Hello everyone,
In a response spectrum analysis, we first obtain the individual modal response to an input response spectrum.
Let’s say that we have obtained 20 modes for a given structure; this means that we have 20 responses.
Note that we only have the modal response but no phase information,
so have you ever wondered how the peak response for the structure is estimated?
The concept of mode combination methods comes into picture here,
where the selected method is used to approximate the maximum response of the structure.
In this video, we will discuss the different mode combination methods available in Ansys Mechanical, their formulation and usage,
along with their applications followed by a walkthrough example to help you select the appropriate method.
Ready?
Let’s get started.
To conduct a response spectrum analysis of a structure, we first need to obtain modal results,
including the frequencies, mode shapes and the participation factors of each mode.
With the given input response spectrum, we can find the mode coefficient or each mode, which is denoted by A_i.
Note that the subscript I indicates that the mode coefficient is found for each mode.
Knowing the mode coefficient, frequency, and mode shape, we can then calculate the response of each mode.
Now we have N sets of response, depending on how many modes we obtained for the structure.
Our goal in Response Spectrum analysis is to approximate the peak response of the structure.
Unfortunately, we don’t have phase information, so we can’t directly add the modes together,
so our next step is to find another way to combine these modal contributions.
There are multiple mode combination methods that have been developed for response spectrum analysis,
but most of them evolved from one general equation.
In this general form, we are looking at a linear combination of the product of mode pairs Ri and Rj,
and the square root of this value is taken.
Note that i and j can be different or can be the same mode number.
Epsilon is a number between 0 to 1, and we call it the modal combination coefficient or coupling coefficient.
It decides the degree at which the two modes are correlated with each other.
If epsilon is 0, then the term RiRj becomes 0 and it doesn’t contribute to the combined response.
On the other hand, if epsilon is 1, the entire product RiRj are added in the combined response.
Use a simple example may give a better explanation here.
Say we only have two mode responses, R1 and R2.
Then the total response can be fully written as this,
The first two terms can be thought of as the individual contributions of modes 1 and 2 while,
the latter two terms can be considered as the interaction between modes 1 and 2.
If we define epsilon12 and epsilon 21 to be 0, then the coupled terms are removed in the response Ra.
We may ask ourselves, when should we consider the coupling effect of two modes in mode combination?
It all depends on the “distance” between the two modes.
When we say distance, we mean the frequency difference between the two modes.
If the frequencies of two neighboring modes are very close to each other, we call them closely spaced modes.
You might ask, how close is closely spaced?
Actually,there is no hard rule, but a guideline, especially for seismic applications, is when the damping ratio is less than 2 percent,
modes may be considered closely spaced if the frequencies are within 10% of each other.
If there’s no closely spaced modes for a structure,
we can neglect the dependency between the modes.
This means we can remove the coupled terms in the mode combination equation.
And this leads to a simplified expression. We call this SRSS method, basically it’s the square-root sum of the squares of each mode.
Now if we have two mode responses R1 and R2, by SRSS, the total response will be quite straightforward.
On the other side, if there are closely spaced modes for a structure,
we may consider using a mode combination method that can take the coupling effect account in the total response.
In Ansys Mechanical, there are two available methods, Rosenblueth Method and Complete Quadratic Combination Method.
Let’s check Rosenblueth method first
The Rosenblueth method does not change the general equation for mode combinations.
It just defines how the coupling coefficients should be calculated.
We don’t need to remember exactly the expression of coupling coefficient here, but we can try to understand it.
Here omega_i and omega_j represents natural frequencies of mode I and J.
Xi prime is related to the modified damping ratio.
So, if the frequencies of two modes are very close to each other, this term tends to be 0,
and the coupling coefficient will approach to 1.
On the other hand, if the difference of the frequencies is extremely large, the denominator tends to be infinite,
and the coupling coefficient epsilon will approach 0, meaning little or no coupling effect will be considered.
Note that xi prime is a modified damping ratio.
For Rosenbluth method, even when we specify 0 damping for the structure,
there is an internally calculated damping included in the mode combination.
Now let’s move to the Complete Quadratic Combination Method.
We’ll just call it CQC method.
This method takes a slightly different format.
First of all, there’s an absolute bracket to make sure that the term under the square root is positive.
And the coefficient k is 1 for I = j and 2 for I does not equal to j .
The definition of coupling coefficient epsilon is a little bit complicated here.
But we don’t need to worry about it too much.
Ansys calculates the coefficient automatically for us.
Although it cannot be directly read from the expression,
this epsilon definition also approaches unity when the modes are closely spaced and becomes 0 when they are sufficiently spaced.
Also note that,for the CQC method, damping is required.
Both CQC method and Rosenblueth methods originated from seismic applications.
Now let’s go to a workshop model to check how mode combination methods is specified, and how different methods affect the results.
In this demo,
we will use a simplified tower with two eccentric masses to compare different mode combination methods for Response Spectrum analysis.
The simplified geometry is for demonstration purposes only.
Now,let’s open the archived Ansys workbench file of the simplified tower.
The project is already set up to perform the modal analysis.
We can expand the branches in the Project Tree to become familiar with the model setup.
The tower structure is represented with line bodies, these line bodies are already meshed with beam elements.
The beam elements have been assigned cross sectional properties.
Material properties of all beams are assigned with linear elastic structural steel.
Two eccentric distributed masses have been scoped to the tower beams and assigned a mass to represent a specified design load for the tower.
Expanding the Modal Branch, we see that The tower is fixed at the bottom to the ground.
The modal has already been set up to solve for the first 160 modes.
Now, we can solve the Modal Analysis
Since in this case, we know our Input spectrum is given in X direction,
so we will check the frequencies and the Participation Factors in X direction,
Let’s go to solution information, Solution output, change it to participation factor summary.
We can see that the frequency results include some close-spaced modes with relatively large participation factors.
With modal results abtained, then we will run the Response Spectrum anlaysis with SRSS and Rosenblueth method respectively.
For Rosenblueth method, we will run the model with and without damping.
Go to Workbench, Drag, and drop a Response Spectrum analysis system over to Solution Cell of the Modal analysis and rename system B to “SRSS”.
Go back to Mechanical window.
Go to Analysis settings of system B, choose SRSS for Modes Combination method.
Note that SRSS doesn’t use any damping input.
Now, we will apply an Response Spectrum Acceleration load, right click over in the Response Spectrum B, insert RS Acceleration,
change Boundary Condition to All Supports and Direction to X Axis.
We will use the seismic data from an excel file, copy the spectrum data, and paste it into the Tabular Data.
Now, we can solve the Analysis.
Insert Total Deformation and Direction Deformation for X,Y and Z direction and evaluate all results.
Here. notice that the contour plot does not show a deformed shape,
since the values are always positive from a response spectrum analysis, it would be misleading to show a deformed shape on such values.
We will set up another two scenarios, Roseblueth and Roseblueth with 2% damping.
Let’s go back to Workbench, Duplicate system B twice.
Click yes for the prompt to keep the upstream connections.
Now rename the newly created system C to “Rosenblueth”, and system D to Rosenblueth 2% damping.
Let’s go to Analysis settings of system C, choose Rosenblueth for Modes Combination method.
And do the same thing for system D but here we put 0.02 for the damping ratio in damping control.
Now we can solve the Analysis C and D.
Let’s compare the results from these three different scenarios.
For total deformation, there is small differences between SRSS and Rosenblueth without damping.
Rosenblueth with 2% damping shows obvious difference from the other two cases.
For directional deformation in the excitation direction the x direction, Rosenblueth predicts higher deformation than SRSS.
For Y and Z deformations, Rosenblueth method shows lower deformation compared with SRSS
Here, SRSS may overpredict response since it seems unintuitive that deformation in the z direction would be larger than in the x direction, which is the excited direction.
which is the excited direction.
Rosenblueth considered the interaction between closely spaced modes.
The total estimated response is lower since close modes may interact in such a way to negate effects due to sign of modes.
In the solver output, we can check model combination coefficients.
Since we required 160 modes, solver output file is too large to be shown in Mechanical worksheet.
We can open the solve.out file located in the Solver files Directory.
If we open solve.out file and take the 7th mode of this tower as an example, it is coupled with multiple modes, indicated by the coupling coefficient.
This is expected because the frequencies of these modes are quite close.
This concludes the demo.
Now, let’s summarize what we learned in the video.
Once Individual responses are available, to estimate a maximum response, these responses need to be combined and analyzed,
and this is done with the help of mode combination methods.
There, are multiple mode combination methods that have been developed for response spectrum analysis,
but most of them evolved from one general equation.
Here, epsilon is the coupling coefficient that considers the interaction of modes,
where it can range from zero, indicating no interaction, to a value of one, indicating a strong interaction.
In Ansys Mechanical, 3 types of Mode Combination methods are available-
​SRSS​
CQC​
Rosenblueth
The SRSS method does not account for modal coupling between different modes whereas,
the CQC and Rosenblueth method account for the interaction of modes that are close to each other.
I hope that you find this video informative,
don’t forget to visit courses.ansys.com to discover more useful courses.