SIZE AND STRAIN RATE EFFECTS ON THE
FRACTURE OF REINFORCED CONCRETE
Gonzalo Ruiz1, Xiaoxin Zhang1,2 and Rena C. Yu1
ETSI Caminos, C. y P., Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain.
2
College of Mechanical and Electrical Engineering, Harbin Engineering University, 150001Harbin, China.
1
ABSTRACT
This paper presents very recent results of an experimental program aimed at disclosing size and strain rate effects on the
fracture behavior of reinforced concrete beams. Eighteen reinforced beams of three sizes (75mm, 150mm and 300mm in
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depth), were tested under four strain rates (1.05×10 /s, 1.25×10 /s, 1.25×10 /s and 3.75×10 /s). The results show that the
peak loads increase with increasing strain rate; the rate dependence of the peak load is stronger for larger specimens than for
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smaller ones. Moreover, size effect is only shown under the nominal strain rate 1.05×10 /s, while it is inconspicuous to the
higher strain rates. These results present themselves as an apparent physical inconsistency, the reason behind is sought
numerically using an explicit cohesive model.
Introduction
Our purpose is to study the dynamic behavior of reinforced concrete in a wide range of strain rates. We present very recent
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experimental results at low strain rates, from 10 /s to10 /s. As opposed to the technological standpoint, lightly reinforced
beams are chosen, so that only one crack is generated. This will help us to isolate the physical phenomena and facilitate
posterior analysis. In addition, we will provide complete material characterization for validating numerical models.
In the past few decades, impact behavior of beams and slabs have been studied by Seabold [1], Takeda et al. [2], Hughes and
Beeby [3], Banthia et al. [4], Miyamoto et al. [5], Ragan [6] and Kishi [7], etc. The results showed that the absorbed energy
increases with the loading rate increasing as a tendency, while under certain circumstances, it showed the opposite conclusion
due to a different mode of failure [1, 3, 6].
Compared with impact/impulsive tests, much less information is available on the higher border of the quasi-static region.
Kulkarni and Shah [8] tested reinforced concrete beams at varying strain rates whose upper limit was 10/s, the results showed
that the peak load and the area under the load-deflection diagram increase with the increasing of strain rates. Some earlier
research results were reported by Bertero et al. [9], Wakabayashi et al. [10], Limberger et al. [11], Mutsuyoshi and Machida [12]
and Takeda and Tachikawa [13].
In the research mentioned above, the strain rate effect on reinforced concrete has been studied in detail, while the size effect
has not been revealed. As an exception, Bažant and Gettu [14] presented some results on CMOD (Crack Mouth Opening
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Displacement) rate lower than 10 m/s and size effects for notched plain concrete beams. They showed that plain concrete
has size and rate effects, namely, the peak loads increase with increasing rate of loading. A decrease of the loading rate
causes a shift towards a more brittle behavior.
Under the static loading conditions, various experiments on lightly reinforced concrete beams [15-21] have been conducted.
These experimental programs showed that brittle collapse of lightly reinforced beams is size dependent. Ruiz, Elices and
Planas [20] made a set of tests which disclosed the influence of several parameters —size, steel ratio, steel yield strength and
bond-slip properties—on the fracture behavior of lightly reinforced concrete. They made a complete material characterization
by direct testing that made objective numerical modeling possible [22, 23]. So, as an extension of this particular research and
aiming at obtaining information of size and strain rate effects on reinforced concrete beams, we designed our experimental
program. Three point bend tests over three sizes of specimens (small, medium and large specimens; 75 mm, 150 mm and 300
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mm in depth, respectively) were conducted, by using a servo-hydraulic testing machine under various strain rates, from 10 /s
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to 10 /s. The results show that the peak loads increase with increasing strain rate; the rate dependence of the peak load is
stronger for larger specimens than for smaller ones. Moreover, we can anticipate here that size effect is only shown under the
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slowest nominal strain rate 1.05×10 /s, while it is inconspicuous to the higher strain rates. As this is apparently inconsistent,
we seek for a feasible explanation with the help of an explicit cohesive model.
The paper is structured as follows: a brief overview of the experimental program is given in section 2. The material properties
and control tests are described in section 3. Section 4 summarizes the reinforced beam tests under static and dynamic loading.
The experimental results are presented and discussed in section 5. Finally, in section 6, some conclusions are extracted.
Overview of the experimental program
The experimental program was designed to study the combined size and strain rate effects on the fracture behavior of
reinforced concrete beams.
D
0.85D
We chose the beam sketched in Figure 1 as a convenient geometry for this research. We used the constant reinforcement
ratio (ρ = 0.15%, where ρ = As/Ac. As and Ac being the area of reinforcement and the area of the beam’s cross section
respectively) and the same thickness (B=50 mm) for all the specimens. The rebar was kept in the same relative position in the
beams.
4D
B
4.5D
Figure 1. Schematic diagram of the beams
Regarding the scale of the specimens, Hillerborg’s brittleness number βH was used as the comparison parameter. As a first
approximation, two geometrically similar structures will display a similar fracture behavior if their brittleness numbers are
comparable [24, 25]. βH is defined as:
β H = D / lch , where lch = EcGF / f t 2
(1)
D is the depth of the beam and lch is Hillerborg's characteristic length; Ec, GF and ft are the elastic modulus, the fracture energy
and the tensile strength, respectively. According to this, a relatively brittle micro-concretes was selected with characteristic size
of approximately 60 mm (the details of the micro-concrete are given in the next section), the beams were to be 75, 150 and
300 mm in depth. Since the characteristic length of ordinary concrete is 300 mm on average, so laboratory beams of 150 mm
depth are expected to simulate the behavior of ordinary concrete beams 750 mm in depth, which is considered a reasonable
size for the study.
With respect to the loading rate, the relationship between the loading rate V (V= δ& , where δ is the displacement of the loading
point) and the nominal strain rate ε&N of the lower surface of the beam is as follows:
V =
S2
ε&N ; S=4D
6D
(2)
where S and D are the span and depth of the beam, respectively. From now on, we use the word “strain rate” instead of
“nominal strain rate” for simplicity.
In order to keep the same strain rate for beams of different sizes, we scaled the loading rate in proportion to the beam size.
Table 1 arrays the values chosen for ε&N and the resulting loading rate for the three sizes. The slowest
static region, whereas the fastest
large beam.
ε&N
ε&N falls in the quasi-
is determined by the maximum velocity of the actuator of the machine when testing a
Standard characterization and control tests were performed to determine the compressive strength, tensile strength, elastic
modulus and fracture energy of the concrete. Likewise the mechanical parameters of the rebars and the steel-concrete
interface were also measured in our Laboratory.
Table 1 Relationship between V and
small
D=75 mm
V1
V2
V3
V4
0.0021
0.25
2.5
7.5
V (mm/s)
medium
D=150 mm
0.0042
0.5
5.0
15.0
large
D=300
mm
0.0084
1.0
10.0
30.0
ε&N
ε&N
(/s)
1.05×10-5
1.25×10-3
1.25×10-2
3.75×10-2
Characterization and control tests
(1) Micro-concrete
A micro-concrete mixes was used throughout the experiments, made with a lime aggregate of 5 mm maximum size and
Pozzolanic Portland cement (ASTM type II/A). All the cement used was taken from the same cement container and dry-stored
until use. The mixing proportions by weight was 3.2:0.45:1 (aggregate: water: cement).
All the specimens were made from 2 batches of 80 liters. There was a strict control of the specimen-making process, to
minimize scattering in test results. We made characterization specimens of all batches.
Compressive tests were conducted according to ASTM C39 and C469 on 75 mm×150 mm (diameter × height) cylinders.
Brazilian tests were also carried out by using the same dimensional cylinders following the procedures recommended by
ASTM C469.
The fracture energy GF was measured by using notched plain concrete beams following the procedures devised by Elices,
Guinea and Planas [26-28], all the beams were 50mm in thickness, 75mm in depth and 337.5 mm in length. The span was 300
mm. During the tests the beams rested on the anti-torsion supports like the ones used to do the reinforced concrete beam
tests (see Section 4). The tests were performed in position control. We used three linear ramps at different displacement rates:
10 µm/min during the first 15 minutes, 50 µm/min during the following 15 minutes and 250 µm/min until the end of the test.
Table 2 shows the characteristic mechanical parameters of the micro-concrete determined in the various characterization and
control tests.
Table 2 Micro-concrete characteristics
ft
Ec
GF
fc
(MPa) (MPa) (GPa) (N/m)
Mean
40.37 4.83
29.57 51.70
Std. Dev
1.58
0.33
5.70
6.98
lch
(mm)
65.53
-
(2) Steel
For the beam dimensions selected and desired steel ratio, the diameter of the steel bars had to be smaller than that of
standard ones, so commercial ribbed wires with a diameter of 2.675 mm were used as reinforcement. The elastic modulus is
133.2GPa, the standard yield strength at the strain of 0.2% is 434.1 MPa, the ultimate strength is 465.1MPa, and the ultimate
strain is 0.9%.
(3) Steel-concrete interface
Pull out tests were conducted by pulling the wire at a constant velocity while keeping the concrete surface compressed against
the steel plate. Figure 2a sketches the experimental setup, a prismatic specimen of 50 mm×50 mm×75 mm with a wire
embedded along its longitudinal axis. The bonded length was 25 mm to allow a constant shear stress at the interface of the
reinforcement wire [29, 30]. A stiff frame hitched to the machine actuator holds the concrete specimen while the wire was
protruded through a hole across the upper steel plate. The relative slip between the wire and the concrete surface was
measured at the bottom end to avoid the steel elastic deformation. The tests were carried out at a constant displacement rate
of 2 µm /s.
Figure 2b shows the typical bond-slip curve of the material. The bond strength τc deduced from these tests (8 specimens) is
7.00±0.79 MPa.
9
τ (MPa)
6
3
0
-3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
s (mm)
(a)
(b)
Figure 2 (a) Experimental setup for pull out tests (b) the typical τ-s curves
Reinforced beam tests under dynamic and static loading conditions
As we have already described in Section 2, the specimens for static and dynamic tests were beams reinforced with
longitudinal bars, the reinforcement ratio was kept as 0.15% for all the beams. Figure 1 sketches the shape and dimension of
the reinforced beams. Table 1 shows the various loading rates for the small, medium and large beams.
All reinforced beams were tested in three-point bending as sketched in Figure 3. The beam rests on two rigid-steel cylinders
laid on two supports permitting rotation out of the plane of the beam and rolling along the beam longitudinal axis with negligible
friction. These supports roll on the upper surface of a very stiff beam fastened to the machine actuator. The load-point
displacement is measured in relation to points over the supports on the upper surface of the beam.
For small specimens, it is easier to measure the displacement due to the beam’s dimensions, so, two LVDT (linear variable
differential transducer) fixed on the steel beam are directly used to measure the displacement between the loading rod and the
steel beam. For medium or large specimen, a reference frame is laid over the beam, resting on conical supports located over
the beam. A LVDT installed inside the central loading rod measures the loading-point displacement relative to the frame.
For loading, a hydraulic servo-controlled test system was employed. The tests were performed in position-control.
(a)
(b)
Figure 3. Experimental set-up for three-point bending tests on reinforced concrete beams:
(a) medium and large beams (b) small beams
Results and discussions
(1) Experimental results
Figure 4 shows the comparison of the typical displacement-time (δ-t) curves recorded by the internal device of the machine
which indicates the position variable and the external LVDTs. The δ-t curve recorded by the internal device is a straight line,
the slope is the loading rate as we set; while it shows three main stretches (before point A, AB and after point B) recorded by
the LVDT. Point A is in correspondence with the peak load; line AB represents that the load applied on the specimen drops
quickly in load-displacement (P-δ) curves, which is shown in Figure 5; following the point B, the slope of the straight line is the
same as the set loading rate. These differences in the δ-t curves stem from the flexibility of the experimental set-up, specially
the load cell, which is not negligible compared with the flexibility of the specimen. Therefore, the data recorded by the external
LVDTs should be interpreted as the loading-point displacement.
1.5
Recorded by device of machine
1.0
δ (mm)
B
Recorded by LVDT
A
0.5
0.0
0
100
200
t (s)
Figure 4. Comparison of the typical δ-t curves
All the experimental load-displacement (P-δ) curves for the reinforced beams are shown in Figure 5. The plots are arranged so
that the strain rate is kept constant for each figure.
A typical P-δ curve starts with a linear ramp-up; there is a loss of linearity before reaching the peak load, which indicates the
initiation of the fracture process. Right after the peak the beam loses resistance, the load transfer between the concrete and
reinforcement enables the beam to recover. Moreover, with the loading rate increasing, the load shows some undulation due
to the vibration of the system composed by the specimen and the set-up (mainly the load cell). We recorded such vibration just
after the peak load as the main crack propagates and regardless the size of the specimen. Only for the large beams loaded at
fast velocities (10 and 30 mm/s) the vibration affected the reference frame shown in Figure 3a, which invalidates the
recordings after the peak. This is why the post-peak curves in Figure 5c-d are dashed.
In addition, the initial slope of the P-δ curves of the beams show variations, which are due to the sensitivity of the elastic
flexibility of the beam to the boundary conditions in the application of the concentrated load [31].
(a)
(b)
(c)
(d)
Figure 5. Load-displacement curves of the reinforced beams under various strain rates
With respect to the crack pattern, all of the specimens broke with only one crack located around the center of the specimen
due to flexural failure. Furthermore, the curvature of the crack path has no relation with the loading rate.
(2) Influence of strain rate and beam size on the peak load
Figure 6 shows the effect of loading rate on the peak load (Pmax) for each size. As it is clear from the figure, the measured
values of the peak load suffer scattering, which could be explained by the fact that the specimens were cast from different
batches of reinforced micro-concrete. Nevertheless, some general trends can still be observed: (1) the peak loads increase
with increasing strain rate; (2) the rate dependence of the peak load is stronger for larger specimens than for smaller ones.
Figure 7 represents the mean cracking load of all the kinds of beams tested versus their corresponding beam depth under
various strain rates; both are expressed in dimensionless form: the load level by means of the nominal strength divided by the
2
concrete tensile strength (σNmax/ft=3PmaxS/(2BD ft)) and the size by the Hillerborg’s brittleness number, Equation 1. It is obvious
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that size effect is only shown under the strain rate 1.05×10 /s, while it is inconspicuous to the higher strain rates.
Figure 6. Effect of loading rate and specimen size on the peak load
Figure 7. Non-dimensional cracking strength versus their corresponding depth under various strain rates
(3) Numerical analysis
As shown above, for higher strain rates, the size effect of the tested beams is not discernible, which presents an apparent
physical inconsistency. As seen from Table 1, in order to achieve the same strain rate, the large specimen has to be loaded
four times faster than the small specimen. This means that in a larger specimen the crack is forced to open faster. The
increase in the CMOD rate would generate higher loads than expected, which may hide the size effect. Since the CMOD rate
was not recorded in the experiments, we turn to an advanced numerical model to clarify this point.
Figure. 8 Comparison of the CMOD rate between the large and small specimen
A 3D cohesive model accompanied by an insertion algorithm, see [32-35], is adopted. In this numerical approach, the concrete
matrix and steel rebar are modeled explicitly. The fracture behavior of concrete is described by a 12-node cohesive element,
the debonding between concrete and steel rebar is simulated through an interface element. The feasibility of this model has
been validated for static and dynamic fracture propagation in plain and reinforced concrete [34-37]. The material parameters
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presented in Section 3 were fed directly to the numerical model. We analyze the case for a strain rate of 3.75 ×10 /s, and
extract results for the small and large specimens respectively. Figure 8 shows the comparison of the CMOD rate between the
large and small specimen. Note that the small specimen was loaded at a speed of 7.5 mm/s, while the large specimen was
loaded at a speed of 30 mm/s in order to achieve the same strain rate. In order to facilitate the comparison, we subtracted the
crack initiation time, 0.0174s and 0.0042s for the small and large specimen respectively, for the time axis. For both cases, a
turning point (point A and B) for CMOD curve which signals the attainment of the peak load is observed. It is noteworthy that
the model indicates that the crack opens stably before the peak load is attained. The stain rate is not meaningful anymore
under such circumstances. Before the peak load, the CMOD rate (2.1E-2 m/s) for the large specimen is actually about 10
times of that (2.1E-3 m/s) of the small specimen (Please, note that this range of CMOD rate in our experiments is more than
one hundred times larger than that in the experiments by Bažant and Gettu [23]). Therefore the growth of the peak load due to
the growing CMOD rate might have been balanced with its decrease due to size effect. This may explain why we were not able
to observe the size effect for the specimens loaded at the same strain rate. Moreover, the peak load maybe more sensitive to
the CMOD rate than to the strain rate.
Conclusions
We have presented recent experimental results on reinforced concrete beams to study combined size and rate effects. Two
types of micro-concretes were adopted to cast the specimens. Geometrically similar beams of three sizes and with constant
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steel ratio (0.15%) were tested under various strain rates, from 10 /s to 10 /s. The following conclusions can be drawn from
the study.
(1) The peak loads increase with increasing strain rate.
(2) The rate dependence of the peak load is stronger for larger specimens than for smaller ones.
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(3) Size effect is only shown under the nominal strain rate 1.05×10 /s, while it is inconspicuous to the higher strain rates.
(4) The apparent absence of size effect at higher strain rates is explained numerically. The growth of the peak load due to the
growing CMOD rate might have compensated its decrease due to size effect.
Further experimental as well as numerical studies are needed to disclose and quantify the combined size and strain effects, as
well as the sensitivity of the dynamic response to the CMOD rate.
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