 2 in stress diagram
gion 1 tu at concrete ultimate tensile stain
l normalized residual tensile strength (rp/rcr) cu at concrete ultimate compressive strain
lcrit the critical normalized residual tensile strength that 1 at concrete compressive strain approach infinity
change deflection-softening to deflection-hardening
q steel reinforcement ratio per effective area Superscripts
qbal steel reinforcement ratio per effective area at balance 0
normalizing symbol
failure
qg steel reinforcement ratio per gross area

crack approach by Hillerborg et al. [16] has been modified by many width of localization and prevents snap-back and other numerical
researchers [17–19]. It does not address crack formation and prop- instabilities [26]. In the present paper the length of localization
agation, but instead uses a stress–crack width (r–x) response as zone has been used as a constant length parameter that affects
an input parameter in the post peak tensile zone [20,21]. A repre- the postpeak descending response of the load deformation curve
sentative volume element of a cracked section of a flexural beam where cracks are localized. The r–e approach is more suitable for
with length Lp and depth h is shown in Fig. 1. The section is char- HRC elements since distributed cracking and tension stiffening
acterized by compression and tensile zones. The tensile zone is are expected [27]. For example application of superposition to
represented by two regions; an elastic tensile strain as well as a add the contribution of reinforcement and fibers by updating the
bridged crack in opening mode. The stresses carried by fibers stress crack width relationship in the tensile zone of multiple
across the crack in tension are represented as a function of crack cracks in under-reinforced flexural sections is challenging.
opening and the method is widely used in simulation and design Furthermore, reinforcement ratio affects rebar stress and affects
of quasi-brittle materials [11,22,23]. One of the main parameters crack opening which will in turn affect fiber phase’s contribution.
of these models is a characteristic length parameter defined as Lp, Development of a serviceability design approach based on
which prevents mesh dependency of the results in finite element deflection, ductility or allowable stress would require the compu-
models as it relates the crack width to strain [24,25]. In smeared tation of load capacity of a cracked section based on a given curva-
crack models, characteristic length parameter determines the ture or crack width. Such solutions would keep track of the strain
166 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

the residual tensile strength is taken as fres = 0.37ft,eq, where ft,eq is

the average equivalent bending strength recorded between 0.5
and 2.5 mm deflection. Factor 0.37 expresses the ratio between
the tensile stress in the uncracked section and the equivalent tensile
stress in the cracked section assuming the validity of plane sections
remaining plane, and further assuming a depth of the compressive
zone in the cracked stage as 10% of the original depth [33].
Empirical methods are limited by their inability to be extended to
back-calculation approaches or hybrid reinforcement; hence devel-
opment of an equivalent residual strength method is not possible.
This is partially because of failure to incorporate the strain parame-
ter hence the constitutive model cannot be used for serviceability
criteria, deflection calculation, hybrid reinforcement, or shear
strength calculations. Indeterminate tests such as the round panel
specimens on three or more supports however provide insight
beyond the determinate beams by developing multiple cracking
thus reflecting the structural response more appropriately [22].
Analytical solutions for serviceability based nonlinear design
address a variety of structural HRC systems. For example, sustain-
ability, serviceability, and durability perspectives for design of ele-
vated slabs, structural vaults, retaining walls, and pump and lift
stations for environmental structures are proposed by limiting
the curvature, and crack width. Strain based serviceability limit
states can be specified using short and long-term deformations,
Fig. 1. Schematic presentation of localized zone for a beam section as a non-linear cracking, shrinkage, and verified to address ultimate limit states
hinge, normal stress distribution and strain distribution in steel rebar. requirements [34,35].

and curvature distribution and enable the measurement of effec- 3. Derivation of analytical moment–curvature response
tive deflection and ductility requirements. Moreover, analytical
equations can be used for selection of variables using a design 3.1. Material models
automation procedure; hence gradient-based optimization algo-
rithms can be conducted much faster. Soranakom and Mobasher Fig. 1 shows the schematic 2-D representation of the represen-
used a parametric material tensile and compression constitutive tative element of a cracked beam section as a nonlinear hinge dur-
model and derived analytical flexural load–deflection behavior ing an incremental state of cracking. The element is represented by
from closed form moment–curvature expressions [28,29]. characteristics of length Lp, depth h, crack length a, angle of rota-
Constitutive properties are then obtained by inverse analysis of tion u, nominal curvature j, normal stress distribution, and steel
load–deflection response. This approach was used by Van Zijl and strain distribution. As the flexural crack extends, the steel rebar
Mbewe [30] for an analytical flexural model for hybrid SFRC, how- debonds and carries more stress at the flexural crack. However in
ever they employed a single mode of failure which limits the appli- order to convert the 2-D representation into a 1-D cross sectional
cability to strain softening, deflection hardening SFRC. Taheri et al. model, it is assumed that the average strain in the steel rebar
[12] used a similar approach to develop a design model for hybrid can be represented by the nominal strain distribution at the rebar
SFRC with steel and FRP bars using the constitutive model of level of the section using the assumption of plane section remain-
Soranakom and Mobasher and investigated post-cracking strength, ing plane. The cross section may be of a variable shape and by inte-
and reinforcement ratio. grating stresses over the area forces, bending moments, and
In the present work analytical solutions for moment–curvature, neutral axis kh can be computed. The next step is to use the
load–deflection relationships, and minimum flexural reinforce- moment–curvature formulation in the analysis of a specific struc-
ment ratio are derived to address the synergy between continuous tures by means of analytical solutions or finite element approach.
and fiber reinforcements. Derivations are presented as analytical Templates for predicting load–deflection of elements with differ-
flexural behavior of beam and slab systems and support equivalent ent boundary conditions are then developed.
design charts based on a given deformation of composite systems Fig. 2 presents three distinct material models used in the
for conventional, fiber reinforced, and hybrid reinforced concrete. derivation of parametric response of HRC beams. Material param-
eters are described as two intrinsic parameters: tensile modulus E
2. Existing design approaches for FRC materials and the first cracking tensile strain ecr while other variables are
normalized with respect to these intrinsic parameters. Fig. 2a
Several design guides address the contribution of fibers to the shows an idealized tension model with an elastic range of stress
post-cracking region by means of a residual strength approach. increases linearly with E up to the first cracking tensile strength
The flexural data obtained from beam tests include three point of coordinates (ecr, rcr). In the post-crack region, the stress is con-
bending (3PB) by RILEM, EN 14651, or four point bending (4PB) stant at rp = lrcr = lecrE and terminates at the ultimate tensile
test used by JCI and ASTM C1609 are used in back-calculation of strain etu = btuecr. Fig. 2b shows the elastic-perfectly plastic com-
tensile properties. In the RILEM TC 162-TDF [15] test, the tensile pression response with a modulus Ec = cE. The plastic range initi-
r–e relation is obtained from the load capacity at certain deflec- ates at strain ecy = xecr corresponding to yield stress rcy = xcecrE
tions based on closed loop controlled bending tests on notched and terminated at ecu = kcuecr. Fig. 2c is the elastic-perfectly plastic
beams, and calibrated using finite element method. steel model using yield strain and stress of esy = jecr and fsy = jnecrE
Residual tensile strength is also obtained from simplifications as defined by normalized parameters: j and n. No termination
proposed by RILEM, or fib Model Code 2010 [31,32]. For example, level is specified for steel strain. Geometrical parameters are also
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 167

Fig. 2. Material model for single reinforced concrete design (a) tension model; (b) compression model; (c) steel model; (d) beam cross section.

normalized with the beam dimensions of width b and full depth h

as shown in Fig. 2d with steel parameters defined as area
As = qgbh = qgbd/a at the reinforced depth d = ah. The reinforce-
ment ratio qg is defined per gross sectional area bh, and differs
slightly from the conventional definition based on term bd used
in reinforced concrete nomenclature. The material models for ten-
sion and compression of FRC and the model for steel rebar are pre-
sented as:
8 8
< Eet
> 0 6 et 6 ecr
rt ðbÞ <b
> 06b61
rt ðet Þ ¼ lEecr ecr < et 6 etu ; ¼ l 1 < b 6 btu ð1Þ
>
: Eecr >
:
0 et > etu 0 b > btu
8 8
< Ec e c
> 0 6 ec 6 ecy
rc ðkÞ < ck
06k6x
>
rc ðec Þ ¼ Ec ecy ecy < ec 6 ecu ; ¼ cx x < k 6 kcu
>
: Eecr >
:
0 ec > ecu 0 k > kcu
ð2Þ
 
Es es 0 6 es 6 esy f s ðvÞ nv 06v6j
f s ðes Þ ¼ ; ¼ ð3Þ
Es esy es > esy Eecr nj v>j
where normalized strains are defined as b = et/ecr, k = ec/ecr and
v = es/ecr. Variable k as top compressive fiber ectop is used in the
derivation of moment–curvature diagram and other variables such
as tensile strain in concrete and steel strain are obtained using the
expressions derived based on the present formulation.

3.2. Moment–curvature diagram

In derivation of moment–curvature for a beam with rectangular

cross section, the assumption of plane section remaining plane is
assumed. By applying linear strain distribution across the depth,
ignoring shear deformation, and using material models of Eqs. Fig. 3. Strain and stress diagram at three stage of applied compressive strain at top
(1)–(3) and Fig. 2a–c, the stress distributions as shown in Fig. 3 fiber (k); (a) stage 1 (0 < k 6 kR1) elastic compression–elastic tension; (b) stage 2
(kR1 < k 6 x) elastic compression – post crack tension; (c) stage 3 (x < k 6 kcu)
are obtained. The normalized compressive strain at the top con-
plastic compression – post crack tension.
crete fiber k is used as an independent variable to incrementally
impose flexural deformation for three distinct stages. The first
stage (0 < k 6 kR1) corresponds to elastic range until tensile strain post-crack region. Finally, stage 3 (x < k 6 kcu) corresponds to the
at the bottom fiber reaches ecr. Stage 2 (kR1 < k 6 x) corresponds plastic compressive strain while the tensile strain is in post-crack
to an elastic compressive strain and the tensile strain in range. For stages 2 and 3 two possible scenarios exist: the steel
168 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

Table 1
Normalized height of compression and tension zones for each stage of normalized compressive strain at top fiber (k).

Zone Normalized height Stage 1 Stage 2 Stage 3

0 < k 6 kR1 kR1 < k 6 x x < k 6 kcu
2.1 es 6 esy 2.2 es > esy 3.1 es 6 esy 3.2 es > esy
Compression hc2 – – kðkxÞ
h k
hc1 k k xk
h k

Tension ht1 1-k k

k
k
k
h
ht2 – kðkþ1Þk kðkþ1Þk
h k k

Table 2
Normalized stress at vertices in the stress diagram for each stage of normalized compressive strain at top fiber (k).

Zone Normalized stress Stage 1 Stage 2 Stage 3

0 < k 6 kR1 kR1 < k 6 x x < k 6 kcu
2.1 es 6 esy 2.2 es > esy 3.1 es 6 esy 3.2 es > esy
Compression f c2 – – cx
Eecr
f c1 ck ck cx
Eecr

Tension f t1 ð1kÞk 1 1
Eecr k
f t2 – l l
Eecr
fs nkðakÞ nkðakÞ nj nkðakÞ nj
Eecr k k k

Table 3
Normalized force component for each stage of normalized compressive strain at top fiber (k).

Zone Normalized force component Stage 1 Stage 2 Stage 3

0 < k 6 kR1 kR1 < k 6 x x < k 6 kcu
2.1 es 6 esy 2.2 es > esy 3.1 es 6 esy 3.2 es > esy
Compression F c2 – – cxkðkxÞ
bhEecr k
F c1 1 1 cx2 k
bhEecr 2 ckk 2 ckk 2k

Tension F t1 k 2 k k
bhEecr 2k
ðk  1Þ 2k 2k
F t2 – l l
bhEecr k ðk  kk  kÞ k ðk  kk  kÞ
Fs qg nk qg nk qgnj qg nk qgnj
bhEecr k
ða  kÞ k
ða  kÞ k
ða  kÞ

Table 4
Normalized moment arm of force component for each stage of normalized compressive strain at top fiber (k).

Zone Normalized moment arm Stage 1 Stage 2 Stage 3

0 < k 6 kR1 kR1 < k 6 x x < k 6 kcu
2.1 es 6 esy 2.2 es > esy 3.1 es 6 esy 3.2 es > esy
yc2 kðxþkÞ
Compression h
– –
2k
yc1 2 2 2xk
h 3k 3k 3k
yt1 2
Tension h 3 ð1  kÞ 2k
3k
2k
3k
yt2
h
– kþð1kÞk kþð1kÞk
2k 2k
ys
h
ak ak ak

is either elastic, or yielding, therefore stages 2 and 3 are each The net section force is obtained as the difference between the
divided into two sub-stages, 2.1, 2.2, or 3.1, and 3.2 where term tension and compression forces, and solved for internal equilib-
1 represents elastic and term 2 represents plastic response. rium to obtain the normalized location of neutral axis, k. When
Three stages of stress distribution in Fig. 3, show the height of steel is elastic in stages 1, 2.1 and 3.1, the expressions for net force
compression and tension zones normalized with respect to the are in the quadratic forms and result in two possible solutions for
beam depth h, while stresses are normalized with respect to the k. With a large scale of numerical tests covering the practical range
first cracking strength Eecr and presented in Tables 1 and 2, respec- of material parameters, only one solution yields the valid value in
tively. Forces and their lines of action are normalized with respect the range 0 < k < 1. During stage 1, the singularity of k1 for c = 1,
to cracking tensile force bhEecr and beam depth h as shown in requires an asymptotic expression. When steel is in yield condition
Tables 3 and 4. in stages 2.2 or 3.2, there is a unique solution for k as presented in
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 169

Table 5
Normalized neutral axis, moment, curvature and stiffness for each stage of normalized compressive strain at top fiber (k).

Stage k M0 /0 K0
h i
1 k1 ¼
18qg aþ1
M 01 ¼ 2k 3
C 1 k1 þ
2
C 2 k1
þ C 3 k1 þ C 4 /01 ¼ k
2k1 K 01 ¼
M 01
18qg þ2 k1 /01
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i
2.1 k21 ¼ Bk1 B2 þ B3 þ 2aqg nB1 M 021 ¼ k21k
3 2
C 5 k21 þ C 6 k21 þ C 7 k21 þ C 8 /021 ¼ 2kk21 K 021 ¼
M 021
21 /021

2.2 k22 ¼ BB41 2

M 022 ¼ k12 ½C 5 k22 þ C 9 k22 þ C 10  /022 ¼ 2kk22 K 022 ¼
M 022
/022
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i
3.1 k31 ¼ Bk5 B2 þ B3 þ 2aqg nB5
3 2
M 031 ¼ k21k C 11 k31 þ C 6 k31 þ C 7 k31 þ C 8 /031 ¼ k
2k31 K 031 ¼
M 031
/031
h31 i
3.2 k32 ¼ BB45 2
M 032 ¼ k12 C 11 k32 þ C 9 k32 þ C 10 /032 ¼ 2kk32 K 032 ¼
M 032
/032

where, the coefficients are:

B1 ¼ k2 þ 2lðk þ 1Þ  1; B2 ¼ l  9qg k; B3 ¼ 9qg ðqg 9k2  2lkÞ þ l2 ; B4 ¼ 2kð9qg j þ lÞ; B5 ¼ 20k  101 þ 2lðk þ 1Þ

C 1 ¼ 0; C 2 ¼ 27qg þ 3; C 3 ¼ 3  54qg a; C 4 ¼ 1 þ 27qg a2 ; C 5 ¼ 2k3 þ 3lðk2  1Þ þ 2; C 6 ¼ 6k2 ð9kqg  lÞ; C 7 ¼ 3k2 ðl  36qg akÞ;
C 8 ¼ 54qg a2 k3 ; C 9 ¼ 6k2 ð9qg j þ lÞ; C 10 ¼ 3k2 ð18qg aj þ lÞ; C 11 ¼ 30k2 þ 3lðk2  1Þ  998:

The yield condition for tensile steel is checked by first assuming

that it yields and then using k22 or k32 in Table 5 for k in Eq. (9) to
calculate the steel strain es:
ak
es ¼ kecr ð9Þ
k
If es is greater than esy, the assumption is correct, otherwise steel
has not yielded and one has to use k21 or k31. Using the values in
Table 5 and Eqs. (4)–(6) analytical expressions for moment–curva-
ture response and flexural stiffness are calculated.
By considering an under-reinforced section, one can solve for
the balanced reinforcement ratio qg,bal, representing compression
failure and steel reaching its yield limit defined as (ec = ecu and es =
esy). The strain gradient in stage 3.2 of Fig. 3c, represents a plastic
compressive strain and tensile strain in the post-crack region as:
Fig. 4. Normalized moment–curvature diagram and approximate bilinear model for
deflection hardening (l > lcrit).
kcu ecr jecr
¼ ð10Þ
kh ða  kÞh
By substituting kcu in the expression for k32 in Table 5 and fol-
Table 5. Internal moment is obtained by integrating the force com-
lowing with k in Eq. (10), one can solve for the balance reinforce-
ponents using the distance to the neutral axis as the moment arm,
ment ratio as:
and the curvature is represented as the ratio of compressive fiber
strain (ectop = kecr) to the depth of neutral axis kh. Effective flexural 2lðkcu ða  1Þ þ a  jÞ þ acxð2kcu  xÞ  a
qg;bal ¼ ð11Þ
stiffness is defined as the ratio of the moment to the curvature at 2njðkcu þ jÞ
any given imposed k. By normalizing the moment Mi, curvature
For a majority of under-reinforced design procedures the
/i and stiffness Ki for each stage i, using the cracking values Mcr,
moment capacity in the zone 2.1 defined by tensile yield and com-
/cr and Kcr are expressed as analytical expressions Mi0 , /i0 and Ki0
pression elastic is given by Eq. (12), whereas in the zone 2.2 with
as presented in Table 5.
tension and compression yielding, the moment–curvature rela-
1 2 tionship are given by Eq. (13):
M i ¼ M 0i Mcr ; M cr ¼ bh Eecr ð4Þ
6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 
k21 ¼ B2 þ B3 þ 2aqg nB1 and M 021
2ecr B1
/i ¼ /0i /cr ; /cr ¼ ð5Þ
h 1 h 3 2
i
¼ 2 C 5 k21 þ C 6 k21 þ C 7 k21 þ C 8 ð12Þ
k k21
1 3
K i ¼ K 0i K cr ; K cr ¼ bh ð6Þ
12 B4 0 1h 2
i
k22 ¼ M22 ¼ 2 C 5 k22 þ C 9 k22 þ C 10 ð13Þ
The compressive strain corresponding to end of elastic region 1 B1 k
(kR1) is determined from the strain gradient diagram shown in
The coefficients of Bi, Ci of these equations are provided in
Fig. 3a.
Table 5.
kR1 ecr ecr
¼ ð7Þ
kh ð1  kÞh 3.3. Simplified analytical solutions for load–deflection response

By substituting k1 from Table 5 for k in Eq. (7) and solving for

Load–deflection response of various geometries are obtained
kR1, one obtains:
from the analytical moment and curvature distribution expres-
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> 1þq n q2g n2 þ2qg nð1aþacÞþc sions for a few loading cases. The first step is to simplify and rep-
<  g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi when c – 1
kR1 ¼
2 2
cþqg n qg n þ2qg nð1aþacÞþc
ð8Þ resent the normalized moment–curvature as a bilinear response as
>
: 2qg naþ1 shown by the dashed line in Fig. 4 for the case of a deflection hard-
2qg nða1Þ1
when c ¼ 1
ening beam [28]. By applying the moment-area method to the
170 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

8 16
ρg=0.00 (b) μ=0.00
(a)

Normalized Moment, M'

Normalized Moment, M'

6 12

4 8

2 4

0 0
0 25 50 75 100 0 25 50 75 100
Normalized Curvature, φ' Normalized Curvature, φ'

8 16
μ=0.33
(c) (d)
Normalized Moment, M'

Normalized Moment, M'

6 12

4 8

2 4

ρg=0.01
0 0
0 25 50 75 100 0 25 50 75 100
Normalized Curvature, φ' Normalized Curvature, φ'

Fig. 5. Parametric studies of normalized moment curvature diagram for different levels of post crack tensile strength parameter l and reinforcement ratio qg.

1.25 1.25
Normalized Secant Stiffness, K'

Normalized Secant Stiffness, K'

(a) μ=1.00 (b) ρ=0.03

1 μ=0.67 1 ρ=0.02
μ=0.33 ρ=0.01
μ=0.00 ρ=0.00
0.75 0.75
ρ=0.01 μ=0.33
0.5 0.5

0.25 0.25

0 0
0 10 20 30 40 0 10 20 30 40
Normalized Curvature, φ' Normalized Curvature, φ'

Fig. 6. Parametric studies of normalized secant stiffness for different levels of reinforcement ratio qg and residual tensile strength parameter l.

bilinear moment–curvature diagrams, mid-span deflection can be 1 2

dbcr ¼ L /bcr ð14aÞ
derived explicitly. For 3PB, additional parameter for plastic length 12
Lp at the vicinity of the load is needed to simulate the zone
undergoing localization in postpeak response while the non- L2 h    i
du ¼ 2M2u  Mu Mbcr  M 2bcr /u þ M2u þ M u Mbcr /bcr
localized zone is elastically unloading. For the 4PB, the distance 24M 2u
between the two load points was used as the plastic length Lp.
The load–deflection response is affected by the residual tensile l > lcrit ð14bÞ
strength. The transition from deflection softening to deflection
hardening is obtained at a threshold postpeak tensile capacity
/u Lp M u /bcr L
lcrit = x/(3x  1)  0.35, and equations for mid-span deflection du ¼ ð2L  Lp Þ þ ðL  2Lp Þ l < lcrit ð14cÞ
d of 3 PB at first bilinear cracking dbcr, and at ultimate du are 8 12M bcr
presented in Eqs. (14a–c) and (15a–c), [7,28].
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 171

15 material models for SFRC and steel rebar were used that include:
Grade 60 btu = 160, c = 1, x = 8.5, kcu = 28, n = 8.33, j = 16 and a = 0.8. The
Normalized Ultimate Moment, M' ( λ=λcu)
variables of the study were: residual tensile strength parameter
12 Grade 80 0.0 6 l 6 1.0 and reinforcement ratio 0.0 6 qg 6 0.03.
Grade 40 Fig. 5 illustrates the effects of parameters l and qg on the nor-
malized moment–curvature diagram. Fig. 5a shows the effect of
μ=0.00
9 increasing the residual tensile strength from brittle (l = 0) to duc-
μ=0.33 tile (l = 1) in plain FRC. Noted that at a level l = 0.33 which is suf-
μ=0.67 ficiently close to lcrit = 0.35, the flexural response is almost
6 μ=1.00 perfectly-plastic, beyond which the deflection softening shifts to
hardening. The elastic–plastic tensile response of FRC (l = 1) yields
balance
failure an upper bound normalized moment capacity of 2.7. With a main
flexural reinforcement of qg = 0.01 (Fig. 5b), the normalized
3
fc'=30 MPa, γ=1.00, ω=8.5, moment capacity of 5.8 is achieved. Note that as qg increases,
the response eventually changes from a ductile under-reinforced
α=0.80, κ=16.0, λcu=28.0
to over-reinforced. Fig. 5c reveals the effect of residual tensile
0
0 0.01 0.02 0.03 0.04 0.05 strength (l = 0.0–1.0) for a fixed reinforcement ratio of 0.01 while
Fig. 5d shows the marginal benefit of FRC with l = 0.33 compared
Reinforcement Ratio, ρg
to the reinforced concrete system. The moment capacity slightly
Fig. 7. Design chart of normalized ultimate moment capacity (determined at
increases in comparison with the reinforced concrete without
k = kcu) for different levels of post crack tensile strength l and reinforcement ratio any fibers (Fig. 5b). The present analysis ignores the contribution
qg. of the fiber phase to the compression response in the context of
internal confinement, however that can be easily incorporated in
the input parameters.
Similarly, a set of equations for 4PB can be written as:
The neutral axis depth ratio k and the normalized secant stiff-
23 2 ness K0 are also affected by changes in l and qg. The neutral axis
dbcr ¼ L /bcr ð15aÞ
216 starts at a slightly higher value than 0.5 for a conventional rein-
forced concrete system (l = 0, qg P 0), since a larger compressive
L2 h i zone is needed to balance the summation of tensile forces of con-
du ¼ ð23M2u  4Mu Mbcr  4M 2bcr Þ/u þ ð4M2u þ 4M u Mbcr Þ/bcr crete and steel. The neutral axis location, k decreases as the com-
216M 2u
pressive strain at top fiber k increases as functions of l and qg.
l > lcrit ð15bÞ This shift diminishes as qg or l increase, indicating the role of fiber
and reinforcement in maintaining the tensile force after cracking.
5L2 /u M u L2 /bcr For plain FRC with low fiber contents, the normalized secant stiff-
du ¼ þ l < lcrit ð15cÞ
ness K0 equals to 1.0 in elastic range (/0 6 1.0) while K0 is larger
72 27M bcr
than 1.0 in conventional reinforced concrete systems as shown in
Fig. 6. Fig. 6a shows that for the same reinforcement ratio, the
4. Parametric studies rate of stiffness degradation decreased with addition of fibers
(l increased) as the curvature increases since the crack is
Parametric studies of post-crack tensile strength and reinforce- bridged by distributed fibers through its depth. Fig. 6b shows that
ment ratio as two main reinforcing factors were conducted. for a given fiber residual tensile strength, l = 0.33 higher qg
Changes in the location of neutral axis, moment–curvature levels in conventional reinforced concrete efficiently reduces the
response, and stiffness degradation of a beam are normalized with rate of stiffness reduction and retains the post-crack stiffness.
respect to first cracking parameters of plain FRC. In addition to the More details on the effect of parameters have been discussed
two baseline parameters: E = 24 GPa and ecr = 125 lstr, typical elsewhere [36].

Reinforcement Ratio, ρ=As/bd Reinforcement Ratio, ρ=As/bd

0 0.025 0.05 0.075 0.1 0.125 0 0.025 0.05 0.075 0.1 0.125
20 20
Coefficient of Resistance, R (MPa)

Coefficient of Resistance, R (MPa)

fc' = 55 MPa
(a) (b) fc' = 55 MPa

15 fc' = 43 MPa 15 fc' = 43 MPa

fc' = 30 MPa fc' = 30 MPa

10 10
ρ
g,bal ρ
g,bal
Grade 40 Grade 60
5 5 fsy = 420 MPa
fsy = 280 MPa
ACI 318-11 ACI 318-11
Eq. (18) Eq. (18)
0 0
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
Reinforcement Ratio, ρg=As/bh Reinforcement Ratio, ρg=As/bh

Fig. 8. Comparison of coefficient of resistance using ACI stress block method and the Eq. (18); (a) for steel grade 40 (280 MPa); (b) for steel grade 60 (420 MPa).
172 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

5. Design Table 6
Beam test series.

5.1. Design charts Beam Mix Fiber content (kg/m3) Span (m) Rebar
B1 NSC 25 1.0 –
The ultimate moment capacity as a function of residual tensile B2 NSC 25 2.0 –
strength and reinforcement ratio can be used as a convenient B3 NSC 50 1.0 –
B4 NSC 50 2.0 –
design tool for combinations of reinforcements. A limiting case of
B5 HSC 60 1.0 –
ductile moment–curvature response of under-reinforced section
B6 HSC 60 2.0 –
(Fig. 5) is obtained at (k ? 1) by applying L’Hopital’s rule in the
B7 NSC 25 1.0 2-/8
limit case of compressive strain failure (k = kcu = 1). Thus, the B8 NSC 25 2.0 2-/8
ultimate moment Mu is reasonably approximated by the moment B9 NSC 50 1.0 2-/12
at infinite compressive strain M1 for under-reinforced section B10 NSC 50 2.0 2-/12
(qg < qg,bal). The yielding condition of steel is obtained by B11 HSC 60 1.0 2-/16
B12 HSC 60 2.0 2-/16
comparing it to the reinforcement ratio at balance failure as
defined by Eq. (11). Normalized moment at infinite M0 1, is found
by substituting the expression for k32 into the M0 32 in Table 5,
followed by taking the limit of k to 1, which results in: shows that as the steel grade increases from 40 to 80 (280–
550 MPa), the balanced failure is obtained at much lower rein-
6qg njðla  l þ axÞ þ 3xl  3ðqg njÞ2 forcement ratio, from about 0.035 to 0.015. To design flexural
M 01 ¼ lim M032 ¼ ð16aÞ HRC members with this chart, the ultimate moment Mu due to fac-
kcu !1 xþl tored load is determined and then normalized with cross sectional
And the corresponding ultimate moment capacity Mu: geometry while the cracking moment of the plain matrix Mcr is
employed to obtain demand ultimate moment capacity Mu0 . The
M u  M 01 M cr chart is then used to select any combination of normalized residual
tensile strength l, grade of steel, and reinforcement ratio qg that
6qg njðla  l þ axÞ þ 3xl  3ðqg njÞ2
¼ Mcr ð16bÞ meets the demand for Mu0 .
xþl As a comparison with the customary design approach, one can
For a plain FRC beam without any flexural reinforcement develop a parameter representing coefficient of resistance R as a
(qg = 0) and modulus of FRC are equal in compression and tension design chart [38], and proceed to determine a beam size for a given
(c = Ec/E = 1), Eq. (16) reduces to M01 = 3xl/(x + l) reported previ- required moment. The normalized moment design chart in Fig. 8 is
ously [28]. The applicability of Eq. (16) is limited to the sections equivalent to the well-established R-chart for single
that fail in a ductile manner only when flexural steel reinforcement under-reinforced concrete design nominal moment capacity Mn as:
ratio is below the balance failure qg,bal defined in Eq. (11). !
2 f sy 2
Fig. 7 shows a design chart for the numerical model used in the Mn ¼ Rbd ¼ qf sy 1  0:59q 0 bd ð17Þ
parametric studies with various grades of steel as defined by ASTM fc
A615 [37]. The moment capacity is strongly dependent on the
where d is the effective depth, q = As/bd is the reinforcement ratio.
amount of reinforcement ratio whereas the residual tensile
For the proposed model, the moment equations are represented
strength provides extra capacity. Under-reinforced sections are
as ratio of ultimate moment to cracking moment and reinforcing
shown by the curves below the balance failure points (qg 6 qg,bal,
depth to full depth a = d/h as:
shown as hollow circles), as the moment capacity increases pro-
portional to the reinforcement ratio. When qg > qg,bal, the strength Mðkcu Þ M 0 ðkcu ÞMcr M0 ðkcu Þ 1 2 ecr E
R¼ ¼ ¼ bh ecr E ¼ 2 M0 ðkcu Þ ð18Þ
of all curves marginally increases as the steel fails to reach yield bd
2
bd
2
bd
2 6 6a
strength. Effect of fiber contribution becomes negligible as the fail-
Therefore, R is the normalized moment M0 (kcu) by a factor of
ure is governed by compression failure of concrete. Fig. 7 also
ecrE/(6a2). In order to use equivalent set of input parameters, the
compressive constitutive relationship is calibrated using parabolic
0.6 stress–strain curve of Hognestad [39] up to the ultimate strain
ACI-318: α = 0.5 ecu = 0.003 to obtain equivalent areas under both curves:
Minimum Reinforcement Ratio, ρmin (%)

Grade (ASTM A615) α = 0.6 Z    

0.5
ecu
ec ec 2 1
40 α = 0.7 2  dec ¼ ecy rcy þ ðecu  ecy Þrcy ð19Þ
α = 0.8 0 ec0 ec0 2
0.4
60 α = 0.9 By substituting ecu = 1.5ec0 and rcy = 0.85fc0 in Eq. (19), the com-
EC 2: pressive yield strain ecy and compressive modulus Ec can be esti-
Grade (EN 10025) mated as.
0.3 80
6 rcy 289f 0c
S275N ecy ¼ ec0 ; Ec ¼ ¼ ð20Þ
0.2 17 ecy 120ec0
S420N Three concrete strength fc0 = 30, 43 and 55 MPa and two grades
S550Q
0.1
γ=0.90, ω=8.5 μcric of steel 280 and 420 MPa with Young’s modulus Es of 200 GPa were
used to compare the coefficient of resistance defined by the ACI
n=8.33, κ=16.0 approach [Eq. (17)] and the proposed method [Eq. (18)]. Other
0 assumed parameters were the first cracking strain ecr = 0.0001,
0 0.1 0.2 0.3 0.4
compressive strain at peak stress ec0 = 0.002, normalized depth of
Normalized Post Crack Tensile Strength, μ
steel reinforcement a = 0.8, and assumption of no softening range
Fig. 9. Comparison of minimum reinforcement ratio qmin between proposed for plain concrete (l = 0). For the proposed method, a set of mate-
method and design codes: ACI 318-11 and Eurocode 2 (EC 2). rial parameters of concrete and steel are used to calculate qg,bal by
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 173

Fig. 10. Material stress strain model for RILEM method [45]; (a) tension and (b) compression model for SFRC; (c) steel model.

Table 7
Steel fiber reinforced concrete parameters for RILEM and proposed models.

Beam type Mix Fiber content E RILEM Proposed l

(kg/m3) (GPa)
fc 0 r1 r2 r3 e1 (%) e2 e3 Ec rcy rcr rp ecr etu
(MPa) (MPa) (MPa) (MPa) (%) (%) (GPa) (MPa) (MPa) (MPa) (%) (%)
SFRC (Plain) NSC 25 31.8 30 3.5 1.1 0.8 0.011 0.21 2.5 22.6 30.2 3.5 1.0 0.011 2.5 0.273
NSC 50 30.6 26 4.2 2.0 1.2 0.014 0.24 2.5 20.0 26.6 4.2 1.6 0.014 2.5 0.382
HSC 60 38.4 53 6.2 3.1 3.1 0.016 0.26 2.5 39.7 52.9 6.2 3.1 0.016 2.5 0.501
SFRC (with NSC 25 30.5 26.4 3.2 1.3 0.9 0.011 0.21 2.5 20 26.4 3.2 1.1 0.011 2.5 0.345
Rebar) NSC 50 30.3 26.1 3.8 1.8 1.1 0.013 0.23 2.5 20 26.1 3.8 1.5 0.013 2.5 0.383
HSC 60 39.0 55.4 6.3 3.8 3.2 0.016 0.26 2.5 41 55.4 6.3 3.5 0.016 2.5 0.557
a
Strain at peak stress, ec0 = 0.2%, at compressive yield stress, ecy = 0.133%, and ultimate compressive strain, ecu = 0.35% for all mixes.

Eq. (11) and compared it to the reinforcement ratio qg used in a present formulation that is based on the gross section bh to the
beam section. For qg < qg,bal, the expression k31 and M031 in effective cross section bd. The equation is further simplified as an
Table 5 are used to determine moment at ultimate compressive analytical minimum reinforcement ratio for conventional rein-
strain M(kcu)0 . For qg > qg,bal, the expression k32 and M032 in forced concrete system by substituting parameters: l = 0, c = 3/4
Table 5 are used instead. Finally, by substituting the calculated and x = 6 into Eq. (21).
M(kcu)0 in Eq. (18) one obtains the R value for the proposed method. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fig. 8 compares the two methods showing excellent agreement 9a  81a2  6
for the reinforcement ratio up to the ACI balance failure [Eq. (17)].
qmin ¼ ð22Þ
2anj
These points are generally lower than the balance failure points
Fig. 9 shows minimum reinforcement ratio as a function l and
qg,bal defined by Eq. (11) as marked by a circle symbol. The discrep-
a, and compared to specifications of ACI 318-11 and Eurocode 2
ancy between these two balanced failures is due to the fact that
(EC2) with varying grades of steel. For an assumed value of
ACI approach uses a conservative empirical parameter b1 in the cal-
a = 0.5–0.9, the trend shows that as the residual tensile strength
culation of the reinforcement ratio at balance failure while the qg,bal
l increases, the required minimum reinforcement qmin,rc decreases
is analytically determined by Eq. (11). Note that the applicable
indicating the role of steel fibers in substitution of reinforcement.
range of the R by ACI approach is terminated at the balance failure
Additionally, the effect of a is diminishing gradually and all the
whereas the current method predicts a wider range in both under-
curves converge when l ? lcrit = 0.35 in accordance with the onset
and over-reinforced beam sections.
of deflection hardening, where no longitudinal reinforcement is
required to meet the minimum strength requirement.
5.2. Minimum reinforcement ratio

A reinforced concrete beam can fail abruptly if its residual 6. Experimental verification of flexural model
strength is less than the cracking moment of unreinforced concrete
section computed from its modulus of rupture. In order to prevent Full scale beam tests from the Brite/Euram project
such failures, the minimum reinforced ratio is defined as level of BRPR-CT98-0813 ‘‘Test and design methods for steel fibre rein-
reinforcement to ensure that residual capacity is equal to the forced concrete’’ by Dupont were used for model verification
cracking moment, and is determined in accordance with ACI [45]. The experimental program studied the effects of four vari-
318-11 Section 10.5 [40] and Eurocode 2 [41]. The minimum ables: concrete strength, fiber dosage, span length and longitudinal
required reinforcement is empirically stipulated to be a function reinforcement ratio. Table 6 provides the details of the 12 beam
of concrete strength, yield limit of steel, as well as the beam size series, each with 2 replicates, of two grades of normal (NSC), and
[42–44]. An analytical expression for minimum reinforcement high strength concrete (HSC). Normal strength concrete used fiber
ratio qg,min is derived explicitly by setting the moment from Eq. type RC 65/60 BN at 25 and 50 kg/m3 while HSC used fiber type RC
(16a) at infinity to unity, M0 1 = 1. A quadratic equation is obtained 80/60 BP at 60 kg/m3. All beams had a cross section of
such that the root satisfying qmin 6 qbal is valid and expressed as: 0.20  0.20 m, with two different span lengths of 1.0 and 2.0 m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and tested under four point bending set up with a constant spacing
3G1  G2 þ 9G21 between the two point loads at 0.2 m. The first half of the series
qmin ¼ ð21Þ (B1–B6) contains no rebar and the other half (B7–B12) contained
3anj
two rebars of size 8, 12 and 16 mm. Steel parameters were
where G1 = l(a  1) + acx, G2 = 3cx(3l  1)  3l and parameter a Young’s modulus of 200 GPa, yield strength of 560 MPa, and a con-
in the denominator is introduced to express and correlate the crete cover of 15 mm.
174 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

30 12

(a.1) (a.2)

20 8

Load (kN)

Load (kN)
NSC, Beam-B2
Fiber content 25 kg/m3,
NSC, Span 2 m
Fiber content 25 kg/m3,
10 Span 1 m 4
Experiment (Avg.)
Experiment (Avg.)
RILEM
RILEM
Proposed Model
Proposed Model
0 0
0 1 2 3 4 0 2 4 6 8
Mid Span Deflection (mm) Mid Span Deflection (mm)

40 20
(b.1) (b.2)
16
30
Load (kN)

Load (kN)
12
20
NSC, NSC, Beam-B4
Fiber content 50 kg/m3, 8 Fiber content 50 kg/m3,
Span 1 m Span 2 m
10 Experiment (Avg.) Experiment (Avg.)
RILEM 4 RILEM
Proposed Model Proposed Model

0 0
0 1 2 3 4 0 2 4 6 8
Mid Span Deflection (mm) Mid Span Deflection (mm)

30

60 (c.1) (c.2)

20
Load (kN)

Load (kN)

40
HSC, HSC, Beam-B6
Fiber content 60 kg/m3, Fiber content 60 kg/m3,
Span 1 m Span 2 m
10
20 Experiment (Avg.) Experiment (Avg.)
RILEM RILEM
Proposed Model Proposed Model

0 0
0 1 2 3 4 0 2 4 6 8
Mid Span Deflection (mm) Mid Span Deflection (mm)

Fig. 11. Load deflection responses of SFRC beams at three levels of fiber contents (25 kg/m3, 50 kg/m3 and 60 kg/m3). RILEM refers to the updated RILEM stress–strain model
[45].

The load–deflection responses of the 12 beam series were sim- method. This is attributed to the differences between the tensile
ulated by the algorithm proposed and compared with updated responses used by the two models. The RILEM model specified
RILEM method [45] as reference. The material model parameters two points (r2, e2) and (r3, e3) to express the descending branch
used were obtained by fitting the tension and compression models (Fig. 10a) as opposed to a constant residual strength rp with a
shown in Fig. 2 to the models shown in Fig. 10 and summarized in lower bound estimation of the residual strength in the proposed
Table 7. The residual tensile strength l corresponds to average model with a single step drop from rcr to rp < r2 (Fig. 2a). This is
response of the RILEM method. Ultimate tensile strain of 0.025 also shown in Table 7 and results in lower predicted load at fiber
was used for both models [15]. contents of 25 and 50 kg/m3 (Fig. 11a and b). For the high fiber con-
Fig. 11 shows the simulations of the 6 plain SFRC beams with- tent, residual strengths used in two models are identical and thus
out flexural reinforcement representing the effect of concrete similar load–deflection responses are obtained (Fig. 11c). It is also
strength, fiber content, and span length. Average test results of noteworthy that both models underestimate the post-crack
two replicate samples of each series is also shown and compared response of beam B1 but overestimate the response of beam B2.
to the simulation curves. The simulations compare favorably to Since the same model parameters are used for beams of different
the experimental results while underestimating the RILEM lengths, the simulations demonstrate the nature of size effect of
 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177 175

80
(a.1) (a.2)
30
60

Load (kN)
Load (kN)
20
40 NSC, Beam-B7 NSC, Beam-B8
Fiber content 25 kg/m3, Fiber content 25 kg/m3,
Span 1 m, Span 2 m,
2-φ8 mm 2-φ8 mm
10
20 Experiment (Avg.) Experiment (Avg.)
RILEM RILEM
Proposed Model Proposed Model

0 0
0 1 2 3 4 5 6 0 2 4 6 8 10 12 14
Mid Span Deflection (mm) Mid Span Deflection (mm)

150 60
(b.1) (b.2)

100 40
Load (kN)

Load (kN)
NSC, Beam-B9 NSC, Beam-B10
Fiber content 50 kg/m3, Fiber content 50 kg/m3,
Span 1 m, Span 2 m,
50 2-φ12 mm 20 2-φ12 mm
Experiment (Avg.) Experiment (Avg.)
RILEM RILEM
Proposed Model Proposed Model
0 0
0 1 2 3 4 5 6 0 2 4 6 8 10 12 14
Mid Span Deflection (mm) Mid Span Deflection (mm)

300 120
(c.1) (c.2)

200 80
Load (kN)
Load (kN)

HSC, Beam-B11 HSC, Beam-B12

Fiber content 60 kg/m3, Fiber content 60 kg/m3,
Span 1 m, Span 2 m,
100 2−φ16 mm 40 2-φ16 mm

Experiment (Avg.) Experiment (Avg.)

RILEM RILEM
Proposed Model Proposed Model

0 0
0 1 2 3 4 5 6 0 2 4 6 8 10 12 14
Mid Span Deflection (mm) Mid Span Deflection (mm)

Fig. 12. Load deflection responses of HRC beams at three levels of fiber contents (25 kg/m3, 50 kg/m3 and 60 kg/m3). RILEM refers to the updated RILEM stress–strain model
[45].

properties obtained from smaller specimens. Larger specimens limit-state load capacity is insensitive to the length of localization
indicate the apparent size effect observed in experiments with zone as shown by Bakhshi et al. [46], and the size effect due to span
the descending parts that behave differently from smaller beams. in HRC beams is not as pronounced as the FRC samples. Finally, the
Fig. 12 presents load–deflection responses for the 6 HRC beams difference between the predictability of the two methods dimin-
with flexural reinforcement of q = 0.13–0.20%. Both models simu- ishes since they both use the same elastic-perfectly plastic steel
late the experimental results with the discrepancy in the flexural model.
stiffness after cracking for the HSC beams in Fig. 12c. The present Since the present analytical model keeps track of the curvature
model assumes cracks to be uniformly distributed throughout distribution and allows the determination of load capacity at given
the mid-zone between the two loading points used as the localized deformation, it can be used to compare the results of beams of var-
zone. However additional cracks form outside the mid-zone in HSC ious sizes for a serviceability based criterion such as maximum
beams in the shear span region due to tension stiffening effects allowable curvature, deflection, ductility or stress. Various param-
which results in a larger localized zone. Since the deflection corre- eters have been proposed and used to characterize the flexural
lates with the double integration of curvature, additional cracking toughness and residual strength of FRC from experimental data.
over a larger section will inherently result in a larger localization For example, EN14651 uses equivalent flexural tensile strength
zone, and higher deflections at the same loading level. The feq,3 determined at a specific deflection level of d = 2.5 mm; ASTM
176 B. Mobasher et al. / Engineering Structures 100 (2015) 164–177

25 The derivations are used in terms of design charts representing

the normalized ultimate moment capacity as a function of residual
1 δ = L/150
tensile strength and reinforcement ratio and are applicable to
20 79 kg/m Hooked end
3
conventional-, fiber-, and hybrid-reinforced concrete. Results are
Maximum Curvature, 10-5/mm

2 79 kg/m3 Twisted further converted to coefficient of resistance R by stress block

13, 26, and 39 kg/m3
Hooked end approach, nominal strength, and minimum reinforcement ratio.
Numerical tests covering materials and geometrical ranges as a
15
well as comparison with available experimental data confirm the
50 kg/m3 FRC
60 kg/m3 FRC proposed equations against the original equations.
3 25 kg/m3 HRC
10
3
60 kg/m3 60 kg/m3 HRC
45 kg/m3
30 kg/m3
Hooked end References
1. Kim et. al [48] 50 kg/m3
5 HRC
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ate for the design of full scale structures due to the size effect [47]. SFRC panels. Constr Build Mater 2014;73:289–304. http://dx.doi.org/10.1016/
j.conbuildmat.2014.09.043.
A study of size effect on the curvatures which correlates with max- [6] Singh B, Jain K. Appraisal of steel fibers as minimum shear reinforcement in
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