5. Treatment of modal results#

In this case, the command takes as input the dynamic modes of the structure (conceptmode_meca). The operations carried out are as follows:

  1. Calculation of modal torsors.

  1. Calculation of responses mode by mode from modes and spectra.

  2. Combination of modal responses using method CQC (signed or not): this provides the maximum probable response for each earthquake direction \(X\), \(Y\), \(Z\) (in the global system).

  1. If requested, combination of responses direction by direction (Newmark or quadratic combination)

Operation 1) is performed as described in section 4, mode by mode.

Operations 2 - 4 are detailed below.

We consider spectra in mono-support: the supports all see the same imposed displacement.

5.1. Calculating the oscillator response#

5.1.1. Transitional calculation reminders on a modal basis#

The structure studied is represented by its spectrum of real natural modes at low frequency \(\mathrm{\phi }\) in an embedded base, a solution of \(\left(K-M{\mathrm{\omega }}^{2}\right)\mathrm{\phi }=0\).

In the dynamic equation of the system we can introduce a new transformation \(x=\mathrm{\phi }q\), the system is then written, using the matrix of modal participation factors \(P\):

(5.1)#\[ \ ddot {q} +\ frac {{\ mathrm {\ phi}}} ^ {T} C\ mathrm {\ phi}} {{\ mathrm {\ phi}} ^ {T} M\ mathrm {\ phi}}}\ mathrm {\ phi}}}\ dot {\ phi}} ^ {2} q=-\ frac {{\ mathrm {\ phi}}}} ^ {T} MO} {{\ mathrm {\ phi}}} ^ {T} M\ mathrm {\ phi}}\ ddot {s} =-P\ ddot {s} =-P\ ddot {s}\]

where \(\mathrm{O}\) are rigid body modes.

It is also assumed that for industrial studies involving seismic analysis by spectral method, we limit ourselves to the case of proportional damping, called Rayleigh, for which we can diagonalize the term \(\frac{{\mathrm{\phi }}^{T}C\mathrm{\phi }}{{\mathrm{\phi }}^{T}M\mathrm{\phi }}=2\mathrm{\xi }\mathrm{\omega }\). Damping is then represented by a modal damping \({\xi }_{i}\) that may be different for each specific mode.

With these hypotheses, the system of equations () is composed of independent equations, each relating to an eigenmode of pulsation \({\mathrm{\omega }}_{i}\). Each equation describes the behavior of a simple parameter oscillator \(({\omega }_{i},{\xi }_{i})\) whose behavior is represented in single support by:

(5.2)#\[ \ ddot {{q} _ {i}} +2 {\ xi}} +2 {\ xi} _ {i} {\ omega} _ {i}} _ {i}} + {\ omega} ^ {2} {q}} {q} _ {i} _ {i} =- {p} _ {i}\ ddot {s}\]

where \(\ddot{s}\) is the training acceleration and \({p}_{i}\) is the modal participation factor for the \(i\) mode:

(5.3)#\[ {p} _ {i} =\ frac {{\ mathrm {\ phi}}} _ {i} ^ {T} MO} {{\ mathrm {\ phi}} _ {i} ^ {T} M {\ mathrm {\ phi}} M {\ mathrm {\ phi}} M {\ mathrm {\ phi}}} _ {i}} ^ {T} MO} {{\ mathrm {\ mu}} _ {i}}\]

where \({\mu }_{i}\) is the generalized modal mass, which depends on the normalization of the eigenmode.

Suppose that the earthquake acts only in one of the three directions of the global coordinate system \(X,Y,Z\): if for example we take it in the direction \(X\), the vector \(O\) will have \(1\) for the degrees of freedom DX and and \(0\) for the others. For example, always in the \(X\) direction, we will then have:

\[\]

: label: eq-4

{p} _ {mathit {iX}}} =frac {{mathrm {phi}}} _ {i} ^ {T} MO} {{mathrm {phi}}} _ {i} ^ {T} ^ {T}} M {mathrm {phi}} ^ {T}} M {mathrm {phi}} ^ {T}} _ {i}} ^ {T}} M {mathrm {phi}} ^ {T}} {T} {T}} =frac {{{mathrm {phi}}} _ {i}} ^ {T}} T} M {O} _ {X}} {{mathrm {mu}} _ {i}}

\({p}_{\mathit{iX}}\) being called a modal participation factor in the direction of \(X\). For more details on transient analysis on a modal basis see documentation R4.05.01.

5.1.2. Reminders of the spectral method#

The maximum relative displacement response of the parameter oscillator \(({\mathrm{\omega }}_{i},{\mathrm{\xi }}_{i})\) for a direction \(X\) is determined by reading from an absolute pseudo-acceleration oscillator spectrum \(\mathit{SRO}(\ddot{{x}_{X}}({\mathrm{\xi }}_{i},{\mathrm{\omega }}_{i}))\) the value \({a}_{\mathit{iX}}\) that corresponds to cuts \(({\mathrm{\omega }}_{i},{\mathrm{\xi }}_{i})\) and by dividing by \({\mathrm{\omega }}_{i}^{2}\), from where:

\[\]

: label: eq-5

{q} _ {mathit {ixMax}} = {p}} = {p} _ {mathit {ix}}frac {mathit {SRO}} ({ddot {x}} _ {x}} ({mathrm {xi}}} _ {xi}} _ {i}))} {{mathrm {omega}}} _ {i} ^ {2}} = {p} _ {mathit {iX}}}frac {{a} _ {mathit {iX}}}} {{mathrm {omega}}} _ {i} ^ {2}}

Remember that we set \(x=\mathrm{\phi }q\) so \({x}_{i}={q}_{i}{\mathrm{\phi }}_{i}\) if we consider only one mode. The \({x}_{\mathit{iXmax}}\) contribution of this oscillator to the relative displacement of the structure for an earthquake in the \(X\) direction depends on the participation factor and on the modal deformation \({\mathrm{\phi }}_{i}\) in physical space:

(5.4)#\[ {x} _ {\ mathit {ixMax}}} = {\ mathrm {\ phi}}} _ {i} {q} _ {\ mathit {ixMax}} = {\ mathrm {\ phi}}} _ {i} {p}} _ {i} {p}} _ {\ mathit {iX}}} {{\ mathrm {\ omega}}} {{\ mathrm {\ omega}}} _ {i} ^ {2}}\]

Suppose we have \({r}_{i}\), a quantity (field or scalar variable) derived by a linear application of \({\mathrm{\phi }}_{i}\) modes, for example an effort component on a node. The contribution \({r}_{\mathit{iXmax}}\) where \({R}_{\mathit{iX}}\) of this oscillator to the response of the structure for an earthquake is also written:

(5.5)#\[ {R} _ {\ mathit {ix}}} = {r}} = {r} _ {\ mathit {ixMax}} = {r} _ {\ mathit {ixMax}} = {r} _ {i} {i} {p} _ {p} _ {\ mathit {iX}} _ {\ mathit {ix}}} {\ mathit {ixMax}}} = {\ mathit {ixMax}}} = {r} _ {r} _ {r} _ {r} _ {r} _ {r} _ {r} _ {r} _ {i} _ {\ mathrm {\ omega}}} _ {i} ^ {2}}\]

In the case of the CALC_COUPURE command, the \({r}_{i}\) components are:

  • the shell force components \(\mathit{NXX}\), \(\mathit{NYY}\), \(\mathit{NXY}\), etc. at a point on the AB cut line

  • the resultants/resulting moments on the AB cut line, detailed in section Error: Reference source not found

For more details see documentation R4.05.03.

5.1.3. Combining modes for one direction#

The first modal combination carried out produces the probable maximum response in the directions \(X,Y,Z\) for which the spectra will have been provided at the input of the command.

The method used is the signed Complete Quadratic Combination (or signed CQC), in mono-support. Efforts \({R}_{\mathit{iX}}\) (scalar component on a node or on a resultant) are therefore combined in the following way to give the answer \({R}_{\mathit{mX}}\):

(5.6)#\[ {R} _ {\ mathit {mX}}} =\ frac {{R}} _ {\ mathit {kX}}} {\ left| {R} _ {\ mathit {kX}}\ right|}\ right|}}\ sqrt {{\ sum}} _ {\ sqrt {\ sum} _ {\ mathit {ij}}} {R} _ {\ mathit {iX}} {R} _ {\ mathit {jX}}}\]

The amounts are based on the \(N\) modes considered. We define the correlation coefficient between modes \({\mathrm{\rho }}_{\mathit{ij}}\):

(5.7)#\[ {\ rho} _ {\ mathit {ij}}} =\ frac {8\ sqrt {{\ xi} _ {i} {\ xi} _ {\ omega} _ {i} {\ omega} _ {\ omega} _ {j} _ {j}} _ {j}} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j}} _ {j} _ {j} _ {j} _ {j} _ {j}} _ {j} _ {j} _ {j} _ {j}} _ {i} {\ omega} _ {j}}} {{({\ omega}}} {{i} ^ {2} - {\ omega} _ {j} ^ {2})} ^ {2} +4 {\ xi} _ {i} _ {i} {\ xi} _ {i} {\ xi} _ {i} {\ xi} _ {i} {\ xi} _ {2} _ {i} {\ xi} _ {i} {\ xi} _ {2} _ {i} {\ xi} _ {2} _ {i} {\ xi} _ {2} _ {i} ^ {2} _ {2} + {\ omega} _ {j} ^ {2}) +4 ({\ xi} _ {i} ^ {2} + {\ xi} _ {j} ^ {2}) {\ omega} _ {i} _ {i} ^ {2} {i} ^ {2}} {i} ^ {2}}\]

or, by introducing the pulsation or frequency ratio between two \(\mathrm{\eta }={\mathrm{\omega }}_{j}/{\mathrm{\omega }}_{i}\) modes:

(5.8)#\[ {\ rho} _ {\ mathit {ij}}} =\ frac {8\ frac {8\ eta\ sqrt {{\ xi} _ {j}\ eta} ({\ xi} _ {i} + {\ xi} _ {i} + {\ xi} _ {i}} + {i} _ {i} + {i}} + {i} _ {i} + {i} _ {i} + {i}} + {i} _ {i} + {i} + {xi} _ {i} + {i} _ {i} + {\ xi} _ {i} + {\ xi} _ {i} + {i}} + {\ xi} _ {i} + {i}} + {\ xi} _ {i} + {i}} + {\ xi} _ {i} {\ xi} _ {j} (1+ {\ eta} ^ {2}) +4 {\ eta} ^ {2} ({\ xi} _ {i} ^ {2} + {\ xi} + {\ xi} _ {j} ^ {2})}\]

For more details see documentation R4.05.03.

5.1.4. Combination of directions#

5.1.4.1. Newmark combinations#

For each of the directions \(X,Y,Z\), we choose a main direction and calculate the following eight values. For example, for \(X\):

(5.9)#\[ {R} _ {l} =\ pm {R} _ {\ mathit {mX}}\ pm\ mathrm {0.4} {R} _ {\ mathit {mY}}\ pm\ mathrm {0.4} {0.4} {R}} _ {\ mathit {mZ}}\]

24 combinations are then obtained. These combinations are identified in the table at the output of the command.

5.1.4.2. Quadratic combination#

We calculate the quadratic combination of the contributions from each direction:

(5.10)#\[ R=\ sqrt {{R} _ {\ mathit {mX}}} ^ {2}} + {R} _ {\ mathit {mY}} ^ {2} + {R} _ {\ mathit {mZ}}} ^ {2}}\]

The output table contains 4 order numbers: 1, 2 and 3 corresponding to the contributions of the earthquake in the X, Y and Z directions, respectively. As for the order number 4, it corresponds to the quadratic combination of the contributions of the 3 directions.

{R} _ {m} =text {max} ({R} _ {l})