7th Asia-Pacific Workshop on Structural Health Monitoring November 12-15, 2018 Hong Kong SAR, P.R. China

A Model-Based Fatigue Damage Estimation Framework of Large Scale

Structural Systems

D. Giagopoulos 1*, A. Arailopoulos 1, S. Natsiavas 2

1 Department of Mechanical Engineering, University of Western Macedonia, Greece Email: dgiagopoulos@uowm.gr; aarailopoulos@uown.gr

2 Department of Mechanical Engineering, Aristotle University of Thessaloniki, Greece Email: natsiava@auth.gr

KEY WORDS: FE model Updating; Large Scale; Vibration Measurements; Fatigue Estimation

ABSTRACT

A model-based fatigue damage estimation framework is proposed for online estimation of fatigue damage, for structural systems by integrating operational vibration measurements in a high fidelity, large-scale, finite element model and applying a fatigue damage accumulation methodology. To proceed with fatigue predictions, one has to infer the stress response time histories characteristics based on the monitoring information contained in vibration measurements collected from a limited number of sensors attached to a structure. Predictions, like the existence, the location, the time and the extent of the damage are possible if one combines the information in the measurements with information obtained from a high-fidelity FE model of the structure. Such a model may be optimized with respect to the data, using state of the art FE model updating techniques and Uncertainty Quantification and Propagation (UQ&P) methods. These methods provide much more comprehensive information about the condition of the monitored system than the analysis of raw data. The diagnosed degradation state, along with its identified uncertainties, can be incorporated into robust reliability tools for updating predictions of the residual useful lifetime of structural components and safety against various failure modes taking into account stochastic models of future loading characteristics. Fatigue is estimated using the Palmgren-Miner damage rule, S-N curves, and rainflow cycle counting of the variable amplitude time histories of the stress components. Incorporating a numerical model of the structure in the response estimation procedure, permits stress estimation at unmeasured spots. The proposed method is applied in a steel frame of a real city bus.

1. Introduction

Model-based methods for structural health monitoring incorporate an analytical or numerical model of the mechanical or structural system, typically of the Finite Element (FE) type, that can integrate and reproduce ultimate limit states and failure modes. Use of a finite element model in combination with operational vibration measurements and state of the art FE model updating techniques, allows for prediction of the existence, location, time and extent of the damage, providing much more comprehensive information about the condition of the monitored system than the analysis of raw data[1-3]. Damage detection schemes that use only structural modal identification methods, track changes in the stiffness of several parts of the structure or an overall loss of damping. Although such changes are efficient indicators of a damaged state, there are other major manifestations of damage in structural systems that cannot be identified. Loss of strength, fatigue, loss of ductility capacity and residual stresses and deformations in several parts of a structural system are all damage scenarios that cannot be addressed by modal identification-based structural health monitoring systems[4]. This is an

* Corresponding author.

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important safety-related issue in metallic structures since information on fatigue damage accumulation is valuable for designing optimal, cost-effective maintenance strategies and for structural risk assessment. Predictions of fatigue damage accumulation at a point of a structure can be estimated using available damage accumulation models that analyse the actual stress time histories developed during operation[5, 6]. To proceed with fatigue predictions, one has to infer the stress response time histories characteristics based on the monitoring information contained in vibration measurements collected from a limited number of sensors attached to a structure. Such predictions are possible if one combines the information in the measurements with information obtained from a high fidelity finite element model of the structure. To optimize the FE model of a structure, structural model updating methods[7], have been proposed in order to reconcile the numerical (FE) model, with experimental data[8], measuring the residuals between measured and model predicted modal characteristics. Standard optimization techniques are then used to find the optimal values of the structural parameters that minimize a single-objective function[9-13]. The main objective of the present work is to estimate the full strain time histories characteristics at critical locations of a complex mechanical assembly using operational vibration measurements from a limited number of sensors. To achieve this, appropriate numerical and experimental methods were applied, to identify, update and optimize the model parameters. In this process, many issues are considered, related to the development of FE model, the experimental modal analysis procedure and the development of effective computational model updating techniques[7, 14-19]. The discrepancies between experimental and analytical time domain dynamic responses for the same imposed excitation are utilized as an overall measure of fit[4, 20-22].

2. Finite element model updating formulation based on strain response

time histories In this work, parameter estimation is based on response time history measurements of strains. This formulation has the advantage of applicability over both linear and non-linear systems; it compares the measured raw data of the experimental arrangement to the equivalent predictions of the analytical model. In this way, all available information is preserved and systematic errors of the identification procedure are alleviated. Assuming that the prediction errors, quantified as the difference between the measured response time histories and the model predicted response time histories, are independent between different sensors and between different time steps within a response time history, the overall measure of fit takes the form[22].

J q( ) =1

n

gij

qm

| M( ) - yij( )

2

j=1

m

å

yij( )

2

j=1

m

åi=1

n

å (1)

where ( )|ij mg M is the analytical time-history of the introduced FE model and ˆijy is the respective

experimental signal. Subscripts i correspond to the sensor (accelerometer) location and measurement direction, and j corresponds to the time-step instant. n is the total number of measured sensor

locations and directions, whereas m is the total number of measured time-steps (number of observations).

3. Estimation of fatigue damage accumulation using stress

reconstruction The process of inception and spread of cracks through a structural member due to action of fluctuating stress is also known as fatigue[23]. When dealing with a uni-axial stress state with axial stress time history , 1,..., =k tk N , the Palmgren-Miner rule[5, 24] is commonly used to predict damage

accumulation due to fatigue. According to this rule, fatigue damage at a point in the structure

subjected to variable amplitude stress time history k , is defined as the ratio of the number of cycles

of operation to the number of cycles to failure at a given stress level. When dealing with multiple stress levels, according to the Palmgren-Miner rule the sum of fatigue cycles at various levels yields the total damage

D =n

i

Nii=1

k

å (2)

where in denotes the number of cycles at a stress level ( ) i of the stress time history k , iN stands

for the number of cycles required for failure at a stress level ( ) i , and is the number of stress levels

identified in a stress time history at the corresponding structural point. When dealing with arbitrary stress time histories at a point of interest on a structure, the number of cycles at a stress level is usually obtained by applying cycle counting methods, such as the rainflow cycle counting[25-27]. Design codes determine the stress levels for fatigue of commonly used engineering materials; furthermore, the so-called S-N curves are included in the design codes to specify the number of cycles iN required for

failure in terms of the stress level ( ) i [23, 27]. The S-N curves are obtained via experimental tests on

real size specimens. In doing so, uniaxial, constant amplitude cyclic stresses applied to various structural members with different shapes. In design codes, the S-N curves are expressed by log-log curves, in which plots the number of cycles for fatigue failure versus its associated stress range. Each curve is designated with a number that specifies the function, shape and built of the considered structural member. From what precedes, it is evident that accurate estimate of the fatigue damage accumulation at a point is contingent upon accurate prediction or measurement of stress time histories, as well as accurate cycle counting procedures for determining the stress range spectrum. It is noteworthy that, the number of cycles to fail depends also on the mean stress; as the mean stress increases for a given level of alternating stress the fatigue life decreases[28]. Therefore, the fatigue accumulation model must be revised to account for a non-zero mean stress according to the Goodman relationship[29, 30]

Rt R

u

(1 )

= − (3)

where Rt stands for the modified stress cycle range, R signifies the original stress cycle range,

denotes the mean stress, and is calculated by the cycle counting algorithm; and u is the ultimate

tensile strength of the material.

Once the stress range spectrum for a structural member is obtained, and the relevant detail category is determined, S-N curves are used for estimating fatigue strength. In this regard, Miner summation is employed, and the fatigue damages pertinent to the stress ranges are linearly summed. The parametric representation of fatigue damage reads[23]

(4)

where D denotes fatigue limit for constant amplitude stress ranges at 65 10= DN cycles; L stands

for the cut-off limit; i and j

are the thi and th

j stress ranges; in and j

n are the number of cycles

in each i and j block; 1 and 2 represent the number of different stress range blocks above or

below the constant amplitude fatigue limit D , respectively.

4. Experimental application 4.1 Development of the Finite Element model The proposed method is applied in a linear steel frame of a real city bus as presented in Fig. 1, whereas Fig. 2. presents the detailed FE model of the chassis of the bus.

Figure 1. Real city bus Figure 2. Finite Element model of the chassis of the bus

including applied masses. The FE model of the chassis is developed and discretized using both shell and solid (tetrahedral) elements resulting in about 4,000,000 DOFs. Dead loads that come from the engine, the fuel tank, the air-conditioning, as well as live loads coming from the passengers, along with various other loads are applied as masses on the FE model using special elements as presented in Fig. 2. For the development and solution of the finite element model appropriate software[31, 32] was used. All parts of the examined structure are made of steel with nominal material parameters of Young’s modulus 210=E GPa , Poisson’s ratio 0.3=v and density 37850 / = kg m . The following Fig. 3

presents indicative eigenmodes of the bus frame using nominal material parameters colored by spectrum colors of normalized deformations.

Figure 3. Indicative eigenmodes of the bus chassis.

4.2 Operational vibration measurements Vibration measurements during real operating conditions were collected comprising the experimental data. Specifically, six (6) tri-axial accelerometers were placed at the points where the six wheels are mounted on the chassis of the bus and ten (10) strain gauges with three bridges at 120° angle rosette each, were placed at characteristic points of the frame. During real operating conditions the accelerometers were recoding the induced accelerations on the frame and the stain gauges were

monitoring the stress rate at the same time. Fig. 4 and Fig. 5 present the accelerometer and strain gauges locations respectively.

Figure 4. Accelerometer locations.

Figure 5. Strain gauge locations.

4.3 FE model updating The model updating methodology is conducted by parameterizing the FE model as shown in Fig 5.

Figure 5. FE model parameterization.

All parts (P1-P15) are modelled using shell and solid elements, while the Young’s modulus and material density are used as design variables initiating from the nominal values of steel. Beyond material properties parameters, Rayleigh damping parameters

a , b are used as parameters resulting in

a total number of 47 design parameters. The structure is modeled with free boundary conditions at all nodes. The real operating acceleration measurements from the six tri-axial accelerometers are used as enforced ground motion representing the imposed excitation of the FE model, whereas the strains monitored from the experimental procedure are compared to the strains predicted from the numerical (FE) model, comprising the measure of fit between the physical structure and the FE model. A comparison between time histories of stresses at strain gauge locations 1 and 7 is presented in the following Fig. 6 proving very good match. Specifically, the black continuous line corresponds to the experimental data and the red continuous line corresponds to the predicted response of the optimized updated FE model.

Figure 6. Comparison between numerical – FE model results and experimental measurements at

reference locations SG1 and SG7.

5. Fatigue monitoring using operational vibrations Using the updated FE models of the bus frame, the stresses under real dynamic load conditions are calculated. The ultimate aim is the identification of those points in the frame where the larger stresses appear. Fig. 7 presents Von-Mises stresses of the updated FE model loaded by real operating time histories of accelerations.

Figure 7. Predicted maximum stress locations using the updated FE model.

Incorporating representative operational measurements available for a relatively short time interval, the stress time histories at the most critical points ST1 – ST8 are computed for the same time duration. The stress pattern during this short time interval is assumed to repeat, thus extrapolating the stress time histories to the fatigue time. Using the calculated stress time histories at locations (ST1-ST8) and utilizing the available S-N fatigue curves, the Miner’s rule is applied to estimate the fatigue damage accumulation. The frame is made of steel and the fatigue detail category 36 is adopted to illustrate the method. The static strength of steel is assigned the value 440 =u MPa. According to Eurocode 3 for

detail category 36, the following values of the parameters of the design S-N curves are recommended: 3=m , D 26.5 = MPa and 14.5 =L MPa. Using equation (4) and the repeatable pattern of stress

time histories measured during a short monitoring period of minutes. The calculated fatigue life for the eight locations is presented in Table 1.

Table 1. Calculated fatigue life

Location Calculated Fatigue Life (Days) from all Mobility Tests

Min Max

ST1 382.51 550.13

ST2 288.29 380.34

ST3 265.73 332.65

ST4 3353.15 4724.91

ST5 1422.87 1961.43

ST6 735.69 872.51

ST7 1393.73 1555.42

ST8 5142.89 5352.37

The results of numerical analysis were confirmed from the real structure. More specifically, the photos in Fig. 8, present the crack in the frame, exactly in the same locations (ST2-ST3) where the numerical analysis results show. Also, this crack appeared only in 280-305 days from the time that the bus was put into operation, which is very close to the calculated fatigue life. These results indicate that the methodology applied gives accurate results and provides a useful tool in predicting the fatigue damage accumulation and for designing optimal fatigue-based maintenance strategies in a wide variety of structures.

Figure 8. Crack locations in real frame.

6. Conclusions A computational framework is proposed for estimating fatigue damage accumulation in a linear steel frame of a real city bus. A FE model of the frame is developed and updated to match the dynamic characteristics measured in real operating conditions. This is achieved through coupled use of numerical and experimental methods for identifying, updating and optimizing a high fidelity FE model. The full stress time histories of the bus frame are estimated, at critical locations, by imposing operational vibration measurements from a limited number of sensors in the updated FE model. Fatigue damage and remaining lifetime is subsequently estimated via commonly adopted engineering approaches, such as the Palmgren-Miner damage rule, S-N curves, and rainflow cycle counting. Incorporation of a numerical model of the structure in the response estimation procedure, permits stress estimation at unmeasured locations, thereby enabling the drawing of a complete and substantially dense fatigue map consistent with the vibration measurements. Fatigue predictions via the proposed framework compare favorably to experimental fatigue results, - proving the efficiency and applicability of the framework.

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