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  Journal of Civil Engineering and Construction Technology Vol. 3(5), pp. 159-178, May 2012 Available online at http://www.academicjournals.org/JCECTDOI: 10.5897/JCECT11.064ISSN 2141-2634 ©2012 Academic Journals   Full Length Research Paper  Development of a wind tunnel model compared withresults of method of initial parameters and BS6399models for dynamic analysis of multi-storey buildingsubjected to aerodynamic loadings Onundi L. O. 1 *, Elinwa A. U. 2 and Matawal D. S. 2   1 Department of Civil and Water Resources Engineering, University of Maiduguri, Borno State, Nigeria. 2  Abubakar Tafawa Balewa University, Bauchi, Nigeria.  Accepted 3 April, 2012 The research investigated the development of a wind tunnel model compared with results of method of initial parameters and BS6399 models for dynamic analysis of a multi-storey building subjected toaerodynamic loadings. The dimensional analysis which based its concept on the law of motion andenergy conservation was used as wind tunnel experimental data assessment tool, for the determinationof the aerodynamic loadings for a multi-storey building. Although, this problem was earlier solvedusing dimensional analysis with different approaches by many Scholars but such solutions did notinclude an important quantity; the influence of the structure’s deflection (δ) and foundation infinitesimalrotation. The study observed that, the non-stationary component is also dependent of the stationarycomponent of the aero-dynamic force. It also explained how the product of the Strouhal number (St)and model relative displacement (L o number),   is equal to the Bernoulli universal constant, 0.5. Tests onmost important parameters capable of influencing the assessment and design of multi-storey buildingswith varying height-breadth ratio, basic wind speeds, relative displacement and aspect ratio increasedwith logarithmic laws as a function of model height; but the reduced frequency, Etha (   and Landa (   decreased with polynomial and power laws respectively. It was therefore concluded that, aero-elasticdamping property of the structure is influenced by the L o number and in particular, λ. Finally, thecumulative base moment from the mathematical (method of initial parameters) model is 2.108% higher than the physical model result and the results from the BS6399 model was less by 1.732%. In general, adifference varying between ±2.1% is within acceptable level of deviation for most conventionalengineering and scientific design results.Key words: Dimensional analysis, Strouhal number, Bernoulli universal constant, wind tunnels, initialparameters, BS6399, multi-storey buildings, aero-elastic damping, aero-dynamics loadings. INTRODUCTION Recent research experiences have shown that there aremany situations where analytical methods cannot beused to estimate certain types of wind loads andassociated structural response (Onundi, 2012). For  *Corresponding author. E-mail: onundii@yahoo.co.uk.  example, when the aerodynamic shape of the building israther uncommon or the building is very flexible so that itsmotion affects the aerodynamic forces acting on it.Buildings are usually considered flexible when any of thesmaller plan dimensions (length or width) divide by heightis greater than five or when the minimum frequency isless than unity. In such situations, more accurateestimates of the pulsating wind effects on such tall  160 J. Civ. Eng. Constr. Technol.structures can be obtained through aero-elastic modeltesting in a boundary-layer wind tunnel. The oscillator of the structural system is naturally wind forces or itsgustiness; but for the purpose of the physical model for the investigation, the wind tunnel served as the oscillator.Therefore, the final result of such a study will be a hybrideffort from the theoretical and laboratory investigations.Therefore, the present study is a comparative study of the pressure and consequently the load distribution on afull-scale building and a wind tunnel tested model. Theconcept of the mathematical model used was based onthe method of Initial parameters (MIP) and a substitutecantilever, whereas the physical model (laboratoryexperiment of wind tunnel tested model) was analyzedbased on the method of dimensional analysis. According to Kramadibrata et al. (2001), dimensionalanalysis has been used widely in solving engineeringproblems. Its application is dependent on listing of alldimensional variables affecting the process in questionand the dimensionless groups obtained. This method canalso be a means of correlating experimental data anddeveloping functional relationships between dimensionalvariables. It has been of immeasurable value in analyzingcomplex engineering problems in many fields’ , notablyfluid mechanics and heat transfer. As far as themechanics of solids are concerned, dimensional analysishas been used in the study of the elastic deformation andvibrations of complex engineering structures(Kramadibrata et al., 2001). It has also been used toestablish the modeling criteria for the scale model testingof coal-face production system (Roxborough andEskikaya, 1974), in subsidence, modeling was referred toby Whittaker and Reddish (1989), and more recently, itsuse in rock excavation and lifting of boring machine wasmentioned by Kramadibrata and Jones (1996) andKramadibrata, et al. (2000), respectively.In fluid mechanics and many other disciplines of science and engineering, similitude, dimensional analysisand modeling can be seen as a conglomeration of usefultools for solving many problems through laboratoryinvestigations. The application of the Buckingham pitheorem is popularly employed for the development of aset of dimensionless variables for a given flow or other engineering phenomena. The use of dimensionlessvariables in data analysis is most importantly applied for the concepts of modeling and similitude to developprediction equations that satisfactorily describe manyinteracting physical, mechanical and chemicalphenomena. Many fluid mechanics and other engineeringproblems are solved by equations and analyticalprocedures. But some rely solely on experimental data(Schmidt and Housen, 2011).Because it is often impractical to conduct experimentsunder the specific conditions desired, one may wish toredesign the experiment to make it more manageable.For example, testing at reduced size scale can provide asignificant cost savings, as is often the case in studies of aerodynamics or fluid mechanics. In other instances, it isnot only impractical but impossible to perform the desiredexperiment. In these cases, one can simulate a prototypeexperiment by designing a model experiment withappropriate test conditions, which may be less expensiveor at least attainable. Dimensional analysis identifies theconditions required for similarity and provides theframework within which the results can be applied to theactual problem of interest. Therefore, dimensionalanalysis is a unique technique used in many fields of engineering to facilitate correlation and interpretation of physical, mechanical and chemical phenomena andexperimental data. It provides a means of combining themany parameters of an experiment into a lesser number of dimensionless groups. This technique greatly reducesthe amount of experimental work needed to determinethe effect of parameter variation on the dependentparameter of the experiment (Kramadibrata, et al., 2001).By definition, similitude involves the measurementsmade on one system or phenomena (for example in theLaboratory) to be used to describe the behavior of other similar systems or phenomena (that is, outside thelaboratory). Laboratory studies use models and thesemodels can be useful to develop or formulate empiricalequations. Models are widely used in studying andsolving fluid mechanics and other engineering problems.The basic definition of engineering model is a repre-sentation of a physical system or phenomena that maybe used to predict the behavior of the system in somedesired respect. The physical system for whichpredictions are to be made is called the prototype. Although computer models conform to this definition, for the purpose of this study; we are interested in physicalmodels, which are herein referred to as models thatresemble the prototype but of different size. A physicalmodel is usually smaller than the prototype.It is useful to highlight certain features that distinguishcommon phenomena. Some typical model studies suchas fluid (wind) flow around the model in the wind tunnel or flow around immersed bodies such as around aircraft;Strouhal number and drag force are important. Fluid flowwith a free surface such as in river, spillways; Fr number is important. All these criteria are referred to as modeldesign conditions, also known as similarity requirementsor modeling laws. Although, the problem of aero-dynamic analysis of force (and moment) acting on multi-storey buildings wasearlier solved using dimensional analysis in differentways by many Scholars and Engineers in the past, for example, Simiu and Scanlan (1978), Simiu and Scanlan(1996) etc; but such solutions did not include animportant quantity which is the influence of the buildings deflection (δ). Hence, their findings are slightly or significantly different from the present result.Therefore, the main objective of the present work is acomparative study of the pressure and consequently theload distribution on a full-scale building and a wind tunnel   tested physical model (that is, laboratory experiment of wind tunnel tested model) analyzed based on the methodof dimensional analysis, with results obtained from amathematical model that used the method of initialparameters (MIP). The secondary objective is anexperimentally measured deflection at the top (top drift)of a physical model tested in an Eiffel-type boundarylayer wind tunnel to determine all other relevantparameters that affect the dynamism of multi-storeybuilding. The dimensional analysis was use for theassessment of the ensuing parameters. Attempt was alsomade to determine the stationary (static) andcorresponding non-stationary (dynamic forces) along themodel height. The experiment shows that wind pressuredistribution on the bluff building can be adequatelyrepresented by the result of the model studies in anEiffel-type boundary layer wind tunnel with elegantturbulence simulation. Therefore, the current researchused the boundary layer wind tunnel for a more accurateprediction of various parameters to which the theories of fluid computational dynamics   and dimensional analysisare being applied. These results will be physically andstatistically compared with values obtained through themathematical model by using the method of initialparameters on a substitute cantilever. GENERAL THEOREM OF TALL STRUCTURESDYNAMICS Targ (1976) in his work on general theorem of particledynamics remarked that in solving many problems of dynamics, the so called general theorem representingcorollaries of fundamental law of dynamics are moreconveniently applied than the integration of differentialequations of motion. The importance of general theoremsis that, they establish visual relationships between theprincipal dynamic characteristics of motion of materialbodies, thereby presenting broad possibilities of analyzing the mechanical motions widely employed ingeneral human and practical engineering developments.This approach makes it possible to study a specificaspect of a given phenomenon without investigating thephenomenon as a whole, thereby, simplifying thesolution. Applying the above definitions, the basic dynamiccharacteristics of a multi-storey building are momentum(or linear momentum) and kinetic energy. The momentumof a multi-storey building is defined as a vector quantity(mu) equal to the product of linear modal mass (m) of amulti-storey building and its velocity (u). The vector, mu isacting along the same directrix as the velocity u (that istangent to the path of the multi-storey building).Targ (1976) came up with the conclusion that, thepractical implications of the above definitions is that, theunits of measurements of the quantities must be definedusing both the system international (SI) [kg-m s -1 for muand kg-m 2 S -2 for 0.5 mu 2 ] as well as the continentalOnundi et al. 161metric [(mkg (f) s) kgf-s   for mu and kg-m for 0.5 mu 2 ]system units. He also recommended the following that:1. Knowing the initial kinetic energy of the tall structure or multi-storey building and the force developed due to theinfluence of direct induced or aero-elastic damping, it ispossible to determine the travelled distance and notpossible to simultaneously determine the damping period.2. That, the mass of the tall structure or multi-storeybuilding is constant and its acceleration a, whichexpresses the fundamental law of dynamics as thederivative of the linear momentum of the tall structure or multi-storey building with respect to time is equal to thegeometric sum of the forces acting on the building. Thechange of momentum of the structural system during anyinterval of time is equal to the geometric sum of theimpulses of all forces acting on the system during thatinterval of time.3. If the motion of a tall structure is constrained, the workdone by the given reactive force of a fixed smoothsurface or curve in any displacement of the tall structureis zero. Therefore, in a displacement of a fixed smoothsurface or curve the change in kinetic energy of the tallstructure is equal to the sum of work done in thisdisplacement by the active force (wind) applied to the tallstructure. THE MATHEMATICAL MODEL The mathematical model used in this work was based on Equations1(a - d), the method of initial parameters (MIP) (Christev et al.,1974; Onundi, 2012):(1a)(1b)(1c)(1d)Where, y o , α o , M o and Q o are respectively, the generalizedamplitudes of deflections, rotations, moments and shear forces atthe initial section (x=0); y i and α i are respectively, the deflectionsand rotations at the sections where the lumped mass are located.This approach enables the dynamic analysis to be conducted for astructural system as infinite degrees of freedom.(2a)  162 J. Civ. Eng. Constr. Technol. Where,K = the stiffness of the bar (beam), ω = the natural frequency of the system,g = the acceleration due to gravity, 9.81 m/sec 2 ,q = the weight per unit length of the beam, including theuniformly distributed load ( if any) and E is modulus of Elasticity for the material used for the structural element(beam).(2b)Where m i is the linear modal mass of the beam.Therefore:(2b) A kmx , B kmx , C kmx and D kmx are the influence function given by: A kmx = 0.5 (cosh kmx + cos kmx) (3a)B kmx =0.5(sinh kmx + sin kmx) (3b)C kmx =0.5(cosh kmx  – cos kmx) (3c)D kmx = 0.5 (sinh kmx - sin kmx) (3d)The method of initial parameters was initially applied to dynamicbehavior  of flexural members’ that is, beams on elastic foundationand not for multi-storey buildings. In order for this method to beapplicable to the dynamic analysis of multi-storey buildings, thestiffness value (k) and moment of inertia (I) were modified. Thisenables us to reduce the design of a deformed structure to that of an elastic column in which several point masses are fixed to it, and time dependent displacement Δ(t) of the upper end of the substitute cantilever was measured for a model with an infinite degree of freedom. A typical model of an infinite degree of freedomcantilevered system with applied loads along and at the top of themodel is shown for the multi-storey building in Figures1(a, b, c andd) given as: The Influence functions for multi-storey building The influence functions are coefficients which depend on structuralcharacteristics such as stiffness, mass and the overall length or height of the structural system been investigated. The deformablestructural system with infinite degrees of freedom has the frequencyequal to (Varvanov, 1975):(4a)Since, the period (T), for multi-storey building, Figures (1 (a, b, cand d) is 0.026 L (Nakashima et al., 1992), therefore: And: Figure 1a and b. Description of the model.(4b)Where isthe minimum natural frequency of the multi-storeybuilding and Equations 4a and b, give the minimum naturalfrequency for a system with infinite degrees of freedom as:(4c)Finally, since U = 1.875 for a substitute cantilevered multi-storeybuilding:(5a)Equation 6 is the generalized moment of inertia (I g ), correspondingto the odd modes of vibration of a multi-storey building under asteady state vibration; m i is the linear modal mass which is givenas:(5b)The generalized moment of inertia for the multi-storey building isdetermined from Equations 5a and b by substituting, say: , 13000 2500072000 a. Front elevation. b. Side elevation.
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