Dynamic Analysis - Eigensolution

The dynamic analysis calculates the modes and frequencies of vibration for the model.

“Eigensolution” refers to the process used to calculate the modes. The frequencies and mode shapes are referred to as eigenvalues and eigenvectors. Refer to the Program Limits section for information on the maximum number of modes that can be solved for in RISA. The program can also solve for an approximate type of eigenmode called a Ritz Vector.

Perform a Dynamic Analysis / Eigensolution

To Perform a Dynamic Analysis / Eigensolution:

Suggestion: You may wish to solve a static analysis first to verify that there are no instabilities.

1. Go to the Home ribbon.

   

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2. Click the Solve icon down-arrow.

   

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3. Choose Dynamics from the solution options that appears, to open the Dynamics window.

   

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4. Click the Load Combinations for Mass down-arrow and choose the load combination to use as the mass and the number of modes to solve.

   The dynamic analysis uses a lumped mass matrix with inertial terms. Any vertical loads that exist in the Load Combination for Mass is automatically converted to masses based on the acceleration of gravity entry on the Solution tab of the Model Settings window. However, you must always enter the inertial terms as Nodal Masses.  

Eigensolutions In a Combined RISAFloor/RISA-3D model

When a dynamic solution is run from within a combined RISAFloor / RISA-3D model, the Eigensolution portion of the dialog includes eccentric mass and Floor Diaphragm mass options as described in RISAFloor Diaphragm Mass Options.

Required Number of Modes

You can specify how many of the model’s modes (and frequencies) are to be calculated. The typical requirement is that when you perform the response spectra analysis (RSA), at least 90% of the model's mass must participate in the solution.  Mass participation is discussed in the Response Spectra Analysis topic.

The catch is you first have to do a dynamic analysis in order to know how much mass is participating, so this becomes a trial and error process. First, pick an arbitrary number of modes (5 to 10 is usually a good starting point) and solve the RSA. If you have less than 90% mass, you'll need to increase the number of modes and try again. Keep in mind that the more modes you request, the longer the dynamic solution will take.

Dynamic Mass

The eigensolution is based on the stiffness characteristics of your model and also on the mass distribution in your model.  There must be mass assigned to be able to perform the dynamic analysis. Mass may be calculated automatically from your loads or defined directly.

In order to calculate the amount and location of the mass contained in your model, RISA takes the vertical loads contained in the load combination you specify for mass and converts them using the acceleration of gravity defined in the Model Settings.  The masses are lumped at the joints and applied in all three global directions (X, Y and Z translation).

You can also specify mass directly. This option allows you to restrict the mass to a direction. In addition, you can apply a mass moment of inertia to account for the rotational inertia effects for distributed masses. See Loads - Nodal Load / Displacement to learn more about this.

RISAFloor Diaphragm Mass Options

Models that get generated by RISAFloor automatically assign dynamic mass and mass moments of inertia (MMOI) to each of the RISAFloor diaphragms. This mass and MMOI is based on the self weight and loading information defined in RISAFloor and specified in RISAFloor's Model Settings.

There are 4 different sets of Mass defined for each diaphragm. The Centered Mass solution is when the mass is located at the mass centroid of the diaphragm. The Plus X, Minus X, Plus Z, and Minus Z solutions are when the mass has been offset to account for accidental eccentricity as specified in the Floor Diaphragms spreadsheet.

Whether this diaphragm mass / mass moment of inertia is included in the definition of dynamic mass is determined by the Include Floor Diaphragm Mass checkbox.

If the Include Floor Diaphragm Mass option is included in the analysis, then a Load Combination in most cases should not be considered for mass. The reasons you may wish to check the Include Load Combination box are:

• If you have RISA-3D only elements that you want to also include their mass in the eigensolution.

  To do this you would need to create a load combination that either contains only this additional seismic load, or that one that includes ALL seismic weight. If you choose to include ONLY the additional seismic weight in this load combination then you would check both the Include Floor Diaphragm Mass and the Include Load Combination checkboxes. If you include ALL seismic weight in your load combination then you would only choose the Include Load Combination checkbox.

• If you had a specific reason to not wish to use the RISAFloor calculated masses.  

  In that case you would uncheck Include Floor Diaphragm Mass and have only the Include Load Combination checkbox checked.

Eccentric Mass Solutions (for Accidental Torsion)

The eccentric mass options give an automated way to consider accidental torsion.

When more than one eigensolution option is selected (centered mass or one of the eccentric mass options), the program solves for a different set of frequencies and mode shapes for each of the mass options included in the solution. For information on how this affects response spectra analysis results, see RISAFloor Spectra Results.

Modeling Accidental Torsion

Most design codes require an assumed accidental torsion that is in addition to the natural torsion created by the location of mass with respect to rigidity.  While RISA-3D calculates the natural torsion you may want to model additional accidental torsion. This can be done easily by taking advantage of the rigid diaphragm feature.

If you have modeled the dynamic mass at the center of mass only, then you may simply move the nodes that specify the center of mass. For example, if the required accidental eccentricity is 5% of the building dimension, then move the nodes that distance, perpendicular to the applied spectra. You can then run the dynamic/rsa solution and combine the results with a static solution to check your members and plates for adequate capacity. You need to do this by running solutions for all four directions to capture the controlling effects on all frames. You will not be able to envelope your results since you are changing the dynamic results each time you move the mass. This means you’ll probably want to check all your load conditions one additional time after all your member sizes work to make sure that any force redistribution in your frames hasn't caused other members to fail.

Note that when you lump all your floor mass to the center of mass, you should also enter a Mass Moment of Inertia for your diaphragm as well by applying it as a Nodal Mass to the center of mass node. The rotational inertial effects of the diaphragm mass will contribute to your torsion shears and should not be ignored in most cases. See Loads - Nodal Load / Displacement to learn more about this.

If you have not modeled the mass at discrete points that can be easily moved, then you have to apply the accidental torsion as a static load that is be part of a static analysis solution which includes the response spectra or equivalent lateral force procedure results. The magnitude of the torque is the product of the story force and the accidental offset distance.

The accidental offset distance is usually a percentage of the building dimension perpendicular to the assumed earthquake direction. The story force is the story mass times the acceleration at that story level. If you are using an equivalent lateral static force procedure, you have already calculated your story forces. If you are performing a response spectrum analysis, you can get the story forces exactly as the difference between the sum of the shears below and above the floor. Alternatively, you could simply use the full weight of the floor as the story force and then apply the scaling factor for your normalized spectra to this value as well. This simplified method can be unconservative if your floor accelerations have large amplifications as compared to your base acceleration due to the dynamic characteristics of your building.

The torsion can be applied as a point torque that you can apply to a node on the diaphragm.  The torque can also be applied as a force couple, with the magnitude of the forces determined by the distance between them to make the needed torque value. Often it is convenient to apply the forces for the couple at the ends of the building. One advantage of applying the accidental torsion as a static force is that you can set up all your required load combinations and let RISA-3D envelope them for you in one solution run.

Eigensolution Convergence

The eigensolution procedure for dynamic analysis is iterative, i.e. a guess is made at the answer and then improved upon until the guess from one iteration closely matches the guess from the previous iteration. The tolerance value is specified in the Model Settings and indicates how close a guess needs to be to consider the solution to be converged. The default value of .001 means the frequencies from the previous cycle have to be within .001 Hz of the next guess frequencies for the solution to be converged. You should not have to change this value unless you require a more accurate solution (more accurate than .001?).  Also, if you're doing a preliminary analysis, you may wish to relax this tolerance to speed up the eigensolution. If you get warning 2019 (missed frequencies) try using a more stringent convergence tolerance (increase the exponent value for the tolerance).

Saving Dynamic Solutions 

After you’ve done the dynamic solution, you can save that solution to file to be recalled and used later. 

Work Vectors

When you request a certain number of modes for dynamic analysis (let's call that number N), RISA tries to solve for just a few extra modes.  Once the solution is complete, RISA goes back to check that the modes it solved for are indeed the N lowest modes. If they aren't, one or more modes were missed and an error is reported.

Dynamics Modeling Tips

Dynamics modeling can be quite a bit different than static modeling. A static analysis almost always gives you some sort of solution, whereas you are not guaranteed that a dynamics analysis will converge to a solution. This is due in part to the iterative nature of the dynamics solution method, as well as the fact that dynamics solutions are far less forgiving of modeling sloppiness than are static solutions. In particular, the way you model your loads for a static analysis can be very different than the way you model your mass for a dynamic analysis. 

The term “dynamics solution” is used to mean the solution of the free vibration problem of a structure, where we hope to obtain frequencies and mode shapes as the results.

In general, the trick to a “good” dynamics solution is to model the structure stiffness and mass with “enough” accuracy to get good overall results, but not to include so much detail that it take hours of computer run time and pages of extra output to get those results. Frame problems are simpler to model than those that include plate elements. “Building type” problems, where the mass is considered lumped at the stories, are much easier to successfully model than say a cylindrical water tank with distributed mass. It is often helpful to define a load combination just for your dynamic mass case, separate from your “Dead Load” static case (You can call it “Seismic Mass”). Your seismic mass load combination will often be modeled very differently from your “Dead Load” static case.

Modes for discretized mass models with very few degrees of freedom may not be found by the solver, even if you know you are asking for fewer modes than actually exist. In this case it may be helpful to include the self weight of the model with a very small factor (i.e. 0.001) to help the solver identify the modes.

Self-Weight Considerations

Distributed mass models with plate elements, like water tanks, often require special consideration. You will want to use a fine enough mesh of finite elements to get good stiffness results. Often though, the mesh required to obtain an accurate stiffness will be too dense to simply model the mass with self-weight or surface loads. You will want to calculate the water weight and tank self-weight and apply it in a more discrete pattern than you would get using surface loads or self-weight.  This method of using fewer nodes to model the mass than to model the stiffness is often referred to as "discretizing" the mass. You want to lump the mass at fewer points to help the solution converge faster, however you have to be careful to still capture the essence of the dynamic behavior of the structure.

Whenever you perform a dynamic analysis of a shear wall structure, and the walls are connected to a floor, you must be careful to use a fine mesh of finite elements for each wall. Each wall should be at least 4 elements high between floors. This gives you at least 3 free joints between them.

Beam Vibration Considerations

When you perform a dynamic analysis of beam structures, such that you are trying to capture the flexural vibrations, (i.e., the beams are vibrating vertically or in the transverse direction), you must make sure that you have at least 3 free joints along the member between the points of support. If you use a distributed load as the mass, you must remember that some of the load will automatically go into the supports and be “lost” to the dynamic solution. In general, you get the best results by applying your mass as joint loads to the free nodes.

If you are trying to model dynamic effects on a 2D frame, you’ll want to make sure that you restrain the out-of-plane degrees of freedom. See Boundary Conditions at ALL Nodes to learn how to do this.

Calculate Residual Mass

The mass of the structure that has not been activated by the solved number of periodic modes may still be included in the response spectra analysis results. This is done by checking the Calc. Residual Mass box. This checkbox calculates a residual mass mode shape which helps to show where the missing mass in the structure is located. It also includes the response of this missing mass as a static correction to the response spectra results. The acceleration used to calculate this static correction vector is based on largest acceleration from the specified response spectra which occurs between the user entered Cut Off frequency (see below) and the zero period acceleration.

Common Application for Missing Mass Vectors

If you apply your dynamic mass with distributed loads or surface loads on members/plates that are adjacent to supports, remember that the some of the load goes directly into the support and is lost to a traditional dynamic solution. The mass that can actually vibrate freely is your “active mass”, as opposed to your “static mass” which includes the mass lost into the supports. If you are having trouble getting 90% mass participation, you should roughly calculate the amount of mass that is being lost into your support to determine if that is the cause of the problem.

Since this mass is directly associated with boundary conditions and cannot be activated by a traditional eigensolution mode, it is referred to as the Residual Rigid Response of the structure. The common solution to account for this response, is to check the Calc. Residual Mass? checkbox. This calculates a "Missing Mass Vector" or "Residual Mass Vector" to account for this residual mass in the dynamic analysis.

Alternatively, you can switch to the Ritz Vector dynamic solver because Ritz modes inherently include correction for the rigid response.

Modal Frequency Results

To access the Frequencies spreadsheet:

1. Go to the Explorer panel Results section.

   

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2. Click on Frequencies.

   The Frequencies spreadsheet opens.

   

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   The Frequencies spreadsheet shows the calculated model frequencies and periods. The period is simply the reciprocal of the frequency. These values are used along with the mode shapes when a response spectra analysis is performed.  The first frequency with a high participation is sometimes referred to as the model's natural or fundamental frequency. These frequency values, as well as the mode shapes, are saved and remain valid unless you change the model data, at which time they are cleared and you need to re-solve the dynamics to get them back.

   Also listed on this spreadsheet are the participation factors for each mode for each global direction, along with the total participation. If the participation factors are shown in red (as opposed to black) then the response spectra analysis (RSA) has not been performed for that direction. If the RSA has been done but a particular mode has no participation factor listed, that mode shape is not participating in that direction. This usually is because the mode shape represents movement in a direction orthogonal to the direction of application of the spectra.  See Dynamic Analysis - Response Spectra for more information.

   Note:

   • If you have not solved an RSA for a given direction, the participation factor shows in red as an indicator. This is not an indication that there is anything wrong with the solution.
   • If you are using the Calc Residual Mass? checkbox, you get three "residual" mode shapes that account for the missing mass vector. For more information on this see the Residual Rigid Response section.
   • At this time residual / missing mass modes are not considered in Time History analysis.

   

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Mode Shape Results 

To access the Mode Shape spreadsheet:

1. Go to the Explorer panel Results section.

   

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2. Click on Mode Shapes.

   The Mode Shape spreadsheet for the first Mode Shape Period.

   

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   These are the model's mode shapes. Mode shapes have no units and represent only the movement of the nodes relative to each other. The mode shape values can be multiplied or divided by any value and still be valid, so long as they retain their value relative to each other. To view higher or lower modes you may select them from the drop-down list of modes on the Window Toolbar.

   Note:

   • Keep in mind that the X, Y, and Z mode shapes do not, in and of themselves, represent model deflections. They only represent how the nodes move relative to each other. You could multiply all the values in any mode shape by any constant value and that mode shape would still be valid.  Thus, no units are listed for these mode shape values.
   • The rotation is in units of radians.

   These mode shapes are used with the frequencies to perform a Response Spectra Analysis. The first mode is sometimes referred to as the natural or fundamental mode of the model. The frequency and mode shape values will be saved until you change your model data.  When the model is modified, these results are cleared and you will need to re-solve the model to get them back.

   
   

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3. (Optional) To plot and animate the mode shape of the model, got to the ‘View‘ ribbon and click the Results icon in the Animate section, as shown in the following image.

   

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   This allows you to verify the mode shapes that were obtained and highlights local modes making them easy to troubleshoot.  See Results View Settings - Deflections for more information.

Dynamics Troubleshooting – Local Modes

A common problem you may encounter are “localized modes”. These are modes where only a small part of the model is vibrating and the rest of the model is not. A good example of this is an X brace that is vibrating out of plane.  Localized modes are not immediately obvious from looking at the frequency or numeric mode shape results, but they can be spotted pretty easily using the mode shape animation feature. Just plot the mode shape and animate it. If only a small part of the model is moving, this is probably a localized mode.

The problem with localized modes is that they can make it difficult to get enough mass participation in the response spectra analysis (RSA), since these local modes don’t usually have much mass associated with them.  This will show up if you do an RSA with a substantial number of modes but get very little or no mass participation. This would indicate that the modes being used in the RSA are localized modes.

Quite often, localized modes are due to modeling errors (erroneous boundary conditions, members not attached to plates correctly, etc.).  If you have localized modes in your model, always try a Model Merge before you do anything else.  See Model Merge for more information.

Eliminating local modes Using Boundary Conditions

To get rid of localized modes that are not the result of modeling errors, you can sometimes use boundary conditions to restrain the mode shape. 

For example, if your localized mode is at a weak X brace (as mentioned before), you could attach a spring to the center of the X brace to restrain the mode shape. Alternatively, in this instance you could also:

• Make the braces a weightless material. In that case no mass would be assigned to the X-brace intersection and it would not participate in the eigensolution.
• Delete the node at the crossing of X-braces. If there is no node here there is no degree of freedom. This location will not likely be crucial for a dynamic solution anyway, so eliminating the node eliminates a flexible mode with minimal mass participation.

Using Ritz Vectors to Minimize Local Modes

Another cause of local modes is including the self-weight in models with walls or horizontal diaphragms modeled with plate/shell elements. These walls and floors can have many modes that will tend to vibrate out-of-plane like drums, but will have very little effect on the lateral seismic response of the structure. For cases like this it will often be better to switch from the standard or accelerated dynamic solvers to the Ritz Vector dynamic solver because Ritz vectors are inherently biased to avoid modes with little mass participation in the desired direction.