microsoft word - spie paper final.doc - spiepaperfinal

7/29/2019 Microsoft Word - Spie Paper Final.doc - Spiepaperfinal

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Equivalent Static vs. Response Spectrum

A Comparison of Two Methods

David T. Finley*a, Ricky A. Cribbs,*b

aVertexRSI, 1700 International Parkway, Suite 300, Richardson, TX, USA 75081:bHLA Engineers, Inc., 7267 Envoy Ct., Dallas, TX, USA 75247

ABSTRACT

Developments in computer hardware and software have made analysis techniques that were formerly too expensive

within the reach of most project budgets. Foremost among these has been seismic response spectrum analysis. This

method yields much more accurate results than the equivalent static approach. The problem with using response

spectrum analysis exclusively in the design of deflection controlled structures, such as astronomical telescopes, is that

the nature of the structure minimizes the benefits of the approach. A typical response spectrum from Eurocode 8 deals

with a range of natural periods between 0.1 and 5 seconds. These correspond to a frequency range of 0.2 to 10 Hz. The

typical telescope structure has a minimum frequency of around 10 Hz. or greater. The result is that the response

spectrum analysis involves only a narrow band of frequencies and accelerations. This result could be reliably obtainedusing an equivalent static analysis approach.

1. OVERVIEW

In recent years the state of the art of seismic analysis has increased along with the development of technology. The

developments in computer hardware and software have made analysis techniques which were formerly too expensive

within the reach of most project budgets. Foremost among these has been the response spectrum analysis method for

seismic analysis. This method can give more accurate results than an equivalent static approach. This increase in

accuracy is largely due to combining specific vibratory modes from the structure with the spectral accelerations

determined for the site. The result is a much more accurate description of the seismic effects on a given structure, for a

specific site location.

However, this increase in accuracy comes at a steep price. The analysis technique requires a deep understanding of thestructure being analyzed, the nature of the spectral data, and of mathematical methods of mode superposition for

dynamic analysis. The final solution results must also be carefully interpreted, especially in the case of reactions. For a

simple analysis model these issues are not critical. The importance of these issues increases as the complexity and size

of the model increases.

In this paper we will review the basics of equivalent static and response spectrum analyses, and their specific use in the

analysis and design of an optical telescope. We will compare and contrast response spectrum analysis with the more

typical equivalent static analysis, and point out conditions in which one would be preferred over the other. In addition,

correct interpretation and use of analysis results will be discussed. These issues will be investigated and discussed using

the ongoing VISTA optical telescope structure as an example. The current VISTA analysis model is shown in Figure 1.

VISTA is funded by a grant from the UK Joint Infrastructure Fund, supported by the Office of Science and Technology

and the Higher Education Funding Council for England, to Queen Mary University of London on behalf of the 18

University members of the VISTA Consortium of: Queen Mary University of London; Queen's University of Belfast;

University of Birmingham; University of Cambridge; Cardiff University; University of Central Lancashire; University of

Durham; University of Edinburgh; University of Hertfordshire; Keele University; Leicester University; Liverpool John

Moores University; University of Nottingham; University of Oxford; University of St Andrews; University of

Southampton; University of Sussex; and University College London. The authors are grateful to the VISTA Program

Office and the organizations mentioned above for allowing the use of data for this paper.


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Figure 1Current VISTA Analysis Model

2. EQUIVALENT STATIC ANALYSIS: A BRIEF OVERVIEW

In recent years the topic of seismic loads and analysis has become of increasing importance in both Europe and the

United States. This is due largely to the frequency of large magnitude seismic events that have been witnessed, often in

large metropolitan areas, typically resulting in tragic loss of life. As a direct result greater efforts have been made to

understand and quantify loads that might be experienced during an earthquake.

This interest also extends to the expanding boundaries of science. Optical and radio telescopes are being continuously

used to increase and improve humanitys knowledge of the universe surrounding us. By their very nature these

instruments are extremely sensitive to vibratory disturbances. They are also located in remote regions such as northern

Chile or Hawaii which are active seismic zones. Proper consideration of seismicity is important in guaranteeing a long

design life for the telescope.

Historically, seismic loads were taken as equivalent static accelerations which were modified by various factors,

depending on the locations seismicity, its soil properties, the natural frequency of the structure, and its intended use.

The method was refined over the years to enable increasingly adequate designs. The underlying design philosophy was

basically unchanged; some modifications were made to the coefficients as a result of strong earthquakes. Other

modifications to account for new information were introduced by specifying acceptable structural details for different

construction materials.

However, this method was developed in order to design buildings and not telescopes. These two applications have some

important differences. Buildings have longer periods of vibration. They are also designed as regular frames and can be

simplified as two-dimensional frames. Telescopes, on the other hand, are deflection controlled structures with short

periods of vibration, composed largely of orthogonal, closely spaced modes.


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3. RESPONSE SPECTRUM ANALYSIS: A BRIEF OVERVIEW

With the advent of personal computers and improved structural analysis techniques, the use of more precise methods

increased. One of the most popular was response spectrum analysis. The method requires the determination of a

response spectrum from measured seismic activity. This data was then reduced into a spectrum of seismic action versus

natural frequency. The seismic action could be displacement, velocity, or acceleration, although the typical value used

was acceleration. Detailed information from the structural model was coupled with the corresponding spectral values foreach specific mode of vibration. The independent results were then combined using an appropriate technique to

determine the response of the overall structure. A typical response spectrum is shown below in Figure 2.

100 year Response Spectrum

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6

Period (s)

SpectralAcce

leration

(g)

Figure 2

100 Year Response Spectrum

The response spectrum is a function of period, the reciprocal of circular natural frequency, and damping ratio. It is

developed using Duhamels integral for a single degree of freedom harmonic oscillator to develop equations for

displacement, velocity and acceleration. The values in the appropriate spectrum are the maximum absolute values from

these equations. It can be seen that the spectral acceleration drops exponentially with increasing period after leaving the

plateau region. This means that for structures with low first frequencies, the resulting spectral accelerations can be quite

low. However, structures with high fundamental frequencies fall within either the sharp initial linear region or plateau

region of the spectrum.

The major problem with using response spectrum analysis exclusively in the design of deflection controlled structures,

such as astronomical telescopes, is that the nature of the structure minimizes the benefits of the approach. As stated

above, the response spectrum is a plot of maximum absolute spectral value versus period. The spectral values are

determined from Duhamels integral, a method used to determine response of a single degree of freedom oscillator in the

time domain. Most current structural analyses are performed using multiple degree of freedom models, so from the

outset care must be taken in interpreting results from the response spectrum analysis. In the VISTA analysis artificial

methods had to be used to obtain correct base reaction information for the design of the telescope foundation.

A typical response spectrum from Eurocode 8 deals with a range of natural periods between 0.1 and 5 seconds. These

correspond to a frequency range of 0.2 to 10 Hz. The typical telescope structure has a minimum frequency of around 7

Hz or greater. The result is that the response spectrum analysis involves only a narrow band of frequencies and


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accelerations. Because of this narrow banding of frequencies, perfectly adequate results for the overall structure could

be reliably obtained using an equivalent static analysis approach.

For a large telescope such as VISTA there are obviously other considerations than the overall structural design. Primary

among these are the accelerations imparted to the optical instruments. These values can be quite easily determined using

the response spectrum analysis technique.

4. RESPONSE SPECTRUM ANALYSIS CASE STUDY: VISTA

For an example of seismic analysis for an optical telescope, we present some of the effort for the VISTA program.

Figures 3 and 4 below show the version of the mount structure used for the analysis. The analysis model contained

12836 nodes and 13377 elements. Various element types were used in the analysis. There were approximately 77000

active degrees of freedom in the model.

Figure 4 gives the constraints at the base of the mount structure representing the connection between the mount and the

foundation. The pattern of the connection was a ring of anchor bolts. At each anchor bolt location the model was

constrained in all six degrees of freedom. While the actual constraint can be approximated to be similar to that used in

deriving Duhamels integral, and thus the spectral values, the number of points of constraint are greater than one. The

direct result of this will be shown below in the details of the analysis results.

Figure 3

FEA Model of the VISTA Telescope


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Figure 4

FEA Model of VISTA Telescope(Boundary Conditions Shown)

The first step in the response spectrum analysis is the modal analysis. It is important that the majority of the structural

mass be included in the analysis. Table 1 shows for each mode used the natural frequency, spectral acceleration,

participation factor, effective mass and other values from the VISTA analysis. This data is for the X direction of

excitation only.

Table 1

Response Spectrum Data

ModeFrequency

(Hz)

Spectrum

Value (m/s2)

Participation

Factor

Mode

Coefficient

M.C.

Ratio

Effective

Mass

2 9.942 11.029 234.3 0.6621 1 54888.7

3 10.42 10.866 -2.495 6.33E-03 0.009552 6.22421

4 12.93 10.287 -0.4395 6.85E-04 0.001035 0.193133

6 14.52 9.9871 1.613 1.94E-03 0.002922 2.60184

7 16.83 9.6201 25.65 2.21E-02 0.033334 657.883

8 17.88 9.4732 22.24 1.67E-02 0.025209 494.441

10 18.59 9.3798 -8.088 -5.56E-03 0.008398 65.4116

13 20.52 9.1468 -9.191 -5.06E-03 0.007635 84.4795

16 22.75 8.9105 -4.966 -2.17E-03 0.00327 24.6635

17 26.39 8.5813 -7.307 -2.28E-03 0.003445 53.3864

20 36.1 7.9242 87.62 1.35E-02 0.020376 7677.23

23 39.42 7.7493 -42.89 -5.42E-03 0.008183 1839.72


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27 41.81 7.634 -7.01 -7.75E-04 0.001171 49.1431

30 42.83 7.5875 -45.05 -4.72E-03 0.007127 2029.35

35 47.03 7.4093 10.45 8.87E-04 0.00134 109.304

37 47.8 7.3788 -102.4 -8.38E-03 0.012651 10487.4

44 50.8 7.2656 -21.35 -1.52E-03 0.002299 455.735

52 56.16 7.083 19.98 1.14E-03 0.001717 399.10856 57.78 7.0319 17.21 9.18E-04 0.001387 296.247

62 60.33 6.9551 16.41 7.94E-04 0.001199 269.184

64 60.94 6.9373 29.33 1.39E-03 0.002096 860.394

Sum of Effective Masses 80751

Total Structural Mass 100400

Percent of Total Mass

Effective80.4%

The data shows that out of 100 modes extracted, only 21 excited enough of the structure to be counted as significant inthe seismic analysis. These 21 modes represent 80.4% of the total structural mass. Two-thirds of this 80% comes from

one mode, mode 2. In fact, 77% of the total structural mass comes from 5 modes: modes 2, 20, 23, 30, and 37. In other

words, 5% of the total modes extracted account for 77% of the effective mass.

This strongly suggests that an equivalent static approach would very closely approximate the effect of the response

spectrum on the structure. Figure 5 shows the response spectrum of Figure 2 with the above natural frequencies plotted.

It is obvious that for the VISTA structure the results are grouped in a specific portion of the spectrum.

100 year Response Spectrum

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6

Period (s)

SpectralAcceleration

(g)

Figure 5

Response Spectrum with VISTA Frequencies


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The method used to combine results in the response spectrum method is also critical to obtaining the correct answer.

The results for each of the effective modes used in the analysis must be combined using an appropriate technique. The

method most commonly used is the square-root-of-sum-of-squares (SRSS) method. This simply sums the squares of all

of the individual spectral results, and then takes the square root of the result. The most accurate technique is known as

the Complete Quadratic Combination (CQC) method. This combines the spectral results using structural damping and a

weighted ratio of relative frequencies. The act of combination removes the sign of the result, thus leaving only a

magnitude for the final answer. This must be assumed to act as amplitude on the structure, and can vary from positive to

negative. The practical result is that each method deals with the magnitude of the load at each support, but not the sign

of that load. Combining two moments on a bolt pattern would normally result in tension and compression loads

canceling the loads on some bolts. By treating everything as tension, the resultant bolt loads are unrealistically high.

As mentioned in the overview, the spectral responses are maximum values of the spectral acceleration obtained using a

time integration of Duhamels integral. Since these response values are for a single degree of freedom model, they

cannot be used directly on a multiple degree of freedom model such as VISTA. In the case of base reactions, to

correctly obtain a maximum base reaction, the individual reactions must first be summed about a common point, then

combined using either of the two methods described above. In the case of VISTA this point was the intersection of the

horizontal plane of the foundation, and the optical axis when the telescope was pointed to zenith. This effectively

simulates the base support as a single degree of freedom support. If results are obtained directly from the analysis

program for each individual support point, then summed the resulting magnitude will be extremely large.

To illustrate this point, the following tables show reaction results from VISTA. Table 2 shows results obtained bysumming the individual results from the SRSS combination in the analysis. Table 3 shows results obtained by first

summing the individual results, the combining these using the SRSS method.

Table 2Base Seismic Reactions (using direct SRSS method)

Seismic Load Forces at Base (N) Moments at Base (N*m)

Resp.

Spect.Direction Fx Fy Fz Mx My Mz

X 1,072,300 928,530 2,136,000 4,141,244 4,782,458 1,0001

MLE 1% Y 601,750 818,870 1,535,400 3,652,157 2,683,805 747

Table 3Base Seismic Reactions (first summing, then combining using SRSS)

Seismic Load Forces at Base (N) Moments at Base (N*m)

Resp.

Spect.Direction Fx Fy Fz Mx My Mz

X 613776 5413 10333 46295 4354681 17491

MLE 1% Y 5535 528837 25475 3179410 38382 21327

It can be seen from the two tables that the results are significantly different. This is primarily the result of summing the

results from data which were determined from the SRSS combination. As stated above, the results must first be

converted into results resembling that of a single degree of freedom system, then combined using either the CQC or

SRSS method.


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The alternative to this for a deflection controlled structure is to perform the overall structural analysis using the static

equivalent acceleration approach. In the absence of specific code requirements, this can be approximated using the

natural frequency, effective mass, and response spectrum data provided. Taking the product of Spectrum Value (the

spectral acceleration), and the Effective Mass for each mode, then summing the products we obtain the Effective Shear

for each mode. The sum of the Effective Shears divided by the total Effective Mass then yields the Equivalent Static

Acceleration for the effective modes. Using the data from Table 1 we obtain an equivalent static acceleration of 1.01 g

for the telescope. In comparison, taking the value of 613,776 N for the X direction shear from Table 3 and dividing it by

the effective mass of 80751 N we obtain an equivalent static acceleration of 0.8 g. Though there is a difference of 20%

in the two values, the approximated equivalent static acceleration is on the high side and would provide a good value for

foundation and base detail work. The amount of effort to perform the analysis with the equivalent static approach is

significantly less than that required for the full response spectrum approach, even including evaluating the modal and

spectrum data.

The real benefit of performing a response spectrum analysis comes in determining accelerations at specific locations

within the structure. In the case of VISTA it was required to know what the seismic accelerations were at the primary

and secondary mirror centroids, and at the cassegrain instrument. Because the response spectrum method shows degree

of freedom results as accelerations, determining that information was simply a matter of listing results at nodes located at

the centroids of those components.

5. CONCLUSIONS

For many reasons the two methods discussed are important for the analysis and design of displacement controlled

structures. It is often tempting to discard an old approximate method when the state of the art reaches a level where the

latest technology can be effectively used. This is, however, not always the best approach. With tight financial and

delivery budgets, it is advisable to use the best of both worlds to streamline the design process, while insuring that

accurate data is developed at each step.

An optical telescope can be described as a displacement controlled structure because it usually has extremely tight

tolerances on any deflections of the system. Because of this, the natural frequencies of vibration of the system are

typically quite high relative to a usual structure, and fall in the linear region of a response spectrum. As a result, for

overall structural design an equivalent static approach is typically the most efficient. This enables rapid development of

foundation loadings, as well as approximating the final design stiffness of the mount structure. This design will almost

certainly be revised to enable it to meet the tight tolerances for local deflection criteria. For sensitive local regions,

response spectrum analysis may then be used on the near final design to determine accelerations.

In the absence of an equivalent static acceleration from a governing code or specification, one can be developed using a

modal analysis, and the response spectrum. This acceleration should be larger than that obtained using the full response

spectrum method. This can be checked by performing a full response spectrum analysis on the final design of the

structure.

If it is decided that a full response spectrum analysis is required for each step of the process, it is advisable to use

personnel who are very experienced with this type of analysis, and with the finite element analysis approach in general.

REFERENCES

1. R. Clough,, and J. Penzien,Dynamics of Structures,McGraw-Hill, New York. 19932. Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary,

Structural Engineers Association of California, Sacramento, 1996

3. A. Williams,. Seismic Design of Buildings and Bridges, Engineering Press, Austin., 1998

4. M. Paz,. Structural Dynamics, Van Nostrand Reinhold, New York, 1985

*a [email protected]*b

[email protected]


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