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{Test Problem

This page describes a simple beam example of how you could set up the stepping loading in a time history analysis. Please note that we have not used realistic parameters for this example as its purpose is simply to show you how to set up the model, not how to determine the parameters, such as load magnitude, stride length, time between steps, etc.

The four steps required are:

(1) Define as many load cases as you have number of foot fall positions that you want to simulate in your model. If you have 100 foot fall load points, then you will need 100 load cases. You will want to number these so it is obvious as to the order; I have used 'Step 1', 'Step 2', etc.

(2) For each load case apply a point load at the position at which the load will be applied for the respective step. We like to use a unit load at each position so that we can adjust the overall magnitude more easily elsewhere.

(3) Define a single time history function that represents the impulse of the foot fall. This could be a unit impulse or if all of the foot fall positions will receive the same load, you can set the magnitude in the function definition. Note that if you more than one foot fall impulse function shapes, there is no reason why you cannot define more than one.

(4) Finally you need to define a time history analysis case. There are three analysis types that you can use, as follows:

(a) Modal time history based on Eigen modes - this option uses an Eigen modal analysis to carry out a modal time history analysis. You need to ensure that you have captured enough modes for the structure under consideration.

(b) Modal time history based on Ritz modes - Ritz modes are a better option for modal time history analysis, but when used in a modal time history analysis, require you to use each of the foot fall loads defined in (1) as a starting load vector and you will need a mode for each of these loads, so if you have 100 load cases, you will need at least 100 modes.

(c) Direct integration time history - this is not based on modes, but instead works on a step-by-step basis. Direct integration tends to be slower than modal time history analysis, but if the number of modes required for modal time history analysis is getting high, this could be just as good of an option.

For any of these three analysis types you need to add each of the foot fall load cases to the Loads Applied section, as shown below. For each load case, specify the impulse function, a scale factor (this is optional depending on whether you included the full magnitude in the function; ultimately the overall result will be Load * Function * Scale Factor), and an arrival time. The arrival time is the important thing here as it will define when each of the loads is applied, so you need to work out the timing between steps. Finally, make sure that the Time Step Data part of the form is getting enough time steps to cover the duration of the time history for which you want results.

Attachments

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Modal time-history analysis is well-suited for the modeling and analysis of human-induced vibrations which result from such impacts as pedestrian footfalls. Guidelines are presented in this test problem. Please note that these parameters are not intended to be realistic. It is up to the engineer to determine load magnitude, stride length, load frequency, and other input values.

Human-induced vibrations may be modeled as follows:

1. Define a load case for each footfall location, and for convenience, name each according to the loading sequence, such as Step 1, Step 2, etc. For example, 100 footfall locations would require 100 load cases.
   
   
2. For each load case, apply a point load in the location of the corresponding footfall. It may be convenient to use a unit load, then adjust magnitude when setting the scale factor.
   
   
3. Define a single time-history function which will represent the footfall sequence. If a unit load is applied to each load case, set the magnitude in the function definition. Additional functions may be defined to consider multiple impulse-function shapes.
   
   
4. Define a time-history analysis case from the following options:
   
   
   • Modal time history based on Eigen modes, where a sufficient number of modes should be captured for the given structure.
     
     
   • Modal time history based on Ritz modes, which may be a better option as far as analysis goes, though Ritz formulation requires as many modes as there are footfall locations and load cases, since each position must be considered as a starting load vector. For example, 100 footfalls require 100 modes.
     
     
   • Direct-integration time history, which is not based on modes, but rather a step-by-step process. Direct integration typically demands more time, but if a large number of modes are necessary for modal analysis, direct integration may be more viable.
     
     
5. Add each footfall load case to the Loads Applied section (Figure 1), and specify the impulse function, scale factor, and arrival time. The timing of load sequence should be carefully considered and calculated, since arrival time and the application of impact is critical to dynamic response. Ensure that sufficient time steps are provided in the Time Step Data section to cover the duration of the time history.

Image Added

Figure 1 - Loads Applied

See Also

• Power-spectral-density FAQ article

• Floor vibration due to human footfalls ETABS article

Attachments

• SAP2000 V11.0.8 model (zipped .SDB file)

Metadata

• Name: Human-induced vibrations
• Description: The modeling and analysis of human-induced vibrations due to footfalls or another type of impact.
• Program: SAP2000
• Program Version: 14.1.0
• Model ID: na