Dynamic LoadingSolid Mechanics Anders Ekberg

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Dynamic loading

More difficult to measure, analyze and estimate dynamic loading compared to static loading

Loads can vary in time and space

Load range (or more accurately stress range) is the most important parameter in fatigue analysis

This calls for a definition of a load cycle (or rather stress cycle)

There can be large statistical scatter in the loading

High frequency content of the loading is difficult to measure and/or analyze. This content may have an effect on the fatigue behaviour

Standardized load spectra are often used as input to the fatigue models.

Dynamic LoadingSolid Mechanics Anders Ekberg

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Varying amplitudes

Residual stresses

Overloads may introduce residual stresses due to plastic deformations.

These stresses may supress the initiation of fatigue cracks and/or lead to closure of existing cracks during (parts of) the load cycle

This is normally beneficial

Add influence from different load cycles

This is far from obvious

In continuum approaches (initiation), this is done by damage accumulation

In crack propagation analysis, it is done by the use of a crack propagation law.

Dynamic LoadingSolid Mechanics Anders Ekberg

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Load cycles

A load cycle is a closed loop in “load space”

For harmonic loading, the load cycle• starts from a certain load magnitude• moves through a max-value and a min-value back to

the start magnitude (or the other way around)

The load cycle is then completely defined by the amplitude and mid value

t

F

t

F

The problem in identifying a load cycle comes when we are not dealing with harmonic loadpaths

Dynamic LoadingSolid Mechanics Anders Ekberg

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σ

t

σ

t

σ

Rainflow counting

Depict the loading sequence as a function of time. For convenience

• start with largest maximum or smallest minimum• use straight lines between (local) minima and maxima

Start from the top and let a “drop” start from every maximum and minimum. A drop stops if:

• it starts from max and passes a larger or equal max• it starts from min and passes a larger or equal min• it reaches the run of another drop

Identify closed loops by joining drops

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σ

t

1

2

35

4

678

Rainflow counting Ð example

1 passes an equally large maximum

2 passes a larger minimum

3 passes a larger maximum

4 reaches the run of drop 2

5 reaches the run of drop 1

6 “falls out”

7 “falls out”

8 reaches the run of drop 6

1 and 6, 2 and 5, 3 and 4; 7 and 8 are running the same distances in the opposite directions.

These couples are forming closed loops in “load space” and can thus be identified as stress cycles with minimum and maximum magnitudes and a mid value.

Dynamic LoadingSolid Mechanics Anders Ekberg

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Rainflow counting Ð notes

Rainflow counting is used to identify the most damaging stress cycles

When applying the rainflow count method, the chronological order of applied loads diminishes. In other words, you no longer know at what instant of time the “overloads” and (their pertinent residual stresses) occur

More than one load component ð rainflow counting should be carried out on the stress

The rainflow count method is only applicable for uniaxial loading. There are theories for how to extend the method to multiaxial loading (however not generally accepted)

The rainflow count method is fairly easy to implement in a computer code. Several such codes exist and are in practical use

Dynamic LoadingSolid Mechanics Anders Ekberg

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A Uniaxial Stress Cycle

Pulsating compression

Alternating tension/compression

time

Pulsating tensionσ a

σa

σa

σ a

σ a

σ a

R = σmin

σmaxσm = 1

2(σmax + σmin ) σa = 1

2(σmax − σmin )

Stress ratio Mid Stress Stress amplitude

R = −1

σm = 0

σa = σmax

R = 0

σm = σmax

2

σa = σmax

2

R

m

a

= −∞=

=

σ σ

σ σ

min

min

2

2

(half the stress range)

Dynamic LoadingSolid Mechanics Anders Ekberg

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10-20 -15 -10 -5 0 5-5

-4

-3

-2

-1

0

1

2

3

4

5R-ratio

R=σ m

in/σ

ma

x

σmin

The R-ratio

R

RR

R

=

= −( )⇒= ⋅

−= ⋅ ⋅

−

σ σ

σ σ σ

σ σ

σ σ

min max

max min

max

max

a

a

a

12

21

21

The stress cycle is defined by the stress amplitude and the R-ratio

σ σmax max= +10