Frequency domain application of the Hot-Spot method for the fatigue assessment of the weld seams

Transcription

1 Frequency domain application of the Hot-Spot method for the fatigue assessment of the weld seams Dr. Ing. Sauro Vannicola 1 sauro.vannicola@ch.abb.com Dr. Ing. Luigi De Mercato 2 luigi.de-mercato@ch.abb.com Summary Welding is extensively used in all sectors of manufacturing, since it represents an efficient and economic method for joining metals together. However one of the main problems associated with welded joints is that their fatigue strength is lower than the relevant base material. The result is that even in a welldesigned structure, the weld seam is the most vulnerable element to fatigue loads. Therefore any fatigue calculation should focus on the assessment of the weld seams in the first instance. Many methods were developed for this purpose such as the nominal stress, effective notch and the hot-spot method. In addition to this, the real structure is often subject to input loading that are random in nature such as offshore platform, railway equipment and so on. The fatigue behaviour of the structure must thus be analysed in the frequency domain in terms of statistical moments. This paper shows the application of the hot-spot method for the weld assessment in a random environment. The stress tensor components (Re and Im components), coming from the harmonic analysis, are used in fact to calculate the absolute maximum principal stress, at certain location at a fixed distance from the weld toe, together with their relevant stochastic variables. The fatigue damage is then verified by comparison with the one calculated on the same specimen following the more accurate effective notch approach and analysed by means of the ncode DesignLife software. Keywords Welding fatigue analysis, Random fatigue, Hot-spot stress method, Railway equipment 1 Structural Engieer, TRASFOR SA - ABB Group, Molinazzo di Monteggio, CH, 2 R&D Senior Engineer ABB CHDC Dry Type Tranformers, Molinazzo di Monteggio, CH,

2 Introduction Welding is used in many sectors of manufacturing since it represents an efficient and economic method for permanently joining similar metals together. However one of the main problem associated with welded joints is in that their fatigue strength is lower than those of the material being joined together. The result of this is that even in a well-designed structure the weld seams are the weakest point for fatigue. Any fatigue calculation should therefore focus on the assessment of this type of connection. Many methods were developed for this purpose: nominal stress, effective notch and hot-spot. The nominal stress is the simplest and most widespread approach, where just overall quantities are considered from a global point of view. This is also used as the basis of all standards and design codes. It implies the use of empirical S-N curves, detail categories and corrective factors. The selection of a detail category may be not straight forward, especially for complex structures. The effective notch approach is instead a local method where the variation of the weld shape parameters is taken into account by replacing the real weld shape with an equivalent one. The hot-spot stress is an intermediate approach between global and local methods and is suitable for the assessment of fatigue failure occurring at the weld toes. It is based on an idealized stress distribution in the thickness of joined member. It includes, in fact, all stress raising effect of a structural detail excluding that due to the local weld profile itself. The structural hot-spot stress can be determined using reference points for the extrapolation to the weld toe. Fatigue analysis in the frequency domain The fatigue analysis in the frequency domain is the estimation of expected life of the component when the input loading, or the stress history obtained from the component, is random in nature and therefore best specified using statistical information about the process. This is usually in the form of Power Spectral Density (PSD) function. A random process is a phenomenon that unpredictably varies to some degree as time goes on. It is not possible in fact to predict the signal magnitude at time t+1 from time t, even if the signal magnitude at time t is known. Wherever the loadings are random and the dynamic response of the structure is present it is usually desirable to perform the structural analysis in the frequency domain using PSD and transfer function. The first step of this analysis is the calculation of the frequency response by means of the Harmonic Analysis Workbench module. It provides the transfer function between the stress tensor components and the input load in the frequency domain. The transfer function has then to be squared in order to be used for a PSD analysis. The second step is the product between the input PSD and the squared transfer function. This gives the PSD of the stress tensor that the components of the structure will experience when subjected to the considered input loads. On the other hand operations between squared quantities lead to a PSD signal in which the information about the phase between each frequency components is completely missed. If that information was available the response signal could be reproduced in the time domain and a rain-flow cycle counting could be applied for the evaluation of fatigue cycles. A different method to predict the distribution of the rain-flow cycles is therefore needed. Several methods were developed to compute the stress cycles distribution as a function of the response PSD. Dirlik [5], Lalanne [3], Steinberg [6] and Rayleigh [7] are the most popular methods for the evaluation of stress cycles distribution starting from the stress PSD. Each of these methods is suitable for a specific field of application since was developed taking into account different reference signals. The Rayleigh method, for example, is suitable only for narrow band signals and can be too conservative if applied to a wide band signal. However they all assume that the stress history, represented by the PSD, is stationary, random, Gaussian and ergodic. Once a suitable rain-flow counting method is chosen, the statistical distribution of the stress cycles can be estimated based on the shape of the stress PSD in terms of its spectral moments. The stress PSD spectral moments give the expected number of peaks and zero crossing of the stress signal. All of these statistical quantities can be used for the calculation of the expected damage by means of the Palmgren- Miner rule. Methodology The finite element analysis is widely used during the design phase as a powerful tool for the development of light and reliable structures. Once the finite element model is available, the stress hotspot regions can easily be detected in the structure, and accurate values can be obtained providing that the model is created as recommended in [1], especially in the hot-spot locations. As defined in this guideline the structural hot spot stress can be determined either by measurement or by calculation, as it is for this study. The non-linear stress peak is cancelled by extrapolation of the stress at the surface

3 to the weld toe. This is a suitable procedure for both type a and b hot spots as reported in the guideline. If we deal with deterministic fatigue, at least two loading cases should normally be analyzed, one for the maximum and one for the minimum principal stresses. In case of random excitation, the stress components at each node are available only from a statistical point of view. Therefore is not possible to proceed as for the deterministic fatigue. The following procedure has thus been developed to cover this gap. A harmonic analysis shall be firstly performed in order to calculate steady-state response of the structure and the relevant stress tensor components at each extrapolation point away from the weld toe, as shown on Figure 1. Figure 1: Example of extrapolation path for a type "a" weld The stress tensor components have a real and imaginary component, since the response of an oscillating system can be always represented as the real part of a rotating vector on the complex plane. As mentioned above, for the weld seam assessment, we are interested in the principal stresses. They are normally determined from the eigenvalues of the stress tensor. The determination of the absolute maximum principal stress from the Frequency Response Function obtained by the finite element analysis is difficult, because there is no general solution for the eigenvalues of a complex matrix. As per [3] an alternative way for the calculation of the principal stresses is to find the eigenvalues of the Real and Imaginary part of the stress tensor separately, and taking the absolute maximum principal FRF to be composed of the real and imaginary eigenvalues with the largest magnitude. This is an approximation, but is likely to be reasonable unless critical parts of the frequency range have significant contribution from more than one mode of vibration. The phase of each stress tensor component can be alterate, in fact, by the contribution of the additional mode of vibration. This can produce a variation of the principal directions and therefore of the principal stress along each frequency. The eigenvalue problem steps are briefly summarized on the equations below:

4 = ( ) = d et ( [ ]) = max ( ) = ( ) =0 = d et ( [ ]) = max ( ) (1.1) (1.2) (1.3) (1.4) = + (1.5) This calculation is carried out by means of a MATLAB routine. Once the absolute maximum principal stress is calculated at each distances, it can be extrapolated to the hot-spot based on one of the extrapolation formulas given in [1]. The absolute maximum principal stress at the hot-spot must be doubled, since the stress range must be used for FAT classes, and squared to be compatible with the input spectrum. This is then multiplied by the input load spectrum. Figure 2: Example of input spectrum (left) and hot-spot stress response PSD (right) This calculation gives the response spectrum in terms of MPa 2 /Hz (see Figure 2 - right). This is the PSD of the absolute maximum principal stress at the weld toe. Its shape and amplitude, through its spectral moments, are used to compute the probability density function of the maximum absolute stress ranges as shown on Figure 3, which represents the loading cycles distribution of the stress ranges according to the rainflow cycle counting method. The most famous empirical formula for the estimation of the rainflow amplitude distribution is that proposed by Dirlik [5], which results from a fitting procedure over a large set of data from numerical simulation. It should be noted that this approach gives an amplitude distribution depending on four spectral moment only. Many works showed how Dirlik method is far superior to other existing methods in estimating rainflow fatigue damage.

5 Figure 3: Example of probability density function of the stress ranges (Dirlik) The spectral moments, and, are also used to calculate the expected number of peaks [] as per (1.6). The probability density function of stress ranges, the number of peaks, the test duration, and material properties (Basquin curve parameters) can now be used to calculate the estimated damage as per (1.7). [] = (1.6) = [] = = [] [] (1.7) Where: [] is the expected number of peaks, and are respectively the 4 th and 2 nd order spectral moment, [] is the expected damage, is the absolute maximum principal stress, is the test duration, and are the Basquin curve parameters Methodology validation The described methodology has been adopted to evaluate the fatigue damage on a welded plate test specimens with transverse attachments. This test specimens consist of 70 mm wide main plate with two transverse secondary plates welded to the main plate by means of fillet welds (see Figure 4).

6 Figure 4: Test specimen dimensions A 150 kg point mass has been attached to the main plate end (point A). The specimen is then ground fixed on point C as shown on Figure 5. All 6 nodal d.o.f.s have been set to zero on the corresponding surface. The FE model is composed of: - 104,328 total nodes, contact elements, - 21,600 solid elements, - 21,762 total elements. Figure 5: Boundary conditions and acceleration load direction

7 Solid186 elements have been adopted. This is a 20 nodes element having three degrees of freedom per node and is suitable to model irregular geometry. Figure 6 shows the elements quality as mesh quality index. The best possible element quality being 1. Figure 6: Mesh element quality The load has been defined as an acceleration spectrum acting in longitudinal direction (See Figure 5) for 4 hours with the following parameters: Frequency [Hz] Acceleration [g 2 /Hz] 150 5e e-4 Table 1: Load input spectrum parameters The structure steady state response to a sinusoidal input loading has been determined by means of a harmonic analysis. The analysis was run using the mode-superposition technique, since it allows solution to be clustered about the structure natural frequencies and is less time and computational expensive. The response, in terms of stress component, has been extrapolated from the reference points at respectively d2 = 6 mm and d1= 2.4 mm away from the weld toe as recommended in [1] for type a hot- spot with fine mesh and element length not more than 40% of the plate thickness. Each stress component tensor has been used for the calculation of the maximum principal stress at points d1 and d2 as described above. Then, the maximum principal stress component at the weld toe has been extrapolated using the following interpolation formula: = (1.8) Figure 7: Maximum absolute principal FRF [MPa/g] at the weld toe

8 Strictly speaking, the correct method would be first to extrapolate each stress component to the weld toe, and then to resolve the principal stress and its direction at the hot spot. However in practice it is sufficient to extrapolate either the maximum principal stress or the stress component normal to the weld toe. The absolute maximum principal stress steady state response at the weld toe, properly squared, was then multiplied by the input spectrum in order to obtain the PSD of the maximum principal stress. The spectral moments have been used to define the shape and the amplitude of the probability density function of the stress ranges according to Dirlik (see Figure 8). Figure 8: Pdf of stress ranges The S-N curve parameters used for the assessment of the fatigue life of a detail, on the basis of structural hot-spot stress, are given in the table below: FAT class [MPa] Stress ranges (S) Stress range at knee point (1E+7 cycles) [MPa] For stress ranges above the knee point Values of constant C N =C/S m For stress ranges below knee point m = 3 Constant amplitude m = e e e15 Table 2: Basquin curve parameters for the hot-spot approach Variable amplitude m = 5 The damage summation, according to the Miner s rule, has been carried out considering the effect of stress above and below the knee point. In particular the variable amplitude coefficients have been used for the calculation of the cumulative damage for the stress range below the knee point. At that point the evaluated cumulative damage should be compared with those calculated considering the stress ranges cycle distribution of a recorded strain signal on an experimental specimen. The methodology should be in fact validated comparing the results with those resulting from specimen tested on a vibrating table. This will be done as a follow up of this work. As an alternative the cumulative damage calculated with this methodology has been compared with the one evaluated using the effective notch stress approach, being this considered as the most reliable method for the fatigue assessment of the weld seams. The basic idea of this concept is that the stress on the weld toe and root is calculated on an equivalent geometry where a fictitious enlargement of the notch radius is present as shown on Figure 9. The real weld contour is thus replaced by an effective notch root radius of 1 mm. The method is restricted to the assessment of welded joints with respect to potential fatigue failures from the weld toe or weld root. For fatigue assessment, the effective notch stress is compared with a single fatigue resistance curve having the following parameters: FAT class [MPa] m Table 3: Effective notch fatigue resistance for the effective notch approach

9 The FE model used for the assessment is shown on the figure below: The FE model is composed of: Figure 9: Effective notch FE model - 450,828 total nodes, contact elements, - 149,597 solid elements, - 150,332 total elements. The mesh contains SOLID186 and SOLID185 which are respectively 20 and 8 nodes solid elements. The transition between fine and coarse mesh are modelled with degenerated SOLID186 elements. Figure 10 shows the elements quality as mesh quality index. Figure 10: Mesh element quality The harmonic response analysis has been performed using the ncode DesignLife vibration module. The same parameters as for the hot-spot analysis (input spectra, test duration and cycle counting method) have been defined. The resulting damage on the weld toe has been taken as reference for the comparison with those calculated before. Cumulative damage Hot-Spot Routine Effective Notch (ncode DesignLife) Table 4: Cumulative damage comparison for random analysis The damage predicted using the methodology at the weld toe is higher than those calculated using the effective-notch approach. This is mainly due to the differences in the S-N curve parameters and in the

10 FE model in terms of geometric features and mesh parameters. This has been demonstrated with a further analysis where a deterministic fatigue calculation was performed on both of the FE models using the TimeSeries module of ncode DesignLife. The FE models have been, in fact, ground fixed and loaded with the same longitudinal sinusoidal force. The resulting cumulative damage contour plots are shown on the pictures below: Figure 11: Cumulative damage for deterministic fatigue Hot-spot model (left) and Effective-notch model (right) This analysis clearly shows that the difference between the cumulative damage calculated with the model used for the stress extrapolation to the weld root and the one calculated with the effective-notch approach is exactly of the same order of magnitude as for the previous random analysis. Discrepancies between the two methods cannot be therefore attributed to the formulation of the method for the adaption of the hot-spot criteria to a random analysis as described in this work. At the same time the consistency of the differences for the two analyses demonstrate that no significant approximation is introduced when a random analysis is performed instead of a deterministic one. This shows in a way the robustness of method, which can be used as a reliable tool for the fatigue assessment of the weld seams when the crack is expected to occur at the weld toe. Conclusions A methodology has been proposed for the fatigue assessment of the weld seams applying the principle of the hot-spot method in a random environment. The steady state response to an input load of a test specimen has been calculated in ANSYS Mechanical (Harmonic Response) and the stress tensor components at specific location exported. These components have been processed by means of a MATLAB routine for the calculation of the absolute maximum principal stress at the weld toe. The statistical moments of the absolute maximum principal stress response have been used for the calculation of the probability density function of the counted cycles according to the rain-flow technique. For this case the Dirlik approach have been adopted. The cumulative damage according to the Miner summation have been finally calculated. The comparison with the effective notch approach carried out using the ncodedesignlife vibration module shows that the predicted damage is higher than the one calculated with the effective-notch method. However, this discrepancy can be explained by the differences in the FE model, i.e. geometry features and mesh, and in the S-N curve parameters, as shown by the comparison of results from the two models when a deterministic analysis is performed instead. The proposed approach still needs to be validated experimentally. A comparison with the effective-notch approach has been made instead for verification purposes. This shows that the predicted damage always leads to a conservative estimation of the expected damage. It can be concluded that the proposed approach can be used as an efficient tool for the preliminary estimation of the fatigue life of the weld seams, since it does not require any particular model idealization except for those recommended in [1]. It can also be adopted to detect the most critical seams as the ones having the highest damage summation. Another advantage of the proposed method is the easy of implementation into ANSYS Mechanical interface by means of an ACT application, being it based only on the analytical formulation of the fatigue problem. The end used should in fact define only the material S-N curve parameters and select the extrapolation points location.

11 References [1] International Institute of Welding: "Recommendations for fatigue design of welded joints and components", A. Hobbacher, [2] Henning Agerskov: "Fatigue in steel structures under random loading", [3] HBM-nCode: DesignLife Theory Guide, [4] Niemi, Fricke and Maddox: Fatigue analysis of welded components Designer s guide to the structural hot-spot stress approach, [5] Dirlik T. Application of Computers in Fatigue Analysis PhD Thesis University of Warwick, 1985, [6] Steinberg D.S. Vibration Analysis for Electronic Equipment John Wily & Sons, New York, [7] Dr NWB Bishop & Dr F Sherratt Finite Element Based Fatigue Calculations NAFEMS.