Quasi-static superposition, modal superposition, and transient analysis

This and the following sections are intended for the analysis of structures under non-proportional loading conditions. In these sections, we analyze the damage in the time domain.

Whenever a local analysis is done, the first step of the analysis is to get to the local pseudo stress tensor history:

To understand the term pseudo stress, refer to the topic on Multiaxial fatigue. It is the local stress tensor that may be determined by linear elastic finite element codes and may be adjusted for local plasticity by the individual methods.

You have to decide how these local stress tensors are determined:

• For stiff components that show a high first natural frequency and are subjected to low frequency loading, the method of quasi-static superposition is applicable.

• Alternatively, if the component shows low natural frequencies or the loading shows frequencies high enough to induce resonance in the structure, then modal superposition is required.

• If the stress history is already calculated from a transient FE-analysis the method transient analysis may be used.

Quasi-static superposition

For each loading history channel one matching load case has to be calculated by a linear static finite element analysis. It is recommended that the load cases are calculated for normalized loads and the corresponding load histories are scaled in the same unit.

Before solving an analysis case, sometimes it is required to scale the FE results to unit load case results. To achieve this, Specialist Durability allows defining a scaling factor for the load time histories and also allows entering the load applied in the FE calculation.

Two Unit Load Finite Element Calculations Define the Load Influence Factors

Applying the Load History the Local Pseudo Stresses May be Calculated

If we denote by Lk(t) the k-th (calibrated) load history and by (cij)(k)(x) the local stress tensor calculated in the k-th unit load case, the local pseudo stress tensor history is calculated as

The factors (cij)(k)(x) define the load influence factors for the k-th load case.

Modal superposition

For a damage calculation based on modal superposition, some preliminary steps have to be done. First, stress mode shapes have to be calculated by the finite element code for a given number of natural frequencies. The number of modes to be calculated depends on the component and the loading. In a second step, the modal contribution factor histories have to be calculated from the load histories.

To sketch the background of the procedure we consider the simplified equation

Mü + Ku = f

where u denotes the displacement, M the mass matrix, K the stiffness matrix and f the loads. (Damping is neglected for simplicity.)

Let the frequency content of f be contained in an interval which is covered by the first p modes. Then we can express u by the first p mode shapes:

where ϕk denotes the k-th mode shape and βk the modal contribution factors.

By plugging the latter into the original equation we get

Multiplying with from right and noting that the are orthogonal, we get a system of decoupled ordinary differential equations for the contribution factors:

Those contribution factors may be calculated by a modal transient analysis and must be made readable to Specialist Durability.

Once the mode shapes (cij)(k)(x) and the contribution factors βk (t) are available the procedure to calculate the local pseudo stresses is similar to the quasi-static case:

Note that the mode shapes define the load influence factors for the modal superposition (the (cij)(k)(x) terms) whereas the modal contribution factors are treated like "load histories" (the βk (t) terms).

Some guidelines

In this section we want to give some guidelines on when to choose which approach. However, you should check the conditions in The two basic finite life approaches before deciding on which approach to take.

The excitation frequencies that are relevant for durability of vehicles (that is, the amplitudes leading to damage) are between 5 Hz and 30 Hz.

For suspension components like knuckles, control arms, and so on, the natural frequencies usually are above 40 Hz, such that the quasi- static approach is well suited for these cases.

For chassis components like the exhaust system or heavy truck cabins, the natural frequencies usually lie below 15 Hz so that the modal superposition approach is recommended here. Other chassis components like frame cross members may show natural frequencies around 25 Hz such that in these cases the decision as to the approach to use should be made on the individual load conditions and components.

If a transient analysis is performed, then there is no more stress recovery done in Specialist Durability. The quality of the results in this case depends on the quality of the stresses as calculated inside the FE solver.

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Quasi-static superposition, modal superposition, and transient analysis, Simcenter 3D 2021.1 Series

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