A STATISTICAL MEASURE OF THE NON-PROPORTIONALITY OF STRESSES

In mechanical engineering, fatigue assessment based on Finite Element analysis of components, where complex non-proportional loads are acting on, plays an important role to identify critical spots and predict the lifetime till crack initiation. In this contribution the lifetime analysis of a dynamically loaded crankshaft of a V6 diesel engine is demonstrated. Finite Element method, multi-body system simulation and an elasto-hydrodynamic simulation of the oil film have been combined for a modal analysis in the frequency domain. Based on these results a fatigue analysis follows to identify the critical engine speed range. In notches rotating principal stresses can be found, which reduce the lifetime significantly. To quantify the non-proportionality of stresses and the resulting decrease of the fatigue limit, a statistical measure originally proposed by Chu et al. is applied. The critical plane criterion has been used in conjunction with a stress based method (concept of synthetic S/N-curves, influence parameter concept) and linear damage accumulation according Miner’s rule.

KEYWORDS

Non-proportional loads; non-proportional stresses; degree of non-proportionality of stresses; synthetic S/N-curves; fatigue life prediction; FEMFAT

INTRODUCTION

In mechanical engineering, computational simulations of stress and strain distributions on mechanical components with the Finite Element Method (FEM) are widely used. Nevertheless, much user experience and know-how are still necessary to obtain good results of lifetime simulations, which are based on stress/strain analysis results.

In this contribution, we will focus mainly on the load situation. It is demonstrated on a practical example, namely a dynamically loaded crankshaft of a diesel engine, that rotating principal stresses occur, which affect the component’s fatigue limit significantly. Therefore rotating principal stresses have to be considered particularly in numerical fatigue analysis. The software FEMFAT of the Engineering Center Steyr, which is able to do fatigue life prediction from Finite Element Analysis results, takes into account a lot of influences, which affect the lifetime (notches, mean stress, surface roughness, surface treatments, temperature, etc., so-called “Influence Parameter Concept” [1-4]). Also rotating principal stresses are an important influence and can be taken into account . For its quantification a statistical measure is needed. In FEMFAT a so-called degree of non-proportionality of stresses (DNS) will be calculated. In dependence on DNS and material the local fatigue limit will be reduced appropriately.

A STATISTICAL MEASURE OF THE NON-PROPORTIONALITY OF STRESSES –

INVESTIGATIONS AND APPLICATIONS

H. Dannbauer, C. Gaier and M. Steinbatz

MAGNA STEYR, Engineering Center Steyr GmbH & Co KG,

Steyrer Strasse 32, A-4300 St. Valentin, Austria

HOW TO DEFINE A DEGREE OF NON-PROPORTIONALITY OF STRESSES (DNS)

In technical practice, measured load histories must always be digitized to be processed by computers. A Finite Element analysis will deliver a discrete history of local stress states. Considering these stress states in stress space (which is 3-dimensional for plane stress states and 6-dimensional for spatial ones), one can easily distinguish qualitatively between proportional and non-proportional loading (Fig. 1).

Fig. 1. Local stress history.

Of course there is a smooth transition between proportional and non-proportional load cases. Our main goal is looking for a procedure, which delivers a characteristic number for the history of the local stress state, and which is mathematically sound. This characteristic values should be 0 for proportional loading (in this case the three principal stresses are always proportional during the load period or sequence, and the principal stress directions are fixed in respect to geometry) and 1 for strongly non-proportional loading (the three principal stresses are not always proportional and/or the principal stress directions are not constant). Such a procedure has been proposed by Chu et al. [5], where an inertia ellipsoid has been constructed, which is determined by the discrete history of stresses in stress space. The aspect ratio of this ellipsoid has been taken as a measure for the non-proportionality of stresses (= degree of non-proportionality of stresses DNS).

There is an important demand on the DNS for theoretical consistency and practical applicability, which is the invariance from the coordinate system. Independency can be achieved by multiplying the shear stress components with 2, which is a consequence of the symmetry of the stress tensor. As shown in Fig. 2, the two dimensions in stress space which are related to the corresponding shear stresses τxy and τyx, can be replaced by one dimension, which is stretched by a factor of 2. Consequently, for combined tension/torsion loading with 90° phase shift, the DNS reaches its maximum value of 1, when the loading ratio is σxxτxy=2 (see Fig. 3).

Fig. 2. Dimension reduction. Fig. 3. Combined tension/torsion loading with 90° phase shift.

DYNAMIC SIMULATION OF A V6 DIESEL ENGINE CRANKSHAFT

In order to perform an accurate fatigue life prediction of a crankshaft it is necessary to take a couple of different effects into account. The crankshaft itself is a linear reacting structure that undergoes large nonlinear displacements. Additionally, an elasto-hydrodynamic (EHD) oil film model is required, being capable to consider the stiffness and damping properties of the oil film inside the journal bearings.

An efficient computation of all these effects requires different algorithms (e.g.: FEM for linear reacting structure, MBS for large nonlinear movements and EHD software for the oil film dynamics). The FE, MBS and EHD software contribute to this integrated simulation process with their particular advantages. The modal representation of the crankshaft (component modes, Craig Bampton Theory [6]) is imported into an MBS Software (e.g. ADAMS, RecurDyn). The reaction force and moment of the elasto-hydrodynamic oil film is computed in a co-simulation process using a user-written subroutine within the MBS software. The result of the time integration of the MBS solver are, among many other result sets, the modal coordinates of the crankshaft, representing the amplitude of each mode shape at each time step. A superposition of all mode shapes weighted by the corresponding modal coordinates gives the total deformation of the flexible structure.

Aside the flexible crankshaft, the MBS model of the V6 diesel crank train consists of rigid parts (dual mass fly wheel, belt pulley, torsional damper, connecting rods and pistons; see Fig. 4. The system is excited by the gas forces using a 3D spline function depending on crank angle and rotational speed of the crankshaft. For each time integration step of the solver, the pressure distribution inside the journal bearings is computed, considering the global stiffness of the bearing chairs. The oil film model is based on the well-known Reynolds differential equation.

Fig. 4. MBS model with exemplary in- and outputs

The results of the dynamic simulation can also be used to generate Campbell diagrams to identify possible system resonances (order analysis). An example is given in Fig. 4. Since resonances are likely to occur, each mode is selectively damped. The intrinsic damping of the oil film is included by the EHD model.

FATIGUE ANALYSIS OF A V6 DIESEL ENGINE CRANKSHAFT

Within the MBS the position of the elastic crankshaft is computed by superimposing its rigid body motion and elastic deformation. The elastic deformation of all degrees of freedom is approximated by a linear combination of suitable modes as outlined in Fig. 5. The so-called ‘Component Modes Synthesis’ turned out to be a reliable technique in order to determine such a set of modes [7, 8] and contains both, the FE structure’s static and dynamic properties. This ‘Component Modes Synthesis’ is performed by FEM software.

Each deformation of a FE structure is related to a clearly defined stress distribution. Consequently, each component mode shape (deformation) corresponds to a clearly assigned stress distribution (modal stress), which is an output of a subsequent FE analysis. The resulting stress state of the FE structure is again computed by a linear combination of the modal stresses. The single modal stress shape’s modal coordinate is the same as the modal coordinate of the corresponding component mode and a result of the MBS (Fig. 5).

The durability analysis is performed with the software FEMFAT, developed at the Engineering Center Steyr, using the multiaxial fatigue module MAX. Fig. 5 outlines the procedure of the multiaxial and channel-based fatigue lifetime prediction. The modal stresses and modal coordinates are the input for the durability analysis. Each channel consists of a modal stress and the time history of the corresponding modal coordinate. FEMFAT then computes the resulting stress states for each time step (stress history).

Further it allows to define influence parameters like surface roughness, surface treatment and much more. For parts under a high non-proportional load (uncorrelated load history), FEMFAT has an option to consider the rotating principal stresses. This influence due to multiaxiality of loads (yielding non-proportionality of stresses) will influence the fatigue result to lower endurance safety factors (see Fig. 6). FEMFAT is also able to compute the DNS (=

Fig. 5. Proceeding of the durability analysis based on modal stresses

degree of non-proportionality of stresses) according to the channel-based stresses and load histories.

It has to be mentioned, that the modal based approach provides the technique for fatigue life prediction of any vibration dominated problems or dynamic loaded parts, like components of an engine.

RESULTS AND DISCUSSION

Splitting up the modal coordinates into equidistant rpm intervals, a rpm sensitive fatigue analysis can be performed. The results for different locations (bearing surface and notches) are plotted in Fig. 6. The FEMFAT analysis have been done with and without the influence of the DNS. From the results it can be seen that the effect of the non-proportionality of stresses on the endurance safety factor is between 7 to 18%, depending on the position and rotational speed of the crankshaft and thus on the loading situation.

Fig. 8 shows that the distribution of minimum safety factors and maximum DNS do not match. This is clear due to the fact that minimum safety factors appear in notches, where the directions of maximum principal stresses are geometrically fixed. Therefore the influence of the DNS is smaller in bearing notches than at the bearing surface, as it can be seen in Fig. 6. In Fig. 7 the maximum DNS over rotational speed is shown, as computed by FEMFAT. It is between 0.95 and 0.99, which indicates a similar loading situation as shown in Fig. 3.

Fig. 6. Minimum endurance safety factors against rotational speed of crankshaft

Fig. 7. Max. DNS over rotational speed (the positions are different from Fig. 6, see Fig. 8)

Fig. 8. Detail of the with endurance safety factors and DNS distributions from FEMFAT

CONCLUSIONS

A fully flexible crankshaft fatigue simulation was performed using modal stresses. The result accounts for the dynamic effects of the crankshaft and the dynamic properties of the oil film inside the lubrication gap as well as the elasticity of the crank housing, resulting in a very accurate bending line of the crankshaft. Consequently, the performed fatigue life prediction, capable of considering the degree of non-proportionality of stresses, yields more accurate results than other commonly applied techniques. With the presented example, a safety factor of endurance larger than 1 has been obtained, which is in agreement with test results. No crack initiation has been observed. Future efforts take aim at improving lifetime prediction by gaining further test results to derive more accurate values for material dependent sensitivities of non-proportionality of stresses (which appear e.g. with combined tension/torsion out of phase loadings).

REFERENCES

[1] G. Steinwender, C. Gaier, B. Unger, Improving the Life Time of Dynamically Loaded Components by Fatigue Simulation, SAE-paper 982220, pp. 465-470. [2] C. Gaier, G. Steinwender, H. Dannbauer, FEMFAT-MAX: A FE-Postprocessor for Fatigue Analysis of Multiaxially Loaded Components, NAFEMS-Seminar Fatigue Analysis, 8.-9. November 2000, Wiesbaden, Germany.

[3] C. Gaier, G. Pramhas, W. Steiner, An Extended Critical Plane Criterion for General Load Situations, Proc. of the 8th International Fatigue Congress, 2002, pp. 259-266.

[4] C. Gaier, H. Dannbauer, Fatigue Analysis of Multiaxially Loaded Components with the FE-Postprocessor FEMFAT-MAX, ESIS Publication 31, 2003, pp. 223-240. [5] C.C. Chu, F.A. Conle and A. Huebner, An Integrated Uniaxial and Multiaxial Fatigue Life Prediction Method. VDI Berichte Nr. 1283, Germany, 1996. pp. 337-348 [6] M. Prandstötter, H. Riener and M. Steinbatz, Simulation of an Engine Speed-UP Run: Integration of MBS – FE – EHD - Fatigue. ADAMS User Conference 2002 - Europe [7] CRAIG, R. R; BAMPTON M. C. C., Coupling of Substructures for Dynamics Analysis AIAA Journal, Nr. 6, 1968, pp. 1313 – 1319

[8] CRAIG R. R., Structural Dynamics – An Introduction to Computer Methods John Wiley & Sons, ISBN 0-04499-7, New York, U.S.A, 1981.

c.gaier@ecs.steyr.com Contact address: