Vibration of floors and footfall analysis

Transcription

1 Webinar Autodesk Robot Structural Analysis Professional 20/04/2016 Vibration of floors and footfall analysis Artur Kosakowski Rafał Gawęda

2 Webinar summary In this webinar we will focus on the theoretical background and present floor vibrations and footfall analysis in Robot. 2

3 Topics covered in this Webinar and what we plan for the next one: This webinar: Vibration of floors and footfall analysis Dynamic analysis of structures Modal analysis definition Dynamic mass definition Forced harmonic analysis in the frequency domain (FRF) Footfall analysis Frequent mistakes Tips & tricks Next webinar: Time History Analysis 3

4 Basis of dynamic analysis M * a(t) + C * v(t) + K * d(t) = F(t) where: M - mass matrix K - stiffness matrix C - damping matrix d - displacement vector v - velocity vector a - acceleration vector F - load vector t - time 4

5 Dynamic modal analysis For this type of analysis the previous general equation simplifies to the following form: M * a(t) + K * d(t) = 0 This equation defines the eigenvalue problem and by solving it we can obtain natural frequencies (eigenfrequencies) and determine associated shapes (modes, eigenvectors) of free vibrations of a structure. Mind that the mass matrix M Robot can be influenced by defined added masses or load to mass conversion. 5

6 Harmonic analysis in the frequency domain (FRF) This analysis is intended for specific type of forced vibrations. It treats all loads defined in the associated load case as amplitudes of harmonic force from some range of frequencies. The obtained results (displacements, accelerations, forces, moments, etc.) are also amplitudes corresponding to steady-state sinusoidal harmonic vibrations. 6

7 Harmonic analysis in the frequency domain (FRF) In case of a model with many dynamic degrees of freedom and with non-zero damping the typical diagram of response in the function of frequency may look as on the picture below: Peaks on this diagram correspond to resonance observed for natural frequencies of the structure. In case of no damping these peaks would go to the infinity. 7

8 Harmonic analysis in the frequency domain (FRF) The same type of diagram from FRF analysis in Robot obtained for the slab with using 1% damping declared. 8

9 Harmonic analysis in the frequency domain (FRF) Inflence of damping value on the reduction of the resonace peaks 1% damping 3% damping 9

10 Footfall analysis The implementation of the footfall analysis in Robot is based on: The Steel Construction Institute SCI P354, Design of Floors for Vibration, 2007 rev SCI AD 253 SCI AD 254 SCI AD 254 The Concrete Centre, A design Guide for Footfall Induced Vibration of Structures, 2006 AISC DG11, Floor Vibrations Due to Human Activities, 2003 SCI and Concrete Center publications additionally refer to BS , BS 6472, BS 6841, EN 1990:2002, ISO 2631, ISO

11 Footfall analysis Footfall analysis is to some extent similar to harmonic FRF analysis it is also based on harmonic sinusoidal force input but instead of single harmonic a complex continuous founction is used instead. This function can be broken down into a series of sine waves, each of which has a frequency at an integer multiple (or harmonic) of the forcing frequency. Each harmonic will have an associated amplitude and phase shift, and the set of harmonics are known as a Fourier series. The first harmonic, with the lowest frequency, corresponds to the frequency of footsteps. This frequency for floors in various methods is in the following range: 1.8 to 2.2 Hz for SCI P to 2.8 Hz for Concrete Centre 1.6 to 2.2 Hz for AISC DG11 In all these methods 4 harmonics are used. In case of footfall analysis for stairs (SCI P354) the range of footstep frequency is higher (1.2 to 4.5 Hz) and only 2 harmonics are used 11

12 Footfall analysis The example of representing low impact aerobics as Fourier series Walking activity on floors is approximated in SCI P354 by 4 harmonics with amplitudes F h and frequencies and phase angles given in this formula and table 12

13 Footfall analysis Response of a structure to footfall depends both on the frequency of force function (footfall) and eigenfrequencies of the floor. Types of responses: resonant when the response for the consecutive footsteps builds up (cumulates), especially for these harmonics of footsteps which have the same frequencies as the natural frequencies of the floor. impulsive (transient) significant mainly for high frequency floors (having the fundamental frequency higher than the 4th harmonic of walking). In such situation the response from one footstep will fade away before the next one starts. 13

14 Footfall analysis The resonant response is calculated in the way similar to harmonic FRF analysis by composing the response from different harmonics The impulsive (transient) response is calculated as for the time history analysis for single footfall impulse It is observed that: the resonant response is dominating for low frequency floors the impulsive (transient) response is dominating for high frequency floors. The code defined limit can be different e.g.: the 4th harmonic of walking frequency; 8 Hz; 4.2 times the maximum walking frequency; 10 Hz Mind that Robot for Concrete Centre and for SCI provisions provides results for both responses as especially that for irregular floors resonant response may dominate in some parts of the floor while impulsive (transient) response may dominate in other parts. 14

15 Footfall analysis Presentation of variable acceleration (response): Peek value root-mean-square (rms) value In case of sinusoidal signal the rms value is 2 times less than the peak value. root-mean-quad (rmq) value: It gives more emphasis to higher values 15

16 Footfall analysis Acceptance criteria for human comfort Human perception of vibrations depends on their frequency. This diagram shows the base curve for human perception of continuous vertical vibration according to BS It uses rms acceleration and logarithmic scale. In the most sensitive frequency range, between 4 and 8 Hz, the criterion is constant acceleration. Above 8 Hz it is linearly increasing acceleration which corresponds to constant velocity. Such base curve is directly used by Concrete Centre and refered to in SCI P354 and AISC DG11. 16

17 Footfall analysis Acceptance criteria for human comfort The basic value, calculated by footfall analysis for SCI P354 and for Concrete Centre provisions, is the response factor (R). It is a multiplier for the level of vibration compared to the average threshold of human perception. Mind that the response factor of 1 corresponds to the magnitude of vibration just perceptible by typical human while R=2 corresponds to vibration twice stronger, R=4 corresponds to vibration 4 times stronger than perceptible and so on. In case of the resonant response R is calculated for each harmonic and then combined as SRSS (square root of the sum of squares) combination. In case of impulsive (transient) response R is based on RMS values calculated from time histories using 1-second averaging period. 17

18 Footfall analysis Acceptance criteria for human comfort code defined limits: SCI P354, based on BS 6472: AISC DG 11 : 18

19 Footfall analysis Recommendations from references: Using the dynamic value of the modulus of elasticity for the concrete, 38 GPa for normal weight concrete, 22 GPa for lightweight concrete Recommended values of relative damping: Load to mass conversion: unfactored self weight and other dead loads live loads: it is recommended to convert to masses only 10% of nominal imposed load (EN 1990 mentions 30% but it is considered as conservative) 19

20 Footfall analysis Footfall analysis parameters: 20

21 Frequent mistakes Not all required loads are converted to masses Doubled self weight due to unchecked Ignore/Disregard density switch while converting to mass the load case containing the self weight load Comparing eigenfrequencies obtained for Footfall analysis with only mass in the Z direction active with modal analysis case where all directions are active. 21

22 Tips & tricks The default maximum frequency limit of 15 Hz in Footfall analysis may result in having no results for a structure with vibration frequencies higher than this limit. Depending on the method some results of footfall analysis such as e.g.: velocities for SCI P354, RMS acceleration for Concrete Centre provisions, transient/impulsive results, velocities and RMS acceleration for AISC DG11) may be not available. In case of large number of modes below 15 Hz limit running Footfall analysis may take long time. Starting from RSA 2016 SP3 it is possible to speed up analysis several times by setting the Lanczos Method in Job Preferences. 22

23 Tips & tricks The Self excitation method in Footfall analysis calculates and stores the response only for the same nodes where excitation forces were applied. For the Full excitation method in Footfall analysis the solver can calculate results for selected nodes which are not the ones where excitation forces are applied. These results are available in Footfall analysis tables and Footfall analysis maps.... Footfall analysis diagrams... are calculated on line and can be used to access results for all nodes for both above methods regardless of their original node selection. 23

24 Tips & tricks The footstep excitation forces are applied independently for each of the nodes and there is no interaction between them. In Harmonic FRF analysis excitation forces can be aplied simultaneously in different locations but they are limited to the same frequency in all these locations with no phase angles among them. Footfall analysis is limited to standard walking activities. In this type of analysis it is not possible to calculate e.g. the influence of crowd movement or aerobics activities or interaction between footstep excitation forces. In case you need more use Time History Analysis 24

25 Useful links Robot webinars Robot discussion forum Robot troubleshooting articles on AKN 25

26 Questions? Please feel free to ask questions using «GoToMeeting» Questions tab now We may not be able to answer all questions during the webinar. Please post them on the Robot forum after the presentation. 26

27 Next webinar session on 25/05/2015 on the following topic : Time history analysis Autodesk is a registered trademark of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries. All other brand names, product names, or trademarks belong to their respective holders. Autodesk reserves the right to alter product and services offerings, and specifications and pricing at any time without notice, and is not responsible for typographical or graphical errors that may appear in this document Autodesk. All rights reserved Global Customer Support & Operations