A meth 1 od sla for bs the oad A.Fatemi-Ardakani,BSc(Eng), PhD Bovis ConstructionLtd.(formerly,QueenMaryCollege,Universityof London) E.Burley, BSc(Eng), PhD, CEng, MICE QueenMaryCollege,University ofLondon Professor L.A. Wood,BSc, PhD, CEng, MICE South Bank Polytechnic Synopsis A basicmethodfor thedesignof ground-bearingwarehouseslabssubjected to point loads is proposed,based on theuseof design charts. Thesecharts have beendrawn fromthecomputedresultsof atruly3-dimensional analysis,inwhichboththesoilandtheslabaretreatedas behaving elastically. The method utilisesplate bending finite elements and a layered continuum model for the soil (Wood').The method is suitable for usein a design office, allowingconsiderationof the additional moments imposed underone load by any other pointload onthe slab.Theaccuracy ofthis methodis shownto bewithin2%of the more rigorous full solution. Notation ais the radius ofrelative stiffness ofslab and subgrade (m) a =E,(l +:)h3 6Es(1-uc2) I E,is the elastic modulus ofconcrete slab(MN/m2) E,is the elastic modulus ofsubgrade (MN/m2) his the thickness ofslab (m) Kis the modulusofsubgradereaction(MN/m3) Pis theappliedload(kN) Sis the side of loaded area (m) ucPoisson'sratio of concreteslab usPoisson'sratio ofsubgrade M, is the moment intensityabout x-axis (kNm/m),sagging (+ve), hogging ( - ve) M,,is the moment intensity about y-axis (kNm/m),sagging (+ ve), hogging (- ve) 4is anon-dimensionalcoordinate=x/ a is a non-dimensionalcoordinate=y / aIntroduction The design of ground-bearingconcretefloorslabs is not explicitly covered by recommendations in British Codes of Practice, and is often based on rule ofthumbandpast experience ratherthana formalanalysis. Concrete ground floors2 is a useful guide to the design and construction of ground-supportedconcrete slabs for all types of building,from small- scale domestic through commercial and institutional to large-scale industrial floors. This publication is based on the design principle that plain concrete can resist flexural tensile stresses andthatreinforcement is providedto control shrinkage,notto withstandflexural tensile forces. The ability to design ground slabs subjected to a series of pointloads is becoming increasingly important as more warehouses use racking systems which impose loads of theorderof 100 kN in a regular grid over the floor. There is also agrowth in the useof steel framed mezzanine floors to create greaterfloor area in existing buildings; these also invariably impose a grid of pointloadsontotheground slab. A Cement & ConcreteAssociation technical report Design of floors on ground3 goes some way to dealing with the problem, but enables only the TheStructuralEngineerNolume67/No.19/3 October 1989 de ed 0 S1 b Y' Pint additional stress due to adjacent loads on theappropriateaxis to be included. Thus,for example, in Fig ltheadditional bending moment M,at position 5 ,due to loads1, 3, 7and 9,cannot be considered. Y 2.02.0 IInternall oad 2' ol l l X Fig1.The positionof loads f or example 2(dimensions in m) Theory The resultshavebeenobtainedusingthe computer programLAWRAFT (Wood4) in which the slab is represented asan assemblage of orthotropic rectangularplatebendingfiniteelements of displacement type, exhibiting threedegreesoffreedom,i.e.twoorthogonalrotationsandvertical translationat each node (thin plate assumption). The subgrade is modelled as a tranversely isotropic layered continuum. The stress distribution within the layered continuum duetovertical loads applied at thesurface is taken, with acceptableaccuracy, to be identical with that occurring within ahomogeneous semi-infinite continuum. Thus for theisotropiccase, this reduces to theBoussinesq,stressdistribution. In this paper the thickness of the subgrade has been taken as 10 times the slab length in all cases, thereforeapproximating the subgrade to be a semi- infinitecontinuum. Theoretical results using thisprogramhave previously been compared with measured results for two warehousefloorslabs.Wood& Larnachs showed goodagreementbetween predictedand measuredmoments and 341 Paper: Fatemi-Ardakani et a1 settlementsofaheavily loaded warehousefloor, and Wood,Burley & Harvey6 also showed goodagreement for awarehouse floor constructed on highly compressible soil. Method ofdesign (a)Stress dueto pointloads The magnitude of flexural moments (M, and My)immediately beneath the pointofapplicationoftheload is the decisive factorforthe design of a slab. Toobtainthemagnitude of the flexural momentsunderthesubjected point load, considerationmustbe given tothe loading areaand position of theload within the slab. A full parametricstudy has been carried out, assumingaconcrete slab ofelastic modulus 20 GN/m2and Poissonsratio 0.15,for a rangeof thicknesses 0.15,0.3 and 0.4m.The subgrade hasbeen takento have a Poissonsratio of 0.3 and values ofelastic modulusof 20,100 and 3 0 0MN/m2. I t is convenient to combine the stiffness of the slab andthe subgrade intoasingletermcalledthe radiusofrelativestiffness. Thishasthe dimension oflength and is defined as: a = [ ~,(1-~,2)h3 1 ....(1) 6Es(1-U,) The range of radius of relative stiffness of slab and subgrade covered by theaboveparameters varies between0.35 and 2.5m. Pointloadscannotexist in practice, as theymust always be distributed over a small area. This area is assumed to be asquareof side S for internal and comer loads and a rectangle of side S and S/2 for edge loads, thereby allowing theloadtobe applied close tothe edge of the slab (see Fig 2). The value ofSforanedge load baseplate of equalareatoaninternal baseplate will, therefore, be squareroot of 2, times theinternal value. Five values ofS are used in theparametricstudy, ranging from 0.2m tolm 0.4v ~ c 0 . 2xszo.2 h 0,351+s=O4 aOS=O. 6 Int ernalload Edge Load I-- Corner Load Fig2. Definitionof pointloadareas instepsof0.2m.Loads withbaseplatesofothershapesmayalsobe introduced intothe analysis by representingthe loadedareaasasquare or rectangle ofequivalent area. Variationof the position of the load relative to the edges of theslab showed that, if the load is within adistance of0.7a froman edge, the loadshould be consideredas an edge load,and if it is within aradius of 0.5a fromthe cornerthisshouldbe considered to be acornerload;all other positionsare assumed to be internal loads. The graphs of maximum flexural moment divided by the load, plotted against the radius of relative stiffnessa for thedifferent loading areas, are shown in Figs 3-6forthethreecases of internal load, edge load,and Radi us of r el at i vest i f f ness(a) Fig3.Variationof moment(M,)under pointloadagainstradiusofrelative stiffness(a) for internal loads 0.7 0.1 0.00.20.40.60.81.01.21.41.61.82.02. 22.4 Radius ofrel ati vesti ffness(a) Fig4.Variation of moment(M,)under pointloadagainstradiusofrelative stiffness (a) for edge loads 342TheStructuralEngineer/Volume67/No.19/3 October 1989 Paper: Fatemi-Ardakani et al -I Radi us of rel at i vest i f f ness(a) Fig5.Variation of moment(M,)under pointloadagainst radiusofrelative stiffness(a) for edge loads Fig6.Variation of moment(M,)under pointloadagainst radiusofrelative stiffness(a) for corner loads cornerload,respectively.I n allcasesexceptfor thecornerload, the maximum momentoccursunder the pointload and is a sagging moment. In the case of the corner load,the maximumoccurssomedistance from the corner,and this is a hogging moment. (b) Stress dueto additionalloads In order to cope with additional loads in any position,moment contours are required covering the whole area ofinfluence. It was foundfromthe results obtained fromthe parametricstudy thatthe moments M,and My duetoapointloadextend over anareaof 4a in each direction (i.e. outside 4a bending moments canbe ignored). Thisagrees with the resultsof Gorbunov-Posadov7 and Gorbunov-Posadov& Serebrjanyis. The momentcontours ofM, and M, for the rangeofloading areas considered andforthethree loading positions were plotted, yielding atotal of 23contourcharts, which are sufficient to enablea designer to assess correctly the twoprincipalmomentsintheslab, regardless of the number ofpoint loads appliedor of their positionon the slab. Eachcontour represents the percentage value of the peak momentoccurringunder the load, plottedona non-dimensionalgridofcoordinates 5 =x/ aand v =y/ a , with the origin atthecentre of the loaded area. Forinternal and cornerloads, thevaluesfor M,onlyare produced,since My maybe obtained by interchanging axes. Examples are shown for internalloading (Figs 7 and 8)and cornerloading (Fig 11).ContoursforbothM, and M,are required foran edge load,and typical results are shown in Figs 9 and 10. In these cases, contoursareplotted with the y-axis coincident with the edge of the slab. TheStructuralEngineedVolume 67/No.19/3 October1989 v Loadingar ea 5 ~ 0 . 4/ 4 a- Fig7. Percentage contourof momentM,duetointernal load atorigin 343 Paper: Fatemi-Ardakani et all q tLoading area S