E2. Quantities in Dynamics

E2. Quantities in Dynamics

1. Time-dependent analyses

Post-processing of time-dependent quantities does not present any major difficulty as long as one is not interested in targeting a precise instant or in characterizing a single representative value at all times of the analysis.

For cases where the loading is controlled and not random, a single analysis may be enough. If it is preferred to extract a scalar quantity representative at all times of the analysis, the choice of the standard is the responsibility of the engineer who analyses the results, and it must be justified. A statistical characterization may also be relevant (typically expressed as a percentage).

If the engineer is interested in extracting a set of quantities constituting a vector, the problem of the concomitance of the normalized quantities (absolute, maximum, minimum values, etc.) arises. The time t at which a bending moment is maximal in a point for a torsor does not necessarily coincide with the point of maximal shear stress.

When the loading has a random character, it is important to multiply the number of calculation cases by integrating the random character of the loading. As an example, if we consider N cases of calculations and we construct for a quantity of interest g_i(t) the corresponding dimensioning value G_i for a case i (it can be the maximum positive value and also the maximum negative value to then incorporate them into an approach of actions concomitance). We consider:

• The used dimensioning value, which could be:

• Its average:  

• Its standard deviation:  
  
  

• The estimation of the mean value of G for the N results:

With λ(N) calculated from the Student variable for N samples (calculation cases) for a 95% confidence level (one-tailed) such that:

The table below provides estimations of the mean for various calculation cases.

2. Seismic spectral analysis

The seismic spectral analyses provide representative quantities of the average of an extremum or period.

The purpose of this paragraph is to draw the reader's attention towards important precautions that must be taken to avoid making significant errors in the calculation of the results coming from a quadratic accumulation.

As mentioned in the dynamic chapter, it is possible to quadratically cumulate simply or completely the spectral responses of each mode. The result of these cumulations is a positive value.

It is important to consider that any operation that one wishes to perform on a quantity of interest that is the subject of a quadratic cumulation (simple or complete) of the spectral responses of each mode must be performed before the cumulation.

Let us take a simple illustration with the difference between the displacement of two nodes A and B cumulated simply on modes 1 and 2 (an illustration with a CQC combination would just be more troublesome to write using cross-terms):

Strictly speaking, the difference evoked as an example can be written:

Therefore, it is obvious that estimating the difference afterward is largely erroneous:

The first value is always positive, but it integrates algebraic differences of quantities in the same mode. The second expression might lead to a drastic underestimation of an extremum response. It can also be demonstrated by using the Cauchy-Schwartz inequality that the first estimate is the upper bound of the absolute value of the erroneous calculation presented in the second formula.

A more complex illustration can also be used with the estimation of a von Mises stress based on the main stresses for 2 modes :

An ‘a posteriori’ estimation of the accumulation of the von Mises stresses is largely erroneous:

Example of a specific application: estimation of the opening of a joint between two buildings

In seismic engineering, it is necessary to estimate the opening of a joint between two buildings under seismic conditions to ensure that there is no risk of collision. The accepted practice is to independently calculate the extreme values of displacement of the envelope of each building based on a quadratic accumulation and then to evaluate the maximum opening of the joint by calculating the difference between these two positive values. Thus, an unfavorable phase opposition of the maximum responses considered.