Rayleigh Damping Parameters of a Gravity Dam
Outline
1 Description 3
1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Finite Element Model 9
2.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Structural Eigenvalue Analysis 11
3.1 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Results: Rayleigh Damping Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Appendix A Additional Information 16
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1 Description
1.1 Model
In this tutorial we derive the Rayleigh damping parameters for the gravity dam presented in Figure 1 using DianaIE. The modeling requires the following steps:
• generation of the foundation and dam geometry
• assignment of the material properties
• application of the constraints
• assignment of the deadweight as the only load acting on the dam.
The geometry of the foundation and the dam and the corresponding material properties are the same as in the tutorial “Response Spectrum Analysis of a Gravity Dam”. As shown in Table 1,
all materials are linear elastic. Since we focus on the calculation of the Rayleigh damping parameters of the dam, the density of the foundation is assumed equal to zero. By assuming a
massless foundation the wave speed becomes infinite and the excitation is applied at the dam at the same time as it is applied at the bottom of the model1 .
Foundation Dam
Foundation
Young’s modulus 8e+9 Pa
Poisson’s ratio 0.2
Density 0 kg/m3
Dam
Z Young’s modulus 3.2e+10 Pa
Y
Poisson’s ratio 0.2
X Density 2400 kg/m3
Figure 1: Geometric model of the dam Table 1: Material properties
1
The assumption of a massless foundation is for simplicity of this specific tutorial in which the user is shown how to perform an eigenvalue analysis to determine the Rayleigh damping parameters for the dam as preparation
for a (transient) dynamic analysis. In reality the foundation is not linear elastic and certainly not massless. For more realistic (transient) dynamic analysis results it is better to use mass in the foundation and apply a
non-linear, preferably cyclic, soil material. When the mass of the foundation is also included, the user can also apply absorbing boundary conditions for a (transient) dynamic analysis
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1.2 Rayleigh Damping
When performing a transient analysis of structural models using Rayleigh damping, the damping matrix C is calculated as the linear combination of the mass matrix M and the stiffness
matrix K:
C = aM + bK (1)
where a and b are the Rayleigh damping parameters related to the mass matrix and the stiffness matrix, respectively. As described in detail in Chopra (2007) 2 , damping ratios ζn depend on
the eigenfrequencies fn of the structure and on the Rayleigh parameters a and b [Fig. 2]. More specifically, the damping ratio for the nth eigenfrequency is:
1
ζn = a + bπfn . (2)
4πfn
Typically, Equation (2) is expressed as a function of angular speed ωn instead of the eigenfrequencies fn . Thus, since ωn = 2πfn , Equation (2) can be rewritten as:
1 ωn
ζn = a +b . (3)
2ωn 2
2
Chopra, Dynamics of structures – Theory and applications to earthquake engineering, 2007
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Therefore, the Rayleigh damping parameters a and b “can be determined from specified damping ratios ζi and ζj Rayleigh damping
for the ith and j th modes, respectively”(Chopra 2007)[1]: c = am + bk
ζn = 2ωan + bω2n
1 1/ωi ωi
a ζi
2 1/ωj ωj b = ζj . (4)
Damping ratio ζn
ζj
ζi c = bk
DianaIE calculates the Rayleigh damping parameters based on Equation (4) by choosing a set of damping ratios (ζi ζn = bωn /2
and ζj ) and eigenfrequencies (fi and fj ). The set of damping ratios are defined by the user while the eigenfrequencies
fi and fj can be:
• explicitly specified, c = am
ζn = a/2ωn
• determined by providing the corresponding eigenmodes (e.g., mode 1 and mode 3 )
• determined based on the eigenfrequencies specified by a cumulative effective mass percentage in the global ωi ωj
XY Z directions (e.g., 90%, 90% and 90%). In case the requested cumulative effective mass percentages Angular speed ωn
cannot be reached for the calculated number of eigenfrequencies, the second frequency is the highest calculated
eigenfrequency. Figure 2: Variation of modal damping ratios ζn with the angular
speed ωn
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In this tutorial we choose the eigenfrequencies fi and fj such that the cumulative effective mass percentage in all the three directions is higher or equal than 90%. Consequently, the first
frequency (fi ) is the one associated to the first eigenmode (i.e., f1 ) while the second corresponds to the smallest eigenfrequency associated to a cumulative effective mass percentage higher or
equal than 90%.
As shown in Figure 3 for the first ten eigenmodes in the X direction (see the tutorial “Response Spectrum Analysis of a Gravity Dam”), the cumulative effective mass percentage at the seventh
eigenmode is higher than 90%. The same holds for the cumulative effective mass percentage in the Y and Z directions as shown in Figure 4 and Figure 5, respectively.
In Figure 3 to Figure 5 we highlighted in red the eigenfrequency corresponding to the first mode and the first eigenfrequency associated to a cumulative effective mass percentage higher than
90%.
Figure 3: First ten eigenmodes in the X-direction and corresponding cumulative effective mass percentage
Therefore, the two angular speeds used for the calculation of the Rayleigh damping parameters are ω1 = 2π · f1 = 2π·3.1357 = 19.70 rad/s and ω7 = 2π · f7 = 2π·7.6036 = 47.77 rad/s.
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Figure 4: First ten eigenmodes in the Y direction and corresponding cumulative effective mass percentage
Figure 5: First ten eigenmodes in the Z direction and corresponding cumulative effective mass percentage
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It is noteworthy that the choice of the frequencies fi and fj should also depend on the base acceleration used in the transient analysis.
In Figure 6 we show the accelerogram used in the tutorial “Time History Analysis of a Gravity Dam”. From the corresponding Discrete Fourier Transform in Figure 7, obtained through
the Fast Fourier algorithm, we observe that most of the energy content of the earthquake is between 0.5 and 4 Hz. Therefore, fi and fj should be chosen such that main frequencies of the
earthquake are not filtered out.
In the present case, the eigenfrequency f7 = 7.61 Hz complies with this requirements since it is higher than 4 Hz. On the other hand, f1 = 3.14 Hz is too high as we do not want to filter
out the frequencies of the input signal above 0.5 Hz. This can be achieved by using a smaller value of the damping ratio ζ1 (by default, DianaIE assumes ζ1 = 0.05 and ζ7 = 0.05). Thus, we
assume ζ1 = 0.025 and Equation (4) becomes
1 1/19.70 19.70 0.025 0.2082
a a
2 1/47.77 47.77 b = 0.05 ⇒ b = 0.0020 (5)
We will later observe in Section 3.2 that the chosen set of eigenfrequencies and damping ratios will not filter out the relevant frequencies of the earthquake.
Figure 6: Accelerogram for base acceleration Figure 7: Discrete Fourier Transform of the accelerogram on the left
In this tutorial, the values of a and b in Equation (5) are calculated automatically in DianaIE.
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2 Finite Element Model
The finite element model here employed is the same used in the tutorial “Response Spectrum Analysis of a Gravity Dam”. The user is referred to the latter to create the geometry of the dam
and the foundation and to assign the material properties, the constraints and the deadweight. For simplicity, compared to the aforementioned tutorial, we do not consider the reservoir, the
fluid-structure interaction interface and the hydraulic pressure and fixed head potential.
Therefore, when starting the new project in DianaIE, we select to perform only a structural analysis as shown in Figure 8 (in the aforementioned tutorial structural and fluid-structure interaction
analyses were performed).
The model used in the present tutorial is shown in Figure 9.
Figure 8: New project dialog Figure 9: Geometry of the finite element model
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2.1 Mesh
We set the mesh properties such that the elements size is equal to 40 m. Then, we generate the mesh.
DianaIE
Main menu Geometry Assign Mesh properties [Fig. 10]
Main menu Geometry Generate mesh [Fig. 11]
Figure 10: Mesh properties Figure 11: Finite element mesh
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3 Structural Eigenvalue Analysis
3.1 Commands
We set up a structural eigenvalue analysis that, based on the calculation of the eigenfrequencies of the structure, allows to the determine the Rayleigh damping parameters of the dam. The
deadweight load is considered for the geometric stress stiffness matrix in the eigenvalue analysis since it influences the eigenfrequencies of the dam.
DianaIE
Main menu Analysis Add analysis
Analysis browser Analysis1 Rename Rayleigh Damping [Fig. 12]
Analysis browser Rayleigh Damping Add command Structural eigenvalue [Fig. 13]
Analysis browser Rayleigh Damping Structural eigenvalue Define eigenvalue type Free vibration Edit properties [Fig. 14] [Fig. 15]
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Figure 13: Analysis browser - add
Figure 12: Analysis browser command Figure 14: Analysis browser Figure 15: Edit free vibration properties
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We choose to compute the first ten eigenmodes. As observed in the tutorial “Response Spectrum Analysis of a Gravity Dam”, the first ten modes are enough to have a cumulative effective
mass percentage above 90%.
DianaIE
Analysis browser Rayleigh Damping Structural eigenvalue Execute eigenvalue analysis Edit properties [Fig. 16] [Fig. 17]
Figure 16: Analysis browser Figure 17: Execute eigenvalue analysis properties
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We set up the eigenvalue analysis for the calculation of the Rayleigh damping parameters a and b. As mentioned in the previous section, we determine a and b such that the cumulative
effective mass percentage is above 90% in all three global directions for the highest eigenfrequency used.
Then, we can run the analysis.
DianaIE
Analysis browser Rayleigh Damping Structural eigenvalue Add... Calculate Rayleigh parameters - effective mass [Fig. 18]
Analysis browser Rayleigh Damping Structural eigenvalue Calculate Rayleigh parameters Edit properties [Fig. 19] [Fig. 20]
Main menu Analysis Run selected analysis
Figure 19: Analysis browser - add Rayleigh Figure 20: Edit calculation of
Figure 18: Analysis browser parameters calculation Rayleigh parameters
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3.2 Results: Rayleigh Damping Parameters
The calculated Rayleigh damping parameters are written in the standard DianaIE output file rayleigh Rayleigh Damping.out as shown in Figure 21.
Figure 21: Rayleigh damping parameters Figure 22: Damping ratio vs frequency
These values are in agreement with the hand calculations in Equation (5) performed earlier on slide 8.
Substituting the obtained values of a and b and using ζ1 = 0.025 and ζ7 = 0.05 in Equation (3) we can plot the curve in Figure 22. This graph shows that the earthquake frequencies between
approximately 1 and 7.5 Hz are not overdamped for the calculated Rayleigh damping coefficients. Thus, the main frequencies of the input signal in Figure 7 are not being filtered out due to
too much damping during the transient analysis.
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Appendix A Additional Information
Folder: Tutorials/DamRayleigh
Number of elements ≈ 1500
Keywords:
analys: eigen.
constr: suppor.
elemen: hx24l py15l solid te12l tp18l.
load: weight.
materi: elasti isotro.
option: direct.
post: binary ndiana.
pre: dianai.
result: displa eigen total values.
References:
[1] A. K. Chopra. Dynamics of structures – Theory and applications to earthquake engineering. Prentice-Hall, 2007.
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