Calculation of Rayleigh Damping Coefficients for a Transient
Structural Analysis
ANDREY GRISHIN1, VITALIY GERASCHENKO2
1Department of Structural and Theoretical Mechanics,
Institute of Industrial and Civil Engineering,
Moscow State University of Civil Engineering,
Yaroslavskoye Shosse 26, Moscow, 129337,
RUSSIAN FEDERATION
2Department of Reinforced Concrete and Masonry Structures,
Institute of Industrial and Civil Engineering, Moscow State University of Civil Engineering,
Yaroslavskoye Shosse 26, Moscow, 129337,
RUSSIAN FEDERATION
Abstract: - Direct numerical integration of differential equations of motion is widely used by engineers to
describe the behavior of structures under dynamic loading. The method entails directly integrating the motion
equations over time. In the direct method, the damping matrix is formed as a linear combination of mass and
stiffness matrices multiplied by the Rayleigh damping coefficients α and β, respectively. The Rayleigh damping
coefficients have a significant effect on the response of building structures under dynamic loading. Therefore,
the design values of the damping coefficients α and β have crucial importance to ensure accurate and reliable
results in a dynamic analysis. The paper presents a time domain analysis for a building subjected to seismic
excitations using the modal superposition and direct integration methods. The direct method considers the
damping properties of building structures by Rayleigh damping coefficients obtained using various approaches.
The building's response to seismic load is compared by response spectra. The authors proposed the least
conservative approach for calculating the Rayleigh damping coefficients for analyzing a building in the time
domain.
Key-Words: - Rayleigh damping coefficients, transient analysis, time-history analysis, choosing, direct method,
response spectra.
Received: April 22, 2023. Revised: February 21, 2024. Accepted: March 19, 2024. Published: April 23, 2024.
1 Introduction
Time history analysis is used for actual time-varying
loads (such as earthquakes) to the structure and
predicting its response over time. The direct
integration and modal superposition methods are the
main methods that used in structural analysis
considering inertial forces and damping. The direct
integration method solves the equations of motion
for the entire structure directly in the time domain.
The modal superposition method builds the solution
on the scaled mode shapes and sums them to capture
the dynamic response.
When the dynamic analysis of a structure using
the direct integration method is performed, the
proportional damping (Rayleigh damping) can be
applied to account for the damping properties of the
structure, [1]. This includes creating a damping
matrix [C] using a linear combination of mass [M]
and stiffness [K] matrices, which are multiplied by
proportional coefficients α and β as follows [2], [3],
[4], [5], [6]: (1)
where α and β are Rayleigh coefficients that decide
how much the system damps. Engineers can change
α and β to better match the true way the building
moves. The values of α and can be determined
based on the modal damping ratios i, which
represent the actual damping compared to critical
damping for a specific mode shape i. If i is the
natural circular frequency of mode i, α and can be
related as follows, [1]:
(2)
Rayleigh damping can accurately match modal
damping values at one or two natural frequency
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.5
Andrey Grishin, Vitaliy Geraschenko
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points, making it suitable for structures with
dominant frequencies. However, for structures with
numerous modes across a wide range of natural
frequencies, Rayleigh damping may result in
significant deviations in response compared to
modal damping.
The paper aims to determine the most suitable
method for calculating Rayleigh damping
coefficients for a structure experiencing dynamic
loads. This will be achieved through a comparison
of response spectra generated from both direct and
modal superposition analyses, ensuring the accurate
consideration of the structure's damping
characteristics.
2 Problem Formulation
The objective is to determine Rayleigh damping
coefficients for a dynamic structural analysis that
most accurately corresponds to the data. The
accuracy of the structure's fit to a data point is
evaluated through its residual, which is the variance
between the prescribed damping ratio () and the
estimated value from the chosen method within the
designated frequency range:
(3)
The least squares method determines the best
parameter values by reducing the objective function,
S, which is the sum of squared residuals for each
mode shape i:
(4)
Setting the gradient to zero helps to find the
minimum of the sum of squares. In the Rayleigh
damping model, which has two coefficients, there
are two gradient equations that need to be
considered:
(5)
In the following approaches to various types of
objective functions, S is taken into consideration,
[7], [8], [9], [10].
2.1 Least Squares Method Approaches
The method of least squares is commonly used to
estimate the solution of over-determined systems,
such as sets of equations with more equations than
unknowns. Given that, real structures typically have
more natural modes than unknowns, the least
squares method is applied to calculate Rayleigh
damping coefficients (α and β).
2.1.1 Conventional approach (CA)
In this method, by inputting ξ, ωmin, and ωmax, two
simultaneous equations (2) can be solved to
determine α and β:
;
, (6)
where ξ represents the damping ratio (specified in
the National Regulatory Guides); ωmin is the lowest
undamped circular frequency of the structure; and
ωmax is the highest undamped circular frequency that
affects the structure's response.
2.1.2 Pure approach (Pure)
When expression (2) is replaced by (4), the
objective function, S, simplifies as follows:
+
-2
N
i=1 (7)
2.1.3 Inverse Frequency Weighted Approach
(IFWA)
S is considered as the objective function:
+
-2
N
i=1 (8)
2.1.4 Mass Participation Weighted Approach
(MPWA)
S is considered as the objective function:
+
-2
N
i=1 (9)
2.2 Finite Element Model of a Building
The design of a Shielded Control Room building
(SCR) is intended for the management of a nuclear
plant during both regular operations and
emergencies. Constructed primarily with reinforced
concrete, the SCR building includes concrete block
partitions and features a concrete strength of 17
MPa and an elasticity modulus of 32.5 GPa. The
reinforcing bars have a modulus of 200 GPa.
Anchored partly in soil, the influence of soil-
structure interaction effects on Rayleigh damping
coefficients is not considered in this study.
A three-dimensional finite-element model of the
SCR building, depicted in Figure 1, is meshed with
BEAM188 and SHELL181 finite elements based on
the building's design. An additional mass of 300
kg/m2 is distributed evenly on the floor elements,
while 250 kg/m2 is placed on the roof elements. The
foundation elements of the model are constrained
with a fixed support boundary condition.
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Fig. 1: Finite Element Model of the SCR building
2.3 Earthquake Ground Motion
Figure 2 and Figure 3 display the vertical and
horizontal spectra components of the seismic input
ground motions for the Safe Shutdown Earthquake
(SSE). The predicted intensity of the seismic input
ground motions is 0.201g for the horizontal
component and 0.134g for the vertical component.
In Figure 4, a three-component accelerogram is
presented for the SSE seismic input ground motions.
The components of the accelerogram are statistically
independent, with a coefficient of mutual correlation
not exceeding 0.1. The accelerogram time
discretization is 0.005 s, allowing for the
consideration of frequencies up to 100 Hz.
Fig. 2: Horizontal and vertical spectra components
of the seismic input ground motion
Fig. 3: Ground motion three-component
accelerogram
3 Problem Solutions
The process of selecting the optimal method for
establishing Rayleigh damping coefficients
involved:
a) conducting modal analysis on the SCR
building;
b) computing Rayleigh damping coefficients for
different methods;
c) performing transient analysis on the SCR
building using the direct and modal superposition
methods with specified damping coefficients;
d) creating response spectra for both direct and
modal superposition analyses;
e) comparing the resulting response spectra.
3.1 Modal Analysis
Based on the results of the modal calculation, Table
1 shows the vibration characteristics of the SCR
building.
Table 1. Vibration characteristics of the SCR
building
Mode
fi, Hz
mxi, kg
myi, kg
mzi, kg
1
15.6
0
41
2350
2
15.7
2
30
118000
3
17.4
6
700
23400
5
21.1
696
2830000
318
…
11
27.2
2240000
263
123
…
108
50.8
1250
2480
79200
…
500
106.8
34
127
11
Total mass of the building M = 0.505E+07 kg
Note: fi and ωi represent the phase and circular natural
frequencies respectively for mode shape i; mxi, myi, mzi
indicate the effective horizontal and vertical masses
corresponding to mode shape i; mi=(mxi2 + myi2 + mzi2)0.5 is
the total mass associated with mode shape i.
3.2 Rayleigh Damping Coefficients for the
Approaches Considered
The values of Rayleigh damping coefficients are
provided in Table 2 based on the approaches
described in Section 2.
Table 2. Rayleigh damping coefficients for the
considered approaches
Approach
Rayleigh Damping
Coefficients
Formula
CA
11.8909
0.000193
(6)
Least squares method:
- Pure
21.397
0.000187
(11)
- IFWA
17.9289
0.000209
(13)
- MPWA
15.3401
0.000242
(15)
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3.2.1 Pure Approach (Pure)
The system of equations (5) representing the
objective function (7) is written as:
+
-
+
-
(10)
The solution to the system of equations (10) for α
and β will be written in the form:
(11)
3.2.2 Inverse Frequency Weighted Approach
(IFWA)
The system of equations (5) representing the
objective function (8) is written as:
+
-
+
-
(12)
The solution of the system of equations (12) for
and is:
(13)
3.2.3 Mass Participation Weighted Approach
(MPWA)
The system of equations (5) representing the
objective function (9) is written as:
+
-
+
-
(14)
The solution of the system of equations (14) for
and is:
(15)
3.3 Rayleigh Damping Curves for the
Approaches Considered
Figure 4 displays the Rayleigh damping curves for
the suggested methods.
Fig. 4: Rayleigh damping curves for the suggested
methods
3.4 Response Spectra
Response spectra of 2% are generated for the points
illustrated in Figure 1. The reference response
spectrum is represented by the MSUP curve as it
accurately incorporates damping through the modal
superposition method.
a) along the horizontal axis x
b) along the horizontal axis y
c) along the vertical axis z
Fig. 5: Response spectra at point 1
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a) along the horizontal axis x
b) along the horizontal axis y
c) along the vertical axis z
Fig. 6: Response spectra at point 2
The CA, Pure, IFWA, and MPWA curves
display response spectra generated by the direct
method with the Rayleigh damping coefficients
listed in Table 2. The response spectra for points 1
and 2 are depicted in Figure 5 and Figure 6.
4 Conclusion
Selecting Rayleigh damping coefficients is crucial
in a direct dynamic analysis. Four methods were
suggested to determine these coefficients. The SCR
building underwent transient analyses using both the
direct and modal superposition methods. Among the
methods studied, MPWA is closest to the ideal
solution (MSUP). The spectral acceleration values
are similar across all methods, possibly due to the
building's frequency high-frequency first mode of
vibration.
More research into soil-structure interaction and
flexible building designs will lead to better
recommendations for selecting Rayleigh damping
coefficients.
Future studies will focus on more flexible
buildings and consider soil-structure interaction to
identify the best approach for Rayleigh parameters.
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that are relevant to the content of this article.
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