Linear static analysis of perforated plates with round and staggered holes under their self-weights

Transcription

1 Liner sttic nlysis of perforted pltes with round nd stggered holes under their self-weights Mustf Hlûk Srçoğlu*, Uğur Albyrk Online Publiction Dte: 8 Nov 2015 URL: DOI: Journl Abbrevition: Res. Eng. Struct. Mt. To cite this rticle Srçoğlu MH, Albyrk U. Liner sttic nlysis of perforted pltes with round nd stggered holes under their self-weights. Res. Eng. Struct. Mt., 2016; 2: Disclimer All the opinions nd sttements expressed in the ppers re on the responsibility of uthor(s) nd re not to be regrded s those of the journl of Reserch on Engineering Structures nd Mterils (RESM) orgniztion or relted prties. The publishers mke no wrrnty, explicit or implied, or mke ny representtion with respect to the contents of ny rticle will be complete or ccurte or up to dte. The ccurcy of ny instructions, equtions, or other informtion should be independently verified. The publisher nd relted prties shll not be lible for ny loss, ctions, clims, proceedings, demnd or costs or dmges whtsoever or howsoever cused rising directly or indirectly in connection with use of the informtion given in the journl or relted mens.

2 Reserch Article Liner sttic nlysis of perforted pltes with round nd stggered holes under their self-weights Mustf Hlûk Srçoğlu *1, Uğur Albyrk 2 1 Deprtment of Civil Engineering, Dumlupınr University, Küthy, Turkey 2 Deprtment of Civil Engineering, Eskişehir Osmngzi University, Eskişehir, Turkey Article Info Article history: Received 10 Sep 2015 Revised 26 Sep 2015 Accepted 1 Nov 2015 Keywords: Finite element method, Mid-point deflection, Perforted plte, Round nd stggered holes. 1. Introduction Abstrct Perforted pltes with round holes which re mnufctured esily offer the wide re of pplictions such s metl ceiling tiles, ir condition grilles, etc. In this study, optimum hole size nd shpe of perforted pltes with round nd stggered holes investigted using mid-point deflections of perforted pltes under their self-weights. Anlyzed pltes re simply supported on their four sides. Anlysis results re obtined by using APDL codes for ANSYS. The squre perforted pltes re exmined in terms of mid-point deflections by chnging numbers, rdii nd loctions or their holes. The grphics showing the reltionship between the number of holes nd mid-point deflections of the pltes were presented to indicte the effects of geometric discontinuities. The grphics nd formultions presented in the study could be useful to determine the design prmeters nd perforting opertions of the perforted squre pltes with round nd stggered holes MIM Reserch Group. All rights reserved. Perforted pltes with round nd stggered holes offer the wide re of pplictions such s metl ceiling tiles, decortive bnisters, ir condition grilles, etc. (Fig. 1) According to the ltest surveys, most perforted sheets re produced with round holes. Becuse of the round roles re mnufctured reltively esier with esthetic effects. The circulr die for punching sheet cn lst longer nd esy to be mnufctured which mkes the round hole perforted sheet cheper thn ny other perforted sheet with other hole ptterns. Bending of squre plte with circulr hole nd circulr plte with circulr hole t the center exmined in the most importnt reference for pltes nd shells [1]. Perforted metl is mde through the metl stmping nd sheet metl mnufcturing process [2]. Round holes re the most economicl nd extreme open re proportions tend to increse distortion [3]. Round roles re mnufctured reltively esier with esthetic effects. The circulr die for punching sheet cn lst longer nd esy to be mnufctured which mkes the round hole perforted sheet cheper thn ny other perforted sheet with other hole ptterns [4]. Young nd Jo presented equivlent mteril properties of perforted plte with tringulr or squre penetrtion pttern for dynmic nlysis [5]. Chen et l. developed the boundry element lternting method to study the stress concentrtion of two-dimensionl perforted plte [6]. Rens et l. studied the mechnicl behvior of perforted pltes nd simulte by the nlysis of homogenized (unperforted) pltes [7]. In this study, Ly-In nd Ly-On Systems in both Ly-In nd Ly-On type suspension systems were exmined. The sme exposed grid system is utilized with the difference stemming from the tiles nd the wy the tiles re plced inside the grid. Metl ceiling tiles * Corresponding uthor: mhluk.srcoglu@dpu.edu.tr DOI: Res. Eng. Struct. Mt. Vol. 2 Iss. 1 (2016)

3 tht re widely produced s thin squre pltes with multiple circulr holes re mde of steel or luminum pltes with punching process (Fig. 1). Fig. 1 Architecturl ceiling system exmples - metl ceiling tiles 2. Anlysis of Thin Squre Plte with Multiple Circulr Holes In clssicl plte theory, for trnsversely loded plte without xil deformtions, bending of rectngulr thin plte, the governing eqution hs the form 4 w w + 4 w = q(x,y) x 4 x 2 y 2 y 4 D (1) where; D = E h3 12(1 ν 2 ) is the flexurl rigidity, h is the thickness of the plte, E is the Young s modulus, ν is the Poisson s rtio nd q(x,y) is the externl trnsverse lod. This eqution is n equilibrium eqution in direction z of plte crrying uniform distributed lod q. In this eqution, w is the elstic surfce function showing the deflection to be mde depending on the x nd y coordintes of the mid-plne of the plte. The boundry conditions for simply supported edges re: w=0 w=0 2 w x 2 w y 2 = 0 for x=0 nd x= 2 = 0 for y=0 nd y= For n infinitely lrge plte in uniform stte of stress defined by the bending moments Mx=M nd My=0 deflection surfce of the plte cn be defined s; w = M(x2 ν y 2 ) 2 D (1 υ 2 ) to obtin the deflection produced in such stte of pure bending by circulr hole with rdius (Fig. 2), we ssume the mteril to be removed inside the periphery of the circle. w = M 2 2 D 2 [A log r + (B + C r2) cos 2θ] (2) We cn fulfill conditions by choosing the dditionl deflection in Eq (2) w = w + w (3) 40

4 Finl deflection of the plte with circulr hole cn be defined s in Eq (3) 0 θ r x y Fig 2. Plte with circulr hole This expression lso stisfies the homogeneous differentil Eq (1) nmed s Clssicl Plte Eqution [1]. 3. Finite Element Anlysis A plte is the flt construction element whose length nd width re multiply lrger thn the thickness loded minly in bend. Metl ceiling tiles nlyzing in this study cn be modeled s simply supported squre pltes with multiple circulr holes under selfweights. All of the plte exmples in this study were nlyzed stticlly under self-weights using ANSYS 13.0 tht is verstile pckged softwre use finite element method. The pltes mde out of steel ccording to ASTM A992, with modulus of elsticity N/mm 2, Poisson s rtio of 0.3 nd density ρ = 7.697e-8 N/mm 3. The ANSYS element librry contins more thn sixty elements for sttic nd dynmic nlyses but in this cse Shell93 element is chosen to solve the problem. Shell93 is prticulrly well suited to model curved shells becuse there re 8 nodes, ech of the element sides re prbols rther thn stright lines [8-10]. The reltionship between input dt nd output results produced by the progrm, nd is essentil for thorough understnding of theoreticl descriptions of Shell93 is described below: Shell93 is 8-node structurl finite element in the progrm element librry. The element hs six degrees of freedom t ech node: trnsltions in the nodl x, y, nd z directions nd rottions bout the nodl x, y, nd z-xes. The deformtion shpes re qudrtic in both in-plne directions (Fig. 3). The element hs plsticity, stress stiffening, lrge deflection, nd lrge strin cpbilities [8] The geometry, node loctions, nd the coordinte system for this element re shown in Fig. 3. The element is defined by eight nodes, four thicknesses, nd the orthotropic mteril properties. A tringulr-shped element my be formed by defining the sme node number for nodes K, L nd O. Pressures my be input s surfce lods on the element fces s shown by the circled numbers on Fig. 3. Positive pressures ct into the element. Edge pressures re input s force per unit length. 41

5 Nodes, degrees of freedom, thickness s rel constnts, modulus of elsticity, Poisson s rtio, density s mteril properties re element inputs. Fig. 3 Geometry of SHELL93 finite element in ANSYS The solution output ssocited with the element is in two forms; one is nodl solution the other is element solution. Severl items re illustrted in Fig. 4. Printout includes the moments bout the x fce (MX), the moments bout the y fce (MY), nd the twisting moment (MXY). The moments re clculted per unit length in the element coordinte system. The element stress directions nd force resultnts (NX, MX, TX, etc.) re prllel to the element coordinte system. Fig. 4 Stress output of SHELL93 finite element in ANSYS Volume of elements, stresses, strins, moments, displcements re some outputs of SHELL93. Some of ssumptions nd restrictions of SHELL93 element is s follows: Sher deflections re included in this element. The out-of-plne (norml) stress for this element vries linerly through the thickness. The trnsverse sher stresses (SYZ nd SXZ) re ssumed to be constnt through the thickness. The trnsverse sher strins re ssumed to be smll in lrge strin nlysis. (ANSYS, 2005). APDL (Ansys Prmetric Design Lnguge) is powerful lnguge for optimizing the FEM workflow nd provides the user with flexibility nd power in creting complex mcros. 42

6 APDL codes re developed to crete nd nlyze the plte models systemticlly in this study. The pltes in this study re modeled with simply-supported boundry conditions on 4 sides nd tringulr free mesh is used in free meshing opertions. The used element edges were depend on meshing res nd enclosed only tringulr elements. Finite element mesh dimensions re similr in ll of the exmples pproximtely. One side of ech mesh element is equl to thickness of the plte in this wy totl numbers of mesh elements re similr in ech model (Fig. 5). Fig. 5 Mesh geometry of common squre perforted pttern lyout in ANSYS The squre pltes with round nd stggered holes nlyzed with respect to different number, rdius nd loctions. One side of the perforted squre thin plte ws chnged then the plte thickness ws lso chnged ccording to plte dimensions. Totl open res of holes re exctly sme for ech model. The volume nd weight of the pltes re exctly sme, becuse of totl hole re (open re) is equl in ll models which were nlyzed under the sme lods (self-weights). () Model No 1 (2 holes) (b) Model No 2 (8 holes) (c) Model No 3 (18 holes) (d) Model No 4 (32 holes) (e) Model No 5 (50 holes) (f) Model No 6 (72 holes) (g) Model No 7 (98 holes) (h) Model No 8 (128 holes) (i) Model No 9 (162 holes) (j) Model No 10 (200 holes) x = y y z r 1 x t 43

7 Fig. 6 Perforted squre plte models with different number of round nd stggered holes. The min importnt fctor in perforting is the hole pttern, the others re the shpe nd the size of the holes. The most common ptterns used in perforted ceiling tiles re stright line nd stggered ptterns. In first model for perforted pltes with round nd stggered holes, the squre thin plte with 2 holes tht re plced symmetriclly long the lines x=y re exmined (Fig. 6). The center of this two holes re lso symmetric nd the rdius of ll holes re r=/8, where is one side of the squre plte nd r is the rdius of the hole shown in Fig. 6. The new plte models with different number of holes nd rdius re creted by converting into series on regulr bsis up to model with 200 holes (Fig. 6). Chnging the length of squre plte, the plte thickness t is not to be constnt nd chnges depends on while /t rtio is constnt nd equls to 150 [11]. For ech model, the totl hole re of ech plte is constnt nd equl to π 2 /32 tht re independent of r nd number of holes, where is one side of the squre plte. In such problems, the min prmeter used for investigtion of the influence of holes on deflection is the volume frction of considered holes, which is defined s rtio between the volume of ll the holes vi the volume of plte without holes. In this pper the volume frction is constnt s 9.82%. The rdius of the holes re the design vrible furthermore the rdius is constrined between minimum vlue of r=/80 nd mximum vlue of r=/8 for 10 different models (Tble 1). Tble 1 Number of holes in perforted squre plte models Model No Number of holes Rdius (r) Totl hole re 1 2 / 8 2..(/8) / (/16) / (/24) / (/32) / (/40) / (/40) / (/40) / (/40) / (/40) / (/40) 2 n 2.n 2 / 8.n. 2 /32 4. An Exmple to the Anlysis of Thin Squre Plte with Holes Model No.4 plte which the dimensions re both 300mm nd thickness is 2mm nd hs 32 circulr holes exmined stticlly under self-weights with grvity of g = 9810 mm/sn 2. The plte is mde out of steel ccording to ASTM A992. The plte is modeled nd nlyzed by Ansys Prmetric Design Lnguge codes. Liner elstic isotropic mteril properties of plte re defined. The progrm systemticlly cretes the models nd meshes the plte with tringulr free mesh using Shell93 element 44

8 0 w mm Srçoğlu nd Albyrk / Reserch on Engineering Structures & Mterils 2 (2016) nd the mesh round the holes refined. The boundry conditions for simply supported edges re defined. The self-weight of the plte is ssumed s the sttic lod distributed over the surfce uniformly. The deflections for the whole plte fter nlysis of the plte in this section re given by contour plots shown s Fig. 7. Also the deflections on the pths pssed from x=/2 nd y=/2 re lso given in Fig. 7. The mximum deflection vlue is mm occurs t the midpoint of the given plte. Fig. 7 Anlyze results of 300x300x2mm squre plte with 32 circulr holes The solved plte size of 300x300x2mm squre plte with 32 circulr holes for Model No.4 is lso nlyzed for 10 models with different number of holes nd the results of ll the nlysis re plotted in grphic (Fig. 8) number of holes : Model No wo = Fig. 8 Midpoint deflections of squre plte tht size of 300x300x2mm for 10 models 5. Results Midpoint deflections for perforted pltes with round nd stggered holes under their selfweights investigted with Ansys. Models re developed systemticlly with different rdius nd number of holes. The results of the nlysis re presented with grphics for the midpoint deflection of pltes with multiple circulr holes. The nlysis procedures reproduced for different vlues which is one dimension of squre plte. For different plte length 10 model with different number of holes re nlyzed nd midpoint deflections respect to number of holes nd plte width re presented (Tble 2). 45

9 In this study, for ny size of squre plte, the number of holes increses the midpoint deflection lso increses lthough totl hole re is equl in ll models. Number of holes for ny plte is proportionl to midpoint deflection but it is not liner. In models for less hole numbers; deflections increse rpidly up to 50 holes, beyond this point hole numbers increse, the deflections lso begin to increse slightly. Tble 2 Midpoint deflections (w0 ) of perforted squre pltes (mm) Model No Number of Holes (mm) Conclusions In this study, mid-point deflections of perforted pltes with round nd stggered holes under their self-weights is investigted. Sttic nlysis in Ansys is performed to evlute the influence of rdius nd number of holes. APDL codes re developed to crete nd nlyze the 10 plte models systemticlly in the study. The pltes mde out of steel ccording to ASTM A992, with modulus of elsticity N/mm 2, Poisson s rtio of 0.3 nd density ρ = 7.697e-8 N/mm 3 nd simply-supported on four sides. After meshing of the problem, number of elements nd number of nodes re pproximtely sme for ll models. The squre pltes exmined to different number of holes s 2, 8, 18, 32, 50, 72, 98, 128, 162 nd 200 respectively nd rdius nd loctions re lso different. Totl surfce res of holes re exctly sme for ech model. Plte thickness is dependent to one side of the squre plte so the nlysis results were compred properly. For different plte length s 100 mm, 200 mm, 300 mm, 400 mm nd 500 mm respectively; 10 models with different number of holes re nlyzed nd midpoint deflections respect to number of holes nd plte width re presented. The present study demonstrtes the effects of geometric discontinuities on thin squre perforted pltes with round nd stggered holes. As result of the nlysis for 10 models, for ny size of squre plte, the number of holes increses the midpoint deflection lso increses lthough totl hole re is equl in ll models. Number of holes increse, the vrition of the midpoint deflections decrese symptoticlly fter 50 holes (Fig. 8). The results of the nlysis re used to develop useful nd prcticl grphics for the midpoint deflection of perforted pltes with round nd stggered holes. For required squre plte dimension or number of holes is chosen thn midpoint deflection cn be esily determined from the grphic. References [1] Timoshenko S, Woinowsky-Krieger S. Theory of pltes nd shells. 2nd edition, Singpore McGrw-Hill, 1959, ISBN [2] 46

10 [3]. [4] [5] Young MJ, Jo JC. Equivlent mteril properties of perforted plte with tringulr or squre penetrtion pttern for dynmic nlysis. Nucler Engineering nd Technology, 2006; 38(7): [6] Chen KT, Ting K, Yng WS. Stress nlysis of two-dimensionl perforted pltes using boundry element lternting method. Computers nd Structures, 2000; 75: [7] Rens BJE, Brekelmns WAM, Bijens FPT. Homogeniztion of the eltoplstic behvior of perforted pltes. Computers nd Structures, 1998; 69: [8] ANSYS (2005). ANSYS commnds reference. [9] ANSYS (2005). APDL progrmmer's guide. [10] ANSYS (2005). Relese 10.0 Documenttion for ANSYS. [11] Albyrk U, Srçoğlu MH. Anlyzing of thin squre pltes with multiple circulr holes. Proceedings of the Interntionl Symposium on Advnces in Applied Mechnics nd Modern Informtion Technology (ISAAM&MIT'11), Bku, Azerbijn, 79 83,