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9/ 16/13 “Missing Mass ” in Dy na mic Analy sis - St ructural An aly sis and Des ign Wik i - Structu ral Ana ly sis a nd Design - Be Communities by Bentley communi ti e s.bentle y.com/pr oducts/stru ctur a l/ str uctur al _anal ysi s desi gn/w /structur al _anal ysi s and_design w i ki /9114.aspx 1/3 Structural Analysis and Design This is w here you c an find and c ontribute to di scus sions, ideas, and other informati on about Bentl ey Structural Analysis and Design products. “Missing Mass” in Dynamic Analysis  Applies To Product(s): STAAD.Pro  Version(s): All  Environment: N/A  Area: Technotes  Subarea: Structural Analysis (Dynamics)  Origina l Author: Sudip Narayan Choudhury, TSG (Structural), Bentley Kolkata  “Missing Mass” in Dynamic Analysis  Introduction: The common practice of performing a dynamic analysis of a structure is to evaluate the response of a structure mode by mode and then combining the results from each mode. While in a time-history the Modal Superposition method is employed, which involves de-coupling of the dynamic equation of motion into an individual equation for each mode involving Generalized Mass, Generalized Stiffness, Generalized Damping and Generalized Force for that mode and evaluating the response per mode, the response spectrum method involves finding the spectral acceleration corresponding to the period of each mode and evaluating the response by considering that the structure in each mode is subjected to an equivalent static force involving the modal mass and the spectral acceleration.  As can be un derstood , t o ev alua te the complete respon se of the structure, the respon se from all possible mod es need to be combin ed. How ev er, fro m the point of view of practicality and the computer run-time and memory considerations, it is not always possible to run a structure for all possible modes. Hence the truncation of modes is employed. However, as can be understood the modal truncation introduces errors in evaluating the complete response of the structure. “Missing Mass ” is the method emplo yed to ass ess t he app roximate respo nse of the truncated mod es. Theory: The total number of modes in a structure can be classified in two categories: 1. The Flexible Modes and 2. The Rigid Modes. While the flexible mode s is s ubjected to dynamic amplifications the respon se from the rigid mode s are essentially static . The total response from a flexible mode is evaluated as {Φ n }Y n , where {Φ n } is the mode shape vector for mode n and Yn is the modal co-ordinate of the nth mode. The response fro m the rigid modes can ev aluated as the st atic al response o f the st ructure due to the applied load {s}f(t) minus the static al response for the flexible modes.

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“Missing Mass” in Dynamic Analysis

 

Applies To

Product(s): STAAD.Pro

  Version(s): All

  Environment: N/A

  Area: Technotes

  Subarea: Structural Analysis (Dynamics)  Original Author: Sudip Narayan Choudhury, TSG (Structural), Bentley Kolkata

 

“Missing Mass” in Dynamic Analysis

 Introduction:

The common practice of performing a dynamic analysis of a structure is to evaluate the response of a structure mode by mode and then

combining the results from each mode. While in a time-history the Modal Superposition method is employed, which involves de-coupling of the

dynamic equation of motion into an individual equation for each mode involving Generalized Mass, Generalized Stiffness, Generalized Damping

and Generalized Force for that mode and evaluating the response per mode, the response spectrum method involves finding the spectral

acceleration corresponding to the period of each mode and evaluating the response by considering that the structure in each mode is subjected to

an equivalent static force involving the modal mass and the spectral acceleration.

 As can be understood, to evaluate the complete response of the structure, the response from all possible modes need to be combined. However,

from the point of view of practicality and the computer run-time and memory considerations, it is not always possible to run a structure for all

possible modes. Hence the truncation of modes is employed. However, as can be understood the modal truncation introduces errors in evaluating

the complete response of the structure. “Missing Mass” is the method employed to assess the approximate response of the truncated modes.

Theory:

The total number of modes in a structure can be classified in two categories: 1. The Flexible Modes and 2. The Rigid Modes. While the flexible

modes is subjected to dynamic amplifications the response from the rigid modes are essentially static .

The total response from a flexible mode is evaluated as {Φn}Yn, where {Φn} is the mode shape vector for mode n and Yn is the modal co-ordinate

of the nth mode.

The response from the rigid modes can evaluated as the statical response of the structure due to the applied load {s}f(t) minus the statical

response for the flexible modes.

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Response Spectrum Analysis and Missing Mass:

Response Spectrum Analysis is widely used for the seismic analysis of structures. The consideration of the contribution of the truncated modes

is very important for the proper assessment of the structure under seismic loads. However, the most important decision is the number of modes

that needs to be considered for dynamic analysis.

Most of the seismic design codes around the world have suggested that at least 90% of mass participation in either direction be achieved with the

assumption that the significant modes will be captured considering this. While this assumption holds true for a class of building structures, this is

not applicable for all type of structures. So, the general guideline is to consider all the flexible modes for dynamic analysis and do a missing

mass correction to include the effect of the rigid modes.

There are different views on what should be a rigid mode. The general conception is that the modes are rigid corresponding to modal frequencies

of 33 Hz frequency and above. The corresponding spectral acceleration is called the Zero Period Acceleration (ZPA) and this will be the maximum

value of the recorded ground acceleration considering that the modes are rigid.

So, the contribution of the residual mode can be expresses by modifying equation 3 by substituting equation (4) in it.

{X0} = -[K]-1 [M]{Φr}(ZPA)

 

The best way would be to ensure a 90% mass participation as suggested by the code and then ensuring that all the modes the frequencies of which are below ZPA are included in the dynamic analysis. The inclusion of flexible modes in the residual mode can cause substantial errors to

occur.

STAAD Implementation of Missing Mass theory in Response Spectrum Analysis:

The well known command lines to invoke the response spectrum analysis in STAAD.Pro is as follows:

SPECTRUM { SRSS | ABS | CQC| ASCE | TEN | CSM | GRP } *{ X f1 | Yf2 | Z f3 } { ACC | DIS } (SCALE f4)

{DAMP f5 | CDAMP | MDAMP } ( {LIN| LOG} ) (MIS f6) (ZPA f7) (FF1 f8) (FF2 f9)( { DOMINANT f10 | SIGN } ) (SAVE) (IMR f11)

(STARTCASEf12)

The specification of the MIS will invoke the missing mass calculation representing the mass of the residual mode. If the value of f6 is specified

then this value is used as the corresponding spectral acceleration for the residual mode. If 

not specified f7 is used as the spectral acceleration which is used to define the ZPA frequency. If we do not have the value of f7 specified, a

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default value of 33 Hz is used.

Once the residual mode analysis in invoked, the mass participation output in the output file will invoke that mode as ZPA.

However, at his point of time we do not have a way of extracting the response of the residual mode into a load case as we can do for a normal

mode.

Time History Analysis and Missing Mass:

Going by the same logic as we have discussed for a response spectrum analysis we need to ensure that all the flexible modes are included in

dynamic analysis, a mass participation of 90% is ensured and all the modes having frequencies less than the exciting frequency is considered in

the dynamic analysis.

STAAD Implementation of Missing Mass theory in Time History Analysis:

Starting from the forthcoming SS5 release, STAAD will be equipped to consider the rigid mode contribution in a time history analysis. More

contents will be added on this once the SS5 is released.

 A PDF version of this article is available at the following link:

http://communities.bentley.com/products/structural/structural_analysis___design/m/structural_analysis_and_design_gallery/258859.as

 

References:

1. ASCE 4-98: Seismic Analysis of Safety Related Seismic Structures.

2. Dynamics of Structures: Theories and Applications to Earthquake Engineering; Anil K. Chopra; Prentice Hall

3. Dynamics of Structures; Ray W. Clough, Joseph Penzien; Computers and Structures Inc.

4. “Missing Mass” Correction in Modal Analysis of Piping Systems; G.H.Powell

5. On the use of residual shapes in Modal Analysis; R. Alvarez, J.J Benito

6. An Alternative Cut-Off Frequency for the Response Spectrum Method of Seismic Analysis; M. Dhileep, N.P. Shahul Hameed and S. Nagan

7. Derivation of Mode Acceleration Method for MDOF Systems (Proportional Damping or Light Damping); Luis San Andres