Natural frequency of Cobiax flat slabs. Lukas Wolski

Transcription

1 Natural requency o Cobiax lat slabs Lukas Wolski Thesis subitted in partial ulilent o the requireents or MSc in Civil and Structural Engineering Coventry University Septeber,

2 Abstract The proble o huan discoort due to low-level vibrations o concrete slabs is an iportant actor o consideration during any design process. The continuing trend towards large open loors, ree o partitions, and increased slenderness in design aesthetics increases the likelihood o annoying loor vibration induced by sall ipacts such as huan ootalls. The present research covers several areas concerned with addressing this proble. A basic literature review o previous work in the ield o loor vibrations is presented and provides an introduction into this general topic. At irst soe recoendations o acceptance liits by national standards as well as by independent authors are presented. The probles hindering the proper evaluation o loor vibration are also shown. The next area o study involves sipliied hand calculation ethods or the approxiate estiation o concrete slabs undaental requencies. Dierent approaches are presented and aterwards copared and evaluated against several exaple solutions ro accurate inite eleent sotware. The third chapter ocuses on the undaental requency o a speciic biaxial hollow concrete slab: the Cobiax lat slab. An investigation o its vibration behaviour under dierent paraeters was carried out using inite eleent sotware. The data o seven dierent loor designs was obtained and copared to conventional solid slabs, leading to a inal evaluation o the vibration perorance o Cobiax lat slabs.

3 Acknowledgents First o all I would thank y supervisor Dr. Messaoud Saidani or his support, patience and guidance throughout the developent o this thesis. Thanks are also due to Dr John Davies and John Karadelis or their advice and their help to realise this research. I would like to express sincere gratitude to Pro. Dr.-Ing. Andrej Albert or his generous support and guidance throughout this dissertation. His suggestions and encourageent has been greatly appreciated. Many thanks are due to Cobiax Technologies GbH in Darstadt, Gerany and Cobiax Technologies Ltd in London, United Kingdo. Special thanks go to Dr. Karsten Peer and Daniel Ptacek who helped to establish this project and supported it during its copletion. I ust also thank Christian Roggenbuck, or his constant support and helpul advice not only throughout this research but or the entire last year. Finally, I need to thank y aily and especially y girlriend or their never ending belie in e and y ability to accoplish what I set out to do. Their support and love have allowed e to achieve this goal.

4 Notation a b C D E F g G h I I x I y k S α β γ ν ω Length o plate Width o plate Claped edge Cobiax lat slab Plate rigidity Young s odulus Natural requency Fundaental requency Free edge Gravity Modulus o rigidity Plate thickness Moent o inertia Moent o inertia in x-direction Moent o inertia in y-direction Stiness Mass (per unit area) Siply supported Solid slab Length-width ratio Daping ratio Density o plate aterial Poisson s ratio Circular requency ( π x requency)

5 Contents 1. Introduction and Background Knowledge Introduction Natural Frequency Cobiax lat slab Ais & Objectives Need o research Literature review Huan Response to Floor Vibration Case Studies Consideration o Vibration in Design Two Way Hollow Decks Coparison o FEM and Field Tests Deterination o Frequency...1. Huan Response and Acceptance Criteria Recoendations in Codes British Standard Geran Standard.... Recoendations in Literature....3 Suary Sipliied Hand-Calculation Methods Coon Matheatic Techniques Forulas and Tables or the Calculation o Fundaental Frequency Equivalent Bea Method Equivalent Plate Approach Concrete Society Method Static Delection Method Approxiation Presented by Hearon Approxiation Presented by Jänich Estiation or Pin Supported Plates Copilation o Forulas by Bachann...

6 3..9 Copilation o Forulas by Blevins Analysis o Results Conclusion...7. Nuerical Analysis Sotware...9. Veriication o Sotware Accuracy...5. General Settings Analysis o Results...5. Conclusion Conclusions and Recoendations General Conclusions Areas o Future Research...1 Reerences... APPENDIX A (Sipliied Hand Calculation)...7 APPENDIX B (Calculated Values by FEM)...93 APPENDIX C (Cobiax Inoration)...11

7 Figures Figure 1.1 Cobiax cage odules...3 Figure 1. Cobiax sei-precast slabs... Figure.1 BS 7 (199): Coordinate systes or vibration inluencing huans...1 Figure. BS 7 (199): Building vibration z-axis curves or acceleration (r..s.)...3 Figure.3 DIN 15- (1999) progression o assessent procedure...5 Figure. Reiher-Meister scale... Figure.5 Graph o reduced huan response...7 Figure. Annoyance criteria by Allen and Rainer... 8 Figure 3.1 Frequency paraeter or continuous slabs...3 Figure 3. Coparison o hand-calculated and coputed requencies...5 Figure 3.3 Individual accuracy o approxiations... Figure 3. Average accuracy o approxiations...7 Figure.1 Cobiax odule...9 Figure. 1.Mode shape obtained by Tornow-Sotware...51 Figure.3 1.Mode shape obtained by RFEM...51 Figure. Fundaental requencies or a single span slab...5 Figure.5 3-D view o requency dependency...55 Figure. Cobiax advantages against loading...5 Figure.7 Accuracy o critical load or continuous slab...58 Figure C.1 Coparison o spans and concrete quantity...1 Figure C. Coparison o spans and loads...1

8 Tables Table.1 BS 7 (199): Multiplying actors... Table. Extract o DIN 15- (1999): Reerence values A or residential and siilarly used buildings... Table.3 Values o K and β...9 Table. Huan perception criteria by Bolton...3 Table.5 Overall acceptance levels or various types o environent...3 Table 3.1 Frequency paraenter provided by Hearon...38 Table 3. K and N paraeters... Table 3.3 Frequency paraeters or pin supports...1 Table 3. Frequency paraenter provided by Bachann... Table 3.5 Frequency paraenter provided by Blenvis... Table.1 Cobiax advantage related to loads and thickness...5 Table. Accuracy o critical load or one span slab...58 Table.3 Extract o iposed loadings in BS (199) and DIN ()...59 Table A.1 Suary o approxiate hand calculation...9 Table B.1 Siply supported slab, one-way spanning...9 Table B. Siply supported slab, two-way spanning...95 Table B.3 Two span slab, two-way spanning...9 Table B. Three span slab, two-way spanning...97 Table B.5 1x1 bay slab, supported by coluns...98 Table B. x1 bay slab, supported by coluns...99 Table B.7 3x3 bay slab, supported by coluns...1 Table C.1 Existing Cobiax projects...1 Table C. Factors considering reduced stiness...13 Table C.3 Cobiax paraeters...1

9 1. Introduction and Background Knowledge Lukas Wolski 1. Introduction and Background Knowledge 1.1 Introduction In the last years, the nuber o loor vibration coplaints in residential buildings and oices increased signiicantly (Hanagan 5, Willias and Waldron 199, Naei F. 1991). The two usual causes or this annoying proble are huan activities such as walking, running, juping or dancing and echanical oveent ro, or exaple, air-conditioning systes, heating, and washing and drying achines. In rarer cases, indirect excitations ro autoobiles on parking levels below a loor, or transitted vibration through building coluns ro other loors or the ground are to blae. The psychological eect o the up-and-down otion caused by loor vibration can be iense. Generally, it gives people an unpleasant eeling and propts ear o structural collapse. This eeling increases even ore i a person is not actively involved in inducing the acting load. People s quality o lie and working conditions, then, are negatively aected by perceptible vibration and so it is usually considered undesirable. However, loor vibration does not only aect the inhabitants o a building; in extree cases it can also lead to atigue ailures or daage structural eleents which results in costly reodelling. Additionally, buildings housing sensitive equipent such as hospitals, laboratories and anuacturing plants that use odern icro- and nanotechnologies are in especial need o protection. The proble o vexatious loor vibration is not new. Civil engineer Thoas Tredgold (188) wrote: "girders should always be ade as deep as they can to avoid the inconvenience o not being able to ove on the loor without shaking everything in the roo." In the past, a siple delection criterion (delection o less than span/x under distributed live load) usually ensured structures against heavy vibration, but because o the current trend towards longer spans and lighter loor systes (the result o ore aesthetical and eicient constructions), this approach no longer works and the need to reconsider loor vibration has increased. Slender structural ors and decreased loor ass reduce natural requency as well as structural daping and so loor vibration has becoe an area o concern. 1

10 1. Introduction and Background Knowledge Lukas Wolski There exist several ways to prevent or at least reduce this proble. The siplest and ost eective ethod or achinery-induced loor vibration is to isolate the source ro the ground. This could be by eans o springs, insulating plates or other elastic bodies. However, or huan-induced vibration it is ipossible to isolate the source ro the loor syste. In this case huans are both the source and receiver o vibrations which akes the situation very diicult. Thus, the structure itsel ust be considered and odiied to prevent annoying loor vibration. One way o addressing this proble is to increase natural requency to a level which can hardly be perceived by a building s occupants. 1. Natural Frequency Natural requency is one o the undaental paraeters used in the deterination o a structure s response to dynaic loads. It is the requency at which an elastic object naturally vibrates when hit, struck, or otherwise disturbed. Every syste able to oscillate has its own natural requencies. A pendulu, or exaple, always oscillates at the sae requency when set in otion. Its requency depends only on physical properties such as the ass, length or stiness o the spring. Furtherore, the aount o natural requencies or a syste depends on its degree o reedo and thus on its coplexity. The lowest natural requency o a syste is called its undaental requency. I a orced vibration is applied to a syste, at its natural requency only a iniu o energy is required to keep it in vibration. It is iportant to know the natural requency o an object to predict its behaviour in relation to vibration. The ost iportant reason or this is resonance. I a varying orce with a requency equal to the natural requency is applied to a syste, the oscillation will becoe violent. Its aplitude will increase highly and daages ay occur. Although rare, total collapse is possible due to overloading or ailure in atigue (this scenario predoinantly aects bridges). The undaental requency or the siplest odel o a dynaic syste which has only one degree o reedo and no daping is given by: 1 π k

11 1. Introduction and Background Knowledge Lukas Wolski In this case the natural requency siply depends on the stiness k and the total ass o the syste. This equation indicates the iportance o these two qualities or a dynaic syste. For concrete loors, the stiness is coposed o urther three actors: it depends on Young's odulus, Poisson s ratio and the oent o inertia o the considered structure. To iniise the perception o loor vibration it is iportant to achieve as high as possible values or the syste s natural requencies. This will occur i the stiness is very high and the ass is by contrast very low. This is the ideal case which will result in very high requencies. The opposite eect happens i a high ass in cobination with a low stiness starts to vibrate and thereore this situation should be avoided. 1.3 Cobiax lat slab Cobiax is an international operating copany which has developed a special solution in the lightweight lat slab syste sector. Their product, the Cobiax lat slab, consists o hollow plastic spheres which are placed between the upper and lower static reinorceent o the slab (Figure 1.1, published by Cobiax Technologies AG). Each o these sphere are located in odules which consists o a steel cage including several balls. The cage avoids a contact between ball and static reinorceent which leads to an ipairent o its bond. Additionally, a buoying upwards during concreting is avoided. The balls replace the concrete on its area with the lowest beneit. The ain idea o this syste is to reove the useless Figure 1.1 Cobiax cage odules concrete which just produces dead load without iproving the static qualities o the slab. The concrete 3

12 1. Introduction and Background Knowledge Lukas Wolski ors a hard shell with struts by using appropriately located cavities ored by hollow spheres. Nevertheless the slab has the sae load bearing behaviour as traditional solid slabs and brings along soe iproveents to the. First o all Cobiax slabs weighs up to 35% less than solid slabs o equivalent diensions. This has a positive eect on the nuber o necessary vertical bearing eleents (up to % less colun usage). It is also easible to create large spans, up to ore than, without using beas. These two actors increase the possibilities o open areas in buildings, aking ore alterations possible. Furtherore, the ass reduction is noticeable in designing oundations, leading to savings in the aount o aterial used. Other advantages include reductions in CO eissions, savings in the aount o reinorceent needed, the application o all coon standard designs and the sooth botto view. This, the coon orwork and the biaxial load bearing ade possible by the hollow section s spherical shape are also advantages over coon hollow concrete slabs such as wale decks. During the design process a ew changes concerning the slab s speciic qualities should be considered, including the decrease in stiness or Cobiax slabs caused by the reduced oent o inertia copared to a solid slab. For this purpose nueric actors have already been deterined and can be easily used or conversion (see Appendix C). Besides sall odiications the whole design requires no other variation. At the building site the Cobiax syste arrives in the or o cage odules or on-site use or as seiprecast slabs (Figure 1., published by Cobiax Technologies AG). Alternatively it can be used in cobination with precast or coposite slabs. The available ball diaeters o the spheres range between 18 and 5 which Figure 1. Cobiax sei-precast slabs allows or the production o slabs ro c to upwards o c.

13 1. Introduction and Background Knowledge Lukas Wolski 1. Ais and Objectives The general ai o this dissertation is to investigate the ipact o Cobiax lat slabs speciic qualities on their undaental requencies in coparison to solid concrete loors. As in both ain nuerical actors or the undaental requency the stiness as well as the ass o the subject is decreased, the consequence or the slabs behaviour in case o natural requency should be researched. The result should clariy i and how these dierent qualities aect the slabs perorance. Furtherore it should deliver an overview o the treatent o concrete loors natural requencies in structural engineering. This study investigates three objectives. Firstly, due to varying estiations concerning the liits o structures natural requencies, dierent evaluations ollowing national codes and independent recoendations will be presented. Aterwards a copilation o sipliied ethods or hand calculations will be provided. They will be tested in soe exaple calculations and their accuracy when copared with inite eleent values will be investigated. The third objective is the ain area o this research. A precise coparison o Cobiax lat slabs and traditional solid slabs will be undertaken using inite eleent sotware. The data yielded will then be thoroughly analysed. This will result in a detailed evaluation o the characteristic qualities o a Cobiax lat slab. 5

14 1. Introduction and Background Knowledge Lukas Wolski 1.5 Purpose o Research This research is developed in collaboration with Cobiax Technologies GbH, Darstadt, Gerany and Cobiax Technologies Ltd, London. To arket their product eectively Cobiax ust possess all necessary inoration concerning its structural behaviour. Soe areas o perorance are yet to be investigated, including the Cobiax slabs behaviour in relation to natural requency; a detailed coparison with a conventional concrete solid slab is also required. As already explained, natural requency in this context depends on two ajor actors. The irst is the stiness o a slab which depends on its basic aterial as well as geoetry. Copared to a solid slab the aterial property is the sae but due to its hollow inside section, the Cobiax lat slab has a dierent geoetry. This results in a lower stiness, which indicates a decrease in its natural requency. The second key coponent o the equation or natural requency is the ass. Contrary to the stiness, where the hollow sections o the spheres have a negative inluence, in the case o ass they are an advantage. The one-third reduction in concrete becoes perceivable and increases the value o natural requency. The decrease o both values has an opposite eect; one will iprove the solution and the other will ipair it. The question is how uch these characteristics ipact on the product s perorance and so which is to be the decisive actor in the inal evaluation and recoendations. The research will carry out a clear investigation o this proble and deliver an unabiguous judgeent. It is iportant to explore all possible advantages o the Cobiax lat slab as, in the uture, this research ay help to convince clients worried about the eects o the slabs lower stiness on its dynaic qualities.

15 1. Introduction and Background Knowledge Lukas Wolski 1. Literature review 1..1 Huan Response to Floor Vibration Arthur Bolton (199) explained the iportance o dynaics in structural engineering with the ollowing sentence: Crowds go to airgrounds to be subjected to quite large accelerations and enjoy the sensation, but i subjected to a tiny raction o that excitation in a building they ight becoe sick or anxious. The psychological eect o vibrating structures, then, can be proound. It can cause people discoort, nausea or anxiety. O course these are all extree reactions, but ilder eects present a diicult proble because o wide variations in huan sensitivity. There is also dependence on whether subjects are alone or in a group; one particularly sensitive person aongst other people can soeties trigger o a collective belie that a barely perceptible vibration is dangerous or uncoortable. Another criterion o percipience vibration is the activity which is being pursued at the tie. Usually people are ore sensitive to vibration when they are in a quiet and untroubled area rather than, or exaple, a busy region. Furtherore the direction o vibrations aects the percipience. Investigations show that a translation in a horizontal direction has ore eect than one in a vertical direction. As a result o all these actors it is diicult to set an accurate borderline between acceptable and unacceptable levels o vibration. It is only possible to provide data with ranges which are deinitely unacceptable and thus should be prevented. Reiher and Meister (1931) perored investigations to obtain such ranges. People o dierent ages, proessions and provenances had to stand and lie on a vibrated plator. The tests covered sinusoidal vertical as well as sinusoidal horizontal vibration. The requencies used started ro 3 Hertz up to 7 Hertz and aplitudes ro.1 to 1. c, igures which approach realistic values. During the tests, noise was an iportant actor. It was necessary to iniize noise as uch as possible so as not to disturb hearing and thus the results. Ater a vibration ipact o 5 inutes every subject had to evaluate their sentiences. They had to organise their sentiences into six groups which ranged ro not perceptible to very disturbing. Ater inishing all tests, the 7

16 1. Introduction and Background Knowledge Lukas Wolski results were plotted in charts showing requency against aplitude. All results were included in these charts and aterwards it was possible to draw borderlines or each category. These charts are one possible eans o evaluating the consequences o vibration or the huan body. In general Reiher and Meister recoend avoiding the last two categories. Residential areas should also avoid vibration ro category which translates as keenly noticeable vibration. A siilar investigation was carried out by Wiss and Parelee (197). They extended the aount o subjected persons ro 1 to and conined the scope o their tests to a standing position. All persons had to assess their perception using ive classiications. For a steady-state condition ( daping) this investigation showed a lower perceptible or a particular requency and displaceent copared o those perored by Reiher and Meister. However, the perorance and analysis o the studies were not exactly the sae and so could explain these dierences. Further results concerned the eect o changing the daping or the perception o vibration. It was assued that i the daping was increased ro. to. o critical, the product o requency and displaceent would approxiately double. In general Bachann (1987) recoends avoiding requencies below 7.5Hz or oice buildings ade o reinorced concrete. This value ensures that even the third haronic is taken into consideration, which eans, that besides to the undaental requency its integer ultiples are regarded. For exaple, i the requency is, the haronics have requency, 3,, etc. Copared to pedestrian structures such as gynasia or sport halls where it is suicient to regard the second haronic, or oice buildings it is necessary to consider the third haronic as the occupants are ore sensitive. Not only the occupants, however, should be protected against vibration. Huan otions such as walking, running, dancing and skipping are suicient to cause overstressing o the structure, and in extree cases the loss o structural integrity, daage to nonstructural eleents (e.g. claddings), and developent o cracking or excessive noise (e.g. due to reverberating equipent). 8

17 1. Introduction and Background Knowledge Lukas Wolski Brownjohn (1) investigated the energy dissipation ro vibrating slabs due to huan-structure interaction. It was clariied how the presence o people located on a vibrating structure aects its dynaic behaviour. For this purpose a siply supported 7 x 1 x.75 prestressed concrete plank was orced to vibrate while a subject was standing on it. Five dierent sets o test were perored, including the subject sitting on a plastic chair, standing erect, with knees slightly bent, with knees very bent and inally with a solid ass equivalent to the subject. The results conired that the huan body acts dynaically with the structure by decreasing its natural requency. This was explained by the act that the eective ass is increased as well, but it was also identiied that the huan body has a beneicial eect concerning daping ratio because, depending on posture, daping can increase signiicantly. 1.. Case Studies The iportance o both vibration probles and proper undaental requencies is shown by Hanagan (5), who has published a paper containing several case studies on walking-induced loor vibration in existing buildings. This type o vibration is shown to aect dierent types o buildings including oices, a classroo and a clothing store. In each o these cases, the cause o vibration was people walking around the space. In one case, occupants o an oice building started to report annoying loor vibration in. Interestingly, this building was constructed in 197 and there had been no previous coplaints concerning loor vibration. Ater an investigation easuring the acceleration o the aected slabs, it was detected that the undaental requency o this loor syste was about.7 Hz. This value is not acceptable nowadays but it shows that even low requencies ight work well in special circustances. Ater alost 3 years vibration probles occurred due to the reoving o partitions which resulted in reduced daping and a higher vibration. Another case study in this paper shows the iportance o coplying with current recoendations or natural requencies. In this study structural engineers suggested a ore substantial loor syste, including a thicker slab, to eet 9

18 1. Introduction and Background Knowledge Lukas Wolski recoendations designed to avoid vibration probles. However, as a result o previous experience with this building type, the higher costs involved and the absence o vibration probles in the past, the developer elected or a thinner solution. Unortunately he was wrong and the occupants perceived otions close to walk paths. The coplaints stopped iediately ater additional support and daping was created through the use o ull-height partitions. Further case studies were presented by Bachann (199). He published ten cases o vibration probles produced by huan activity. One exaple speciied serviceability probles in a two-story gynasiu. Every tie the upper hall was used by itness classes, loor vibration was noticed in the hall below and glazed exterior walls started to vibrate horizontally. Additional eects included rattling o doors and shutters and clattering o equipent. An investigation was carried out to deterine the dynaical qualities o the loor. It was established that the undaental requency was about.9 Hz. When people juped with a requency o approxiately.8 Hz, resonance was excited by the second haronic. In order to avoid annoying eects and possible atigue daage the undaental requency was iproved to 7.3 Hz by increasing the loor s stiness. Other cases discussed in this paper showed siilar probles caused by low undaental requency and excitation by huans which resulted in signiicant odiications being ade Consideration o Vibration in Design Fisher and West (1) divided the consideration o huan response to loor vibration into our ajor steps. First o all the natural requency should be calculated which is aected by acceleration due to gravity, Young s odulus, oent o inertia, supported weight and the span o the structure. Aterwards the initial aplitude should be calculated. Daping o a loor is another essential actor as it aects the duration and nature o the vibration: physical tests showed daping percentages ranging ro 3% or bare loors to % or inished loors and up to 13% or inished loors with partitions. The ourth actor is a standard o easure involving the previous three igures. For this purpose graphs are used, 1

19 1. Introduction and Background Knowledge Lukas Wolski with axes showing the requency and displaceent aplitude. The huan response to these actors is then plotted. The adverse eects and thereore the necessary avoidance o resonance were illustrated in a report by Cooney and King (1988). It was claied that, due to resonance, the otion o a loor ay be agniied by up to ties its static load condition. A signiicant increase in acceleration, velocity and displaceent occurs; an eect which should be avoided by all eans. For this purpose the authors provided a design ethod to identiy a possible risk o resonance or loors: speciically, vibration induced by huan activities. First the expected load o the area including all participants and their activities had to be assessed. Occupants activities will lead to an appropriate orcing requency and their total load in cobination with a particular actor will give the dynaic load. Special literature, or exaple the BS 7 (199), will provide values or the acceptable liiting o acceleration. The inal steps are the deterination o the total loor load including the dynaical load coponent and urther the calculation o the undaental requency or the structure. With the help o these data and a special equation presented by the authors an initial check o potential resonance ay be ade. Where the acceptable level o acceleration was exceeded, increasing the stiness was suggested along with relocating or controlling the activity or just accepting the discoort. The Canadian Institute or Scientiic and Technical Inoration (198) published a paper to provide a better understanding o vibration due to dynaical loads. The paper was speciied to huan induced vibration and gave a general overview o this topic. The axiu walking requency or a person was given as 3 Hz; the approxiate requency or juping was 5 Hz, but as also told, these values were unlikely to be reached. Also included were approxiated equations or the undaental requency o siply supported, claped and cantilever beas and uniaxial plates. Furtherore the relationships between static and dynaic deoration and vibration behaviour under periodic and single loads were shown by equations and exaples. The exaple given o a group juping in a gynasiu clariied that static delection reains unchanged or dierent 11

20 1. Introduction and Background Knowledge Lukas Wolski undaental requencies. However, the dynaical delection increased or a reduced requency and reached extree values in the case o resonance. Crist and Shaver (197) coplained about insuicient investigation o loor vibration in national codes. The accuracy o an evaluation or loor vibration can be coplicated by such actors as a lack o necessary coponents. In addition the structure location, the type o structure, the type o occupancy and daping should be considered in literature. An additional cause or concern was the insuicient provision o data or evaluation which ust then be qualiied through urther research. Furtherore, this publication explained the coplexity surrounding the deterination o huan activity and occupant response. Both are rando variables. The dynaic load caused by the orer depends on varying characteristic actors including walking gait, variation in weight, heel-to-ball o oot contact and ootwear. Inluences on the perception go beyond the technical values o requency, direction and duration to encopass psychological actors in or o ental state, otivation and experience and the physical actors o sound and sight. 1.. Two Way Hollow Decks An overview o the general structural perorance o biaxial hollow section slabs o the type Cobiax produces is presented by Peer (), who investigated the slabs bond between reinorceent and concrete, lexure load-bearing capacity, delection and punching behaviour. Initial tests showed that due to the contact o spheres with reinorceent the bond between reinorceent and concrete decreased in these areas. This led to a developent o reduction actors and urtherore to a suggestion or iproving slab design by relocating the spheres ro the reinorceent. However, the lexure load-bearing capacity o a two-axis hollow slab is coparable to a solid slab. I the concrete copression zone is above the sphere, it can be diensioned as a rectangular cross-section by eans o the usual ethods. As a result o the decreased sel-weight the bending perorance o a hollow section slab is better than a solid slab. This occurs up to 1

21 1. Introduction and Background Knowledge Lukas Wolski an external load-to-sel weight ratio o 1:5. In case o punching it was discovered that load capacity is approxiately 5% lower i spheres are included. To avoid this disadvantage, it is recoended that spheres are reoved ro inside the punching area. This results in a siilar punching capacity to solid slabs and allows or a coon punching design. Another structural behaviour, the transverse orce capacity, was investigated by Schnellenbach-Held (3). A coparison between biaxial hollow section slabs and conventional concrete solid slabs showed the dierences in their shear orce perorance. Though the load-bearing capacity beore and ater shear crack oration was siilar, the ailure load o the lightweight slabs was about 5% lower than the breaking load achieved with solid slabs. This can be explained by the reduced concrete area which decreases the transission o tensile stresses. For slabs without shear reinorceent, this is the ain ipact on transverse orce capacity Coparison o FEM and Field Tests The results discussed in this research can be copared to the real behaviour o Cobiax slabs. Ead El-Dardiry et al. () ran an investigation which yielded good results. He and his colleagues copared easured natural requencies o an existing building with values calculated by dierent inite eleent odels. For this purpose and as a part o the European Concrete Building Project (ECBP) a realistic oice building was constructed inside the BRE Cardington Laboratory. It was a seven-storey in-situ concrete building consisting o long-span lat slabs supported by coluns designed to Eurocode. Each loor was 3.75 high, giving a total height o.5. The building had three bays o 7.5 constituting a width o.5 and our bays o 7.5 aking a length o 3.. All slabs were designed as reinorced concrete lat slabs with.5 thickness. The intended iposed load was.5 kn/². Ater inishing the construction, Building Research Establishent Ltd conducted dynaic tests on the loors. The tests involved onitoring the acceleration o the centre o each loor area in response to a heel-drop. The response was then 13

22 1. Introduction and Background Knowledge Lukas Wolski converted to an autospectru using a Fast Fourier Transor procedure and the doinant natural requency was identiied. All seven loors were covered by easureents ro 11 dierent locations on each loor. The easureents provided a basis or evaluating the quality o dierent FE odels. Consequently, FE analysis o several coonly used odels was conducted, and the nuerical and experiental results copared. The engineers used the FE sotware LUSAS and odelled dierent approaches to loor-colun connection. One result o this investigation was that, while the dierent odels used in this study give dierent requencies, the ode shapes are siilar in a global sense. All approaches had a variance o between % and 17% ro the easured values. The average dierence was 1%. Another conclusion o a prior investigation was the negligible eect o esh size on dynaic behaviour. In case o natural requency all three eshes considered had no signiicant ipact. However, they aected the appearance o the ode shapes and so a ine esh was used. A siilar coparison was perored by Willias et al. (1993). Tests were carried out on reinorced and prestressed concrete loors o various conigurations, covering the ull range o spans and thicknesses encountered in typical structures. Newly cast, bare loors as well as already inished loors including alse loors and services were tested. The building types tested included oices and car parks. These types are structurally quite siilar with the exception o the lack o any inishes on the loor o the car park, which results in lower daping values. The experiental set-up used a haer test, in which a sot-tipped haer generates the input excitation through a striking otion. By using other experiental equipent general vibration qualities such as natural requencies, ode shapes or daping ration were deterined. A single bay within the test loor was chosen as the test panel and divided into a 5 x 5 grid o equally spaced points. Aterwards every point was investigated ive ties to obtain an averaged response or each speciic point. Later a inite eleent odel was created using I- DEAS inite eleent sotware to copare the gained values. A detailed coparison was given here or the speciic exaple o a car park in Wycobe. The car park consisted o a.1 thick slab, supported by posttensioned beas along colun lines. 1

23 1. Introduction and Background Knowledge Lukas Wolski The results o this coparison show that the coputer odel gives very good estiates or the irst three requencies. All three requencies are quite siilar to those investigated. The averaged dierence between both results is about %. It was supposed that, due to the increasing iportance o accurately representing the boundary condition, natural requencies o higher odes would exhibit less siilarity. However, as when assessing potential huan discoort due to vibration only the irst ew requencies are iportant, the investigation concluded that it is possible to obtain a reasonable estiate o the dynaic characteristics o a loor by using inite eleent sotware. Osborne and Ellis (199) have presented a study o vibration design and testing o long-span lightweight loors, ocussing on the estiation and evaluation o loor design. One ajor objective was the coparison between sipliied hand calculations, coputer supported calculations and accurate tests on-site. It was shown that all three, and especially the latter cases, predict siilar values; the estiated requency o a coputer analysis was just.1 Hz (approxiate 3%) higher than easured requency. Another interesting inding o this study was the change in dynaic behaviour ro the bare loor to a inished loor including a alse loor, service installations and ire protection. Although the inished loor showed only a sall increase o daping and stiness, qualitative observation by people peroring a heel drop test agreed an iproved perception. The vibration assessent loor ro Ove Arup & Partners () provides particularly useul inoration because o its strong reseblance to the Cobiax lat slab syste. The report includes the results o an investigation into the vibration behaviour o a loor or a typical hospital. For this case an idealised area o hospital loor was assued. Its properties included thickness, 315 ball size and 3 x 3 square bays. Each bay had a span o 9 x 9. The iposed loads were estiated as realistic in-service values averaged over the entire loor area. Using the inite eleent sotware MSC NASTRAN, a odel was created to analyse the loor s dynaic perorance. The slab was odelled as a thick solid slab and its speciic qualities were considered by a reduced stiness 15

24 1. Introduction and Background Knowledge Lukas Wolski and ass. The analysis showed that the undaental requency o this loor is 11.8Hz. Furtherore a ootall response analysis was carried out to obtain the root-ean-square (r..s.) velocity o the loor. Aterwards all results were copared with a loor o solid concrete. The irst natural requency reduces to 1.Hz, a decrease o 1%. The responses or Bubbledeck slabs are 1% higher than those or a solid slab o the sae thickness. 1.. Deterination o Frequency Mazudar (1971) deterined the undaental requency o elastic plates o arbitrary shape by aid o constant delection lines. For this purpose he assued the classical sall-delection theory to be valid. His ethod or the case o elliptical plates was illustrated speciically because o its increased coplexity copared to other shapes. The assuption that the lines o equal delection also had an elliptical shape was ade in response to the proble o deterining the resulting tie-dependent delection ield. This approxiation is then only valid or slender elliptical plates, aking this ethod only practical or thin plates. Ater the deterination o all necessary dynaical equations, two exaples were calculated. One plate was supposed to have claped edges and the second was siply supported, an estiation which had previously only been published in one work. Furtherore the author copared his ethod with results already established in literature. For sall ratios o both sei-lengths this coparison showed very siilar outcoes to the other present values. Jones (1975) used this ethod and extended the coparison. He investigated sipliied calculations or the undaental requency o structures with dierent shapes and boundary conditions such as equilateral triangular, rectangular or seicircular plates. Aterwards he also copared these approxiations with coputed and ore exact values. As beore, the results o this coparison were very good. For the exaple o a claped quadratic plate, the dierence between the two estiations was.5%. 1

25 1. Introduction and Background Knowledge Lukas Wolski Magrab (197) adopted a dierent approach to estiating the natural requencies or plates. He derived an expression or orthotropic rectangular plates with siply supported, elastically supported or claped boundary conditions. Instead o using existing estiation ethods and thin-plate theory which relies on a length-tothickness ratio he solved the proble with another atheatical technique: the Mindlin-Tioshenko theory. This theory is an iproveent on the Euler- Bernoulli bea theory, which condensed a bea to a 1-D structure. Another assuption o this theory is that the plane cross-section o a bea reains plane and noral to the reerence line when the bea deors due to bending. In addition to this hypothesis Tioshenko's theory considers sheer and rotational inertia eects and the resulting deoration. Coparing an exaple with other estiations which use the thin-plate theory yielded analogical values with dierences between.8% and 3.7%. Excepting the estiate values o Elishako (197), all other undaental requencies are higher than those calculated by the author. This results ro the consideration o transverse sheer and rotary inertia which also ibibes vibration energy additional to bending as required by the thinplate theory. The inluence o Tioshenko s additional consideration, rotational inertia and sheer deoration or rectangular plates, was orerly investigated by Mindlin, Schacknow and Deresiewicz (195) who deterined a ethod to obtain natural requencies with coupled odes. Special regard was given or the case o a plate with one pair o parallel ree edges and the other pair siply supported. Leissa (1973) presented a study o approxiate orulas or ree vibration o rectangular plates. It was the irst copilation o all 1 cases which involved all possible cobination o classical boundary conditions, like claped, siply supported and ree edges. Aongst other techniques he used the Ritz ethod or the bea unction or this purpose. This led to the production o a set o 1 tables or the estiation o the irst 9 odes or each plate including dierent length-towidth ratios. Furtherore the eect o changing Poisson s ratio on the natural requencies was presented. In every case the requency depends on Poisson s ratio. An exaple o a plate supported on two parallel edges by siply-supports 17

26 . Huan Response and Acceptance Criteria Lukas Wolski and on the other a pair o ree edges showed that increasing Poisson s ratio caused a decrease in natural requency. Other objectives o this investigation were the evaluation o accuracy copared to the reerenced Warburton s orulas or natural requencies and the eect o changing edge condition upon the requencies and their accuracy.. Huan Response and Acceptance Criteria Evaluation o easured or calculated values o loor vibration ust be carried out in order to predict its inluence on the surrounding environent. This requireent creates the need or speciic acceptance criteria. It is possible to classiy the eects loor vibration has on its environent into three ain areas: - Overstressing o structural ebers - Physiological eect on people - Ipact o production processes with sensitive equipent or susceptible achinery in general (Bachann and Aann, 1987) O these three, ost attention is paid to huan response. This is because daage and atigue ailure o structural eleents due to walking-induced loor vibration are unusual, and every dierent type o achinery has its own very speciic requireents. Dierent acceptance criteria and recoendations have been developed to easure huan response. Unortunately, though, it is not possible to provide exact liit values and this can obviate perception o otion. As the variety o huan responses to loor vibration varies greatly, these criteria can only utilise reerence values gained by experience or ield tests. The coplexity o both perception levels and huan sensitivity to vibration is illustrated by a high nuber o interrelated actors. Aong the are: Direction o otion: Huans evaluate every direction o otion dierently. Generally vertical oot-to-head vibration is considered ore annoying than horizontal chest- 18

27 . Huan Response and Acceptance Criteria Lukas Wolski to-back vibration (Cooney and King 1988). However, every direction o otion has to be considered because o its potential occurrence. While horizontal vibration causes only sall concern in oices and other workplaces, its iportance increases in the design o residences and hotels where sleeping coort ust be considered. Personal characteristics: Dierent responses are given depending on the age, sex and level o concentration o the subjects as well as those o surrounding counity. Tiing and duration: Motions at night are less tolerated than those occurring during the day. Furtherore continuous otion (steady-state) is ore annoying than otion caused by inrequent ipact (transient). Expectation: I subjects are orewarned o vibration, their perception will be less sensitive Current activity: Dierent levels o acceptance exist or oice work, physical work, resting, dining and dancing. Acceptance levels are also aected by the surrounding environent (e.g. hoe, oice or gynasiu). Since the pioneering work o Reiher and Meister (1931), ost vibration criteria provide graphs deining regions o acceptable and unacceptable vibration. Usually these are plotted in requency versus peak acceleration due to gravity o the loor vibration, but other nuerous paraeters such as velocity or displaceent o the treated loor can be included. On the graph, single lines represent a constant level o huan reaction (isoperceptibility lines) with the region above a line denoting unacceptable vibration. These act as boundaries between dierent levels o perception. 19

28 . Huan Response and Acceptance Criteria Lukas Wolski.1 Recoendations in Codes.1.1 British Standard In British Standard BS399-1:199 Annex A, two dierent approaches are recoended or the design o doestic and residential structures, especially single aily buildings. In areas subjected to dancing or juping there can be an increased risk o unpleasant loor oveent and even resonance ay occur. In order to avoid this phenoenon, it is recoended that vertical natural requency is liited to at least to 8.Hz and horizontal natural requency to a iniu o. Hz. These requencies should be calculated or the epty structure. Another approach is to consider dynaic loads as well as dead and static iposed loadings during the design stage. Deoration due to dynaic loads should not exceed liits appropriate to the building or structure type. No detailed speciications are provided or lightweight and long span structures. Only the general advice o taking loor vibration into account and the recoendation o specialist guidance docuents are given: Where lightweight and long span structures are used as concourses and public spaces, they are likely to be subjected to inadvertent or deliberate synchronized oveent by people, causing dynaic excitation. The design provisions should take account o the nature and intended use o the structure, the potential nuber o people and their possible behaviour. Structural design should be undertaken with the help o specialist advice and specialist guidance docuents, as required by the appropriate certiying authority. A ore detailed treatent o loor vibration is covered by the British Standard BS 7:199 Guide to evaluation o huan exposure to vibration in buildings (1 Hz to 8 Hz). This guide has a general approach, or application to any vibratory environents. It is applicable to vibrations transitted through the supporting surace to the body as a whole by considering dierent positions and all three axes, as deined in igure.1. Within the range o 1 to 8 Hz, the guide also considers dierent types o structures including oices, residential buildings or critical working areas such

29 . Huan Response and Acceptance Criteria Lukas Wolski x-axis: back to chest y-axis: right side to let side z-axis: oot to head Figure.1 BS 7 (199) Coordinate systes or vibration inluencing huans as operating theatres. All allowable vibrations are provided in curves o annoyance or huans in ters o direction o transission, requency and acceleration or velocity. While acceleration is given as r..s. acceleration (rootean-square acceleration), velocity is speciied as a peak value. In ters o huan response the British Standard divides vibrations into two classes: ipulsive and continuous vibration. Ipulsive vibration is deined as a rapid build-up to and decrease ro a peak; or exaple vibration caused by the ipact o a single heavy object on a loor. This type ay also consist o several cycles o vibration providing that duration is short (less than approxiately seconds). The other category describes continuous vibration which reains uninterrupted over a certain tie period (or exaple vibration caused by a group o people walking). Their dierent consideration is given by individual ultiplication actors shown in table.1. These actors are used to ultiply the base curves and obtain the according curve or a speciic case. 1

30 . Huan Response and Acceptance Criteria Lukas Wolski Place Critical working areas (e.g. hospital operating theatres, precision laboratories (see notes 3 and 1) Tie Exposure to continuous vibration [1 h day, 8 h night] (see note and Appendix B) Multiplying actors (see notes 1 and 5) Day 1 1 Night 1 1 Ipulsive vibration excitation with up to 3 occurrences (see note 8) Residential Day to (see note ) to 9 (see notes and 9, and Appendix B) Night 1. Oice Day Day 18 (see note ) Night 18 Workshops Day 8 (see note 7) 18 (see notes and 7) Night 8 18 NOTE 1 Table 5 leads to agnitudes o vibration below which the probability o adverse coents is low (any acoustical noise caused by structural vibration is not considered). NOTE Doubling o the suggested vibration agnitudes ay result in adverse coent and this ay increase signiicantly i the agnitudes are quadrupled (where available, dose/response curves ay be consulted). NOTE 3 Magnitudes o vibration in hospital operating theatres and critical working places pertain to periods o tie when operations are in progress or critical work is being perored. At other ties agnitudes as high as those or residences are satisactory provided there is due agreeent and warning. NOTE Within residential areas people exhibit wide variations o vibration tolerance. Speciic values are dependent upon social and cultural actors, psychological attitudes and expected degree o intrusion. NOTE 5 Vibration is to be easured at the point o entry to the entry to the subject. Where this is not possible then it is essential that transer unctions be evaluated. NOTE The agnitudes or vibration in oices and workshop areas should not be increased without considering the possibility o signiicant disruption o working activity. NOTE 7 Vibration acting on operators o certain processes such as drop orges or crushers, which vibrate working places, ay be in a separate category ro the workshop areas considered in Table 3. The vibration agnitudes speciied in relevant standards would then apply to the operators o the exciting processes. NOTE 8 Appendix C contains guidance on assessent o huan response to vibration induced by blasting. NOTE 9 When short ter works such as piling, deolition and construction give rise to ipulsive vibrations it should be borne in ind that undue restriction on vibration levels can signiicantly prolong these operations and result in greater annoyance. In certain circustances higher agnitudes can be used. NOTE 1 In cases where sensitive equipent or delicate tasks ipose ore stringent criteria than huan coort, the corresponding ore stringent values should be applied. Stipulation o such criteria is outside the scope o this standard. Table.1 BS 7 (199) Multiplying actors Figure. shows one exaple o a ultiplied curve where the requency is plotted against the r..s. acceleration. It is recoended that the requencyacceleration cobination is kept below the line which corresponds to the relevant case, thereore iniising adverse coents or coplaints o vibration.

31 . Huan Response and Acceptance Criteria Lukas Wolski Figure. BS 7 (199) Building vibration z-axis curves or acceleration (r..s.) Another ethod or speciying satisactory vibration agnitudes is provided in Appendices A and B. By calculating and coparing the vibration dose value with liit values presented in tables it is also possible to evaluate a structure s vibration; however, this approach is exclusively provided or residential buildings and is thereore only applicable to a sall aount o proble areas. 3

32 . Huan Response and Acceptance Criteria Lukas Wolski.1. Geran Standard The Geran Institute or Standardisation published a siilar code entitled DIN 15- (1999) Erschütterungen i Bauwesen; Einwirkungen au den Menschen in Gebäuden. This code provides recoendations concerning huans vibration perception in residential and siilarly used buildings and is applicable to periodic as well as non-periodic vibrations. It also deals with requencies ro 1 to 8 Hz and considers all three axes o a huan body as well as dierent types o buildings; but in contrast to the English code, the Geran DIN uses a odiied paraeter called KB value, which depends on the requency o otion and was established to assess the acceptance o otion by liiting the requency to speciied values (see table.). As in the BS399-1(199), the liit criteria in the DIN 15- (1999) depends on occupancy and tie o day. Place Areas with coercial buildings and as an exception residencies or occupants or directors o these copanies Day Night A u A o A r A u A o A r Areas with coercial buildings predoinantly Areas without predoinance o coercial buildings as well as residences Areas with residential buildings predoinantly Critical areas (e.g. hospitals) Table. Extract o DIN 15- (1999); reerence values A or residential and siilarly used buildings

33 . Huan Response and Acceptance Criteria Lukas Wolski For the coparison o easured and recoended liit values two dierent KB paraeters are used. These are represented by KB Fax or the axiu otion and KB FTr, which is an averaged value spread over the assessent tie. These two values ust be estiated or each o the three axes in which otion ay occur. The worst case becoes decisive. Once both critical values are known, a ixed procedure shown in igure.3 can be applied. KB Fax and in special cases KB FTr need only be copared to reerence values A u and A o and a inal evaluation is then given. Figure.3 DIN 15- (1999) progression o assessent procedure This ethod o predicting vibration acceptance is ore coplex than that o the British code. The calculation o all necessary values requires a lot o tie and precise knowledge o the circustances, such as duration o ipact. It sees to be a ethod or evaluating easured values rather than calculated values ro the design stage. 5

34 . Huan Response and Acceptance Criteria Lukas Wolski. Recoendations in Literature Besides national codes and standards, any independent and individual recoendations are available. These are partly developed ro practical tests on subjects and partly gained by experience o existing buildings. The ollowing paragraphs will deliver an overview o soe o the recoendations ound in literature. The ost requently cited reerence in the ield o huan acceptance and loor vibration is Reiher and Meister (1931). These authors carried out the irst research on this topic by investigating how horizontal as well as vertical vibration aects huans. Subjects were placed on shaking tables which varied in aplitude and requency. Aterwards, they had to rate the otion using one o six categories. This inoration then ade it possible to plot the relationship between aplitude and requency in relation to huan perception. It should be entioned here, though, that due to the long duration (approxiate 5in.) o each test the results should be applied to continuous rather than ipulsive vibration. The Reiher Meister scale or vertical vibration is shown in igure.. Figure. Reiher-Meister scale

35 . Huan Response and Acceptance Criteria Lukas Wolski Lenzen (19) deterined that daping and ass, and not stiness, were the ost iportant paraeters in preventing unacceptable loor vibration caused by walking. He suggested that i vibration is reduced by daping to a negligible quantity in 5 cycles the huan will not respond, whereas i it persists beyond 1 cycles a steady-state vibration is noticeable. He also odiied the Reiher-Meister scale by increasing displaceent by a actor o 1 (Figure.5). The dierence results ro discriinative huan sensitivity against the duration o vibration. In contrast to Reiher and Meister s steady-state vibration, Lenzen developed this criterion or transit vibrations which have a reduced eect on subjects. Figure.5 Graph o reduced huan response 7

36 . Huan Response and Acceptance Criteria Lukas Wolski Allen and Rainer (197) developed annoyance criteria or walking vibrations in ters o acceleration and daping based on tests using long-span loor systes. These were then incorporated into the Canadian Standards Association s national code. The proposed criterion (igure.) is an extension o Lenzen s work and considers continuous vibration (1 to 3 cycles) as well as walking vibration. They are suggested or use with quiet huan occupancies, or exaple residences, oices or schoolroos. Pernica and Allen (198) odiied the criteria or active occupancies such as shopping centres and car parks by increasing the liits by a actor o 3. Figure. Annoyance criteria by Allen and Rainer Interpretation o this graph requires care, because both types o line are a criterion or loor vibration. The continuous vibrations are caused by a person walking on a loor, as are the walking vibrations. The dierence between these types is that instead o the continuous vibration line which uses average peak acceleration to assess acceptability, the walking vibration lines represents the initial peak acceleration resulting ro a heel drop test. 8

37 . Huan Response and Acceptance Criteria Lukas Wolski Allen and Murray (1993) entioned the necessity o the irst three haronics in avoiding resonance. The third haronic should be considered or a single person walking with noral velocity. For jogging or ore than one person, only the irst two haronics are iportant. I the nuber o persons walking on a structure increases then the dynaic loading does increase, but at the sae tie lack o coherence at higher haronics increases. However, generally such cases are rare enough to not be a proble in practice. The proposed design criterion or the acceptance o loor vibration is provided through dierent approaches. One o the can be expressed in ters o undaental requency and is given by: K.8 ln βw Where is the undaental requency, W is the weight and K is a constant given in table.3 which also provides approxiate values or the daping ratio β. K kn β Oices, residences, churches 58.3* Shopping Malls. Footbridges 8.1 *.5 or ull-height partitions,. or loors with ew non-structural coponents (ceilings, ducts, partitions, etc.) as can occur in churches Table.3 Values o K and β Bolton (199) ephasised the iportance o acceleration or the perception o loor vibration by using the exaple o passengers in an aircrat. As the crat lies with high and constant speed, the huans inside do not eel any oveent. It is the change o velocity, that is, the acceleration, which is perceived. The acceleration is directly proportional to the square o the requency and to its aplitude o displaceent. Acceleration - (π)² x δ 9

38 . Huan Response and Acceptance Criteria Lukas Wolski Provided with knowledge o the undaental requency and acceleration due to gravity, it is possible to evaluate the eect o loor vibration on huans. To this end, Bolton proposed to divide the range o requencies into two areas: ro to 1 Hz and above 1 Hz. The relationship between acceleration and huan perception is presented in table. below: 1 Hz acceleration [/s²] > 1 Hz barely perceptible.3.5 clearly elt.1.13 unpleasent.5.7 entirely unacceptable..133 Table. Huan perception criteria by Bolton A ore general recoendation was proposed by Bachann and Annann (1987). They argued that due to the lack o exact knowledge concerning various loor paraeters and their inter-relations, it would be ore practical to provide rough liit values or the designer s use. Their recoended values or natural requency as well as the acceleration o loor vibration are shown in table.5. The act that the requencies increase or dierent construction aterials is due to their decrease in stiness, ass and daping. reinorced concrete prestressed concrete [Hz] coposite steel acceleration [/s²] Oices > 7.5 > 8. > 8.5 > Gynasia and sport halls Dancing and concert halls > 7.5 > 8. > 8.5 > 9.. >.5 > 7. > 7.5 > Table.5 Overall acceptance levels or various types o environent Siilarly to Allen and Murray (1993), Bachann and Annann also entioned observing ore than requency; while ore active areas like sport halls, dancing or concert halls should be considered using the second haronic, they propose 3

39 . Huan Response and Acceptance Criteria Lukas Wolski high tuning with respect to the third haronic o load-tie unction or oices or other quiet working places. Other literature presents this topic with less accuracy and provides only rough liit values or consideration during the design o loor systes. For the prevention o resonance Fisher and West (1) recoend avoiding natural requencies o between 1 and Hz or walking areas and 5 Hz or dancing areas. Cooney and King (1988) entioned that crowds involved in activities such as dancing or gynastic can be synchronised by usic or other eans up to requencies o Hz. Beyond this liit they becoe uncoordinated and a rando orcing unction results. Thereore they suggest checking loors with natural requency below Hz and possible support o assebly occupancies or resonance. Furtherore, Hanes (197) reported that studies using autoobile and aircrat passengers showed that the natural requency o huan internal organs is between 5-8 Hz. Thereore, loor systes with natural requencies in that range could possibly cause huan discoort and should be avoided. Morrison () speciied the interering requencies o individual sub systes within the body. Soe exaples are the abdoen-thorax region, 3 Hz, the spine, 5 Hz or the heart, 7 Hz. The requencies at which the whole huan body is ost sensitive are 3 - Hz and 1-1 Hz..3 Suary The eects o vibration on huans vary so widely that evaluation is very coplex and depends on any actors. In general, it is diicult to set accurate liits to any paraeters. However, past research has attepted to obtain boundaries or dierent kinds o structures and activities. These should result in iproved abience in new structures as well as decreasing the chance o justiiable coplaints ade by users. 31

40 3. Sipliied Hand-Calculation Methods Lukas Wolski 3. Sipliied Hand-Calculation Methods In the past, there have been any attepts to ind sipliied ethods or calculating the undaental requency o structures. Beore inite eleent sotware and coputer packages, which predict all necessary inoration accurately within a ew seconds, this inoration needed to be estiated. Even today, when coputers are used universally, additional hand calculations are still recoended because they help to give an initial assessent and are able to orecast critical areas beore or during the design stage. Alternatively, they ight be used as an additional check on results calculated by a coputer (Weber ). Siple equations and tables ake it possible to estiate natural requencies in a very short space o tie. Furtherore, only a ew predictions are necessary, these being aterial properties such as the Young's odulus or the Poisson ratio and geoetrical properties such as thickness o the structure or the length o span. Equipped with this data and a siple calculator, undaental requency can be predicted in seconds, provided the structure is not too coplex. As the requencies o coplex structures would require huge eorts to hand-calculate, ethods are only published or siple odels such as one-span or continuous beas, one-span slabs or chineys and pylons. The sipliication is usually based on bea theory and can easily be adopted or these kinds o structures. Another sipliication o these ethods is the assuption o boundary condition. In reality, the grade o restraint or each support is unclear. Obviously in practice the support conditions are rarely siply supported or truly ixed, but in sipliications all supports are assued to be 1% siply supported or claped. This produces only sall inaccuracies copared with the real dynaic behaviour o structures. Other actors can also aect the accuracy between odel and real behaviour. There are soe inluences which a sipliication cannot or will not predict. An exaple o one such inluence is the dissipation o energy due to contributions ro coating or a suspended ceiling. This increases the daping ratio and thereore counteracts the vibration. Brownjohn (1) also highlighted that a certain aount o vibration is reduced by occupants standing on the loor, a actor which cannot be predicted in hand calculations. Another diiculty is the nonlinearity which ay occur ater cracking. Once the concrete starts to crack, the 3

41 3. Sipliied Hand-Calculation Methods Lukas Wolski eective stiness is in lux. Due to the saller aount o concrete cross-section, the oent o inertia decreases, resulting in a higher natural requency. These are a ew circustances which prevent 1% accurate values ro being ade; nevertheless, in any cases sipliied estiations can provide good practical results and thereore initial ideas concerning the dynaical behaviour o the structure. Thus, probleatical areas can be identiied in an early design stage. This chapter deals with sipliied ethods or calculating the undaental requency o slabs. For this purpose a suary o soe existing literature providing equations or tables is presented and is checked against their accuracy. Thereore, calculations or a set o coon structural slabs considering ree undaped natural requency or rectangular isotropic plates are carried out (Appendix A). Aterwards each exaple is copared against ore exact values calculated by inite eleent sotware, enabling a general evaluation o their practical use to be undertaken. 3.1 Coon Matheatic Techniques When estiating the necessary values o eigenvalues, natural requency or ode shapes, sipliied ethods usually involve dierent atheatical techniques. The ain ideas o two dierent approaches, the Stodola ethod and the Rayleigh ethod, are described briely below. Reerring to Caverson., Waldron and Willias (199) the best-known approach is the Stodola ethod. This is an iterative ethod, in which the shape ode o an eleent is estiated and the initial orces associated with this ode shape are deterined. A static analysis is carried out concerning these orces, providing a delected shape or the condition. Instead o the previous estiated ode shape, the new delection shape is now used or the sae procedure. These iterations are then repeated until a suiciently accurate solution is achieved. The Rayleigh and Rayleigh-Ritz ethods are also widely used to predict the natural requency o structures. The Rayleigh ethod is based on the principle o 33

42 3. Sipliied Hand-Calculation Methods Lukas Wolski energy conservation: i an undaped reely vibrating spring-ass syste is assued, no absorption o energy will take place and so the aount will reain constant. The relationship between the involved energies, potential and kinetic, requires their axiu values to be equal. The equations or both energies can then be coputed and an equation or the undaental requency o a one-asssyste ored. This principle is then used when analysing systes with a greater degree o reedo. In this case, the ode shape o the undaental requency needs to be assued irst. Further on, it should be noted that, while the aplitude o otion varies with tie, the shape o vibration does not, leading to an expression by a shape unction. The assued shape unction sipliies the structure to a single degree o reedo syste. The initial ethod o equating the energies can then be applied and the undaental requency calculated (Clough and Penzien 1993). The Rayleigh-Ritz ethod is an extension o Rayleigh's ethod and deterines the natural requency in the second or higher order. It uses the basic concept o Rayleigh's ethod and iniises the total energy to its original aount ater adding the assued shape unction to gain the lowest requency. This ethod can be applied iteratively as well as partially. The ost iportant step within the ethod is to ake a realistic assuption o the ode shapes in order to avoid a high nuber o iterations. 3. Forulas and Tables or the Calculation o Fundaental Frequency 3..1 Equivalent Bea Method A widely used technique or deterining the undaental requency o a loor is the equivalent bea ethod (Lenzen 19). However, the assessent o beas is only recoended or the estiation o one-way spanning systes and not or two-way spanning loors. For this reason it is ore oten used or coposite loors than concrete slabs. Reerring to Caverson, Waldron and Willias (199), this equation should provide an accurately estiated requency or coposite as well as one-way spanning concrete loors. The irst natural requency is deterined by: π a E I (1) 3

43 3. Sipliied Hand-Calculation Methods Lukas Wolski 3.. Equivalent Plate Approach In addition to the equivalent bea ethod only applicable to one-way spanning concrete slabs, Willias and Waldron (199) as well as Jeary (1997) presented an equation or the undaental requency o siply supported two-way spanning slabs. Thanks to the inluence o all three diensions and the lexure rigidity o the plate D, it incorporates biaxial slab properties and provides good solutions. π D l x l y () 3..3 Concrete Society Method Another calculation procedure or biaxial slabs was proposed by the Concrete Society (5). As with the equivalent plate approach, it uses an approxiation o the equivalent bea ethod and considers the increased stiness o a two-way spanning loor. This leads to two independent orthogonal odes occurring or both directions individually. The lower o both natural requencies can be considered as the undaental requency or this slab. A general advantage o this ethod is its ultiarious application to dierent types o slabs including solid, ribbed and wale, and its additional consideration o several bays or each direction. However, the ollowing equations are just presented or solid slabs in one direction (x-direction). The characteristics o the second direction ode are deterined by interchanging the x- and y-subscripts in these equations. The irst step is to deine the eective aspect ration o the slab by: nx l λx l y x EI EI y x 1 (3) Aterwards it is necessary to calculate the odiication actor k x : k x λ x () 35

44 3. Sipliied Hand-Calculation Methods Lukas Wolski The natural requency or slabs with perieter supports is: ' x π k EI y x ly (5) For slabs without perieter supports, the latter equation has to be odiied by calculation o an additional requency b: b π 1 + EI l EI EI x x xly ylx () The inal natural requency can now be obtained by: x ' x ' ( ) x b 1 n x 1 + n y (7) 3.. Static Delection Method As seen in paragraph 3.1, it is possible to calculate the natural requency by aid o the kinetic and potential energy within the structure. As a result o the associations o both, kinetic energy with the otion o ass and potential energy with the strain energy stored in the elastic structure during deoration, a relationship between the natural requency and the delection o a structure exists. This relationship can be used to calculate the undaental requency or structures whose static delection is known. The derivation o an equation or an exaple o the spring-ass-syste deonstrates the relationship well. The static delection o a ass attached on a spring is given by: δ S g k (8) 3

45 3. Sipliied Hand-Calculation Methods Lukas Wolski The natural requency is already known as: 1 π k (9) Now it is possible to incorporate equation 3..1 into equation 3.1 which gives: 1 π g δ S (1) The inal equation shows that only the acceleration due to gravity and the axiu static delection are required to obtain the undaental requency. But due to an additional actor, the length o span is involved now, this ethod should be ore accurate than that o equation 1. O course, this depends on the accuracy o the estiated static delection but the higher aount o necessary data used in the calculation o static delection allows this initial conclusion. A odiied version o this ethod was published by Blenvis (1979), who used the expression derived by Mazudar (1971) and odiied by Jones (1975) or calculating undaental requency. These earlier authors developed a ethod to estiate natural requency aided by the constant delection lines o an eleent. Although this equation was developed or claped elliptic plates, it also predicts the irst requency o plates o various shapes and boundary conditions. The new equation is now: 1.77 π g δ S (11) 3..5 Approxiation Presented by Hearon Hearon (1959) presented approxiations or the estiation o rectangular orthotropic plates using the already derived expressions ro Warburton (195) and extending the to orthotropic plates with a cobination o claped or 37

46 3. Sipliied Hand-Calculation Methods Lukas Wolski supported edges. These approxiations are based on the Rayleigh ethod and assue that the nodal lines o the delection are approxiately parallel to the sides o the slab. Furtherore, it is supposed that all three axes o a plate are rightangled to each other. With this criterion, and the assuption that the thickness o the slab and its delection are sall, it is possible to apply a two-diensional treatent. This produces a sipliied expression because the elastic properties o third syetry can be disregarded. Equation 1 and table 3.1 are odiied or non-orthotropic plates and lead to a value or their natural requency. 1 π A a D + B D b + CD a b ' (1) With 3 ' Eν h D + 1 (1 ν ) G h (13) boundary conditions A B C n β β ε.73 ε ε ( ε ) 1.3 β ( β ) β ε ( β )( ε ) 3,,5,... 3,,5,... 3,,5,... 3,,5, β ε 1 ε 1 ε ( ε 1) β ε ( β )( ε 1) 1 1,3,,...,3,,... 3,,5, β ε ε 1.3 ε β ε ( β ) 3,,5,...,3,,...,3,,... β 1 ε 1 βε ( β )( ε 1) ,3,,...,3,,... β 1 ε β ε ( β 1) 1 1,3,,...,3,,... β ε β ε,3,,...,3,,... β ( 1)π β1 (. 75)π β (. 5)π ε ( n 1)π ε1 ( n. 75)π ε ( n. 5)π Table 3.1 Frequency paraenter provided by Hearon 38

47 3. Sipliied Hand-Calculation Methods Lukas Wolski 3.. Approxiation Presented by Jänich Another approxiation that uses the Rayleigh ethod to solve the calculation o undaental requency is presented by Jänich, who odiied the double integral o the potential as well as kinetic energy to gain a inal equation or the undaental requency. This equation also considers additional loadings. Excepting static loads, this approxiation akes it possible to include additional unior loads (e.g. loor paveent or tiling), point loads or even iposed loads. These considerations are covered in an extra paraeter deterined by the kinetic energy and are given by: N p P + + w g g N 1 P (1) Where p is the additional load (coating + iposed load), g is the ass o a slab, P is the point load and w p is the displaceent below the point load P. A second paraeter was derived ro the potential energy to get a unior calculation. The K value consists o: K1 K K + + a a b K b 3 (15) The necessary values or N, K 1, K and K 3 can be taken ro table 3., which shows an extract providing eight dierent exaples o support condition and their corresponding paraeters. The original table presented by Jänich includes 18 dierent set-ups o boundary condition with cobinations o ree, claped and siply supported edges. With both paraeters, N and K, and the equation presented below, the irst natural requency ay be estiated. π D g K h γ N (1) 39

48 3. Sipliied Hand-Calculation Methods Lukas Wolski Another advantage o this ethod is its capability o estiating the requencies o two-span slabs. Provided that both jointed edges o a slab have equal support conditions, equation 1 can be odiied to estiate the undaental requency with: π D g ( h γ ( a K ) i i i i a N ) (17) boundary condition K 1 K K 3 N Table 3. K and N paraeters

49 3. Sipliied Hand-Calculation Methods Lukas Wolski 3..7 Estiation or Pin Supported Plates Reed Jr. (195) published a NASA report in which he presented and copared two dierent approaches to calculating the natural requency o rectangular plates supported by isolated pins in each corner. The two approxiate ethods were then developed using the Ritz ethod and a series solution to the dierential equation o otion. The general solution or this type o support can be gained by treating a plate as those with supports along their entire perieter. Aterwards superpositions o the initial solutions ust be done to satisy the speciic conditions o the isolated pin supports. Coparing both ethods revealed a higher accuracy or the series solution but also ore diiculties in coputing this approach. However, because o the need or sipliied hand estiation and the already existing requency paraeter, this disadvantage can be disregarded here. With both ethods the natural requency can be obtained by: λ π a γ h D g 1 (18) Mode α requency paraenter λ Ritz solution Series solution Table 3.3 Frequency paraeters or pin supports 1

50 3. Sipliied Hand-Calculation Methods Lukas Wolski 3..8 Copilation o Forulas by Bachann Walter Aann and Hugo Bachann, one o the ost well-known authors in the ield o loor vibration issues, provided in their book a set o tables and charts or estiating the natural requencies o several structural systes. These were inored by the partial dierential equation o otion or ree vibration. All plates considered in these tables are assued to be two-way spanning and have boundary conditions o siply supported, claped or ree edges. One extract o these tables is given in table 3.. It includes siply supported, claped and a ix o both conditions and allows the calculation o the irst two natural requencies o these saples. ϕi i a E h 3 1 (1 -ν ) (19) support conditions ϕ ϕ ϕ 1,1 1,1 1, (1 + α ).8 (1 +.5α ) 1.57 (1 + α ) ϕ ϕ 1,1, α + 5.1α 1 +.5α +.31α ϕ 1, α + 39.α ϕ 1.57 ϕ ϕ 1,1,1 1, α + 5.1α 1+.98α +.13α α + 39.α ϕ ϕ 1 1,1 ; ϕ ϕ in ϕ Table 3. Frequency paraenter provided by Bachann 1,,1

51 3. Sipliied Hand-Calculation Methods Lukas Wolski For the case o continuous plates a chart is presented in igure 3.1, which helps to estiate the irst three natural requencies o a two-span slab. However, because this chart was developed or continuous beas, it is only applicable to onespanning slabs. One indication o this liitation is the siple assuption o paraeters in equation without reerence to plate characteristics such as the plate rigidity or the width o the slab. n λ n π EI 1 () Figure 3.1 Frequency paraeter or continuous slabs 3

52 3. Sipliied Hand-Calculation Methods Lukas Wolski 3..9 Copilation o Forulas by Blevins Robert D. Blevins (1979) published a huge nuber o dierent sets or sipliied hand calculation in Forulas or Natural Frequency and Mode Shape. The book is intended as a reerence or engineers and provides tables or any dierent structures and shapes. It also considers dierent boundary conditions and diverse ode shapes. The tables are copiled ro a variety o sources. In the case o rectangular plates (Page 5-78) the layout is siilar to that previously considered with the addition o pin supported slabs. The natural requency can be estiated by: i i λ π a 3 E h 1 (1 -ν ) 1 (1) Siply Supported - Free - Siply Supported - Free a / b requency paraeter λ i 1. Mode. Mode 3. Mode. Mode 5. Mode. Mode / Siply Supported - Siply Supported - Siply Supported - Siply Supported a requency paraeter λ / i b 1. Mode. Mode 3. Mode. Mode 5. Mode. Mode / Rectangular Plate, Corner Supports a / b λ 1 λ N-Bay Plate Nuber o Bays λ 1 λ λ bay square plate Table 3.5 Frequency paraenter provided by Blenvis

53 3. Sipliied Hand-Calculation Methods Lukas Wolski 3.3 Analysis o Results The undaental requencies o 1 dierent Cobiax lat slabs were estiated by sipliied hand calculation. Aterwards each solution was copared with its corresponding value calculated with inite eleent sotware. A suary including all ethods and all calculated exaples is contained in Appendix A, which also provides the ratio o requency derived ro sipliied calculations to an exact inite eleent solution or each exaple. The evaluation o accuracy is inored by igure 3. where requencies estiated with sipliied hand calculations are plotted on the x-axis and the solution ro FEM on the y-axis. Each arking represents one calculated value with its particular ethod. The dashed line stands or the FEM values and sybolises the ideal position or the calculated values. The closer a point is located to the FEM line (vertical or horizontal distance), the higher its accuracy copared to the inite eleent solution. obtained by FEM [H z] FEM Equivalent bea ethod Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Estiation or pin supported plates Approxiation by Bachann Approxiation by Blevins obtained by hand calculation [H z] Figure 3. Coparison o hand-calculated and coputed requencies 5

54 3. Sipliied Hand-Calculation Methods Lukas Wolski Overall, there is a reasonably good correlation between the hand-calculated approxiations and the coputed values. Nevertheless it is noticeable that soe values have higher inaccuracies. The static delection ethod (agenta cross) is the ost inaccurate estiation ethod, underestiating all its requencies. In contrast, the odiied static delection ethod (green plus sign) tends to overestiate its solutions. This becoes clearer i the ratio o approxiation to coputed value is plotted or each exaple, as in igure 3.3, aking it possible to see their higher variation in coparison with all other ethods. The act that both ethods consist o the sae paraeters and only dier in a odiication actor explains their siilar pattern. 13 Equivalent bea ethod 1 Equivalent plate approach ratio %: hand calculation / FEM exaple 1 exaple exaple 3 exaple exaple 5 exaple exaple 7 exaple 8 exaple 9 exaple 1 exaple 11 exaple 1 Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Estiation or pin supported plates Approxiation by Bachann Approxiation by Blevins Figure 3.3 Individual accuracy o approxiations This graph also clariies the higher dierences o Jänisch s ethod (blue triangle) or exaple 1 (5%) and exaple 1 (9.%). As these exaples involve continuous slabs with two varying span lengths, it ight be supposed that this approxiation is ore qualiied or continuous slabs with equal span length. This assuption is conired by exaples 9 and 11 calculated using Jänisch s ethod, which ulil this condition and have variances o only 1.5% and.%.

55 3. Sipliied Hand-Calculation Methods Lukas Wolski 3. Conclusion Ten dierent ethods or estiating the undaental requency o concrete loors have been presented. The general ai has been to provide an overview o their accuracy and thereore their serviceability or an initial estiation. Their integrity was checked by coparing the solutions o concrete exaples with accurate values calculated with inite eleent sotware. Although soe variations occurred aong each ethod, all ethods provided good predictions and suice or an initial assessent. It should always be considered that these ethods are only approxiations and are conducive to rather than conclusive in evaluating slabs natural requency. The average ratio o hand calculated values to coputed values is plotted or each respective estiation ethod in igure 3., which indicates the coon high accuracy o all ethods while also coniring the relative inaccuracy o the static delection and odiied static delection ethods. 1. % arithetic ean o ratio hand calculation / FEM 1. % 99.3 % 99. % 98. % 17.8 % 99. % 1. % 11.1% 1.3 % 99.3 % 8. % 8. %. %. %. %. % Equivalent bea ethod Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Estiation or pin supported plates Approxiation by Bachann Approxiation by Blevins estiation ethods Figure 3. Average accuracy o approxiations This inaccuracy could occur as a result o these ethods general application. In contrast to all other tested approxiations speciied or one ixed set-up, both the delection ethods ay be used or all kinds o boundary condition. An even 7

56 . Nuerical Analysis Lukas Wolski ore iportant actor or their less accuracy is their derivation, as shown in 3... Because their origin is derived or the equation o a one-degree-o-reedo syste considering just one ass, it is an extreely approxiation. However, even with variations o 15.% and 7.8%, the estiated values still can be used or a rough prediction o vibration perorance. The use o adapted equations or speciic boundary conditions provides very close values copared to a inite eleent solution. Methods considering dierent kinds o slab paraeters in the present investigation yielded an accuracy o.% (overestiation) and 1.8% (underestiation), results ore than good enough or an approxiate assessent o the undaental requency o slabs.. Nuerical Analysis The speciic qualities o a Cobiax lat slab as copared to a traditional solid slab include a decreased stiness and ass. As these paraeters are two ain actors inluencing natural requency, this change has an ipact on vibration perorance. However, because o the contrary eect o the decreased values it is as yet undeterined how the inal results are aected. An investigation will be perored to clariy this lack o knowledge. A series o detailed investigations will be carried out to evaluate the speciic behaviour o Cobiax slabs in relation to natural requency. Calculations with inite eleent sotware will be undertaken considering a whole range o coon situations in the designs o loors. Paraeters such as geoetry, boundary conditions and loadings will change or each exaple. This variety guarantees the easibility o an overall and universally valid assessent. To evaluate the Cobiax lat slab syste, every exaple is also carried out or conventional solid slabs which allows or a coparison o the slab types and assesses the quality o Cobiax slabs in these conditions. 8

57 . Nuerical Analysis Lukas Wolski.1 Sotware Because this research was undertaken in collaboration with Cobiax Technologies GbH, Gerany, special sotware was provided or the investigation, naely the inite eleent sotware Tornow-Sotware, established in Gerany since The sotware is subdivided into several packages or individual scopes. For this investigation the FEM-Tripla package, developed or the design o loor systes, was applied. A big advantage here was its additional Cobiax odule (Figure.1), specially generated or the design o Cobiax lat slabs. Ater the input o all necessary data, including thickness o the slab and ball diaeter, the corresponding decreasing o ass and stiness are taken into consideration. Figure.1 Cobiax odule The package has dierent set-ups or investigating natural requency and is capable o calculating up to 1 natural requencies using dierent approaches. The ollowing investigation considers the irst three requencies with the ain ocus on undaental requency. Analysis o the natural requencies is calculated using the Lanczos algorith, an iterative algorith which eploys the Lanczos recursion. This is a process o deining or expressing a unction or the solution to a proble in ters o itsel, by producing a recursive unction. However, it is also a very powerul solver and well known as an eicient ethod o inding eigenvalues and eigenvectors o atrices. The Lanczos procedure is generally 9

58 . Nuerical Analysis Lukas Wolski used or large sparse atrices. Cullu and Willoughby (1985) suarised the basic steps in any Lanczos procedure as shown below. 1. Transor a given syetric atrix A into a aily o syetric tridiagonal atrices o varying sizes. Copute eigenvalues and eigenvectors o certain ebers o this aily 3. Take soe or all o these eigenvalues as approxiations to eigenvalues o atrix A and ap the corresponding eigenvectors o the tridiagonal atrix into Ritz vectors or atrix A. Use these Ritz vectors as approxiations to the eigenvectors o A The accuracy o the calculated requencies is set to 1-5 and is also iproved by a sall and detailed esh.. Veriication o Sotware Accuracy In ters o the obtained values accuracy, an initial test is carried out. A proo in or o a coparison between Tornow-Sotware and a second inite eleent sotware will clariy that Tornow-Sotware provides exact and accurate values. For this purpose a slab was odelled with both types o sotware and its undaental requency as well as its ode shape was calculated. The used sotware is called RFEM.1 established by Dlubal Sotware. It is a inite eleent sotware applicable or a wide range o tasks in structural engineering. The coparison consists o an exaple considering ollowing paraeters: - One span solid slab - All edges are siply supported - 1 x 1 x.3 - E 8,3 N/² ; ν. (C3/37) - Iposed load q 5. kn/² 5

59 . Nuerical Analysis Lukas Wolski The results gained o both calculations are presented in igure. and igure.3 below. Fundaental requency : Hz Figure. 1.Mode shape obtained by Tornow-Sotware Fundaental requency : 7.18 Hz Figure.3 1.Mode shape obtained by RFEM 51

60 . Nuerical Analysis Lukas Wolski The conclusion o this coparison is unabiguous. Both estiated undaental requency are nearly the sae value. Only a very little dierence o less than.% is realised which could be inluenced by dierences in eshing the slab or dierent calculation approaches. Nevertheless, this initial coparison leaves no doubt in the exactness o values described in this chapter. Furtherore the good correlation between coputed vales used Tornow-Sotware and hand-calculated vales in chapter 3 is an additional indicator o the well perored accuracy o Tornow-Sotware..3 General Settings The ost iportant actors in the provision o a correct evaluation are realistic and coparable values. Soe initial settings were carried out with the intention o ensuring these necessary conditions. By applying coon aterial properties, geoetries and construction ors which are used in practice the need to be realistic is satisied. As ar as the coparability requireent is concerned, constant estiations throughout the entire investigation will provide a good solution. The investigation includes seven coon types o slabs in construction: Exaple 1: Exaple : Exaple 3: Exaple : Exaple 5: Exaple : Exaple 7: Siply supported slab, one-way spanning Siply supported slab, two-way spanning span slab, two-way spanning 3 span slab, two-way spanning 1x1 bay slab, supported by coluns x1 bay slab, supported by coluns 3x3 bay slab, supported by coluns All line supports used in this investigation are considered to be siply supported without any restraints. Coluns odelled in exaples 5 7 are assued to be pinned, also without any restraints. 5

61 . Nuerical Analysis Lukas Wolski For each o these seven systes a range o dierent geoetries are regarded. These include changes in length o spans (-17) and dierent width o spans (-17). The proper thickness was obtained with inoration oered on Cobiax s website () and is provided in appendix C. One good resource here was a diagra presenting aongst other things the interrelationship between slab length and necessary thickness as well as ball diaeter. Furtherore, a list o already existing projects including loor geoetries and ball diaeters was used to estiate an appropriate deck thickness. Depending on this thickness, which ranges ro between 3c and c, a proper Cobiax hollow sphere (Ø.5c- Ø5c) was used. A detailed description including all paraeters is given in the corresponding tables in appendix B. As regards aterial properties, realistic ters are assued by using a concrete providing a quality o C3/37. As this is an averaged concrete quality, coon in design, it will deliver useul values. Reerring to DIN15-1, 9.1.7, its ean Young's odulus is 8,3 N/². The second aterial quality, the Poisson ratio, is supposed to be. or the concrete used in both types o slabs. According to DIN 155-1, 5.1, the unit weight or reinorced concrete is 5. kn/³, the value used to calculate the dead load. In addition, each slab is also loaded with 1.5 kn/² to represent loading due to possible use o coating. For urther loadings it is iportant to consider the wide range o structures Cobiax lat slabs are suitable or, including residences, oices and car parks. Thus, rather than liiting this investigation to one design situation, as wide an application ield as possible is regarded. The iposed loads are staged according to this purpose. A loading o 5. kn/² is stepwise decreased by degrees o 5%, leading to urther loads o 3.75 kn/²,.5 kn/², 1.5 kn/², and inally kn/². In addition to the dead load these ive dierent loadings are used or each slab, providing a variety o possible set-ups. 53

62 . Nuerical Analysis Lukas Wolski. Analysis o Results The overall result o this investigation is that or all types o slabs, including their entire range o varying diensions, Cobiax lat slabs reached higher and thereore better natural requencies than traditional solid slabs with the sae geoetries. The absolute values range ro. Hz (x1 bay slab, 3c, 5. kn/²) to Hz ( span slab, 3c, kn/²). The trend o undaental requencies or each exaple o a single span, two-way spanning slab is plotted in igure.. To present a better view, only two load situations ( kn/² and 5. kn/²) are displayed. It is noticeable that by increasing the applied load, the dierence o the absolute values or natural requency between Cobiax slabs and solid slabs decreases but, as will be shown shortly, their ratios stay constant. Furtherore a decreasing and siultaneously a narrowing o requencies is observable i the slab systes becoe larger undaental requency [Hz] x x 8x 8x8 1x 1x8 1x1 1x1 1x1 15x1 15x1 17x15 17x17 diension [] : q kn/² : q kn/² : q 5. kn/² : q 5. kn/ ² Figure. Fundaental requencies or a single span slab With larger spans, the natural requency becoes reduced or both types o slabs, resulting in saller dierences between the values. This is explained by the constant ratio o requency o Cobiax slabs to solid slabs. Provided that the slab thickness and ball diaeter o a Cobiax slab stay constant, the reduction o 5

63 . Nuerical Analysis Lukas Wolski stiness and ass will not change either. This leads to a speciic and constant value or each slab in which both types dier regardless o their geoetry. This eans that the ratio cs / ss o a x slab is equal to any other slab diension, as long as its deck is 3c thick and a sphere with a diaeter o.5c is included. I the relative values are regarded, it is not necessary to consider all exaples o a slab but only its dierent thicknesses. This leads to an iproved view o the results as shown in igure.5. The relationship between deck thickness, applied loads and the resulting dierence in undaental requency is plotted. This consideration has the advantage o allowing the possibility o evaluating vibration perorance coparatively. 1.% increased o copared to 1.% 8.%.%.%.% q kn/² q 1.5 kn/² q.5 kn/².% c q 3.75 kn/² iposed loading c deck thickness 3 c q 5. kn/².%-.%.%-.%.%-.%.%-8.% 8.%-1.% 1.%-1.% Figure.5 3-D view o requency dependency The three-diensional shape provides inoration on how the vibration behaviour o Cobiax lat slabs as copared to conventional slabs iproves by increasing load and thickness. The sloped surace with increasing gradients towards higher thicknesses as well as lesser loadings indicates that Cobiax slabs increase their advantage o higher requencies or these two changes. In the best case, when a c thick loor is loaded according only to its own sel-weight, the undaental 55

64 . Nuerical Analysis Lukas Wolski requency o a Cobiax slab is 11.9% higher than its solid equivalent. Its iniu advantage o 3.% is obtained ro the Cobiax systes or a 3c deck with a 5. kn/² applied load. These two values deine the range along which all others coparisons are located. An extensive suary o all dierences is shown in table.1. 3c (Ø.5c) c (Ø31.5c) c (Ø5c) q kn/² 1.1% 11.% 11.9% q 1.5 kn/² 7.7% 9.% 1.3% q.5 kn/².% 7.5% 9.% q 3.75 kn/².%.% 7.9% q 5. kn/² 3.% 5.% 7.% Table.1 Cobiax advantage related to loads and thickness The values o this table are also plotted in the shape o soothed curves in igure.. In addition to the latter igure and table these three curves and thereore the predoinance o Cobiax slabs also clariy their decrease with increental loading. The point o intersection o the curves and the x-axis is worthy o consideration. This point would provide the aount o loading at which Cobiax slabs will achieve the sae natural requencies as solid slabs. For any point below the ratio would change and the solid slabs would gain higher natural requencies. 1.% 1.% 1.% advantage o 8.%.%.%.%.% kn/² 1 kn/² kn/² 3 kn/² kn/² 5 kn/² kn/² loading 3c (Ø.5c) c (Ø31.5c) c (Ø5c) Figure.: Cobiax advantages against loading 5

65 . Nuerical Analysis Lukas Wolski I an approxiation is applied, it is possible to express the alling trend with the help o a cubic equation able to deliver knowledge concerning the urther run. Aterwards these equations can be used to calculate the approxiate intersection o curve and x-axis. This will be the boundary or which Cobiax lat slabs have an iproved vibration behaviour copared to solid loors. h 3c : y -.1x 3 +.x -.15x +.15 h c : y -.1x x -.195x h c : y -*1-5 x 3 +.1x -.137x These equations yield axiu values or unior loadings o (including 1.5 kn/² due to coating): h 3c : ax. q 15.1 kn/² h c : ax. q 1. kn/² h c : ax. q 1.8 kn/² Again, this can be expressed by ratios o applied load to sel-weight: h 3c : q/g.95 h c : q/g 1.83 h c : q/g 1. These values are valid or all Cobiax slabs with thicknesses o 3c, c or c including a sphere diaeter o.5c, 31.5c or 5c, regardless o their boundary condition. Because these values are gained by extrapolated calculations, a coniration is necessary. For this purpose additional inite eleent calculations were carried out, including the critical loadings. All exaples showed very good high siilarity between both undaental requencies. Exeplary table. present results or a one span slab siply supported on all our edges. With the utost probability, the slightly variance o these nubers occur due to the approxiation during the extrapolation procedure. 57

66 . Nuerical Analysis Lukas Wolski Diension [Hz] [Hz] ratio / 8 x 8 x x 1 x x 17 x Table. Accuracy o critical load or one span slab Another check o the accuracy is given in igure.7. This chart has plotted the direct relationship o applied load and resulting undaental requency or both types o slab. In this case, the values relate to a continuous slab with two equal spans and the sae length-width ratio. Like beore, it is noticeable that the obtained critical value or 3c slabs provides the highest accuracy which is indicated by the very close intersection o both upper curves relating to the predicted value (dashed line). But even i the approxiation according to c and c slabs are less accurate, the variance o % is still accurate enough to evaluate the relationship between both types o slab undaental requency [Hz] applied load [kn] : 8 x 8 x 3c : 8 x 8 x 3c : 1 x 1 x c : 1 x 1 x c : 17 x 17 x c : 17 x 17 x c Figure.7 Accuracy o critical load or continuous slab The change in ratio between the dierent types o slab is caused by the decreasing relevance the Cobiax slabs reduced sel-weight. I the applied loadings are siilar to the sel-weight, the reduction o approxiately 3% ass is an advantage or Cobiax slabs. However, once the applied loads increase, the selweight becoes just a sall aount o the overall load and is thereore negligible. I this occurs, the only dierence on the part o Cobiax slabs is the reduced stiness which has a negative ipact on the vibration behaviour and thereore leads to lower natural requencies. 58

67 . Nuerical Analysis Lukas Wolski.5 Conclusion An evaluation o Cobiax lat slabs () copared to traditional solid slabs () was required. For this purpose an investigation o 9 dierent slabs including changes in type (/), diensions and boundary condition was carried out. The overall conclusion is that Cobiax lat slabs possess higher undaental requencies or all investigated cobinations. However, it was also presented that this only occurs i a speciic load to sel-weight ratio exists. Indeed, Cobiax slabs lose their advantage o ass reduction ater a certain point; but due to the aount o loadings required to realise this, its general perorance is not aected. More precisely, the usual ields o application or Cobiax slabs are oices, public buildings and car parks which could all be expected to bear a typically lesser iposed load than the critical values. Table.3 shows soe ranges o loadings which should be considered in these areas according to the British as well as Geran Standard. Type o structure Uniorly distributed load [kn/²] BS (199) DIN () Residence Oice and siilar use Public areas Car parks (vehicles 5 kn) Table.3 Extract o iposed loads in BS and DIN The provided values in both national Standards are all less than the iniu critical value o q 1. kn/² or a c thick Cobiax slab. For this reason the Cobiax syste is supposed to gain higher natural requencies than conventional solid slabs in all its projects. 59

68 5. Conclusions and Recoendations Lukas Wolski 5. Conclusions and Recoendations 5.1 General Conclusions This research presents an overall view o natural requency o concrete slabs in structural engineering. Although it is a very large ield o consideration, already existing literature and research done in past provide good knowledge. In case o huan acceptance due to loor vibration a variety o recoendation is available. By people like Reiher and Meister (1931), Wiss and Parelee (197) or Brownjohn (1) any investigations were carried out to iprove the knowledge o this topic. Nowadays, with help o the wide range o recoendation, no signiicant coplaints should occur. This conclusion is supported by dierent case studies including annoying loor vibration, as shown by Bachann (199) and Hanagan (5). All cases describing vibration coplaints conir very low natural requencies o their loors and thereore a high risk o perceptible otions. Approxiate hand-calculations presented in chapter three are a very well-know area. Due to the act that coputer supported calculations were not available, natural requencies had to be estiated by hand calculations in the past. The high aount o existing literature gives an easy access and considers coon slab types. In ters o accuracy o these sipliications perored coparisons o hand-calculated values and solutions obtained by inite eleent sotware conired their high quality. Even i soe ethods include sall inaccuracy, it is still possible to use the as an initial estiation. With all existing coputer sotware nowadays these ethods are used or rough estiations anyway. Chapter our containing the ain issue o this research shows the iproved vibration perorance o Cobiax lat slabs copared to conventional solid slabs. A detailed investigation clariied the dierent eects o the reduced weight and stiness o Cobiax slabs. It is shown that due to their lower dead load Cobiax lat slabs achieve higher natural requencies or coon practical use. However, with an increasing o iposed load this advantage decreases and ater a certain point

69 5. Conclusions and Recoendations Lukas Wolski Cobiax slabs show lower and thereore worse natural requencies. This change happens because the negative reduction o stiness reains constant while ater increasing the ratio o applied load to sel weight the positive reduction o ass decreases constantly until it becoes negligible. An estiation o these critical values indicates that because o the high aount o iposed loadings necessary to achieve this change, the vibration behaviour is still passable. All application areas Cobiax lat slabs used to ocus show less iposed load than the critical values obtained. 5. Areas o Future Research Because the act that vibration perorance o structures covers very large ield o consideration, urther investigation are possible. According to Lenzen (19) one urther iportant actor is the daping. A high daping value is necessary to reduced loor vibration during its irst cycles to avoid huan perception. Due to its reduced weight, the daping ratio is changed and leads to other dierences between Cobiax slabs and traditional solid slabs. Another point which should be regarded is the thickness variety o Cobiax slabs. Because each sphere diaeter ay be used or dierent thicknesses, the load reduction changes as well. Again, this leads to individual critical values or each slab-thickness ratio. I, or exaple, a.5c diaeter ball is located inside a c slab instead o a 3c, the load reduction decreases ro approxiate 3% to %. According to chapter our this would decrease the critical value at which Cobiax slabs lose their advantage copared to solid slabs. 1

70 Reerences Lukas Wolski Reerences Bachann H. and Aann W. (1987) Vibration in structures: Induced by an and achines, International Association or Bridge and Structural Engineering (IABSE) Bachann H. (199) Case studies o structures with an-induced vibration, Journal o Structural Engineering, 118(3), 31-7 Bachann H. et al. (1995) Vibration probles in structures: practical guidelines, Brinkhäuser Verlag Bilbao S. () Tioshenko's bea equations [8 August ] Blevins R.D. (1979) Forulas or natural requency and ode shape, Krieger Publishing Copany Bolton A. (1978) Natural requencies o structures or designers, The Structural Engineer, 5A(9), 5-53 Bolton A. (199) Structural dynaics in practice: A guide or proessional engineers, McGraw-Hill British Standards Institution (199) BS 7, Guide to evaluation o huan exposure to vibration in buildings (1 Hz to 8 Hz) British Standards Institution (199) BS Loading or buildings; Part 1: Code o practice or dead and iposed loads Brownjohn J.M.W. (1) Energy dissipation ro vibrating loor slabs due to huan structure interaction, Shock and Vibration, 8(), Clough R.W. and Penzien J. (1993) Dynaics o structures, nd edn. McGraw-Hill

71 Reerences Lukas Wolski Cobiax () Planungshile: Cobiax Flachdecken, Cobiax Technologies AG Cobiax () Steiigkeitsaktoren zur Berücksichtigung der Verinderung durch Hohlkörper, Cobiax Technologies AG Cobiax () Lightweight solutions in concrete [8 August ] Concrete Society Working Party (5) Post-tensioned concrete loor; Design handbook, The Concrete Society Cooney R.C. and King A.B. (1988) Serviceability criteria or buildings, Branz study report, Building Research Association o New Zealand Crist R.A. and Shaver J.R (197) Delection perorance; Criteria or loors, U.S. Departent o Coerce, National Bureau o Standards Cullu J.K. and Willoughby R.A. (1985) Lanczos algoriths or large syetric eigenvalues coputations, Brickhäuser-Boston Deutsches Institut ür Norung (1) DIN 15-1, Tragwerke aus Beton, Stahlbeton und Spannbeton; Teil 1: Beessung und Konstruktion Deutsches Institut ür Norung () DIN 155-3, Einwirkungen au Tragwerke; Teil 3: Eigen- und Nutzlasten ür Hochbauten Deutsches Institut ür Norung (1) DIN 15-1, Erschütterungen i Bauwesen; Teil 1: Vorerittlung von Schwingungsgrößen Deutsches Institut ür Norung (1999) DIN 15-, Erschütterungen i Bauwesen; Teil : Einwirkungen ür Hochbauten Elishako I.B. (197) Vibration analysis o claped square orthotropic plate, Aerican Institute o Aeronautics and Astronautics Journal, 1,

72 Reerences Lukas Wolski Ead El-Dardiry E. et al. () Iproving FE odels o a long-span lat concrete loor using natural requency easureents, Coputers and Structures, 8, Fisher J.M. and West M.A. (1) Serviceability design considerations or lowrise buildings, Aerican Institute o Steel Construction Griin M.J. (199) Handbook o huan vibration, Acadeic Press Ltd. Hanes R.M. (197) Huan sensitivity to whole-body vibration in urban transportation systes: A literature review, Applied Physics Laboratory, The Johns Hopkins University, Maryland. Hearon R.F.S. (1959) The requency o lexural vibration o rectangular orthotropic plates with claped or supported edges, Journal o Applied Mechanics,, Jeary A. (1997) Designer s guide to the dynaic response o structures, E & FN Spon Leissa A.W. (1973) The ree vibration o rectangular plates, Journal o Sound and Vibration, 31(3), Linda M.and Hanagan L.M. (5) Walking-Induced Floor Vibration Case Studies, Journal o Architectural Engineering, 11(1), 1-18 Magrab E.B. (1977) Natural requencies o elastically supported orthotropic rectangular plates, Journal o the Acoustical Society o Aerica, 1(1), Mazudar J. (1971) Transverse vibration o elastic plates by ethod o constant delection lines, Journal o Sound and Vibration, 18(), Mindlin R.D., Schacknow A. and Deresiewicz H. (195) Flexural vibration o rectangular plates, Journal o Applied Mechanics, 3, 3-3

73 Reerences Lukas Wolski Morrison J.B. () Huan response to vibration [15 August ] Osborne K.P. and Ellis B.R. (199) Vibration design and testing o a long-span lightweight loor, The Structural Engineer, 8(1), Ove Arup & Partners () Typical hospital loor; Vibration assessent, Bubbledeck UK Ltd. Pernica G. and Allen D.E. (198) Floor vibration easureents in a shopping centre, Canadian Journal o Civil Engineering, 9(), Pernica G. and Allen D.E. (1998) Control o Floor Vibration [11 August ] Peer K. () Untersuchung zu Biege- und Durchstanztragverhalten von zweiachsigen Hohlkörperdecken, VDI Verlag Reed R.E.Jr. (195) Coparison o ethods in calculating requencies o corner supported rectangular plates, NASA Technical Note, NASA TN D-33 Reiher H. and Meister F.J. (1931) Die Epindlichkeit des Menschen gegen Erschütterungen, Forschung au de Gebiet des Ingenieurwesens, (11), Schnellenbach-Held M. (3) Transverse orce capability o the BubbleDeck, Expert Report, Institute or Solid Construction, Technical University Darstadt Tredgold T. (188) Eleentary Principles o Carpentry, nd Ed., J. Taylor Warburton G.B. (195) The vibration o rectangular plates, Proceedings o the Institution o Mechanical Engineers, 18,

74 Reerences Lukas Wolski Weber B. () Vorlesung Tragwerksdynaik, Institut ür Baustatik und Konstruktion, ETH Zürich Wikipedia () Rayleigh-Ritz-Prinzip [15 August ] Wikipedia () Rayleigh-Ritz ethod [15 August ] Willias M.S. et al. (1993) Coparison o inite eleent and ield test results or the vibration o concrete loors, Int. Con. on Structural Dynaics Modelling: Test, Analysis and Correlation, Cranield, Wiss J.F. and Parelee R.H. (197) Huan Perception o Transient Vibrations, Journal o the Structural Division, ASCE, 1(ST), Young Benidickson V. (1993) Dealing with Excessive Floor Vibrations [11 August ]

75 APPENDIX A Lukas Wolski APPENDIX A General Assuption...8 Exaple Exaple...71 Exaple Exaple...7 Exaple Exaple...8 Exaple Exaple Exaple Exaple Exaple Exaple Suary o results...9 7

76 APPENDIX A Lukas Wolski General Assuption The accuracy o the sipliied hand calculation ethod proposed in chapter 3. was checked by the ollowing exaple. To obtain coparable values, all ethods within each exaple dealt with the sae set-up. It was assued that all loors were built using the Cobiax lat slab syste. For the calculation o the undaental requency no urther loadings besides the sel-weight were iplied. The aterial qualities were chosen or a C3/37 concrete with a Young's odulus o 8 3 N/² and a density o 5 kn/³. Depending on the geoetry o each syste, the ball size diaeter was.5c or 31.5c. For these two Cobiax lat slab systes speciic qualities such as stiness reduction and deal load were considered according to Appendix C. The boundary conditions included siply supported one-way as well as two-way spanning loors, and pin supported 1x1 and 1x bay slabs. The only value which changed during this coparison was the Poisson ratio. In general all exaples used a Poisson ratio o., but because two o the tables used or the estiation o pin supported slabs iply a Poisson ratio o.3 these exaples had to be odiied and the Poisson ratio was increased. Furtherore, it is iportant that every exaple used only suitable estiations or its boundary condition. A coparison o speciic ethods to provide appropriate solutions or dierent boundary conditions was waived on this occasion. 8

77 APPENDIX A Lukas Wolski Exaple 1: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Equivalent bea ethod: π a E I π 8. N 3 8,3 1 (.3 1.) 1 51 kg Hz Static delection ethod: g 9.81 s π δ stat π.98 Hz Modiied static delection ethod: g 9.81 s π δ stat π 8.91 Hz 9

78 APPENDIX A Lukas Wolski Approxiation by Jänich: N π D g K π s h γ N.3 17,33 N Hz Approxiation by Blevins: N λ E h ,3 1.3 π a 1 (1 -ν ) π kg ( 1 -. ) Hz Coparison: Equivalent bea ethod Static delection ethod Modiied static delection ethod Approxiation by Jänich Approxiation by Blevins FEM calculated value 8.9 Hz.98 Hz 8.91 Hz 8. Hz 8. Hz Hz Ratio %: hand calculation / FEM 99. % 85. % 19. % 11.3 % 98.8 % --- 7

79 APPENDIX A Lukas Wolski Exaple : General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Equivalent bea ethod: π a E I π 15. N 3 8,3 1 (.3 1.) 1 79 kg Hz Static delection ethod: g 9.81 s π δ stat π.7 Hz Modiied static delection ethod: g 9.81 s π δ stat π 3.5 Hz 71

80 APPENDIX A Lukas Wolski Approxiation by Jänich: N π D g K π s h γ N. 1,5 N Hz Approxiation by Blevins: N λ E h ,3 1. π a 1 (1- ν ) π kg ( 1 -. ) Hz Coparison: Equivalent bea ethod Static delection ethod Modiied static delection ethod Approxiation by Jänich Approxiation by Blevins FEM calculated value 3.9 Hz.7 Hz 3.5 Hz 3.15 Hz.99 Hz 3.1 Hz Ratio %: hand calculation / FEM 99.5 % 8.9 % % 11. % 9.3 % --- 7

81 APPENDIX A Lukas Wolski Exaple 3: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Equivalent plate approach: D 1 π a 1 + b π N 1 51 kg Hz Concrete Society ethod: λ x n x a EI b EI y x N N k x λx. ' x k EI π y π x. b N 51 kg Hz 73

82 APPENDIX A Lukas Wolski Static delection ethod: g 1 π 9.81 s.1 1 π δ stat 13.1 Hz Modiied static delection ethod: g 1.77 π 9.81 s π δ stat 1.78 Hz Approxiation by Jänich: N π D g K π s h γ N.3 17,33 N Hz Approxiation by Hearon: 1 π A 1 a D D B + b CD3 + a b 1 π π N π N π N kg 1.5 Hz Approxiation by Bachann: ϕ a N E h 3.1 8, (1 -ν ) kg ( 1-. ) n 1.51 Hz 7

83 APPENDIX A Lukas Wolski Approxiation by Blevins: N ,3 1 λ E h π a 1 (1 -ν ) π kg ( 1 -. ) Hz Coparison: Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Approxiation by Bachann Approxiation by Blevins FEM calculated value 1.5 Hz 1.5 Hz 13.1 Hz 1.78 Hz 1.5 Hz 1.5 Hz 1.51 Hz 1.5 Hz 1.7 Hz Ratio %: hand calculation / FEM 1.3 % 1.3 % 79.8 % 11.9 % 1.3 % 1.3 % 1.3 % 1.3 %

84 APPENDIX A Lukas Wolski Exaple : General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Equivalent plate approach: π D 1 a 1 + b π N 1 79 kg Hz Concrete Society ethod: λ x n x a EI b EI y x N N k x λx 1. ' x k EI π y π x 1. b N 79 kg Hz 7

85 APPENDIX A Lukas Wolski Static delection ethod: g 9.81 s π δ stat π 8.3 Hz Modiied static delection ethod: g 9.81 s π δ stat π 1.51 Hz Approxiation by Jänich: N π D g K π s h γ N. 1,5 N Hz Approxiation by Hearon: 1 π A 1 a D D B + b CD3 + a b 1 π π N π N π N kg 1. Hz Approxiation by Bachann: ϕ a N E h 5.1 8, (1 -ν ) kg ( 1-. ) n 1.3 Hz 77

86 APPENDIX A Lukas Wolski Approxiation by Blevins: N ,3 1 λ E h 3.8. π a 1 (1 -ν ) π kg ( 1 -. ) Hz Coparison: Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Approxiation by Bachann Approxiation by Blevins FEM calculated value 1. Hz 1. Hz 8.3 Hz 1.51 Hz 1. Hz 1. Hz 1.3 Hz 1. Hz 1.3 Hz Ratio %: hand calculation / FEM 99. % 9.9 % 79.7 % 11.8 % 99. % 99. % 99.1 % 99. %

87 APPENDIX A Lukas Wolski Exaple 5: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Equivalent plate approach: D π a b π N 1 79 kg Hz Concrete Society ethod: λ x n x a EI b EI y x N N k x λx. ' x k EI π y π x. b N 79 kg Hz 79

88 APPENDIX A Lukas Wolski Static delection ethod: g 9.81 s π δ stat π 5.5 Hz Modiied static delection ethod: g 9.81 s π δ stat π. Hz Approxiation by Jänich: N π D g K π s h γ N. 1,5 N Hz Approxiation by Hearon: 1 π A 1 a D D B + b CD3 + a b 1 π π N π N π N kg.3 Hz Approxiation by Bachann: ϕ a N E h 3.1 8, (1 -ν ) kg ( 1 -. ) n.9 Hz 8

89 APPENDIX A Lukas Wolski Approxiation by Blevins: N ,3 1 λ E h π a 1 (1 -ν ) π kg ( 1 -. ).88.3 Hz Coparison: Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Approxiation by Bachann Approxiation by Blevins FEM calculated value.3 Hz.18 Hz 5.5 Hz. Hz.3 Hz.3 Hz.9 Hz.3 Hz.35 Hz Ratio %: hand calculation / FEM 99. % 97.3 % 79.5 % 11. % 99. % 99. % 99. % 99. %

90 APPENDIX A Lukas Wolski Exaple : General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :.3 Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Static delection ethod: 9.81 s.85 1 g 1 π δ S π 5.3 Hz Modiied static delection ethod: 9.81 s g 1.77 π δ S π.8 Hz Estiation or pin supported plates: λ π γ h 1.71 π 8. 17,33 N N 9.81 s a D g 1.1 Hz 8

91 APPENDIX A Lukas Wolski Approxiation by Blevins: λ 3 E h (1- ν ) π 8. N 3 8, kg ( 1-.3 ).89 π a 3.1 Hz Coparison: Static delection ethod Modiied static delection ethod Estiation or pin supported plates Approxiation by Blevins FEM calculated value 5.3 Hz.8 Hz.1 Hz.1 Hz.1 Hz Ratio %: hand calculation / FEM 87.8 % 11.1 % 1.3 % 1.3 %

92 APPENDIX A Lukas Wolski Exaple 7: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Static delection ethod: 9.81 s g 1 π δ S π. Hz Modiied static delection ethod: g 9.81 s π δ S π 3.3 Hz Estiation or pin supported plates: λ π γ h π 15. 1,5 N N 9.81 s a D g 1 3. Hz 8

93 APPENDIX A Lukas Wolski Approxiation by Blevins: λ 3 E h (1- ν ) π 15. N 3 8, kg ( 1-.3 ).88 π a 3.93 Hz Coparison: Static delection ethod Modiied static delection ethod Estiation or pin supported plates Approxiation by Blevins FEM calculated value. Hz 3.3 Hz 3. Hz.93 Hz.93 Hz Ratio %: hand calculation / FEM 88.7 % % 13.1 % 1. %

94 APPENDIX A Lukas Wolski Exaple 8: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :.3 Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Static delection ethod: 9.81 s.59 1 g 1 π δ S π. Hz Modiied static delection ethod: 9.81 s g 1.77 π δ S π.1 Hz Estiation or pin supported plates: λ π γ h 1.71 π 15. 1,8 N N 9.81 s a D g 1.33 Hz 8

95 APPENDIX A Lukas Wolski Approxiation by Blevins: 8,3 1 N 1 79 kg λ E h 7.1 π a 1 (1 ν ) π 15. (1.3 ).3 Hz Coparison: Static delection ethod Modiied static delection ethod Estiation or pin supported plates Approxiation by Blevins FEM calculated value. Hz.1 Hz.33 Hz.3 Hz.331 Hz Ratio %: hand calculation / FEM 87.5 % 11. % 1. % 1. %

96 APPENDIX A Lukas Wolski Exaple 9: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Approxiation by Bachann: λ n π.1 π 51 8,3 1 kg N 1 μ 3 EI Hz Approxiation by Jänich: 1 π D g ( ) N ai Ki π s h γ ( a ).3 17,33 N i Ni Hz Coparison: Approxiation by Jänich Approxiation by Bachann FEM calculated value 5.9 Hz 5.5 Hz 5.11 Hz Ratio %: hand calculation / FEM 11.5 % 1.7 %

97 APPENDIX A Lukas Wolski Exaple 1: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Approxiation by Bachann: λ n π μ EI 1.55 π 79 kg 8,3 1 N Hz Approxiation by Jänich: N s. 1,5 N π D g ( a ) i Ki π h γ ( ai Ni) 1.73 Hz Coparison: Approxiation by Jänich Approxiation by Bachann FEM calculated value.73 Hz 3.87 Hz Hz Ratio %: hand calculation / FEM 15. % 1. %

98 APPENDIX A Lukas Wolski Exaple 11: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: 3 c Cobiax slab properties: Ball size Ø :.5c Factor o stiness reduction:.89 Dead load reduction:.39 kn/² Approxiation by Jänich: N s.3 17,33 N 3 5. π D g ( a ) i Ki π h γ ( ai Ni) 1.58 Hz Coparison: Approxiation by Jänich FEM calculated value 11. Hz Hz Ratio %: hand calculation / FEM 19. % --- 9

99 APPENDIX A Lukas Wolski Exaple 1: General properties: Concrete: C3/37 Young's odulus E : 8,3 N/² Poisson's ratio ν :. Density γ: 5 kn/³ Thickness h: c Cobiax slab properties: Ball size Ø : 31.5c Factor o stiness reduction:.88 Dead load reduction: 3.3 kn/² Approxiation by Jänich: π D g K π N s. 1,5 N h γ N Hz Coparison: Approxiation by Jänich FEM calculated value 1.58 Hz Hz Ratio %: hand calculation / FEM 1. %

100 APPENDIX A Lukas Wolski Equivalent bea ethod Equivalent plate approach Concrete Society ethod Static delection ethod Modiied static delection ethod Approxiation by Hearon Approxiation by Jänich Estiation or pin supported plates Approxiation by Bachann Approxiation by Blevins FEM calculated values [Hz] Exaple Ratio %: hand calculation / FEM 99. % 85. % 19. % 11.3 % 98.8 % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 99.5 % 8.9 % % 11. % 9.3 % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 1.3 % 1.3 % 79.8 % 11.9 % 1.3 % 1.3 % 1.3 % 1.3 % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 99. % 9.9 % 79.7 % 11.8 % 99. % 99. % 99.1 % 99. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 99. % 97.3 % 79.5 % 11. % 99. % 99. % 99. % 99. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 87.8 % 11.1 % 1.3 % 1.3 % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 88.7 % % 13.1 % 1. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 87.5 % 11. % 1. % 1. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 11.5 % 1.7 % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 15. % 1. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 1. % calculated values [Hz] Exaple Ratio %: hand calculation / FEM 19. % arithetic ean in % 99.3 % 99. % 98. % 8. % 17.8 % 99. % 1. % 11.1 % 1.3 % 99.3 % standard deviation Table A.1 Suary o approxiate hand calculation 9

101 APPENDIX B Lukas Wolski APPENDIX B (Calculated Values by FEM) Siply supported slab, one-way spanning...9 Siply supported slab, two-way spanning...95 span slab, two-way spanning span slab, two-way spanning x1 bay slab, supported by coluns...98 x1 bay slab, supported by coluns x3 bay slab, supported by coluns

102 h sphere Ø a q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode Table B.1 Siply supported slab, one-way spanning b 9

103 sphere h a b Ø q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode Table B. Siply supported slab, two-way spanning

104 h q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² sphere a b Ø 1 a [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode Table B.3 Two span slab, two-way spanning 9

105 h q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² sphere a Ø 1 a b [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode ,798 1,9 1,853 1,3 9,57 8,5 11,159 8,89 8,3 7,515 5,8 7,99,8 1,373 15,815 15,85 1,89 1,1 1, 11,18 9,9 9,37 7,77,1 8,9 7,5 3,89 18,957 1,83 17,577 1,1 1,889 15,99 11,55 1,551 1,83 7, 1,9 8,959 19,73 13,9 11,787 11,7 8,9 7,755 1,391 8,59 7,7,998 5,89 7,597,83 19,1 1,5 1,38 11,37 9,8 9,9 1,819 9,5 8,97 7,1 5,785 7,99 7,88 8,39 17,385 15,8 1,119 11,15 9,98 1,19 1,853 9,85 1,,951 1,3 8,517 17,717 1,35 1,99 1,7 7,88 7,3 9,73 7,7 7,1,575,99 7,5,191,9 13,7 13,31 1,89 8,1 8,55 1,15 8,79 8,3,83 5,35 7,35,79 18,7 1,19 1,331 1,973 1,3 9,7 13,735 1,197 9,31 9,51,53 9,93 8,13 1,1 11,558 1,7 9,797 7,395,755 9,37 7,31,83,,71,95 5,93 17,7 1,3 1,53 1,13 8,8 8, 9,1 8, 7,977,3 5,1 7,3,9,59 15,1 13,39 1,1 9,717 8,98 1,99 9,7 8,733 8,9,178 9,5 7,799 15,95 1,918 9,99 9,5,98,381 8,787,98,91 5,917,7,9 5,71 1,19 11,93 11,83 9,575 7,37 7,578 9,18 7,31 7,589,13,89 7,1,3 3,9 1,3 1,95 13, 9,18 8,17 1,31 9,177 8,38 8,5 5,877 9,13 7,5 18,898 13,17 11,79 11,13 8,1 7,83 1, 7,9 7,1,77 5,1 7,1,9 19, 1,37 1,5 11,53 9,195 9,1 1,3 8,7 8,53,98 5,578 7,515,3 8,9 17, 15,8 15,971 11,53 9,89 1,9 1,5 9,7 9,7,7 9,779 8, 17,78 1,319 1,9 1, 7,88 7, 9,51 7,53 7,9,8,83,88 5,87 18,198 13, 13,355 1,8 8,1 8,55 9,97 8,5 8,18,31 5,97 7,5,,83 1,11 1,3 1,9 1,357 9,71 13,387 9,939 8,997 9,1,35 9,9 7,719 1,718 11,3 1,33 9,858 7,1,797 9,81 7,18,78,115,,55 5,79 17,18 1,713 1,8 1,19 8,135 8,7 9,55 7,887 7,83,38 5,55 7,3,9,81 15,39 13,53 1,19 9,778 8,753 1,775 9,85 8,58 8,791,7 9,113 7,1 15,878 11, 9,81 9,3 7,7,55 8,71,91,7 5,859,9,7 5,51 1,31 1,7 11,97 9,87 7,7 7, 9,59 7,557 7,51,3,8,783,1 3,5 1,7 1,83 13,19 9,87 8,313 1,1 9,88 8,7 8,3 5,8 8,88 7,7 15,15 1,51 9,3 8,93,7,1 8,35,9,179 5,33,58,5 5,339 15,571 11,5 11,7 9, 7,373 7,31 8,79 7,5 7, 5,89,57,583 5,837,89 13,811 1,5 1,85 8,8 7,933 11,78 8,737 7,99 8,98 5,595 8,57 7,1 Table B. Three span slab, two-way spanning 97

106 h sphere Ø q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode ,71 9,55,339 5,,37 3,939 3,33 3,8 3,19,591,1,3 3,193,51 9,5 1,33 15,7 1, 11, 9,1 7,89 8,93 7,3,59 5,18,715 8,71 5,55 3,5 1,31 1,9 1,19 13,18 1,8 7,9 9,188 7,3 7,559,17,718 1,191 5, 1,77 8,87 5,813,77 3,98 3,1 3,57 3,3,971,1,8 1,93 3,3,33 7,9 19,57 13,8 11,9 1,78 8, 7,51 7,99,85,8 5,3,391 8,8 5,375 31, 19,59 15,53 11, 1,89 9,33 7,5 8,55,85 7,39 5,79,393 9,87 5,38 1, 7,88 5,,35 3,8 3,355,8 3,5,791,,11 1,788,899,5 5,18 18,17 1,81 1, 9,919 7,85,55 7,31, 5,71,91,1 7,911 5,13 9,17 18,197 1,31 1,39 11,1 8,57,55 8,39,1,13 5,,18 9,5 5,138 9,379 7,393 5,,159 3,57 3,1,3 3,51,1,1 1,998 1,91,78,13 3,1 17, 1, 9,59 9,3 7,319,1 7,3,9 5,,5 3,93 7,585,9 7,357 17, 13,533 9,1 1,879 8,,1 7,5,9,57 5,11 3,95 8,871,97 8,8,98,78 3,99 3,5,97,51,9,51, 1,91 1,9,7,5,95 1,11 11,375 9,59 8,787,91 5,8,88 5,79 5,11, 3,713 7,9,735 5,8 1,1 1,78 9,7 1,7 7,597 5,8 7,35 5,797 5,95,8 3,715 8,533,739 1,9 8,9 5,7,731 3,93 3,579 3,9 3,31,85,18,18 1,835,85,19,85 19,387 13,97 1,98 1,58 8,35,987 7,,9 5,99 5,5,3 7,787 5,53 31,117 19,1 15,393 1,91 1,37 9,18,991 8,5,1,37 5,5,3 9,17 5,58 9,997 7,88 5,397,33 3,8 3,35,838 3,13,71,7,59 1,73,751,11 5,155 18,17 1,83 1,1 9,91 7,81,57 7,3,7 5,3,791,1 7,58,87 9,158 18,188 1, 1,3 11,595 8,57,551 7,835,78,7 5,5,3 8,78,87 9,38 7,39 5,9,185 3,78 3,1,8 3,,59 1,978 1,95 1,3,59,1 3,79 17,151 1,117 9,5 9,3 7,35,181,91 5,99,98,57 3,837 7,57,79 7,58 17,171 13,17 9,1 1,97 8,93,185 7,77 5,991 5,77 5, 3,839 8,8,713 8,93 7,5,839 3,97 3,33 3,7,55,87,88 1,895 1,88 1,593,57 1,977,555 1,89 11,58 9,1 8,889,995 5,87, 5,739,775,381 3,77 7,9,51,1 1,37 1,933 9,17 1,39 7,8 5,87 7,1 5,7 5,59,81 3,79 8,,55 8,55,73,18 3,793 3,15,89,9,73,39 1,8 1,81 1,53,5 1,919 1,5 15,5 1,98 8,7 8,83,75 5,,37 5,517,591,1 3,535,8,7,99 15,5 1,3 8,75 9,91 7,335 5,5,887 5,519 5,315,8 3,537 7,978,31 Table B.5 1x1 bay slab, supported by coluns a b 98

107 h q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² sphere a b Ø 1 a [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode ,3 5, ,, ,1 1, ,817 5, ,73 5, ,8 11, ,333 5, ,5 5, ,79 1, ,938, ,87, ,893 9, ,1, ,17, ,18 9, , , , , , , , , , , , , , , , Table B. x1 bay slab, supported by coluns 99

108 h q kn/² q 1.5 kn/² q.5kn/² q 3.75 kn/² q 5. kn/² sphere a b Ø 1 a [Hz] [Hz] [Hz] [Hz] [Hz] [c] [c] [] [] [] 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode 1.Mode. Mode 3. Mode Table B.7 3x3 bay slab, supported by coluns 1

109 APPENDIX C Lukas Wolski APPENDIX C (Cobiax Inoration) Existing Cobiax projects...1 Suary o stiness reduction...13 Design considerations

110 APPENDIX C Lukas Wolski project Zollvereinschool Essen (D) span diension h Ø [] [c] [c] ax / 5 3 Hessischer Landtag Wiesbaden (D) ax. 17 / 5 / 5 18 / 31,5 / 3 Mainova Frankurt (D) ax. 1, 3 / 5 / 3 / 35 / 39 / 18 /,5 / 7 / 31,5 Shopping Mall Palladiu Praga (CZ) 8, x 8, / 18 / 7 Novartis Basel (CH) x Newcastle College, Newcastle Tyne & Wear (UK) 7.5 / 8.5 / 7.5 x / 3,5 18 Residential Ubiale Bergao (IT) 1 5 Parking BRG Freistadt (AT) 1, x 5, 55 / 5 Spedition Gebrüder Weiss Maria Lanzendor (AT) 8, in one direction 3 18 SF Swiss Television Zürich (CH) 9, x Peugeot Center Moosseedor (CH) 15 x 1 / 5 31,5 / 3 Wylerpark Bern (CH) 1, x 9,8 3.5 Eclipse Park, Maidstone Kent (UK) x 3 18 Sheield University LCR Sheield (UK) 9 x Iprona Lana (IT) 9, x 7,5 / 31,5 / 5 Coercial Centre Settevalli Perugia (IT) 1,5 5 / 7 3 Table C.1 Existing Cobiax projects 1

111 APPENDIX C Lukas Wolski Steiigkeitsaktoren zur Berücksichtigung der Verinderung durch Hohlkörper Zur Berücksichtigung der Verinderung der Steiigkeit inolge der eingebauten Hohlkörper werden nacholgend Steiigkeitsaktoren ür die Hohlkörperdecke angegeben. Die Werte beruhen au Berechnungen ür den Zustand I bei zentrischer Kugellage. Die Einlüsse ür den Zustand II wurden anhand von Biegeversuchen überprüt. Geäß der Auswertung dieser Versuche ist die Abinderung inolge Zustand I aßgebend. Mit diesen Faktoren kann eine Verorungsberechnung der Decken durchgeührt werden, wobei die reduzierte Eigenlast zu berücksichtigen ist. Deckenstärke h cb [c] 3 * Hohlkörper D cb [c] Verhältniswert h cb /D cb [-] 1,8 1,33 1,39 1, 1,5 1,5 1,1 1,7 1,7 1,78 1,83 1,89 1,9,,11,17, Verhältniswert Ι cobiax /Ι assiv [-],88,89,9,9,9,93,9,9,95,95,9,9,97,97,97,97,97,98 18 Deckenstärke h cb [c] 8 * Hohlkörper D cb [c],5 Verhältniswert h cb /D cb [-] 1, 1,9 1,33 1,38 1, 1,7 1,51 1,5 1, 1, 1,9 1,73 1,78 1,8 1,87 1,91 1,9 Verhältniswert Ι cobiax /Ι assiv [-],87,88,89,9,91,9,93,93,9,9,95,95,95,9,9,9,97,97 Deckenstärke h cb [c] 3 * Hohlkörper D cb [c] 7 Verhältniswert h cb /D cb [-] 1, 1,3 1,33 1,37 1,1 1, 1,8 1,5 1,5 1,59 1,3 1,7 1,7 1,7 1,78 1,81 1,85 1,89 Verhältniswert Ι cobiax /Ι assiv [-],87,88,89,9,91,9,9,93,93,9,9,9,95,95,95,9,9,9 Deckenstärke h cb [c] * Hohlkörper D cb [c] 31,5 Verhältniswert h cb /D cb [-] 1,7 1,3 1,33 1,37 1, 1,3 1, 1,9 1,5 1,5 1,59 1, 1,5 1,8 1,71 1,75 1,78 1,81 Verhältniswert Ι cobiax /Ι assiv [-],88,88,89,9,91,91,9,9,93,93,9,9,9,95,95,95,95,9 Deckenstärke h cb [c] 5 * Hohlkörper D cb [c] Verhältniswert h cb /D cb [-] 1,5 1,8 1,31 1,33 1,3 1,39 1, 1, 3 1,7 1,5 1,53 1,5 1,58 1,1 1, 1,7 1,9 1,7 Verhältniswert Ι cobiax /Ι assiv [-],87,88,89,89,9,9,91,9,9,9,93,93,9,9,9,9,95,95 Deckenstärke h cb [c] 5 * Hohlkörper D cb [c],5 Verhältniswert h cb /D cb [-] 1,8 1,31 1,33 1,3 1,38 1,1 1,3 1, 1,8 1,51 1,53 1,5 1,58 1, 1,3 1,5 1,8 1,7 Verhältniswert Ι cobiax /Ι assiv [-],88,89,89,9,9,91,91,9,9,93,93,93,9,9,9,9,95,95 Deckenstärke h cb [c] 58 * Hohlkörper D cb [c] 5 Verhältniswert h cb /D cb [-] 1,9 1,31 1,33 1,3 1,38 1, 1, 1, 1,7 1,9 1,51 1,53 1,5 1,58 1, 1, 1, 1,7 Verhältniswert Ι cobiax /Ι assiv [-],88,89,89,9,9,91,91,9,9,9,93,93,93,9,9,9,9,9 * epohlene Mindestdeckenstärke Wert bei exzentrischer Hohlkörperlage Cobiax Technologies AG Postach 1 Oberallendstrasse A CH-31 Zug Tel: Fax: Ino.cobiax.co Zweiachsige Hohlkörperdecke Verorungsberechnung Steiigkeitsaktor Anlage zur allgeeinen Bauausichtlichen Zulassung vo xx.xx.xxxx Table C. Factors considering reduced stiness 13

112 APPENDIX C Lukas Wolski Sphere diaeter [c] Min. centre distance [c] Max. aount o spheres [1/²] Recoended deck thickness [c] Dead load reduction per sphere [kn] Max. dead load reduction [kn/²] Stiness actor [-] Shear actor [-] Table C.3 Cobiax paraeters Figure C.1 Coparison o spans and concrete quantity Figure C. Coparison o spans and loads 1