Harmonic Response - Problem Description
Tutorial 5: Spectral Response Analysis - Earthquake Load
Response Spectrum Curve
The earthquake load is acting horizontally in the X-X direction and it is given with a typical Acceleration Response Spectrum Curve as shown in Figure 5.4 below.
Figure 5.4 - Acceleration Response Spectrum Curve for Damping Ratio of 5.0%
An earthquake excitation of the ground can be given in the form of a time history of the ground acceleration, or in the form of a Response Spectrum. The spectral response approach is more common, and it is utilised by almost every modern design code. The Response Spectrum curve used in this tutorial is taken from the Australian Earthquake Loading design code AS 1170.4 - 1993. Any other national code will have a spectral curve similar to this. Sometimes, in the design codes, the Response Spectrum curve is referred to as a Spectral Curve or Design Spectrum. Whatever the terminology used, the analytical approach is identical. In the design codes, the values of the Response Spectrum curves are always normalised using non-dimensional values or some other convenient units. There is a scaling factor or multiplier, which multiplies the values on the vertical axis of the curve. After scaling, these values usually have the dimension of acceleration, i.e. m/s^2 or mm/s^2. The horizontal axis represents the frequency content of the design earthquake and this excitation is applied according to the natural frequency of the structure. The units for the
horizontal axis are seconds. Sometimes, in mechanical engineering applications, such as in shock analysis or machine vibration, the higher frequencies are of interest and the horizontal axis represents the natural frequencies of the structure, (in Hz). This is simply the reciprocal of the natural period.
maximum response of a single degree of freedom system, defined with its natural period and its damping. For different damping there will be different response curves. Only this type of curve is used in the design codes. In these design codes, the local soil conditions determine the damping and are commonly referred to as the governing site factor. The other type, known as Spectral curve or simply as Spectrum, is a Fourier Transform of time process. When the time process is an acceleration record of an earthquake, the Spectral curve is only a mathematical transformation of this data from the time domain into the frequency domain. This type of Spectral curve provides information about the frequency content of the time process. It is not used in design, and it is usually produced by vibration monitoring instruments.
As mentioned earlier, the design codes give the Response Spectrum curve in a non-dimensional form. A separate scaling factor is provided which multiplies the curve. For this tutorial the scaling factor for the vertical axis of the Response Spectrum curve is defined below.
Where:
Note: Values taken from Australian Earthquake Loading design code AS1170.4-1993 Part 4.
The Response Spectrum curve should be multiplied by the Scaling Factor and Gravity.
After this multiplication, the units of the vertical axis are m/s2
(i.e. acceleration). In different national design codes there are different expressions for this scaling factor. But in the end, it is only one number that multiplies the vertical axis of the Response Spectrum curve.
Scaling Factor - multiplier of the spectral curve
a = 0.09 - acceleration coefficient (See note below) I = 1.0 - importance factor (See note below) Rf = 4.0 - structural response factor (See note below)
The Response Spectrum curve shown in Figure 5.4 above is defined with the following expression.
In order to determine the location of intersection of the flat part and the curved part of the Response Spectrum curve (i.e. X), we substitute a spectral acceleration value of 2.5 into the second of the above expressions. Solving the
equation for T we obtain 0.35355. Now the Response Spectrum curve can be expressed in terms of period T as follows.
Where:
Strand7 allows you to input the Response Spectrum curve in terms of either period or frequency. Thus the above expressions are suitable for entering directly into Strand7 to form a Spectral Table.
1. Choose Tables/Factor vs Frequency/Period.
2. Type Factor vs Period.
3. Click OK.
4. Under Table type select Period.
5. For the first data point, enter the Period to be 0.
6. Enter the Factor to be 2.5.
7. Click the Equation Editor button.
8. The Factor vs Period dialog box appears.
9. Enter 1.25/x^(2/3) in the Factor(x) box.
Sa = 2.5 for 0 < T < X
11. Under Parameters enter 2 and 20 in the Sampling range boxes.
12. Click OK.
13. A Confirm dialog appears, click Yes.
This produces the response for the period 0<T<3.0 s. The graph should look like Figure 5.5 below:
Figure 5.5 - Definition of Response Spectrum Curve
For your convenience the Standard Australian Code Response Spectrums (for both 1993 and AS1170.4-2007) are included with the Strand7 installation. You can use these tables by importing the appropriate *.txt file.
Before executing the Spectral Response Solver, you need to perform a Natural Frequency Analysis.
Natural Frequency Analysis
The Spectral Response Solver uses a mode superposition technique that requires the mode shapes and corresponding natural frequencies. To get these we need to run the Natural Frequency Solver first.
1. Choose Solver/Natural Frequency.
2. Under Parameters type or select 5 in the Modes box.
3. Leave all other values as default.
4. Click Solve.
After several iterations the solution converges and the results for the first five natural frequencies are displayed at the end of the solution file as shown in Figure 5.6 below.
Figure 5.6 - Frequency Listing for First Five Natural frequencies
Spectral Response Analysis
At this stage we do not know whether the first five natural frequencies are sufficient enough to adequately represent the dynamic behaviour of the structure. Running the Spectral Response Solver and looking at the Total Mass
Participation factor inside the spectral solver log file (SRL) can check this. A Total Mass Participation factor of greater than 90% indicates that the number of modes is sufficient. If it were less than 90%, we would need to recalculate a few more mode shapes and repeat the spectral analysis.
The Response Spectrum Curve, as given in Figure 5.5 above, is normalised using gravitational acceleration and is given in g's (gravitational acceleration). It will be multiplied by a scaling factor and g, which will be entered in the Spectral Response Solver panel as a direction factor. After this multiplication, the values of the curve will be in m/s2
. Units are consistent with all other units used in the analysis.
You are now ready to execute the Spectral Response Solver.
1. Choose Solver/Spectral Response.
2. Under Load Type select Base Acceleration. This setting indicates that the acceleration will be applied to the fixed nodes.
3. Under Damping select None.
5. The 'Tutorial 5.nfa' file should already be active automatically, if not, locate the file and open it.
6. Include all 5 modes.
7. Click OK.
8. Click Direction Vectors.
9. For the column Spectral Table choose the 'Factor vs Period' table you created earlier.
10. For Factor: X enter 0.22 (See note below).
11. Leave Factor: Y and Factor: Z to be 0.
12. Click OK.
13. Under Results make sure only Modal and SRSS are selected.
14. Leave all other values as default.
15. Click Solve.
Note
This value is obtained by multiplying the Scaling Factor and Gravity.
X = Scaling Factor x Gravity
X = 0.0225 x 9.81
X = 0.22
The Direction Vector 'X=0.22' will multiply the spectral values from the Factor vs Period table, and it will apply the seismic action in the X direction. The other two factors, 'Y' and 'Z' are zero, meaning that the earthquake does not act in the 'Y' and 'Z' directions.
As explained above, the first thing you should check after running a spectral response analysis is the Total Mass Participation factor. Checking in the Tutorial-5.SRL file, we see that the Total Mass Participation is 99.851%, which indicates that the number of included modes is sufficient.