Crack sensor with embedded optical fibre Report D5.3
PRIORITY 6
SUSTAINABLE DEVELOPMENT GLOBAL CHANGE & ECOSYSTEMS
INTEGRATED PROJECT
This report is one of the deliverables from the Integrated Research Project “Sustainable Bridges - Assessment for Future Traffic Demands and Longer Lives” funded by the European Commission within 6
thFramework Programme.
The Project aims to help European railways to meet increasing transportation demands, which can only be accom- modated on the existing railway network by allowing the passage of heavier freight trains and faster passenger trains.
This requires that the existing bridges within the network have to be upgraded without causing unnecessary disrup- tion to the carriage of goods and passengers, and without compromising the safety and economy of the railways.
A consortium, consisting of 32 partners drawn from railway bridge owners, consultants, contractors, research insti- tutes and universities, has carried out the Project, which has a gross budget of more than 10 million Euros. The European Commission has provided substantial funding, with the balancing funding has been coming from the Pro- ject partners. Skanska Sverige AB has provided the overall co-ordination of the Project, whilst Luleå Technical Uni- versity has undertaken the scientific leadership.
The Project has developed improved procedures and methods for inspection, testing, monitoring and condition as- sessment, of railway bridges. Furthermore, it has developed advanced methodologies for assessing the safe carrying capacity of bridges and better engineering solutions for repair and strengthening of bridges that are found to be in need of attention.
The authors of this report have used their best endeavours to ensure that the information presented here is of the highest quality. However, no liability can be accepted by the authors for any loss caused by its use.
Copyright © Authors 2007.
Figure on the front page: Photo of a crack sensor mounted on the Övik Bridge
Project acronym: Sustainable Bridges
Project full title: Sustainable Bridges – Assessment for Future Traffic Demands and Longer Lives Contract number: TIP3-CT-2003-001653
Project start and end date: 2003-12-01 -- 2007-11-30 Duration 48 months Document number: Deliverable D5.3 Abbreviation SB-5.3 Author/s: Paulo J. S. Cruz & A. Diaz de León; UMINHO
Date of original release: 2007-11-30 Revision date:
Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006)
Dissemination Level
PU Public X
PP Restricted to other programme participants (including the Commission Services)
RE Restricted to a group specified by the consortium (including the Commission Services)
CO Confidential, only for members of the consortium (including the Commission Services)
Summary
This document reports the development of a distributed optical fibre crack sensor. This sensor can be employed for detection of cracks and measurement of crack widths in concrete struc- tures. The basic principle of operation for the crack sensor is based on intensity variation of the optical power within the optical fibre due to the initiation and opening of cracks.
This technique does not require prior knowledge of the crack locations, which is a significant advancement over existing crack monitoring techniques. Moreover, several cracks can be de- tected, located and monitored with a single fibre. However, the crack direction needs to be known.
An ideal application of the sensor is in the monitoring of flexural cracks in bridges, which may appear at arbitrary locations along the deck, but essentially perpendicular to the spanning direc- tion. A method for installing the sensor on existing structures was recently proposed. It involves the embedment of the optical fibre into a polymeric plate to form a Plate Sensor
5, which is then attached to the surface of the structure.
The present report summarizes the improvements introduced in the sensor to make it appropri- ate for crack monitoring in bridges, performed by researchers of the University of Minho, in Por- tugal.
The primary objectives of the work were:
1) To select the proper polymeric material for making the plate;
2) To optimize the manufacture process;
3) To assure the bonding between the plate sensor and concrete members;
4) To determine the corresponding calibration curves of intensity losses versus the crack aper- ture;
5) The application of the plate sensor to the monitoring of single crack RC beams and masonry slabs and the monitoring of multiple cracks on RC beams;
6) The field implementation of the plate sensor.
The first objective of this investigation was to identify a suitable polymeric material from which to make the sensor plate. Experiments were performed with the incorporation of different volumes of accelerant and catalyst, to identify the right polymer composition.
The failure behaviour of the polymeric plate can be controlled by the incorporation of fine parti- cles (granite, calcareous, metakaolin, quartz, river sand, and abrasives) and measured with direct tensile tests. The results of these tests provided guidelines for modifying the polymeric material to attain different grades of resistance and thus different sensibility levels of the sensor, i.e. the plate sensor breaks for thinner or wider cracks.
Instead of pour the polymer with the fibre placed inside, as it was proposed in the original manu- facture process, a 254μm diameter steel wire, coated with releasing agent, was placed in the mould at the desired angle. After that polymer was cast into the mould and allowed to cure.
Once the polymer was fully cured it was removed from the mould. The steel wire was removed and the optical fibre can be placed into the hole left by the steel wire. This way the inopportune break of the optic fibre related with a possible adherence between the optical fibre and the sur- rounding polymeric matrix was avoided.
The bonding procedures to fix the plate sensor to concrete members were deeply studied and are presented in this report.
There are different ways in which the calibration of prototype of crack sensor can be done. In
this study was used a power meter to measure the loss in forward power transmission, instead
of use an Optical Time Domain Reflectometer (OTDR). The reasons for this choice were: the simplicity and accuracy of the power meter and the fact that the location of the crack was known in advance.
The experimental setup of the calibration tests and the corresponding calibration curves of the intensity losses versus the crack aperture are presented and discussed.
The applicability of the sensor to detect and measure crack opening, in concrete and masonry elements, was deeply studied. The detection capabilities under a single crack or multiple cracks were evaluated as well as the functional characteristics in terms of spatial resolution, and sensi- tivity.
The field implementation of the plate sensor in the destructive test carried out in the Övik
Bridge, in the north of Sweden, is also included. The main purpose of this test was to demon-
strate the capability of the sensor plate to detect cracks and measure the crack opening in con-
crete railway bridges. This report provides the description of the bridge and gives an overview of
the procedures for wiring and installing the sensor plate.
Acknowledgments
This report has been drafted on the basis of Contract No. TIP3-CT-2003-001653 between the
European Community represented by the Commission of the European Communities and the
Skanska Teknic AB contractor acting as Coordinator of the Consortium. The authors acknowl-
edge the Commission of the European Communities and the University of Minho for its financial
support.
Table of Contents
1 Introduction...7
2 Cracking detection and monitoring ...9
3 Novel crack sensing concept based on fibre optics...11
3.1 Sensing principles ...11
3.2 Thermal and mechanical properties of the polyester...12
3.3 Mechanical properties of modified polymeric materials...15
3.4 Sensor plate fabrication process ...17
3.5 Experimental analysis of the bond properties...18
4 Calibration of the Plate Sensor ...22
4.1 Calibration curves...23
4.2 Statistics for the curve calibration...25
4.3 Probabilistic model for curve calibration...31
4.4 Static performance characteristics ...31
5 Single crack monitoring with Low Resolution OTDR ...33
5.1 Wiring and data acquisition of the Sensor Plate...33
6 Practical implementation of the Plate Sensor ...44
6.1 Monitoring of single crack RC beams and masonry slabs...44
6.2 Monitoring of multiple cracks on RC beams ...47
7 Field implementation of the Sensor Plate ...51
7.1 Description of the bridge and destructive test ...51
7.2 Wiring configuration and acquisition results ...53
8 Conclusions ...59
9 References ...60
Annex I – Results of the calibration tests ...61
Annex II – Comparison of PMCC and calibration curves ...71
1 Introduction
Existent bridges, particularly those made of reinforced concrete, are deteriorating at a rapid rate, faster than they are being repaired, strengthened or replaced. Many structures built in the 1960's and 1970's are now considered deficient by today's design standards. The deficient be- haviour of some bridges has highlighted the importance of effective monitoring systems, which are able to identify structural problems at an early stage. Apart from the safety concern, a finan- cial problem is also arising. The potential of monitoring systems to reduce operational mainte- nance costs, by identifying problems at an early stage and by verifying the effectiveness of re- pair procedures, is clearly significant.
The healthy condition of many concrete structures can be assessed through the detection and monitoring of cracking. The cracking as an adverse effect in the performance of bridges particu- larly those made of reinforced concrete.
Cracking performance is one of the structure condition data that need to be surveyed. Once initiated, cracking increases in severity and extent and allows water to. The water will further accelerate the rate of structure deterioration. Thus, to determine the timing and cost of structure maintenance, the information on structure crack condition is needed. Collection of cracking data is difficult and time consuming because a manual and expensive survey has to be involved in the process, which always requires the intervention of specialized equipment and operators.
Due to the nature of the subjective survey, it is very difficult to obtain results that are accurate, repeatable, and reproducible. Thus, there is a need to automate the cracking survey process to improve safety and achieve more objective and consistent data of structure cracking.
The types and severity of cracks in civil structures are varied; the cracking can be classified as inactive or active. Inactive cracks (such as early shrinkage cracks in concrete structures), once located, usually require corrective action to prevent moisture infiltration. Depending on the se- verity of cracking, it can indicate or represent serious problems to the structure. Active cracking is related to foundation settling, inherent design flaws, structural degradation and deterioration conditions. It always requires active monitoring, evaluation and corrective action.
In Reinforced Concrete (RC) structures, the reaction between cementitious material and aggre- gates (Alkali-Silica Reactivity, ASR) leads to random cracking at the surface, and requires sur- face repair in a maintenance process. It belongs to the severity corresponding to deterioration conditions.
A crack in an element can be subjected to three different types of loading. These loading condi-
tions are illustrated in Figure 2.1. The opening mode involves loads that produce a displace-
ment of the crack surfaces perpendicular to the plane of the crack. The stress intensity factor,
indicated as K
I, is due to tension or flexure loading. The shearing mode of loading is due to in-
plane shear loads which cause the two cracks surfaces to slide on one another. The displace-
ment of the crack surfaces is the crack’s plane and is perpendicular to the leading edge of the
crack. The stress intensity factor, indicated as K
II, is due to shear mode loading. The tearing
mode of loading is due to out-of-plane shear loading. The displacement of the crack surfaces is
in the crack’s plane and is parallel to the leading edge of the crack. The stress intensity factor,
indicated as K
III, is due to tearing mode of loading.
a) Opening mode b) Shearing mode c) Tearing mode Figure 2.1: Different types of loading subjected on a crack in an element.
While the superposition of the three modes of loading gives the most general loading condition, the survey usually is interested in tensile opening mode or mixed I-II mode since they occur more frequently under active cracking (under mixed I-II mode, both K
Iand K
IIare present).
The fracture control is commonly and frequently used in mechanics and aeronautics, but its im-
plementation in civil structures is not common and very rare for bridges. The complexity and
extent of bridges make the collection of crack data difficult and time consuming as it involves a
manual and expensive survey in the process, which always requires the intervention of special-
ized operators and equipment. Thus, there is a need to automate these survey processes to
improve the safety of bridges and reduce the bridge's operation and maintenance cost.
2 Cracking detection and monitoring
The condition of many important concrete structures can be partially assessed through the de- tection and monitoring of cracking. Usually the crack detection in bridges is based on visual in- spections. This procedure is time consuming, expensive and unreliable, therefore the use of cracking sensors is highly recommended. Nevertheless, most existing sensors/transducers are quite limited in their ability to detect and monitor cracks.
Advances in the production of optical fibres made possible the recent development of innovative sensing systems for health monitoring of civil structures. The main reasons of this development are the reduced weight and dimensions of fibre optic sensors, the strong immunity to electro- magnetic interference, the improved environmental resistance and the scale flexibility for small- gauge and long-gauge measurement.
This report provides an overview of the characteristics of the fibre optic sensors for cracking monitoring and the description of the improvements introduced in a sensor recently developed.
The proposed technique does not require prior knowledge of the crack locations, which is a sig- nificant benefit over existing crack monitoring techniques. Moreover, several cracks can be de- tected, located and monitored with a single fibre. An ideal application of the sensor is in the monitoring of flexural cracks in bridges, which may appear at arbitrary locations along the deck, but essentially perpendicular to the spanning direction.
Recently, various researchers have developed fibre optics based crack sensors for concrete structures. Existing optical crack sensors are, however, limited in their applications. For exam- ple, sensing based on fibre breakage
1can distinguish between the presence and the absence of cracking but cannot provide information on gradual structural degradation. Point sensors, based on measurement of intensity loss due to deformation, developed by Ansari and Navalur- kar [1] can detect and monitor the opening of a crack only if the cracking occurs in a small re- gion that is known in a priori. In real structures where the crack location is not known, the sen- sor is not applicable.
Zako et al. [2] used an OTDR to measure the cracking point by Fresnel reflection of four optical fibres, which have been bonded to the surface of a mortar beam with epoxy resin. In addition, the crack propagation in the mortar beam can also be monitored by the breaking sequence of four optical fibres. For this approach to work in a real structure, a very large number of fibres have to be incorporated, which may make the sensing scheme overly costly and impractical.
Gu et al. [3] developed a distributed fibre optic sensor consisting of individual segments spliced on one line. By measuring the Fresnel reflection at each splice between two pieces of fibre, the average strain within each piece can be obtained. Based on the strain reading, the severity of cracking within a certain region can be assessed. An OTDR was employed for interrogation of the sensor signal. Structural monitoring capability of the sensor was evaluated through experi- ments with reinforced concrete beams. For accurate determination of crack location, the splices need to be close. However, if the splices are placed very close to one another, costs will be high and the forwarded signal may drop rapidly with distance (due to the presence of many reflection points), making the sensor inapplicable to real structures where a long sensing length is re- quired [4].
Cai et al. [5] applied the distributed optical fibre sensing technology to detect the cracks in a
small-scale plaster model of an arch dam. By using OTDR the real time monitoring of cracks
can be achieved. The practice of this technology shows that the sensor network bonded to the
downstream surface of the dam will not affect the stiffness of the model, but it must be correctly
distributed.
A sensor for the reliable detection and monitoring of cracks in concrete structure has recently been developed by Leung et al [6] The sensor is based on the principle of distributed optical fibre microbending [7]. An optical fibre is embedded in the concrete element in a “zigzag” shape (Figure 3.1). Using an OTDR, the light intensity distribution along the fibre is measured. Before the formation of cracks, the backscattered signal along the fibre should follow a relatively smooth curve. In the straight portions of the fibre, the small loss is due to absorption and scat- tering. In the curved portion (where the fibre turns in direction), macrobending loss may occur depending on the radius of curvature.
Figure 3.1: Principle of operation of the “zigzag” sensor.
When a crack opens in the structure, a fibre intersecting the crack at an angle other than 90º has to bend to stay continuous. This perturbation in the fibre is very abrupt, and thus can be considered as microbending. This microbending results in a sharp drop in the optical signal.
This intensity loss is detected and located by means of the OTDR equipment. In addition, from the magnitude of the drop, the crack opening can be obtained if a calibration relation is avail- able.
When the optical fibre is bent, the light wave hits the core-cladding interface at an angle smaller than the critical angle and consequently perfect internal reflection is violated. Hence, part of the light is leaked into the cladding and it is lost. A power meter can monitor the loss in forward power intensity. However, if only forward power is measured, the crack location cannot be found. The back-scattered light, which is a small fixed portion of the forward light, is also re- duced at the bend. In addition to loosing intensity, part of the light wave is sent back down the fibre to the source. Optical Time Domain Reflectometer (OTDR) can be used to detect this back-scattered wave. By the time of flight, information one can determine the location of the bend. The intensity loss of the signal can also be found and can be correlated to crack size
4. This technique does not require prior knowledge of the crack locations, which is a significant advancement over existing crack monitoring techniques. Moreover, several cracks can be de- tected, located and monitored with a single fibre. However, the crack direction needs to be known. An ideal application of the sensor is in the monitoring of flexural cracks in bridges, which may appear at arbitrary locations along the deck, but essentially perpendicular to the spanning direction. A method for installing the sensor on existing structures was recently proposed. It in- volves the embedment of the optical fibre into a polymeric plate to form a Plate Sensor
4, which is then attached to the surface of the structure.
Crack sensors designed with the aforementioned principles need to create a fibre bend when the crack forms. There are different ways in which this can be accomplished and some of the different approaches will be reviewed.
Position Change in fibre direction
Fibre optic Signal in and out
Cracks
Fibre bends to stay continuous as crack
propagates
I
After cracking Crack location
Before cracking
3 Novel crack sensing concept based on fibre optics
3.1 Sensing principles
The plate sensor is a polymeric plate with an embedded optical fibre and it works as a trans- ducer; the principle is that once a crack forms in a structural element, the bonded polymeric plate will crack in the same location and direction of the crack.
Special attention must be given to the following aspects:
1) The polymeric matrix of the transducer plate surrounding the optical fibre should allow a reli- able sliding of the fibre when the crack is opening;
2) The process of fabrication should guarantee a similar boundary condition when the micro- bending of the fibre happens;
3) The optical sensor performance is influenced by the optical and mechanical properties of the fibre, and by its inclination angle to the crack.
For the crack sensor to work properly, the bonding between the sensor and the concrete mem- ber has to be assured. If debonding occurs the sensor plate will not be able to pick up the crack opening; likewise, the polymeric material implemented to build the transducer plate should have the necessary brittle mechanical behaviour to break right after cracking occurs in the concrete structure.
Moreover, the polymeric matrix of the transducer plate surrounding the optical fibre should allow a reliable sliding of the fibre when the crack is opening. In addition, to guarantee a similar boundary condition when microbending of the fibre happens, similar interfacial conditions be- tween the matrix and the polymeric coating around the fibre should be assured; any imperfec- tion of the matrix around the fibre could cause unequal light loss intensity for the same crack aperture.
There are many different design parameters that influence the sensor performance, such as the optical properties of the fibre and its inclination angle to the crack, as well as mechanical prop- erties of the fibre, polymeric coating and matrix. In view of the fact that it is desirable to detect small cracks (between 0.2 and 1.0 mm) the sensitivity needs to be sufficiently high. However, to detect many cracks requires that the sensitivity not be too high. Otherwise, only a limited num- ber of cracks could be detected since the dynamic range of any OTDR system is not unlimited.
Therefore, to optimise the design of the crack sensor, the understanding of the optical and me- chanical behaviour of the sensor is required.
Recently, crack simulating specimens, which are 5.08 × 5.08 × 0.95 cm
3epoxy blocks with an optical fibre placed inside a hole (254 μm diameter) at different angles (30º and 45º), were used to perform controlled experimental tests
5, to simulate the measuring power loss of the sensor during the opening of a crack.
In the present research, the plate sensor was made from a thermosetting polyester resin. The
resin curing was studied and different additives were added to the polymer (granite, calcareous,
metakaolin, quartz, river sand and abrasives), to obtain different grades of resistance and brittle
mechanical behaviour of the sensor. The obtained results provide guidelines for modifying the
polymeric material to assure that the plate breaks for thinner or wider cracks. In this way, the
transducer can be designed to assess the crack aperture for different tensile stress along the
crack, following the mechanical tension-softening laws of different kind of concretes.
3.2 Thermal and mechanical properties of the polyester
A medium reactive unsaturated polyester resin (UP) based on an isoftalic-saturated acid (Resi- pal VUP 4686/62) was used to produce the sensor plate by casting moulding. A methyl-ethyl- ketone peroxide initiator (Butanox M50) and a cobalt octoate accelerator were added to the resin to promote curing. As the relative proportions of components in the mixture have great influence on curing itself and on the final sensor plate mechanical properties and transparency, curing experiments were made to achieve the best combinations to be used.
A thermocouple was embedded in different catalyst mixtures to follow-up the UP resin curing and determines the maximum temperature reached (exothermal peak) and curing time. The knowledge of these parameters is extremely important to develop a suitable manufacture pro- cedure, preventing to reach the gel phase during the casting of the sensor plate. High exother- mal peaks and short curing times may also cause undesired resin shrinkage and sensor plate cracking. Furthermore, these tests allow detecting the strong influence that different initiator and promoter proportion has on the cured sensor transparency/opacity.
The curing experiments were carried out using 25 different combinations of initiator and accel- erator at 20ºC. Cobalt octoate concentrated at 10% in a solvent was used as promoter in the tests.
Figure 4.1 shows the exothermal peaks and colours obtained in the tested sensors. As ex- pected, the increment of the amount of initiator and accelerator made the exothermal peak and the sensor opacity to increase and the curing time to decrease. From the components used in the resin formulation, the cobalt accelerator has shown to be the product most affecting the resin transparency and colourless due to its purple colour. For this reason small percentages of accelerator should be used in the resin formulation.
Accelerator (%)
2.5 2.0 1.5 1.0 0.5
2.5
134 130 101 51 47
2.0
132 122 85 47 43
1.5
130 121 73 43 38
1.0
123 112 62 38 32
Initiator (%)
0.5
118 66 48 35 25
Figure 4.1: Exothermal peak determined in the curing process and colour obtained in different combina- tions of accelerator and peroxide in 10 cc. specimens of polyester resin.
The tests also demonstrated that in small volume the initiator peroxide could not by itself assure
the UP resin curing at typical room temperature. The peroxide radical decomposition, which is
necessary to start the polymerisation reaction, is too slow at room temperatures. To speed up
the radical decomposition in controllable way, the volume of peroxides must therefore be in-
creased. However, the pre-mixing of a suitable amount of cobalt accelerator with the UP resin
has proven to be the most effective method to promoting the decomposition of the organic per-
oxide and, therefore, the curing reaction at room temperature.
From the curing tests was conclude that could be possible to produce the transducer plate with the desired characteristics, using volume fractions of initiator and accelerator ranging from 1.5%
to 2.5% and 0% to 1.5% respectively. The curing process became too slow for lower percent- age of accelerator and initiator. Several drawbacks can be reported when high fractions of ac- celerator and initiator are used, such as: considerable shrinkage and high resin opacity.
As it is illustrated in Table 4.1, five different combinations of peroxide and accelerator in the range of percentages established in the curing tests have been tested to produce the casting polyester.
Table 4.1: Combinations of the initiator and accelerator tested.
Material Initiator (%) Accelerator (%)
Pol. 1 2.50 1.50
Pol. 2 2.50 0.50
Pol. 3 2.50 0.05
Pol. 4 1.67 0.33
Pol. 5 1.50 0.50
To gather the necessary parameters and to characterize the mechanical behaviours of the polymeric transducer plate a series of tensile tests were carried out using a servo-controlled universal Instron 4505 testing machine (Figure 4.2), following the recommendations of ISO 527- 110 [8]. The tests were conducted at 20ºC and 40ºC to analyse the influence of the environ- mental temperature on tensile properties.
Figure 4.2: Tensile tests of the polymeric materials.
Because it will desirable to apply sensor plates and bonding adhesives presenting similar me- chanical behaviours, the epoxy resin (Sikadur® 32N) chosen to be use for bonding the polyes- ter plate was also submitted to the same tensile tests for comparison.
Figures 4.3-4.5 show the tensile properties of the sensors made with the five polyester formula- tions studied and of the epoxy resin that shall be applied as adhesive between the bronze disk and the polyester plate. Results obtained from the 96 tests (6 per material) performed are shown in those figures. In the figures U is the maximum value; { is the mean value; − is the minimum value; and
The obtained results show that the tensile properties of the polyester and epoxy resin varied significantly with the temperature. This could result from incomplete cure of the liquid resin, which is controlled by the amounts of peroxide and accelerator. Moreover, it also may be seen that both polyester and epoxy resin presented lower modulus of elasticity and higher strength in tension than any conventional concrete.
A more detailed analysis of the obtained results allows concluding that the polyester sensors
presented a much more yielding and plastic behaviour than the epoxy ones. Thus, for trying to
increase polyester brittleness the resin formulation reference as Pol. 2, which was the one that
presented most similar mechanical behaviour to epoxy, was subjected to several modifications.
One of the modifications made was to use ethanol for reducing the concentration of the octoate cobalt in the accelerator.
A significant change in the UP resin mechanical behaviour occurred when a concentration of 1% octoate cobalt was employed as accelerator (material referenced as Pol. 2a1) in the per- centage defined for Pol. 2. When submitted to tensile testing, this material fails immediately af- ter the peak load being reached with minimal signs of yielding. Figure 4.6 shows the stress- strain relation curve in tension of Pol. 2a1, compared with the original Pol. 2 and Epoxy. Such behaviour probably resulted from the increasing of sub-micron voids in the polyester resin gen- erated during curing by the evaporation of the larger amount of solvent.
200 400 600 800 1,000 1,200 1,400 1,600
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 E poxy
Material
Y oun g' s M od ulu s ( M Pa )
200 300 400 500 600 700 800 900
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 Ep ox y
Material
Y oung' s Modul us (M Pa )
Figure 4.3: Young’s modulus determined at 20ºC (a), 40ºC (b).
0 10 20 30 40 50 60 70
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 E poxy
Material
St re ss at M ax . L oad (M Pa )
8 13 18 23 28 33 38
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 E poxy
Material
St re ss at M ax . L oad (M Pa )
Figure 4.4: Stress at maximum load determined at 20ºC (a), 40ºC (b).
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 Ep ox y
Material
St ra in at M ax . L oad (m m /m m )
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Po l. 1 Po l. 2 Po l. 3 Po l. 4 Po l. 5 E pox y
Material
St ra in a t M ax . L oa d (mm/ mm)
Figure 4.5: Strain at maximum load determined at 20ºC (a), 40ºC (b).
a) b)
a) b)
b) a)
Although identical mechanical behaviour was found between the epoxy and polyester resin ref- erenced as Pol. 2a1, values significantly different of Young’s modulus and strength were deter- mined in both materials. For trying to reduce these discrepancies in mechanical properties dif- ferent additives were added to Pol.2a1. The additives used and their influence on the mechani- cal properties of Pol. 2a1 are described in detail in next paragraph.
Figure 4.6: Example of typical σ-ε curves resulted from uniaxial tensile tests.
3.3 Mechanical properties of modified polymeric materials
To adjust the mechanical properties, the following additives were added to Pol. 2a1 UP resin to produce sensors submitted to tensile testing: four fillers (granite, calcareous, metakaolin, and quartz), two river sand with a classification by size of particles of 200 (0.16 mm) and 80 (0.08 mm) and two kinds of abrasives commonly used to determine the resistance to surface abra- sion, referenced as A1200A and A600.
Figures 4.7 to 4.9 show the results obtained in the 132 tests (6 per material) made. For com- parison the mechanical proprieties obtained for epoxy (Sikadur® 32N), Pol. 2a1 and Pol. 2 have been also plotted in the figures. The mass fraction of additive used in the tested sensors corre- sponds approximately to 10% (Figures 8a to 10a) and 20% (Figures 9b to 11b).
200 700 1200 1700 2200
G ran ite Ca lc ar eo us M et aka ol in S and 20 0 S and 8 0 Qu ar tz A 1200 A 600 Po l. 2 a1 Po l. 2 E poxy
Y oung' s M odul us , M P a
200 700 1200 1700 2200
G ran ite Ca lc ar eo us M et aka ol in S and 20 0 S and 8 0 Qu ar tz A 1200 A 600 Po l. 2 a1 Po l. 2 E poxy
Y oung' s M odu lu s, M P a
a) 10 % of the addition material. b) 20 % of the addition material Figure 4.7: Young’s modulus at 20ºC of Pol. 2a1 with different additions.
Epoxy 40 Pol. 2 - 40 Pol. 2 - 20 Pol. 2a1
Epoxy 20
σ (MPa)
ε (mm/mm)
0 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0
10
20
30
40
50
Table 4.2 gives a summary of the results in comparison to Pol. 2a1. This table lists the incre- ment of Young’s Modulus and, respectively, the reduction of stress and strain values obtained at maximum load. The Pol. 2a1 has a Young’s Modulus of 813 MPa, with a stress and strain at maximum load of 41 MPa and 0.07 mm/mm respectively. (e.g. in comparison to Pol. 2a1, the Epoxy exhibits mean increment of Young’s modulus of 43% and mean reduction of stress and strain of 41% and 59% respectively).
0 10 20 30 40 50 60
Gr an ite Ca lc ar eous M et aka ol in S and 200 Sa nd 8 0 Qu ar tz A 120 0 A6 00 Po l. 2 a1 Po l. 2 Ep ox y
St re ss a t Ma x. L oa d, MPa
0 10 20 30 40 50 60
Gr an ite Ca lc ar eous M et aka ol in S and 200 Sa nd 8 0 Qu ar tz A 120 0 A6 00 Po l. 2 a1 Po l. 2 Ep ox y
St re ss a t Ma x. L oa d, MPa
a) 10 % of the addition material. b) 20 % of the addition material
Figure 4.8: Tensile stress at maximum load at 20ºC of Pol. 2a1 with different additions.
0.00 0.02 0.03 0.05 0.06 0.08
G ra nite C alc ar eo us M eta ka olin S and 200 S and 80 Qu ar tz A 120 0 A6 00 Po l. 2 a1 Po l. 2 E poxy
S tra in a t M ax. L oa d, m m /m m
0.00 0.02 0.03 0.05 0.06 0.08
G ra nite C alc ar eo us M eta ka olin S and 200 S and 80 Qu ar tz A 120 0 A6 00 Po l. 2 a1 Po l. 2 E poxy
S tra in a t M ax. L oa d, m m /m m
a) 10 % of the addition material. b) 20 % of the addition material
Figure 4.9: Tensile strain at maximum load at 20ºC of Pol. 2a1 with different additions.
Table 4.2: Mean comparison to Pol. 2a1, values in percentage of the increment of Young’s modulus, E
t, and the reduction of stress and strain at maximum load, σ
tand ε
t.
10% of DVMA 20% of DVMA Material
Et
σ
tε
tσ
tε
tσ
tGranite 50 17 48 77 40 67
Calcareous 64 2 46 79 26 67
Metakaolin 27 39 59 68 49 76
Sand 200 75 26 54 96 46 75
Sand 80 98 48 73 129 57 83
Quartz 42 7 26 154 27 68
A1200 40 33 62 128 38 78
A600 63 12 39 144 17 57
Mean 57 23 51 109 38 71
Maximum 98 48 73 154 57 83
As can be seen in Table 4.2, the size of particles affected significantly the UP resin mechanical behaviour. The overall effect of the change in size of particle becomes minimal when the Den- sity Volume of the Material Added (DVMA) is increased because the polymeric resin ap- proaches the Maximum Incorporable Quantity (MIQ). Values of columns 2 and 5 are in percent- age of the increment of Young’s modulus, and values of columns 3-4 and 6-7 are in percentage of the reduction of stress and strain at maximum load.
It can also be seen that additives referenced as Sand 80 and A 1200 at 10% of DVMA are those presenting increment and reduction values more similar to those above mentioned for Epoxy. In fact as Figure 4.10 shows, the stress-strain curve of Sand 80, this is the material with least ten- sile stress and strain at maximum load (river sand with a classification by size of particles of 0.08 mm). This material exhibited mean values for the Young’s modulus and stress and strain at maximum load of 1865 MPa, 18 MPa and 0.01 mm/mm, respectively. These values approach quite well the results obtained for the epoxy bonding adhesive.
With the exception of Sand 80, abrasive A1200 and metakaolin all other materials added to Pol.
2a1 exhibit ductility after the elastic region in the tensile tests.
0 10 20 30 40 50 60
0 0.02 0.04 0.06 0.08 0.1 0.12
T ensile strain, mm/mm
Tensile stress, MPa
Pol. 2a1 Sand 80
Figure 4.10: Stress-strain curves at 20ºC of Pol. 2a1 with river sand 80 added.
3.4 Sensor plate fabrication process
The sensor plate fabrication was the most important issue to be solved before the determination of calibration curve of the sensor.
The sensor plate fabrication procedure started as a continuation of the research done by Olson [4]. In this previous research, the sensor was applied to existing structures as well as to new buildings in the form of the sensor plate. The description of the principle and graphical illustra- tion of the design of the sensor plate were published together with the process and details in the fabrication. Nevertheless, it was found that the fabrication process which involved the pouring of polymer over an optical fibre wrapped around some pegs placed in zigzag in a cast made of polypropylene was too complicated.
The experiments done to propose the new fabrication process were based on the improvement
of the design of the cast to avoid unnecessary expenses allowing the manufacture of sensor
plates with different sensitivity using the same cast. Figure 4.11 shows the original cast pro-
posed by Olson [4] and the equipment developed in this work to fabricate the plate sensors. The
benefits of the modifications start from cast levelling, auto sealing and releasing, including eas-
ier fabrication and reproducibility control with different operators or in different locations. The key of the fabrication process is the implementation of a steel wire (from guitar n.8) which is coated with releasing agent, and wrapped around the pegs in a zigzag manner. The wire was held tight before casting.
The implementation of steel wire rather than the optical fibre allows the manufacturing of a transducer plate in isolation, with the optical fibre post-installed as a secondary step. The fabri- cation process is easy and repetitive (well defined), and it is easy to produce a stock of sensor plates with the same mechanical properties under humidity and temperature control.
Figure 4.11: The original cast proposed by Olson [4] and the and the equipment developed in this work to fabricate the plate sensors.
Figure 4.12 shows the first prototype of sensor plate produced on April 2004. It corresponds to the fibre angle of 45º. The first prototypes were fabricated with the depth of 2 mm and two sides perfectly plane with shiny surfaces. The objective was to establish the process to manufacture solid matrix sheets of polyester without evidence of air bubbles.
Figure 4.12: The first prototype of sensor plate (April 2004).
To get solid matrix in polyester sheets several problems were encountered. These problems were: the inclusion of air when the substances are going through a certain mixture process, the pouring process in a recipient that contains air, the air in the gap when the polyester resin is covered. The solution to solve the first problem was combining a sonic vibration process of the resin with the extraction of air with a vacuum system.
3.5 Experimental analysis of the bond properties
The establishment of a strong bond of the plate to the concrete surface is one of the most im-
portant pre-requisites for the successful performance of the sensor. In a successful bonding
application the strength of the substrate surface is not the only important concern. It is equally
important to ensure a clean and dry surface, the absence of contaminants and the best profile
that can be achieved.
Surface blasting with hand held mechanical equipment was used to attain an uniform surface texture and to remove the laitance (the weak alkaline surface residue), dirt and dust until coarse aggregates are exposed.
Any oil and/or grease contamination on the concrete must also be removed prior to bonding.
Cleaning concrete with steam is an effective mean of removing heavy deposits of oils and greases. It consists of cleaning the surface with a jet of high-pressure steam sufficient to re- move contaminants.
The importance of cutting back the concrete to a clean sound surface cannot be overempha- sized since adhesion relies partly on mechanical interlock by penetration of the surface pores, and partly on the physical forces of attraction to clean high energy aggregate surfaces [9].
Moreover, to have better compatibility with the adhesive a surface pre-treatment to the sensor plate is recommended. An effective pre-treatment of the plate includes a slight roughening with sand paper and cleaning with pure acetone (C3H60), to remove any contaminant like oils, dirt and mould release agents.
To empirically investigate the global bonding behaviour of the sensor plate, initially a modified three-point bending test was carried out over three beams of plain concrete of low resistance (Figure 4.13).
a) Specimens with polymeric plate b) Bending test c) LVDT measurement Figure 4.13: Modified three-point bending tests.
To obtain a localized crack opening at mid-span in all tests, a notch of 200x3 mm was intro- duced at mid-span on each side of the beam. The material of the plate was Pol. 2 and adhe- sives corresponding to Pol. 1, Pol. 2 and Pol. 5 were employed. As can be observed in Figure 4.14 the bonding between the sensor and the concrete was properly assured when Pol. 2 was used as adhesive since the cracks in the plate and beam seems to occur at the same location.
Nevertheless, the results for Pol. 1 and Pol. 5 show peeling of the sensor plate caused by the weak bonding and/or the mechanical properties of the plate. Because of these results, it was proposed to measure the adhesion and figure out ways to improve it.
Figure 4.14: Specimens after failure.
A number of tests were performed in the laboratory for measuring adhesion. In this research pull-off tests were used. To simulate the bonding condition of the sensor plate to the concrete, the pull-off test (Figure 4.15) involves the following steps. Firstly a bronze disc is glued to a small part of a sensor plate. Then, the disc and polyester plate is bonded to a partial core of concrete drilled perpendicular to the surface. The disc is pulled off in direct tension using a hy-
Pol. 1 Pol. 2 Pol. 5
draulic precision instrument with controlled loading rate (1 MPa/s). The peak force in kN is re- corded and transformed to adhesion/tensile strength by dividing with the core area. To compare an adhesive material which spreads and adheres well to the substrate with an adhesive when cured is a highly cross-linked structure possessing significant cohesive strength in a short time, two epoxies (MBrace Resin 55 and Sikadur® 32N) and two polyesters (Pol. 2, and Pol. 2a1) were studied as adhesive between the polyester and the concrete.
Polyester plate
adhesive Adhesive
Polyester plate Force
a) Pull-off test. b) Cohesive failure
in the concrete. c) Adhesive failure at adhesive/concrete interface.
d) Adhesive failure at poly- mer/adhesive interface.
Figure 4.15: Failure modes in pull-off test.
Figure 4.16 illustrates the results of the 72 pull-off tests carried out in a fixed-alignment adhe- sion tester (DYNA, T.F. 16kN), illustrated in Figure 4.17. In this figure Su is the pull-off stress at maximum load of three different concretes (C1, C3 and C5). Theoretically the tensile strength corresponds to the applied stress when cohesive failure occurs in the concrete. In the course of tests when the epoxies were used, cohesive failure was always observed. The polyester resins, in spite of being more resistant to tension compared to epoxies, do not achieve enough me- chanical interlock by penetration into the concrete.
0 1 2 3 4 5 6 7 8
Su-Pol.2 Su-Pol.2a1 Su-Mbrace Su-Sikadur Su-Pol.2 Su-Pol.2a1 Su-Mbrace Su-Sikadur Su-Pol.2 Su-Pol.2a1 Su-Mbrace Su-Sikadur
Stress at Max. Load Vs. Resistance to tension, MPa
Concrete C1 Concrete C2 Concrete C3
Figure 4.16: Pull-off stress of adhesion/tensile strength.
Pol. 2a1 has longer curing time compared to Pol. 2, and consequently has more time for im-
pregnation to enhance the adhesion. The epoxies with long time of curing (24 hours) facilitate
the flow of the material over the concrete surface together with increased molecular mobility
which enhances the wetting potential. For the MBrace Resin 55, failure occurs along the
resin/plate interface in a number of tests. Through the implementation of the sensor plate for the
detection and monitoring external flexural crack opening, on reinforced concrete beams, and
masonry slabs, the epoxy Sikadur® 32N performed the best as the adhesive, as it had a strong
bond with the polyester plate.
a) After Pull-off test b) Specimen rectification c) Specimen preparation
Figure 4.17: DYNA, T.F. 16kN, specimen preparation and rectification in the pull-off tests.
4 Calibration of the Plate Sensor
There are different ways in which the calibration of prototypes of crack sensor can be made. In this study, instead of using an Optical Time Domain Reflectometer (OTDR) a power meter to measure the loss in forward power transmission was used. The reasons for this choice were:
the simplicity and accuracy of the power meter and the fact that the location of the crack was known in advance.
Figure 5.1: Specimen used in the calibration tests
The basic principle of operation for the crack sensor is based on intensity variation of the optical power within the optical fibre due to the initiation and opening of cracks. Crack sensors de- signed with the aforementioned principles need to create a fibre bend when the crack forms.
Figure 5.1 illustrates the specimen used in the calibration tests. The specimen size is represen- tative of the conditions predominant in the sensor, i.e. the span between the intersections of the fibre in a “zigzag” shape.
Table 5.1: List of the equipment used in the calibration tests.
Laser 1550nm stabilized Light Source
Optical Head Optical Head 0.38 to 1.15µm (MA9412A) Power Meter Optical Power Meter Model ML9001A Transducer Amplifier ATA-101-Analog Transducer Amplifier LVDT 100-MHR Voltmeter Multi-meter 45-05 (GPIB interface)
Several calibrations were performed to understand and to predict the behaviour of the sensor under different operating conditions. The calibration tests were done using the mechanical simulator of cracking developed by Olson [4]. This simulator was designed so that the crack opening could be measured using a Linear Variable Differential Transformer (LVDT). The test- ing stage has a fixed part and a moving part. The moving part rests on two hardened steel rods and has four precision ball bearings to keep it aligned and moving with very little friction. The specimen is clamped onto the stage by tightening the screws. The engine turns the reaction nut that moves against the reaction block and therefore pulls the main rod that opens the testing stage.
Figure 5.2: Photograph of the mechanical simulator of cracking developed by Olson [4].
The experimental set-up can be seen in Figure 5.2. Table 5.1 summarizes the equipment used.
It must be noticed that this setup is monitoring the crack opening using forward power transmis- sion (instead of an OTDR). The LVDT measurement and the power loss measurement are of primary interest, to determine the curves of calibration of the corresponding intensity loss versus crack aperture.
Laser
Power Meter Transducer Amplifier
Computer
Optical Head Optical Fiber
LVDT
GPIB Cable Interface
Voltmeter
Figure 5.3: Experimental set-up for calibration tests.
The measurement data is gathered in digital form in the computer using LabVIEW. The optical power loss measurement path is in dashed lines and grey shading. The displacement meas- urement path is the other path. The LVDT path and optical measurement path were controlled with a GPIB interface. The LabVIEW interface controls the GPIB card (PCI-1200) as well as collects the data from it.
4.1 Calibration curves
Several calibrations were performed to understand and to predict the behaviour of the sensor under different operating conditions, the procedures and results were deeply commented in a previous report [10]. In this report additional results are presented regarding the observed per- formance characteristics [11]. such as input and output range, resolution, noise, and precision.
Figures 5.4 to 5.6 show the representative results of the calibration tests of 45º specimens, with Pol. 2a1, sand 200 and abrasive A1200, respectively (five calibrations per specimen with a crack opening between 0.20 mm and 1 mm).
The maximum intensity loss is close to 20 dB, and the change of material seems not affect its behaviour, nevertheless the reproducibility is better with abrasive A1200. High intensity loss limits the application of any OTDR through the monitoring of cracks; if this sensor is imple- mented, at this moment, with the best OTDR it will limit the monitoring to only one crack.
Although the proposed procedure to build the Plate Sensor allows the sliding of the optical fibre, a small instability after 1mm of crack aperture was detected.
Power Meter Voltmeter
Laser
Crack aperture, mm
Instensity loss, dB
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 4 8 12 16 20 24 28 32 36
Figure 5.4: Calibration of Pol.2a1 sensor at 45º
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 4 8 12 16 20 24 28 32 36
Crack aperture, mm
Instensity loss, dB
Figure 5.5: Calibration of Sand 200 sensor at 45º
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 4 8 12 16 20 24 28 32 36
Crack aperture, mm
Instensity loss, dB
Figure 5.6: Calibration of abrasive A1200 sensor at 45º
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 2 4 6 8 10 12 14 16
Crack aperture, mm
Instensity loss, dB
Figure 5.7: Calibration of Pol2a1 sensor at 30º
The instability corresponds to the non-uniform sliding of the optical fibre when the coating of the fibre starts to be deformed and stripped by the friction with the material that is holding the opti- cal fibre at the transducer’s failure plane. The asymmetry of the hole at this location depends on the inclination of the fibre and influences the noise exhibited by the sensor. The noise is directly dependent on this inclination [11].
Figures 5.7 and 5.8 show the results of the calibration representative of sensors with Pol2a1 at 30º and 15º, respectively.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Crack aperture, mm
Instensity loss, dB
Figure 5.8: Calibration of Pol2a1sensor at 15º
The sensors at 30º and 15º measure the crack opening starting from 0.20 mm and 0.40 mm respectively. The maximum intensity loss is close to 12 dB and 1.2 dB respectively. The behav- iour of both sensors is good. The sensor at 15º has lower maximum intensity loss, and can be used with an OTDR to determine the position of cracks greater than 0.40 mm along concrete elements. The sensor at 30º has good resolution, but its intensity loss is considerable and could limit the number of cracks to be monitored.
Figures 5.4 to 5.8 exhibit good precision of the sensor producing the same results when meas- uring the same quantity under the same operating conditions, however, when the calibration curves from different sensors are compared at almost same crack direction, it was found that the precision of the Plate Sensor is significant affected by changes in the crack direction. To be consistent with this phenomenon it was done a deep statistic analysis of results about the cali- bration of the sensor. The objective of the statistic analysis was fitting a calibration model for the sensor, and be able to determine the behaviour of the intensity loss versus crack aperture of the sensor at different crack direction.
4.2 Statistics for the curve calibration
The need for the statistical analysis of curve calibration arises from the observation that the
calibration curves show large variations when the angle between the crack and is slightly
changed. Figure 5.9 illustrates the physical phenomenon of bending for a given optical fibre
based on the results of described in the current and in the following section. The idealization of
the bent fibre within the sensor plate (Figure 5.9) exhibits a constant radius of curvature that is
dependent on the angle between the crack and the fibre, over a particular length of a bent opti-
cal fibre which depends on the crack opening. This idealization works only between the meas-
urement range of implemented version of the sensor plate, outside the range the optical fibre is
not fully constrained allowing a variable radius of curvature which is reflected in variations of the
optical power loss versus crack opening.
Radius of curvature (R)
Length of a bent optical fiber (Plnorm)
Figure 5.9: Idealization of physical bending for a given fibre within the sensor plate (not to scale).
After employing several approaches to fit single curves of calibration to three versions of sen- sors (15º, 30º, 45º) it was found that the calibration curves can be simply described by Equation 1, which is the product of the guided wave’s amplitude-coefficient 20/ln(10). α (R) for a given fibre curvature with a lognormal cumulative probability distribution function (Plnorm) to represent the total length of bent fibre for a given crack opening.
( μ σ ) ( ) ( ) α Plnorm ( , μ , σ )
10 ln , 20 , ,
Loss x R = ⋅ R ⋅ x (1)
where, x is the crack opening, R is the radius of curvature of the bent fibre, µ, and σ, are the mean and standard deviation parameters which define the mean and the variation of crack opening parameters in the Plnorm distribution that gives the total length of bent fibre.
The bases of the statistical analysis are:
1) The coefficient of variation of length distribution of bent fibre is the independent variable; it is determined by the division of the standard deviation (σ) with the mean (µ); parameters which characterize the variation and the mean crack opening parameters of the bent optical fibre’s length distribution. It is important to note that the coefficient of variation is simply τ / μ for the best fit Plnorm function, and is not derived from statistical parameters.
2) The amplitude-loss coefficient of the guided wave (in dB/mm) is a function of radius of curva- ture of the bent fibre; it affects the sensitivity only. A mean radius was accepted that would re- main constant when the crack is opening between the measurement range; the radius might change for different angle between crack and fibre. The radius of curvature was related to the coefficient of variation of length distribution of bent fibre.
3) The lognormal cumulative probability distribution function (Plnorm) represents the total length of the bent optical fibre that is a function of the crack opening. It affects the range of crack open- ing and the sensitivity in terms of the mean and standard deviation crack opening parameters.
Both parameters are affected by the coefficient of variation of length distribution of bent fibre.
The implementation of a probability distribution function in Equation 1 is not related to statistics associated with the total length of the bent optical fibre that is a function of the crack opening.
This section has the objective to make a statistical analysis of the parameters characterizing Plnorm function and the radius of curvature. The names of the parameters characterizing Plnorm function were defined following the concept of a probability function.
The statistical analysis involves the definition of a least-square function (Equation 2) for deter-
mining goodness of fit among 54 curves of calibration. The calibration curves were obtained
from a selection of repeated calibration process with transducers of Pol. 2a1 at 15º, 30º and 45º.
There were 19 calibration curves at 15º, 16 at 30º and 19 at 45º. The selected range of crack opening was not bigger than 2 mm. The nonlinear approximate solution of Equation 2 by Quasi- Newton process gave the solution for all R, μ , and σ can be found in León [12].
( ) ( ) Plnorm ( , , ) 0
10 ln
1
20
0
2
⎟⎟ =
⎠
⎜⎜ ⎞
⎝
⎛ − ⋅ ⋅
∑
−= n i
i
i
R x
OPL α μ σ (2)
Figure 5.10 shows the optical power loss in dB/mm versus the bend radius in mm. The dots correspond to the loss for a given R calculated from Equation 2. It is observed that the mean constant bend radius (R) is in the range of 1.8 to 5.7 mm corresponding to optical power loss of 0.5 to 37 dB/mm. The bend radius is smaller when sensors at 45º or 30º were evaluated, in comparison to the sensors at 15º. For sensors at 45º and 30º the bent radius is approximately between 2 and 3 mm, while for the sensor at 15º is between 5 and 6 mm.
1 2 3 4 5 6 7
0.1 1 10 100
Bend radius, mm
Optical power loss, dB/mm
Solutions of R in Equation 2
Equation
ln ( ) ( ) 10 ⋅ α
R20
Figure 5.10: Solution of mean constant radius of curvature R of the bended fibre.
In Figure 5.10, the results are in good agreement with the natural behaviour of the sensor from previous research [4, 13]; because, to achieve high optical power loss it is always necessary to develop a reduced radius of curvature by increasing the angle between the fibre and the crack.
Figure 5.11 shows the mean constant radius of curvature for the bent fibre versus the coefficient of variation of length distribution of bent fibre. Notice that for the prediction of the mean value a fit regression line is shown with the confidence and prediction intervals at 95% of probability.
Confidence limits are interval estimates for the mean value. Interval estimates are often desir- able because the mean’s estimate varies from sample to sample. Instead of a single estimate for the mean answer such as the fit regression line, a confidence interval generates a lower and upper limit for the mean. The interval estimate gives an indication of the uncertainty in our esti- mate of the true mean value. The narrower the interval, the more precise is the estimate. While confidence interval estimate presents population characteristics, the prediction interval esti- mates future values from present or past background samples taken. The prediction interval attempts to determine future values with probability (1- α )100%, just as in the confidence inter- vals.
The difference between confidence and prediction interval is that the confidence interval is re-
lated to present values, and the prediction interval with future values; in Figures 5.11 to 5.13,
both use the same probability of 95% ( α /2 = 0.025). The confidence interval is an interval based
in the mean answer (the fit regression line) while the prediction interval is an interval based on past or future values of the random variable. The probability will establish the limits for certain parameter: the mean answer in the case of confidence interval, while for prediction interval is a future value of the random variable. Through the analysis is used the inverse cumulative prob- ability student’s t distribution, both, normal and t-distribution become equivalent by analogy be- tween the normal and ji-square distribution, t-distribution is used as discrete function for small group of sample representatives of the population (52 = n -2, in our case), the normal distribu- tion is a continuous function for large group of samples equivalent to the population. The t- distribution becomes similar to the normal when n increased to the infinite.
Mean bend radius, mm
Coefficient of variation, adim Prediction interval
Confidence interval
Fit regression line
0.5 1 1.5 2 2.5 3
1 2 3 4 5 6 7
Figure 5.11: Mean constant radius of curvature of the bended fibre.
From Figures 5.11 to 5.13 it can be concluded that the variables R, µ and σ have a relation with the coefficient of variation of length distribution of bent fibre. To achieve good precision and ac- curacy for a proposed calibration model, it is important that the model should predict future val- ues with a variable of probability associated with the coefficient of variation.
0.5 1 1.5 2 2.5 3
0.4 0.55 0.7 0.85 1 1.15 1.3 1.45 1.6
Mean parameter, mm
Coefficient of variation, adim Prediction interval
Confidence interval Fit regression line
Figure 5.12: Mean crack opening value of length distribution of the bended optical fibre.
Figure 5.12 shows the mean crack opening parameter of the bended optical fibre’s length distri-
bution versus the coefficient of variation of length distribution of bent fibre. As previously, the
figure includes the fit regression line with the confidence and prediction intervals at 95% of probability.
Figure 5.13 shows the standard deviation crack opening parameter of bent optical fibre’s length distribution versus the coefficient of variation of length distribution of bent fibre. As previously, the figure includes the fit regression line with the confidence and prediction intervals at 95% of probability.
0.5 1 1.5 2 2.5 3
0.7 0.85 1 1.15 1.3 1.45 1.6 1.75 1.9 2.05 2.2
Standard deviation parameter, mm
Coefficient of variation, adim Prediction interval
Confidence interval
Fit regression line
