Structural analysis

When the stochastic input loading acting on the structure is exceeding limits of the

material properties, damage and a failure of the structure can occur. These limits

are analyzed for ultimate and fatigue loading, and are described by the ultimate limit

state (ULS) and the fatigue limit state (FLS), respectively. In ULS-analyses, response

loads of the structure are investigated for the exceeding of elastic deformations,

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while in FLS-analyses, the impact of cyclic loading on the lifetime of the structure is

calculated.

Stress calculation

The stress calculation for tubular beam elements is in general a simple task. Axial

stresses are calculated by the division of axial force and structural area; in-plane and

out-of-plane bending stresses are calculated by the division of bending stress and

moment of inertia, multiplied by the radius of the beam cross section. However,

due to the connection of several members in one tubular joint, the calculation of

stresses is more complex in the joints of lattice tower structures. 4 or 6 members are

typically connected in X- and K-joints (cf. Fig. 2.2), respectively. The interaction

of several members in one joint leads to stress concentrations. Therefore, stress

concentration factors (SCF) have to be calculated by dedicated formulas for tubular

joints, as for example described in DNV-RP-C203 [66]. SCFs are defined as the ratio

of hot spot stress (HSS) to local nominal stress.

The superposition of stresses in tubular joints is calculated at in total eight

different points around the circumference of the intersection. HSS σ

are derived

by the summation of the single stress components from axial, in-plane and out-of-

plane action. Therefore, the structural analysis of a multi planar joint such as a

K-joint in lattice tower structures is a comprehensive task. More details about the

number of stresses to be evaluated in such structures are described in Paper 2 in

Appendix A.

Ultimate loads

Each HSS time series obtained by the stress calculation is analyzed on its maximum

absolute stress, representing the ultimate load of the limited time series. Expected

ultimate loads for longer periods, as for example the 50 year recurrence period,

are calculated based on statistical extrapolation of ultimate loads, adapted to the

procedure described in Annex F of IEC 61400-1 [67]. This method extracts a selection

of extreme loads from the response time series and calculates their probabilities

of occurrence. By the use of fitting functions, the expected ultimate load for the

probability of the 50 year recurrence period can be calculated.

Fatigue assessment

For cyclic loading with a constant stress range, fatigue damage occurs when the

number of applied load cycles is exceeding the maximum allowable number of cycles

before fatigue failure for this specific stress range. Fatigue properties of materials

are documented in standards and are typically given as S-N-curves (S-stress range,

N-number of cycles), as for example in DNV-RP-C203 [66]. The fatigue investigation

of stochastic responses is based on the same principle; however, each individual

stress range of the time series has to be analyzed. This is done by performing a

rainflow counting analysis [68]. This analysis extracts all load cycles with its stress

ranges from the response time series. Due to the large amount of data for longer

time series, results are normally collected in a histogram for stress range bins with

its summed up number of cycles.

A common method for fatigue damage calculation has been defined by Palmgren

[69] and Miner [70]. The method is based on the definition that a fatigue failure

occurs when the number of applied load cycles divided by the maximum allowable

number of cycles exceeds D = 1 (D-damage). This simple definition is valid for cyclic

loading with a constant stress range, and has to be extended for the application of

stochastic loading with various stress ranges (Eq. 2.1).

D =



n

N

(2.1)

where D is the accumulated fatigue damage, k is the number of stress range

bins, n

is the number of stress cycles in stress range bin i and N

is the number of

cycles to failure at constant stress range Δσ

.

The above mentioned histogram data is used for the total fatigue damage

calculation based on the Palmgren-Miner rule. Therefore, each stress range bin

contributes with a fraction of fatigue damage to the total fatigue damage D of the

analyzed time series. The result obtained by the total fatigue damage D can be put

into context to the expected lifetime of the cyclic loaded material by dividing the

length of the analyzed time series by the fraction D. For results with D < 1, the

material can survive a corresponding loading longer than expected; for D≥ 1, the

lifetime is actually shorter than expected.

For analyses with focus on relative comparisons only, a simplified fatigue assess-

ment can be performed based on a so-called damage equivalent load (DEL) [71].

This method is not following the complete fatigue assessment with stress analyses

and lifetime calculations, but is only focusing on the fatigue characteristics of the

stochastic response load time series. The basis of this method follows the Palmgren-

Miner rule, too. However, results are treated differently in the DEL-analysis. For

a defined reference number of cycles, the fraction of total fatigue damage can be

transferred back to a specific load cycle range, assuming that this constant load cycle

range (=DEL) causes the same amount of fatigue damage as the stochastic response

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load time series. The so found DEL can be used as a fatigue characteristic of the

response time series. By this, the DEL allows for example for direct comparisons of

several simulation results in a parameter study.