Kurt Pötter

Multiaxial Loads and Stresses

Many design tasks are concerned with structural components or systems subject to multicomponent loads. In this case, several forces and moments, which are partially time-independent, can act on the structural component to be evaluated. The local stress state and its variation with time at relevant locations result from the superposition of the local stresses. These stresses are caused by external loads acting on the component.

Supplementary References to Publications on Load Assumptions in Various Specialised Fields

In two publications from 1858 and 1861, A. Wohler already described measurements performed on railway axles under service conditions. By means of a self-developed compound lever system, the deflection of the axle was scratched on a zinc plate by a scriber. This was accomplished for a number of four-wheeled and six-wheeled freight and passenger cars on trips between Breslau and Berlin and between Frankfurt an der Oder and Berlin. Only the maximum moments for bending and torsion per trip could be measured [Wohl1858, Wohl1860, Engi1867, Schu96, Zenn15].

Application of the Counting Methods

In the preceding chapters, the various counting methods have been described and assessed. In the following, the criteria for selection are summarised. For this purpose, the graphical representation of the counting results is first considered. Subsequently, the suitability of these results for analytical fatigue-life predictions is discussed.

Analytical Fatigue-Life Estimate

In the preceding chapters, the procedure for transforming measured stress-time functions to yield a stress spectrum and a frequency matrix has been described. This transformation is a prerequisite for the load assumption, which is necessary for designing and dimensioning of structural components with the required fatigue strength under variable stress amplitude. Dimensioning of components in this manner constitutes a comparison between the loading (stress) and the capacity to withstand loads (strength). For static and endurance-strength designing, characteristic parameters can be simply compared. For designing and dimensioning of components with a defined fatigue life under variable stress amplitude, however, characteristic functions must be employed. For the loading capacity, for instance, such a characteristic function is the S-N curve for the component, and for the stress, the spectrum or frequency matrix is such a function. These characteristic functions are the input parameters for damage-accumulation hypotheses by means of which an allowable number of load cycles to failure is calculated.

Description of the Counting Methods

In the following, the most important counting methods applied for fatigue strength calculation are described and assessed. Recommendations are given for application.

Comparison of the Counting Methods for Exemplary Load Time Functions

For the comparison of various counting methods, four different load time functions are evaluated by means of the following counting methods: level-crossing counting, range-pair counting, transition matrix (transition counting), and rainflow counting.

Design and Dimensioning Load Spectra

For designing and dimensioning a structural component or a structure, a design and dimensioning spectrum which covers the design life is required. The data which are acquired from a measurement constitute only an initial basis. Since the measurement extends only over a short period of time in comparison with the design life, an extrapolation is necessary. In particular, however, a preview over the so-called operational profile during the design life is necessary. If it is assumed that different operational states can occur, an assumption is also required for determining the frequency with which these states occur, for example start and stop procedures, idle time, or different states of operation. For making an assumption of this kind, many years of operational experience are required; the results of a single measurement are often not sufficient. For this purpose, partial spectra can be analysed for individual operational states, and the associated damage content can be evaluated by means of a Palmgren-Miner calculation. The purpose of the fatigue-life estimate is to obtain a relative estimate of the damage contribution from each state of operation, rather than a qualitative lifetime estimate. This can be accomplished with the use of a fictitious S-N curve, for instance, with the slope k = 5 for bending and k = 8 for torsion (Miner elementary, steel component). In correspondence with the severity of the risk associated with a case of damage, the operational profile can be varied in such a way that the load assumption results in a reasonable partial safety factor. A second partial safety factor results from the specification of the tolerable loading capacity (the strength).

Time-at-Level and Level-Distribution Counting

Besides cycle counting of amplitudes further statistical analysis are useful for the analysis of load-time-functions. The time-at-level counting methods are applied in case the duration of a process or operation condition shall be evaluated. Although they are not applicable for life-time prediction directly the time-at-level counting methods are very often used to describe the distribution of different working conditions. In the following, one- and two-parameter time-at-level counting methods are described.

Load Spectra and Matrices

As shown in Sect. 3.1, two-dimensional frequency distributions, so-called load spectra, are the results of one-parameter counting. The frequency of the amplitude or range (double amplitude) is thus represented graphically. The results of two-parameter counting are three-dimensional frequency matrices. With the use of a transition matrix, every half load cycle is recorded in a field of the matrix as a counting event, from a starting class all the way to a target class. From these min-max and max-min values, the amplitude of the stress as well as the mean value of each half cycle can be determined.

Load Assumptions in Various Special Fields

As already pointed out in the preceding chapters, the procedures applied for making load assumptions differ from one specialised field to another in correspondence with the specific boundary conditions involved. Examples include such fields as automotive construction, machine and plant designing, as well as high-volume manufacture and individual constructions. In the following chapter, the load assumption in the series of technical rules is first considered as an example. The procedure applied for deriving a load assumption in the absence of such technical rules is illustrated for the example of automobile construction in Sect. 12.2. Further important publications on load assumptions in various specialised fields are indicated in Chap. 13.