DEFORMATION OF ROLLING SHUTTERS UNDER UNIFORM WIND PRESSURE

Transcription

1 48 Journal of Marne cence and Technolog, Vol. 9, No., pp () DEFORMATION OF ROLLING HUTTER UNDER UNIFORM WIND PREURE Nan-Nong Huang* and an-mng Chang* Ke words: rollng shutter, wnd pressure, deflecton, closed-form soluton. slat ABTRACT In ths stud, a model s developed to anale the structural behavors of rollng shutters subjected to wnd pressure. In the current shutter theor, the elastc propertes of shutters are charactered b two bendng rgdtes, whch can be determned from the fleural tests of slats and shutters. Close form solutons can be obtaned for the deflecton of shutters wth two opposte smpl supported edges. To assess the valdt of the present theor, rollng shutters wth varous ses and supportng condtons under varous magntudes of loadngs are tested. It s noted that the deflectons and aal dsplacements based on the current theor agree ver well wth epermental results. I. INTRODUCTION A rollng shutter conssts of a seres of slats that form a curtan wth both sdes of slats beng nserted nto gude rals. The shutter curtan can be rolled onto an ale. The shutter curtan and gude rals of a rollng shutter are schematcall shown n Fg.. The cross secton and the jont of a partcular tpe of slats are depcted n Fg.. hutters can provde securt protecton and prvac, save energ, and even serve as sound barrers. Eteror rollng shutters wth varous ses have been wdel used n structural applcatons snce 9s. Made of materals of the tme, manl wood and steel, the were heav and cumbersome. Recentl, lght-weghted slats made of alumnum and/or plastcs have become popular, whch have been streamlned to be more aesthetcall pleasng, more functonal, and easer to nstall. Despte the ncrease n the use of shutters, especall the lght-weghted shutters, rgorous structural analses for shutters attract lttle attentons. The major concern for the structural ntegrt of shutters s wnd load, especall n the areas afflcted wth tropcal cclones. trong wnds ma rp an Paper submtted /6/9; revsed 6/3/9; accepted 3/7/. Author for correspondence: Nan-Nong Huang (e-mal: b@mal.ntou.edu.tw). *Department of Mechancal and Mechatronc Engneerng, Natonal Tawan Ocean Unverst, Keelung, Tawan, R.O.C. gude ral a Fg.. chematc dagram of slats and gude rals for a rollng shutter. Fg.. Cross secton and the jont of slats. entre shutter off the gude rals or cause an ecessve deflecton to damage the nteror wndows or doors. Most analses focus on the deflecton of slats based on the beam theor. For eample, Chen [3] conducted a large deflecton analss on the alumnum slats. However, results from the analss of a sngle slat can not be appled to shutters. The comple shapes of slats and the varous tpes of jonts make t dffcult to conduct a comprehensve analss on the structural behavors of shutters. In, Ach and Alart [] treated shutters as mult-jonted structures wth mperfect jonts between slats. The slat s smplfed as a flat plate whch s nserted nto an dealed channel shape slde, as shown n Fg. 3. Both clearance and rotatve frcton were taken nto consderaton. A hbrd fnte element formulaton was developed. The contact condtons and frcton law result n a ver comple algorthm. In ther work, the authors dd not provde an verfcaton eample (ether from publshed results or from epermental works) to assess the accurac of ther numercal results. In the current work, a modfed model from the classcal plate theor s developed to eamne the structural responses of shutters subject to wnd pressure. In ths model, onl two effectve stffnesses (namel, bendng rgdtes) are requred to charactere the elastc propertes of an knd of shutters regardless of the shapes of slats and the varet of jonts. These two bendng rgdtes, whch can be obtaned easl b

2 N.-N. Huang and.-m. Chang: Deformaton of Rollng hutters Under Unform Wnd Pressure 49 Q M slat slat clearance mperfect jont Fg.3. The slats and the jont of shutters studed b Ach and Alart []. M M N M Q Q Q N M Fg. 4. Resultant forces on an nfntesmal shutter element. M from the fleural tests of slats and shutter curtans, presumabl can account for the effects of clearance and frcton of the jonts. Close form solutons based on Lev s method can be obtaned for the shutters wth two opposte smpl supported edges. Deformaton tests are conducted for the shutters wth varous dmensons and dfferent tpes of supports under varous magntudes of loadng. The deflectons and aal dsplacements are measured. Results based on the current model agree ver well wth the epermental results. II. THEORY The coordnate sstem for the shutter analss s shown n Fg.. Coordnate s n the longtudnal drecton of the slat, s n the rollng drecton, and s n the transverse drecton of the shutter. The length of the shutter n the and drectons are a and b, respectvel. A modfed theor based on the classcal plate theor s ntroduced hereafter. The jonts of slats, as ndcated n Fg., serve as fleble lnkages so that slats are able to revolve around the reel of shutter. In ths respect, a shutter s free to stretch and roll up n the drecton and provdes no n-plane shear rgdt. Therefore, stress resultants N, N, N, and M are assumed to be neglgble. The nonero stress resultants are depcted on an nfntesmal shutter element as shown n Fg. 4. For the case of small dsplacement gradent problems, the n-plane responses are decoupled from the out-of-plane responses. Hereafter, onl out-of-plane behavors of shutters are concerned. The moments are related to curvatures and bendng rgdtes b [6] M = D κ, M = D κ, M = D κ () Followng the Krchhoff constrant, the curvatures are related to the deflecton b κ = w, κ = w, κ = w (),,, Consder a dfferental shutter element subject to transversel dstrbuted loadng p(, ) as shown n Fg. 5. The stress resultants actng on the element are also ndcated n the fgure. The moment equlbrum condtons n the and drectons are M M Q M Q Q + dq p d Q + dq d M + dm M + dm M + dm Fg. 5. Forces eertng on a dfferental shutter element. Q = M, Q = M + M (3),,, Equatng the forces n the drecton elds Q + Q + p= (4),, ubsttutng Eqs. ()-(3) nto the above equaton gves Dw + ( D + D ) w = p (5),, Upon ntroducng the followng non-dmensonaled parameters w W =, X =, Y = a a b D D pa a d = = λ = (6) D D b 3 ( + ), p, force equlbrum Eq. (5) can be rewrtten as W + λ dw = p (7), XXXX, XXYY III. LEVY OLUTION AND AXIAL DIPLACEMENT The shutter curtan s assumed to be smpl supported b gude rals. For shutters wth two opposte smpl supported edges, the close form solutons to Eq. (7) are avalable. Con-

3 5 Journal of Marne cence and Technolog, Vol. 9, No. () sder edges = and a (X =, ) are smpl supported. Followng the Lev approach [6], the deflecton and loadng are represented b the Fourer sne seres: s ds dw l deformed confg. W( X, Y), p( X, Y) = sn( π X) W( Y), p( Y) (8) [ ] = w It s noted that the assumed deflecton satsfes the followng smpl supported condtons on edges X =, : d orgnal confg. a Fg. 6. Deflecton curve and aal dsplacement (Δ = l a) of a shutter. W(, Y) = W(, Y) = M (, Y) = M (, Y) = (9) The wnd pressure dstrbuton on a wall, whch s at rght angle to the wnd drecton, was reported b Dalglesh and chrever [4]. The mamum wnd pressure (.e., stagnaton pressure) develops near the center of the wall. The pressure contours reveal that there s an ncreasngl steep pressure gradent towards the edges of the wall [4]. For the purpose of shutter desgn, the wnd pressure s assumed to be unform over the entre shutter curtan. The Fourer coeffcents for unform loadng, pxy (, ) = q, are 4 q/( π ) ( =,3,5, ) p ( Y) = ( =,4,6, ) () B substtutng Eqs. (8) and () nto Eq. (7), the partal dfferental equaton s reduced to the followng set of ordnar dfferental equatons: n whch W β W = f () ( π ) p, β = f = () λ d ( π) λ d The general soluton to Eq. () s W( Y) = c cosh( β Y) + c snh( βy) + f / β (3) where c and c are arbtrar constants. In addton to the knowledge of deflecton, the calculaton of aal dsplacements of slats s also ver mportant for the desgn of shutters. A slat wth an ecessve aal dsplacement could rp off gude rals. For an netensble slat, the aal dsplacement can be determned from the length of the deflecton curve between supports. Consder the deflecton curve of a shutter n the drecton (.e., the longtudnal drecton of slats) as shown n Fg. 6. Let s be the coordnate along the deflecton curve. The length of the deflecton curve between supports s a ds a l( ) = d= + ( w, ) d d (4) Introducng L = l/a elds the non-dmensonaled epresson for the above equaton: ( ) = + (, X ) d LY W X (5) The aal dsplacement for the slat can then be determned as Δ ( ) = l a= a + ( W, X ) dx (6) The mpson rule s emploed n ths stud to evaluate the above ntegral. The partcular solutons correspondng to two boundar value problems are presented hereafter. Frst, consder a shutter wth all edges smpl supported (desgnated as ). The second problem corresponds to a shutter havng three smplsupported edges and one free edge (desgnated as F).. hutters The coordnates for an shutter are shown n Fg. 7(a). The smpl supported condton along edges Y = ±/ s W(X, ±/) =, whch leads to W ( ± /) = (7) Coeffcents c and c n Eq. (3) can be determned from the above condton and b consderng the smmetr of the structural responses wth respect to X as: c = f, c = (8) β cosh( β / ) Therefore, the deflecton soluton of the shutter s W X Y f = cosh( β Y) (, ) sn( π X) =,3,5, β cosh( β / ) (9)

4 N.-N. Huang and.-m. Chang: Deformaton of Rollng hutters Under Unform Wnd Pressure 5 / Y Y slats X X -/ F (a) (b) Fg.7. Coordnates for (a) shutter and (b) F shutter.. F hutters The coordnates for an F shutter are shown n Fg. 7(b). The smpl supported condton along edge Y = s W(X, ) =, whch leads to support Fg. 8. chematc dagram of the setup for shutter tests. W () = () Along the free edge Y =, the boundar condton s Q + M, = [5]. B ntroducng moment equlbrum condton Eq. (3), the boundar condton becomes M, M, + = () ubsttutng moment-curvature relatons Eq. () nto the above equaton elds κ, =, whch subsequentl results n W, XYY (X, ) =. Therefore, the boundar condton along Y = s W () = () Undetermned coeffcents c and c n Eq. (3) can be obtaned b satsfng both boundar condtons Eqs. () and (): c = f, c = (3) β cosh( β) Then the deflecton soluton of an F shutter s f cosh( βy ) W( X, Y) = sn( π X) =,3,5, β cosh( β ) IV. EXPERIMENT AND REULT (4) In the current theor, two bendng rgdtes D and d are requred for the shutter analss. Frst, rgdt D s determned from the clndrcal bendng test of slats. For the slat consdered n ths stud, a value of 37. Kg-m s adopted for D [3]. Then, the value of d can be determned from the bendng test of a shutter assembl. Fg. 9. Deformaton test of a shutter assembl loaded b unforml dstrbuted sand bags. To eamne the valdt of the current model, a seres of tests on the rollng shutters wth varous ses and supportng condtons are conducted. The shutters wth followng dmensons (a, b) are tested: (.68 m,.4 m), (.68 m,.5 m), (.68 m,.9 m), (.3 m,.4 m), and (.3 m,.5 m). Meanwhle, the F shutters wth dmensons (a =.68 m, ) and (a =.68 m, b =.5 m) are also tested. The setup of the shutter test s schematcall shown n Fg. 8. The unform loadng s appled b usng p plastc bags flled wth sand, whch s llustrated n Fg. 9. The detals of test procedures were descrbed n Chang []. The shutters are subjected to varous magntudes of loadngs. For most shutters tested, the mamum pressure s up to Kg/m, whch s appromatel equvalent to Force 6 n the Beaufort wnd force scale. The mamum number of the Beaufort scale s Force 7. The deflectons and aal dsplacements are measured usng a dal gauge and a calper, respectvel. The test results for mamum deflecton and aal dsplacement are plotted

5 5 Journal of Marne cence and Technolog, Vol. 9, No. () Central Deflecton (mm) hutters wth a =.68 m deflecton (theor) aal dsp. (theor) deflecton (ep.) aal dsp. (ep.) b =.9 m b =.5 m b =.9 m b =.5 m Mamum Aal Dsplacement (mm) Central Deflecton (mm) F hutters wth a =.68 m deflecton (theor) aal dsp. (theor) deflecton (ep.) aal dsp. (ep.) b =.5 m b =.5 m Mamum Aal Dsplacement (mm) 5 5 Pressure (Kg/m ) Fg.. Comparson between the theoretcal and epermental results of the central deflectons and mamum aal dsplacements for the shutters wth wdth a =.68 m and varous heghts (b =.4 m,.5 m,.9 m). 5 5 Pressure (Kg/m ) Fg.. Comparson between the theoretcal and epermental results of the mamum deflectons and mamum aal dsplacements for the F shutters wth wdth a =.68 m and varous heghts (b =.4 m,.5 m). Central Deflecton (mm) hutters wth a =.3 m deflecton (theor) aal dsp. (theor) deflecton (ep.) aal dsp. (ep.) b =.5 m b =.5 m 5 5 Pressure (Kg/m ) Fg.. Comparson between the theoretcal and epermental results of the central deflectons and mamum aal dsplacements for the shutters wth wdth a =.3 m and varous heghts (b =.4 m,.5 m). aganst loadngs n Fgs. -. Analtcal results based on the current model are also ncluded n the fgures for comparson. Fg. shows the responses of shutters wth a =.68 m and,.5 m, and.9 m. The responses of shutters wth a =.3 m and,.5 m are Mamum Aal Dsplacement (mm) gven n Fg.. The responses of F shutters wth a =.68 m and,.5 m are shown n Fg.. For the shutters, the mamum deflecton occurs at the center of curtan (.e., central deflecton) and the central slat possesses the mamum aal dsplacement. For the F shutters, the mamum deflecton s produced at the center of the free edge and the slat along the free edge has the mamum aal dsplacement. The epermental data for the central deflecton of an shutter wth a =.68 m and b =.5 m s emploed to determne the value of bendng rgdt d. The shutter assembl conssts of 48 slats. The central deflecton and mamum aal movement are shown n Fg.. It s noted that the deflecton behavor s almost lnear for a loadng up to 5 Kg (96 Kg/m n pressure). The deflectons and aal dsplacements calculated from the Lev solutons wth d =.486 ft the test data ver well. The effectve stffnesses D = 37. Kg-m and d =.486 are taken to be the elastc constants for shutters consstng of ths partcular tpe of slats. All the calculated results of shutters wth varous ses and dfferent tpes of supports are based on these two effectve stffnesses. The convergence rate of the close form solutons s ver fast. For eample, consder an shutter wth a =.68 m and b =.5 m under a pressure of 56.6 Kg/m. The numercal data of the central deflectons and mamum aal dsplacements of the shutter, correspondng to varous numbers of terms used n the seres soluton, are lsted n Table. It s noted that even one-term soluton produces near-converged results for both the deflecton and aal dsplacement. The results presented n ths paper are based on the 6-term seres solutons.

6 N.-N. Huang and.-m. Chang: Deformaton of Rollng hutters Under Unform Wnd Pressure 53 Table. Convergence rate of the Lev soluton ( shutter wth a =.68 m and b =.5 m, under pressure 56.6 Kg/m ). # of terms Dsp central deflecton (mm) w(a/, b/) aal dsp (mm) Δ( = ) -term ( = ) term ( =, 3) term ( =, 3, 5) term (( =, 3, 5, 7) term ( =, 3,, 9) term ( =, 3,, ) term ( =, 3,, 3) A close eamnaton of the results presented n Fgs. - reveals that the theoretcal results of deflectons agree ver well wth the epermental data for all the shutters consdered. A notceable dscrepanc n aal dsplacement ests n some shutters when the loadng s small. However, the dscrepanc between the theoretcal values and test data dmnshes as loadng s ncreased. For the cases of small loadngs, the aal dsplacement s trval wth a value of onl a few mllmeters, whch makes t dffcult to get an accurate measurement b usng a calper. Also, the dfference ma be partl due to the estence of frcton between slats and gude rals, whch could have a sgnfcant effect on the dsplacements n case of a small loadng. Overall, the dscrepanc n both deflectons and aal dsplacements for all the shutters tested s less than 4% when pressure s hgher than Kg/m. V. CONCLUDING REMARK A model coupled wth two bendng rgdtes s developed to predct the structural behavors of shutters under wnd pressure. The current theor, wthout the necesst of consderng the detals of the shapes of slats and the clearance of jonts, can be vewed as a smplfed model of the classcal plate theor. The two effectve stffnesses can be determned from the bendng tests of the slat and shutter. Close form solutons based on the current theor can be obtaned for the shutters wth two opposte smpl supported edges. Deflectons and aal dsplacements are calculated for the shutters wth four smpl supported edges and for the shutters wth three smpl supported edges and one free edge. To evaluate the lmtatons and accurac of the current theor, bendng tests are performed for the shutters wth varous dmensons and supportng condtons. The plastc sand bags are dstrbutvel stacked on the shutter to smulate the applcaton of unform wnd pressure. The shutters are subjected to varous magntudes of pressure, wth mamum pressure up to Kg/m for most of the tests. Comparsons between theoretcal predctons and epermental results are made. It s noted that the dscrepanc s small for both the deflecton and aal dsplacement. For all the cases consdered, the dfference between theoretcal and epermental results s less than 4% when loadng pressure s hgher than Kg/m. The lght-weghted shutters are wdel used n houses, shops and factores. However, due to the complet of the shapes and the jonts of slats, a smple but practcal shutter theor has not been reported before. Most desgns of shutters are based on past eperence, whch poses a great threat to publc safet, especall n tphoon-afflcted areas lke Tawan. The present theor wth ts smplct and accurac can be adopted to make shutter desgn much easer and safer. NOMENCLATURE a, b lengths of shutter c, c coeffcents n the soluton to the dfferental Eq. D, D bendng rgdtes D, d bendng rgdtes f parameter appearng n the dfferental Eq. () l(), L(Y) length of the deflecton curve M, M moments M, M moments N, N n-plane forces N, N n-plane forces p, pq, pressure p ( Y ) Fourer coeffcents for pxy (, ) Q, Q shear forces s coordnate along the deflecton curve w, W deflecton W (Y) Fourer coeffcents for W(X,Y),, coordnates X,Y non-dmensonaled coordnates β parameter appearng n the dfferental Eq. () Δ() aal dsplacement of the slat κ, κ, κ curvatures λ geometrc aspect rato (a/b) REFERENCE. Ach, K. and Alart, P., Numercal smulaton of the behavour of a mult-jonted structure, Phlosophcal Transactons of the Roal ocet of London A, Vol. 359, pp ().. Chang,. M., Deformaton of Alumnum Rollng hutters under Wnd Load, Master Thess, Department of Mechancal and Mechatronc Engneerng, Natonal Tawan Ocean Unverst, Keelung, Tawan (). 3. Chen, H. C., Deflecton Behavor of Alumnum Rollng hutter Leaves, Master Thess, Department of Mechancal and Mechatronc Engneerng, Natonal Tawan Ocean Unverst, Keelung, Tawan (). 4. Dalglesh, W. A. and chrever, W. R., Wnd pressures on buldngs, Canadan Buldng Dgest, CBD34, NRC-IRC (968). 5. Rvello, R. M., Theor and Analss of Flght tructures, McGraw-Hll, New York (97). 6. Tmoshenko,. P. and Wonowsk-Kreger,., Theor of Plates and hells, nd ed., McGraw-Hll, New York (959).