On the Evaluation of Prestress Loss in PRC Beams by Means of Dynamic Techniques
International Journal of Concrete Structures and Materials
On the Evaluation of Prestress Loss in PRC Beams by Means of Dynamic Techniques
Marco Breccolotti 0
0 Department of Civil and Environmental Engineering , Perugia , Italy
In the last few decades, prestressing techniques have been used to build very important structures and infrastructures. Since the serviceability and the safety of prestressed reinforced concrete (PRC) elements rely on the effective state of prestressing, development of tools and procedures capable of estimating the effective prestress loss would be very useful. Amongst other techniques, dynamic identification has proved to be an economical, quick and reliable method to evaluate structural integrity. However, the influence of prestressing in the dynamic behavior of PRC elements is not completely clear. In fact, while many references in the literature state that the prestressing force does not affect the frequencies of vibration, almost every experimental test carried out on PRC beams shows an increase in the eigenfrequencies for increasing value of the prestressing force. This paper aims to contribute to the debate, investigating the dynamic behaviour of PRC beams taking into account properties, such as nonlinearity, softening, confinement and micro-cracking of concrete subjected to compression and tension stress states, and the variation of the flexural stiffness of the PRC beam along its length according to bending stress distribution. Non-linear discrete modelling, in combination with system identification and optimization was used to define the dynamic properties of PRC beams, taking into account the effect of prestressing level. The proposed model was applied to four PRC beams with known mechanical and dynamic properties in literature. The influence of the prestressing force on the trend of frequencies of vibration was closely captured for each beam, with errors less than 3% of the estimated frequencies. The results of this investigation thus indicate that dynamic identification techniques can potentially be used to identify the prestressing force level, and consequently the prestress loss, provided the complete concrete mechanics is taken into account.
prestressing; prestress loss; dynamic identification; micro-cracking; concrete softening
List of Symbols
r Concrete stress
fc Concrete strength
fcc Confined concrete strength
fco Unconfined concrete strength
e Concrete strain
eco Unconfined concrete peak strain
ecc Confined concrete peak strain
Ec Concrete elastic modulus
Eco Micro-cracked concrete elastic modulus at the origin
yci Distance between the i-th concrete fiber and the
ysj Distance between the j-th steel fiber and the concrete
Aci Area of i-th concrete fiber
Asj Area of j-th steel fiber
x Angular natural frequency and eigenvalue
Mass per unit length of the entire beam (concrete,
prestressing and non-prestressing steel)
Beam stiffness matrix
Beam mass matrix
Element stiffness matrix
Element mass matrix
Concrete age (days)
Concrete age at the beginning of drying (days)
Since the first applications of prestressing to reinforced
concrete structures dates back to the first half of the
twentieth century by Eugene Freyssinet and Gustave Magnel,
prestressing has continuously increased its application to
more ambitious and challenging structures. Today
prestressing is widely used in many applications, ranging from
small elements, such as railway sleepers, to more substantial
constructions such as long span bridges, wide thin flat slabs,
long and light precast flooring and roofing elements for
commercial and industrial buildings. Both the serviceability
and the safety of prestressed reinforced concrete (PRC)
elements rely on the effective state of prestressing
(Breccolotti and Materazzi 2015)
. In fact, the prestressing force is
used to control cracks formation, to reduce deflections and to
partially counterbalance the effect of dead and live loads. As
a consequence, an excessive prestress loss may jeopardize
the performance of PRC elements, especially in existing
aging structures. It would thus be highly desirable to have
tools and procedures capable of estimating the actual loss of
prestressing. Besides other emerging methods
(Bartoli et al.
2009; Kim et al. 2010; Lan et al. 2012)
identification techniques have proved to be economical, quick and
reliable in order to evaluate structural integrity
. Nevertheless, as far as the influence of prestressing on
the dynamic properties of PRC elements is concerned, the
debate is still on going. While many experimental
investigations indicate that prestressing affects the dynamic
properties of PRC beams, there is no definitive evidence to
confirm this behaviour from a theoretical point of view.
This paper is set out to investigate the dynamic behaviour
of PRC beams, taking into account, as far as possible, the
most comprehensive mechanical behaviour, including
nonlinearity, softening, confinement and micro-cracking of the
concrete subjected to compression and tension stress states
and variation of the flexural stiffness of the PRC beam along
its length, as per bending stress distribution. At first,
experimental investigations and theoretical analysis are
reviewed. The next stage focuses on factors that could affect
the bending tangent stiffness of PRC sections such as
materials nonlinearity, concrete cracking and
micro-cracking, concrete confinement and axial and bending stresses
acting along the PRC elements. Finally, dynamic
identification techniques were applied to the results of several
experimental tests found in literature and the main features
of concrete mechanics that influence the dynamic behaviour
of PRC beams have been identified.
2. Literature Review
Research efforts have been devoted to investigate the
effect of prestressing on the dynamic behaviour of beams
made of different materials since late 1960s. A brief review
of the main contributions of this topic is presented herein,
including experimental tests and theoretical investigations.
2.1 Experimental Tests
James et al. (1964)
were among the first to investigate the
dynamic response of a prestressed beam in 1964. Although
the study was mainly focused on comparison between
flexural rigidity of reinforced and prestressed beams as
opposed to evaluating the effect of different prestressing
levels applied to the same beam, they observed that the
modulus of elasticity of prestressed concrete beams should
be 20–30 % higher than that of the companion reinforced
concrete beams in order to obtain the same dynamic features.
carried out several experimental tests
on a steel cantilever beam with a continuously supported
barycentric tendon. Based on the results and theoretical
investigations, the author found that the natural frequencies
were not affected by the magnitude of the prestressing force.
A few years later,
monitored the dynamic
behaviour of several PRC beams made from normal and
lightweight concrete. Investigation looked at the effect of
prestressing levels on frequency and damping of concrete
beams. The author found that applying an increasing
prestressing force, acting unevenly on the beam, would increase
the frequency of vibration. In several cases, it has been
recorded that application of further degrees of prestress
increase, would result in drop of frequency of vibration.
Similar experimental results obtained by
Saiidi et al.
who tested a PRC beam with a concentric ungrouted
strand and prestressing force varying from 0 to 0.5 times the
compression strength of the concrete section. Investigations
demonstrated an increase in the first eigenfrequency from
11.41 Hz for the case of null prestressing force, to 15.07 Hz
(?32.1%) for maximum value of prestressing.
Miyamoto et al. (2000)
tested the dynamic behaviour of
prestressed beams, strengthened with external tendons.
According to their findings, prestressing forces introduced to
external tendons, affect the natural frequency of vibration of
the girder. In particular, in the case of slightly eccentric
tendon arrangement, the authors found that the natural
frequency decreased for the dominance of axial force.
Lu and Law (2006)
tested a 4.0 m long RC beam with an
ungrouted seven-wire straight strand, located at the centroid
of the beam cross-section. Two conditions were tested: with
and without a prestressing force of 66.7 kN. The authors
observed that the application of the prestressing force
produced an increase in the first three eigenfrequencies in range
Xiong and Zhang (2009)
tested three externally
prestressed, simply supported concrete beams with different
paths of the external tendons. The authors observed that the
natural frequency of beams increased in the first stage with
the increased prestressing force. Inversely, the natural
frequency decreased after the cracks produced by the
prestressing forces occurred in the beams.
Kim et al. (2010)
tested a laboratory-scaled PRC girder for
several damage scenarios of prestress loss in the tendon.
Starting from a state of no prestress loss, the external
prestressing force has been gradually reduced to Zero. During
this unloading process, dynamic measurements allowed to
identify reductions of the first four eigenfrequencies up to
values of 4.0–4.4% from the initial stage to the final one.
Jang et al. (2010)
tested six scaled post-tensioned concrete
beams with bonded tendons. By applying a continuously
increasing prestressing force from 0 to 523 kN, the authors
observed a progressive increase of the first eigenfrequency
from 7.6 to 8.7 Hz (?15.7%).
Maas et al. (2012)
observed that in an undamaged beam
subjected to compressive stress by tensioning a central
reinforcement with a hydraulic jack, the increase in the first
three eigenfrequencies is roughly equal to 1% for each MPa
increase in the concrete compression stress. They also
assumed that this stiffness increase was due to the closure of
small micro-cracks in the concrete produced by compression
Noh et al. (2015)
carried out experimental tests
on 3 PRC beams with different strand configurations. They
detected that generally, natural frequency increased as tensile
force in the prestressing steel was increased. They also noted
that frequencies of vibrations of internally prestressed
concrete beams were affected by other parameters, such as
tendon profile and boundary conditions.
The results of the most relevant experimental
(Saiidi et al. 1994; Kim et al. 2010; Zhang 2007; Jang
et al. 2010)
are summed up in Figs. 1 and 2. These
investigations have been selected from those mentioned above as
they are related to tests on reinforced concrete elements and
in the interest of data completeness, reported in the source
papers. In the graphs, ratios between vibration frequency of
prestressed beam fp and that of the non-prestressed beams f0
are plotted against the ratios between prestressing force
P and the nominal compression strength Nu of the concrete
cross-section for the first two frequencies of vibration. The
ratio between the tendon eccentricity and the height of the
beam is also demonstrated. Some general trends can be
extracted from these figures. Firstly, it can be stated that for
low levels of prestressing force, a prestress increase
produces an increase in eigenfrequencies, especially for
fundamental frequency of vibration. For higher values of the
prestressing force the rate of increase of eigenfrequencies
tends to decrease. Furthermore, it can be observed that
variations in frequencies of vibrations are higher for smaller
values of the prestressing cable eccentricity.
From the abovementioned references it can be claimed that
prestressing affects the dynamic properties of PRC beams.
Jang Zhang 1.35 1.3
P / Nu
Nevertheless, relevance of these modifications depends on
many aspects (concrete cracking, concrete nonlinearity,
strands bonding, strands eccentricity,. . .) that sometimes
produce counterbalancing effects making it difficult to
identify a clear link between dynamic properties and level of
2.2 Theoretical Investigations
Several theoretical investigations have been carried out in
recent years addressing the influence of prestressing on the
dynamic properties of PRC beams.
During the development of a moving force identification
method that takes into account the effects of prestressing,
Chan and Yung (2000)
observed, that the natural frequencies
of a prestressed bridge decrease with the increase of the
prestress forces due to the compression softening effect.
Kim et al. (2004)
developed a non-destructive method to
detect prestress-loss in beam type PRC bridges based on
measurement of changes in the natural frequencies.
Comparison between the experimental results obtained by
et al. (1994)
and the previsions of their model allowed for
validation of this method.
From the theoretical point of view,
Law and Lu (2005)
analyzed the time-domain response of a prestressed
EulerBernoulli beam under external excitation. By comparing the
results of numerical simulations with the theoretical findings,
the authors were able to identify the prestress force in the
time domain by measuring displacements and strains.
According to their findings, the frequency of vibrations
decreases as prestressing force increases.
Hamed and Frostig (2006)
developed a nonlinear
analytical model for dynamic behavior of prestressed beams with
bonded and unbonded tendons. Based on the derived
governing equations authors proved that the magnitude of
prestressing force does not affect the natural frequencies of
bonded or unbonded prestressed beams.
Looking at the abovementioned references it can be
observed that, from the theoretical point of view, there is no
agreement on the effect of prestressing on dynamic
behaviour of PRC beams.
3. Dynamic Behaviour of PRC Beams
To assess the effect of the prestressing force on the
dynamic behavior of PRC beams, an accurate modelling of
concrete mechanical behavior, both in tension and in
compression, is necessary. For this purpose, the following
aspects have been investigated:
1. Compression softening;
2. Concrete softening under increasing compressive
stresses produced by the damaging process inside the
3. Concrete hardening under increasing compressive loads
due to the confinement effect produced by the
transversal and longitudinal steel reinforcements;
4. Concrete early-age cracking due to the restrained
shrinkage and thermal deformations.
3.1 Compression Softening
Based on existing literature
(for instance the work by
Hamed and Frostig 2006)
it is demonstrated that, assuming
linear elasticity, the magnitude of the prestress force does not
affect the natural frequencies of PRC beams in both cases of
bonded and unbonded strands that undergo the same
transversal displacement of concrete sections during
vibration. In the case of unbonded straight tendons, concrete
element could vibrate without dragging in vibration
prestressing tendons. In this case, neglecting small rotations at
the beam ends, prestressing force can be considered as an
external compression force. Consequently, the natural
frequency of simply supported axially compressed beams can
be derived from Eq. 1 that includes the compression
np 4 EI
np 2 N
where n is the mode number, L is the beam length, N is the
axial compressive force (positive), m is the beam mass per
unit length and EI is the uniform flexural stiffness of the
beam. Nevertheless, this case seldom occurs in real
situations. In fact, in parabolic or draped tendons configurations,
the strands are in contact with the ducts and, thus, with the
concrete, hence they vibrate together with concrete beams.
For straight tendons configurations of pre-tensioned beams,
tendons are bonded to the concrete, thus vibrating together
with the concrete beam. In straight tendons configurations of
post-tensioned beams, strands are generally grouted inside
the ducts to reduce the risk of corrosion and to improve the
load bearing capacity at ultimate limit state. Also, in this
case tendons vibrate together with the concrete beam.
Consequently, the compression softening effect rarely occurs in
real prestressed structures and will be disregarded in the
3.2 Concrete Softening
Compression behavior of concrete is markedly non-linear
with deformations that increase faster than stresses, even at
low stress levels. The basic stress–strain parabola provided
has been one of the first constitutive
laws to model this behavior (Fig. 3):
r ¼ fco
From this graph it can be noted that material tangent stiffness
decreases for increasing values of compressive stress.
Consequently, it is expected that an increase of the prestressing
force in an uncracked PRC section will produce a reduction
of tangent flexural stiffness.
As pointed out by
and by other
researchers, concrete lateral expansion influences the behavior of RC
elements since the confinement effect generated by the
stirrups and by the longitudinal reinforcement that opposes
the concrete expansion can result in a significant strength
and stiffness enhancement. The lateral deformation is
generally small for low values of the axial stresses while it
becomes more relevant when the concrete undergoes axial
plastic deformation. In PRC beams where compressive
stresses and strains affect the entire cross-section this
behavior can potentially influence their dynamic properties.
According to several researchers
(Park et al. 1982; Fafitis
and Shah 1985)
the effect of confinement can be disregarded
for low confinement pressures thus leaving the ascending
branch of the stress–strain curve unchanged. Other authors
Fig. 3 Nonlinear stress–strain relationships for concretes with
different strengths according to
proposed stress–strain relationships where confinement
effect modifies the initial branch as well
Madas and Elnashai 1992)
. Among them, the model
proposed by Mander et al. (1988) has been adopted for
investigating the influence of confinement on the dynamics of
PRC beams. In this case the ascending branch of the
constitutive relationship is described by the following equation:
r ¼ fcc 64
r ¼ Ec
7 for ec
1 þ ecc
The ratio between the confined strength fcc and the
unconfined strength fco can be obtained by graphical interpolation
(see Mander et al. 1988)
as a function of the parameter ke
that depends on the geometry of the cross-section and on the
reinforcement configuration, and of the lateral confining
strengths flx and fly in the x and y directions, respectively.
A comparison between the stress–strain relationships for
the unconfined and confined concrete is show in Fig. 4. The
comparison is relative to a RC section made with concrete
having an unconfined strength fco of 25 MPa. The dimension
of the beam cross-section is 0:3 0:3 m, the concrete cover
is equal to 20 mm, the longitudinal reinforcement is made up
of four rebars with 16 mm diameter and the stirrups have
diameter 6 mm and are placed with 100 mm spacing. It can
be noted that whilst the main confinement effect takes place
during approach to peak strength and in the post-peak
region, in the pre-peak region a very small hardening and
strengthening effect occurs. It is therefore assumed that the
effect of the confinement on the dynamics of PRC beams is
negligible and will be neglected in the subsequent analysis.
Fig. 4 Comparison between the stress–strain relationships
for the unconfined (Hognestad parabola) and confined
1. Volume contraction due to drying and autogenous
shrinkage in the cement paste;
2. Volume contraction in cooling phase of the hardened
3. Presence of restraints that limit the contraction of
In fact, after casting the concrete mixture, whilst still in fluid
state and cannot sustain any loads. Deformations can occur
without hindrance. From the moment a supporting structure
is formed, hindrance of deformations may cause stresses.
Those caused by shrinkage are dependent on development of
the stiffness. Whether these stresses lead to cracking depends
on the development of strength. In cementitious materials
the stiffness generally develops faster in comparison to the
strength and, thus, a high crack risk occurs at early-stages.
Tensile stresses can also be caused by prevented thermal
deformations in cooling phase of hardened concrete. The
heat generated by cement hydration in concrete produces a
temperature increase and a volume expansion. At the end of
the hydration process the temperature decreases and concrete
tends to shrink.
Restraints can be caused by other structural elements
against which the new element is cast, such as aggregates,
formworks or steel reinforcements. The role of
reinforcement in formation of cracks is twofold. From one side, it is
responsible for the crack formation due to restraint effect,
and conversely an increase in the amount of reinforcement
can postpone the formation of a through-crack with the
formation of several micro-cracks
(Sule and van Breugel
. In concrete structures higher free shrinkage, higher
hydration heat, higher restraint degree, higher elastic
modulus and lower creep generally lead to a higher restrained
stress and, consequently, to a worse micro-cracking tendency
(Gao et al. 2012)
Cracking produced by hindered concrete contraction can
be divided in two types (see Fig. 5):
Main cracks that affect large areas of concrete section;
Local secondary cracks (micro-cracking) that affect the
concrete around a single steel rebar.
While the effect of the first type of cracking on the
dynamic behavior of RC beams has already been
(Breccolotti et al. 2008; Breccolotti and Materazzi
, that of the second type is not easily predictable and
no relevant references have been found in literature.
Nevertheless, some indications have been found on the effect of
shrinkage induced micro-cracking on static behavior of RC
beams. Tanimura et al. (2003), for instance, found that the
use of expansive additive and shrinkage reducing agent
produced an increase of flexural cracking load and a
reduction of deflection for the same load condition, thus
indicating an increase of flexural stiffness of the RC beam.
Kaklauskas et al. (2009)
investigated the influence of
shrinkage on stress–strain state of RC members using test
data reported in literature. The results of their investigation
demonstrated that shrinkage might reduce the cracking
resistance and the stiffness of RC members significantly. It
is, therefore, conceivable that shrinkage induced
microcracking could also affect the dynamic behavior of PRC
In order to evaluate this influence, a smeared
microcracking will be assumed in subsequent analysis. It will be
modelled by means of a suitable modification to the original
concrete stress–strain curve shown in Fig. 6. This
modification is composed of a translation esh of the original
constitutive relationship in the direction of higher compressive
strains, a reduction of tensile strength and a connecting
stretch between translated compression stress–strain curve
and reduced tensile strength expressed by means of an
exponential law that depends on the exponent a.
The mathematical formulation of this behavior is
expressed by Eq. 5 for the compression and the tensile stresses,
where the elastic modulus Eco has been set equal to the
concrete modulus for compressive stresses at origin to
ensure continuity in the behaviour from compressive to
tensile stresses. The value of Eco (Eq. 6) has thus been
calculated as the value assumed by the first derivative of
Eq. 5 for e 0 at origin:
Eco ¼ oe e¼0 ¼ fco
þ 2 e2sh
This general behaviour would allow for initial lower
concrete stiffness in compression up to the closure of internal
micro-cracking and reduced tensile strength of
microcracked concrete to be taken into account.
4. Flexural Stiffness of PRC Beams
The flexural behavior of RC beam, with and without
prestressing action has been evaluated, considering the
features listed in the previous paragraph through a discrete
approach. The beam was firstly discretized in the
longitudinal direction in a suitable number of finite elements. The
dimension of the element has been chosen in order to
properly model the discrete location of the main cracks, if
they occur. To this end, the element dimension in the
longitudinal direction has been set to one third of the distance
between two consecutive cracks in the stabilized cracking
(fib International Federation for Structural Concrete,
Model code 2010)
. The flexural and axial stiffness of these
elements have been calculated through a fiber modeling
(Taucer et al. 1991)
dividing each cross-section
into finite areas (each fiber characterized by constant stress
and strain). Constitutive relationships have been defined in
terms of stress–strain relations for every fiber. Hognestad
parabola with the proposed modification to take into account
micro-cracking has been used for the concrete. Where the
original Hognestad law is parabolic only in the pre-peak
branch and it is linear in post-peak, the parabolic stress–
strain relationship has been used for both the ascending and
the descending branches. This assumption did not introduce
any inaccuracies in analysis for the level of compressive
stresses involved in the case studies that are always smaller
than the concrete peak strength. Classical elastic—perfectly
plastic stress–strain law is used for reinforcing steel while
the power formula proposed by
Devalapura and Tadros
was used for prestressing steel:
r ¼ e 6A þ n
1 þ ðC eÞDo1=D57
where A, B, C and D are suitable constants that depend on
the type of low-relaxation prestressing strands. The two
constitutive laws are shown in Fig. 7.
The classic assumption of Bernoulli (plane cross-sections
perpendicular to the axis of the beam before bending remain
plane and perpendicular to the bent axis of the beam after
bending) has been used to calculate the strains distribution
inside the beam cross-section.
The construction of relationship between bending moment
M and the curvature v was accomplished for each value of
the axial force N (which is equal to the prestressing force) by
integrating the stresses distribution over the beam cross
X Aci rðeci Þ þ
X Asj r esj
X Aci rðeci Þ yci þ
X Asj r esj
with eci ¼ v yci and esi ¼ v ysi .
Several examples of moment-curvature relationships for
the same cross-section but with different values of
prestressing force are shown in Fig. 8. It is evident that
prestressing action not only alters the limits of the uncracked
stage but also slightly changes its bending stiffness. For each
value of axial load Ni and each value of bending moment Mi
the tangent flexural stiffness EIi of the beam can be
EIiðNi; MiÞ ¼
where the stiffness [K] and mass [M] matrices of the entire
beam has been assembled from element stiffness ½Kie and
consistent mass ½M e matrices:
Mass per unit length q comes from the concrete,
non-prestressing and prestressing steel. It must be noted that the
mass of prestressing steel is considered also in the case of
null prestressing force since prestressing strands are not
removed from the beam, but simply they are non-tighten.
The flexural stiffnesses EIi of Eq. 12 of each element have
been obtained from the moment-curvature relationships
recalled in par. 4 and shown in Fig. 8 as tangential stiffness,
measured on the curve for the specific prestressing force and
for bending moment produced by external loads and by
prestressing force on the specific element (Eq. 10). The
numerical procedure to calculate the stiffness [K] and mass
[M] matrices of the PRC beam is summed up in the
flowchart of Fig. 9.
(i) Discretization of the beam in the
longitudinal direction in n elements.
(ii) Determination of axial Ni and bending Mi stresses
in each element (i = 1 : n) taking into account all relevant
loads (selfweight, dead loads, live loads, prestressing).
(iii) Construction for each element i = 1 : n
of the moment-curvature relationship for
the relevant axial stress Ni (Fig. 8).
(iv) Calculus for each element i = 1 : n of the flexural
stiffness EIi corresponding to the axial Ni and
bending Mi stresses in the element (Eq. 10).
(v) Assemble global stiffness K and mass M matrices.
6. Identification of the Unknown Parameters
The proposed numerical modeling of dynamic behavior of
PRC beams has been applied to several experimental
investigations, results of which are available in literature
(Saiidi et al. 1994; Kim et al. 2010; Zhang 2007; Jang et al.
in order to check correctness of assumptions made in
numerical modeling. Nevertheless, direct application of the
proposed method is not viable since the parameters esh and
a that describe the correction of concrete stress–strain
relationship (which takes into account the concrete
microcracking) are unknown. Furthermore, even if the nominal
concrete strength was evaluated by means of conventional
tests on concrete samples, the effective concrete strengths of
tested beams may differ from those of small size samples.
Nominal concrete strength obtained by the 28 days
compressive tests on small dimension samples can in fact, be
different from that of the real structure. For instance, it can
be smaller than that of tested PRC beam for hardening
duration of concrete belonging to the beam at the time when
dynamic tests are carried out in excess of 28 days.
Conversely, it can be greater than that of the tested PRC beam,
where inadequate curing and compaction procedures were
applied to the PRC element, as opposed to the standard
procedure applied to small size concrete samples
European Committee for Standardization, EN 13791 2007)
For this reason, concrete strength fc has been considered an
unknown parameter too.
The problem has been solved by considering concrete
mechanics described by Eqs. 5 and 6. Tangent stiffness of
each beam elements has been calculated as per procedure
described in Sect. 4. The eigenvalues problem represented
by Eq. 11 has been thus expressed as a function of
parameters fc, esh and a. Identification of these unknown
parameters has been carried out essentially as an optimization
process by minimizing the values of error function expressed
in terms of unknown parameters and the first
eigenfrequencies of experimental tests f1;exp;i and, numerical
simulations f1;num;i. The optimization problem can consequently
be formulated as follow:
ðfc; esh; aÞ i¼1
with constraints: 0:6fc;nom fc 1:4fc;nom, 10 6 esh 10 4
and 103 a 105 and where n is the number of prestress
load values applied in each experimental test. The choice of
using only 1st frequencies of vibration in the optimization
process was based on data found in references and on
previous investigations carried out by the author
et al. 2009)
. It is in fact generally true that the effect of
prestressing in modifying frequencies of vibration is much
more relevant in the 1st mode rather than the higher ones, as
visible by comparing Figs. 1 and 2. Data about 2nd
frequencies of vibration were initially added in the optimization
process but the error achieved in this case turned out to be
higher than that obtained with only 1st one, probably due to
lower accuracy in evaluating the higher modes of vibration.
Optimization problem was solved by means of the genetic
algorithm toolbox of Matlab (2007a). Populations of 100
terns of the unknown parameters fc, esh and a were generated
in each step up to a maximum of 100 generations. An
example of convergence error for this optimization process
is shown in Fig. 10 for the case of Saiidi et al. experimental
tests. It can be noted that the convergence error is reduced to
an acceptable value, starting from a few tens of generations.
Fig. 10 Error convergence in the GA optimization for
et al. (1994)
Similar results have been also obtained for other
experimental tests. The results of identification process for the
above mentioned experimental investigations are presented
and discussed in the next paragraphs.
6.1 Comparison with Tests by Saiidi et al.
Saiidi et al. (1994)
tested a prestressed beam with
clear 3.66 m length between supports, a cross section of
0.102 m wide and 0.127 m high. Prestressing was achieved
by means of a 0.5 in. strand, placed centrally in a duct that
remained ungrouted. At one end of the beam a load cell was
placed between the beam and the dead anchorage. At the
other end, a hydraulic jack applied prestressing forces to the
strand, as per Table 1. For each value of prestressing force,
free-vibration tests were conducted, measuring the
eigenfrequencies listed in the same table.
Dynamic identification procedure identified the unknown
parameters listed in Table 5. Accuracy on the evaluation of
the unknown parameters can be argued by looking at Fig. 10
where the convergence of fitness value (error) calculated by
means of objective function (Eq. 14) for investigated
generations are highlighted. Distributions of flexural stiffness
corresponding to the identified unknown parameters are
shown in Fig. 11. An abrupt stiffness reduction at midspan
for the curve corresponding to non-prestressed beam can be
noted amongst them. This reduction is caused by formation
of a flexural crack in the concrete. Formation of this crack
was also noted by the authors who highlighted the
development of a small crack at the beams midspan under its own
weight. The effect of midspan crack can be identified by
looking at the way the first two eigenfrequencies increase at
the beginning of prestressing. 1st frequency of vibration
increases by 18% from 11.41 to 13.47 Hz, while the 2nd
increases by only 2% from 43.99 to 44.89 Hz. This behavior
is caused by the midspan crack which forms in
non-prestressed case and closes after prestressing. By considering
the first two mode shapes, it can be noted that closure of
midspan crack, affects the 1st mode of vibration while
leaving the 2nd one practically unaffected for the point of
contraflexure at midspan in the second flexural mode shape.
Presence of the crack does not become more visible when
prestressing force is applied to the beam. It can, thus, be
assumed that prestressing force is able to close the crack
during dynamic tests.
Comparison between experimental and numerical
fundamental eigenfrequencies for different level of prestressing is
show in Fig. 12. In this figure, relative errors (%) between
experimental and numerical frequencies are reported. A
good correlation between experimental and numerical values
can be observed. From this figure, it can be argued that for
very high prestressing force concrete softening becomes
predominant with very small increments of the 1st frequency
6.2 Comparison with Tests by Kim et al.
Kim et al. (2010)
tested a T-shaped girder having
an overall height of 0.6 m and a clear length of 6.0 m simply
supported by thin rubber pads. A 0.6 in. straight strand was
used as an ungrouted, prestressing tendon. Starting from an
initial value of 117.6 kN, prestressing force was
progressively reduced down to a null, in 5 discrete steps. For each
prestressing value, the first eigenfrequencies were extracted
with a frequency-domain decomposition technique from the
free response of the beam after an impact excitation. Values
of prestressing forces and corresponding frequencies of
vibration are shown in Table 2. Also, in this case it can be
noticed that the natural frequency of vibration increases for
higher values of prestressing force.
Dynamic behavior of the beam was simulated with the
methodology established in previous paragraphs and
unknown parameters have been determined by minimizing
the objective function. The identified unknown parameters
are listed in Table 5. Distributions of flexural stiffness for
each value of prestressing force are shown in Fig. 13. The
hardening effect produced by prestressing force on the
elements flexural stiffnesses is visibly evident. In Fig. 14 the
comparison between the first experimental frequencies of
vibration and the corresponding values obtained by the
numerical simulation is demonstrated. It can be clearly
observed that in this case that the 1st frequency of vibration
continuously increases as the value of prestressing force
increases. This behavior can be attributed to the small value
of maximum prestressing force P in comparison to the
ultimate compression strength Nu of the beam cross section
(P=Nu ¼ 0:04). Nevertheless it must be pointed out that in
this case, relative errors up to 1.5% (for the maximum
prestress force) have been obtained between frequencies of
vibration evaluated with the proposed methodology and the
experimental ones. These errors can probably be ascribed to
some missing information (such as the amount of
non-prestressed reinforcement) not available in the reference source.
40 60 80
Prestress force (kN)
6.3 Comparison with Tests by Zhang and Li
Zhang and Li (2007)
tested a rectangular concrete beam
with a linear unbonded tendon. The beam was simply
supported with a cross section of 0:12 0:24 m with a total
length of 3.9 m and a distance between the supports of 3.7
m. The 28-day cube crushing strength of the concrete was
45.2 MPa. Prestressing steel was made with an eccentric 0.6
in. tendon while the reinforcing steel was made with 2 bars
of 8 mm diameter in the upper part of the beam and 2 bars of
12 mm diameter in the lower part of the beam. Stirrups with
diameter of 6 mm were placed with 150 mm spacing along
the beam. Dynamic tests performed with a DH5938
vibration analyzer and a sampling frequency of 1000 Hz were
repeated with increasing values of applied prestressing force.
The measured values of the 1st and 2nd eigenfrequencies,
shown in Table 3, increased for increasing values of
Dynamic identification procedure allowed for
determination of unknown parameters listed in Table 5. Distributions
of flexural stiffness for each value of prestressing force are
shown in Fig. 15. Even in this case, it is possible to see the
initial hardening effect produced by prestressing force in the
element flexural stiffnesses with the 1st frequency of
vibration that increases as the value of prestressing force
increases (Fig. 16). Conversely, for higher prestressing force,
concrete softening starts playing a relevant role with very
small increments of the 1st frequency of vibration.
Fig. 15 Distribution of the flexural stiffness along the beam
Zhang and Li (2007)
for different values of
prestressing force P.
40 60 80
Prestress force (kN)
6.4 Comparison with Tests by Jang et al.
Jang et al. (2010)
tested a total of six equal
scalemodel prestressed beams with bonded strands, each of them
having different values of prestressing force. The beams
were 8.0 m long with a clear length between the supports of
7.4 m. Cross section was rectangular with dimension of
0:3 0:3 m. Nominal concrete compressive strength of the
tested beams was 35.0 MPa. Four rebars with 16 mm
diameter were used for longitudinal reinforcement and 10
mm diameter stirrups were placed with spacing of 100 and
150 mm. Three 0.6 in. strands placed centrally, were used to
apply six different prestress force values. Experimental
modal tests with impact hammer and Multi-Input
MultiOutput sweep tests were used to extract the eigenfrequencies
from acceleration time histories, recorded at several points of
the prestressed beams by piezo-electric sensors. The results
observed in dynamic tests and summed up in Table 4 show
that as prestress force increases, natural frequencies also
The proposed minimization procedure allowed to identify
unknown parameters listed in Table 5 and the bending
stiffness distributions shown in Fig. 17. It can be noted that
dynamic identification requires a sudden decrease in flexural
stiffness of the beam at midspan for the load case with null
prestressing force. Also in this case, the stiffness decrease
could be associated to concrete cracking. In places where
cracking was not identified in the report of the experimental
tests, stress calculations have verified that concrete area of
0:3 0:3 m section beam on a clear length of 7.4 m
undergoes tensile stresses higher than its tensile strength due
to its own weight. In fact, under the self-weight, the midspan
section is subjected to a flexural moment of 15.3 kNm that
would produce a maximum tensile stress on the concrete
equal to 3.4 MPa. This value is greater than the mean tensile
strength fctm equal to 3.2 MPa for a concrete that has a
compressive strength of 35.2 MPa. It can be reasonably
assumed that the presence of a crack contributes to the
reduction of flexural stiffness. Comparison between
experimental and numerical fundamental eigenfrequencies
for different levels of prestressing is show in Fig. 18. A good
agreement between experimental and numerical values and
an overall behaviour similar to that noticed for the tests
conducted by Zhang and Li can be observed.
7. Comments to the Results of the Identification Procedure
Some comments can be made on the results obtained by
dynamic identification in terms of unknown parameters fc,
esh and a. The corresponding stress–strain relationships were
normalized with respect to the nominal value of concrete
strength in each test and shown in Fig. 19 together with a
reference stress–strain curve, representing the basic
Hognestad parabola, Eq. 2 (or similarly Eq. 5 with esh equal
The identified concrete compressive strengths resulted in
two cases (Kim et al. and Jang et al.) very similar to the
concrete nominal strength obtained by testing concrete
samples at 28 days. In the other two cases, the identified
values are higher than the corresponding nominal values.
This could be ascribed to the age of concretes at the time of
dynamic testing greater than 28 days.
The identified values of parameter esh resulted in the range
of 0.055 10 3–0.091 10 3 (Table 5). These values are
much smaller than those estimated based on the provision of
fib Model Code
(fib International Federation for Structural
Concrete, Model code 2010)
ecsðt; tsÞ ¼ ecasðtÞ þ ecdsðt; tsÞ
For instance, the final total shrinkage predicted by the fib
Model Code for structural concretes in different atmospheric
condition and with different notional sizes is in the range of
0.26 10 3–0.57 10 3 and thus 3–10 times bigger than
those resulting from the identification. This result reflects the
fact pointed out in par. 3.4 that micro-cracking produced by
hindered shrinkage strains only affect the concrete where
contraction is prevented. Thus, it happens around the steel
rebars and close to the formworks and not on the entire
concrete section as assumed in numerical simulations.
The last parameter, the exponent a, assumed values in a
limited range of 5015–7716 (Table 5). These high values
identified by optimization procedure, produce fast transitions
of stress–strain curves between unloaded condition and
translated stress–strain curve of micro-cracked concrete.
This means that micro-cracks are completely closed by the
application of small compressive stress, letting the concrete
behaves as though it is uncracked, but with increased strain
values after the closure of the micro-cracks.
Looking at the results of experimental tests, some
preliminary comments can be made on the prestressing level at
which concrete softening becomes significant, although at
the current state of research, it is not easy to define this value
accurately since many factors, such as the eccentricity of
prestressing force, the external bending moment, etc. affect
concrete softening in different position of the beam cross
section. In Table 6 the ratios between average compressive
stresses and identified concrete strengths for the four
experimental tests are reported. In the same table, the
relevance of concrete softening in reducing frequencies of
vibrations for increasing values of prestressing force inferred
from Figs. 12, 14, 16 and 18 are indicated. For instance it
can be noted that in tests by Saiidi et al. the rate of increase
of the 1st frequency of vibration decreases as prestressing
force increases, while it is almost constant in the tests carried
out by Kim et al. (see also Fig. 1). It can be observed that
generally concrete softening increases its relevance as the
ratio between average compressive stress and concrete
strength becomes higher.
A further interesting observation can be made on the data
shown in Fig. 20 where for each experimental work the
relationships between identified parameter esh and
reinforcement ratio of tested beams are plotted. An almost
linear relationship between these two parameters with higher
micro-cracking effects expected for higher reinforcement
ratios can be observed.
Nevertheless, it must be pointed out that in real structures
other aspects should be considered in order to identify
prestress losses accurately. For instance, in pretensioned
elements, the beam is subjected to a relevant variation of
the prestressing force from a 0 value at each end of the
beam to the full prestressing force at the end of the
development lengths. In post-tensioned parabolic strand
configurations, friction between tendons and ducts reduces
the pretension force along the beam. Thus, in both cases a
non-uniform prestress force is present along the beam and
its variation should be considered in dynamic identification
In this work, several aspects that could influence the effect
of prestress force on the dynamic behavior of prestressed
reinforced concrete (PRC) beams have been analyzed.
Whereas many references found in the literature state that
prestressing force does not affect the frequencies of
vibration, almost every experimental test carried out on PRC
beams showed an increase in eigenfrequencies for increasing
value of prestressing force. This trend is generally true for
low values of prestressing level (5–10% of the ultimate
compression strength of the section) while for higher values
of prestressing force, the rate of change of frequencies of
vibration is reduced. For even higher prestressing force, the
eigenfrequencies tend to decrease.
This overall behavior is a consequence of several features
of concrete mechanics such as micro-cracking produced by
hindered volume contraction, its nonlinear compressive
stress–strain behavior with concrete softening and its low
tensile strength. In particular the effect of the concrete
microcracking has been proven to have a significant effect on the
dynamic behaviour of PRC beams with varying prestressing
Taking these features into account, inverse analysis has
been carried out to identify the effective stress–strain
relationship of concrete in several experimental works
found in literature, expressed in terms of some unknown
parameters. The identified parameters that model the
influence of micro-cracking produced by concrete
shrinkage are in good agreement with the expected values.
Obtained results confirm that prestressing force level affects
the dynamic behavior of PRC beams. Dynamic
identification, therefore, can potentially be used to identify
prestressing force level and consequently the prestress loss,
provided that the complete concrete mechanics and the
environmental effects, such as temperature and humidity,
are taken into account appropriately
(Breccolotti et al.
2004; Xia et al. 2006)
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