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The paper focuses on damage detection of civil engineering structures and especially on concrete bridges. A method for structural health monitoring based on vibrational measurements is presented and discussed. Experimentally identified modal parameters (eigenfrequencies, mode shapes and modal masses) of bridge structures are used to calculate the inverse stiffness matrix, the so-called flexibility matrix. By monitoring of the stiffness matrix, damage can easily be detected, quantified and localized by tracking changes of its individual elements. However, based on dynamic field measurements, the acquisition of the flexibility matrix instead of the stiffness matrix is often the only choice and hence more relevant for practice. But the flexibility-based quantification and localisation of damage are often possible but more difficult, as it depends on the type of support and the location of the damage. These issues are discussed and synthetized, that is an originality of this paper and is believed useful for engineers in the damage detection of different bridge structures. First the theoretical background is briefly repeated prior to the illustration of the differences between stiffness and flexibility matrix on analytical and numerical examples. Then the flexibility-based detection is demonstrated on two true bridges with real-time measurement data and the results are promising.

Nowadays the traffic is permanently increasing and many bridges are operating beyond their initially planned lifetime. A high number of reinforced concrete bridges were built after the Second World War and are suffering from ageing and today’s traffic load. But their safety must be guaranteed and efficient methods for structural health monitoring and damage detection are urgently needed.

Traditional visual inspections are cost intensive but not always sufficient, because they investigate only the surface. Small cracks for instance covered by coating, as well as corroded internal tendons or reinforcement may remain undiscovered.

Dynamic testing can provide an alternative or addendum for damage detec- tion, where the structure is either excited by a measured input or by unmeasured ambient forces such as wind or traffic. The modal parameters can then be identi- fied and changes of eigenfrequencies, mode shapes, damping ratios and modal masses may be caused by damage that reduces before all the stiffness and its dis- tribution.

Based on identified modal parameters, transfer functions can be computed, which describe the relationship between input (normally forces) and output (often accelerations) signals of a system [_{pq} is the transfer function between a force at DOF p and a system answer at DOF q. If the frequency of the input force is tending towards zero, it is equivalent to static loading and the transfer function [G] becomes equal to the inverse stiffness matrix, the so-called flexibility matrix. Theoretically this latter allows calculating the stiffness matrix just by inversion. However, this inversion is only feasible, if the quality of the measurements is good and the flexibility matrix is calculated based on a complete set of N identified modes. But in most practical cases of real bridges, the number of clearly identifiable modes is typically much lower than the number of measured DOFs. A decrease of the number of DOFs to the number of observable modes typically leads to an insufficient spatial resolution of the stiffness matrix.

Over the last years, several sophisticated approaches were investigated for damage assessment based on flexibility that can be found in literature.

In Yan and Golinval (2005) [

Duan et al. (2005) [

In a model updating approach, Perera et al. (2007) [

A multi-criteria approach for damage assessment of beam and plate structures was proposed in Shih et al. (2009) [

Reynders and De Roeck (2010) [

Another variation of flexibility matrix was proposed by Yan et al. (2010) [

Weng et al. (2013) [

Furthermore, Chen et al. (2014) [

Masoumi et al. (2015) [

Stutz et al. (2015) [

Along these lines, it is shown that flexibility-based methods are mostly validated by numerical models or laboratory tests. Far more seldom a paper dealing with in-situ tests at real bridge structures can be found. The obvious reasons may be the high costs and logistic effort for such tests as well as the fact, that a still used structure cannot be artificially damaged for test purposes. On the other hand, a real structure has to be monitored over several years or even decades in order to see effects of accumulating damage. Only the demolishing of a structure may offer the opportunity for extensive tests. But for the sake of transfer from academic researches to practical application, such real-life tests are indispensable. Therefore, a relevant contribution of this paper is the performance of in-situ tests and the analysis based on flexibility for two bridges in Luxembourg.

The present paper discusses practical possibilities and limitations of damage localisation based on the flexibility matrix. The necessary theoretical background is repeated and it is shown that an inversion of flexibility matrix to obtain the stiffness matrix with sufficient resolution is often impossible in practice, as the number of observable measured modes is limited. However, from vibrational measurement, the monitoring of identified flexibility matrices may be useful for damage detection. The efficiency depends on several factors, namely the bearing conditions and the influence of environmental temperature.

After some numerical examples, two real bridges are examined in the following. The first bridge is a new real structure without damage that the influence of temperature on stiffness is very important. The second real structure is a pre- stressed concrete bridge. Increasing artificial damage in multiple steps is identified and localized correctly by the changes in the flexibility matrix.

Cracking of concrete leads to reduction of local rigidity or stiffness. Hence, iden- tification and localisation of stiffness changes compared to the healthy reference state is therefore a logic strategy for damage detection. Structural health moni- toring may therefore be done by permanent monitoring or by repeated tests at distinct time intervals.

Static loading tests with the analysis of deflection line show a long tradition in civil engineering. However, these tests may become cumbersome, as a bridge should be loaded for at least a few hours by known mass until a steady state is reached. During this time, the traffic must be stopped that is inconvenient and may generate considerable costs. Additionally, heavy load testing (going beyond the service loading) may increase already present damage. Therefore, alternatives or amendments are welcome, e.g. dynamic tests either by permanently or periodically installed monitoring systems.

The discretisation of a continuum, e.g. by the finite element method, including N degrees of freedom (DOF), leads to the following differential equation of motion.

where [M], [C] and [K] is respectively the mass-, damping- and stiffness-matrix.

The Frequency Response Function (FRF) matrix

It is obvious that for

The partial fraction decomposition of this transfer function matrix can be formulated as follows (cf. [

where index r refers to an individual mode.

The last equation reveals that the FRF matrix can be represented by the ensemble of eigenvalues _{r}, which is called Modal A. It can be obtained from the state space formulation of the system as described by Wang et al. (2011) [

By evaluating Equations (3-2) and (3-3) for the case

For the special case of proportional damping, i.e.

The quantity _{r} = 1, this factor is implicitly con- tained in the UMM-normalized modes-shapes.

For low damped systems as for instance bridges, it is common to assume proportional damping and to use this simplified formula for the calculation of the flexibility matrix.

Theoretically, the stiffness matrix can hence be calculated by just inverting the flexibility matrix. But if the modal parameters were identified experimentally, the number of well identified modes M is often smaller than the number of measured DOFs N, as higher modes are more difficult to excite and to measure with precision. Hence, the sum in Equation (3-3) is truncated after r = M ≤ N. Due to the square of angular eigenfrequency in the denominator, the contribution of higher modes to the flexibility is quite small compared to lower modes. Therefore, adequate results for [F] may be obtained even with M < N.

Since the flexibility matrix is calculated from the dyadic products of M mode shape vectors, a N × N flexibility matrix has the rank M. If M < N, the resulting flexibility matrix cannot be inverted, meaning that the stiffness matrix cannot be calculated using this approach. Nevertheless, the identified flexibility matrices can still be useful for damage localisation for typical bridge structures. It is shown in the following by several numerical and analytical examples as well as in-situ measurements on two real bridges.

In this section, three theoretical examples with different damage locations are analysed to discuss the usability of the flexibility matrix for damage detection and localisation. It involves a simply supported concrete beam, a cantilever beam and finally a continuous beam. The beam models are simulated in ANSYS and consist of BEAM3 elements (2 translational and 1 rotational DOFs). The following material properties and geometry were used:

・ Young’s modulus: 3 × 10^{10} N/m^{2};

・ Poisson’s ratio: 0.18;

・ Density: 2300 kg/m^{3};

・ Cross section: 0.3 × 0.3 m^{2};

・ Length: 4 m.

The beam model consists of 20 BEAM3 elements with N = 63 DOFs in total [

Different damage scenarios are simulated by reducing the bending stiffness EI by 5% and 30% respectively. The damage is provoked in the middle (elements 10 and 11) and in an eccentric location (elements 4 and 5) as indicated in

For every damage state, the flexibility matrix was calculated according to Equations (3-4) with the mode shapes simulated in ANSYS and normalized to Unit Modal Mass (UMM). Like mentioned above the model contained 63 DOFs in total, where 3 are constrained due to the supports. Hence a set of 60 modes was considered for the 60 × 60 flexibility matrix.

A diagonal element of the flexibility matrix can be interpreted as the deformation at the DOF where a single unit force (1N) is applied. The diagonal elements are shown in

Additionally, the contributions of the first 10 bending modes in the flexibility matrix are plotted as dashed lines. It is obvious that the first two bending modes have predominant influence on the flexibility and that the influence of B2 is already much smaller than the one of B1. As the contribution to the flexibility from the third mode seems insignificant, it seems even possible to attain a good approximation for the flexibility matrix, if the sum in Equations (3-4) is truncated after r = 2. This is important, because in the experimental modal analysis it is impossible to excite and measure accurately all modes as will be shown in sec- tion 0.

The approach to use only lower modes for the flexibility matrix calculation is convenient for the experimental procedure in practice. The diagonals of the flexibility matrices based on the first 2 bending modes are presented in

E30% as well as in the middle as M5% and M30%) are also highlighted in

The difference between the flexibility matrices of a damage state and the intact state is presented in Equation (4-1) [

These differences are shown in

simply supported beam, it is possible not only to detect the presence of damage, but also to localize it based on the flexibility evaluation.

In order to improve the visualisation of changes especially due to small dam- ages, a relative calculation of the differences between the flexibility matrices can be useful. Following row number i and column number j in the flexibility matrices, the relative difference is defined as:

Considering in _{1}. Damage is introduced in section 2 with length b by reducing the moment of inertia to I_{2}.

For each section the deflection w(x) is calculated analytically by integration of the following equation:

Taking into account the boundary conditions, the bending line of the three sections is formulated:

At a discrete location x the flexibility can be determined for a chosen set of a and b:

In the following, the same material and cross section properties as for the

simple beam in the first example are used. By choosing

Alternatively, a finite element model of 20 BEAM3 elements, with the same properties as for the simple supported beam, is examined as for the cantilever beam but here by using ANSYS. The flexibility matrix is calculated based on UMM normalized mode shapes in accordance with Equation (3-4).

Two damage scenarios are investigated again by a 30% reduction of Young’s modulus: first, near the fixed support named B30% and second, near the free end named E30% as presented in

continuous line. Again it is obvious that the contribution of the first bending mode to the flexibility matrix is by far predominant and higher modes have no visual effect and so are more or less negligible. Therefore, only the first two modes are taken into account for flexibility calculation.

The absolute and the relative differences of the diagonal elements of the flexi- bility matrix are shown in

we see in the relative differences a deviation if we include all modes or only two bending modes.

The last theoretical case is investigated for a continuous beam as sketched in

In

From the two graphs in

・ Damage within the smaller span (EL4): The absolute difference shows a dominant peak at element 4 and another second peak in the middle of the long span (elements 14 - 15). This second peak is considerably lower in the relative difference in

・ Damage on the intermediate support (EL8): The absolute difference in

・ Damage within the long span (EL11, EL15, EL18): The absolute difference points out accurately the locations of damages. The relative difference shows again false side lobes, especially for (EL11), (EL18).

・ Damage on end supports (EL1, EL20): The absolute difference increases near the end supports, while the relative difference reaches a maximum at node 2 and 20 respectively. Between nodes 8 to 16 there is no visible difference. Therefore, good damage detection is not really feasible taking into account the noise on real measurements.

Summarily, it is stated that the relative difference works well, but may lead to inaccurate interpretations when damage occurs close to the supports.

In the following the flexibility based results according to Equation (3-4) are compared to the detection based on the stiffness matrix, provided the latter is known from an FE-model. The absolute and relative differences are calculated for the stiffness matrix for every state and are presented in

Of course the stiffness matrix allows a straightforward damage localisation, even for damage close to the intermediate support (EL8).

The above illustrations show that damage detection based on the flexibility is feasible, though the localisation may become difficult close to the supports due to the small absolute values. But the presence of damage is still detectable by an increase of flexibility.

For a cantilever beam damage can be localized by the shape of the flexibility variation: the absolute difference starts to deviate from zero at the location of damage then increases monotonously toward the free end. On the other hand, the relative difference shows a peak close to the damage location, if it is close to

the clamped end. If damage is close to the free end the flexibility curve leaves the zero line at the damaged location and increase towards the free end.

The stiffness matrix would be better in any case; but this is practically not possible by inversion of the experimentally determined dynamic flexibility matrix as explained in section 2.

In this section, several in-situ tests on real roadway bridges are presented. The reliability and feasibility of flexibility for damage detection is discussed. Furthermore, the influence of environmental effects, namely temperature changes, is also considered.

It is a small roadway bridge crossing a creek in Useldange, a town in Luxem- bourg. This new bridge is monitored for several years with multiple sensors. Though there was no damage in the test period, considerable differences of measured modal parameters and the deduced flexibility matrices are observed between summer and winter.

In total six measurements were done, whereof three in summer (August 2012) and three in winter (February 2013). The bridge was always excited with an unbalanced mass exciter system with a swept sine force of constant force amplitude of 2.5 kN, while the frequency was varied from 3 Hz to 10 Hz at a sweep rate of 0.02 Hz/s. The response of 21 accelerometers of type PCB393B04 was captured

at a sample rate of 2500 Hz. In order to calculate FRFs the signals were processed using a Hanning window on separated time intervals of 100 000 samples with an overlapping of 80%. With this setting a frequency resolution of 0.0125 Hz is achieved. Two measurements were done in winter (named W1 and W2) within a time interval of 20 minutes. Furthermore, measurement W3 was performed 1.5 h later. In summer, the first measurement S1 was followed by two consecutive measurements 3 hours later, referred as S2 and S3.

Additionally output-only modal analysis was performed for a permanent monitoring of this bridge for several years [

Date/Temperature [˚C] | Air* | Concrete | Asphalt | Steel |
---|---|---|---|---|

15.2.2013 (Winter: W1, W2) | −1˚C | 0˚C | 4˚C | 0˚C |

15.2.2013 (Winter: W3) | 0˚C | 0˚C | 8˚C | 0˚C |

28.8.2012 (Summer: S1) | 20˚C | 18˚C | 20˚C | 16˚C |

28.8.2012 (Summer: S2, S3) | 22˚C | 20˚C | 25˚C | 18˚C |

elastic modulus is drastically varying with temperature [

As in previously shown examples, the flexibility is calculated based on measured mode shapes and eigenfrequencies and then the diagonal elements are analysed. The modal decomposition of the flexibility matrix is illustrated in

By comparing the diagonal elements of the flexibility matrices in

Therefore

Although there was no damage at the bridge, important variations of the measured flexibility can be observed. But the variations obtained in the same season (e.g. W2-W1, S3-S2) are quite low (maximum 10%) compared to the big differences between summer and winter measurements. All summer-winter curves show a constant parallel shift, as the asphalt temperature and hence its stiffness changed constantly over the length of the bridge. Damage would be reflected by local changes of the flexibility, so that the observed global shift is due to changed ambient conditions.

The bridge Champangshiehl [

The prestressed concrete bridge had two spans of 65 m and 37 m as sketched in ^{2}, compressive strength of 37 N/mm^{2} and tensile strength of 3.6 N/mm^{2}. Furthermore, the tendons were made of Steel ASTM A416 57T with Young’s

modulus of 194,000 N/mm^{2} and tensile strength of 1730 N/mm^{2}. They were pre- tensioned with a force of 1177 kN (σ = 1050 N/mm^{2}). The superstructure was supported by two abutments and one intermediate column made of reinforced concrete. The West abutment at the end of the large span was equipped with a roller bearing, whereas the East abutment was fixed. In 1987, 56 external pre- stressed steel cables were added into the box girder of the large span for safety reasons.

The dynamic tests described here were done by an unbalanced mass exciter with a force amplitude of F = 2.5 kN. A swept sine force excitation with a sweep rate of 0.02 Hz/s was applied and the introduced forces were measured by force transducers of type HBM U10M with a range 12.5 kN. The response was re- corded with 20 accelerometers of type PCB 393B04 with a sample rate of 1000 Hz.

The accelerometer positions are shown in

Damage was introduced by cutting tendons to create local cracking of the concrete. Different scenarios are shown in

The dominating three bending and four torsional modes were identified for all damage states. The change of the eigenfrequencies is shown in

In the following the changes of flexibility matrix due to the increasing damage are examined. The flexibility-matrix is calculated with the above mentioned 7 modes, whose individual contributions are shown in

Damage scenario | Cutting tendons | Percentage cutting (100% equals all prestressed tendons in the defined section) | |
---|---|---|---|

Cutting line 1 29.25 m from the East | Cutting line 2 63.5 m from the East | ||

#0 | Undamaged state | 0% | 0% |

#1 | 20 straight lined tendons in the lower part of the bridge | 33.7% | 0% |

#2 | 8 straight lined tendons in the upper part of the bridge over the column | 33.7% | 12.6% |

#3 | 56 external tendons | 46.1% | 24.2% |

#4 | 16 straight lined and 8 parabolic tendons in the upper part of the bridge | 46.1% | 62.12% |

Damage state | Description of cracks |
---|---|

#1 | Shear cracks due to the new anchorage of the prestressed cables by cutting the lower 20 straight lined prestressed cables After mass loading with 250 tons for the static test, a vertical bending crack in the girder at position of the cutting line 1 occurred + some new shear cracks |

#3 | Growing of the existing cracks and formation of new cracks at cutting line 1 + small cracks between the holes on the upper side by the loading of static test |

#4 | More bending cracks above the column |

It is obvious that the highest flexibility was found in the largest span, which can be clearly recognized in

The contribution of the first bending mode B1 is predominant in the sum, what is revealed by the modal decomposition in

The computation of the flexibility was done for all the damaged states and the diagonal elements are presented in

For a better comparison of the changes, the differences of flexibility matrices are presented in

Cracks inside concrete lead to a stiffness reduction in structure. Therefore, damage may be detected and even localized by identifying local reduction of the stiffness matrix or by a local increase of the flexibility.

The stiffness matrix itself would naturally be more efficient for damage local- isation. But an inversion of the flexibility matrix deduced from experimental vi- brational measurements is often impossible. It is usually rank deficient, because the number of identifiable physical modes is in most practical applications lower than the number of measured DOFs. In the last example of the Champangshiehl bridge, totally 7 modes could be clearly identified from the measurements of 20 sensors. But 7 sensor-positions for a bridge of approximately 100 m length does not lead to reasonable resolution in space.

Nevertheless, damage detection based on the flexibility matrix is feasible and its informative value depends on the position of damage relative to the supports. One originality of the paper is that it synthetizes diverse revelations of flexibility change identified from practical measurements. The revelation depends on type of bridge structure (cantilever, continuous beam...) and location of damage (among 2 supports, toward a free end...). Since the behaviour of the flexibility change relates to these conditions, such a synthesis is believed useful for engineers in the damage detection of different bridge structures.

As examined in section 5.1, flexibility measurements of the undamaged bridge in Useldange show important differences between summer and winter, though there is no damage. In summer, the eigenfrequencies of the bridge are much lower than in winter and hence the calculated flexibility is much higher. Additionally, the scatter of the results in summer is higher than in winter, which can be explained by the local temperature variation due to local solar irradiation. As the eigenfrequencies of this bridge depend strongly on temperature, a compensation is necessary prior to the extraction of modal parameters and hence prior to the calculation of the flexibility matrix for condition monitoring. A very simple compensation would be to measure only on cloudy days without solar irradiation in order to keep the temperature constant along the bridge and close to the reference measurements.

For the Champangshiehl Bridge with severe artificial damages, the results are similar to the continuous beam analysed in section 4.3. The first cutting line was located in the middle of the large span and hence well identified with the absolute and the relative differences of flexibility. But the second cutting line above the intermediate pylon was not detected due to the low absolute flexibility or the high stiffness.

The theoretical examples as well as the in-situ tests presented in this paper show that it is possible to detect or even localize damage by interpreting changes to the flexibility calculated from measured modal parameters. But it has to be considered that these changes are often very small, as can be seen for instance in the example of a damage near the free end of the cantilever beam in section 4.2.2. A correct interpretation for damage detection can be complicated in case of measurement noise and in case temperature effects interfere whose impact can be higher than structural damage.

The authors declare that there is no conflict of interest regarding the publication of this paper.

The authors acknowledge the high value contribution of Administration des Ponts et Chaussées Luxembourg, specially Mr. Gilles Didier et Gilberto Fernandes.

Schommer, S., Mahowald, J., Nguyen, V.H., Waldmann, D., Maas, S., Zürbes, A. and De Roeck, G. (2017) Health Monitoring Based on Dynamic Flexibility Matrix: Theoretical Models versus In-Situ Tests. Engineering, 9, 37- 67. https://doi.org/10.4236/eng.2017.92004

[M], [C], [K]: Mass, damping and stiffness matrix;

Ω: Angular frequency of the input signal;

[R]: Residua matrix;

a: Scaling factor modal A;

[F]: Flexibility matrix.

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