Appl. Mth. Inf. Sci. 7, No., 579-585 (013) 579 Applied Mthemtics & Informtion Sciences An Interntionl Journl c 013 NSP Nturl Sciences Pulishing Cor. Anlyticl Solution to the Roof Bending Deflection with Mixed Boundry Conditions under Uniform Lod Shuren Wng 1, nd Zhongqiu Wng 1 1 School of Civil Engineering nd Mechnics, Ynshn University, Qinhungdo, 066004, Chin Key Lortory of Mechnicl Reliility for Hevy Equipments nd Lrge Structures, Qinhungdo, 066004, Chin Received: 8 Aug, 01; Revised 10 Oct, 01; Accepted 0 Oct, 01 Pulished online: 1 Mr. 013 Astrct: Anlyticl solution to the roof ending deflection is the key to conduct the stility evlution nd risk prediction of the shllow mined-out res. The engineering mechnics model of the roof ws uilt through the generliztion engineering model of the prcticl shllow mined-out res. Bsed on Reissner s thick plte theory, the roof ending deflection with mixed oundry conditions under uniform lod ws nlyzed through the reciprocl theorem method. The function of roof ending deflection ws derived nd the nlyticl solution ws verified y the numericl solution. Then, comprtive nlysis ws conducted for the difference chrcteristics of roof ending deflection with mixed oundry conditions nd simply-supported oundry conditions. Keywords: Mined-out res, roof, mixed oundry conditions, Reissner s thick plte theory 1 Introduction The deformtion properties nd instility mechnism of the roof in the shllow mined-out res re the current chllenging prolems which should e worked out urgently in engineering prctice [1]. Some scholrs tret the roof s elstic rock em for nlyticl nlysis nd reserch of the roof deformtion chrcteristics of the mined-out res. For exmple, X.Y. Zhng et l. [] simplified the roof s elstic em nd nlyzed the creep process on the se of the creeping dmge theory. X.L. Jing et l. [3] improved the em model nd discussed the influence to the roof thickness cused y horizontl stress nd rock frctures sed on structure stility theory nd dmge theory. S.Q. Qin et l. [4] regrded the roof s elstic em nd nlyzed the instility process of mechnicl system of stiff roof nd col pillr with ctstrophe theory. G.M. Swift et l. [5] considered the roof s elstic em nd nlyzed the stility fctors to the roof in the mined-out res. The differences etween the elstic em hypothesis nd the prcticl roof mde it difficult to reflect the ctul conditions of the roof stress nd its deformtion. Other scholrs treted the roof s elstic thin plte in order to do such nlysis nd reserch. For exmple, R.H. Lin et l. [6] nlyzed the overlying strt nd otined the strength condition of the key lyer reking instntly through the elstic thin sl theory nd plstic limit nlysis method. J.A. Wng et l. [7] regrded the roof s thin plte, nlyzed the roof frcture process nd its effect to the collpse of the mined-out res. H. Li et l. [8] simplified the overlying strt of the steeply-inclined sem s thin plte, otined the prediction deformtion formul sed on the thin plte ending theory with elsticity mechnics. X.Y. Lin et l. [9] treted the stiff roof s elstic thin plte to study the stress distriution rules nd the frcture mechnism of the roof while considering the initil oundry conditions nd the periodicl ground pressure. H. Liu et l. [10] ssumed the roof s thin plte nd explored the stility of the stiff roof nd pillr system. In prcticl engineering, it ws limited to do the reserch using thin plte theory ecuse the roof of the mined-out res ws usully in the stte of the thick plte. Although some scholrs regrded the roof s thick plte nd chieved some conclusions [11-13], the numerous prolems need to e solved urgently with the development of the engineering prctice. Therefore, the uthors ssumed the roof s thick plte nd nlyzed the chrcteristic of roof ending deflection with mixed oundry condition Corresponding uthor: e-mil: w sr88@163.com c 013 NSP Nturl Sciences Pulishing Cor.
580 Shuren Wng et l. : Anlyticl Solution to the Roof... under uniform lod y the reciprocl theorem method sed on Reissner s thick plte theory in order to provide the technologicl supports for prcticl projects. Building computtionl model In the northest edge of Antio Surfce Mine, there ws the mined-out res of Jingyng Mine exploited y the room-nd-pillr stoping method. Bsed on the field investigtion nd the collected informtion, the engineering model of the mined-out res ws uilt s in Fig.1. The hypothesis of the roof ws usully regrded s the thick plte with simply supported or with four edges fixed, which were different with the rel oundry conditions. Thus, it ws more pproprite to simplify the roof s the thick plte with mixed oundry conditions (Fig.1) nd the engineering mechnics model ws showed in Fig.. The engineering mechnics model ws ssumed tht the long side of the thick rectngulr plte is, the short side is ), the supporting length long the long side direction of the plte is 1,the free length is s, the supporting length long the short side direction is 1, the free length is l. The thickness of the roof is h, its elstic modulus is E, nd the poisson s rtio is v. The ded lod of the overlying strt is regrded s uniform lod q which is distriuted on the upper surfce of the roof. Thus, the engineering mechnics model ws generlized s in Fig.. Fig. The sketch of engineering mechnics model (Rel system of thick plte with mixed oundry). 3 Deriving formuls s for the roof ending deflection with mixed oundry conditions 3.1 Bsic eqution On the se of Reissner s thick plte theory, the sic sttic equtions re showed s follows: h D 4 W = q(x, y) ν 1 ν 10 q(x, y) (1) φ ( 10 ) / h φ = 0 () where D is the flexurl rigidity of the roof, is Lplce opertor, W is the deflection of the roof, q(x,y) is the uniform lod distriuted on the roof, v is the Poisson s rtio, h is the thickness of the roof, nd φ is the stress function. D = Eh 3 / [ 1 ( 1 υ )] (3) where E is the elstic modulus of the roof. 3. Bsic system of the thick plte Fig.3 shows the sic system of the thick rectngulr plte with simply supported under concentrted lods, tht is, the trnsverse two-dimensionl Dirck-Delt function (x ξ, y η) cts in the convective coordinte (ξ, η) of the thick plte with four edges simply supported. And w 1,x0, w 1,x, w 1,y0, nd w 1,y re the four-sided rottion ngles of the sic system respectively. V 1x0, V 1x, V 1y0, nd V 1y re the equivlent sher forces respectively. R 100, R 10, R 1, nd R 10 re the corner lods respectively. 3.3 Rel system of the thick plte Fig. 1 The sketch of engineering model.()section mp of the mined-out res;()overhed view of the mined-out res Fig. shows the thick plte with the centrl free nd the rest simply supported of four edges under uniform lod, tht is, the rel system of thick rectngulr plte with mixed oundry. And deflections nd twist ngles re w x0, w x, w y0, w y, ω x0, ω x, ω y0, nd ω y respectively. For the rel symmetric system, the deflection w y0 nd twist ngle ω y0 c 013 NSP Nturl Sciences Pulishing Cor.
Appl. Mth. Inf. Sci. 7, No., 579-585 (013) / www.nturlspulishing.com/journls.sp 581 1 Q 1x w x dy 1 Q 1x0 w x0 dy M 1xy ω xy dx M 1xy0 ω xy0 dx 1 1 Fig. 3 Bsic system for the thick rectngulr plte. on free oundry y=0 re equl to w y nd ω y on free oundry y=, nd the deflection w x0 nd twist ngle ω x0 on free oundry x=0 re equl to w x nd ω x t free oundry x=. Thus, the deflection formul is written s follows: W = W (ξ, η) (4) The deflection equtions re w y0 = w y = A i sin iπ(x 1) s w x0 = w x = The twist ngles re ω yx0 = ω yx = ω xy0 = ω xy = B j sin jπ(y 1) l sin kπ(y 1) l sin fπ(x 1) s where x [ 1, ], y [ 1, ], = 1 s, = 1 l. The stress function is (5) (6) (7) (8) φ(ξ, η) = [E n cosh δ n ξ F n cosh δ n ( ξ)] cos β n η (9) [G m cosh γ m η H m cosh γ m ( η)] cos α m ξ where α m = mπ, β n = nπ, γ m = α m 10 h, δ n = β n 10 h A i, B j,,, E n, F n, G m, nd H m re undetermined coefficients respectively. 3.4 Driving the deflection eqution The reciprocl theorem method is used etween the sic system nd the rel system, nd the result is written s follows: W(ξ, η) Q 1y w y dx Q 1y0 w y0 dx 1 1 1 M 1yx ω yx dy 1 M 1yx0 ω yx0 dy = qw 1 (x, y; ξ, η)dxdy (10) 0 0 Sustituting the relted oundry vlues Eq. (5) to Eq. (8) of the rel system, the relted oundry vlues Q 1y0, Q 1y, Q 1x0 Q 1x, M 1xy0, M 1xy, M 1yx0, M 1yx, nd the sic solution W 1 (x, y, ξ, η) [14] of the sic system into Eq. (10). Then the deflection eqution is reched s follows fter clculting: W (ξ, η) = 4q D {1 1 cosh 1 α m [α m(η ) sinh α m (η ) ( 1 α m tnh 1 α m) cosh α m (η )]} 1 αm 5 sin α m ξ l π [sinh β n ( ξ) sinh β n ξ] sin β n η sinh β n ia i (1 υ) π jb j ϕ jn (, l) s π [sinh α m ( η) sinh α m η] ϕ sinh α m im (, s) {β n coth β n [sinh β n ( ξ) sinh β n ξ] β n ξ cosh β n ξ β n ( ξ) cosh β n ( ξ)} n sin α m ξ (1 υ) H nk (, l) α m sinh α m π {[sinh α m ( η) sinh α m η]α m cot α m α m ( η) cosh α m ( η) α m η cosh α m η} m sin β nη H mf (, s) (11) β n sinh β n where ϕ im (, s) = ( 1)i sin 1s 1 mπ sin mπ (1) i ( s m) ϕ jn (, l) = ( 1)j sin 1l nπ sin 1 nπ (13) j ( l n) H mf (, s) = ( 1)f sin 1s mπ sin 1 mπ m ( s f) (14) H nk (, l) = ( 1)k sin 1s nπ sin 1 nπ (15) n ( l k) c 013 NSP Nturl Sciences Pulishing Cor.
58 Shuren Wng et l. : Anlyticl Solution to the Roof... The oundry conditions of the rel system re s follows : η = 0, nd ξ [ 1, ] : M ξη =0, Q η =0, M η =0; ξ = 0, nd η [ 1, ]: M ηξ =0,Q ξ =0, M ξ =0. Due to the twist ngles ssumed in the corners of ech edge, tht is ω x0, ω x, ω y0, nd ω y. The relted equtions on the oundry should e stisfied η = 0, nd ξ [ 1, ] : ω ξ =ω ξη0 ; ξ=0, nd η [ 1, ]: ω η =ω ηξ0. where M η =0, M ξ =0 hve een stisfied. For the symmetric conditions E n = F n, G m = H m. Therefore, we cn get six equtions ccording to the ove mentioned six oundry conditions nd otin six undetermined coefficients. The deflection, sher, moment nd rottion ngel re showed s follows: 4q tnh βn β n sin mπ1 16D(ν 1) π β n mα m α m β n 4D(ν 1) (1 cosh β n )nβ n m ( f s ) π sinh β n sin nπ 1 n ( E nβ n (1 cosh δ n ) k l ) 4 G m β n γ m sinh γ m = 0 α m β n n = 1, 3... (16) 4q tnh αm 16D(ν 1) α m π sin nπ1 n ( k l 4 4D(ν 1) ) π sin mπ1 β n nα m α m β n (1 cosh α m )mα m sinh α m m ( f s ) G mα m (1 cosh γ m ) E n α m δ n sinh δ n α m δ n = 0 m = 1, 3... (17) 16 π 16h 5sπ mα m 4h 5Dπs(1 υ) (sin mπ 1 ) ia i [i ( sm ) ][m ( f s ) ] m α (sin mπ1 ) m [m ( f s ) ] mg m γ m sinh γ m sin mπ 1 m ( f s ) = 0 f = 1, 3... (18) 16 π 16h 5lπ nβ n jb j (sin nπ 1 ) [j ( nl ) ][n ( k l ) ] n βn (sin nπ 1 ) [n ( k l ) ] 4h 5Dπl(1 υ) ne n δn sinh δ n sin nπ 1 n ( k = 0 l ) k = 1, 3... (19) q υ) {(1 βn 3 (tnh β n β n cosh β n υh tnh βn } = D(1 υ) 16s 5β n π ) α m 3 α m β n sin mπ 1 4l ia i D(1 υ) i ( sm ) π tnh β n β n sin nπ 1 jb j j ( D(υ 16 nl 1) ) π mα m β n (α m β n ) 16h 5π mα m 4 α m β n sin mπ1 D(1 υ) m ( f s ) sin mπ 1 m ( f s ) D(1 υ) 4h 5π nβ n 3 tnh β n sin nπ 1 n ( k l ) D(υ 1) n π [β n β n (1 β n coth β n ) tnh β n ] sinh δ n h 5 sin nπ1 n ( k l ) h 5 E n(β n δ n ) G m α m γ m γ m β n γ m sinh γ m n = 1, 3 (0) q υ) {(1 αm 3 (tnh α m α m cosh α m ) υh tnh αm } = D(1 υ) 4s 5α m π tnh α m α m sin mπ1 A 16l i i ( sm D(1 υ) ) π β n 3 α m β n i sin nπ1 jb 16 j j ( nl D(υ 1) π ) nα m β n sin nπ 1 (α m β n ) C k D(1 υ)16h n ( k l ) 5π nβ n 4 α m βn 4h 5π mα m 3 tnh α m sin nπ 1 C k n ( k D(1 υ) ) l sin mπ1 D f m ( f (υ 1) s ) c 013 NSP Nturl Sciences Pulishing Cor.
Appl. Mth. Inf. Sci. 7, No., 579-585 (013) / www.nturlspulishing.com/journls.sp 583 D m π [α m α m (1 α m coth α m tnh α m ] sin mπ 1 D f m ( f h 5 G m(γ m α m ) s ) sinh γ m h δ n β n E n 5 α m β β n sinh β n n m = 1, 3 (1) Solving the equtions from Eq. (16) to Eq. (1) to get the coefficients A i, B j,,, E n nd G m, nd then the deflection, sher, moment nd rottion ngel cn e worked out. As numericl exmple, the model ws ssumed tht side length of squre thick plte is ==10 m, the length of simply supported edge is 1 = 1 =3 m, the length of free section is s=l=3 m. The poison s rtio of the plte is v=0.3, nd its elstic modulus is E =50 GP. The uniform lod cing on the plte is q=0.3mp. The deflection is supposed to otin when the width-thickness rtio is h/=0.1, 0., 0.3. According to the ove equtions, it is esy to get the deflection vlues of ny plce y using softwre MAT- LAB. Under the softwre MATLAB environment, the deflection vlues long the z direction cn e reched when the points x/=0.5, y/=0.1, 0., 1.0; x=0, y/l=0.1, 0., 1.0 re chosen respectively. In the solving process of coefficient equtions set, the pproximtion solution with sufficient ccurcy cn e otined through picking up the finite terms s shown in Tle 1. So, the stisfctory results re otined to keep reltive errors less thn 0.05 through clcultion when c ( c mens the coefficients m, n, i, j, k, f ) is equl to 80 respectively. Tle 1 The mximum deflection vlues nd the reltive errors of thick plte t x/=0.5 nd y/=0.5 with different c. c =40 c =60 c =80 ANSYS W Error W Error W Error (mm) (mm) (%) (mm) (%) (mm) (%) 3.45 3.58 3.97 3.54.77 3.5.1 0.51 0.54 6.69 0.53 5.48 0.53 4.54 0.0 0.1 5. 0.1 4.06 0.1 3.55 under the uniform lods eing pplied, nd the displcement of z direction cn e otined, tht is, the deflection vlues of the roof of the mined-out res. As showed in Fig.4 nd Fig.5, the deflection curves of the thick plte t x/=0.5 nd t x=0 with different thicknesswidth rtios hd een worked out respectively. The result indictes tht whether on the oundry or in the plce of the deflection mximum, the error is less thn 0.05 etween the numericl solution nd the nlyticl solution, which is cceptle in the prcticl engineering. 4.0 3.5 3.0.5.0 1.5 1.0 0.5 0.0 h/=0.1(anlyticl solution) h/=0.(anlyticl solution) h/=0.3(anlyticl solution) h/=0.1(numericl solution) 0.0 0. 0.4 0.6 0.8 1.0 Fig. 4 The deflection curves of thick plte t x/=0.5. 0.5 0.0 0.15 0.10 0.05 0.00 h/=0.1(anlyticl solution) h/=0.(anlyticl solution) h/=0.3(anlyticl solution) h/=0.1(numericl solution) 0.0 0. 0.4 0.6 0.8 1.0 Fig. 5 The deflection curves of the free section of thick plte t x/=0. 4 Results nd nlysis 4.1 Verifiction for nlyticl solution The thick plte with mixed oundry condition under uniform lod is simulted y using ANSYS. A squre plte with side length ==10 m is selected, nd the thicknesswidth rtio is h/=0.1, 0., nd 0.3 respectively. The numericl model ws computed under the sme condition 4. Comprtive nlysis of two different oundry conditions The squre pltes of side length ==10 m with simplysupported of four edges nd with mixed oundry conditions re selected, nd the length of free sections nd the simply-supported sections of the thick plte with mixed oundry conditions re 1 = 1 =3 m, nd s=l=3 m respectively. Uniform lod q eing pplied on the upper surfce c 013 NSP Nturl Sciences Pulishing Cor.
584 Shuren Wng et l. : Anlyticl Solution to the Roof... of the plte is equl to 0.3 MP. The poison s rtio of the plte is v=0.3, nd its elstic modulus is E=50 GP, The thickness-width rtio of the plte is h/=0.1, 0., nd 0.3 respectively. The results re showed in Fig.6,Fig.7 nd Fig.8. 4.0 3.5 3.0.5.0 1.5 1.0 0.5 0.0 h/=0.1(mixed oundry conditons) h/=0.1(simply-supported conditions) 0.0 0. 0.4 0.6 0.8 1.0 y (10mm) Fig. 7 The deflection surfce of h/=0. thick plte with simplysupported oundry conditions. 0.6 0.5 0.4 0.3 0. 0.1 0.0 h/=0.(mixed oundry conditons) h/=0.(simply-supported conditions) 0.0 0. 0.4 0.6 0.8 1.0 0.1 c 0.18 0.15 0.1 0.09 0.06 0.03 0.00 h/=0.3(mixed oundry conditions) h/=0.3(simply-supported conditions) 0.0 0. 0.4 0.6 0.8 1.0 Fig. 6 The deflection curves of thick plte t x/=0.5. () h/=0.1; () h/=0.; (c) h/=0.3 5 Conclusion Bsed on Reissner s thick plte theory, the roof ending deflection with mixed oundry conditions in the minedout res under uniform lod ws nlyzed through the reciprocl theorem method. The function of roof ending deflection ws derived nd the nlyticl solution ws worked out. Fig. 8 The deflection surfce of h/=0. thick plte with mixed oundry conditions. Compring the numericl solution with the nlyticl solution, the thick plte theory is pproprite for nlyzing the chrcteristic of roof ending deflection under the sme condition. Through nlyzing the difference chrcteristics of roof ending deflection with two different oundry conditions, it is showed tht roof ending deflection with mixed oundry conditions is igger thn tht with simply-supported oundry conditions, nd the deformtion scope of roof ending deflection with mixed oundry conditions is pprently greter thn tht of simply-supported oundry conditions with increse of the thickness-width rtio. Acknowledgement This work ws finncilly supported y the Ntionl Nturl Science Foundtion of Chin (No:51074140). c 013 NSP Nturl Sciences Pulishing Cor.
Appl. Mth. Inf. Sci. 7, No., 579-585 (013) / www.nturlspulishing.com/journls.sp 585 References [1] Wng, S.R., Ji, H.H. nd C.F. Wu, Determintion method of roof sfety thickness in the mined out regions under dynmic loding nd its ppliction, Journl of Chin Col Society, 35, No. 8, 163-168 (010). [] Zhng, X.Y. Anlysis of creep dmge frcture of upper roof, Journl of Lioning Technicl University (Nturl Science), 8, No. 5, 777-780 (009). [3] Jing, X.L., Co, P., Yng, H. nd H. Lin, Effect of horizontl stress nd rock crck density on roof sfety thickness of underground re, Journl of Centrl South University (Science nd Technology), 40, No. 1, 11-16 (009). [4] Qin, S.Q. nd S.J. Wng, Instility leding to rockursts nd nonliner evolutionry mechnisms for col-pillr-ndroof system, Journl of Engineering Geology, 13, No.4, 437-446 (005). [5] Swift, G.M. nd D.J. Reddish, Stility prolems ssocited with n ndoned ironstone mine, Bulletin of Engineering Geology nd the Environment, 61, No. 3, 7-39 (00). [6] Lin, H.F., Li, S.G., Cheng L.H. Key lyer distinguishing method of overlying strt sed on the thin sl theory, Journl of Chin Col Society, 33, No. 10, 1081-1085 (008). [7] Wng, J.A., Shng X.C. nd H.T. M, Investigtion of ctstrophic ground collpse in Xingti gypsum mines in Chin, Interntionl Journl of Rock Mechnics & Mining Sciences, 45, No.8,1480-1499 (008). [8] Li, H., Ling, B., Li, G., Bi Y.P. nd C.M. Zhng, Prediction on ending deflection of overlying strt cused y steeply-inclined col sem mining in mountinous re, The Chinese Journl of Geologicl Hzrd nd Control, 1, No. 3, 101-104 (010). [9] Li, X.Y., Go F. nd W.P. Zhong, Anlysis of frcturing mechnism of stope roof sed on plte model, Journl of Mining & Sfety Engineering, No., 180-183 (008). [10] Liu, H., Hu, Q.T., Wng J.A. nd J.G. Li, Anlysis on stility of pillr nd stiff roof system in the go re, Journl of Col Science & Engineering, 15, No., 06-09 (009). [11] Zhu, F.C., Co P. nd W. Wn, Determintion of sfe roof thickness of underground shllow openings sed on xisymmetric thick plte mode, Ground Pressure nd Strt Control, 3, No. 1, 115-118 (006). [1] He, G.L. Determintion of criticl thickness of stiff roof in col mine sed on thick plte theory, Chinese Journl of Underground Spce nd Engineering, 5, No. 4, 659-663 (009). [13] Tn, W.F., Liu J.J. nd B.L. Fu, Reciprocl theorem method for ending of thick rectngulr pltes supported t four corner points, Journl of Engineering Mechnics, 13, No. 4, 49-58 (1996). [14] B.L. Fu nd W.F. Tn, Reciprocl theorem method for solving the prolem of thick rectngulr pltes, Journl of Applied Mthemtics nd Mechnics, 16, No. 4, 367-379 (1995). Shuren Wng is one professor from the School of Civil Engineering nd Mechnics, Ynshn University, Chin. He otined his Ph.D. degree from University of Science nd Technology (Beijing). And he hs een worked s visiting scholr t the School of Mining Engineering, the University of New South Wles, Austrli. He is n ctive resercher in mining engineering, geotechnicl engineering, soil mechnics nd numericl simultion, nd hs pulished more thn 50 reserch rticles in journls of the relted fields. Zhongqiu Wng received the B.S. degree from the Shool of Engineering Technology, Chin University of Geosciences (Beijing), Chin. Now he is mster student t School of Civil Engineering nd Mechnics, Ynshn University,Chin. His interested reserch is mining engineering nd risk prediction of the shllow mine-out res. c 013 NSP Nturl Sciences Pulishing Cor.