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The purpose of this plan is to have an outline of procedure and method in place so work can begin straight away in Semester 3. After the plan, this document contains a preliminary background review describing how objectives can be achieved. As this is a preliminary survey of the literature it can be updated as the project progresses into a full literature review; however, it will provide enough of an understanding to commence the dissertation work. Also in this plan is an intended schedule of work and time required.
1.2 The Problem
Planned new construction of a multi-purpose leisure building was due to be finished and opened in 2009. The new opportunities provided for the seaside town were delayed however due problems with the design. The two storey building acquired planning permission to house a bowling alley along with retail shops, cafes and restaurants etc, and a leisure and conference suite along with roof terraces for views of the sea, as well as function rooms available to hire for weddings etc.
On the first floor, above the bowling alley and shops is where the additional function rooms are to be situated. During construction of this suspended floor over-size vibrations were noticed. The problem arises in that this floor will be receptive of crowd dynamic loading from the various uses and so was thought to be incapable of accepting such loading bearably. The co-ordinated movements of groups of people from dance, aerobic and gymnasium type activities were not properly considered in an adequate design vibration check, leading to problems becoming visible. The University of Sheffield Vibration Engineering Section (VES) were employed to dynamically test the structure leading to a solution of the problem.
1.3 Project Proposal
In short the project proposes to analyse the floor from the design stage, through to the dynamic testing results and come up with a viable solution for the structure's problems.
Firstly research will be required into accurate ways model this type of dynamic loading and any relevant design codes of practise, this will be included in a full literature review. Secondly the construction of an FE model in ANSYS to which this loading simulation can be applied. Thirdly, it is proposed to assess the model's usefulness against the experimental data available from the VES. This data can also be used to update the FE model after comparison. With a working model it is then planned to come up with a resolution to the problem and test the outcome. Lastly, as an extra endeavour, an analytical model of the problem using hand calculation is planned.
The chance to create a fully working FE model and validate against the VES test data for a real life structure provides rationale for the project.
1.4 Outline Plan
This section describes the general approach to the project.
The required main deadlines of the project are:
28th May (09.00)
Preliminary presentation - 4 minutes
14th June (14.00)
Dissertation plan - this document - 15 - 20 pages
3rd September (12.00)
Dissertation submission - 2 bound copies and a PDF
6th - 10th September
7th September (17.00)
Conference paper - maximum 6 pages
10th September (14.00)
Conference presentation submission
Technical conference - 12 minute presentation
Table 1: Deadlines
Before any computer modelling begins a review of the relevant design codes and guidelines needed to be carried out. A brief overview of the findings is given in Section 2. This was a key stage in the project; as progressing with any aspect was otherwise impossible, guidelines being the only way to make sure loading/models are created in a realistic fashion by relying on past research results. By the end of semester two (i.e. given in the background review in section 2) there should be enough researched information into:
Formulations of applying loading
Ways to apply this in ANSYS
The analyses required from the FE software
Methods of quantifying this analysis into 'yes' or 'no' acceptability
Improving vibration characteristics of a structure
The FE modelling of the floor is expected to take a significant amount of time (adding the consideration of negligible experience with the software,) it could be the most important part of the project. To approach this in a methodical way the task was split into four areas;
Start with ANSYS - basic operations/functions tutorials and small engineering problems from internet sites and help tutorials, this was completed by the end of the Easter vacation period.
Structural use of ANSYS -practise/research into different model types and methodologies with hypothetical floor situations, and carry out modal analyses. To be completed by the end of semester 2.
Model leisure centre floor - preliminary attempts at the project model using the given CAD drawings to make .dat input scripts. To be built up in complexity in a sequential manner for discussion with supervisor or phD student advisor at each level if required.
Apply loads - when a reasonable model is made apply walking/ jumping loads and group jumping loads for analysis. These loads are discussed further in section 2.2, as well as the acceptable criteria for design for which they should be measured against in section 2.3.
When the FE model is complete and it is believed that adequate modal results have been acquired the VES modal test data will become available. At this comparison stage it will be possible to understand any short comings of the model and analyse the reasons for them. Also the vibration test results from VES can be compared to the loaded model. Evaluations are likely to be made with acceleration response plots and modal properties.
1.4.5 Remedial Measures
After sufficient validation and updating, it should be clear what the cause of the vibration serviceability problems are. It would be useful here to note how responsible the design consultants were with the design, did they make large misjudgements, ignore guidelines, make understandable errors, or were there problems with the construction/assembly aspect.
From the research done it should also be clear what options are available to solve these problems. If necessary, further structural elements will be designed and added to the FE model to prove their effectiveness.
1.4.6 Time Management
As agreed with the supervisor there will be meetings on roughly a fortnightly basis to review progress but for smaller issues related to ANSYS advice from a phD student will be made available. Over the entire semester it is planned to stay in Sheffield and work a normal working week. This should also allow for a summer break period.
It is useful to see the work schedule in timeline format for which an A3 size Gantt chart is given as Appendix B.
1.5 Present Situation
As there was no prior experience with ANSYS, the starting point was the absolute basics. The built-in tutorials were run through to give an understanding of the general user interface. Appendix A contains screenshots of the models created for training with hypothetical floors. Oasys GSA was also used alongside ANSYS as the program is more intuitive and aided learning in general. Modal analyses were carried out as far as possible to get meaningful results. A great deal was learnt by doing these models (for example the limitations of the academic version of ANSYS). It is currently understood that ANSYS is the only software realistically suitable for the analysis. Alongside the main model it is hoped to produce another working model in GSA. It is heavily marketed as successful vibration analysis software and would be interesting to compare some results, without letting this take up too much significant time. Combined with research this would hopefully develop knowledge of the reasons why only some software packages can give meaningful results and serve as a good general knowledge basis for the future.
The final deliverables of CIV 6020 & 6000 are;
A technical conference paper and presentation and,
A dissertation document of maximum 80 pages,
this is to contain chapter headings based around the following framework in table 2;
2. Outline of Project
Background and Literature Review
1.Current Requirements on Design
2. Vibration Serviceability of Floors.
3. FE Modelling of Floors
4. Modelling Human Dynamic Human Loading
5. Acceptability of Vibrations
6. Mitigation of Vibrations
2. Meshing/ Material Properties/ Boundary Conditions/ Geometry
3. Analysis Results
Original Design Problems
Table 2: Provisional Chapter Headings
2. Preliminary Background Review
This background review gives a brief history of the vibration problem in building floors and then focuses on research directed towards leisure centre type floors and applicable loadings. There is a discussion of vibration acceptability in this scenario and current methods of solving possible vibration issues. At the end of this section, a review of current design guides is given, as applicable to anyone in the UK checking a composite-sectioned floor, with humans as the vibration 'receivers'.
2.1 Vibration of floors
Vibration of floors has most likely been an issue since freestanding floors were first constructed centuries ago. Old references to floors are hard to find but the power of vibration problems has been known for a long time. In 1831, soldiers marching across Broughton Suspension Bridge in Manchester made it vibrate to the point of resonance and complete collapse. Some signs are still around that used to instruct soldiers to break step when crossing the bridge (Debney and Willford, 2009). There have been some high profile cases even with modern bridge designs and the subject is definitely still evolving (Pavic et al, 2002).
With regards to engineering research, it was not until relatively recently that attempts have been made to really understand the mechanics of the problem. Previously, typical engineering judgement of the Victorian era was employed, and solutions were found in over-engineering structures so they could withstand vibration. In the 1960's research was started with basic models being developed giving engineers' guides to find fundamental frequencies of floors with hand calculation such as the plate and shell theories presented by the famous S. Timoshenko. This initiated as at that time other advances in structural engineering meant lighter, more efficient structures were being developed which were more susceptible to vibration. For structural floors, it was thought that if the natural frequency of the floor could be kept above the applied excitation frequency there would be no problem.
However this turned out to be unacceptable. In particular in the last twenty years there has been a surge in research in response to new even more slender floors being constructed. Some such research (Pavic, 1998) was against new guidelines produced to cope with a new trend towards greater spanning office floors (i.e. fewer columns) with less damping (i.e. partitions/paper processing/storage systems etc). The guidelines were deemed over-conservative or also were used incorrectly and the furniture inside would wobble slightly. Over conservative guidelines are a problem because, for example, increasing a slab's thickness by 50mm can add tens of thousands of pounds. Of course there is a trade-off with acceptability.
Experimentation has led to an improvement in the general overall theory and understanding of the fundamentals of the problem. Floors can be excited into motion by harmonics (multiples) of the fundamental frequency and modern floors with lower masses are very sensitive to this phenomenon. More detailed analyses are now used were computer models are made and actual response to loading is calculated and compared to acceptance criteria, described in more detail in section 2.2.
As seen with office floors, similar problems started appearing in shopping centres, health clubs and gymnasiums (Mouring, 1996). A person walking can cause vibration which will annoy users and so this is a serviceability issue. However vibration due to more intense 'crowd' actions can cause greater movements and so actually can become a safety issue (Ji, 1994). Correcting problems after construction is very expensive (Allen, 1990). Secondarily as well as changes in building architecture there was an increase in level of activities such as aerobics with newfound audience participation, so coupled with the aforementioned decrease in building floors stiffness's and damping, problems were discovered.
Troubles of this nature were highlighted in a case study of the dynamic behaviour of a gymnasium floor (Rainer and Swallow, 1986). The floor was designed to the then most advanced vibration guide from Canada; CAN3-S16 Appendix G and passed criteria for design (based on walking). Nevertheless problematic vibrations were detected in the study when exercises of other more rhythmic nature were carried out. In response Ji & Ellis were prompted to co-author two papers in 1994, providing the theory and a verification of the response calculations for floors under dance and exercise-type loading still in use today.
Now it is understood that the fundamental mode of vibration cannot be the only check, but, as this project should highlight there are still problems in industry. However it is expected that this is due to incompetence on the design team's part, as there are various guidelines that try to make engineers aware that floors cannot be excited into resonance (i.e. large amplitude vibrations).
In the Eurocodes (compulsory design guides as of April 2010) the only real valuable reference to vibration is in the National Annex to BS EN 1991-1.1 (Euro code 1). It states that floors may be unaffected by resonance phenomena if its vertical fundamental frequency is above 8.4Hz. This number comes from consideration of the first three harmonics of the average walking force ( 3 x 2.8Hz = 8.4Hz). As mentioned earlier harmonics are important as they can excite a floor into vibration. According to the national annex, if the fundamental frequency of a floor is below 8.4 Hz then a dynamic analysis is required.
For further scrutiny a number of secondary guides are available. The Steel Construction Institute's Guide P354 and the American Institute for Steel Construction's Design Guide 11 are two relevant examples. Further detail covered in Section 2.2 & 2.3.
Pavic and Reynolds (2002) produced two large literature reviews which round together much of the separated research from the past into a single document. As they state, structural vibration isn't solved by any one single process, it is a mix and match of different reviews and guides, "there is no whole-package solution".
2.2 Modelling dynamic human loading
The idea of making a mathematical model of human movement is not new and research covers many disciplines. In civil engineering the movements of people and in particular groups of people are of paramount interest as they can greatly amplify deflections on footbridges, stadia, staircases and floors.
Two loading methods need to be looked at, one considering an individual walking or jumping, and one considering a group of people dancing or exercising. The worst case for the group scenario is jumping (Ellis, 1994), as maximum force arises when loading is both synchronised and periodic (Ellis, 2000). ISO 10137 assumes the overall movement of a group of people doesn't change and characterises the load for different situations values at 0.25 persons/m2 for aerobic and gymnasium activities and 2 persons/m2 for social dancing activity. Also it is assumed one person's weight averages at 746N. (76kg x 9.81ms-2 = 746N).
Loading on a floor is modelled by using half sine pulses, representing footfall contact and the point of zero foot/floor contact. For different activities this force will be transferred in different ways, and so each activity is thus described by a different contact ratio Î±c (which can be thought of a force multiplier), making the force more intense for a vigorous activity.
The SCI method considers frequencies in the range of 1.8 - 2.2 Hz but there are other ranges available for example the method developed by Arup, where walking frequencies considered are 1.0 - 2.8 Hz.
The load function is expressed as a Fourier series summing these contact points:
q = wieght of people per unit area (746N per individual)
Î±h = dynamic load factor (DLF - table 3)
h = harmonic number
fp = frequency of the jumping load
Î¦h = phase lag (time difference) for the activity
Current SCI P354 recommends factors originating from Ji and Ellis 1994.
h = 6
Low impact aerobics
High impact aerobics
Table 3: Phase lags and DLFs
For ordinary walking also tested by the VES, the load function is assumed to be perfectly periodic;
Using different DLF's calculated from table 4
Frequency range (Hz)
Phase lag (rad)
1.8 to 2.2
0.436(hfp - 0.95)
3.6 to 4.4
0.006(hfp + 12.3)
5.4 to 6.6
0.007(hfp + 5.2)
7.2 to 8.8
0.007(hfp + 2.0)
Table 4: Walking DLFs
It is interesting to note there is ongoing research work to improve the above traditional model (Racic, 2009.) Instead of using the periodic half sine footfall Racic makes a new model closer to that of the near periodic reality, by the sum of two Gaussian exponentials combined with the equations of circular motion to generate an artificial jumping force. This makes it more realistic than the traditional method, there is improved curve fitting around the measured jumping force signal, by having two peaks instead of one.
The department for structural engineering at Sheffield University is recording a database of various jumping parameters composed by testing different individuals at varying beat frequencies on metronome. The idea behind this is to develop probability-based stochastic models that can be used to account for the variability of different persons jumping within dynamics in civil engineering.
2.3 Analysis and Modelling of Floors
For any floor of irregular shape hand calculations are not suitable and so the only way forward is to use a more advanced method. Some software programs exist that work on geometric and material properties. Numerous editions exist, including those marketed with floor manufacturers such as Westok. They are especially limited and are really only useful as a guide at a particular set load with and usually making assumptions for a regular square shaped floor plan.
Finite Element (FE) modelling uses computer power to consider many points in space and time and develop large mass and stiffness matrices used to solve the standard vibration Eigen problem, making it possible to yield many more results over iterative steps. A continuous structure is broken into a finite number of elements. Modes higher than the fundamental mode are calculated and complex time-varying loading functions and arrangements on different areas of a floor can be used. This method has gained popularity in structural engineering but evolved from mechanical and aeronautical disciplines of engineering (Reynolds, 2000).
There are several FE packages available to the structural design engineer. LUSAS, Nastran, SAP2000, ABAQUS etc. are capable of modal analysis however for vibration analysis the programs lack general popularity so aren't a major choice for this project. As possibility of saving money was mentioned earlier, it should be noted that an FE analysis can easily expend more than ten thousand pounds.
Oasys GSA is marketed as fully comprehensive for vibration problems, with walking, running, dancing load capabilities, also allowing any geometry to be specified. Different eigenvalue analyses can be performed (Modal/Ritz/Modal P-delta etc). However its finite elements capabilities have only been added relatively recently to the code and its use and options is limited.
The most advanced code is ANSYS. Developed in 1970 in the US this program has vast modelling capabilities and is widely used even outside of structural engineering. The use of varying elements allows different characteristics of solid materials to be modelled. Beams are easily created but for a concrete slab there are a number of considerations. For vibration of a plate, a commonly used element tyoe is 'SHELL63' (Pavic et al 2001) which represents an elastic shell with bending and membrane actions within orthotropic 6 DOF's. Pavic said traditionally there are problems in choosing element type but also how to approach the dynamic modulus of elasticity, Ec,dyn. As, in dynamics generated stresses are much lower than that of a static analysis, and this should be carefully approached (Wyatt, 1989). If required an 'extra' stiffness can be smeared over the top of the shell elements. In a dynamic analysis the connections of a model must also be carefully considered, as pinned connections may behave in a fixed manner if frequencies are high enough.
If one floor of a structure is of concern then the whole building is not required in the model (Middleton, 2008). Also in his literature review Middleton explains that the columns above and below the floor are of importance in the model, and the node density has an effect on the frequency range (El-Dardiry, 2006). SCI P354 also gives specific information on modelling floor decks with profiled steel sheeting.
The broad approach will be either to take spectrum analysis of jumping/dancing frequencies and use Fourier components to see whether it interferes with the natural frequency of the floors (or its harmonics) or use modal superposition in which the contribution from each mode is considered and summed to give total load. For floors subject to rhythmic activity should be analysed for all modes up to the cut off frequency of 24Hz plus 2Hz so all modes up to 26 Hz should be considered according to P354. Modal superposition is anticipated for the project.
For the project, key points have been established so far, in line with SCI P354.
internal: model as fully fixed
facade: fixed but have rotational capacity
method 1: above and below to pinned connections (full storey height)
method 2: above and below to fixed point of inflection (usually half storey height)
sef-weight + permanent loads + a proportion of imposed loads as reasonably can expected to be permanent
no. of elements in mesh
sufficient in quantity when quantity can be doubled without significantly affecting results
38gpa for normal weight concrete
22gpa for lightweight concrete
accelerations calculated for locations that are occupied and connected to, but are not, the area of rhythmic activity
Table 5: Proposed model characteristics
A point to note is also that it is not necessary to use the software's in-built geometry editors, as it is possible to input geometry from CAD drawings into GSA and via converter software into ANSYS. However for the project it is planned to use commands entered into an ASCII file loaded into ANSYS to save cumbersome effort and allow continuous modification.
If required and if possible, validating FE models is done with modal testing, which provides 'real world' results of a structure. The test piece (structure) is excited and its vibration mode properties recorded (natural frequencies, mode shapes and damping ratios). This technology requires sensitive measuring techniques as civil engineering structures' vibration response is very small and competes with a large level on environmental noise (Pavic, 1999) and was not available pre mid 1990's as transducers were not sensitive enough.
When experimentally testing floors with modal analysis there are two methods available (Rainer and Swallow, 1986) however only one will be mentioned here as is relevant to the project. Shaker testing involves placing a reciprocating 'shaking' weight, driven by a known signal. Accelerometers record the movement of the floor and, when these two pieces of equipment are methodically placed at different grid locations on a floor it is possible to record mode shapes and phase lag of the response, i.e. the time difference between the forcing action on the floor and its associated response. After recording mode shapes the excitation can be stopped abruptly and the vibration decay of the floor measured for calculation of the damping ratio (Rainer & Swallow, 1986).
2.4 Acceptability of vibrations
The susceptibility of humans to vibration of floors is a relatively new research field, and although this has been intense most of it is directed to vibration effects from a one person scenario. BS6472 and ISO 10137 describe various methods to which dynamic responses can be evaluated for office floors and floors holding sensitive equipment however there is no generally accepted criterion for structures holding a crowd (SCI P354, 2007). The trouble is the difficulty is quantifying vibrations as perceptible or disturbing for humans varies greatly from person to person. The AISC recommend for a floor subject to rhythmic loading 4 to 7% gravity is a peak acceleration limit (AISC Design Guide 11, 1997), this is equivalent to a response factor of 120 (SCI P354, 2009). However these guides emphasise the lack of certainty on these values. Another method of quantifying tolerance is the vibration dose value (VDV) as described in BS6472 and SCI P354. Mathematically the technique is to take the root-mean-quad of the time integrated acceleration signal (describing as much of the vibration as possible) after it has been weighted against a standard graphical curve. This process can describe vibrations in terms of acceleration, exposure time and frequency of exposure. The guides describe values applicable to people in grandstands that would be disturbing, unacceptable or likely to cause panic. It is assumed people in these structures are in a similar position to those in gyms/dancehalls where jumping as a crowd is the worst case scenario.
In gyms and dance floors people are generally moving and will have a higher level of vibration tolerance. However areas of concern are likely to be those where occupants may not be taking part in the dynamic loading, i.e. there is the possibility with buildings designs that another area of the floor may become excited. For example a function room for weddings could have people sitting and dancing within the same place. This scenario would require careful checking. All guidance must be reviewed all careful choices made on how to judge the vibration limits.
2.5 Mitigation of vibrations
Structures should be designed to accommodate for dynamic loads induced within their lifetime but of course, mistakenly, this does not always happen. The need for retro-fitting of structures with a method of damping vibrations has had some high profile cases. It is costly for the clients and sometimes quite difficult to achieve. There are various methods applicable in different situations, but in general modal mass (i.e. contribution to vibration) must be decreased or damping ratio increased.
ISO 10137 recommends:
a) shift resonance frequency of the transmitting structure or the receiver (change mass or stiffness);
b) add damping to reduce response at resonances (frictional or hydraulic dampers, tuned mass dampers);
c) control the dynamic properties of the source (shift rotational frequency of operation, balance rotating machinery, control traffic speed or route of vehicles, improve surface or subsurface properties of roadway or railway, etc.);
d) isolate the vibration source, such as introduction of barriers to vibration transmission (trenches or piles for ground-transmitted vibrations, proper techniques and air blast control for blasting in the ground);
e) isolate structure by active or passive vibration isolation (servo-controlled mass or reactions, seismic or pneumatic isolation of instruments, spring isolation of building components or of the entire structure).
Care needs to be exercised so that the changes made do not unintentionally deteriorate the vibration levels due to amplifications of hitherto dormant modes of vibration. In general, it is preferable to apply vibration mitigation in the design stages, rather than as a remedial measure after the building is completed and occupied.
One of the simplest but often most impractical methods of adding stiffness to a lively structure is by adding columns supports or bracing to it. This cannot always be accommodated for, especially after construction, without damaging the purpose of the structure. There it necessary to consider some other options but limit these to vibration serviceability problems due to human loading and not seismic loading from earthquakes.
Tuned mass dampers (TMD) function by absorbing the energy of movements through viscosity or stiffness of some fluid or material. In this way damping is increased. An advanced technology in this field is Magneto-Rheological TMDs. They work by adding micro magnetic particles to the oil in a TMD and then placing this fluid under a magnetic field, which, when required can align the particles in the most compressive direction and produce more stiffness. These elements are designed or 'tuned' to work with a structure's natural vibration frequencies.
4. Preliminary References
The references listed below are those used in section 2 and also those planned to be referenced in the literature review of the project.
Papers and Guidelines
Allen, D. E. "Building vibrations from human activities." Concrete International 12.6 (1990):66-73.
Bachmann, H. "Case studies of structures with man-induced vibrations." Journal of structural engineering 118.3 (1992):631-647 New York.
Da Silva, J. G. S. "Vibration analysis of orthotropic composite floors for human rhythmic activities." Journal of the Brazilian society of mechanical sciences and engineering 30.1 (2008):56-65.
Da Silva, J. G. S. "An evaluation of the dynamical performance of composite slabs." Computers & structures 81.18-19 (2003):1905-1913.
Da Silva, J. G. S. "Vibration analysis of orthotropic composite floors for human rhythmic activities." Journal of the Brazilian society of mechanical sciences and engineering 30.1 (2008):56-65.
Debney, P. "Footfall vibration and finite element analysis." Sound and Vibration 43.11 (2009):11-14.
El-Dardiry, E. "Modelling of the dynamic behaviour of profiled composite floors." Engineering structures 28.4 (2006):567-579.
Ellis, B. R. "Floor vibration induced by dance-type loads. Verification." The Structural engineer 72.3 (1994):45-50.
Hanagan, L. M. "Dynamic amplitude prediction for ballroom floors." Engineering journal 39.3 (2002):148-153.
Ji, T. "Floor vibration. Floor vibration induced by dance-type loads. Theory." The Structural engineer 72.3 (1994):37-44.
Littler, J. D. "Frequencies of synchronised human loading from jumping and stamping." The structural engineer vol 81. iss22 (2003):27-36.
Middleton, C. J. "Response of high frequency floors: A literature review." Engineering structures 32.2 (2010):337-352.
Mouring, Sarah E. "Guidelines to Minimize Floor Vibrations from Building Occupants." Journal of structural engineering 120.2 (1994):507-526.
Pavic, A. "Vibration Serviceability of Suspended Cast In-situ Concrete Floors", PhD Thesis, Department of Civil and Structural Engineering, University of Sheffield, UK (1998).
Pavic, A. "Methodology for modal testing of the Millennium Bridge, London." Proceedings of the Institution of Civil Engineers. Structures and Buildings 152.2 (2002):111-121.
Pavic, A. "Dynamic modelling of post-tensioned concrete floors using finite element analysis." Finite elements in analysis and design 37.4 (2001):305-323.
Pavic, A. "Vibration serviceability of long-span concrete building floors. Part 1: Review of background information." The Shock and vibration digest 34.3 (2002):191-211.
Pavic, A. "Vibration serviceability of long-span concrete building floors. Part 2: Review of mathematical modelling approaches." The Shock and vibration digest 34.4 (2002):279-297.
Racic, V. "Mathematical model to generate asymmetric pulses due to human jumping." Journal of engineering mechanics 135.10 (2009):1206-1211.
Racic, V. "Reproduction and application of human bouncing and jumping forces from visual marker data." Journal of Sound and Vibration 329.16 (2010):3397-3416.
Racic, V. "Experimental identification and analytical modelling of human walking forces: Literature review." Journal of Sound and Vibration 326.1-2 (2009):1-49.
Rainer, J. H. "Dynamic behaviour of a gymnasium floor." Canadian Journal of Civil Engineering 13.3 (1986):270-277.
Reynolds. P. "The effects of raised access flooring on the vibrational performance of long-span concrete floor". Thesis (Ph.D.), University of Sheffield, Department of Civil and Structural Engineering, 2000
Reynolds, P. "Modal testing, FE analysis and FE model correlation of a 600 tonne post-tensioned concrete floor." Proceedings of the 23rd International Conference on Noise and Vibration Engineering, ISMA. (1998). :419-426.
Samarajiva, P. "Vibration study of a proposed fitness center adjacent to a server room." Forensic Engineering, Proceedings of the Congress. (2010). :318-324.
Smith A.L., Hicks S.J., and Devine P.J., Design of Floors for Vibration: A New Approach, Steel Construction Institute, 2007
Thornton, Charles H. "Taming structural vibrations." Civil engineering 60.11 (1990):57-59.
Yao, S. "Experimental study of human-induced dynamic forces due to jumping on a perceptibly moving structure." Journal of Sound and Vibration 296.1-2 (2006):150-165.
Dynamics of structures, by Ray Clough W and Penzien J, 2nd edition, McGraw-Hill, New York, 1993. ISBN 0-07-011394-7
Dynamics of Structures, by Chopra A K. Prentice Hall, 3 edition , September 1, 2006, ISBN: 0-1-31-561-74X
Theory of plates and shells, by Timoshenko S P and Woinowsky-Krieger S., Second Edition, McGraw-Hill, 1959. ISBN 0-070-647798
Elements of Vibration Analysis, by Leonard Meirovitch, McGraw-Hill, New York, 1975 ISBN 0-07-041340-1