“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 timehistory 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.
The statical response of the structure due to the applied load = [K]-1{s}f(t)
The statical response for the “m” flexible modes = [K]-1
Thus, the statical response of the rigid modes
= {X0}
= [K]-1{s}f(t) - [K]-1
= [K]-1({s} Now, {
) f(t)
} = Γn. [M]. {Φn}
So, {X0} = [K]-1({s} => [K]{X0} = ({s} -
) f(t)
) f(t)
- Equation (1)
- Equation (2)
While equation (1) gives the total response of the rigid modes, equation (2), which is exactly the
same as equation (1) expresses the force-displacement relationship of a pseudo-mode which
represents all the rigid modes. This pseudo-mode is also called the residual mode.
For earthquakes, {s} = - [M]{i}, {sn} = -
and f(t) =
= Ground Acceleration
Equation (2) can thus be expressed as:
=> [K]{X0} = - [M]({i} -
)
- Equation (3)
Interestingly, Equation (3) is representative of the inertial force of the mass undergoing
acceleration
and hence, the mode shape of the residual mode can be expressed as the
following:
{
} = {i} -
- Equation (4)
The corresponding modal mass of this residual mode is the Missing Mass.
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] {
}(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 | Y f2 | 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) (STARTCASE f12)
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 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.
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