Response Spectrums

The Percent memory allocation on the Analysis Parameters dialog box controls how much of the available RAM is used to read the element data and to assemble the matrices. (When the value is less than or equal to 100%, the available physical memory is used. When the value of this input is greater than 100%, the memory allocation uses available physical and virtual memory.)

Input Spectrums

The input spectrum is defined by pressing the Spectrum Data button on the Analysis Parameters dialog box. Use this screen to enter a user-defined spectrum or to have a spectrum made for you by defining the oscillating frequency and damping ratio.

Since the Response Spectrum analysis uses the results from a modal analysis, specify which design scenario in the current model has the modal results with the Use modal results from Design Scenario field.

Note: The Model Units of the response spectrum analysis must be identical to the Model Units of the modal results.

Input Spectrum Type Then, define the type of spectrum by selecting the appropriate radio button in the Input Spectrum Type section. The spectrum can be defined as Displacement, Acceleration, or g versus the Period. The spectrum is entered in the spreadsheet, where the column heading changes depending on the input spectrum type selected. You can also import spectrum data from a comma delimited file (.CSV) by pressing the Import button.

Tip: The values in the comma-separated file need to match the spectrum type and the active Display Units. Change the Display Units if necessary before importing the data. For example, a value of 314 is imported as 314 in 2 /sec for Acceleration vs. Period spectrum when the Display Units are in English (in), and 314 m 2 /sec per Hz when the Display Units are in metric SI.

Factors Next, select the direction along which the spectrum is applied to the model. This is done in the Factors section. Specify the individual factors for each global direction in the X Dir, Y Dir, and Z Dir fields. A factor of 1 applies the spectrum as defined in that direction. Values other than 1 or 0 can be used, and the spectrum is scaled accordingly.

If you activate the Cluster check box, the modal affects of close frequencies is combined differently (as described below). The value in the Cluster field is used to determine if two frequencies are close together. A value of 0.1 groups together all frequencies that are within 10% of each other. Specifically, two frequencies are close to each other when

(Frequency i - Frequency i-1)/(Frequency i-1) <= Cluster value

where Frequency i is the next mode higher than Frequency i-1. Three (or more) sequential modes can be linked through the equations (see below) if each pair is within the cluster value of each other even though the first and last frequency may not be within the cluster. For example, modes 3, 4, and 5 are handled in the crossover terms in the equations below if frequencies 3 and 4 are within the cluster value and frequencies 4 and 5 are within the cluster value even though frequencies 3 and 5 are not within the cluster value.

Response Spectrum Parameters If you want the program to make a spectrum for you, activate the Generate Response Spectrum check box and specify values in the Oscillating Frequency (Hz) and Damping Ratio (0.01 = 1%) fields. It is helpful to keep in mind that a response spectrum is simply a graph of the maximum value (usually displacement or acceleration) versus the natural frequency (or period) of a single degree of freedom system. So, if you had a single degree of freedom system and vibrated it at some forcing frequency, you could calculate the maximum response using the well known equations from any vibrations book:

Magnification factor = 1 / [ (1-r^2)^2 + (2*damping*r)^2 ]^0.5

Magnification factor for acceleration = Magnification factor for displacement * r^2

where r is the frequency ratio (= forcing frequency/natural frequency). The above equation is often plotted with the frequency ratio r on the abscissa and the magnification factor on the ordinate. This equation is based only on the natural frequency and the damping ratio. Such graphs start with a magnification factor of 1 at a 0 forcing frequency, reach a peak value near resonance (forcing frequency = natural frequency), and then decline towards 0 as the forcing frequency approached infinity.

The previous equations are used to create the spectrum when you choose to generate a spectrum. The Oscillating Frequency that is entered is the natural frequency of the equivalent single degree of freedom system, and the damping ratio controls the peak magnification factor. The difference between the typical magnification factor graph and the spreadsheet/graph input on the Analysis Parameter is that the Analysis Parameters dialog box uses the period on the abscissa instead of the frequency (or frequency ratio). When plotted against the frequency, the above equation starts at a value of 1. So when plotted against period, the above equation ends at a value of 1.

The previous equation has a unit amplitude. You can specify a value in the Scale Factor field which multiplies the magnitude of the spectrum when it is applied to the model to get the total amplitude needed. The graph is not affected by the scale factor. Thus, the maximum amplitude of the spectrum when using the Generate Response Spectrum option is equal to Scale Factor * X Dir factor, Y Dir factor, or Z Dir factor. For example, if the known acceleration magnitude is 100 in/sec^2, the scale factor is entered as 100 when using an acceleration version period spectrum. Using g versus period, the scale factor is entered as 0.2588 (=100/386.4).

Control How Results Calculate

There are three different ways for the processor to calculate the response of the model. The method is specified by selecting the appropriate radio button in the Output section of the Response Spectrum Analysis Input dialog box. The methods available are as follows:

  1. Original Procedure (uses the square root sum of the squares SRSS method).
    • This method provides a reasonable estimate if all the natural frequencies are well separated.
    • The number of results created (number of load cases) equals the number of mode shapes in the natural frequency analysis plus one. [Nload cases = Nmodes + 1]
    • The Original Procedure option uses the following equations to find the response of the model and outputs the following values:

      Response at mode i due to all spectrum directions:

      UiX = XdirFactor*UiXX + YdirFactor*UiXY + ZdirFactor*UiXZ

      UiY = XdirFactor*UiYX + YdirFactor*UiYY + ZdirFactor*UiYZ

      UiZ = XdirFactor*UiZX + YdirFactor*UiZY + ZdirFactor*UiZZ

      Resultant:

      UX = SQRT(U1X**2 + U2X**2 + +UNX**2)

      UY = SQRT(U1Y**2 + U2Y**2 + +UNY**2)

      UZ = SQRT(U1Z**2 + U2Z**2 + +UNZ**2)

  2. NRC Reg. Guide 1.92 (also known as the Ten Percent Method).
    • This method provides a better estimate for structures having closely spaced natural frequencies.
    • When the Cluster input is 0.1, this method is identical to the Ten Percent Method.
    • This method can be used when the spectra in different directions are statistically independent (uncorrelated).
    • The square root sum of the squares (SRSS) of the different spatial components (the factor directions) is done after using the NRC Reg. Guide 1.92 equations.
    • The number of results created (number of load cases) equals the (number of mode shapes in the natural frequency analysis) multiplied by the (number of factor directions) plus one. [Nload cases = Nmodes * Ndirections + 1]
    • The NRC Reg. Guide 1.92 option uses the following equations to find the response of the model and outputs the following values:

      Response at each mode i due to each spectrum direction:

      UiX = XdirFactor*UiXX, YdirFactor*UiXY, or ZdirFactor*UiXZ

      UiY = XdirFactor*UiYX, YdirFactor*UiYY, or ZdirFactor*UiYZ

      UiZ = XdirFactor*UiZX, YdirFactor*UiZY, or ZdirFactor*UiZZ

      Resultant:

      UX = SQRT(U1XX**2 + U2XX**2 + + (|UJXX| + |UKXX|)**2 + + UNXX**2

      + U1XY**2 + U2XY**2 + + (|UJXY| + |UKXY|)**2 + + UNXY**2

      + U1XZ**2 + U2XZ**2 + + (|UJXZ| + |UKXZ|)**2 + + UNXZ**2)

      UY = SQRT(U1YX**2 + U2YX**2 + + (|UJYX| + |UKYX|)**2 + + UNYX**2

      + U1YY**2 + U2YY**2 + + (|UJYY| + |UKYY|)**2 + + UNYY**2

      + U1YZ**2 + U2YZ**2 + + (|UJYZ| + |UKYZ|)**2 + + UNYZ**2)

      UZ = SQRT(U1ZX**2 + U2ZX**2 + + (|UJZX| + |UKZX|)**2 + + UNZX**2

      + U1ZY**2 + U2ZY**2 + + (|UJZY| + |UKZY|)**2 + + UNZY**2

      + U1ZZ**2 + U2ZZ**2 + + (|UJZZ| + |UKZZ|)**2 + + UNZZ**2)

      where any two sequential modes J and K are within the cluster factor of each other.

  3. Modified Procedure (Modified SRSS)
    • Algebraic summation is used through spatial directions if multiple directions are entered for the Factors.
    • The number of results created (number of load cases) equals the (number of mode shapes in the natural frequency analysis) multiplied by the (number of factor directions) plus one. [Nload cases = Nmodes * Ndirections + 1]
    • The Modified Procedure option uses the following equations to find the response of the model and outputs the following values:

      Response at each mode i due to each spectrum direction:

      UiX = XdirFactor*UiXX, YdirFactor*UiXY, or ZdirFactor*UiXZ

      UiY = XdirFactor*UiYX, YdirFactor*UiYY, or ZdirFactor*UiYZ

      UiZ = XdirFactor*UiZX, YdirFactor*UiZY, or ZdirFactor*UiZZ

      Resultant:

      UX = SQRT(U1XX**2 + U2XX**2 + + (|UJXX| + |UKXX|)**2 + + UNXX**2)

      + SQRT(U1XY**2 + U2XY**2 + + (|UJXY| + |UKXY|)**2 + + UNXY**2)

      + SQRT(U1XZ**2 + U2XZ**2 + + (|UJXZ| + |UKXZ|)**2 + + UNXZ**2)

      UY = SQRT(U1YX**2 + U2YX**2 + + (|UJYX| + |UKYX|)**2 + + UNYX**2)

      + SQRT(U1YY**2 + U2YY**2 + + (|UJYY| + |UKYY|)**2 + + UNYY**2)

      + SQRT(U1YZ**2 + U2YZ**2 + + (|UJYZ| + |UKYZ|)**2 + + UNYZ**2)

      UZ = SQRT(U1ZX**2 + U2ZX**2 + + (|UJZX| + |UKZX|)**2 + + UNZX**2)

      + SQRT(U1ZY**2 + U2ZY**2 + + (|UJZY| + |UKZY|)**2 + + UNZY**2)

      + SQRT(U1ZZ**2 + U2ZZ**2 + + (|UJZZ| + |UKZZ|)**2 + + UNZZ**2)

      where any two sequential modes J and K are within the cluster factor of each other.

where

  • N = number of modes
  • U = response to force, moment, translation, and so on
  • UX, UY, UZ = response in the X, Y, and Z directions, respectively
  • UiX, UiY, UiZ = response in i(th) mode in the X, Y, and Z directions, respectively
  • UiXX, UiXY, UiXZ = response in i(th) mode in the X direction, due to X, Y, and Z spectrum input, respectively
  • UiYX, UiYY, UiYZ = response in i(th) mode in the Y direction, due to X, Y, and Z spectrum input, respectively
  • UiZX, UiZY, UiZZ = response in i(th) mode in the Z direction, due to X, Y, and Z spectrum input, respectively
  • XdirFactor = input for the Factor: X Dir
  • YdirFactor = input for the Factor: Y Dir
  • ZdirFactor = input for the Factor: Z Dir
Note: The resultant load cases in each method are square roots of the sum of the squares. Consequently, all components of stress and displacements are missing the negative values. These results may not be physically meaningful.

The Modal Effects check box is applicable only for the NRC Reg. Guide 1.92 and Modified Procedure types of output and only when the Displacement data output is requested (see below). Activating the Modal Effects check box outputs the displacement results at every node in the model for every natural frequency (or mode) due to each response spectrum loading direction. Since this output can be quite voluminous, and the results available in the Results environment are not affected by this setting, it may be better to use the Results environment if only selected results are needed.

Control Data in Output Files

Before the analysis is performed, you can indicate to output additional results. Use the Output Controls section of the Analysis Parameters dialog box to indicate what results to write out.

  • Activate the Displacement data check box or the Stress data check box to get the relevant results in the summary file or stress text file, respectively. Since this output can be quite voluminous, and the results available in the Results environment are not affected by these settings, it may be better to use the Results environment if only selected results are needed.
  • Activate the Stress/strain at midside nodes (binary) check box to get the stress and strain at midside nodes output to the binary result files. These results can then be displayed in the Results environment. (Midside nodes are an option that can be activated for certain element types using the Element Definition dialog box.)