2018-04-10

Modal Contributions and Static Correction

Modal contributions

We analyze the common case in which the dynamic load can be expressed by a constant load vector \(\boldsymbol r\) modulated by an adimensional function of time, \(f(t)\) (e.g., the seismic excitation can be described in such terms).

  • We start our argument introducing the modal partecipation factor: the MPF measures the "tuning" between the loading vector and the modal shape.
  • Reasoning on the definition of the MPF, we find that it is possible to represent the load as a sum of modal load contributions, \(\boldsymbol r = \sum \boldsymbol r_i\).
  • Next, we recognize that it is possible, mode for mode, to represent the modal response in terms of a constant, pseudo-static response, due to the modal load contribution, modulated by a (nearly) adimensional response function.
  • Introducing a) the modal contribution factor, the ratio between the modal response contribution and the static response, b) the dynamic response amplification factor and c) the peak factors, we finally see that it is possible to single out the effects of the static response and the dynamic amplification.

Static correction

We have recognized that, for a given response quantity, we have a static response that, mode by mode, is modulated by a a dynamic amplification and a constant modal contribution factor.

For modal frequencies much larger than the excitation frequency, the dynamic amplification tends to unity and the response is essentially static.

From this consideration, the static correction procedure insert into the response the higher modes contributions in term of an enveloping of a static response that, surprise, can be computed also in terms of the lower modes characteristics!

Numerical Integration

It's the same as Numerical Integration for SDOF system as long as all physical constants are replaced by the corresponding matrices and all the variables and and unknowns are replaced by vectors.

It's different because we cannot use, at least in real problems, numerical procedures that are conditionally stable... we can use only unconditionally stable algorithms

We reviewed the Constant Acceleration algorithm.

We introduced the Wilson's \(\theta\) method, that has the same, or better, accuracy of the Linear Acceleration algorithm but can be made unconditionally stable.

Multiple Support Excitation

To treat multiple support excitation we have to introduce additional degrees of freedom, corresponding the the imposed component of ground motion and partition the augmented coordinates vector in two, the displacements of the superstructure and the (imposed) displacements of the ground. The Superstructure displacement are then represented as the superposition of static displacements and dynamic ones.

In this frame, it is possible to write the equation of motion in a form similar to what we use for earthquake excitation, introducing an influence matrix \(\boldsymbol E\) that describes the static superstructural displacements due to an arbitrary vector of ground displacement, \[\boldsymbol p_\text{eff} = -\boldsymbol M\,\boldsymbol E\, \ddot{\boldsymbol x}_\text{g}.\]

Downloads

Material available for this class:

Giacomo Boffi