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A damage tolerance analysis for complex structures


The assessment of damage is usually performed to determine safety or residual life, which is necessary to define inspection intervals for instance. The concept of a damage tolerance analysis presented in this paper ties on and tries to reveal the possible redundancy that statically indeterminate complex structures provide. Classic strength assessments are not able to map changes in load paths or structural supporting effects and that might lead to a huge waste of potential in weight reduction. The damage tolerance analysis is applied to an exemplary complex structure and shows that the fatigue driven damages are not critical for further operation at the assessed load level. The concept uses a new method for the calculation of equivalent stress—the ‘Modified Mohr Mises’ (MMM)-Hypothesis. The MMM-Hypothesis allows the assessment of non-proportional stresses within a fully automated process, due to its invariant equivalent stress notation. Those non-proportional stresses are most common in complex structures of aircrafts, spacecrafts and vehicles for instance.

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\(\varepsilon \) :


\(\sigma _{\mathrm{eq},a,\mathrm{MMM}}\) :

Equivalent amplitude of MMM stress

\(R_\mathrm{M}\) :

Interim invariant radius of Mohr

\(m_{a}\) :

Macro-supporting effect factor

\(\sigma _1, \sigma _2 , \sigma _3\) :

Principal stresses according to MMM

\(\varepsilon _a\) :

Strain amplitude

\(\hat{{\sigma }}_{a,1}\) :

Highest alternating principal stress

\(\eta \) :

Dimensionless time

\(G_{\mathrm{MMM}}\) :

Normalized stress gradient

\(\hat{{\varepsilon }}_\mathrm{H}\) :

Linear elastic maximum (Hooke-) strain

\(\sigma _{\mathrm{loc}}^*\) :

Local stress in equivalent tension specimen


Lower Neuber stress level

\(\sigma _{\mathrm{eq},m}\) :

Equivalent MMM mean stress

\(N_{\mathrm{alt}}\) :

Iteration load cycle

\(\sigma _{-1,N}\) :

Compression–tension fatigue limit for certain load cycle

\(\sigma _{\mathrm{eq},\mathrm{Mises}}\) :

Equivalent von Mises stress

\(\sigma _x^o , \sigma _y^o , \sigma _z^o\) :

Normal stresses from FEM, BEM calculation

\(\sigma \) :


\(C_\mathrm{M}\) :

Invariant center of Mohr

V :

Sign function

\(n_{el}\) :

\(\hbox {K}_{\mathrm{t}}\)\(\hbox {K}_{\mathrm{f}}\) ratio (micro-supporting factor)

\(k_a\) :

Tension–shear endurance limit ratio

N :

Load cycle

\(\hat{{\sigma }}_{o,1}\) :

Highest maximum principal stress

\(\varphi \) :

Constraint factor

\(\hat{{\sigma }}_\mathrm{H}\) :

Linear elastic maximum (Hooke-) stress

\(\sigma _{\mathrm{loc}}\) :

Local stress in notch

\(\varepsilon _{\mathrm{loc}}\) :

Local strain in notch


Lower Neuber strain level

\(\sigma _{\mathrm{eq},A,\mathrm{MMM}}\) :

Sustainable equivalent amplitude of MMM stress

\(N_\mathrm{E}\) :

Endurance load cycle

\(k_{aN}\) :

Tension–shear limit ratio for certain load cycle

\(\sigma ^{\prime }\) :

Stresses without supporting effects

\(\tau _{xy}^o, \tau _{xz}^o, \tau _{yz}^o\) :

Shear stresses from FEM, BEM calculation


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Pucknat, D., Liebich, R. A damage tolerance analysis for complex structures. Arch Appl Mech 86, 669–686 (2016).

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  • Damage tolerance
  • Fatigue
  • Crack propagation
  • MMM-Hypothesis
  • Mises-Hypothesis
  • Non-proportional loading