2. Principle of the approach#
2.1. Position of the problem under consideration and general principle#
We are in the case of a multi-supported structure, that is to say that the structure has \(m\) degree of freedom of support, each being subject to its own excitation (not necessarily equal everywhere). It is assumed that the structure is represented by a finite element model with \(n\) degrees of freedom. We’re looking for the answer in a finite (and low) number of \(l\) degrees of freedom.
It is assumed that the excitation magnitude is of the imposed movement type and results in a family of \({\mathrm{g}}_{\mathrm{j}}(\mathrm{t})\) accelerograms for each of the degrees of liberty-supports \(j\), \(j=\mathrm{1,}m\).
The absolute movement of the structure is decomposed classically into training movement and relative movement.
The calculation of the response in power interspectra is carried out by modal recombination.
Following this modal calculation, a random dynamic response calculation is divided into three parts:
definition of the exciter power interspectrum,
calculation of the power response interspectrum.
These first two parts are the subject of order DYNA_ALEA_MODAL [U4.53.22].
The reproduction of the response power interspectrum on a physical basis is carried out with the command REST_SPEC_PHYS [U4.63.22].
calculation of statistical parameters from the power interspectrum result.
This last step is processed by the POST_DYNA_ALEA [R7.10.01] [U4.84.04] command.
2.2. Breakdown of the movement#
The following decompositions and projections are detailed in the reference documentation relating to the resolution by transitory calculation of a seismic calculation [R4.05.01]. We have only retained the main points here.
Let \({X}_{a}\) be the absolute displacement vector (of dimension \(n\)) of all the degrees of freedom of the structure.
The total response called absolute \({X}_{a}\) of the structure is expressed as the sum of a relative \({X}_{r}\) contribution and the training contribution \({X}_{e}\) due to anchor movements (subject to accelerations represented by an accelerogram \({g}_{j}(t)\) in each of the degrees of liberty-supports \(j\), \(j\mathrm{=}\mathrm{1,}m\)).
Let \(M,K\) and \(C\) be the mass, stiffness, and damping matrices of the problem, limited to unsupported degrees of freedom.
The equation of motion is then written in the coordinate system linked to relative motion:
\(\begin{array}{c}{{\begin{array}{}\text{M}\ddot{X}\end{array}}_{r}(t)+\text{C {}\dot{X}}_{r}(t)+{\text{K X}}_{r}(t)\text{= -}\text{M}\ddot{{X}_{e}}(t)+{F}_{\text{ext}}\end{array}\)
\({F}_{\text{ext}}\): vector of external forces
In general, the external forces are zero when calculating seismic responses.
2.3. Breakdown on a modal basis#
The response calculation in power interspectra is carried out by modal recombination and uses, in imposed motion, a modal base that includes both dynamic modes and static modes.
Let \(\Phi =\left\{{\phi }_{i,i=\mathrm{1,}n}\right\}\) be the \((n,n)\) matrix of dynamic modes calculated for the associated conservative system, keeping the m supports locked.
Let \(\Psi =\left\{{\psi }_{j,j=\mathrm{1,}m}\right\}\) be the \((n,m)\) matrix of static modes. The \({\Psi }_{j}\) mode corresponds to the deformation of the structure under a unitary displacement imposed at the degree of liberty-support \(j\), the other degrees of liberty-supports being blocked.
The imposed displacement of anchors \({X}_{s}(t)\) is linked to \({X}_{e}(t)\) by the relationship: \({X}_{e}(t)=\Psi {X}_{s}(t)\).
The components of the acceleration of anchor points \({\ddot{X}}_{s}(t)\) are the accelerograms \({g}_{j}(t)\), \(j=\mathrm{1,}m\).
So we can write \(\ddot{{X}_{e}}(t)={\Psi \ddot{X}}_{s}(t)=\sum _{j\text{=1}}^{m}{\psi }_{j}{g}_{j}(t)\text{}\).
We change the variable \({X}_{r}(t)=\Phi \mathrm{.}q(t)\), \(q(t)\) is the vector of the generalized coordinates. By premultiplying the equation of motion by \({}^{T}\text{}\Phi\), we obtain - in the absence of external forces other than seismic excitation - the equation projected on the basis of dynamic modes:
\({}^{T}\text{}\Phi M\Phi \ddot{q}(t)+{}^{T}\text{}\Phi C\Phi \dot{q}(t)+{}^{T}\text{}\Phi K\Phi q(t)=-{}^{T}\text{}\Phi M\Phi {\ddot{X}}_{s}(t)\)
It is assumed that the damping matrix is a linear combination of mass and stiffness matrices (assumption of constant Rayleigh damping on the structure or Basile hypothesis allowing diagonal damping). The base \(\Phi\), which orthogonalizes the matrices \(M\) and \(K\), therefore also orthogonalizes the matrix \(C\).
Given this hypothesis, the previous equation is broken down into \(n\) decoupled scalar equations in the form:
For \(i=\mathrm{1,}n\)
where we noted:
\({\mu }_{i}={}^{T}\text{}{\Phi }_{i}M{\Phi }_{i}\) the modal mass
\({k}_{i}={}^{T}\text{}{\Phi }_{i}K{\Phi }_{i}\) modal rigidity
\({\omega }_{i}=\sqrt{\frac{{k}_{i}}{{\mu }_{i}}}\) the modal pulse
\({\xi }_{i}=\frac{{}^{T}\text{}{\Phi }_{i}\mathrm{.}C\mathrm{.}{\Phi }_{i}}{2{\mu }_{i}{\omega }_{i}}\) reduced modal depreciation »
\({p}_{\text{ij}}=\frac{{}^{T}\text{}{\Phi }_{i}\mathrm{.}M\mathrm{.}{\Psi }_{j}}{{\mu }_{i}}\) the modal participation factor of support \(j\) over dynamic mode \(i\).
Solution \({q}_{i}(t)\) of this equation corresponds to the response of the dynamic mode \(i\) to the entire seismic excitation.
We can further decompose the problem by introducing the unknown \({d}_{\text{ij}}(t)\) solution of the differential equation: \({\ddot{\mathrm{d}}}_{\text{ij}}+2{{\xi }_{i}{\omega }_{i}\dot{\mathrm{d}}}_{\text{ij}}+{\omega }_{i}^{2}{\mathrm{d}}_{\text{ij}}\mathrm{=}{g}_{j}(t)\), this last equation corresponds to the response of the dynamic mode \(i\) to acceleration
. The relative displacement on the physical basis is then expressed:
\({X}_{r}(t)=-\sum _{i=1}^{n}\sum _{j=1}^{m}{p}_{\text{ij}}{d}_{\text{ij}}(t){\varphi }_{i}\)
Information on the position of the fulcrum is contained in the modal participation factor.
2.4. Harmonic response#
The total response of the structure was therefore decomposed into a relative contribution and a differential contribution due to the movements of the anchors such as:
\(\begin{array}{}{X}_{a}(t)\text{=}{X}_{r}(t)\text{+}{X}_{e}(t)\\ \text{avec}\\ \{\begin{array}{}{\ddot{X}}_{e}(t)\text{=}{\Psi \ddot{X}}_{s}(t)\text{=}\sum _{j=1}^{m}{\psi }_{\text{j}}{g}_{\text{j}}(t)\\ {X}_{r}(t)\text{= -}\sum _{i\text{=}1}^{n}\sum _{j=1}^{m}{{p}_{\text{ij}}d}_{\text{ij}}(t){\varphi }_{i}\text{}\text{où}{d}_{\text{ij}(t)}\text{est solution de}\text{}{\ddot{d}}_{\text{ij}}\text{+}2{\xi }_{i}{\omega }_{i}{\dot{d}}_{\text{ij}}\text{+}{\omega }_{i}^{2}{d}_{\text{ij}}\text{=}{g}_{j}(t)\end{array}\end{array}\)
The resolution of this last one differential equation by the Fourier transform method involves modal transfer functions \({h}_{i}(\omega )\) such as:. \({h}_{i}(\omega )=\frac{1}{{\omega }_{i}^{2}-{\omega }^{2}+2i{\xi }_{i}{\omega }_{i}\omega }\)
So we get: \({d}_{\text{ij}}(\omega )={h}_{i}(\omega )\text{.}{g}_{j}(\omega )\text{et}{\ddot{d}}_{\text{ij}}(\omega )\text{=-}{\omega }^{2}{h}_{i}(\omega )\text{.}{g}_{j}(\omega )\)
The total harmonic response of the structure is deduced from the preceding formulas by modal recombination.
\(\begin{array}{}{\ddot{X}}_{a}(\omega )={\ddot{X}}_{r}(\omega )+{\ddot{X}}_{e}(\omega )\\ {\ddot{X}}_{a}(\omega )={\omega }^{2}\sum _{i=1}^{n}\sum _{j=1}^{m}{p}_{\text{ij}}{h}_{j}(\omega ){\varphi }_{j}(\omega ){j}_{i}+\sum _{j\text{=1}}^{m}{\Psi }_{j}{g}_{j}(\omega )\end{array}\)
The following complex matrix \((n,m)\), referred to as the transfer matrix \(H(\omega )\), is then made to appear:
\(H(\omega )\text{=}{\omega }^{2}P\mathrm{.}h(\omega )\Phi \text{+}\Psi\)
where \(P\) is the participation factor matrix, \(h(\omega )\) is the vector of the modal transfer functions \({h}_{i}(\omega )\).
The total response of the structure is \({\ddot{X}}_{a}(\omega )=H(\omega )\ddot{E}(\omega )\) where \(\ddot{E}(\omega )\) is the vector of m lines consisting of Fourier transforms of accelerations
at the \(m\) degrees of freedom - supports.
We can see that this expression determines the acceleration response. This then requires integrating the answer twice to obtain the displacement, this problem is presented in [bib4]. One of the additional advantages of the method that we propose here is to get rid of this difficulty.