A FE-based limit analysis approach for concrete
elements reinforced with FRP bars
D. De Domenico, A. A. Pisano, P. Fuschi
Dept. PAU - University Mediterranea of Reggio Calabria
via Melissari, I-89124 Reggio Calabria, Italy
Abstract
The aim of this paper is to verify the validity of a FE-based methodology,
founded on limit analysis theory, to predict peak load and failure mechanism
of concrete elements reinforced with fiber reinforced polymer (FRP) bars.
Due to dilatancy, which implies the adoption of a nonstandard constitutive
flow law for concrete, such methodology makes use of two different numerical procedures to search for an upper and a lower bound to the actual peak
load. A certain number of experimental tests on FRP reinforced concrete
elements carried out up to collapse are numerically reproduced and the related predictions, in terms of peak loads and failure modes, are critically
discussed. The comparison between experimental findings and numerical
results for six beams and six slabs has proved reasonably good and places
the proposed methodology as a simple design-tool of practical connotation
oriented to FRP-reinforced concrete elements.
Keywords: FRP-reinforced concrete, Limit analysis, FE-based procedure,
Peak loads, Failure modes.
Preprint submitted to Composite Structures
July 23, 2013
1. Introduction
In recent times there has been a growing interest in composite materials
for civil engineering applications. In the field of concrete structures, extensive research work has been directed toward the introduction of innovative
fiber reinforced polymers (FRP) as reinforcement-bars (re-bars) within concrete elements and as valid alternative to traditional steel re-bars. The most
commonly used FRP re-bars within civil engineering applications are made
of carbon-fibers (CFRP), aramid-fibers (AFRP), and glass-fibers (GFRP).
Especially in aggressive environments such as marine surroundings or in
bridge decks requiring deicing salts due to harsh climates, concrete alkalinity
drastically reduces and, as a result, corrosion and degradation of traditional
steel re-bars significantly accelerate. The corrosion of steel re-bars may lead
to problematic consequences from a structural point of view, including cracking and spalling of concrete, reduction in the cross-section of the bar, loss
of bond between re-bars and concrete and so on. Therefore, in all these
situations there is a need of frequent structural interventions, for instance
to guarantee a sufficient concrete cover to prevent or delay the corrosion of
steel re-bars, or even more to repair a corroded concrete structures when
deterioration is already occurred, which results in very high costs of maintenance. Other specific situations where traditional steel re-bars may be
questionable are those involving hospital rooms which host equipments for
diagnostic imaging and/or magnetic resonance. Indeed steel re-bars may interfere with the electromagnetic field created by such devices and lead to a
misinterpretation of their results.
Replacing steel with FRP overcomes all the above disadvantages. In2
deed FRP re-bars offer an outstanding combination of physical and chemical
properties and, in many situations, are much more competitive and convenient to be used than steel re-bars. They also have a lower weight (which
greatly reduces the costs of installation), and are magnetic-permeable and
non-corrosive. Moreover, FRP re-bars, especially GFRP, are non-conductive
for electricity, and this may be a further benefit if used as reinforcement
in electric railways or undergrounds where stray currents may cause serious damage to buried metallic objects by electrolysis accelerating corrosion
of metal objects in touch with the soil. Other undoubted advantages are
related to their higher tensile strength and fatigue resistance compared to
traditional steel re-bars.
On the other hand, FRP RC elements behave quite differently from those
reinforced with traditional steel. For example, FRP re-bars have higher
strength, but lower modulus of elasticity than steel; this causes a substantial
decrease in the stiffness of FRP reinforced concrete elements after cracking
and, consequently, higher levels of deflections under service conditions. Furthermore, steel-reinforced concrete elements are often designed to be underreinforced so as to promote yielding of re-bars, which produces a greater
dissipative capacity and a more ductile failure behaviour at collapse than
crushing of concrete. Ductility of FRP is far lower than steel, not to say
not-existent, and the failure of FRP re-bars is even more brittle than failure
due to concrete crushing. When failure is mainly caused by FRP rupture (especially in under-reinforced elements), the collapse is sudden and drastically
catastrophic. Actually, concrete failure should be preferable to FRP rupture
since it is more progressive and results in a less catastrophic collapse with a
3
higher degree of deformability.
The advantages and the drawbacks above outlined, among other, have
led to a deep reconsideration of design criteria for FRP-reinforced concrete
structures. This has given rise to a number of research papers aimed at
developing suitable numerical methods for predicting mechanical behaviour
(especially flexural performance) of FRP reinforced concrete (RC) structural
elements of common use such, for example, beams (see e.g. [3, 4, 6, 7, 12,
15, 18]) and slabs (see e.g. [3, 8, 13, 33]), both focused hereafter. As for the
traditional steel-reinforced concrete structures, to guarantee a reliable design,
it is also essential to possess codes and guidelines for practical engineering
design purposes (see e.g. [1, 9, 11, 14]), as well as numerical methods able to
catch the mechanical behaviour of the discussed elements beyond the elastic
limit. Post elastic step-by-step analyses are the common tools available to
this aim and the list of contributions could be very long, but it is out of
the scope of the present study. A valid alternative for design purposes can
indeed be given by the so called direct methods, that are able to predict the
load-carrying capacity of a structure in terms of its peak load value. Limit
analysis, belonging to such methods, plays a crucial role in safety assessment
and structural design since it provides the peak load in a direct manner i.e.
without carrying out a complete post-elastic analysis of stress and strain in
a structure so resulting a relatively simple method of practical connotation
for engineering purposes.
In the present study a FE-based limit analysis approach is adopted to
predict both peak load and failure mechanism of concrete structural elements reinforced with FRP bars. A plasticity model for concrete, with a
4
pressure-sensitive yield surface which arise from the failure criterion proposed by Menétrey and Willam [23], is adopted. A cap in compression is also
adopted to limit concrete strength in high hydrostatic compression regime.
FRP re-bars are assumed as elastic members working only in the fibers’ direction and a perfect bond between FRP re-bars and surrounding concrete
is postulated. The nonassociated flow rule, essential to take into account the
volumetric expansion under compression exhibited by concrete (dilatancy),
implies the lack of a unique peak/collapse load and obliges to follow a nonstandard limit analysis approach, [20, 32], to determine an upper and a lower
bound to the actual peak load.
The key-idea of the proposed methodology is the combined use of two wellknown numerical procedures, namely the Linear Matching Method (LMM)
and the Elastic Compensation Method (ECM). The former, originally conceived by Ponter and Carter [30], is related to the kinematic approach of
limit analysis and hence provides an upper bound to the peak load value, allowing also a prediction of the collapse mode. The latter, due to Mackenzie
and Boyle [21], is a procedure based on the static approach of limit analysis
and gives a lower bound to the peak load value. This methodology has been
already used by the authors in a completely different context to predict collapse load and failure mechanism of pinned-joints in orthotropic composite
laminates [24, 25, 27] tackled via a Tsai–Wu-type yield surface. The LMM
and the ECM have also been recently rephrased by the authors to deal with
reinforced concrete elements in a 3D plasticity framework. In [26, 28, 29] the
promoted approach is used to predict the ultimate state, in terms of peak load
and collapse mechanism, in the presence of traditional steel-reinforcements.
5
The main goal of the present paper is indeed the application of the methodology to FRP-reinforced concrete elements.
After a very brief description of the two procedures, twelve experimental
tests, carried out up to collapse by Al-Sunna et al. [3], on typical FRPreinforced concrete elements, precisely six beams and six slabs, are analyzed.
The numerical predictions, in terms of peak loads and failure modes, are then
critically discussed and compared with the experimental findings.
2. Fundamentals of limit analysis numerical procedures
The whole methodology has been thoroughly treated by the authors in
the two, above quoted, papers [28, 29] dealing with steel-reinforced concrete
elements and where theoretical fundamentals, implementation issues, geometrical interpretation and convergence properties of both the LMM and
the ECM are discussed in detail. Hereafter the Reader is provided with a
few concepts concerning the two procedures, also in the form of two flow
charts (see Figs. 1 and 2), one for the LMM and the other for the ECM,
with the aim to synthesize the basic operative steps. The latter are mainly
carried on at Gauss point (GP) level, i.e. at each GP of each element of the
FE mesh assumed for the analyzed structural element.
On taking into account that the presence of FRP reinforcements, as in
the case of traditional steel-reinforcements, injects ductility on the confined
concrete, the latter is assumed to obey, by hypothesis, a 3D plasticity model
derived from the Menétrey–Willam (M–W) failure criterion [23] equipped
with a cap in compression and formulated in terms of three stress invariants,
ξ, ρ, θ, known as the Haigh–Westergaard coordinates (refer again to [28] for
6
more details). With reference to such yield surface, a statically admissible
distribution of stresses and a kinematically admissible distribution of strain
and displacement rates are sought to apply the two fundamental theorems
of limit analysis [10]. Indeed, the main purpose of both the LMM and the
ECM is to simulate, or “to build”, such limit-type distributions by carrying
out FE-based linear elastic analyses during which the elastic moduli and the
imposed initial stresses (for the LMM) within the elements of the FE model
are systematically adjusted. As conventional linear elastic FE analyses are
required, both procedures are applicable even to problems with large numbers
of degrees of freedom where programming techniques (see e.g. [16, 17, 22])
may become more cumbersome.
In particular, the LMM is an iterative procedure based on the kinematic
approach of limit analysis and involving one sequence of linear FE-based
analyses on the structure assumed, by hypothesis, as made of a linear viscous
fictitious material. The kinematically admissible distribution of strain and
displacement rates, together with the associated stresses at yield, i.e. all the
ingredients defining a collapse mechanism and, consequently, an upper bound
to the peak load multiplier, are built with reference to a fictitious structure
within which the elastic material parameters and the imposed initial stresses
are assumed to have different values at different points.
Figure 1 (LMM Flow-chart) should be pasted here
The ECM, based on the static approach of limit analysis, like the LMM
acts in an iterative way, but involving many sequences of linear FE-based
7
analyses carried out for the real discretized structure under study. At each
sequence, defined by a given load value, the real elastic moduli are reduced
within those elements where the elastic stress is greater than the yield one.
This operation “redistributes” the stresses within the structure and attempts
to construct an admissible stress field suitable for the evaluation of a lower
bound to the peak load multiplier.
Figure 2 (ECM Flow-chart) should be pasted here
3. Application to FRP-reinforced concrete elements
The effectiveness of the expounded numerical procedure when applied to
FRP-reinforced concrete elements is investigated by facing some laboratory
tests’ results on large-scale prototypes carried out up to collapse. To this aim,
the experimental study on beams and slabs by Al-Sunna et al. [3] has been
taken into consideration. The main purpose of this study was to investigate
the flexural behaviour of FRP RC elements at service conditions as well as at
ultimate load levels evaluating the load-carrying capacity and the modes of
failure. Obviously, only these latter data fall within the scope of the present
paper.
The experimental programme of [3] comprises 6 series of beams and 6
series of slabs reinforced with GFRP- and CFRP-bars with a wide range
of reinforcement ratios, and all tested under four-point bending. In effect,
24 tests (i.e. 12 beams and 12 slabs) have been performed since each series has included two identical specimens (with same material properties and
8
reinforcement arrangement) to ensure reliability of experimental results. Depending on the amount of longitudinal reinforcement of each element, three
main failure modes have been observed experimentally: rupture of re-bars
(for under-reinforced specimens), compressive concrete failure followed immediately by rupture of the re-bars (for specimens with almost balanced
reinforcement ratio), and concrete crushing (for over-reinforced specimens).
The nominal diameter, elastic modulus and tensile strength of the GFRPand CFRP-bars used as main flexural reinforcement are reported in Table 1.
Table 1 should be pasted here
All the experimental tests have been numerically reproduced by the expounded procedure. The elastic analyses, representing the iterations within
both the LMM and the ECM, have been carried out using the FE-code ADINA [2], with meshes of 3D-solid 8-nodes elements with 2 × 2 × 2 GPs per
element for modeling concrete and embedded 2-nodes, 1-GP, truss elements
utilized for modeling re-bars and stirrups. Such embedded truss elements
(refer to ADINA for details) are 1D FEs connecting the intersections of the
rebar axes with the faces of the 3D-solid concrete elements. Such intersections are “generated nodes” on the 3D-solid FEs faces constrained to the
three closest corner nodes of the 3D element itself. A perfect bond between
concrete and re-bars is so assumed and an indefinitely elastic behavior of
FRP re-bars has been considered. The FRP re-bars have been modelled as
axial members built into concrete elements (beams and slabs) and working
only in the fibers’ direction. The number of finite elements was different
9
for each specimen type and was chosen after a preliminary mesh sensitivity
study to assure an accurate FE elastic solution. Nodal loads equivalent to
the load exerted by the laboratory test equipment are considered. Boundary
conditions consistent with those of the experimental tests will be specified in
the following for each specimen type.
To settle the M–W-type constitutive model, the experimental data of the
quoted reference [3] have been considered. For each beam and slab series,
′
the cube compressive strength fcu , the splitting tensile strength ft and the
elastic modulus Ec are there given. The cylinder compressive strength was
′
derived as fc = 0.83fc u (see [11]). The value of the eccentricity parameter
e of the M–W-type yield surface has been evaluated by the expression e =
′
′
′
′
′
′
[2 + ft /fc ]/[4 − ft /fc ] as suggested by Balan et al. [5], the ft /fc ratio being
assumed as a measure of the material brittleness. Other three values have
been finally fixed to locate the M–W-type yield surface in the principal stress
√ ′
′2
space, namely: ξv assumed as ξv = 3fc /m with m given by m := 3 (f c −
′2
′
′
′
′
f t ) e / fc ft (e + 1), see Pisano et al. [28]; ξa = 0.7923 fc and ξb = 1.8964 fc
as suggested by Li and Crouch [19]. To complete the definition of concrete
as isotropic material, together with the proper Young modulus Ec (given by
the experimental data) an initial Poisson’s ratio of ν = 0.2 has been assumed
for all specimens.
Finally, a Fortran main program has been used to drive both the iterative
procedures here proposed updating, at each GP of each element, the fictitious
elastic parameters and initial stresses when performing the LMM or realizing
the redistribution procedure within the ECM.
10
3.1. Beams
The beams were tested under four-point bending. They have a total
length of 2550 mm and a rectangular cross-section 150 mm wide and 250
mm deep, equal to all tested specimens. The reinforcement arrangement is
specified in the quoted experimental study and it is omitted. It is worth
noting that to avoid shear failure the shear span was reinforced with steel
stirrups with diameter of 8 mm @ 75 mm. The Young modulus of such steel
stirrups was set as Es = 205 GPa. GFRP and CFRP re-bars with nominal
diameter of 6 mm were also used as top reinforcement within the shear span
to hold the stirrups in place.
The assumed mechanical model reporting geometry, loading and boundary conditions of the analyzed beams is sketched in Fig. 3a along with the
adopted FE-model Fig. 3b. The boundary conditions have been imposed by
taking into account the experimental test fixture: all displacements are forbidden to the FE-nodes corresponding to the bearing support, whereas zero
displacements in the y and z directions are assigned to the FE-nodes lying on
the line of the roller support. Due to the symmetry of the problem only half
specimen has been analyzed and this requires to impose zero displacements
in z direction to the FE-nodes lying on the shaded symmetry plane (see again
Fig. 3a). All the beams are subjected to two equal line loads symmetrically
placed about mid-span and denoted as P p̄, where P is the load multiplier
and p̄ is the reference line load assumed equal to 666 N/mm so as to be
equivalent to a total load of 100 kN. Nodal loads equivalent to the line load
exerted by the laboratory test equipment are considered in the FE-model as
shown in Fig. 3b. The adopted FE-mesh consists of 816 3D-solid elements
11
and a number of 1D truss elements ranging from 268 to 332 and depending
on the amount of reinforcement of the beam specimen.
Figure 3 should be pasted here
The six analyzed beam series are designated as BG# and BC# where B
stands for beam and G and C identify the type of reinforcement used, GFRP
or CFRP respectively. The mechanical properties and reinforcement details
of each beam are given in Table 2.
Table 2 should be pasted here
3.2. Slabs
The slabs were tested under four-point bending. They have a total length
(along the main direction x) of 2350 mm, a width of 500 mm and a thickness
of 120 mm, constant for all tested specimens.
The assumed mechanical model showing geometry, loading and boundary
conditions of the analyzed slabs is sketched in Fig. 4a along with the adopted
FE-model Fig. 4b. As before, the boundary conditions have been imposed
by taking into account the experimental test fixture: all displacements are
forbidden to the FE-nodes corresponding to the line of the bearing support,
whereas zero displacements in the y and z directions are assigned to the
FE-nodes lying on the line of the roller support. All the slabs are subjected
to two equal loads symmetrically placed about mid-span at a distance of
12
600 mm between each other and 750 mm from the two lateral supports. The
loads are denoted as P p̄, where P is the load multiplier and p̄ is the reference
line load assumed equal to 100 N/mm so as to be equivalent to a total load
of 100 kN. Nodal loads equivalent to the load exerted by the laboratory test
equipment are considered in the FE-model as shown in Fig. 4b. The adopted
FE-mesh consists of a number of 3D-solid elements ranging from 624 to 720
and a number of 1D truss elements ranging from 144 to 196 and depending
on the amount of reinforcement of the slab specimen.
Figure 4 should be pasted here
As made for beams, the designation assumed for the six analyzed slabs is
SG# and SC#, where S stands for slab and G and C identify the type of
reinforcement used, GFRP or CFRP respectively. The mechanical properties
along with the amount of reinforcement in the two main directions, called
barx and bary , are given in Table 3 for each slab. The slightly different
concrete cover values of each specimen are also reported in Table 3.
Table 3 should be pasted here
3.3. Numerical results and comments
The values of the numerically predicted upper bound (PU B ) and lower
bound (PLB ) to the peak load multiplier are reported in Table 4 against the
experimental one (PEXP ). By analyzing the numerical results, the proposed
limit analysis procedure seems to be quite accurate in defining two close
13
limits to the real—experimentally obtained—peak load value for almost all
the examined FRP RC elements.
Table 4 should be pasted here
In detail, the upper bound values predicted by the LMM are always above
the experimental ones, as it should be when searching for an upper bound.
The difference between the numerical and experimental results are quite small
(with small relative error, in most cases less than 10%) for all elements except
for specimens designated as BG1, BC1, SG1 and SC1, for which the accuracy
of the PU B value is poorer. These cases belong to the class of under-reinforced
FRP RC elements with failure occurring by rupture of FRP re-bars instead
of crushing of concrete. Having postulated an indefinitely elastic behaviour
for FRP re-bars, such failure is practically impossible to be predicted by the
proposed approach in itself. Therefore this wrong/poor prediction, which is
actually a limit of the proposed numerical procedure, is consistent the assumed hypotheses. However, it should be said that such under-reinforced
FRP RC sections are not usually of great interest for design purposes and
are often avoided in engineering practice since the resulting failure is brittle
and drastically catastrophic. With regard to the lower bound values obtained
by the ECM, the predictions are rather accurate for all but three specimens,
namely BG1, SG1 and SC1, whose lower bound multipliers PLB are above,
instead of below as one should expect, the experimental values PEXP . The
same considerations as above can be made about these three wrong predictions, being all under-reinforced elements of no common use in the practical
14
engineering context. However, by excluding these three specimens the average relative error of PLB predictions compared to PEXP values is of 11%,
which is acceptable from an engineering point of view. All in all, for nine out
of twelve analyzed FRP RC elements the experimental peak load multipliers
PEXP fall within the interval numerically located and limited by the computed PU B and PLB values. This predicted interval is quite narrow showing
a good performance of the proposed methodology.
Figure 5 shows, for two of the analyzed specimens, namely beam BG2
and slab SC2, the plots of the upper and the lower bounds to the peak load
multiplier versus the iteration number. Analogous results are obtained for
all the other cases but are omitted for sake of brevity. As shown, only a few
iterations/linear FE-elastic analyses (generally less than fifteen) are sufficient
to obtain a converged solution in terms of both upper and lower bounds. The
monotonic and rapid convergence is assured by a sufficient condition given
by Ponter et al. [31] fulfilled by the assumed M–W-type yield surface.
Figure 5 should be pasted here
A better comprehension of the mechanical behaviour of the FRP RC elements at collapse can be gained by the prediction of the failure modes. As
said, the LMM “constructs” the collapse mechanism the structure exhibits
when the loads attain their peak value or, more exactly, they reach the evaluated upper bound value to such peak. Such mechanism is constructed on a
fictitious structure, i.e. it is located within the analyzed structure made, by
hypothesis, of a material endowed with a fictitious spatially varying distribution of elastic parameters and initial stresses. The prediction of failure modes
15
can therefore be obtained by identifying the plastic zone (collapse mechanism) at the last converged solution of the LMM. To this aim the plots of
the displacement rates (i.e. the deformed configuration), as well as those of
the principal (compressive) strain rates ε̇c3 have been considered on the FRP
RC elements loaded by PU B p̄ and at the final (converged) distribution of
fictitious parameters and initial stresses. Figures 6a and 7a show, for two of
the analyzed FRP RC elements, namely beam BG2 and slab SC2, the principal strain rates ε̇c3 distribution in the deformed (final) configuration attained
by LMM at convergence. The plasticized zones so located appear sufficiently
confined and reasonably close to the damaged zones experimentally detected.
Moreover, the deformed shapes show how around such plasticized zones the
remainder of the structure rotates rigidly as exhibited by the beams’ and
slabs’ collapse/failure mechanisms experimentally observed.
Figures 6 and 7 should be pasted here
To emphasize the significance of such predicted collapse mechanisms, both
the above band plots of ε̇c3 and the deformed shapes obtained by an elastic
analysis on the real elements (i.e. with the real, no spatially varying, material
parameters) under the same load PU B p̄ are shown in Figs. 6b and 7b. It
is clear how the results given by an elastic analysis with real parameters
are quite different from those given by the LMM and are meaningless for
detecting the collapse mode. However, the level of detail in describing the
state of incipient collapse is far to be precise or exhaustive, but rather it can
be useful to localize critical zones or weaker members for example within
reinforced concrete structures of larger dimensions.
16
4. Conclusions
A limit analysis numerical approach grounded on the combined use of
two FE-based procedures, the LMM and the ECM, has been proposed to
evaluate peak load and failure mode of FRP RC elements. The combination
of the two procedures allows to numerically identify an upper and a lower
bound to the peak load multiplier of the examined structural elements. It
also provides some useful information on the expected failure mechanism. A
comparison between the experimental findings, available in the literature, and
the numerical results on real FRP RC beams and slabs has shown capabilities,
reliability of the procedures as well as their limitations.
As in every plasticity-based approach for concrete structures, the most
limiting factor is that the ductility of the critical/weakest sections of the
structure/structural element should be sufficient for the envisaged plastic
collapse mechanism to be formed. The treatment of post-elastic phenomena
that might be exhibited by concrete structures such as: localization, fracturing/damaging mechanisms, creep, interface problems, etc. is not allowed.
Moreover, it is not possible to describe any crack pattern or brittle failure
due to FRP rupture, as the procedure focuses on the prediction of plastic behaviour occurring on concrete and postulates an indefinitely elastic behaviour
of FRP-bars.
On the other hand, it should be noted that such brittle behaviour is
typical of under-reinforced FRP RC elements which are not usually of great
interest for design purposes and are often avoided in engineering practice
since the resulting failure is drastically catastrophic. Although plain concrete
is essentially a brittle material, in practical engineering cases of RC structures
17
re-bars have a stabilizing influence on fracture/damage phenomena and thus
the RC may show some ductile behaviour that can be adequately described by
a plasticity approach as the one here postulated. When the failure mechanism
is mainly dominated by crushing of concrete, the confining effect of re-bars
and the ductile behavior generated by their presence makes then applicable
and effective a limit analysis approach as the one here promoted.
The obtained results witness how the expounded numerical procedures
can be considered a useful computational tool, at least in a first stage of
analysis and/or design process, to acquire preliminary information on the
peak load, failure mode and critical zones of FRP RC elements. Finally, as
the two numerical procedures are based on sequences of linear elastic analyses, the expounded methodology is easily practicable via any commercial
FE-code and potentially applicable even to structures having large dimensions or an intricate geometry and/or boundary conditions and this with
moderate computational efforts.
18
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23
FIGURE CAPTIONS
Figure 1. LMM Flow-chart
Figure 2. ECM Flow-chart
Figure 3. Four-point bending test on FRP-reinforced concrete beams: a)
mechanical model of the half analyzed symmetric specimen showing geometry
(all dimensions in mm), loading and boundary conditions; b) typical FE mesh
adopted with 3D solid FEs modelling concrete and embedded 1D truss FEs
modelling re-bars and stirrups.
Figure 4. Four-point bending test on FRP-reinforced concrete slabs: a)
mechanical model showing geometry (all dimensions in mm), loading and
boundary conditions; b) typical FE mesh adopted with 3D solid FEs modelling concrete and embedded 1D truss FEs modelling re-bars.
Figure 5. Four-point bending test on FRP-reinforced concrete elements:
a) beam #BG2; b) slab #SC2. Values of the upper (PU B ) and lower (PLB )
bounds to the peak load multiplier versus iteration number: LMM prediction,
solid lines with square markers; ECM prediction, solid lines with triangular
markers; collapse experimental threshold (after Al-Sunna et al. [3]), dashed
lines.
Figure 6. Beam #BG2. Band plots of principal (compressive) strain rates
ε̇c3 in the deformed configurations at the ultimate value of the acting loads:
a) result at last converged solution of the LMM giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the beam.
Figure 7. Slab #SC2. Band plots of principal (compressive) strain rates
ε̇c3 in the deformed configurations at the ultimate value of the acting loads:
24
a) result at last converged solution of the LMM giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the slab.
25
TABLES CAPTIONS
TABLE 1. Properties of FRP re-bars used as main flexural reinforcement.
TABLE 2. Four-point bending test on FRP-reinforced concrete beams: specimen number; concrete compressive and tensile strengths; concrete Young
modulus; reinforcement details.
TABLE 3. Four-point bending test on FRP-reinforced concrete slabs: specimen number; compressive and tensile strengths; concrete Young modulus;
reinforcement details; clear concrete cover to the main rebars.
TABLE 4. Peak load multipliers for the analyzed FRP-reinforced elements:
values experimentally detected by Al-Sunna et al. [3] (PEXP ) against the
values of the upper (PU B ) and lower (PLB ) bounds to the peak load multiplier
numerically predicted by LMM and ECM, respectively.
26
INITIALIZATION
Generate FE-mesh
Assign to FEs: fictitious bulk and shear moduli K (0) ; G(0)
fictitious initial stresses (0) , x(0) , y(0)
(0)
Assign loads: PUB pi
(0)
( pi =reference loads; PUB load multiplier)
LMM
START
( k 1)
Set: k 1 ; PUB
PUB(0) 1 ; to start iterations
• START ITERATION LOOP
• START ELEMENTS LOOP
• START GPs LOOP
Perform a FE-elastic analysis on the fictitious structure under loads
( k 1)
PUB
( k 1)
( k 1)
( k 1)
, ( k 1) , x
, y
pi with: K ( k 1) , G
Output linear viscous fictitious strain and displacement rates; associated linear
( k 1)
( k 1)
fictitious stresses: v ( k 1) , d x( k 1) , d y( k 1) , u i( k 1) , ( k 1) , x
, y
Compute associated complementary energy value:
W ( k 1)
1
2
( k 1)
v ( k 1) x ( k 1) d ( k 1) y( k 1) d ( k 1)
x
y
Y ( k 1)
Locate on the M–W-type surface the stress point M ( Y ( k 1) , x
( k 1)
v
of given normal
( k 1)
dx
,
( k 1)
dy
,
Y ( k 1)
, y
)
(matching point)
Impose “matching conditions”, i.e. find the “adjusted” fictitious quantities:
(k )
(k )
(k )
K ( k ) , G , ( k ) , x , y (to be utilized, if necessary, at next iteration) such that the
complementary energy equipotential surface W , , K ( k ) , G( k ) , ( k ) , x( k ) , y( k ) W ( k 1)
matches at M the M–W-type yield surface
The linear fictitious solution can now be interpreted as solution at yield; the fictitious
kinematic quantities in rate form define a collapse mechanism, the linear fictitious
stresses (brought on the M–W-type surface) are the associated stresses at yield.
Set: vc ( k 1) v ( k 1) , dc x( k 1) d x( k 1) , dcy ( k 1) d y( k 1) , uic ( k 1) ui ( k 1)
Compute the upper bound multiplier:
(k )
PUB
V
Y ( k 1)
vc ( k 1) Yx ( k 1) dc x( k 1) Yy ( k 1) dc y ( k 1) dV
Vt
pi uic ( k 1) d(V )
• END GPs LOOP
• END ELEMENTS LOOP
Perform a new
FE analysis with the
“adjusted” fictitious
quantities ( )( k ) .
Set k k 1
CHECK
FOR CONVERGENCE
NO
• END ITERATION LOOP
(k )
( k 1)
|PUB PUB
| TOL
YES
EXIT
END PROCEDURE
INITIALIZATION
Generate FE-mesh
(s)
Assign a design load: PD pi
( pi =reference loads; PD( s ) load multiplier)
Set s 1 to start the sequences of FE analyses
ECM
START
• START SEQUENCE OF ELASTIC ANALYSES
Set k 1 to start the first FE analysis of the sequence
• START ITERATION LOOP
• START ELEMENTS LOOP
• START GPs LOOP
Assign the (real) material parameters: E ( k 1) ; ( k 1)
(s)
Perform a FE-elastic analysis under loads PD pi with E ( k 1) ; ( k 1)
Output: elastic solution, averaged within the eth element, namely e, e, e
e ( k 1)
and locate in the principal stress space the stress point # e
• END GPs LOOP
e ( k 1)
Locate the corresponding stress point at yield, i.e. the intersection O # e
with the M–WY ( k 1)
e
Y
Y
Y
Y
( , , ) with
type yield surface, say: # e
Y ( k 1) e ( k 1) 2
Update the Young moduli within the element # e as: E#( ke ) E#( ke 1) [ | O # e |
/ | O # e |
]
Locate the “maximum stress” in the whole mesh i.e. the stress point farthest away
( k 1)
Y ( k 1)
, and evaluate the pertinent R
from the M–W-type surface, say R
| OR |
• END ELEMENTS LOOP
• END ITERATION LOOP
( k 1)
| OR |
( k 2)
YES
Redistribution failed
(s)
for the current PD
• END ELEMENTS LOOP
• END ITERATION LOOP
NO
( s 1)
Set PLB PLB
Try to redistribute
within the current
sequence s
performing a new
FE analysis with
the updated E#( ke ) .
Set k k 1
NO
( k 1) Y ( k 1)
| OR |
| OR |
END PROCEDURE
EXIT
YES
Compute the lower bound multiplier:
(s)
Y ( k 1)
PD
( k 1)
| OR |
PLB
( k 1)
| OR |
Increase the intensity of
the acting loads setting
( s 1)
( k 1)
and
PD
PLB
perform a new sequence
of elastic analyses.
Set s s 1
• END ELEMENTS LOOP
• END ITERATION LOOP
• END SEQUENCE OF ELASTIC ANALYSES
Pp
Pp
250
75
767
125
766
125
y
x
767
2550
150
25
2Φ 6
(GFRP or CFRP)
steel stirrups
Φ8@75mm
(only along
shear span)
z
main re-bars
(GFRP or CFRP)
250
25
a)
Φ 6 bar
stirrups
y
x
z
b)
stirrups nodes
at the intersection
with the face of
3D-solid elements
half Φ12.7 bar
at plane of
symmetry
of specimen
rebar nodes
at the intersection BC3
with the face of
3D-solid elements
Figure 3 Four-point bending test on FRP-reinforced concrete beams: a) mechanical model of the
half analyzed symmetric specimen showing geometry (all dimensions in mm), loading and
boundary conditions; b) typical FE mesh adopted with 3D solid FEs modelling concrete and
embedded 1D truss FEs modelling re-bars and stirrups.
Pp
Pp
120
125
750
500
600
750
z
x
2350
125
bary
z
concrete
cover
120
y
barx
y
500
a)
z
x
y
b)
Figure 4 Four-point bending test on FRP-reinforced concrete slabs: a) mechanical model showing
geometry (all dimensions in mm), loading and boundary conditions; b) typical FE mesh adopted
with 3D solid FEs modelling concrete and embedded 1D truss FEs modelling re-bars.
2.0
1.5
PUB
PUB
#SC2
PEXP
1.5
PLB
1.0
0.87
0.80
0.75
0.5
0.0
PEXP
load multiplier P
load multiplier P
#BG2
PLB
1.0
0.64
0.56
0.5
0.50
0.0
1
2
4
6
iteration number
8
10
1
2
4
6
8
10
12
iteration number
Figure 5 Four-point bending test on FRP-reinforced concrete elements: a) beam #BG2; b) slab
#SC2. Values of the upper (PUB) and lower (PLB) bounds to the peak load multiplier versus iteration
number: LMM prediction, solid lines with square markers; ECM prediction, solid lines with
triangular markers; collapse experimental threshold (after Al-Sunna et al. [3]), dashed lines.
a)
b)
y
z
x
Beam # BG 2
Figure 6 Beam #BG2. Band plots of principal (compressive) strain rates 3c in the deformed
configurations at the ultimate value of the acting loads: a) result at last converged solution of the
LMM giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the
beam.
a)
b)
z
x
Slab # SC 2
y
Figure 7 Slab #SC2. Band plots of principal (compressive) strain rates 3c in the deformed
configurations at the ultimate value of the acting loads: a) result at last converged solution of the
LMM giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the
slab.
Table 1 Properties of FRP re-bars used as main flexural reinforcement.
Re-bar type
Nominal diameter (mm)
Elastic modulus (GPa)
Tensile strength (MPa)
GFRP
6.35
38.9
600
9.53
42.8
665
12.7
41.6
620
19.05
42.0
670
6.35
133.0
1450
9.53
132.0
1320
12.7
119.0
1475
CFRP
Table 2 Four-point bending test on FRP-reinforced concrete beams: specimen number; concrete
compressive and tensile strengths; concrete Young modulus; reinforcement details.
Series designation
Specimen
fc' (MPa)
ft ' (MPa)
Ec (GPa)
Rebar-details
GFRP beams
BG1
39.6
4.1
29.3
BG2
39.6
3.8
29
2 Φ 12.7
BG3
38.6
3.6
28.2
BC1
46.0
3.9
31.7
BC2
43.7
3.6
30.1
BC3
43.0
3.6
30.7
CFRP beams
a
Two layers, with 25 mm clear spacing between them.
2 Φ 9.53
4 Φ 19.05a
3 Φ 6.35
3 Φ 9.53
3 Φ 12.7
Table 3 Four-point bending test on FRP-reinforced concrete slabs: specimen number; compressive
and tensile strengths; concrete Young modulus; reinforcement details; clear concrete cover to the
main rebars.
Series
designation
Specimen
fc'
(MPa)
ft '
(MPa)
Ec
(GPa)
GFRP slabs
SG1
42.33
3.9
29.3
SG2
38.35
3.4
27.8
SG3
38.10
3.8
27.7
SC1
41.58
3.9
30.3
SC2
42.33
3.4
29.9
SC3
41.33
3.8
29.5
CFRP slabs
barx
(mm)
5 Φ 6.35
5 Φ 9.53
5 Φ 19.05
4 Φ 6.35
4 Φ 9.53
4 Φ 12.7
bary
(mm)
cover
(mm)
14 Φ 9.53
27.5
12 Φ 6.35
40.0
12 Φ 6.35
36.5
14 Φ 9.53
14 Φ 9.53
31.0
12 Φ 6.35
32.0
39.0
Table 4 Peak load multipliers for the analyzed FRP-reinforced elements: values experimentally
detected by Al-Sunna et al. [3] (PEXP) against the values of the upper (PUB) and lower (PLB) bounds
to the peak load multiplier numerically predicted by LMM and ECM, respectively.
Peak load multipliers
Specimen number
PEXP
PUB
PLB
BG1
BG2
BG3
BC1
BC2
BC3
SG1
SG2
SG3
SC1
SC2
SC3
0.45
0.80
1.15
0.76
1.05
1.25
0.20
0.42
0.63
0.38
0.56
0.65
0.83
0.87
1.18
0.95
1.14
1.30
0.46
0.47
0.68
0.60
0.64
0.73
0.64
0.75
1.08
0.69
0.90
1.02
0.40
0.38
0.54
0.50
0.50
0.58
Paolo Fuschi